Wireless Power Transfer: Using Magnetic and Electric Resonance Coupling Techniques [1st ed. 2020] 9811545790, 9789811545795

Table of contents :
Acknowledgements
Contents
1 Wireless Power Transfer
1.1 Types of Wireless Power Transfer
1.1.1 Detailed Differentiation of Wireless Power Transfer Types
1.1.2 Operating Frequency
1.2 Outline of Electromagnetic Induction and Magnetic Resonance Coupling
1.2.1 Difference Between Electromagnetic Induction and Magnetic Resonance Coupling
1.2.2 Types of Circuit Topology
1.3 Outline of Electric Field Coupling and Electric Resonance Coupling
1.4 Radiative-Type Power Transmission
1.4.1 Microwave Power Transmission
1.5 Basic System Configuration
References
2 Basic Knowledge of Electromagnetism and Electric Circuits
2.1 Resistance, Coils, and Capacitors
2.1.1 Resistance
2.1.2 Coils Seen from a Circuit Viewpoint
2.1.3 Capacitors as Seen from a Circuit Viewpoint
2.2 Principle of Electromagnetic Induction
2.2.1 Magnetic Field H, Magnetic Flux Density B, and Magnetic Flux Φ
2.2.2 Ampere’s Law and Biot–Savart Law
2.2.3 Faraday’s Law
2.2.4 Mechanism of Energy Transmission by Electromagnetic Induction (Electromagnetism Viewpoint)
2.2.5 Electromagnetic Induction Described from a Circuit Viewpoint
2.2.6 Self-inductance
2.2.7 Mutual Inductance and Coupling Coefficient
2.2.8 Neumann’s Formula (Derivation of Inductance)
2.3 High-Frequency Loss (Resistance)
2.3.1 Copper Loss, Skin Effect, and Proximity Effect
2.3.2 Iron Loss (Hysteresis Loss and Eddy Current Loss)
2.3.3 Radiation Loss
2.4 Non-resonant Circuit Transient Phenomena (Pulse)
2.5 Resonance Circuit and Transient Phenomena
2.5.1 LCR Series Circuit Transient Phenomena (Pulse)
2.5.2 LCR Series Circuit and Q Value
2.5.3 Magnetic Field Resonance Transient Phenomena
2.6 Root Mean Square (RMS) Values, Active Power, Reactive Power, and Instantaneous Power
2.6.1 Root Mean Square (RMS) Values
2.6.2 Instantaneous Power Versus Active Power and Reactive Power
References
3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics
3.1 Coils and Resonators
3.1.1 Spiral, Helical, and Solenoid Coils
3.2 Summary of Air Gap and Misalignment Characteristics
3.2.1 Efficiency, Power, and Input Impedance with Air Gap
3.2.2 Misalignment Characteristics
3.3 Near Field of Magnetic Field and Electric Field
3.4 Frequency Determinant (kHz to MHz to GHz)
3.4.1 Resonant Frequency
3.4.2 Parameters of Coils for kHz, MHz, and GHz
3.4.3 kHz Coil
3.4.4 GHz Coil
Reference
4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)
4.1 Derivation of Equivalent Circuit
4.1.1 Solution Using Kirchhoff’s Voltage Law
4.1.2 Calculation Using the Z-Matrix
4.2 Equivalent Circuit at the Resonant Frequency
4.2.1 Derivation of the Equivalent Circuit at the Resonant Frequency, and Its Maximum Efficiency
4.2.2 Derivation of Voltage Ratio and Current Ratio
4.2.3 Load Characteristics at the Resonant Frequency
4.2.4 kQ Representation
4.2.5 Maximum Efficiency Considering Coil Performance
References
5 Comparison Between Electromagnetic Induction and Magnetic Resonance Coupling
5.1 Five Basic Types of Circuits: N–N, N–S, S–N, S–S, S–P
5.2 Non-resonant Circuit (N–N)
5.2.1 Verification of Equivalent Circuit (N–N)
5.2.2 Efficiency and Power (N–N)
5.3 Secondary-Side Resonant Circuit (N–S)
5.3.1 Verification of Equivalent Circuit (N–S)
5.3.2 Efficiency and Power (N–S)
5.4 Primary-Side Resonant Circuit (S–N)
5.4.1 Verification of Equivalent Circuit (S–N)
5.4.2 Efficiency and Power (S–N)
5.4.3 Design Using Primary-Side Resonance Conditions (S–N)
5.4.4 Design with Overall Resonance Condition (Power Factor = 1)
5.4.5 Comparison Between Primary-Side and Overall Resonance Conditions (Power Factor = 1)
5.5 Circuit of Magnetic Resonance Coupling (S–S)
5.5.1 Verification of Equivalent Circuit (S–S)
5.5.2 Efficiency and Power (S–S)
5.5.3 Calculations for Magnetic Resonance Coupling (S–S Type Circuit)
5.6 Circuit of Magnetic Resonance Coupling (S–P)
5.6.1 Verification of Equivalent Circuit (S–P)
5.6.2 Efficiency and Power (S–P)
5.6.3 Calculations for Magnetic Resonance Coupling (S–P Circuit)
5.7 Comparison Summary of Five Types of Circuits
5.8 Evaluation and Transition Across Four Types of Circuits Along X1 and X2 Axes
5.9 Comparison of Four Types of Circuits During Magnetic Flux Distribution
5.10 Role of Primary Magnetic Flux
References
6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC
6.1 S–S, S–P, P–S, and P–P
6.2 LCL and LCC, etc.
6.3 Relay Coil and Gyrator Properties
6.4 k = 0 Properties
References
7 Open and Short-Circuit-Type Coils
7.1 Overview of Open- and Short-Circuit Type Coils
7.2 An Intuitive Understanding of the Open-Circuit Type Through the Dipole Antenna
7.3 Lumped-Element Model and Distributed Constant Circuit
7.4 Open-Circuit and Short-Circuit Type Coils from the Perspective of a Distributed Constant Circuit
7.5 The Open-Circuit Type Coil
7.6 The Short-Circuit Type Coil
7.7 Summary of the Open-Circuit Type and Short-Circuit Type Coils
References
8 Magnetic Resonance Coupling Systems
8.1 Overview of Wireless Power Transfer Systems
8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries), and Constant-Power Loads (Motors and Electric Devices)
8.2.1 Resistive Loads
8.2.2 Constant-Voltage Load (Secondary Battery)
8.2.3 Constant-Power Load
8.3 High Power Through Frequency Tracking Control
8.4 Overview of Achieving Maximum Efficiency Through Impedance Tracking Control
8.5 Preliminary Knowledge for Discussing Efficiency Maximization Through Impedance Tracking Control
8.5.1 AC–DC Conversion Through Rectifiers
8.5.2 AC/DC Voltage, Current, and Equivalent Load Resistance
8.5.3 Concept of Impedance Transformation by Using a DC/DC Converter
8.5.4 Impedance Transform in a Step-Down Chopper, Step-up Chopper, and Step up/Down Chopper Using a DC/DC Converter
8.6 Realization of Maximum Efficiency Tracking Control Through Impedance Optimization
8.6.1 Simple Design for Maximum Efficiency
8.6.2 Strict Design for Maximum Efficiency Control
8.7 Realization of the Maximum Efficiency and the Desired Power
8.7.1 Secondary Side Maximum Efficiency and Primary Side Desired Power
8.7.2 The Primary Side Maximum Efficiency and the Secondary Side Desired Power
8.8 Secondary-Side Power on–off Mechanism (Correspondence to Short Mode and Constant Power Load)
8.8.1 Half Active Rectifier (HAR)
8.8.2 Maximum Efficiency Control with HAR
8.9 Maximum Efficiency and Desired Power Simultaneously by Secondary Side Control Alone
8.10 Estimation of Mutual Inductance
References
9 Repeating Coil and Multiple Power Supply (Basic)
9.1 Linear Arrangement of Repeating Coil
9.1.1 Linear Arrangements of Three Repeating Coils
9.1.2 Linear Arrangement of Repeating Coil (N Coils)
9.2 K-Inverter Theory (Gyrator Theory)
9.2.1 Calculation Method Using K-Inverter
9.2.2 Dead Zone
9.3 Calculation Using Z-Matrix Accounting for Cross-Coupling (Three Coils)
9.4 Positive and Negative Mutual Inductance
9.4.1 Vertical Direction and +Lm
9.4.2 Horizontal Direction and -Lm
9.4.3 Combination of Vertical and Horizontal Directions
9.4.4 Multiple Power Supply Equivalent Circuit (Three Coils)
9.5 Calculation with Z-Matrix Accounting for Cross-Coupling (N Coils)
References
10 Applications of Multiple Power Supplies
10.1 Improved Efficiency When Using Multiple Power Supplies
10.1.1 The Model Considered in This Chapter
10.1.2 Deriving the Formula for Total Efficiency and Optimal Load Value
10.1.3 When Mutual Inductance Lm Varies
10.1.4 When Mutual Inductance Lm Is Constant
10.1.5 Graph of Multiple Power Supply Efficiency Increase
10.2 Cross-Coupling Canceling Method (CCC Method)
10.2.1 Formulas for Multiple Loads and Cross-Coupling
10.2.2 Verification of the Effects of Cross-Coupling and Simple Frequency Tracking Method (Method A)
10.2.3 Optimization for Load Resistance Only (Method B) and Limits Thereof
10.2.4 Cross-Coupling Canceling Method (Method C)
References
11 Basic Characteristics of Electric Field Resonance
11.1 Capacitors as Electromagnetics and the True Nature of Displacement Current
11.2 Electric Field Coupling
11.3 Introduction of Electric Resonance Coupling
11.4 Air Gap Characteristics
11.5 Misalignment Characteristics
11.6 Near-Field Electric Field
11.7 Comparison of Electric Field Coupling and Electric Resonance Coupling (N–N, N–S, S–N, S–S, S–P)
11.7.1 Electric Field Coupling Type Coupler Structure and Capacitance
11.7.2 Each Topology of Electric Field Coupling
11.7.3 Derivation of Theoretical Formula
11.7.4 Validation of Theoretical Formula and Analysis
11.7.5 Efficiency Comparison
11.7.6 Conditions for High Efficiency and High-Power Transfer
References
12 Unified Theory of Magnetic Field Coupling and Electric Field Coupling
12.1 Unified Interpretation of Magnetic Field Coupling and Electric Field Coupling and Preparation for Comparison
12.1.1 Magnetic Field Coupling (IPT)
12.1.2 Electric Field Coupling (CPT)
12.1.3 Unified Design Method for Resonance Conditions
12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)
12.2.1 Derivation of Compensation Capacitor Condition ( R1 = R2 = 0 )
12.2.2 Circuit Analysis and Characterization ( R1 =R2 =0 )
12.2.3 Power Transmission Characteristics Evaluation During Load Fluctuation
12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)
12.3.1 Derivation of Compensation Inductor Condition ( R1 = R2 = 0 )
12.3.2 Circuit Analysis and Characterization ( R1 =R2 =0 )
12.3.3 Power Transmission Characteristics Evaluation During Load Fluctuation
12.4 Comparison and Unified Theory of Magnetic Resonance Coupling and Electric Resonance Coupling
12.4.1 Unified Interpretation
References

Citation preview

Takehiro Imura

Wireless Power Transfer Using Magnetic and Electric Resonance Coupling Techniques

Wireless Power Transfer

Takehiro Imura

Wireless Power Transfer Using Magnetic and Electric Resonance Coupling Techniques

123

Takehiro Imura Tokyo University of Science Noda, Chiba, Japan

ISBN 978-981-15-4579-5 ISBN 978-981-15-4580-1 https://doi.org/10.1007/978-981-15-4580-1

(eBook)

[Draft] Original Japanese edition: “Jikai kyoumei niyoru waiyaresu denryoku densou” by Takehiro Imura. Copyright © 2017 by Takehiro Imura. Published by Morikita Publishing Co., Ltd. 1-4-11, Fujimi, Chiyoda-ku, Tokyo 102-0071 Japan © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Acknowledgements

I am deeply grateful to everyone. Prof. Hori, Prof. Fujimoto, Uchida, Okabe, Ote, Koyanagi, Kato, Moriwaki, Beh, Parakon, Koh, Paopao, Tsuboka, Tanikawa, Narita, Kimura, G. Yamamoto, Hiramatsu, Pakorn, Hata, Kimura, Gunji, Nagai, M. Sato, Lovison, Kobayashi, Shibata, Furusato, Takeuchi, Nishimura, Cui, Otuka, Yazaki, Suzuki, Takahashi, Hanajiri, Ji, Helanka, Utsu, Katada, Tajima, Nawada, Tokita, Tantan, Chen, Tomii, Y. Ota, Nakajima, Chandrasekaran, Muruga Prashanth, Sasaki, Kuroda, A. Ota, Kaminuma, S. Yamamoto, Akashi, Ikeda, Ichiyanagi, Suita, Taira, Hanawa, A. Sato, Katsunori, Toshiko, Noriko, Makiko, Midori, Karin, Ema, Ruka. Thank you.

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Wireless Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Types of Wireless Power Transfer . . . . . . . . . . . . . . . . . . 1.1.1 Detailed Differentiation of Wireless Power Transfer Types . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Operating Frequency . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of Electromagnetic Induction and Magnetic Resonance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Difference Between Electromagnetic Induction and Magnetic Resonance Coupling . . . . . . . . . . . 1.2.2 Types of Circuit Topology . . . . . . . . . . . . . . . . . 1.3 Outline of Electric Field Coupling and Electric Resonance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Radiative-Type Power Transmission . . . . . . . . . . . . . . . . 1.4.1 Microwave Power Transmission . . . . . . . . . . . . . 1.5 Basic System Configuration . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Knowledge of Electromagnetism and Electric Circuits 2.1 Resistance, Coils, and Capacitors . . . . . . . . . . . . . . . . . . 2.1.1 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Coils Seen from a Circuit Viewpoint . . . . . . . . . 2.1.3 Capacitors as Seen from a Circuit Viewpoint . . . 2.2 Principle of Electromagnetic Induction . . . . . . . . . . . . . . 2.2.1 Magnetic Field H, Magnetic Flux Density B, and Magnetic Flux U . . . . . . . . . . . . . . . . . . . . 2.2.2 Ampere’s Law and Biot–Savart Law . . . . . . . . . 2.2.3 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Mechanism of Energy Transmission by Electromagnetic Induction (Electromagnetism Viewpoint) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.5

Electromagnetic Induction Described from a Circuit Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Self-inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Mutual Inductance and Coupling Coefficient . . . . . . 2.2.8 Neumann’s Formula (Derivation of Inductance) . . . . 2.3 High-Frequency Loss (Resistance) . . . . . . . . . . . . . . . . . . . . 2.3.1 Copper Loss, Skin Effect, and Proximity Effect . . . . 2.3.2 Iron Loss (Hysteresis Loss and Eddy Current Loss) . 2.3.3 Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Non-resonant Circuit Transient Phenomena (Pulse) . . . . . . . . 2.5 Resonance Circuit and Transient Phenomena . . . . . . . . . . . . 2.5.1 LCR Series Circuit Transient Phenomena (Pulse) . . . 2.5.2 LCR Series Circuit and Q Value . . . . . . . . . . . . . . . 2.5.3 Magnetic Field Resonance Transient Phenomena . . . 2.6 Root Mean Square (RMS) Values, Active Power, Reactive Power, and Instantaneous Power . . . . . . . . . . . . . . . . . . . . . 2.6.1 Root Mean Square (RMS) Values . . . . . . . . . . . . . . 2.6.2 Instantaneous Power Versus Active Power and Reactive Power . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Magnetic Resonance Coupling Phenomenon and Basic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Coils and Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Spiral, Helical, and Solenoid Coils . . . . . . . . . 3.2 Summary of Air Gap and Misalignment Characteristics . 3.2.1 Efficiency, Power, and Input Impedance with Air Gap . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Misalignment Characteristics . . . . . . . . . . . . . . 3.3 Near Field of Magnetic Field and Electric Field . . . . . . 3.4 Frequency Determinant (kHz to MHz to GHz) . . . . . . . 3.4.1 Resonant Frequency . . . . . . . . . . . . . . . . . . . . 3.4.2 Parameters of Coils for kHz, MHz, and GHz . . 3.4.3 kHz Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 GHz Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Equivalent Circuit at the Resonant Frequency . . . . . . . . . . . . 4.2.1 Derivation of the Equivalent Circuit at the Resonant Frequency, and Its Maximum Efficiency . . . . . . . . . 4.2.2 Derivation of Voltage Ratio and Current Ratio . . . . . 4.2.3 Load Characteristics at the Resonant Frequency . . . . 4.2.4 kQ Representation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Maximum Efficiency Considering Coil Performance . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Comparison Between Electromagnetic Induction and Magnetic Resonance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Five Basic Types of Circuits: N–N, N–S, S–N, S–S, S–P . . 5.2 Non-resonant Circuit (N–N) . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Verification of Equivalent Circuit (N–N) . . . . . . . . 5.2.2 Efficiency and Power (N–N) . . . . . . . . . . . . . . . . . 5.3 Secondary-Side Resonant Circuit (N–S) . . . . . . . . . . . . . . . 5.3.1 Verification of Equivalent Circuit (N–S) . . . . . . . . 5.3.2 Efficiency and Power (N–S) . . . . . . . . . . . . . . . . . 5.4 Primary-Side Resonant Circuit (S–N) . . . . . . . . . . . . . . . . . 5.4.1 Verification of Equivalent Circuit (S–N) . . . . . . . . 5.4.2 Efficiency and Power (S–N) . . . . . . . . . . . . . . . . . 5.4.3 Design Using Primary-Side Resonance Conditions (S–N) . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Design with Overall Resonance Condition (Power Factor = 1) . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Comparison Between Primary-Side and Overall Resonance Conditions (Power Factor = 1) . . . . . . . 5.5 Circuit of Magnetic Resonance Coupling (S–S) . . . . . . . . . 5.5.1 Verification of Equivalent Circuit (S–S) . . . . . . . . . 5.5.2 Efficiency and Power (S–S) . . . . . . . . . . . . . . . . . . 5.5.3 Calculations for Magnetic Resonance Coupling (S–S Type Circuit) . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Circuit of Magnetic Resonance Coupling (S–P) . . . . . . . . . 5.6.1 Verification of Equivalent Circuit (S–P) . . . . . . . . . 5.6.2 Efficiency and Power (S–P) . . . . . . . . . . . . . . . . . . 5.6.3 Calculations for Magnetic Resonance Coupling (S–P Circuit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Comparison Summary of Five Types of Circuits . . . . . . . . . 5.8 Evaluation and Transition Across Four Types of Circuits Along X1 and X2 Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Comparison of Four Types of Circuits During Magnetic Flux Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Role of Primary Magnetic Flux . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Feature of P–S, P–P, LCL-LCL, and LCC-LCC 6.1 S–S, S–P, P–S, and P–P . . . . . . . . . . . . . . . 6.2 LCL and LCC, etc. . . . . . . . . . . . . . . . . . . . 6.3 Relay Coil and Gyrator Properties . . . . . . . . 6.4 k = 0 Properties . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Open and Short-Circuit-Type Coils . . . . . . . . . . . . . . . . . . . . . . 7.1 Overview of Open- and Short-Circuit Type Coils . . . . . . . . . 7.2 An Intuitive Understanding of the Open-Circuit Type Through the Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Lumped-Element Model and Distributed Constant Circuit . . . 7.4 Open-Circuit and Short-Circuit Type Coils from the Perspective of a Distributed Constant Circuit . . . . . . . . . . . . 7.5 The Open-Circuit Type Coil . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Short-Circuit Type Coil . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Summary of the Open-Circuit Type and Short-Circuit Type Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Magnetic Resonance Coupling Systems . . . . . . . . . . . . . . . . . . . 8.1 Overview of Wireless Power Transfer Systems . . . . . . . . . . 8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries), and Constant-Power Loads (Motors and Electric Devices) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Resistive Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Constant-Voltage Load (Secondary Battery) . . . . . . 8.2.3 Constant-Power Load . . . . . . . . . . . . . . . . . . . . . . 8.3 High Power Through Frequency Tracking Control . . . . . . . 8.4 Overview of Achieving Maximum Efficiency Through Impedance Tracking Control . . . . . . . . . . . . . . . . . . . . . . . 8.5 Preliminary Knowledge for Discussing Efficiency Maximization Through Impedance Tracking Control . . . . . . 8.5.1 AC–DC Conversion Through Rectifiers . . . . . . . . . 8.5.2 AC/DC Voltage, Current, and Equivalent Load Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Concept of Impedance Transformation by Using a DC/DC Converter . . . . . . . . . . . . . . . . 8.5.4 Impedance Transform in a Step-Down Chopper, Step-up Chopper, and Step up/Down Chopper Using a DC/DC Converter . . . . . . . . . . . . . . . . . .

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Realization of Maximum Efficiency Tracking Control Through Impedance Optimization . . . . . . . . . . . . . . . . . . . 8.6.1 Simple Design for Maximum Efficiency . . . . . . . . 8.6.2 Strict Design for Maximum Efficiency Control . . . . 8.7 Realization of the Maximum Efficiency and the Desired Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Secondary Side Maximum Efficiency and Primary Side Desired Power . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 The Primary Side Maximum Efficiency and the Secondary Side Desired Power . . . . . . . . . 8.8 Secondary-Side Power on–off Mechanism (Correspondence to Short Mode and Constant Power Load) . . . . . . . . . . . . . 8.8.1 Half Active Rectifier (HAR) . . . . . . . . . . . . . . . . . 8.8.2 Maximum Efficiency Control with HAR . . . . . . . . 8.9 Maximum Efficiency and Desired Power Simultaneously by Secondary Side Control Alone . . . . . . . . . . . . . . . . . . . . . . 8.10 Estimation of Mutual Inductance . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

8.6

9

Repeating Coil and Multiple Power Supply (Basic) . . . . . . . . . . 9.1 Linear Arrangement of Repeating Coil . . . . . . . . . . . . . . . . 9.1.1 Linear Arrangements of Three Repeating Coils . . . 9.1.2 Linear Arrangement of Repeating Coil (N Coils) . . 9.2 K-Inverter Theory (Gyrator Theory) . . . . . . . . . . . . . . . . . . 9.2.1 Calculation Method Using K-Inverter . . . . . . . . . . 9.2.2 Dead Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Calculation Using Z-Matrix Accounting for Cross-Coupling (Three Coils) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Positive and Negative Mutual Inductance . . . . . . . . . . . . . . 9.4.1 Vertical Direction and +Lm . . . . . . . . . . . . . . . . . . 9.4.2 Horizontal Direction and −Lm . . . . . . . . . . . . . . . . 9.4.3 Combination of Vertical and Horizontal Directions . 9.4.4 Multiple Power Supply Equivalent Circuit (Three Coils) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Calculation with Z-Matrix Accounting for Cross-Coupling (N Coils) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Applications of Multiple Power Supplies . . . . . . . . . . . . . . . . . . 10.1 Improved Efficiency When Using Multiple Power Supplies . 10.1.1 The Model Considered in This Chapter . . . . . . . . . 10.1.2 Deriving the Formula for Total Efficiency and Optimal Load Value . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 When Mutual Inductance Lm Varies . . . . . . . . . . .

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Contents

10.1.4 When Mutual Inductance Lm Is Constant . . . . . . . . . 10.1.5 Graph of Multiple Power Supply Efficiency Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Cross-Coupling Canceling Method (CCC Method) . . . . . . . . 10.2.1 Formulas for Multiple Loads and Cross-Coupling . . 10.2.2 Verification of the Effects of Cross-Coupling and Simple Frequency Tracking Method (Method A) . . . 10.2.3 Optimization for Load Resistance Only (Method B) and Limits Thereof . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Cross-Coupling Canceling Method (Method C) . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Basic Characteristics of Electric Field Resonance . . . . . . . . . . . . 11.1 Capacitors as Electromagnetics and the True Nature of Displacement Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electric Field Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Introduction of Electric Resonance Coupling . . . . . . . . . . . . 11.4 Air Gap Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Misalignment Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Near-Field Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Comparison of Electric Field Coupling and Electric Resonance Coupling (N–N, N–S, S–N, S–S, S–P) . . . . . . . . . . . . . . . . . 11.7.1 Electric Field Coupling Type Coupler Structure and Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Each Topology of Electric Field Coupling . . . . . . . . 11.7.3 Derivation of Theoretical Formula . . . . . . . . . . . . . . 11.7.4 Validation of Theoretical Formula and Analysis . . . . 11.7.5 Efficiency Comparison . . . . . . . . . . . . . . . . . . . . . . 11.7.6 Conditions for High Efficiency and High-Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Unified Theory of Magnetic Field Coupling and Electric Field Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Unified Interpretation of Magnetic Field Coupling and Electric Field Coupling and Preparation for Comparison . . . . . . . . . . 12.1.1 Magnetic Field Coupling (IPT) . . . . . . . . . . . . . . . . 12.1.2 Electric Field Coupling (CPT) . . . . . . . . . . . . . . . . . 12.1.3 Unified Design Method for Resonance Conditions . . 12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP) . . . . . . . 12.2.1 Derivation of Compensation Capacitor Condition ðR1 ¼ R2 ¼ 0Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12.2.2 Circuit Analysis and Characterization ðR1 6¼ R2 6¼ 0Þ . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Power Transmission Characteristics Evaluation During Load Fluctuation . . . . . . . . . . . . . . . . . . 12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP) . . . . 12.3.1 Derivation of Compensation Inductor Condition ðR1 ¼ R2 ¼ 0Þ . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Circuit Analysis and Characterization ðR1 6¼ R2 6¼ 0Þ . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Power Transmission Characteristics Evaluation During Load Fluctuation . . . . . . . . . . . . . . . . . . 12.4 Comparison and Unified Theory of Magnetic Resonance Coupling and Electric Resonance Coupling . . . . . . . . . . 12.4.1 Unified Interpretation . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Wireless Power Transfer

Wireless power transfer (WPT) refers to the technology of transmitting power without using any wires, such as electric wires, which are normally used to transmit power. Conventional wireless power transfer has been limited to transmitting power over an air gap (transmission distance) of several centimeters; however, in 2007, for the first time, it was proven that highly efficient high-power wireless power transfer is feasible over a large air gap exceeding 1 m [1]. This technology is referred to as magnetic resonance coupling. Before the unveiling of magnetic resonance coupling, it was believed that wireless power transfer was feasible only over a distance of 1/10th of a coil diameter; however, after the emergence of magnetic resonance coupling, it was found that power could actually be transmitted with high efficiency and high power over distances equal to or greater than a coil diameter. Figure 1.1 illustrates the setup for experiments conducted by the authors. We found that the technology works satisfactorily with a large air gap, even when off-center. This is a major boost for research and development in the field of wireless power technology. This technology only recently came to light, making good use of the resonance phenomenon based on coupling through a magnetic field, that is, electromagnetic induction. On the other hand, various other methods of wireless power transfer have been studied in addition to magnetic resonance coupling. In this chapter, we describe the basic setup of wireless power transfer.

1.1 Types of Wireless Power Transfer There are several available methods for wireless power transfer, and one common feature that all these methods share is the wireless power transmission using highfrequency alternating current (AC). Broadly speaking, there are two types of wireless power transfer: coupling and radiative. The coupling type is further categorized into the magnetic field and electric field types, whereas the radiative type is categorized © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_1

1

2

1 Wireless Power Transfer

(a) Directly above

(b) Off-center

(c) To the side

Fig. 1.1 Setup for magnetic resonance coupling light bulb light-up experiment

into microwave (electromagnetic wave) and laser (optical) types. Hence, wireless power transfer can be typically classified into four types.1 However, because there has been relatively little research on lasers, it is often categorized into only three types.

1.1.1 Detailed Differentiation of Wireless Power Transfer Types Figure 1.2 depicts the types of wireless power transfer. First, they are broadly categorized into coupling and radiative types. The radiative type is described in Sect. 1.4,

Fig. 1.2 Types of wireless power transfer 1 Depending

on the focus of attention, the abovementioned types are sometimes referred to as a four-member group consisting of electromagnetic induction, magnetic resonance coupling, electric field coupling, and microwaves; however, this is not systematically balanced. Thus, in this book, we have categorized the types as indicated above.

1.1 Types of Wireless Power Transfer

3

Fig. 1.3 Types of couplings

and the coupling type is described in Sect. 1.1. Figure 1.3 depicts the types of couplings, and Fig. 1.4 illustrates the corresponding diagrams. First, we refer to the power-transmitting side as the primary side and the power-receiving side as the secondary side. Thus, the coupling type can be categorized into ➀ magnetic field coupling (electromagnetic induction) and ➁ electric field coupling (displacement current) depending on whether the coupling is via a magnetic field H or an electric field E. The coupling type is further categorized into a total of four types based on the resonance phenomenon. ➀ The magnetic field coupling type generally uses electromagnetic induction. ➁ The electric field coupling type uses an electric field instead of a magnetic field. Furthermore, introducing a resonant capacitor during electromagnetic induction will cause the capacitor to resonate with the coil, making the resonant frequency on the power-transmitting side the same as that on the power-receiving side. This enables the achievement of high efficiency and high power, as well as a large air gap. In other words, the skillful use of the resonance phenomenon is referred to as ➂ magnetic resonance coupling. Similarly, while the electric field coupling type generally uses electric field coupling, the introduction of a resonant coil, such that it resonates with a capacitor (coupler) and skillfully uses the resonance phenomenon, is referred to as ➃ electric resonance coupling. Put simply, the resonance phenomenon (electromagnetic induction or electric field coupling) is frequently used for power factor improvement. However, magnetic resonance coupling and electric resonance coupling will not work if the resonant conditions are not right. The specific types are discussed in Chaps. 5 and 11. Power transmission is possible whenever a power-transmitting side and power-receiving side are coupled with a magnetic field or electric field. The coupling part is generally referred to as a resonator, or partly as an antenna, although it is often referred to as a coil when coupled with a magnetic field, and a coupler or plate when coupled with an electric field (see in “[Column] Terms”).

4

1 Wireless Power Transfer

Fig. 1.4 Diagrams of magnetic field coupling, magnetic resonance coupling, electric field coupling, and electric resonance coupling

[Column] Terms Wireless power transfer is a field involving interdisciplinary fusion. As a result of the near-simultaneous participation of various persons of different backgrounds, various words with the same meaning are used for certain terms. Unifying the terminology in this book would be difficult; however, because it is important to understand the concept clearly, here, we provide a simple summary of the terms used. • Wireless power transfer Wireless power transfer is sometimes referred to as wireless power supply or noncontact power supply. Although these terms have the almost same meanings, sometimes it can be wireless but in contact. Thus, to avoid any misunderstanding, the term “noncontact power supply” is sometimes deliberately used. However, wireless

1.1 Types of Wireless Power Transfer

5

power transfer generally means no contact in most cases, and hence, the three terms are used interchangeably. • Wireless charging Wireless charging refers to wireless power transfer involving charging, which is also referred to as noncontact charging. • Magnetic resonance coupling Magnetic resonance coupling is a coupling method that uses a magnetic field and resonance. It is also referred to as magnetic resonance. Here, the word resonance may be understood to mean resonating and coupling. More specifically, magnetic resonance coupling is a technology that uses the electromagnetic induction phenomenon and accomplishes wireless power transfer with a circuit topology (circuit structure) consisting of resonant circuits on both sides. • Electromagnetic induction (magnetic coupling) Electromagnetic induction is a term generally used for coupling methods using a magnetic field. It originally referred to the electromagnetic induction phenomenon itself; however, in wireless power transfer, it is used to indicate power transmission. It is also referred to as magnetic coupling or inductive coupling, as well as inductive power transfer (IPT). • Electric resonance coupling (electric resonance) Electric resonance coupling is a coupling method that uses an electric field and resonance. It is also referred to as electric resonance. • Electric field coupling (electric coupling) Electric field coupling is a term generally used for coupling methods that use an electric field. It is also referred to as electric coupling or capacitive coupling, as well as capacitive power transfer (CPT). • Electromagnetic resonance coupling (electromagnetic resonant coupling) This is a term generally used for both electric resonance coupling and magnetic resonance coupling. It is also referred to as electromagnetic resonance. • Coupling components: coil, plate, resonator, coupler, antenna The part of wireless power transfer may be referred to as coupler; however, generally, coupling is often achieved through a magnetic field, and hence, it is frequently referred to as a coil. In addition, a resonator refers to the inclusion of both a coil and capacitor. The background is described below. Initially, coupling components were also called antennas because power was transmitted wirelessly using resonance [2]. However, because an antenna refers to a component that transmits electromagnetic waves, if we focus our attention on the phenomenon of using resonance and coupling to transmit power, then the term resonator is more suitable [3, 4]. Hence, the term resonator

6

1 Wireless Power Transfer

has been increasingly used. Resonators have been used in communication, and thus, are more readily recognized for signals than power. However, because weak power is transmitted during communication, their meaning is unchanged. In addition, it is possible to couple magnetic fields without resonance, and sometimes only coupling components are indicated. Thus, instead of resonator, the word coil is often used. The word coupler is also used. Hence, for the time being, the word coil is generally used in reference to magnetic fields. As for electric fields, because of the image of a capacitor electrode plate, the words plate or electrode plate are sometimes used.

1.1.2 Operating Frequency Figure 1.5 depicts the frequencies and types of wireless power transfer. According to a March 15, 2016 revised ministerial ordinance within Japan, the frequencies for electrical vehicles (EV) are in the 85 kHz band and are generally referred to as being in the 79–90 kHz band at 7.7 kW or less.2 Thus, the 6.78 MHz band for consumer products, such as mobile devices, in the range of 6.765–6.795 MHz at 100 W or less,3 generally referred to as the industrial, scientific, and medical (ISM) band, has been added to type designations for the magnetic field coupling type. Similarly, frequencies designated for consumer products, such as mobile devices,

Fig. 1.5 Frequency and types of wireless power transfer

2 Details of March 15, 2016, revised the ministerial ordinance written as Ministry of Internal Affairs

and Communications No. 15. In ministerial ordinance documents, items for electric vehicles are indicated as “noncontact power devices for electrical vehicles,” and “vehicles fully or partly using electricity as a power supply;” thus, hybrid vehicles are also recognized. 3 ISM is a frequency band made for industry, science, and medicine, and is used relatively freely worldwide. This includes the 6.78 MHz band. However, it is free of restrictions in all countries. In the present revision, the allowable magnetic field strength at a location 10 m away differs between 44 dB for 6.765–6.776 MHz and 64 dB for 6.776–6.795 MHz. Here, 1 µA/m is 0 dB. In addition, in ministerial ordinance documents, the expression “6.7 MHz band magnetic field coupling type general noncontact power supply device” is used.

1.1 Types of Wireless Power Transfer

7

in the range of 425–524 kHz at a frequency of 100 W or less,4 referred to as the 400 kHz band, have been added as type designations for the electric field coupling type. Previously, the use of high-frequency wireless devices exceeding 10 kHz at 50 W required application on an individual basis according to rules governing highfrequency equipment under the Radio Law. However, if the devices are subject to type designations, it is possible for users to use such devices without following the procedures for high-frequency equipment. In addition, in general, wireless power transfer devices may be sold after permission is granted following type-designation procedures proposed by manufacturers or importers. From a technical viewpoint, power transmission is possible at a commercial frequency of 50/60 Hz; however, in terms of device miniaturization and efficiency, in general, devices are operated at high frequencies. High frequencies also vary, including the kHz (103 ) bands, MHz (106 ) bands, GHz (109 ) bands, and THz (1012 ) bands. In general, frequencies up to the MHz bands are of the coupling type, and frequencies in the GHz band or higher are of the radiative type, which is also influenced by the wavelength (Table 1.1). The relationship of frequency f and wavelength λ is described by Eq. (1.1), where c is the speed of light. f =

Table 1.1 Frequencies and wavelengths

4 The

c νc = 299,792,458 ≈ 3 × 108 (m/s) λ Frequency

Wavelength

1 kHz

300 km

10 kHz

30 km

100 kHz

3 km

1 MHz

300 m

10 MHz

30 m

100 MHz

3m

1 GHz

30 cm

10 GHz

3 cm

20 kHz

15 km

85 kHz

3.5 km

6.78 MHz

44.2 m

13.56 MHz

22.1 m

2.45 GHz

12.2 cm

5.8 GHz

5.2 cm

(1.1)

usage frequency is the 400 kHz band, which was established as 425–471, 480–489, 491– 494, 506–517, and 519–524 kHz. In addition, in ministerial ordinance documents, the expression “400-kHz-band electric field coupling type general noncontact power supply device” is used.

8

1 Wireless Power Transfer

For example, for the coupling type 13.56 MHz, one wavelength is approximately 22 m. However, for the radiative type 2.45 GHz, one wavelength is approximately 12 cm. The coupling type requires the wavelength to be sufficiently larger than the coil size; thus, at 1 GHz or higher, the coil size is too small for the coupling type. Hence, GHz coil is generally not used except for special applications. For example, when approximately a five turns coil generates 13.56 MHz, the coil diameter size is approximately 30 cm, which is about 1/100 the wavelength of approximately 22 m. That is, the wavelength is sufficiently larger than the coil size at MHz bands or lower; thus, the coupling type is the main wireless power transfer used. On the other hand, when operating with the radiative type, no electromagnetic waves are emitted at coil sizes that are not close to the wavelength; thus, the wavelength and antenna length are close in size. For example, with a half-wavelength dipole antenna, which is a typical example of a radiative type, the antenna length is approximately 6 cm, which is half the wavelength of approximately 12 cm at 2.45 GHz, making it a practical size. Additionally, the half-wavelength of the wavelength at 13.56 MHz is 11 m; therefore, the radiative type is not a practical size for MHz. [Column] Frequency of wireless power transfer and actual sense of speed Wireless power transfer occurs at approximately 9 kHz to 13.56 MHz, making each cycle very short. For example, if the frequency is 100 kHz, then in terms of time, period is 10 µs. An example of the difference between the switching speed of wireless power transfer and the speed at which we actually move is a car running at 100 km/h takes 36 ms = 36,000 µs to travel 1 m, and in 10 µs only advances 0.000278 m = 0.278 mm.

1.2 Outline of Electromagnetic Induction and Magnetic Resonance Coupling As illustrated in Figs. 1.6 and 1.7, the magnetic resonance coupling method narrows the conditions found in the electromagnetic induction method [5]. Detailed differences between the two methods are described in Chap. 5, but put simply, both the

Fig. 1.6 Relationship of electromagnetic induction and magnetic resonance coupling

1.2 Outline of Electromagnetic Induction and Magnetic Resonance …

(a) Nonresonant electromagnetic induction

9

(b) Magnetic resonance coupling

Fig. 1.7 Schematic diagram of electromagnetic induction and magnetic resonance coupling

magnetic resonance coupling method and electromagnetic induction method involve coupling through a magnetic field, and the principle of electromagnetic induction is used for coupling components in both the methods. However, in magnetic resonance coupling, as depicted in Fig. 1.7b, forming a resonant circuit on both the primary side and secondary side results in a circuit topology (circuit structure) that makes good use of magnetic resonance coupling. Thus, highly efficient high power is realized even with large air gaps. The principle of electromagnetic induction is described in Chap. 2. As depicted in Fig. 1.8, as a result of the magnetic flux created by current I 1 flowing to the primary side passing through (interlinking) a secondary-side coil loop, energy is transmitted to the secondary side. Thus, voltage V is induced in a direction countering the magnetic flux on the secondary side, and power transmission occurs in the form of the current I 2 flow. When this occurs, energy is propagated not just by the magnetic field H alone but also via changes in the magnetic field dH/ dt. In other words, even if immobile magnets and several magnetic fields H with no fluctuations created by direct current are strongly present, if there are no changes in the magnetic fields over time or space, then no energy is transmitted.

Fig. 1.8 Power transmission by electromagnetic induction

10

1 Wireless Power Transfer

1.2.1 Difference Between Electromagnetic Induction and Magnetic Resonance Coupling The difference between electromagnetic induction and magnetic resonance coupling depends on whether resonance is being used. Magnetic resonance coupling (S–S) using resonance with series capacitor can achieve high efficiency and high power over a large air gap, as depicted in Fig. 1.9. On the other hand, with electromagnetic induction (N–N) using no resonance, when an air gap is large, power is hardly transmitted from the power-transmitting side, and the efficiency worsens owing to the inability of the power to be received on the receiving side. Thus, power transmission over a large air gap is not possible. In the example of efficiency η and received power P2 with air gap g illustrated in Fig. 1.9, this difference is self-evident. Conventionally, electromagnetic induction predominantly involved resonating either the primary side or the secondary side. Moreover, these electromagnetic induction methods were used in very close proximity. When the air gaps are small, efficiency is high regardless of the circuit structure, subsequently increasing the power. In such cases, when a resonant capacitor is introduced on the primary side, high power is achieved. In addition, the closeness implies that even if there is no resonance, power can be easily achieved in comparison with large air gaps. If the conditions under which minimum power is achievable are satisfied, then it is possible to introduce a resonant capacitor on the secondary side to improve efficiency. However, in any case, only the power or efficiency is improved, and when the air gap is large, efficiency is poor with primary resonance, and negligible power is received with secondary-side resonance, implying that realistically speaking, electromagnetic induction is not useful.5 In contrast, magnetic resonance coupling having both primary-side resonance and secondary-side resonance are capable of realizing high power and high efficiency over a large air gap. S-S S-S

100

N-N

1000

P2 [W]

80

η [%]

N-N

1200

60 40 20

800 600 400 200 0

0 0

50

100

150

200

0

50

100

150

g [mm]

g [mm]

(a) Efficiency

(b) Received power

200

Fig. 1.9 Comparison of magnetic resonance coupling (S–S) and non-resonant electromagnetic induction (N–N) 5 Strictly speaking, power improves as much as efficiency improves with secondary-side resonance.

However, this value is small.

1.2 Outline of Electromagnetic Induction and Magnetic Resonance …

11

1.2.2 Types of Circuit Topology The mechanism of magnetic resonance coupling achieving high power and high efficiency, as depicted in Fig. 1.9, is described in Chap. 5. Here, we first describe the circuit topology (circuit structure). Magnetic resonance coupling is a method of wireless power transfer for coupling through a magnetic field by making the primary-side resonance frequency and secondary-side resonance frequency the same. Figure 1.10 illustrates five typical circuits [5]. L 1 is the primary-side self-inductance, r 1 is the primary-side internal resistance, L 2 is the secondary-side self-inductance, r 2 is the secondary-side internal resistance, L m is the mutual inductance, RL is the load resistance, C 1 is the primary resonance capacitor, and C 2 is the secondary-side resonance capacitor.

(a) Nonresonance (N-N)

(c) Primary-side resonance (S-N)

(b) Secondary-side resonance (N-S)

(d) Magnetic resonant coupling (S-S)

(e) Magnetic resonant coupling (S-P) Fig. 1.10 Five circuit topologies

12

1 Wireless Power Transfer

Figure 1.10a depicts the electromagnetic induction with no resonance, similar to a transformer. However, coupling coefficient k  1 in a regular transformer; however, for wireless power transfer, the coupling coefficient k becomes smaller than 1. Because there is no resonant capacitor, this is a non-resonant circuit (N–N) with no C. Figure 1.10b depicts an electromagnetic induction method in which a resonant capacitor C 2 is inserted on the secondary side. In other words, it is a C 2 -only secondary-side resonance circuit (N–S: non-resonant–series). Figure 1.10c depicts an electromagnetic induction method in which a resonant capacitor C 1 is inserted on the primary side. In other words, it is a C 1 -only primary-side resonance circuit (S–N: series–non-resonant). Figure 1.10d depicts a magnetic resonance coupling method in which a resonant capacitor C 1 is inserted on the primary side, and a resonant capacitor C 2 is inserted on the secondary side. This is conditioned such that both the transmitting and receiving resonance frequency are the same. Here, both the transmitting side resonant capacitor C 1 and the power-receiving-side resonant capacitor C 2 are connected in series, making this an S–S (series–series) magnetic resonance coupling method. Figure 1.10e depicts a magnetic resonance coupling method similar to that in Fig. 1.10d, in which a resonant capacitor C 1 is inserted on the primary side and a resonant capacitor C 2 is inserted on the secondary side. This is conditioned such that both the transmitting and receiving resonance frequencies are the same. However, while the power-transmitting-side resonant capacitor C 1 is connected in series, the power-receiving-side resonant capacitor C 2 is connected in parallel, making this an S–P (series–parallel) magnetic resonance coupling method. These methods are explained in detail in Chap. 5.

1.3 Outline of Electric Field Coupling and Electric Resonance Coupling Power transmission may be achieved not only by magnetic field coupling but also by coupling a power-transmitting side and power-receiving side through an electric field. Electric field coupling is caused by time change in an electric field. However, coupling through an electric field alone is low in efficiency; thus, electric resonance coupling is used to achieve maximum efficiency and high power. The relationship between electric field coupling and electric resonance coupling is illustrated in Figs. 1.11 and 1.12. This relationship is the same as the relationship between electromagnetic induction (magnetic field coupling) and magnetic resonance coupling. In this book, we describe electric resonance coupling in Chap. 11. As a result of the electric field generated on the primary-side coupling the secondary side, energy is transmitted. In addition, the propagation of energy caused not by the electric field E itself but by changes in electric field dE/ dt is similar to that during magnetic field coupling. In other words, even if several unchanging electric fields E are strongly

1.3 Outline of Electric Field Coupling and Electric Resonance Coupling

13

Fig. 1.11 Relationship between electric field coupling and electric resonance coupling

(a) Nonresonant electric field coupling

(b) Electric resonance coupling

Fig. 1.12 Schematic diagram of electric field coupling and electric resonance coupling

present, if there are no changes in the electric fields over time or space, then energy is not transmitted.

1.4 Radiative-Type Power Transmission Radiative-type power transmission consists of two methods: microwave power transmission and laser power transmission. In this book, only a brief introduction is provided. Figure 1.13 depicts the radiative types, and Fig. 1.14 depicts the microwaveand laser-radiative-type power transmissions, which are the methods that ultimately brought wireless power transfer into being. Unlike the coupling type, the radiativetype power transmission truly sends power flying across large distances. In principle, power can reach any distance even with attenuation and scattering. In other words, it is possible to transmit (send) power even if there is no nearby receiving antenna. Thus, the radiative type is the more advantageous type, with the ability to transmit power over distances exceeding 10 m. Because electromagnetic waves reach into space, the distances they cover are of a completely different dimension from those of the coupling type. Therefore, the radiative type has no comparison with respect to power transmission to flying objects, power transmission from flying objects, and

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Fig. 1.13 Radiative types

(a) Microwave method

(b) Laser method

Fig. 1.14 Two radiation methods

space-based solar power. Microwave power transmission is in short distance technically possible; thus, the coupling type, which presently is superior in terms of cost and efficiency, may be covered by microwave power transmission in the future. Presently, microwave power transmission mainly involves operating within a several GHz band from the viewpoint of beam focusing and demand for device miniaturization. On the other hand, laser power transmission mainly involves operating within a THz band. At present, however, its overall efficiency is just over 50%, and future development is expected.

1.4.1 Microwave Power Transmission Among the radiative types, the electromagnetic wave type generally uses 2.45 GHz or 5.8 GHz microwaves and thus, is referred to as microwave power transmission [6]. Because it does not negatively affect regular communication, its research and development has proceeded on the premise of using the ISM bands of 2.45 and 5.8 GHz, which are special frequency bands that can be used relatively freely. Examples of

1.4 Radiative-Type Power Transmission

15

Fig. 1.15 Technology related to microwave power transmission

familiar frequency bands are the 2.45 GHz band used for microwave ovens and the 5.8 GHz band used for electronic toll collection system (ETC). The electromagnetic waves used here as a form of energy are the same as mobile phone electromagnetic waves; however, when they are transmitted as power, the question arises as to how much to focus the beams such that the power pinpoints the receiving side. On the other hand, microwave power transmission includes the energy harvesting technology that scatters and collects power. Microwave power transmission operates at high frequencies of 2.45 and 5.8 GHz, transmitting power created by a high-frequency power supply from a transmitting antenna as electromagnetic waves, which are then received by a receiving antenna located far away. This receiving antenna is referred to as a rectenna (Fig. 1.15a), which consists of an integration of a rectifier and a receiving antenna. Because the frequencies are high, the wavelength cannot be ignored with regard to the antenna and circuits. Hence, the voltage differs depending upon the circuit position, even on the same line. Separately constructing the receiving antenna and rectifier and then trying to combine them is not suitable. Hence, the receiving antenna and rectifier must be of an integrated type. Furthermore, it is possible to control the beam direction with a phased-array antenna using phase control. This is a form of technology for moving a beam back and forth (Fig. 1.15b). Microwave power transmission consists of a variety of technologies, including one (retrodirective system) that can receive a one-time pilot signal from the location to which power is being transmitted, and thereby transmit power by accurately concentrating the beam in the direction from which the pilot signal originates. Microwave power transmission is expected to be used in solar power satellites (SPS) and space solar power systems. • Laser power transmission Laser power transmission is radiated in the form of electromagnetic waves; however, it is in THz and thus, exists as light [7]. A familiar version of this type of transmission is a laser pointer. Its frequency is higher than that of microwaves, and power is transmitted by a laser transmitter and captured by a solar panel. Thus, improved solar panel efficiency also leads to improved overall efficiency. In addition, when we consider the application of laser power transmission to space-based solar power, although there is attenuation produced by clouds and other factors, owing to its high frequency and lower beam scattering than that of microwaves, it is being explored to supply power to a space elevator.

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Laser power has a unique characteristic in comparison with other forms of wireless power transfer: by using light in the visible range, power transmission locations are visible. This visibility makes it extremely safe. [Column] Interdisciplinary study of antennas, resonators, power electronics, physics, power devices, and high voltage Wireless power transfer technology is a melting pot of various fields, and it is similar to an ideal specimen of interdisciplinary research and development. The magnetic resonance coupling proposed by physicists in 2007 was first explained theoretically by the coupling mode theory and then described as representing wireless power transfer at 10 MHz, three digits higher than before, while the phenomenon itself remained a mystery. One reason why it was a mystery is that open circuit coil type, which can resonate itself, was used. To elucidate the phenomenon, many researchers in antenna engineering and resonators who had studied GHz bands joined the research. At a glance, electromagnetic induction and magnetic resonance coupling are completely different and are also high in frequency; thus, it was surprising that researchers who had previously studied electromagnetic induction were so late in joining this field of research. Gradually, as studies got closer to the general electromagnetic induction theories, including the equivalent circuit theory, and it was found that operation around 100 kHz was possible, many experts in power electronics and control theory, and experts researching electromagnetic induction, began to get involved. In addition, at 100 kHz, power devices also required metal-oxide-semiconductor field-effect transistors (MOSFETs) of silicon carbide (SiC) and not conventional silicon; hence, power device specialists also came aboard. The present reality of the technology is that there has been a momentary drop in frequency to 85 kHz; however, in the future this may return to high frequencies of 6.78 MHz or 13.56 MHz. In addition, it is very likely that gallinium nitride (GaN) and not SiC will be important for power devices. This is because if the problems of the power supply and rectifier circuits are solved, then somewhat higher frequencies will be advantageous in terms of the potential for lightweight coils or larger air gaps. In addition, because there is now a need for high-voltage measures, high-voltage specialists are needed. Thus, there is also a potential growth in the high-frequency power electronics field. For example, high-speed control using field-programmable gate array (FPGA) is also needed. Thus, many fields and industries have come together through cooperation between researchers and engineers.

1.5 Basic System Configuration Although wireless power transfer tends to be focused on the coupling components, it is important to have an understanding of the overall system. Therefore, we present the basic system configuration for wireless power transfer. Details are provided in Chap. 8. Here, we only introduce the basic concept. Figure 1.16 illustrates the conceptual diagram of the basic configuration of a wireless power transfer system.

1.5 Basic System Configuration

17

Fig. 1.16 Conceptual diagram of basic system configuration

The following is based upon a regular household with a 50/60 Hz, 100 V (AC): AC (50/60 Hz) → DC (DC power supply) → DC/DC converter (voltage variation) → AC (inverter, high-frequency generation, RF) → transmit and receive coil → DC (rectification) → DC/DC converter (impedance transformation) → load Because AC → DC from 50/60 Hz represents the generally established technology, it is often omitted in explanations 6 (1) DC generation (DC power supply): Creating DC (direct current) from applicable 50/60 Hz frequencies with an AC/DC converter; in other words, a DC power supply. In the case of batteries, they are DC to begin with; hence, this process does not require them. Power-supply voltage adjustment is sometimes accomplished with a DC/DC converter function included. (2) DC/AC transformation (inverter, high-frequency power supply): DC/AC transformation converts DC performed by a converter or obtained from a battery to AC; in other words, an inverter. At frequencies of MHz or higher, power supply is often referred to as a high-frequency power supply. The frequencies when the DC/AC transformation occurs are those used in wireless power transfer. Frequently used frequencies are 9.9, 20, 85, and 100–200 kHz, and the ISM bands are 6.78 and 13.56 MHz for the coupling type, and 2.45 and 5.8 GHz for the radiative type.

6 DC

is direct current, and AC is alternating current. RF (radio frequency) is a high frequency that is often used at GHz bands or higher. In addition, power-transmitting-side DC/DC converters are often included in DC power supply functions and are thus omitted at fixed voltages. Thus, we shall subsequently explain them as part of the DC power supply. Hereinafter, we explain each part in detail.

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1 Wireless Power Transfer

(a) Square wave

(b) Three-level square wave

(c) Sine wave

Fig. 1.17 Sine wave and square wave

For the coupling type, although the frequency is high, it is often written as AC representing regular alternating current, so whether it is high frequency needs to be determined in context. In addition, in the radiative-type microwave power transmission, this is often written as RF. When drawing an equivalent circuit, it is often written from here. As depicted in Fig. 1.17a, when operating an inverter, waveform is a square wave. However, as depicted in Fig. 1.17b, sometimes a three-level square wave is used with a phase shift to vary the voltage. By creating a 0 V voltage period, it is possible to adjust the time for which the voltage is applied, and vary the voltage. In other words, because the voltage can be varied even without a power-transmitting-side converter, a converter does not need to be used. In addition, when considering only basic wave components or a power supply that creates a sine wave, the power supply is often depicted with a sine wave, as depicted in Fig. 1.17c. In general, square-wave operation is common when using an inverter. A high-frequency power supply for high frequency also sometimes produces a sine-wave output. (3) Power transmission coil and receiving coil: Section of space (air gap) in which wireless power transmission actually occurs. Alternating current is always required and generally needs to be high frequency for high efficiency. A resonant capacitor is connected here. (4) AC/DC transformation (rectifier): Wireless power transferred AC (RF) is converted back to DC with a rectifier circuit; in other words, a rectifier. After passing through a rectifier, the voltage becomes positive and the DC is in a rippling state, which needs to be converted to unrippled DC with a smoothing capacitor. The larger the capacity of the smoothing capacitor, the lesser the rippling; however, because larger capacitor increases its volume, miniaturization is required. Furthermore, there are also methods using synchronous rectification to change switch to active to reduce loss. (5) DC/DC transformation (DC/DC converter): Has optimal load for maximum efficiency. To reach a voltage and current ratio equal to the optimal load, that is, impedance adjustment, a DC/DC converter is often used on front side of the load. Otherwise, DC/DC converter is used for adjustments to reach the desired power. (6) Load components (resistance, battery, etc.): For the load, often, the simplest resistance is depicted; however, in actual products, the load components include batteries, capacitors, and motors. These differ in operation and in the difficulties they pose to the overall system. The resistance is the load with the simplest

1.5 Basic System Configuration

19

operation. The batteries are treated as rated voltages. Capacitors raise the voltage as energy is stored. As for motors, power requirements fluctuate in real time according to operation; hence, their rated power load is momentary and requires accurate transmission of the necessary power, making it the most problematic load. Dividing a system into the six transformation components above is useful system. The simplest division involves three steps: combine (1) and (2) into the powertransmitting-side power-supply components, and divide the rest into the transmit and receive coil components in (3), and the load components from rectification to impedance adjustment in (4)–(6), that is, the three parts of the power-transmitting side, coil components, and power-receiving side. Efficiency can be categorized according to usage as intercoil efficiency (AC– AC efficiency), DC–DC efficiency, and AC–DC efficiency. Considering the system efficiency and overall efficiency originally from 50/60 Hz, the AC–DC (50/60 Hz load) efficiency is AC (50/60 Hz) → DC (DC power supply) → DC/DC transformation (DC/DC converter) → AC (high-frequency power supply) → AC (power transmission coil) → AC (receiving coil) → DC (rectification) → DC/DC transformation (DC/DC converter) → DC (load). However, it is also possible to classify DC–DC (DC power supply-load) into system efficiency and overall efficiency. Control signals are transmitted through wireless communication. Considering power and signals as separate systems is referred to as out-of-band (out-band), and considering them as the same system is referred to as in-band. In-band is not realistic because the signals are treated the same as with wireless equipment and likely require license acquisition. In general, control signals are generally transmitted wirelessly out-of-band in a separate system.7 In Fig. 1.16, there is a deliberate direct change in the DC power-supply voltage on the power-transmitting side, and control signals are entered into a DC/DC converter; however, if a three-level waveform is used instead of a simple square wave, direct inverter control is performed so that the voltage can be changed. In addition, although there is deliberate impedance transformation on the power-receiving side and DC/DC converter control, if the rectifier components are an active circuit, then it is possible to control the rectifier components. [Column] Reflection and efficiency Although not addressed in this book, a phenomenon referred to as reflection occurs at high frequency. A problem begins to occur in which a distribution constant circuit region appears in which the wavelength cannot be ignored. This problem is defined by whether this reflection results in a loss. This phenomenon is not discussed because it exceeds the scope of the book; however, the equations and obtained efficiency characteristics are as follows: In cases in this book in which reflection does not represent loss, the equation for efficiency η is defined by Eq. (1.2) using the S parameter. S 21 7 Control

by estimating and not transmitting control signals is possible.

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1 Wireless Power Transfer

represents penetration, and S 11 represents reflection. η21 is the square of penetration, and η11 is the square of reflection. P1 is the transmitted power, and P2 is the received power. On the other hand, if efficiency is strictly defined as even reflection is loss, then efficiency η is per Eq. (1.3). Equation (1.2) subtracts the reflection from the denominator, and the reflection is not a loss. This is the general formula at low frequency. There are a number of studies that consider reflection to be loss, which may be understood in this case if we consider the relationship in the equations below. Figure 1.18 depicts the efficiency when using these two equations. As in Fig. 1.18a, if g = 150 mm, then the gap is small and the coupling is strong; thus, if reflection is considered as a loss, then efficiency η is split two ways. On the other hand, if reflection is not considered a loss, as in this book, then efficiency η is a peak. In addition, as in Fig. 1.18a, if g = 300 mm, in other words, if the gap is large and the coupling is weak, then there is no splitting in an η21 state and there is a peak. However, because the efficiency is strictly measured by the amount of reflection that is considered as loss, efficiency is deemed low for η considering the reflection. On the other hand, if considered in terms of η, the efficiency is better than η21 . In other words, Based on Eqs. (1.2) and (1.3), the relationship is always η  η . When there is no reflection, the relationship is η = η . When η11 = 0 with no reflection, the maximum efficiency is the same in both the evaluation of η and the evaluation of η [8, 9]. η=

|S21 |2 η21 P2 = = 1 − η11 P1 1 − |S11 |2 

η = η21 = |S21 |2

Fig. 1.18 Reflection presence and efficiency (spiral coil, 50  load)

(1.2) (1.3)

References

21

References 1. A. Kurs, A. Karalis, R. Moffatt, J.D. Joannopoulos, P. Fisher, M. Soljaˇci´c, Wireless power transfer via strongly coupled magnetic resonances. Science 317(5834), 83–86 (2007) 2. T. Imura, T. Uchida, Y. Hori, Experimental analysis of high efficiency power transfer using resonance of magnetic antennas for the near field—geometry and fundamental characteristics, in IEE of Japan Industry Applications Society Conference, vol. 2-62 (2008), pp. 539–542 3. Y. Kobayashi, Y. Kogami, Y. Suzuki, Microwave dielectric filters. IEICE (2007) 4. I. Awai, New theory for resonant-type wireless power transfer. J. Inst. Electr. Eng. Jpn 130(6), 966–971 (2010) 5. T. Imura, Y. Hori, Unified theory of electromagnetic induction and magnetic resonant coupling. Trans. Inst. Electr. Eng. Jpn. 135(6), 697–710 (2015) 6. N. Shinohara, Solar Power Satellite/Station (Ohmsha, 2012) 7. K. Nobuki, The importance of the development of a rover for the direct confirmation of the existence of ice on the moon. Trans. Jpn. Soc. Aeronaut. Space Sci. 43(139), 34–35 (2001) 8. T. Imura, Y. Hori, Determination of limits on air gap and efficiency for wireless power transfer via magnetic resonant coupling by using equivalent circuit. Trans. Inst. Electr. Eng. Jpn. D, Publ. Ind. Appl. Soc. 130(10), 1169–1174 (2010) 9. T. Imura, Y. Hori, Maximizing air gap and efficiency of magnetic resonant coupling for wireless power transfer using equivalent circuit and neumann formula. IEEE Trans. Indus. Electron. 58(10), 4746–4752 (2011)

Chapter 2

Basic Knowledge of Electromagnetism and Electric Circuits

In this chapter, we shall provide basic knowledge of electromagnetism and electric circuits as they relate to wireless power transfer. The technology behind wireless power transfer is premised on much of this basic knowledge; therefore, we shall explain it using the simplest expressions available so that even a complete novice may comprehend it.

2.1 Resistance, Coils, and Capacitors Impedance Z may be written as Z = R + jX. The real part R is the electrical resistance, which is generally simply called the resistance. The imaginary part jX is composed of coils or capacitors, wherein X is referred to as the reactance. When X is positive, it becomes an inductance L component and is referred to as the inductive reactance. When X is negative, it becomes a capacitance C component and is referred to as the capacitive reactance. Inductive refers to coils, and capacitive refers to capacitors. It is possible to consume the energy of the resistance alone. The reactance created by coils and capacitors can change the voltage and current phase but cannot consume energy. Below, we explain resistance, coils, and capacitors.

2.1.1 Resistance Resistance is an element that consumes only power. In addition, the voltage and current phase do not change. This is obvious but very important. Conversely, if there are places where the phase does not change, since they operate as resistance, the impedances in these places act only as a resistance component (Fig. 2.1). Formulas (2.1) and (2.2), respectively, are the equations for the resistance voltage and current. Together they present a complex number. vR and V R are the voltages, and i and I are the currents. R is the resistance in units of  (ohms). [Note: Generally, alternating © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_2

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Fig. 2.1 Load-side impedance Z seen from power supply

(a) Z

(b) In case Z is resistance R

current is written in lowercase text, and complex numbers are written in uppercase. However, in this book, they are both shown in uppercase text, except in special cases. In addition, complex numbers are often shown in bold text with dots (・) above symbols, but we have omitted the dots and do not show them in bold text, except in special cases.] v R = Ri(complex number display :VR = R I )

(2.1)

  vR R i= complex number display : I = R VR

(2.2)

Circuits, when looking at the load side from each point, require awareness of whether they are inductance components such as coils (imaginary positive components of impedance), capacitance components such as capacitors (imaginary negative components of impedance), or resistance components (real components of impedance). At high frequencies, resistance values rise from those during direct current owing to losses peculiar to high frequencies, such as the skin effect or proximity effect not produced with direct current (see Sect. 2.3). In addition, caution is required because what is sold as resistance has an L component at high frequencies (such as the kHz band) that was negligible at low frequencies, and it is often inductive. When the load is only resistance, there is no time delay in operation. That is, time constant τ = 0 s. However, caution is required since the resistance greatly affects the time constant depending on the circuit layout. We will introduce the details of a non-resonant circuit transient phenomena (pulse) in Sect. 2.4.

2.1.2 Coils Seen from a Circuit Viewpoint In this subsection, we describe coils seen from a circuit viewpoint. Using calculus, we present the voltage and current and provide a simple explanation of the characteristics of each. Since differentiation is the amount of change in an instant, the speed of change is fast. On the other hand, integration involves the continued addition of the

2.1 Resistance, Coils, and Capacitors

25

present quantity on a temporal axis, so the speed of change is slow. We explain coils based on the above. Formula (2.3) and formula (2.4), respectively, are the equations for the coil voltage and current. Together they present a complex number. vL and V L are voltages, and i and I are currents. L is the inductance in units of H (henry). di (complex number display : VL = jωL I ) dt   VL 1 i = ∫ v L dt complex number display : I = L jωL vL = L

(2.3) (2.4)

Coils work to prevent “changes in current.” “Changes” are very important. Therefore, even if an attempt is made to send the current instantaneously, if there is a coil, the current gradually increases. In addition, even if you want to reduce the current all at once, the magnetic energy built up in the coil will gradually release, so it is not possible to stop the current instantly. The advantage of this property is that if coils are inserted in series in a place where it appears that the inrush current will be generated, then the current slowly rises. In addition, the current values do not change easily, so if the coil is large, then it is possible to create constant current. The coils used in a constant current source (current source inverters and class-E amplifiers) naturally use this property. We also find slow movement because current formula (2.4) is an integral. On the other hand, changes in voltage are steep. “Changes in current” remain as voltage. The important points are that it does not depend on the direction of the presently flowing current. [Note: This is often misunderstood.] The size and direction of the voltage is determined as much as possible with the size and direction of the “changes in current.” In addition, unlike the current flowing in coils, coil voltage changes in an instant, so caution is required. If the change in the amount of current is large, then the amount of change and counter electromotive force will generate a large voltage. This property may be used in boost operations to raise the voltage. We also find quick movements because voltage formula (2.3) is a differential. As in Fig. 2.2, when we look at the temporal axis waveform, the compared phase of quickly moving voltage leads the current by ¼th period (90°). Figure 2.3 shows how this looks when rendered as a phasor diagram. In a coil, the voltage movement is quick, so it leads the current by a 90° phase. The current movement is slow, so it lags the voltage by a 90° phase. Re is short for real axis, and Im is short for imaginary axis. Next, we look at this from a so-called energy viewpoint. Compared to the capacitors described next, coils are difficult to imagine, but attention may be directed toward how they can hold the magnetic field energy by running current. Otherwise, current flows if there is magnetic field energy. Current I and magnetic flux ϕ appear to correspond with no delay in time such that the magnetic flux increases when the current is increased. This is equivalent to the relationship of the capacitor voltage and charge. It is very easy to imagine a one-to-one magnetic flux loop in which the number of

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Fig. 2.2 Changes in current flowing through coil and voltage

Fig. 2.3 Coil phasor diagram

magnetic fluxes increases when the current is increased. Coils store magnetic field energy in the form of magnetic flux, which then becomes current. Since magnetic flux does not build up instantly, the current increases gradually and does not change instantaneously. This is the same when it is decreasing (Fig. 2.4). Although it is sufficient to understand the basic properties of coils as stated above, beginning in the next section, we shall provide an electromagnetic discussion of magnetic field behavior and the relationship of magnetic fields and electric fields when explaining electromagnetic induction and self-inductance.

Fig. 2.4 Coil and magnetic field energy

2.1 Resistance, Coils, and Capacitors

27

2.1.3 Capacitors as Seen from a Circuit Viewpoint In this subsection, formulas (2.5) and (2.6) are equations for the capacitor voltage and current, respectively, describing capacitors seen from a circuit viewpoint. vC and V C are voltages, and i and I are currents. C is the capacitance in units of F (farad).   1 1 I vC = ∫ idt complex number display : VC = C jωC i =C

dvC (complex number display : I = jωC VC ) dt

(2.5) (2.6)

Capacitor voltage changes are slow. They are able to gradually increase the voltage by building up a charge, so they move slowly unless a large current is flowing. This is the same as when the voltage rises and falls. Hence, if a capacitor is large in capacity, then it can create a constant voltage source. Voltage-stabilizing capacitors used for a constant voltage source (voltage inverters and class-D amplifiers) use this exact property. In addition, smoothing capacitors are used to remove voltage fluctuations after rectification (referred to as ripples). From a design standpoint, it is important to reduce the capacity and volume of the capacitor, but reducing the size produces larger ripples, which requires a limited design. We also find that slow movement occurs because voltage formula (2.5) is an integral. On the other hand, the current changes dramatically. “Voltage changes” are just like current. The important point is that they are not determined in the direction of the presently flowing voltage. The current size and direction are for the most part determined by the size and direction of “voltage changes.” In addition, unlike the voltage applied to capacitors, the capacitor current changes instantly, so caution is required. When the amount of the voltage change is large, then the equivalent charge is released all at once, and a large current is generated. As for rapid movement, we find that current formula (2.6) is a differential. As in Fig. 2.5, when we look at Fig. 2.5 Changes in current flowing through capacitor and voltage

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Fig. 2.6 Capacitor phasor diagram

Fig. 2.7 Capacitor and electric field

the waveform of the temporal axis with rapidly moving current, the phase leads the voltage by ¼th period (90°). Figure 2.6 shows how this looks when rendered as a phasor diagram. In a capacitor, the current movement is quick, so it leads the voltage by 90°. The voltage movement is slow, so it lags the current by 90°. On the other hand, when we look at this from an energy viewpoint, it is possible for capacitors to hold the electric field energy by applying voltage. Otherwise, voltage can be held if the capacitors have electric field energy. Furthermore, capacitors store electric field energy in the form of a charge, which becomes the capacitor voltage. Charge q is easily imagined as discrete grains. Charge does not build up instantly, so the voltage increases gradually and does not change in an instant. This is the same when it decreases. From formula (2.5) as well, we find that the current integration is the voltage, so the fluctuations are small. On the other hand, the current changes in an instant. From formula (2.6), we also find that the current is a voltage differentiation and thus is sensitive to fluctuations in the voltage (Fig. 2.7).

2.2 Principle of Electromagnetic Induction In this section, we summarize points about electromagnetic induction. Starting from the viewpoint of coil electromagnetism, after confirming Ampere’s Law (in which a magnetic field is produced from current), we shall explain Faraday’s electromagnetic

2.2 Principle of Electromagnetic Induction

29

induction, and then describe the mechanism by which energy is conveyed by electromagnetic induction. After that, we shall describe self-inductance, mutual inductance, and Neumann’s formula from a circuit viewpoint.

2.2.1 Magnetic Field H, Magnetic Flux Density B, and Magnetic Flux Φ We shall provide a simple description of magnetic field H, magnetic flux density B, and magnetic flux Φ. When current flows, a magnetic field is produced. If the number of turns is increased, then the magnetic field becomes larger. Here, we shall explain H, B, and Φ after a magnetic field is generated. In discussions of power transmission, these three parameters are treated largely the same and represent similar phenomena. Thus, it may be sufficient to be aware that phase differences are not generated, but an accurate understanding is required when dealing with ferrite and the like. The strength of a magnetic field is expressed in H (A/m) (amperes per meter), and the magnetic flux density is expressed in B (T) (tesla). Often they are not differentiated and are both referred to as magnetic fields. Depending upon permeability μ, the relationship of H and B is as follows: B = μH

(2.7)

Permeability μ is determined by substances in space and is a constant of proportionality expressing the relationship of H and B. That is, magnetic flux density B differs from H in that it is the value of being affected by a substance. Permeability μ, using vacuum permeability μ0 = 4π × 10−7 and relative permeability μr , is expressed as follows: μ = μ0 μr

(2.8)

For air, water, copper, aluminum, and other substances are μr ≈ 1.0, which appears to be the same as a vacuum, but for iron it is μr ≈ 5000. As for ferrite, there are many types, and the frequency dependency is also large. Generally, for many of these types, the permeability is about μr ≈ 10–2000. Magnetic field strength H generates current I. Even when the same amount of current I is flowing, the generated magnetic flux density B differs depending on μr , and magnetic flux density B becomes larger as μr becomes larger. As shown in Fig. 2.8, the relationship of magnetic flux Φ (Wb) (weber), magnetic flux density B, and surface area S is as in formula (2.9). This number of magnetic fluxes is found by multiplying the surface by the density. Φ = BS

(2.9)

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.8 Magnetic flux Φ and magnetic flux density B

2.2.2 Ampere’s Law and Biot–Savart Law The important point in this subsection is that when current flows, a magnetic field is generated. This law is referred to as Ampere’s Law. When current flows through a conducting wire, a right-rotating magnetic field H is generated, as shown in Fig. 2.9a. This is Ampere’s Law. Current and magnetic field generation is comparable to a right-hand screw thread, also called the right-hand screw rule. When a conducting wire is made into a loop, the magnetic flux (magnetic field) intersects the loop, as shown in Fig. 2.9b. This is referred to as linkage. In addition, as shown in Fig. 2.9c, the number of linkages increases as the number of turns increases, making the magnetic field stronger. For example, in the case of an ideal solenoid such as an infinite-length solenoid, when the magnetic field strength made by one turn is H  and there are n turns in the coil, the magnetic field strength H is H = nH  . In other words, the greater the number of coil turns, the stronger the magnetic field generated. When a contour integral is used along the magnetic field in Fig. 2.9a, the relationship between H and current I is the following formula:  H · dl = I

(a) Straight line

(b) Loop

Fig. 2.9 Ampere’s law (right-hand screw rule)

(2.10)

(c) n turns

2.2 Principle of Electromagnetic Induction

31

When there are n turns, the right-hand side is nI. In this formula, the relationship with current I flowing through the center is found when magnetic field H is integrated along closed curve C. This is the formulation of Ampere’s Law. This is an integral representation of the Biot–Savart law described below. Bio–Savart introduced an equation for the current and voltage in a micro area and presented an equation integrating this. For example, when a magnetic field loop around a linear conductor with current flowing through it as in Fig. 2.9a is subject to contour integration, the length for which lines of magnetic force are present is the length of one circuit. Thus, if r is the radius from the center, then the circumferential length is 2πr. The magnetic field strength in this interval is the same since the distance from the current is constant and is determined as follows: 2πr H = I

(2.11)

I 2πr

(2.12)

When this is deformed into H=

we find how much current flows when a magnetic field is generated around a linear conductor. Next, we present the Biot–Savart Law (Fig. 2.10) as follows: dB =

μ0 I ds × r μ0 I ds × rˆ r = (∵ rˆ = ) 3 2 4π r 4π r r

(2.13)

With this law, it is possible to determine the minute magnetic field dB generated by a minute current element Ids. The distance from the current element to the minute magnetic field is r, and the vector is r. Here, × is the cross product. The orientation of the cross product made by ds and r is the direction of dB. For example, determining the magnetic flux density in the center of the circular current is as in the following formula when linearly integrating along the magnetic flux concentration around a current element (circular current). [Note: Generally, considering the angle θ formed by ds and r, ds × r = ds sinθ , but in the case of circular current, θ = 90° and sinθ = 1, so only ds may be considered.] Fig. 2.10 Biot–Savart law

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2 Basic Knowledge of Electromagnetism and Electric Circuits

B=

μ0 I 4π r 2

 ds =

μ0 I μ0 I 2πr = 2 4π r 2r

(2.14)

2.2.3 Faraday’s Law Ampere’s Law explains that a magnetic field is generated when current flows. By contrast, Faraday showed that current flows owing to changes in a magnetic field. More precisely, he showed that voltage is generated by changes in a magnetic field, which is Faraday’s Law. This voltage is referred to as induced electromotive force. Although it is a “force,” it is voltage. What Lenz discovered is that induced electromotive force is generated in a direction opposing the magnetic flux, and this direction property is referred to as Lenz’s Law. Figure 2.11 shows the principle of Faraday’s electromagnetic induction. Figure 2.11a shows an open circuit with open terminals. Changes in the magnetic flux are produced by moving a magnet closer and farther away. [Note: The behavior

(a) Generation of induced electromotive force

(b) Generation of induced current

(c) Schematic diagram of voltage drop

(d) Complete diagram

Fig. 2.11 Faraday’s electromagnetic induction

2.2 Principle of Electromagnetic Induction

33

of magnetic flux Φ and magnetic field H is considered the same (see in Sect. 2.1.1).] When changes in the magnetic flux are produced, they are likely to be produced in the opposite direction (Lenz’s Law). The voltage producing these changes is referred to as the induced electromotive force. This operates as a power supply on the power-receiving side. Normally, if a circuit is connected to form a closed circuit, then the voltage is generated by an induced electromotive force and current flows, so this current is referred to as induced current (Fig. 2.11b). Similarly, it is possible to deliver energy through changes in the magnetic flux (dΦ/dt). Magnetic flux Φ and magnetic field H are energy media, and thus energy can be delivered as a result of temporal or spatial changes in a magnetic field (dH/dt). The changes are important. For example, if a magnet is not moved, then a magnetic flux will be presented, but changes in the magnetic flux are not produced, so no energy is delivered. In the power transmission, the AC power source on the power transmission side generates the fluctuation of the magnetic field. Hence, wireless power transfer is not possible with direct current, so alternating current is always used. The above explanation is sufficient for a conceptual understanding, but we shall continue for a short while longer for the sake of interested readers. We will consider how induced electromotive force V Lm differs from coil end voltage V 2 . First, in the circuit in Fig. 2.11b, load voltage V R equals coil end voltage V 2 . Furthermore, coil end voltage V 2 requires consideration of a voltage drop V L22 in the coil itself, resulting in the following formula: V2 = VLm − VL22

(2.15)

Figure 2.11c depicts a voltage drop only, ignoring the induced electromotive force. Current flows, so the equivalent voltage drop V L22 is shown. Figure 2.11d more accurately depicts the combination of Fig. 2.11b, c. This is easily understood in the form of a circuit diagram. Since it is possible to depict an induced electromotive force as a power supply, Fig. 2.11d is rendered similar to Fig. 2.12a. Furthermore, when the coil position is moved, the result is as in Fig. 2.12b, which is a familiar figure.

(a) Faraday’s electromagnetic induction equivalent circuit

(b) After coil position movement

Fig. 2.12 Faraday’s electromagnetic induction closed-circuit equivalent circuit

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2 Basic Knowledge of Electromagnetism and Electric Circuits

According to Faraday’s Law of Induction, the basic formula for the relationship of voltage V to magnetic flux Φ, magnetic flux density B, and magnetic field strength H, and to inductance L, is the following formula: V =−

dB dH dI dΦ =S = μS =L dt dt dt dt

(2.16)

Coil independence, that is, self-inductance, is as in formula (2.17): V1 =

dB1 dH1 dI1 dΦ1 =S = μS = L1 dt dt dt dt

(2.17)

In formulas (2.16) and (2.17), the following relationship formulas are used: Magnetic flux Φ is the product of magnetic flux density B and surface area S. Φ = BS

(2.18)

Magnetic flux density B is the product of permeability μ and magnetic field strength H. B = μH

(2.19)

Accounting for time variation, changes in magnetic flux density dB/dt are expressed as in the following formula: dH dB =μ dt dt

(2.20)

Accounting for fluctuations in magnetic flux dΦ/dt yields the following formula: dΦ dB dH =S = μS dt dt dt

(2.21)

The relationship of the magnetic flux and current is the relationship to inductance, as in formula (2.22). Inductance is a coefficient that indicates how much magnetic flux can be produced when a current flow. Φ = LI

(2.22)

Formula (2.23) is used when temporal changes are produced. L is determined by the form, so there are no changes over time. dI dΦ =L dt dt

(2.23)

2.2 Principle of Electromagnetic Induction

35

Fig. 2.13 Electric field vortex (magnetic field generation) and rotE

In practical terms, having the above knowledge is sufficient, but it is important to think about the basic Maxwell’s equations in terms of the principles of electromagnetism. Therefore, for the sake of interested readers, we shall re-explain this using the following formula, which is equivalent to one of the four Maxwell’s equations, and the differential form of Faraday’s law: rotE = −

∂B ∂t

(2.24)

In an electric field, the displacement current truly represents the energy transmission, but as described above, in a magnetic field, the changes in magnetic flux density ∂B/∂t constitute the true power transmission, the level of which is equal to voltage V. In addition, in a magnetic field, the physical quantity ∂B/∂t corresponding to changes in magnetic flux density B is not referred to as the “displacement voltage” but an “electric field vortex.” This seems to recall an image of rotE = −∂B/∂t as in Fig. 2.13. Formula (2.24) here defines the relationship of an electric field and magnetic field. rot (rotation) represents the vortex strength, and the vector made by E produced along a loop is rotE. This rote is shown to be equal to ∂B/∂t. A negative indicates Lenz’s Law, in which an induced electromotive force is produced in the opposite direction to which the magnetic flux is produced. We subject the loop to surface integration (formula 2.25). That is, we multiply the vertical component of the linked magnetic flux on a surface by the surface area.   ∂B ∫ rotE · dS = ∫ − dS ∂t S S

(2.25)

We use the theorem of the Stokes contour integral. Directly applying this famous theorem, in which a line integral along a line going around in a circle corresponds to the surface integration of the inside of the circumference thereof, results in formula (2.26) (Fig. 2.14):  E · dl = ∫ r ot E · dS S

C

(2.26)

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.14 Stokes contour integral

Formula (2.27) is obtained from formulas (2.25) and (2.26).  ∂B · dS E · dl = − ∫ S ∂t

(2.27)

C

Assuming the length of one circle to be l, and an electric field to be a constant E, formula (2.27) gives way to the following formula: V = El = S

∂B ∂t

(2.28)

Since integrating an electric field over a distance is voltage V, if E is constant, then the product of E and l becomes the voltage. In the end, the voltage conforms to formula (2.16). In addition, a unit system in which the surface area is multiplied by the amount of change in the voltage and magnetic flux density also conforms to this.

2.2.4 Mechanism of Energy Transmission by Electromagnetic Induction (Electromagnetism Viewpoint) Now that we understand Ampere’s Law and Faraday’s Law of Induction, we shall provide a simple explanation of the mechanism by which energy is transmitted by electromagnetic induction [1]. This is often rendered in a very simple diagram, like that in Fig. 2.15a. Figure 2.15b describes the induced electromotive force V Lm2 (=dΦ 21 /dt) made by dΦ 21 /dt, the voltage V 2 generated at the ends of the coils, and the drop voltage V L22 in the coils. Neither is incorrect, but the concepts of linkage to the primary side and induced electromotive force are missing. Therefore, Fig. 2.16 accounts for the influence on the primary side. In this way, we shall provide a step-by-step explanation keeping in mind the effects on the primary side. First, wireless power transfer starts once voltage V 1 is applied to a powertransmission-side coil connected to a power supply. When voltage is applied, current

2.2 Principle of Electromagnetic Induction

(a) Simplest diagram

37

(b) Diagram accounting for secondary side voltage

Fig. 2.15 Power transmission owing to electromagnetic induction (considering secondary side only)

Fig. 2.16 Power transmission owing to electromagnetic induction (considering primary side also)

I 1 flows. As described previously, when current flows in a conducting wire, a rightrotating magnetic field is generated according to Ampere’s Law (right-hand screw rule), as shown in Fig. 2.9a. When the conducting wire is made into a loop, the magnetic field rotates while intersecting the loop, as shown in Fig. 2.9b. Up to now,

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2 Basic Knowledge of Electromagnetism and Electric Circuits

we have been discussing power-transmission-side coils, as shown in Fig. 2.15, discussing how a magnetic field is generated when a current flows. Later, we shall describe the effects upon the primary side, as shown in Fig. 2.16. Next, we consider the process by which a generated magnetic field moves to the power-receiving side. In Fig. 2.11, changes in the magnetic flux are made by a magnet, but here we use the changes in magnetic flux dB/dt generated by the power-transmission-side coil. First, magnetic field Φ 21 generated by the powertransmission-side coil is linked to the power-receiving coil. When the magnetic flux within the loops of the power-receiving coil changes, voltage V Lm2 is produced in the direction in which the magnetic flux hinders changes, and simultaneously, current I 2 flows. The voltage here is referred to as the induced electromotive force, and the current is referred to as induced current. Although written as induced electromotive “force,” it is still voltage. In addition, magnetic flux is generated by current flowing on the secondary side. Some of this links to the primary side and is Φ 12 , as shown in Fig. 2.16. This series of operations represents power transmission owing to Faraday’s electromagnetic induction. [Note: Electromagnetic induction was first discovered by Joseph Henry (origin of the henry (H) as a unit of inductance L), but likely owing to his success in presenting all electromagnetic induction phenomena, it is often referred to as Faraday’s electromagnetic induction.] For more detailed explanation, Fig. 2.17 shows the relationship between the main magnetic flux and the leak magnetic flux. Here, the direction of the current and magnetic flux on the secondary side is described as being opposite to that of Fig. 2.16

Fig. 2.17 Main magnetic flux and leak magnetic flux

2.2 Principle of Electromagnetic Induction

39

from the relation with the mathematical formula. The magnetic flux that contributes to power transmission in which both power-transmitting and power-receiving coils are linked is the main magnetic flux (Φ 21 , Φ 12 ), and the linkage that does not contribute to power transmission in which coils link to themselves is the leak magnetic flux (Φ 11 , Φ 22 ). The leak magnetic flux itself does not produce loss. In addition, Φ 11 is the magnetic flux produced by the primary-side current and is linked only to the primary side, and Φ 21 is the magnetic flux produced by the primaryside current and is linked to the secondary side. That is, the primary side affects the secondary side, and Φ 21 is the contribution to the power transmission. This is the main element of induced electromotive force V Lm2 produced on the secondary side. When there is no mutual coupling, the magnetic flux contributing to self-inductance was only Φ 1 = Φ 11 ; but when mutual coupling was produced, part of Φ 1 changed to Φ 11 and Φ 21 , and Φ 1 = Φ 11 + Φ 21 . However, the Φ 1 value before and after coupling generally changes. On the other hand, looking at this from the secondary side, Φ 22 is the magnetic flux produced by the secondary-side current and is linked only to the secondary side, and Φ 12 is the magnetic flux produced by the secondary-side current and is linked to the primary side. That is, the secondary side affects the primary side, and Φ 12 plays a role in power transmission. The induced electromotive force V Lm1 produced on the primary side appears as voltage drop V Lm , and is the load voltage relative to equivalent load Z  2 looking at the secondary side from the primary side. If we divide V Lm1 by primary-side current I 1 , then it becomes the load resistance (reflected impedance) Z  2 on the secondary side as seen from the primary side. When there is no mutual coupling, no magnetic flux is produced on the secondary side, and the magnetic flux contributing to self-inductance per numerical formula is Φ 2 = Φ 22 . When mutual coupling is produced here, part of Φ 2 changes to Φ 22 and Φ 12 , and Φ 2 = Φ 22 + Φ 12 . Figure 2.18 shows an electromagnetic induction equivalent circuit. Below, we show the transition from the electromagnetic explanation shown in Fig. 2.17 to the electric circuit explanation shown in Fig. 2.18.

Fig. 2.18 Electromagnetic induction equivalent circuit

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According to this definition, inductance represents the amount of change in the current and magnetic flux, as shown in the formula: Φ = LI

(2.29)

dΦ dI

(2.30)

L=

According to Fig. 2.17, the respective formulas for the primary-side voltage and secondary-side voltage are as follows: V1 = −

dΦ21 dΦ12 dΦ11 − − + Vr 1 dt dt dt

(2.31)

V2 = −

dΦ12 dΦ21 dΦ22 − − + Vr 2 dt dt dt

(2.32)

Using V L11 , V Lm1 , V L22 , and V Lm2 results in the following formulas: V1 = VL11 + VLm1 + Vr 1

(2.33)

V2 = VL22 + VLm2 + Vr 2

(2.34)

When this is done, the relationship of V L11 , V Lm1 , V L22 , and V Lm2 to the magnetic flux, inductance, and current, and the relationship of main magnetic flux Φ m to Φ 21 and Φ 12 , are per the formulas:   dΦ21 dI1 dΦ11 + = L1 VL11 = − dt dt dt dΦ12 dI2 dI2 dI2 dΦ12 =− = L 12 = Lm dt dI2 dt dt dt   dΦ12 dI2 dΦ22 VL22 = − + = L2 dt dt dt

VLm1 = −

VLm2 = −

dΦ21 dI1 dI1 dI1 dΦ21 =− = L 21 = Lm dt dI1 dt dt dt Φm = Φ21 + Φ12

(2.35) (2.36) (2.37) (2.38) (2.39)

In addition, the mutual inductance is the same whether viewed from the primary side or secondary side, so L 12 = L 21 = L m . The sum total of the linked magnetic flux contributing to coupling, that is, the main magnetic flux, is Φ m . However, if Φ m = 0, then when Φ 21 = −Φ 12 , linked magnetic flux Φ 21 and −Φ 12 are present. When this occurs, Φ 21 and I 1 are of the same phase, and −Φ 12 and −I 2 are of the same phase. That is, when this occurs, I 1 and I 2 are antiphase. Thus, the fact that

2.2 Principle of Electromagnetic Induction

41

the main magnetic flux is 0 and no linked magnetic flux is present means there is no equivalence, so caution is required. The above is electromagnetic induction seen in terms of electromagnetism, which we explained as connecting electromagnetic induction to a circuit. Changes in magnetic flux Φ represent voltage V, which may also be described in terms of inductance L and current I. In circuit theory, magnetic flux Φ is not used, and inductance L and current I are used to describe voltage V.

2.2.5 Electromagnetic Induction Described from a Circuit Viewpoint Here, we explain a method for describing electromagnetic induction from a circuit viewpoint. This is what is often done in practice. In a circuit, magnetic flux ϕ is not used, and induction is thought in terms of inductance L. We shall provide an explanation with the equivalent circuit used in Fig. 2.18. From Kirchhoff’s Voltage Law, which says that if a closed circuit is a single loop, then the voltage is 0 V or is the same as the power supply applied to both ends, the formula for electromagnetic induction when there is no resonance is formula (2.40) on the primary side and formula (2.41) on the secondary side, as shown below. In a differential expression, formula (2.42) is on the primary side, and formula (2.43) is on the secondary side. In addition, when operating at a constant frequency sine wave (single frequency), a complex number expression is convenient. When used, this is formula (2.44) on the primary side and formula (2.45) on the secondary side: V1 = VL11 + VLm2 + Vr 1

(2.40)

0 = VL22 + VLm1 + Vr 2 + V2

(2.41)

di 1 di 2 + Lm + r1 i 1 dt dt

(2.42)

di 2 di 1 + Lm + r2 i 2 + R L i 2 dt dt

(2.43)

v1 = L 1 0 = L2

V1 = jωL 1 I1 + jωL m I2 + r1 I1

(2.44)

0 = jωL 2 I2 + jωL m I1 + r2 I2 + R L I2

(2.45)

Here, RL is the resistance load. Formulas (2.42) and (2.43) are calculus expression methods and can be used for such waveforms. On the other hand, the complex number expressions in formulas (2.44) and (2.45) can only be used with a constant frequency.

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[Note: Although we mentioned constant frequency, all waveforms can be expressed as aggregates of multiple frequencies, so they may be expressed with complex number expressions, although this is not used in practice. For example, a rectangular wave can be expressed by adding a fundamental wave, third harmonic wave, fifth harmonic wave, and so on.]. Both expressions are important, but when operating at a constant frequency, they are often treated as complex numbers, where differentiation and integration can be expressed by multiplication and division. On the other hand, when waveforms such as power electronics are often captured on the time axis, calculus expression is used in actual circuit design. When resonated, resonance capacitor C is added to these formulas. Obviously, wireless power transfer is essentially the transmission of energy. In other words, when considering energy in terms of transmission, on the powerreceiving-coil side, the effective power produced by induced electromotive force V Lm2 generated on the secondary side and I 2 flowing on the secondary side becomes the power Pr transmitted to the secondary side. On the other hand, on the powertransmitting-coil side, the induced electromotive force V Lm1 generated on the primary side is equal to the amount of the voltage drop. The effective power (consumption power) produced by this and primary-side current I 1 becomes the power Pt transmitted to the secondary side from the primary side. Pr and Pt are expressed by the formula below, and the two powers are equal, as in formula (2.48). However, θ 2 is the phase made by V Lm2 and I 2 , and θ 1 is the phase made by V Lm1 and I 1 . In addition, in the case of magnetic resonance coupling, θ 2 = θ 1 = 0 and cosθ = 1, and the power factor of the secondary side and primary side in this formula is 1. Pr = VLm2 I2 cos θ2

(2.46)

Pt = VLm1 I1 cos θ1

(2.47)

Pt = Pr

(2.48)

2.2.6 Self-inductance Here, we shall confirm self-inductance L. When voltage is applied to a coil and current flows, magnetic flux is produced such that it links the loops per Ampere’s Law (right-hand screw rule), and the phenomenon produced in this process is referred to as self-inductance. When current is flowing, voltage V L (referred to as the electromotive force) is generated. Self-inductance is also used to describe coils in which the flowing current and generated electromotive force are the same.

2.2 Principle of Electromagnetic Induction

43

Fig. 2.19 Conducting wire and coil

When voltage is applied to both ends of a conducting wire and it is not in a coil, the potential difference between the ends is 0 V, causing a short circuit. However, when the wire is looped into a coil, voltage is induced at the coil ends by the electromotive force V L owing to self-inductance, so there is no short circuit (Fig. 2.19). For example, when t = 0, voltage V in is applied, but in that instant, there is no large current flow. In addition, the integral of the time for which voltage is applied is the current. When this occurs and L is large, the change is small, resulting in a small current. Looking at this another way, self-inductance L is a coefficient that expresses how much voltage is induced in response to “changes in current” flowing in a coil itself. In addition, it may be seen as the coefficient of how much current size is suppressed when applying voltage. Furthermore, the phase is also important. As shown in Fig. 2.20, the voltage phase leads the current by 90°. The formula for the voltage and the formula for the current are as follows: vL = −

Fig. 2.20 Self-inductance waveform

di dΦ = L ⇔ VL = jωL I dt dt

(2.49)

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2 Basic Knowledge of Electromagnetism and Electric Circuits

i=

1 1 ∫ v L dt ⇔ I = VL L jωL

(2.50)

j in formula (2.49) is the phase lead. Current may also be seen to lag the voltage by 90°, the meaning of which is found in 1/j = −j in formula (2.50). When operating with a sine wave, the relationship is as in formulas (2.51) and (2.52). vL = i=

√

2VL cos ωt

√ 2I sin ωt

(2.51) (2.52)

2.2.7 Mutual Inductance and Coupling Coefficient Next, we confirm the mutual inductance L m produced when there is a powertransmitting coil and power-receiving coil. Figure 2.21 shows an equivalent circuit. Figure 2.21a shows a common equivalent circuit, and Fig. 2.21b shows a T-type equivalent circuit. When current flows through a primary-side coil, magnetic flux is produced by Ampere’s Law. This magnetic flux links with the secondary-side coil. When this is done, magnetic flux is produced in the direction that negates changes in the magnetic flux. Voltage V Lm2 (referred to as the induced electromotive force) is generated and

(a) Equivalent circuit

(b) T-type equivalent circuit Fig. 2.21 Mutual inductance (non-resonant circuit N–N)

2.2 Principle of Electromagnetic Induction

45

becomes a secondary-side power source. This voltage V Lm2 is produced by primaryside current I 1 flowing as in the following formula: VLm2 = jωL m I1

(2.53)

On the other hand, the mutual inductance affects the primary side. V Lm1 is produced as in the following formula by current I 2 flowing on the secondary side: VLm1 = jωL m I2

(2.54)

This is also the induced electromotive force. That is, it is possible to look at the secondary-side situation in the form of induced electromotive force V Lm1 . This is perceived as the amount of voltage drop. As in formula (2.55), when divided by primary-side current I 1 , it can be recognized in the form of impedance Z 2  (reflected impedance) that can confirm the secondary-side impedance on the primary side. 

Z2 =

VLm1 jωL m I2 (ωL m )2 = = I1 I1 Z in2

(2.55)

Here, Z in2 is the secondary-side impedance. For example, the following formula is for magnetic resonance coupling involving resonation with resonance capacitors inserted in series on the secondary side: 

Z2 =

VLm1 jωL m I2 (ωL m )2 (ωL m )2 = = = I1 I1 Z in2 r2 + R L

(2.56)

From another point of view, the mutual inductance L m can be said to be a coefficient that expresses how much coil voltage is induced in response to “changes in current” flowing in the other coil. [Note: According to other publications, L m is sometimes described as M, but in this book, L m is adopted according to its relationship with mutual capacitance C m in the electric field resonance.] Here, we confirm the basic characteristics of coupling coefficient k and mutual inductance L m . The coupling coefficient is the percentage of coupling set according to primary-side inductance L 1 , secondary-side inductance L 2 , and mutual inductance L m , as shown in formula (2.57). If primary-side inductance L 1 and secondary-side inductance L 2 are the same L, then the coupling coefficient k is as in formula (2.58). k=√ k=

Lm L1 L2

(2.57)

Lm L

(2.58)

The scope of coupling coefficient k is 0  k  1. When k = 0, there is no coupling. When k = 1, the strongest coupling occurs, and the magnetic flux produced on the

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primary side all links with the secondary side. During wireless power transfer, a magnetic flux is produced that never links, so k = 1. When the distance between coils becomes larger, coupling coefficient k and mutual inductance L m become smaller. This is because as the gap widens, the linking magnetic flux shared by the primary-side coil and secondary-side coil decreases.

2.2.8 Neumann’s Formula (Derivation of Inductance) Inductance L is also the relational expression of magnetic flux Φ and current I. As understood from Φ = LI, it is the coefficient that expresses the relationship of the amount of magnetic flux generated when current is flowing. Therefore, we describe the formula for determining self-inductance and mutual inductance. This formula is referred to as Neumann’s Formula. Mutual inductance is as follows:   dl1 dl2 μ0 (2.59) Lm = 4π D C1 C2

 Here, D is the distance between dl1 and dl 2 . dl expresses a line integral. Although this is a simple formula, it is comprehensive. It is important for values to be determined only at the positions of coils where current is flowing, and not dependent upon the voltage or current size. Integration occurs only along coil lines (line integrals). As for the meaning of the line integral of formula (2.59), we integrate around C2 in relation to minute length dl1 . Then, we integrate around C2 in relation to dl1  at a slightly offset position. By repeating this, the entire C1 circuit is integrated in relation to the C2 circuit. [Note: Here, C1 and C2 are the paths (curves) of coils once around, so they are not capacitors. In general, since a C shape is used, here we use C.] A complex formula is a formula that has been line integrated, and shapes that can be easily expressed as numerical formulas are limited. When there is a pair of single loops, they can be easily written for formula (2.60), but in actuality, multiple windings are often used. Since a pitch is produced between wires, this results in a complex formula, so often formula (2.59) is used for numerical calculation. Lm =

r 2 cos(θ1 − θ2 ) μ0 2π 2π ∫ ∫  dθ1 dθ2 4π 0 0 2r 2 + g 2 − 2r 2 cos(θ1 − θ2 )

(2.60)

2.2 Principle of Electromagnetic Induction

47

Fig. 2.22 Parameters in Neumann’s Formula for one loop

Self-inductance with Neumann’s Formula is per the following formula (see Fig. 2.22): L1 =

μ0 4π

 

dl1 dl1 D

(2.61)

C1 C1

In calculations of self-inductance, places corresponding to path C2 in mutual inductance become self-paths, so part of C2 in formula (2.60) becomes C1. However, if calculations are performed as it is, formula (2.61) diverges for sections where D = 0 is produced, so some (corrective) means are required [2]. Assuming L 1 = L 2 = L, Fig. 2.23 shows the relationship between the mutual inductance and coupling coefficient when the coils have a radius of 150 mm. When there is a single loop, it is possible to use formula (2.60) for calculations, which closely conform to the electromagnetic field analysis results.

Fig. 2.23 Gap, mutual inductance, and coupling coefficient

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2 Basic Knowledge of Electromagnetism and Electric Circuits

2.3 High-Frequency Loss (Resistance) 2.3.1 Copper Loss, Skin Effect, and Proximity Effect In this section, we discuss high-frequency loss. In wireless power transfer, the losses that should be considered are copper loss, iron loss (eddy current loss and hysteresis loss), and radiation loss. Here, we shall discuss copper loss. Copper loss is produced from internal resistance. With direct current, only direct current resistance may be considered. However, in wireless power transfer, the current flowing through the coil used in power transmission is always alternating current; thus, alternating current resistance must be considered. The factors by which resistance increases owing to alternating current are the skin and proximity effects. The skin effect is a resistance component produced in a conducting wire itself when highfrequency current flows through it. The proximity effect is produced as two wires with high-frequency current flowing through them approach each other, producing a loss similar to the skin effect. First, we shall describe the skin effect. As shown in Fig. 2.24, the magnetic flux increases with the current flowing through a conducting wire. As a result, the eddy current is produced in a direction that negates the magnetic flux. On the center side, the eddy current is in the opposite direction of the original current, and they mutually weaken each other; and on the outer side, it is in the same direction as the original current, and they mutually reinforce each other. As a result of this effect, current comes to flow only on the conductor surface. Thus, current is not flowing in the

(a) Principle of skin effect

(b) Full view of skin effect and skin depth Fig. 2.24 Skin effect

(c) How actual current flows

2.3 High-Frequency Loss (Resistance)

49

center section. That is, the flow surface area is decreased; thus, the resistance value increases. With direct current as well, as understood from the formula resistance r = ρ Sl when the surface area narrows, the places where current flows decrease; thus, the reason is the same as when resistance value R () increases. Here, l is the conducting wire length, S is the surface area, and ρ is the electrical resistivity (m). Skin depth δ is defined as the depth from a surface with flowing current. It is the distance to which current flowing on the surface attenuates to e−1 = 1/e (≈1/2.7) ≈ 0.37. [NOTE: e is Napier’s constant equal to e = 2.71828. In addition, e−1 = 0.36788.] As shown in formula (2.62), it is dependent not only on the frequency but also the conductivity and permeability. This is an important point to consider. The skin depth should have a strength of e−1 , with current flowing further inside than the skin depth, as shown in Fig. 2.24c.  δ=

2 ωσ μ

(2.62)

(δ: skin depth, σ : conductivity, μ: permeability). If the wire has a round shape, then the resistance accounting for the skin effect is expressed by formula (2.63): Rohm =

ρl π δ(D − δ)

(2.63)

(ρ: resistivity, l: total length, D: thickness). For example, as an example of a MHz band coil, we shall consider an open type with radius r = 150 mm, number of turns n = 5, and pitch p = 5 mm. This is a self-resonant-type helical coil. In this case, resonance frequency f 0 = 17.5 MHz, and total length l = 4.7 m. However, when this occurs, if copper conductivity ρ = 5.8 × 107 (S/m), relative permeability μr = 1, and element thickness D = 2 mm, and are substituted into formulas (2.63) and (2.62), then the results are δ = 15.7 μm and Rohm = 0.82 . Otherwise, as an example of use with a kHz-band coil, if we assume a short type with r = 200 mm, number of turns n = 20, and pitch p = 5 mm, and resonating at 100 kHz with an external resonant capacitor, then the results are δ = 207.2 μm and Rohm = 0.27 . The total length 18.8 m. Figure 2.25 shows the relationship of the frequency, skin depth, and resistance. In Fig. 2.2a and Fig. 2.2b, the total coil length is changed. The skin depth depends upon the frequency—thus, the skin depth curve is the same; however, the resistance value varies with total length. In this way, current flows only to the surface of the conductor due to the skin effect, so in general, measures are taken to increase the surface area and reduce the resistance value by using a Litz wire and twisting wires with a diameter sufficiently smaller than the skin depth. The Litz wire is a bundle of thin insulated wires and is formed by collecting several hundred wires (element wires) of about 0.1 mm. Figure 2.26 shows a schematic diagram of a single wire and a Litz wire. The current

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2 Basic Knowledge of Electromagnetism and Electric Circuits

(a) MHz coil (4.7 m)

(b) kHz coil (18.8 m)

Fig. 2.25 Skin depth

Fig. 2.26 Single and Litz wire

(a) Single wire

(b) Litz wire

that flows only on the surface of a thick single wire can flow to the center of a thin Litz wire, so the resistance value drops. Next, we describe the proximity effect. Figure 2.27 shows a conceptual diagram of the proximity effect when current flows in the same direction, and Fig. 2.4b shows a conceptual diagram of the proximity effect when current flows in the opposite direction. Similar to the skin effect, eddy current flows in a direction that negates the magnetic flux. Thus, current bias is produced. In the skin effect, magnetic flux produced by the current affects itself; however, the proximity effect influences adjacent wires. I 1 , I 2 , H 1 , and H 2 are the previously flowing currents and magnetic fields, and I  1 , I  2 , H  1 , and H  2 are the currents and magnetic fields that have been excited to negate the magnetic flux. As shown in Fig. 2.27, when current flows in phase, it behaves as if repelling. As shown in Fig. 2.4b, when current flows in antiphase, it behaves as if pulling against each other. In situations in which a Litz wire is used, the current direction is the same; thus, such situations are in phase, as shown in Fig. 2.4a. Under the influence of this proximity effect, the surface area of the flowing current decreases, and the resistance increases. In other words, although a Litz wire may be used to deal with increased resistance owing to the skin effect at high frequencies, the proximity effect increases the resistance. In general, if the coil is a kHz-band coil, then a Litz wire is an effective skin-effect measure; however, if the coil is in the MHz band, depending on which wire is selected, sometimes a single wire is more likely to keep the resistance value down owing to the influence of the proximity effect. Therefore, wire type should be ascertained per each frequency.

2.3 High-Frequency Loss (Resistance)

51

(a) In phase

(b) Antiphase Fig. 2.27 Proximity effect

2.3.2 Iron Loss (Hysteresis Loss and Eddy Current Loss) In order for the magnetic flux to pass through more easily, sometimes an iron core is used, as shown in Fig. 2.28, or a ferrite or other magnetic body is used on the back surface of the coil such that no magnetic flux leaks to the surroundings. However, in cases where such a magnetic body is used, loss is generated in the magnetic body. This is referred to as iron loss. The main forms of iron loss are hysteresis and eddy current loss. Hysteresis loss results from heat being generated within a magnetic body because the direction of the magnetic domain constantly changes when the direction of the magnetic field periodically changes, owing to an alternating current magnetic field, also called an alternating field. Figure 2.29 shows a hysteresis loop. Loss proportional to the surface area of this hysteresis loop is produced. If the energy produced once around the hysteresis loop is W h (J/m3 ), and

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.28 Iron core

Fig. 2.29 Hysteresis loop

we assume that f is the frequency, then the consumed power Ph (W) at f times around in 1 s complies with the formula, and the hysteresis loss is proportional to frequency f. Ph = f Wh

(2.64)

In addition, formula (2.65) for the hysteresis loss was famously determined empirically by Steinmetz. Here, k h is the hysteresis coefficient, and Bm is the maximum magnetic flux density. Ph = kh f Bm1.6

(2.65)

In fact, there are many products with small hysteresis loops. If you choose firmly, hysteresis loss can be made smaller than that of copper or other loss; thus, it can often be ignored. Next, we shall consider eddy current loss. The eddy current loss produced in a magnetic body used to strengthen a coil magnetic field is also small compared

2.3 High-Frequency Loss (Resistance)

53

with copper loss and may often be ignored, similar to hysteresis loss. In contrast, sometimes environments surrounding a coil cannot avoid the use of metal in their design. Such cases similarly produce eddy current loss. The mechanism by which eddy current loss is generated starts with variations in the magnetic flux produced in a metal. Induced electromotive force is produced in a direction that negates the variation in the magnetic flux and eddy current flows. This current becomes loss. Figure 2.30 shows the principle of an eddy current using a magnet. As the magnet approaches, H is generated in the negating direction; thus, eddy current I is generated. The eddy current loss per unit volume (W/m3 ) is per formula (2.66). It is proportional to the square of frequency f. R is the radius, and ρ is the resistivity. [NOTE: k e is a constant of proportionality. Be aware that its definition varies depending on the textbook.] Pe = ke

R 2 Bm2 f 2 ρ

(2.66)

For example, we present the calculation of the eddy current loss produced in a metal disk of radius R (Fig. 2.31). When we use surface area S farther inside than the position of radius r, where a uniform alternating field ϕ links with the metal. The relationship to magnetic flux density B is a formula. ω is the angular frequency. Φ = S B = πr 2 B = πr 2 Bm sin ωt

(2.67)

The voltage v induced in one group of radius r is as in formula (2.68). V =

dB dΦ = = πr 2 ωBm cos ωt dt dt

(2.68)

The resistance Rloop once around radius r is as in the formula. d is the disk thickness.

Fig. 2.30 Explanatory diagram of eddy current

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.31 Eddy current produced by a disk

Rloop = ρ

2πr d

(2.69)

Current I is per the following formula: I =

V dr ωBm cos ωt = Rloop 2ρ

(2.70)

Instantaneous power is per the formula: P = IV =

1 π dr 3 ω2 Bm2 cos2 ωt 2ρ

(2.71)

Accordingly, the power Ploop (W) of one loop in period T is as per formula (2.72). Here, we use the relationship in formula (2.73). Ploop =

1 T 1 T 1 π dr 3 ω2 Bm2 ω 2π/ω ∫ I V dt = ∫ cos2 ωtdt π dr 3 ω2 Bm2 ∫ cos2 ωtdt = T 0 2ρ T 0 2ρ 2π 0 π dr 3 ω2 Bm2 π dr 3 ω2 Bm2 1 = = 2ρ 2 4ρ (2.72) T =

2π 1 = f ω

(2.73)

When we perform an integration in the radial direction and determine the consumed power Pall (W) for the entire disk, not just one loop, it is as in the formula: R

R

0

0

Pall = ∫ Ploop dr = ∫

π dω2 Bm2 R 3 π dr 3 ω2 Bm2 π dω2 Bm2 R 4 ∫ r dr = dr = 4ρ 4ρ 16ρ 0

In contrast, volume V vol is

(2.74)

2.3 High-Frequency Loss (Resistance)

55

Vvol = S · d = π R 2 · d

(2.75)

Therefore, when we determine the power density (W/m3 ), it is as follows: π 2 f 2 Bm2 R 4 Pall π dω2 Bm2 R 4 1 = = Vvol 16ρ π R2d 4ρ

(2.76)

Here, we use the relationship ω = 2π f. We find that this conforms to formula (2.66) for the power density, as shown previously. In this disk example, we find the constant of proportionality k e = π 2 /4.

2.3.3 Radiation Loss Radiation loss refers to unwanted emissions of electromagnetic waves during power transmission (Fig. 2.32). When calculating, radiation resistance Rrad is a value backcalculated from the radiation loss and is expressed using radiation power P0 and the current I 0 flowing through an antenna, as follows: Rrad =

P0 I02

(2.77)

In other words, the radiation power needs to be determined. This is a value that determines the far-field radiating Poynting vector P (=E × H) and is generally determined through electromagnetic field analysis. In addition, the coils dealt with in this book are small relative to wavelength; thus, they have very small radiation losses that almost reach 0%. Thus, Rrad ≈ 0 (). If the radiation loss is high, then the coil is not suitable for wireless power transfer. In addition, when the coil diameter is small relative to the wavelength, the following is sometimes used as an approximation:  Rrad = 120π

Fig. 2.32 Radiation wave

2π 3



kS λ



2 N 2 = 20π 2

2πa λ

4 (2.78)

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2 Basic Knowledge of Electromagnetism and Electric Circuits

However, naturally, this must be strictly determined in electromagnetic field analysis. Here, S is the surface area, a is the coil radius, N is the number of turns, and k is the wavenumber (k = 2π/λ) [3].

2.4 Non-resonant Circuit Transient Phenomena (Pulse) In this section, we will briefly verify the transient phenomena when a pulse wave or its equivalent is applied. An input waveform is not a continuous wave but one that gradually attenuates. Voltage and current waveforms refer to anything prior to settling into constant operation as transient phenomena. When constant operation is reached, this is referred to as a steady state. In general, in wireless power transfer, circuit time constants tend to be overwhelmingly faster in relation to air gap fluctuations, and thus may be regarded as in a steady state. In contrast, there are occasions in which transient phenomena must be considered in load fluctuations. The time constant τ explained using Fig. 2.33 expresses the time until values prior to changing reach e−1 ≈ 0.37; therefore, e ≈ 2.7. When I = I 0 e−αt , if αt = 1, then e−1 . That is, τ = 1/α. For example, in an RL series circuit, α = R/L; thus, τ = L/R. [NOTE: The time constant formula differs depending on the circuit.] In this section, we provide an explanation for the difference between waveforms during resonance and non-resonance. We examine transient phenomena in a setup in which resonance does not occur. To begin, we look at the RL series circuit in Fig. 2.34. Figure 2.35 shows a waveform. We apply a short pulse wave of Vin = 100 V and 5 μs. In this circuit, we adopt the same L value and internal resistance r value used in the explanation of the magnetic field resonance in Sect. 2.5.3; thus, L = 500 μH and r = 1 . Hence, τ = L/r = 500 μs is obtained. When changes begin, I = 1.0 A and t = 5 μs; thus, when we check the value of I at t = 505 μs after 500 μs, it is 0.368 A, and we can verify that it is e−1 . In addition, we find that when we check the waveform, there are no vibrations or resonance. We examine the RC circuit in Fig. 2.36 using another configuration in which no resonance occurs. Figure 2.37 shows a waveform. Current flows through the RC Fig. 2.33 Time constant

2.4 Non-resonant Circuit Transient Phenomena (Pulse)

57

Fig. 2.34 RL series circuit VR

VL

V(in)

VR

VL

I 1.5

0.5

1.0

0

[A]

1.0

100

[V]

[V]

V(in) 200

0.0 -0.5

-100

0.0

-1.0

-200 0

-0.5 0

5 10 15 20 25 30 35 40 t [us]

(a) Immediately after start

0.5

1000 2000 3000 4000 5000 t [us]

(b) Voltage transient phenomena

0

1000 2000 3000 4000 5000 t [us]

(c) Current transient phenomena

Fig. 2.35 RL series circuit transient phenomena

Fig. 2.36 RC circuit I

VR 100

[A]

[V]

100 50 0

0

100

200 300 t [us]

400

500

(a) Voltage transient phenomena

50 0

0

100

200 300 t [us]

400

500

(b) Current transient phenomena

Fig. 2.37 RC parallel circuit transient phenomena

circuit by a procedure in which V in = 100 V is applied, electric charge builds up in the capacitor, and then the circuit switches off 5 μs after the capacitor voltage reaches V C = 100 V. The capacitor voltage and load voltage are equipotential; thus,

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2 Basic Knowledge of Electromagnetism and Electric Circuits

V C = V R . In this circuit, we adopted values significantly larger than the value of C for the resonance used in the magnetic field resonance; thus, the waveform is easy to verify. For a smoothing capacitor, some values are within the scope of common use. Hence, C = 100 μF and R = 1 ; thus, the time constant is τ = Cr = 100 μs. I = 100 A of current at t = 0 becomes I = 36.8 A at τ (μs).

2.5 Resonance Circuit and Transient Phenomena Wireless power transfer using magnetic resonance coupling uses a resonance phenomenon. Therefore, in this section, we shall look at transient phenomena during resonance.

2.5.1 LCR Series Circuit Transient Phenomena (Pulse) Similar to transient phenomena in pulse waves in the non-resonant circuit examined in the previous section, we will look at resonance circuit transient phenomena in pulse waves. Resonance circuits are characterized by having large vibrations. Because applied voltage is a pulse wave, it gradually attenuates, as discussed in the previous section. Figure 2.38 shows a circuit diagram of an LCR series resonance circuit, Fig. 2.39 shows a waveform when R = 1 (), and Fig. 2.40 shows a waveform when R = 10 () (which is 10 times the resistance value). L = 500 μH and C = 5 nF. The 100 kHz period is 10 μs, and we find resonance at 10 μs. When the resistance reaches 10 times the value, the Q value reaches 1/10; thus, the speed of attenuation increases. Fig. 2.38 LCR series resonance circuit (pulse)

2.5 Resonance Circuit and Transient Phenomena

59

VL

I

100

0.5

[A]

1.0

[V]

200 0 -100 -200

-1.0 0

5000 t [us]

10000

(a) Voltage transient phenomena (overall image) VR

VC

VL

0

5000 t [us]

(b) Current transient phenomena (overall image) I

1.0

1.0

100

0.5

0.5

0 -100

[A]

200

0.0 -0.5

-200

-1.0

5 10 15 20 25 30 35 40 t [us]

0

(c) Starting voltage

0.0 -0.5

-1.0 0

10000

VR

[V]

[V]

V(in)

0.0 -0.5

5 10 15 20 25 30 35 40 t [us]

0

5 10 15 20 25 30 35 40 t [us]

(e) Starting current

(d) Starting resistance voltage

Fig. 2.39 LCR series resonance circuit transient phenomena (pulse, R = 1 ) VL

I

100

0.5

[A]

1.0

[V]

200 0 -100 -200 0

5000 t [us]

5000 t [us]

(a) Voltage transient phenomena (overall image)

(b) Current transient phenomena (overall image)

VR

VC

[V]

[V]

100 0

0

0

5 10 15 20 25 30 35 40 t [us]

(c) Starting voltage

I

10

1.0

5

0.5

0

-10

0.0 -0.5

-5

-100

10000

VR

VL

200

-200

-1.0

10000

[A]

V(in)

0.0 -0.5

0

5 10 15 20 25 30 35 40 t [us]

(d) Starting resistance voltage

-1.0 0

5 10 15 20 25 30 35 40 t [us]

(e) Starting current

Fig. 2.40 LCR series resonance circuit transient phenomena (pulse, R = 10 )

2.5.2 LCR Series Circuit and Q Value We will now discuss applying not a pulse wave but a waveform (in other words, a sine wave) similar to that used during actual wireless power transfer. Resonance is the phenomenon of repeated energy reciprocation over a fixed period. Discussing the LCR series circuit in Fig. 2.41, the voltage is per formula (2.79). Because the resonance condition is that the reactance is 0, it satisfies formula (2.80). Hence, as in formula (2.81), the sum of the voltage made by the reactance is V L + V C = 0. Consequently, this may be expressed as V L = −V C .

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.41 LCR series resonance circuit (sine wave)

Vin = VL + VC + VR = jωL I + jωL +

I + RI jωC

1 =0 jωC

VL + VC = jωL I +

I =0 jωC

(2.79) (2.80) (2.81)

When this occurs, input voltage is applied only to load R, as in formula (2.82); thus, the impedance during series resonance reaches a minimum and the current reaches a maximum. Considering that the current I flowing in the circuit is a series circuit, and therefore is the same no matter the coil, capacitor, or resistance, we find that the voltage applied to the coil is V L = jQV in , as shown in formula (2.83), and the voltage applied to the capacitor is V C = −jQV in (in other words, Q times the input voltage V in ). Taking out only Q results in formula (2.84). Vin = VR = R I VL + VC = jωL I +

jωL Vin Vin I = + = j QVin − j QVin jωC R jωC R Q=

1 ωL = R ωC R

(2.82) (2.83) (2.84)

During resonance, voltage V L made by L resonates with voltage V C made by C, and both vibrate at the same time. When the Q value is high, this voltage becomes very large.

2.5 Resonance Circuit and Transient Phenomena

61

VL

I 100

[A]

200

20000

[V]

40000 0

-20000

0 -100

-40000

-200 0

2000 4000 6000 8000 10000 t [us]

0

2000 4000 6000 8000 10000 t [us]

(a) Coil voltage VR

VC

(b) Current I

VR

VL 10

10

1000

5

5

0

[A]

2000

[V]

[V]

V(in)

0

-10

-10

-2000 0

0

5 10 15 20 25 30 35 40 t [us]

(c) Starting voltage

0

5 10 15 20 25 30 35 40 t [us]

(d) Starting load voltage V(in)

VR

VC

VL

20000

100

[V]

200

0

-20000

5 10 15 20 25 30 35 40 t [us]

(e) Starting current V(in)

40000

[V]

0 -5

-5

-1000

VR

0 -100

-40000 9960

9970

9980 t [us]

9990

10000

(f) Steady-state voltage

-200 9960

9970

9980 t [us]

9990

10000

(g) Steady-state applied voltage and load voltage

Fig. 2.42 LCR series resonance waveform (sine wave, R = 1 )

For example, resonance occurs at 100 kHz when L = 500 μH and C = 5 nF, but when this occurs, if R = 1 , then Q = 314.159. Hence, as in formula (2.83), V L = 100 × 314.159 = 31415.9 V. Similarly, when R = 10 , Q = 31.4159; thus, V L = 100 × 31.4 = 3141.59 V. Because in this situation V L = −V C , capacitor withstand voltage should be considered. In addition, during power transmission, there is a load on the secondary side, which is not a problem because R is not that small in relation to the entire circuit. However, if there is no secondary side and no load, then R of the overall circuit becomes small, so if voltage is thus applied, a large voltage is applied to the coil and capacitor, and caution is required. Figures 2.42 and 2.43 show actual waveforms. These occur when R = 1  (Q = 314) and R = 10  (Q = 31), respectively. We find that the higher the Q value during resonance, the longer it takes to reach a steady state. In addition, the coil and capacitor voltage become larger. Furthermore, we find that the coil voltage and capacitor voltage invert by 180° and resonate.

2.5.3 Magnetic Field Resonance Transient Phenomena Figure 2.44 shows a circuit diagram during magnetic field resonance. L = 500 μH and r 1 = r 2 = 1 . Figure 2.45 shows the voltage–current waveform at each point when this occurs. The figure presents overall images, waveforms up to midrange, and waveforms at the start and during the steady state. When this occurs, an optimal load

62

2 Basic Knowledge of Electromagnetism and Electric Circuits I 20

2000

10

[A]

[V]

VL 4000 0

-2000

0 -10

-4000

-20 0

2000 4000 6000 8000 10000 t [us]

0

(a) Coil voltage VR

VC

(b) Current VR

VL

I

40

10

1000

20

5

0

[A]

2000

[V]

[V]

V(in)

2000 4000 6000 8000 10000 t [us]

0 -20

-1000

-40

-2000 0

-10 0

5 10 15 20 25 30 35 40 t [us]

(c) Starting voltage

5 10 15 20 25 30 35 40 t [us]

0

5 10 15 20 25 30 35 40 t [us]

(e) Current

(d) Starting load voltage

V(in)

VR

VC

V(in)

VL

4000

200

2000

100

[V]

[V]

0 -5

0

VR

0 -100

-2000 -4000 9960

9970

9980 t [us]

9990

10000

(f) Steady -state voltage

-200 9960

9970

9980 t [us]

9990

10000

(g) Steady-state applied voltage and load voltage

Fig. 2.43 LCR series resonance waveform (sine wave, R = 10 )

Fig. 2.44 Circuit diagram of magnetic resonance coupling (S–S)

is connected that can realize the maximum efficiency of the load. Compared with Fig. 2.41, which shows LCR series resonance, here the load is large. The capacitor voltage is not as large as that in Fig. 2.42, but nevertheless a large voltage is applied, so caution is required when selecting a capacitor.

2.6 Root Mean Square (RMS) Values, Active Power, Reactive … I1

VL1+VLm1 10

2000

5 [A]

4000 [V]

63

0

0 -5

-2000

-10

-4000 0

1000 [us]

0

2000

1000 [us]

2000

(a) Primary-side coil voltage (overall) (b) Primary-side current (overall) VL2+VLm2

VL1+VLm1

2000 [V]

4000

2000 [V]

4000 0

0 -2000

-2000 -4000

-4000 0

200 [us]

400

0

(c) Primary-side coil voltage (to midrange)

200 [us] I2

I1 10

5

5 [A]

10 [A]

400

(d) Secondary-side coil voltage (to midrange)

0

0 -5

-5

-10

-10 0

200 [us]

0

400

200 [us]

400

(e) Primary-side current (to midrange) (f) Secondary-side current (to midrange) V2 50

5

25

0

0

-100

-5

-200

I1 [A]

10

3 2 1 0 -1 -2 -3

0 -25 -50

-10

0 5 10 15 20 25 30 35 40 [us]

0 5 10 15 20 25 30 35 40 [us]

(g) Starting primary-side voltage and current VL1+VLm1

(h) Starting secondary-side voltage and current

VC1

VL2+VLm2 200

[V]

400

1000 0

0 -200

-2000

-400 0 5 10 15 20 25 30 35 40 [us]

0 5 10 15 20 25 30 35 40 [us]

(i) Starting primary-side coil and capacitor voltage V1

(j) Starting secondary-side coil and capacitor voltage

I1

V2

I2

10

200

10

100

5

100

5

0

0

0

0

-5

-100 -200

I1 [A]

200

-10 1980 1985 1990 1995 2000 [us]

V2 [V]

V1 [V]

VC2

-100 -200

(k) Steady-state primary-side voltage and current VL1+VLm1

-5 -10 1980 1985 1990 1995 2000 [us]

(l) steady-state secondary-side voltage and current

VC1

VL2+VLm2 2000

1000

1000 [V]

2000 0

-1000

-2000

-2000 1980 1985 1990 1995 2000 [us]

VC2

0

-1000

(m) Steady-state primary-side coil and capacitor voltage

I2 [A]

[V]

2000

-1000

[V]

I2 I2 [A]

I1

100

V2 [V]

V1 [V]

V1 200

1980 1985 1990 1995 2000 [us]

(n) Steady-state secondary-side coil and capacitor voltage

Fig. 2.45 Magnetic resonance coupling (S–S) waveform (RLopt = 15.7 , L m = 25 μH)

2.6 Root Mean Square (RMS) Values, Active Power, Reactive Power, and Instantaneous Power In power transmission, power needs to be accurately evaluated. If active and reactive power are not properly handled and there is a lack of understanding of the root mean square (RMS) values and peak values (maximum values), then errors in calculation arise. Here, we describe RMS values and active power, which are the foundation to building knowledge of power calculation. [NOTE: In this section, we take care to

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2 Basic Knowledge of Electromagnetism and Electric Circuits

Fig. 2.46 RMS values and peak values (maximum values)

use different symbols for peak values and RMS values, but be aware that in other sections, there are places where the symbols V and I are used for peak values in order to avoid complications.]

2.6.1 Root Mean Square (RMS) Values The following show the respective formulas for the instantaneous voltage v (t) and instantaneous current i(t), which are measured values: v(t) =

√ 2V sin ωt

(2.85)

i(t) =

√ 2I sin ωt

(2.86)

V (V rms ) and I (I rms ) are root mean square (RMS) values. When the peak values (maximum values) are V m and I m , their relationship to the RMS values is as follows: Vm = Im =

√

2V

√

2I

(2.87) (2.88)

When calculating power, we think in terms of the RMS values of voltage V and current I. Figure 2.46 shows the RMS values and peak values.

2.6.2 Instantaneous Power Versus Active Power and Reactive Power Power has various definitions. Actually, consumed power is active power. For this reason, when calculating power based on efficiency or other factors, instantaneous power is integrated over time and then time-averaged or RMS values of the voltage and current are multiplied by the power factor cosθ . We shall offer an explanation

2.6 Root Mean Square (RMS) Values, Active Power, Reactive …

65

in this section. Assuming the phase difference of voltage v(t) and current i(t) is θ , if the current is leading, then the respective formulas for the voltage and current are as follows: v(t) = i(t) =

√

√ 2V sin ωt

(2.89)

2I sin(ωt + θ )

(2.90)

Figure 2.47 shows graphs of the voltage and current when the current phase is lagging by 60° and leading by 45°. In addition, Fig. 2.48 shows graphs of the active power, reactive power, and instantaneous power corresponding to each. In the graphs, V = 1 and I = 1. We shall explain the graphs using these values. The product of voltage v and current I is instantaneous power p. p(t) = v(t)i(t) =

√ √ 2V sin ωt · 2I sin(ωt + θ ) = V I cos θ (1 − cos 2ωt) +V I sin θ sin 2ωt (2.91)

To determine the instantaneous power at a certain time, p(t) is determined. This formula can be divided into cosθ terms and sinθ terms, and into terms related to the active power and terms related to the reactive power. In addition, power is two times 2

2

1.5

1.5

1

1

0.5

0.5

0 -0.5

0

30 60 90 120 150 180 210 240 270 300 330 360

Voltage Current

0 -0.5 0

-1

-1

-1.5

-1.5

45

90 135 180 225 270 315 360

Voltage Current

-2

-2

(a) θ = -π/3 = -60°

(b) θ = π/4 = 45°

Fig. 2.47 Relationship of voltage v and current i (θ = −60°, θ = 45°) 2

2

1.5

1.5

Active power

1 0.5 0 -0.5

0

30 60 90 120 150 180 210 240 270 300 330 360

Reactive power Instantaneous power

-1

Reactive power

0 -0.5 0

45

90 135 180 225 270 315 360

-1

-1.5

-1.5

-2

-2

(a) θ = -π/3 = -60°

Active power

1 0.5

Instantaneous power

(b) θ = π/4 = 45°

Fig. 2.48 Relationship of active power pe , reactive power pr , and instantaneous power p (θ = − 60°, θ = 45°)

66

2 Basic Knowledge of Electromagnetism and Electric Circuits

the frequency of the voltage and current. Hence, it is 2ω, and only half-period T /2 needs to be considered. Below are the instantaneous values for active power Pe and reactive power Pr. The graph shown in Fig. 2.48 is a rendering of formulas (2.91)–(2.93). pe (t) = V I cos θ (1 − cos 2ωt)

(2.92)

pr (t) = V I sin θ sin 2ωt

(2.93)

The value of integrating the instantaneous power over a time of one period T and then averaging it over the time of one period T is referred to as the active power (effective power/real power) Pe (W), as shown by the formula: Pe = =

1 T 1 T ∫ p(t) dt = ∫ v(t)i(t) dt T 0 T 0

1 T ∫{V I cos θ (1 − cos 2ωt) + V I sin θ sin 2ωt} dt = V I cos θ T 0

(2.94)

In formula (2.94), the instantaneous power includes the reactive power, and a negative is produced. However, if thus integrated, then the average of reactive power Pr in a half period is 0, so the active power portion remains, resulting in the formula for active power Pe . In addition, this becomes the RMS value of the voltage and current multiplied by power factor cosθ . This is frequently used, and is revised into the following formula: Pe = V I cos θ

(2.95)

θ is the phase difference of the voltage and current. Active power Pe is power that is actually consumed. Naturally, formula (2.96), which integrates and time-averages the instantaneous value of the active power, is an equivalent formula. Calculations in these formulas are done in half periods. 1 T /2 1 T /2 ∫ pe (t)dt = ∫ V I cos θ (1 − cos 2ωt)dt = V I cos θ T /2 0 T /2 0

(2.96)

Reactive power Pr (Var) is power that is not consumed. Power equal to that from a power source is returned from the load side and then returned to the power source. Reactive power is simply the back and forth of voltage and current between a power source and reactive elements such as coils and capacitors, the average of which is always 0 W in one period. That is, although it is repetitive, no power consumption occurs. Reactive power is produced when there is a coil or capacitor. Reactive power is the RMS value of the voltage and current multiplied by power factor sinθ , as shown in formula (2.97). The instantaneous value of the reactive power is 0 when it is integrated and simply time-averaged, so formula (2.98) is as follows:

2.6 Root Mean Square (RMS) Values, Active Power, Reactive …

Pr = V I sin θ 1 T /2 1 ∫ pr (t)dt = T /2 0 T /2

T /2 ∫ V I sin θ sin 2ωtdt = 0

67

(2.97) (2.98)

0

In addition, the effect of sinθ resulting from phase θ contributes to the amplitude of the reactive power portion of the instantaneous power. The apparent power Pa (VA) is the product of the RMS values of the voltage and current and can be seen as the product of absolute values. That is, the phases have no relationship, as shown in the formula: Pa = V I

(2.99)

Formulas (2.100) to (2.102) show the relationship of the apparent power Pa , active power Pe , and reactive power Pr . Pa =



Pe2 + Pr2

(2.100)

Pe = Pa cos θ

(2.101)

Pr = Pa sin θ

(2.102)

[NOTE: In general, the active power, reactive power, and apparent power are written as P, Q, and S, but to avoid duplicating the symbols, they are written as above in this book.] Figure 2.49 shows graphs of the phase as it changes from 0° to 90°. According to formulas (2.92) and (2.93), the phase difference θ used in the power factor is related to the amplitude of the active power and reactive power. When the phase difference is 0°, this results in a large value of cosθ = 1, and the instantaneous active power value is large. This gradually becomes smaller as the phase difference approaches 90°. Finally, cosθ = 0, and the active power also becomes 0. In contrast, with the reactive power, at 0° when there is no phase difference, sinθ = 0, so the reactive power is also 0. However, this gradually becomes larger as the phase difference approaches 90°, and finally, sinθ = 1. In general, power P refers to the average active power for the value of the timeaveraged active power, as shown in Fig. 2.50. According to formula (2.92), this value is half the peak value of the active power. For example, in Fig. 2.27a, θ = 0°, so if we consider that V = 1 and I = 1 and compare this to formula (2.31), we find the average power to be 1 W. From the above, when determining the power by efficiency and similar factors, the necessary parameter is the active power. Therefore, we must time-integrate the

68

2 Basic Knowledge of Electromagnetism and Electric Circuits

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0 -0.5 0

45

0 90 135 180 225 270 315 360 -0.5 0

45

0 90 135 180 225 270 315 360 -0.5 0

-1

-1

-1

-1.5

-1.5

-1.5

-2

-2

(c) θ = 30°

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0 -0.5 0

45

0 90 135 180 225 270 315 360 -0.5 0

45

90 135 180 225 270 315 360

-2

(b) θ = 15°

(a) θ = 0°

45

0 90 135 180 225 270 315 360 -0.5 0

-1

-1

-1

-1.5

-1.5

-1.5

-2

-2

45

90 135 180 225 270 315 360

-2

(d) θ = 45°

(f) θ = 75°

(e) θ = 60° 2 1.5

Active power

1 0.5 0 -0.5 0

Reactive power 45

90 135 180 225 270 315 360

-1

Instantaneous power

-1.5 -2

(g) θ = 90°

Fig. 2.49 Changes in relationship of active power pe , reactive power pr , and instantaneous power p 2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0 -0.5 0

45

0 90 135 180 225 270 315 360 -0.5 0

45

90 135 180 225 270 315 360

0 -0.5 0

-1

-1

-1

-1.5

-1.5

-1.5

-2

-2

(a) θ = 0°

Active power

Reactive power

(b) θ = 60°

Instantaneous power

-2

45 90 135 180 225 270 315 360

(c) θ = 90°

Average power

Fig. 2.50 Active power Pe and average power P

instantaneous power, and then determine and time-average it or multiply the RMS values of the voltage and current by cosθ .

References

69

References 1. T. Imura, Relation between magnetic flux and magnetic resonant coupling. IEICE Technical Report WPT2016-9 (2016), pp. 1–4 2. K. Gotoh, S. Yamazaki, Detailed explanation electromagnetics exercises. Kyoritsu Shuppan (2014) 3. C.A. Balanis, Antenna Theory: Analysis and Design (Wiley, New York, 1982)

Chapter 3

Magnetic Resonance Coupling Phenomenon and Basic Characteristics

The purpose of this chapter is to examine the characteristics of magnetic resonance coupling. We start with a basic coil shape, and then introduce an open type and short type. Next, we explain the characteristics of magnetic resonance coupling power transmission. The major characteristic of magnetic resonance coupling is that two peaks are produced. The behavior of the near electromagnetic field is also introduced. In addition, we shall explain that although the operating frequency depends upon the coil size, it can be used across a broad frequency range from kHz to MHz to GHz.

3.1 Coils and Resonators When a capacitor is attached to a coil and achieves a resonant state, it is referred to as a resonator. However, sometimes, resonance can be achieved without a capacitor and with a coil alone. In this case, it is called a self-resonant resonator. In this section, we shall explain coils and resonators.

3.1.1 Spiral, Helical, and Solenoid Coils A magnetic field type coil consists of coils wound around in a loop so that magnetic fields are intensified in the center. These are broadly divided into two types, as shown in Fig. 3.1. In the first type, when the coils are directly facing each other as in the spiral or helical types shown in Fig. 3.1a, the magnetic flux is coupled immediately above (0°). A spiral coil is wound into a single flat spiral-like shape similar to a mosquito coil, also known as circular coil. A helical coil is wound vertically. In the second type, as shown in Fig. 3.1b, when the coils are directly facing each other, as in a solenoid coil, the magnetic flux is coupled with a 180° rotation. Naturally, as in Fig. 3.2, 180° rotation and coupling is possible with spiral coils, and straight (0°) coupling without rotation is possible with solenoid coils too. In addition, © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_3

71

72

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Fig. 3.1 Ways of coupling two types of magnetic flux

Fig. 3.2 Changing magnetic flux coupling by rearranging coils

a high number of helical-style turns, as shown in Fig. 3.3, matches the shape of a solenoid coil. Magnetic flux is created whether the shape is circular or square, but in general, a square is stronger in coupling because of its larger area (Fig. 3.4). In the various multi-applicable arrangements shown in Fig. 3.1, spiral and other coils have symmetrical structures; thus, a misalignment becomes stronger whether to the front and back or left and right. Hence, coils such as spiral coils are superior in situations involving misalignment, regardless of the direction. On the other hand, in the case of an asymmetrical structure such as a solenoid coil, a misalignment is clearly divided into stronger front and back directions and weaker left and right directions. Hence, in terms of deviating in one direction, a solenoid coil is superior.

Fig. 3.3 Helical coil and solenoid coil

Fig. 3.4 Circle versus square

3.1 Coils and Resonators

73

In actual practice, a structure having a ferrite and aluminum plate is often used for kHz bands, as shown in Fig. 3.5 for the purpose of achieving large mutual inductance and reducing leakage flux to the back of the coil. In the case of a spiral structure, an aluminum plate serves to dissipate heat. The reason that ferrite is often used for kHz bands is that the loss with ferrite is largely negligible. The loss becomes larger with MHz bands, and since ferrite is limited in allowing magnetic flux pass through without any loss, ferrite is generally used as a noise measure. Figure 3.6 shows a spiral coil and helical coil analysis model and real examples thereof. Both are self-resonant, so no external resonant capacitor is required. We shall describe the characteristics of open-type self-resonance in detail in Chap. 7. In this example, the spiral coil has a 150 mm radius, 2.75 turns (a total of 5.5 turns between two layers), a 3 mm space between coils (between copper wires), and a 2 mm wire thickness; in other words, it has a 5-mm-pitch double-layer structure

(a) Spiral

(b) Solenoid

Fig. 3.5 Use of ferrite and aluminum plate

(a) Spiral coil electromagnetic (b) Spiralcoil (photograph) field analysis model

(c) Helical coil electromagnetic field analysis model

(d) Helical coil (photograph)

Fig. 3.6 Photographs of electromagnetic field analysis model and coil

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics 10

Fig. 3.7 Spiral coil input impedance (single element)

Re [Ω]

8 6

Zin{Re} Zin{Im}

4 2 0 11

12

13

14

15

16

400 300 200 100 0 -100 -200 -300 -400

Im [Ω]

74

Frequency [MHz]

with 10 mm between layers. The resonant frequency of a single power-transmitting coil or power-receiving coil element is 13.56 MHz. In addition, a helical coil has a 150 mm radius, 5 turns, and a 5 mm pitch; L = 8.5 μH, C = 9.7 pF, and r = 0.82 . The resonant frequency of a single element is 17.6 MHz. We shall examine the characteristics of this single spiral coil element. Figure 3.7 shows the input impedance of a single spiral coil element. We find that the impedance real number component is 1  or less and is the total value of the conductor electrical resistance and radiation resistance, but it is small for both. In addition, generally, the dominant resistance of a coupling type resonator is the electrical resistance of a conductor, and the radiation resistance is slight and often negligible. The fact that the imaginary component of impedance, in other words, the reactance, is 0 , means that resonant frequency f 0 = 13.56 MHz for a single spiral coil element. Using two such resonant state spiral coils as a transmitting coil and receiving coil, power is transmitted by magnetic field coupling. As described below, if there are two elements, then the characteristics of the secondary-side coil also have an influence, which changes the input impedance. In a communication antenna, this structure is generally referred to as a spiral antenna. In the setup used in this book, the coil size is very small relative to the wavelength, and thus matching the impedance to the space is difficult for a single coil even in a resonant state. Hence, the structure cannot emit electromagnetic waves and generally cannot be used as a communication coil. On the other hand, when attempting wireless power transmission, since electromagnetic waves are not emitted, no power is wastefully consumed as radiation loss, and there are few compromising emanations. However, if even small radiation emissions cannot be ignored, then corrective measures are required. However, even if radiation emissions are on a very low level and not large enough to consider protections for the human body, they still affect electronic instruments and should be kept in mind.

3.2 Summary of Air Gap and Misalignment Characteristics

75

3.2 Summary of Air Gap and Misalignment Characteristics In this section, we shall introduce the basic characteristics of magnetic resonance coupling. These include the high efficiency achieved when there is a large air gap, the two-power-peak characteristic of magnetic resonance coupling, and the concurrent two- or three-input impedance resonance points. In addition, we shall mention its strength even with misalignment.

3.2.1 Efficiency, Power, and Input Impedance with Air Gap We use the spiral coil shown in Fig. 3.6. For the mutual inductance and coupling coefficient, we use the values in Table 3.1 and Fig. 3.8. Figure 3.9 shows the power transmission characteristics when the air gap has been changed. Figure 3.9 shows the efficiency, power, and input impedance. η is the power transmission efficiency, ηr1 is the percentage loss in the primary-side internal resistance, and ηr2 is the percentage loss in the secondary-side internal resistance. Pin is the input power, PRL is the receiving power, Pr1 is the power consumption with the primaryside internal resistance, and Pr2 is the power consumption with the secondary-side internal resistance. Z in is the input impedance, which is the ratio of the voltage to the current as seen from the power source looking at the load side, as in formula (3.1). The impedance is divided into a real number component created by factors such as the resistance, and imaginary components created by the coil and capacitor. Z in =

V1 I1

(3.1)

For the load, a 50  fixed resistance is connected. When the air gap is changed, the frequency with the greatest efficiency is always frequency f 0 , which is the same as the resonant frequency of a single coil element. Figure 3.9a, b and c confirm that there are two peaks at which the power is highest when coupling is strong, and the air gap is small. Thus, when the gap becomes larger, the two divided peaks become one. The frequency at which they become one is the same as the resonant frequency f 0 for a single coil element. That is, the lesser the distance between coils, the more the resonant frequency is divided in two. Upon looking at the imaginary component Im{Z in } = 0 of the input impedance, it can be inferred that there are three resonant frequencies when coupling is strong. It is necessary to hold down the characteristics of these three resonant frequencies f m , f 0 , and f e (f m < f 0 < f e ). Resonant frequency f 0 at the center does not move. This is the same as the resonant frequency f 0 of a single element. On the other hand, f m and f e approach the middle and disappear as the coupling becomes weaker. When the coupling is strong, the two power peaks appear close to resonant frequencies f m and f e . The reason that the resonant frequency and

10

0.595

6.56

110

0.139

1.54

210

0.053

0.59

g (mm)

k

L m (μH)

g (mm)

k

L m (μH)

g (mm)

k

L m (μH)

0.54

0.049

220

1.38

0.125

120

5.23

0.475

20

0.42

0.038

250

1.24

0.113

130

4.35

0.394

30

0.29

0.026

300

1.12

0.102

140

3.70

0.335

40

Table 3.1 Air gap g, coupling coefficient k, and mutual inductance L m 50

0.21

0.019

350

1.02

0.092

150

3.19

0.289

60

0.15

0.014

400

0.93

0.084

160

2.78

0.252

70

0.11

0.010

450

0.84

0.076

170

2.45

0.222

80

0.09

0.008

500

0.77

0.070

180

2.16

0.196

90

0.07

0.006

550

0.70

0.064

190

1.92

0.174

100

0.05

0.005

600

0.64

0.058

200

1.72

0.156

76 3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

3.2 Summary of Air Gap and Misalignment Characteristics

77

Fig. 3.8 Coupling coefficient k and mutual inductance L m

power peaks do not exactly match is that the load value affects the frequency at which power peaks. [Note: As the load value approaches 0 , the two power peaks and two resonant frequencies start to match.] These two power peaks are characteristically not visible unless there is magnetic resonance coupling, so they are a major characteristic of magnetic resonance coupling. As understood from Fig. 3.9, the frequency at which the efficiency is largest is f 0 even when using a 50  fixed load, and there is little worsening of the efficiency in response to frequency fluctuations. However, with regard to power, the two peaks from f 0 appearing near resonant frequencies f m and f e tend to be large, as do power fluctuations in response to frequency changes. It should be noted that these facts significantly affect the system settings. It is possible to use the two peaks when wanting to receive power while sacrificing on some efficiency, for example. Here, the load value was fixed at 50 , but during actual use, using the optimal load value for maximally efficient operation whatever the air gap is desirable. In Chaps. 4 and 5, we present details such as the relationships involved in the optimal load for maximum efficiency. The frequency for the maximum efficiency with magnetic resonance coupling is f 0 even in cases of optimal load.

3.2.2 Misalignment Characteristics The magnetic resonance coupling is strong despite air gaps and misalignments. This is essentially a matter of coupling strength or weakness, so large air gaps and misalignments may be thought of in the same way. Let us take a look at this in the example of the square spiral coil shown in Fig. 3.10. This coil is a square of 380 mm on each side with a pitch of 4 mm and 10 turns. It has a self-inductance of 69.4 μH, an internal resistance of 1.39  accounting for the skin effect, and a resonant frequency and operating frequency that are both 10 MHz. Figure 3.11 shows the coupling coefficient, mutual inductance, optimal load, maximum efficiency, and receiving power. There is a null point on the left-right diagonal where the coupling coefficient reaches 0, which is the region where power transmission is not possible. Shift to the plus side or minus side occurs at this null point. It should be noted that the absolute values also increase in the negative direction. On the other hand, if we remove the null point, even if misalignment occurs, we find the efficiency to be high. In addition, the farther is the distance, the greater is the receiving power.

78

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics η[%]

ηr1 [%]

ηr2 [%]

Pin

P RL

P r1

Zin{Re}

P r2

250

80

40 20

150 100 50

0 11

12

13

14

15

Zin [ohm]

200

60

Power [W]

Efficiency [%]

100

0

16

11

12

Frequency [MHz]

13 14 Frequency [MHz]

15

16

500 400 300 200 100 0 -100 11 -200 -300 -400 -500

12

Zin{Im}

13

14

15

16

Frequency [MHz]

(a) g = 100 mm η[%]

ηr1 [%]

ηr2 [%]

Pin

100 80

P r1

Zin{Re}

P r2 400 300 200 100 0 -100 11 -200 -300 -400 -500

200 150

Power [W]

60 40 20 0 11

12

13

14

15

Zin [ohm]

Efficiency [%]

P RL

250

100 50 0

16

11

12

Frequency [MHz]

13 14 Frequency [MHz]

15

16

12

Zin{Im}

13

14

15

16

Frequency [MHz]

(b) g = 150 mm η[%]

ηr1 [%]

ηr2 [%]

Pin

P RL

P r1

Zin{Re}

P r2

80

250

60

200

40 20 0 11

12

13

14

15

Zin [ohm]

300

Power [W]

Efficiency [%]

100

150 100 50 0

16

11

12

Frequency [MHz]

13 14 Frequency [MHz]

15

16

400 300 200 100 0 -100 11 -200 -300 -400 -500

12

Zin{Im}

13

14

15

16

Frequency [MHz]

(c) g = 200 mm ηr1 [%]

ηr2 [%]

Pin

60

Power [W]

Efficiency [%]

80

40 20 0 11

12

13

14

15

16

P RL

P r1

Zin{Re}

P r2

400 350 300 250 200 150 100 50 0

Zin [ohm]

η[%] 100

11

12

Frequency [MHz]

13 14 Frequency [MHz]

15

16

400 300 200 100 0 -100 11 -200 -300 -400 -500

12

13

Zin{Im}

14

15

16

Frequency [MHz]

(d) g = 250 mm ηr1 [%]

ηr2 [%]

Pin

60

Power [W]

Efficiency [%]

80

40 20 0 11

12

13

14

Frequency [MHz]

15

16

P RL

P r1

Zin{Re}

P r2

900 800 700 600 500 400 300 200 100 0

Zin [ohm]

η[%] 100

11

12

13 14 Frequency [MHz]

15

16

400 300 200 100 0 -100 11 -200 -300 -400 -500

(e) g = 300 mm

Fig. 3.9 Efficiency, power, and input impedance frequency characteristics

12

13

Zin{Im}

14

Frequency [MHz]

15

16

3.2 Summary of Air Gap and Misalignment Characteristics

79

Fig. 3.10 Square spiral coil -0.06--0.04 0.06-0.08 0.18-0.2 0.3-0.32 0.42-0.44 0.54-0.56

-0.04--0.02 0.08-0.1 0.2-0.22 0.32-0.34 0.44-0.46 0.56-0.58

-0.02-0 0.1-0.12 0.22-0.24 0.34-0.36 0.46-0.48 0.58-0.6

-7--6 -1-0 5-6

0-0.02 0.12-0.14 0.24-0.26 0.36-0.38 0.48-0.5 0.6-0.62 1000

-6--5 0-1 6-7

-5--4 1-2 7-8

-4--3 2-3 8-9

-3--2 3-4 9-10

900 800 700

g [mm]

300 200

0

100

200

300

400

500

30-40 70-80 110-120 150-160 1000 900

800

800

700

700

600

600 500 400

300

300

200

200

100

100 0

0

100

-600 -500 -400 -300 -200 -100

20-30 60-70 100-110 140-150

900

400

400

10-20 50-60 90-100 130-140

1000

500

600 500

0-10 40-50 80-90 120-130

-2--1 4-5 10-11

0 600

g [mm]

-0.08--0.06 0.04-0.06 0.16-0.18 0.28-0.3 0.4-0.42 0.52-0.54

g [mm]

-0.1--0.08 0.02-0.04 0.14-0.16 0.26-0.28 0.38-0.4 0.5-0.52

displacement [mm]

displacement [mm]

displacement [mm]

(b) Mutual inductance Lm [μH]

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100

100-200 500-600 900-1000

(c) Optimal load RLopt [ Ω] 200-300 600-700 1000-1100

300-400 700-800 1100-1200

1000

1000

900

900

800

800

700

700

600

600

500 400

500 400

300

300

200

200

100

100

0

0

displacement [mm]

(d) Maximum efficiency ηmax

g [mm]

0-10

0-100 400-500 800-900 1200-1300

g [mm]

(a) Coupling coefficient k

displacement [mm]

(e) Receiving power P2

Fig. 3.11 Coupling coefficient, mutual inductance, optimal load, maximum efficiency, and receiving power

80

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Fig. 3.12 Efficiency when load resistance is 50 

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100 1000 900 800

600 500 400

g [mm]

700

300 200 100 0

displacement [mm]

On the other hand, often, 50 , which is the impedance of a measuring instrument, is used for the load. Therefore, Fig. 3.12 shows an efficiency map for when the load resistance is 50 . Although significant differences are difficult to see near coils, distant efficiency drops are more significant compared to the optimal load.

3.3 Near Field of Magnetic Field and Electric Field Let us look at the behavior of magnetic fields and electric fields near a coil. We shall turn our attention to the three resonant frequencies f m , f 0 , and f e (f m ≤ f 0 ≤ f e ). f m and f e are the resonant frequencies at either side, which strongly affect the two power peaks and largely match the two power peaks. The load values exactly match at 0 . On the other hand, f 0 is the middle resonant frequency, which matches the resonant frequency with a single element. Furthermore, f 0 is the frequency of maximum efficiency. Figure 3.13 shows an equivalent circuit for the magnetic resonance coupling (SS). As in Fig. 3.13a, when the direction of magnetic flux is considered, it is easy to see that load-side current I 2 is inward oriented, which we will explain in this section. On the other hand, it is more suitable to consider the circuit manufacture, for example, with I 2 outward oriented, because then it is aligned with the direction of current in a circuit connected to the load side. In Chap. 8, we will deal with systems, which we consider in terms of outward-oriented I 2 . Figures 3.14 and Fig. 3.15 show the efficiency, power, current amplitude, and phase when resistance is fixed at 50 , and the optimal load for maximum efficiency is 78.4 . Here, we used an open-type helical coil with an air gap of 150 mm. First, the three resonant frequencies f m , f 0 , and f e (f m ≤ f 0 ≤ f e ) are known from current phase I 1 in Figs. 3.14d, 3.15d. The ratio of V 1 to I 1 is the impedance. Since V 1 is fixed, we can determine it from I 1 , and the resonant frequency is attained when

3.3 Near Field of Magnetic Field and Electric Field

81

(b) Outward-oriented I2

(a) Inward-oriented I2 Fig. 3.13 Load-side I 2 current direction η[%]

ηr1 [%]

ηr2 [%]

Pin

80 40 20 15

|I1|

16

17

18

19

100 50 0

20

15

16

17

(b) Power

|I2|

arg I1

arg I1

0.5 0.0 19

20

arg -I2

180 135 90 45 0 -45 -90 -135 -180

Phase [deg]

1.0

20

arg I2

Phase [deg]

1.5

18

19

(a) Efficiency

2.0

17

18

Frequency [MHz]

180 135 90 45 0 -45 -90 -135 -180

16

P r2

150

Frequency [MHz]

2.5

15

P r1

200

60

0

Current [A]

P RL

250

Power [W]

Efficiency [%]

100

15

16

17

18

19

20

15

16

17

18

19

20

Frequency [MHz]

Frequency [MHz]

Frequency [MHz]

(c) Current amplitude

(d) Current phase (I2)

(e) Current phase (-I2)

Fig. 3.14 Various current amplitude and phase frequency characteristics, 50 

the phase of I 1 is 0. When the load is 50 , it appears that f m and f e largely match the two power peaks, but true matching is when the load resistance is not 50  but 0 . Whatever the case, the resonance generated at f m and f e strongly influences the power peaks; upon looking at the two power peaks, they may also be considered as two resonant frequencies. As understood from current phases I 1 and I 2 in Figs. 3.14d and 3.15d, they are in phase at low frequencies and antiphase at high frequencies. Checking this against the air gap characteristics, when the coupling is strong, f m and f e are distant from the middle resonant frequency f 0 . Hence, at low resonant frequency f m , the phases are largely the same, and at high resonant frequency f e , the phases are largely opposite. On the other hand, checking the middle resonant frequency f 0 , which matches the resonant frequency of a single element, against the air gap characteristics, there are no changes even when an air gap is produced. However, as found from current phase

82

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics η[%]

ηr1 [%]

ηr2 [%]

Pin

80 60

Power [W]

Efficiency [%]

100

40 20 0 15

16

17

18

19

20

P RL

15

16

Frequency [MHz]

(a) Efficiency |I2|

15

16

17

18

19

20

Frequency [MHz]

(c) current amplitude

16

20

arg I1

180 135 90 45 0 -45 -90 -135 -180 15

17 18 19 Frequency [MHz]

arg I2

Phase [deg]

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

P r2

(b) Power

arg I1

Phase [deg]

Current [A]

|I1|

P r1

180 160 140 120 100 80 60 40 20 0

17

18

19

Frequency [MHz]

(d) current phase (I2)

20

arg- I2

180 135 90 45 0 -45 -90 -135 -180 15

16

17

18

19

20

Frequency [MHz]

(e) current phase (-I2)

Fig. 3.15 Various current amplitude and phase frequency characteristics (optimal load 78.4 )

I 1 in Figs. 3.14d and 3.15d, there is no change even if there are load fluctuations. In addition, f 0 is the frequency for maximum efficiency. As seen from current phases I 1 and I 2 in Figs. 3.14d and 3.15d, there is a 90° phase difference at f 0 , and I 2 advances 90° from I 1 . This is also a characteristic seen in magnetic resonance coupling as related to high-efficiency circumstances. Figure 3.16 shows a vector display figure of the magnetic field behavior near a coil based upon the current phase. Figure 3.17 shows the behavior of the current and the magnetic field. This is the characteristic distribution at the two resonant frequencies f m and f e . When currents are largely in phase, the magnetic field direction in the middle between the coils is aligned vertically (z-axis direction), and when currents are largely antiphase, the magnetic flux is not in the middle between the coils but at the ends aligned horizontally (x-axis direction). Fig. 3.16 Magnetic field vector display

(a) fm

(b) fe

3.3 Near Field of Magnetic Field and Electric Field

83

Fig. 3.17 Current and magnetic field at f m and f e

(a) Magnetic wall, fm

(b) Electric wall, fe

That is, focusing on the state of a magnetic field on the plane of symmetry of a transmitting coil and receiving coil, the plane of symmetry becomes a magnetic wall at f m . A magnetic wall refers to a phenomenon in which a magnetic field is distributed perpendicularly to a plane of symmetry, and an electric field is generated horizontally. Just as the impedance of space becomes infinite, it equals an open state. On the other hand, at f e , the plane of symmetry becomes an electric wall. Here, an electric wall refers to a magnetic field distributed horizontally to a plane of symmetry, and an electric field distributed perpendicularly. Just as the impedance reaches 0  with metal placed in the space, it equals a short-circuited state. This seems to be distributed in the plane of symmetry because the absolute values of the current amplitude of the power-transmitting coil and power-receiving coil are almost equal. In addition, Figs. 3.18 and 3.19 show the near magnetic field and electric field in a scalar display. Figure 3.19 shows the power densities Pm and Pe of a magnetic field and electric field in the plane of symmetry of the power-transmitting and powerreceiving coils, which is standardized at the maximum value. In the case of a magnetic wall, it appears that the magnetic field is strengthened in the middle of the coils, and in the case of an electric wall, the magnetic field is weakened. On the other hand, directly below the coils, in the case of a magnetic wall, it appears that the magnetic field is weakened, and in the case of an electric wall, the magnetic field is strengthened. However, this is merely a way of looking at magnetic fields added up in space and does not indicate superiority or inferiority in terms of the power transmission efficiency, for example. Although this is a magnetic field coupling type helical coil, the electric field coupling does not completely disappear. Figure 3.19a and b show the distribution of a magnetic field and electric field in a plane of symmetry, but directly above and below the windings, there is a slight presence of electric field energy. When we compare the maximum energy density of the magnetic field and electric field in the plane of symmetry of the two coils, we find that the ratio of the electric field energy density to the magnetic field energy density is approximately 4%. In the case of electromagnetic field coupling, the magnetic field coupling and electric field coupling cannot be completely isolated, and in many cases, the coupling is the difference between both the electric field and the magnetic field, as in k = |km − ke | [k m is the coupling coefficient owing to the magnetic field, and k e is the coupling

84

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Fig. 3.18 Scalar display of electromagnetic field

(a) Magnetic field, fm

(a) Electric field, fm

(b) Magnetic field, fe

(b) Electric field, fe

Fig. 3.19 Magnetic field and electric field power density in plane of symmetry

coefficient owing to the electric field]. Here, the magnetic field is dominant and may be regarded as k = k m . Figure 3.20 shows a scalar display of half-cycle changes in the magnetic flux at f m , f 0 , and f e . Figure 3.21 shows a vector display at 0° and 160° when the direction is inverted. At f m and f e , the direction of I 1 and I 2 current is largely in phase (0°) and antiphase (180°) and thus easy to see. f 0 is the phase difference when the I 1 and I 2 current direction is 90°; the behavior is just like that between f m and f e .

3.4 Frequency Determinant (kHz to MHz to GHz)

85

(a) fm

(b) f0

(c) fe Fig. 3.20 Scalar display of half-cycle magnetic field changes

3.4 Frequency Determinant (kHz to MHz to GHz) Coils and other resonators are described as frequency determinants. The initially announced operating frequency in 2007 was approximately 10 MHz. However, a natural question was raised as to whether operation at other frequencies is possible, and later, a relationship with the frequency was shown [1]. Although operating frequency determinants have been explained in terms of the relationship between the wavelength and total coil length, methods based upon equivalent circuits are the most accurate and effective. Therefore, we shall describe a method based on an equivalent circuit comparing both.

3.4.1 Resonant Frequency Figure 3.22 shows a magnetic resonance coupling equivalent circuit. Its derivation and other details will be shown in the next chapter, but here we note that resonant frequency f 1 on the primary side and resonant frequency f 2 on the secondary side, which match when in an uncoupled independent state, are important for magnetic resonance coupling operation. When independently created resonant frequencies f 1 and f 2 match, they match the resonant frequency f 0 of the overall circuit even after coupling by a magnetic field. This resonant frequency f 0 is shown by the following formula:

86

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Fig. 3.21 Vector display of magnetic field changes

0°

160° (a) fm

0°

160° (b) f0

0°

160° (c) fe

f0 = f1 = f2 ,

f1 =

1 , √ 2π L 1 C1

f2 =

1 √ 2π L 2 C2

(3.2)

f 1 and f 2 are determined by values for self-inductances L 1 and L 2 and capacitances C 1 and C 2 . Naturally, in the case of symmetrical coils with the same shape, L 1 = L 2

3.4 Frequency Determinant (kHz to MHz to GHz)

87

Fig. 3.22 Equivalent circuit for magnetic resonance coupling (S-S method)

and C 1 = C 2 , and thus formula (3.2) is satisfied. However, if formula (3.2) is satisfied, magnetic resonance coupling operation is still possible even with asymmetric coils. The derivation of the resonant frequency and related topics shall be covered in the next chapter; here, we shall first consider the obtained resonant frequency formula (3.2). Thinking in terms of a MHz-band coil and capacitor, by making L or C larger, operation at low frequencies such as in the kHz band is possible. On the other hand, by making L or C smaller, operation at high frequencies such as in the GHz band is possible. The coupling strength is determined by mutual inductance L m , and its relationship to coupling coefficient k is as in formula (3.3). In addition, when we consider the efficiency, not only the coupling coefficient k but also the Q value of a coil, which is an indicator of maintaining energy, are required, and the kQ product shown in formula (3.4) serves as an easily understood indicator. We shall describe the kQ product in the next chapter, but as it increases in value, the efficiency increases. k= kQ =

Lm Lm =√ L L1 L2

(3.3)

ωL m L m ωL = L r r

(3.4)

As understood from formula (3.4), ultimately, L m is an important parameter. L m shows how much the primary side and secondary side are coupled within selfinductance L, so if L is small, then L m definitely becomes smaller. Hence, for example, when making use of the kHz band, enlarging L is proves superior to enlarging C. However, the disadvantage of enlarging L is that it increases the number of turns and enlarges the radius, so low resistance value r is increased. In addition, although it is possible to increase L, ferrite is heavy, so sometimes C is enlarged by design for a lighter weight. Moreover, per formula (3.4), we find that large ω is better. In other words, it is better if the frequency is high. However, if the frequency is actually raised, then r

88

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Table 3.2 Frequency and wavelength

Frequency

Wavelength

1 kHz

300 km

10 kHz

30 km

100 kHz

3 km

1 MHz

300 m

10 MHz

30 m

100 MHz

3m

1 GHz

30 cm

10 GHz

3 cm

Table 3.3 Resonant frequency determined from total length and actual resonant frequency Half-wavelength

Wavelength

Resonant frequency calculated from wavelength

Actual resonant frequency

Ratio of diameter to wavelength

243.7 m

487.4 m

615.5 kHz

121.7 kHz

0.0018

9.4 m

18.8 m

15.93 MHz

13.56 MHz

0.0159

52.8 mm

105.6 mm

2.84 GHz

1.49 GHz

0.0474

also increases owing to the skin effect and other factors, so it is not sufficient to raise the frequency without reason. For reference, Table 3.2 shows the resonant frequency calculated not from L and C but from the wavelength and total length. A half-wave dipole antenna resonates at half the wavelength, so the half-wavelength in the table is considered equal to the total length. Table 3.3 compares the actual resonant frequency determined from L and C to the resonant frequency determined using the total length of the spiral coil here as a half-wavelength for the kHz to MHz to GHz bands. Thinking in terms of wavelength in this way, we find that when the number of turns is increased, the resonant frequency computation accuracy drops owing to inductance and the floating capacity between wires. Thus, a method considered in terms of lumped constants L and C is effective. Table 3.3 shows the ratio of the diameter to the wavelength for reference.

3.4.2 Parameters of Coils for kHz, MHz, and GHz Comparing operating in the MHz bands to operating in the kHz bands, there is a concern that the copper loss increases because the coil wire length increases and the efficiency reduces. On the other hand, an effect is expected in which the skin effect is suppressed because the frequency decreases. For a similar reason, there is

3.4 Frequency Determinant (kHz to MHz to GHz)

89

Table 3.4 Coil dimensions (kHz, MHz, GHz) Frequency

Radius (mm)

Number of turns

Wire diameter (mm)

kHz band

121.7 kHz

450

71.5 × 2 layer

2.0

MHz band

13.56 MHz

150

2.75 × 2 layer

2.0

GHz band

1.49 GHz

2.5

4.0 × 1 layer

35 μm, 0.1 (thickness, width)

Table 3.5 Parameters (kHz, MHz, GHz) R ()

L

C (pF)

Q

kHz band

5.9

4.6 mH

370.3

597.3

MHz band

0.8

11.0 μH

12.5

1212.3

GHz band

4.3

60.7 nH

0.2

131.6

a concern that when operated in GHz bands, the frequency becomes higher and the copper loss increases owing to the skin effect. On the other hand, loss suppression owing to the wire length becoming shorter is also expected. Moreover, as originally shown in formula (3.4), if the frequency increases, then the Q value also increases. In this manner, the relationship between the frequency and resistance is a trade-off relationship. We shall construct the design method for kHz-, MHz-, and GHz band coils in the same way so that conditions are as comparable as possible. Specifically, we shall use an open-type coil in which the coil tip has been opened up, and operate the coil with self-resonance. Hence, the value of L in an equivalent circuit changes depending upon the number of turns in the coil. This determines whether to operate it in the kHz, MHz, or GHz band. On the other hand, the value of a capacitor in an equivalent circuit is determined according to the floating capacity between coils, so if the number of turns is increased, C increases, but the value of C does not change significantly compared to changes in L whether the coil is a kHz coil or GHz coil. Thus, the main determinant of the frequency is dependent upon the L value. Tables 3.4 and 3.5 list the kHz, MHz, and GHz coil dimensions and parameters.

3.4.3 kHz Coil Figure 3.23 shows an example of a kHz spiral coil electromagnetic field model and an experimental coil. A MHz coil has also been placed in the experiment photograph for the purpose of size comparison. A spiral shape is used but is double-layered in order to increase the inductance. The double-layered structure has a radius of 450 mm, 71.5 turns (total of 143 turns for both layers), wire gap of 3 mm, and copper wire thickness of 2 mm. The interlayer is 10 mm, and the resonant frequency of a single element is 121.7 kHz. The load is fixed at 50 .

90

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

Fig. 3.23 kHz spiral coil

The characteristic phenomenon of magnetic resonance coupling, such as when two power peaks turn into one, can verify an operation similar to MHz even with this kHz coil (Fig. 3.24). A large inductance is achieved by increasing the radius and number of turns. In addition, the internal resistance is increased by increasing the wire length such that L = 4.62 mH, R = 5.91 , and C = 370.3 pF. Owing to a very large L, the Q value is reasonable at 597.35, regardless of the large internal resistance and low frequency. The kHz coil in Fig. 3.23 has a hollow core but can be similarly composed using ferrite. In this case, the diameter becomes smaller while the weight increases with ferrite, as is understood when the main constituent is iron. ηr2 [%]

Pin

Power [W]

Efficiency [%]

60 40 20 0 110

115

120

125

130

P RL

P r1

η[%]

P r2

ηr1 [%]

60 40

115 120 125 Frequency [kHz]

20 110

130

115

ηr2 [%]

Pin

η[%]

P r2

20 0 110

115

120

125

Frequency [kHz]

130

110

ηr1 [%]

ηr2 [%]

150 100 50

60 40 20 0

0 110

115 120 125 Frequency [kHz]

(c) 1200 mm, k = 0.0154

115 120 125 Frequency [kHz]

Pin

80 Power [W]

Efficiency [%]

40

130

100

200

60

125

P r1

P r2

130

(b) 1000 mm, k = 0.0242 P r1

250

80 Power [W]

Efficiency [%]

100

P RL

120

P RL

200 180 160 140 120 100 80 60 40 20 0

Frequency [kHz]

(a) 800 mm, k = 0.0406 ηr1 [%]

Pin

80

0 110

Frequency [kHz]

η[%]

ηr2 [%]

100

180 160 140 120 100 80 60 40 20 0

Power [W]

ηr1 [%]

80

Efficiency [%]

η[%] 100

130

110

115

120

125

Frequency [kHz]

130

P RL

P r1

P r2

500 450 400 350 300 250 200 150 100 50 0 110

115 120 125 Frequency [kHz]

130

(d) 1500 mm, k = 0.0086

Fig. 3.24 Power transmission efficiency and power with kHz bands according to electromagnetic field analysis

3.4 Frequency Determinant (kHz to MHz to GHz)

91

Fig. 3.25 GHz spiral coil

3.4.4 GHz Coil Figure 3.25 shows an example of a GHz spiral coil electromagnetic field model and experimental coil. It has a radius of 2.5 mm, 4 turns, wire gap of 0.1 mm, wire width of 0.1 mm, and wire thickness of 35 μm. The resonant frequency of a single element is 1.49 GHz. Figure 3.26 shows the results of an electromagnetic field analysis. These results show that the characteristic phenomenon of magnetic resonance coupling, such as two resonant frequencies becoming one, can verify operations similar to MHz even with this GHz coil. The parameters are R = 4.31 , L = 60.7 nH, and C = 0.19 pF. In order to avoid dielectric loss, the coil is made on a polyimide flexible substrate, so the copper wire (copper foil) thickness is only 35 μm, and the resistance is increased by the skin effect. Since the number of turns is low, the inductance is also low, and ηr2 [%]

Pin

60

Power [W]

Efficiency [%]

80

40 20 0

1

1.5

2

200 180 160 140 120 100 80 60 40 20 0

P RL

P r1

η[%]

P r2

ηr1 [%]

1

1.5 Frequency [GHz]

60 40 20

2

1

1.5

ηr1 [%]

ηr2 [%]

Pin

0

P RL

P r1

η[%]

P r2

1

1.5

2

Frequency [GHz]

150 100 50 0

ηr1 [%]

1.5 Frequency [GHz]

(c) 2.5 mm, k = 0.096

1.5 Frequency [GHz]

Pin

80 60 40 20 0

1

1

ηr2 [%]

100 Efficiency [%]

Power [W]

Efficiency [%]

20

50 2

(b) 2.0 mm, k = 0.131

200

40

P r2

100

Frequency [GHz]

250

60

P r1

150

0

2

2

Power [W]

η[%]

P RL

200

(a) 1.0 mm, k = 0.258 80

Pin 250

80

0

Frequency [GHz]

100

ηr2 [%]

100

Power [W]

ηr1 [%]

Efficiency [%]

η[%] 100

1

1.5

2

Frequency [GHz]

500 450 400 350 300 250 200 150 100 50 0

1

P RL

P r1

1.5 Frequency [GHz]

P r2

2

(d) 3.5 mm, k = 0.055

Fig. 3.26 Power transmission efficiency and power with GHz bands according to electromagnetic field analysis

92

3 Magnetic Resonance Coupling Phenomenon and Basic Characteristics

the resistance value is large. Thus, even if the frequency increases, the Q value is 131.65, and significant effects are not really achieved. A GHz coil exhibits low inductance, so the coil diameter is small, and thus the air gap is in mm units. Hence, with GHz bands, radiative-type microwave power transmission is generally used. [Note: Loss arising in a dielectric is referred to as dielectric loss, but loss owing to a dielectric with kHz bands does not occur. With MHz bands, a gradual effect occurs, so a foaming material or similar material that functions electrically as air is used as a retaining material. Furthermore, with GHz bands, the dielectric loss effect produced in substrates that are dielectrics is very large, requiring a solution such as using materials that are small in dielectric tangents (tan δ) at the time of design.]

Reference 1. T. Imura, H. Okabe, Y. Hori, Proposal of wireless power transfer via magnetic resonant coupling in kHz-MHz-GHz. IEICE General Conference, 3, 2010

Chapter 4

Basic Circuit for Magnetic Resonance Coupling (S–S Type)

In this chapter, we derive the equations for magnetic resonance coupling of the S–S type, in which the capacitors in the equivalent circuits, in both the transmitter and receiver, are connected in series. We present distinct methods in which the equations are derived using Kirchhoff’s voltage law and the Z-matrix. Then, after verifying that there are two peaks in the power in the frequency response characteristics, we also verify the characteristics at the resonant frequency f 0 .

4.1 Derivation of Equivalent Circuit In this section, we derive the equations for magnetic resonance coupling. We derive the equations for current from the equations for voltage, and also the ones for impedance, power, and efficiency. The equivalent circuits are shown in Fig. 4.1; the general equivalent circuit is shown in (a), and the T-type equivalent circuit is shown in (b). Coils can be classified into one of two types: the self-resonance open-circuit type, in which the coil produces LC resonance alone, and the excited resonance short-circuit type, in which LC resonance is produced by connecting a capacitor to the coil. They are described

(a) General equivalent circuit

(b) T-type equivalent circuit

Fig. 4.1 Equivalent circuits for magnetic resonance coupling (S–S type) © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_4

93

94

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

in Chap. 7. The equivalent circuits for these two types are the same and are shown in Fig. 4.1 [1–4]. On the primary side, the coil and capacitor resonate in series. The coil has internal resistance; thus, it is possible to represent the primary circuit as a series resonance using L 1 , C 1 , and r 1 . Similarly, on the secondary side, the coil and capacitor resonate in series. Because the secondary side has the load RL and internal resistance, it is possible to represent the secondary circuit as a series resonance using L 2 , C 2 , and r 2, with RL added to it. Furthermore, because the two coils are coupled by a magnetic field, it is possible to represent the coupling by a mutual inductance L m . Therefore, the system can be represented by the equivalent circuits shown in Fig. 4.1. When the resonant frequencies of the primary and secondary sides become equal, it is said that the two sides have been coupled by a magnetic field using magnetic resonant coupling. One common method for analyzing circuits uses Kirchhoff’s voltage law. In this method, equations are constructed by identifying locations in the circuit for which it is known that the voltages must be equal. We will first analyze the circuits using this method. Then we will solve for the equivalent circuit using the Z-matrix (Z parameter), a method that can easily be expanded to cases with increased number of coils.

4.1.1 Solution Using Kirchhoff’s Voltage Law According to Kirchhoff’s voltage law, the equations for the input voltage V 1 and the output voltage V 2 are shown in Eq. (4.1). The relationship between the frequency f and the angular frequency ω is shown in Eq. (4.2). 

V1 = VL1 + VC1 + Vr 1 V2 = VL2 + VC2 + Vr 2 ω = 2π f

(4.1) (4.2)

The relationships between the transmitter and receiver coil voltages V L1 and V L2 , respectively, the transmitter and receiver resonant capacitor voltages V C1 and V C2 , respectively, and the transmitter and receiver internal resistance voltages V r1 and V r2 , respectively, are shown in Eqs. (4.3) through (4.5), respectively. 

VL1 = jωL 1 I1 + jωL m I2 VL2 = jωL 2 I2 + jωL m I1  1 VC1 = jωC I1 1 1 VC2 = jωC2 I2

(4.3)

(4.4)

4.1 Derivation of Equivalent Circuit

95



Vr 1 = r1 I1 Vr 2 = r2 I2

(4.5)

Substituting Eqs. (4.3)–(4.5) into Eq. (4.1) gives the following: 

V1 = jωL 1 I1 + jωL m I2 + V2 = jωL 2 I2 + jωL m I1 +

1 I jωC1 1 1 I jωC2 2

+ r 1 I1 + r 2 I2

(4.6)

However, the voltages across the loads, connected to each coil, can be expressed as 

V1 = V1 V2 = −I2 R L

(4.7)

Because the primary side consists of a voltage source, V 1 does not change. Therefore, by applying Eqs. (4.6) and (4.7), Kirchhoff’s voltage law can be summarized by Eq. (4.8). 

1 V1 = jωL 1 I1 + jωL m I2 + jωC I1 + r 1 I1 1 1 0 = jωL 2 I2 + jωL m I1 + jωC2 I2 + r2 I2 + R L I2

(4.8)

The equations for the voltage are now complete. Further, because there are only two unknowns, I 1 and I 2 , across two equations, it is possible to solve the system of equations given by Eq. (4.8) for I 1 and I 2 using Eqs. (4.9) and (4.10), respectively. Substituting the solution into Eq. (4.7) determines V 2 , as seen in Eq. (4.11). Furthermore, the input impedance Z in1 is given by Eq. (4.12).   r2 + R L + j ωL 2 − ωC1 2    I1 =  r1 + j ωL 1 − ωC1 1 r2 + R L + j ωL 2 −

1 ωC2

jωL m   r2 + R L + j ωL 2 −

1 ωC2

 I2 = −  r1 + j ωL 1 −

1 ωC1

 

+ ω2 L 2m + ω2 L 2m

V1

(4.9)

V1

(4.10)

jωL m R L     V2 =  V1 (4.11) 1 r1 + j ωL 1 − ωC1 r2 + R L + j ωL 2 − ωC1 2 + ω2 L 2m      r1 + j ωL 1 − ωC1 1 r2 + R L + j ωL 2 − ωC1 2 + ω2 L 2m V1   = Z in1 = I1 r + R + j ωL − 1 2

L

2

ωC2

(4.12) Based on the equations above, all the voltages and currents have been derived. Therefore, we can derive the input power P1 and the power consumed by the load

96

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

P2 by substituting into Eq. (4.13). Additionally, the power consumed by the internal resistance in each coil is represented by Eq. (4.14). Power transmission is evaluated in terms of the active power consumed. Therefore, when performing power calculations, only the real component of the complex power value has an impact on the efficiency. Therefore, the efficiency η at the load is given by Eq. (4.15). 

 P1 = Re V1 I 1 

 P2 = PR L = Re V2 −I 2   Pr 1 = ReVr 1 I 1  Pr 2 = Re Vr 2 −I 2

(ωL m )2 R L η=  2  2 1 r1 + (ωL m )2 (r2 + R L ) (r2 + R L ) + ωL 2 − ωC2

(4.13) (4.14) (4.15)

Here, we provide additional explanation regarding the relationship between the active power Pe , reactive power Pr , and apparent power Pa , in regards to the complex power P c used in the equations above. If the complex voltage is represented as V˙ or V, the complex current is represented as I˙ or I, and the active components of the voltage and current are represented as V and I, respectively, then P c is represented by Eq. (4.16). Here, we used the complex conjugate of the voltage and the relationships shown in Eqs. (4.17) and (4.18). θ is the phase of I, where the phase of V is used as a reference. P c = V˙ I˙ = V I = Pe + j Pr = V I e jθ = V I (cos θ + j sin θ )

(4.16)

V = V e j0 = V (cos 0 + j sin 0) = V

(4.17)

I = I e jθ = I (cos θ + j sin θ )

(4.18)

Therefore, the real component of the complex power becomes the active power, as seen in Eq. (4.19), and the imaginary component becomes the reactive power, as seen in Eq. (4.20). Re{ P c } = Pe = V I cos θ

(4.19)

Im{ P c } = Pr = V I sin θ

(4.20)

It is also possible to represent the complex power value using the complex conjugate of the current, as seen in Eq. (4.21). P c = V˙ I˙ = V I = Pe + j Pr = V I e− jθ = V I (cos θ − j sin θ )

(4.21)

4.1 Derivation of Equivalent Circuit

97

Similarly, the real component of the complex power value becomes the active power, as seen in Eq. (4.22), and the imaginary component becomes the reactive power, as seen in Eq. (4.23). In this case, the active power does not change, as seen in Eqs. (4.19) and (4.22). Therefore, we choose the method that is easier to calculate. However, the sign of the reactive power differs between Eqs. (4.20) and (4.23), meaning that the direction is different. Therefore, it is necessary to be careful about this difference. V I represents a leading reactive power when the value of Pr is positive, and a lagging reactive power when the value is negative. On the other hand, V I represents a lagging reactive power when the value of Pr is positive, and a leading reactive power when that value is negative. Re{ P c } = Pe = V I cos θ

(4.22)

Im{ P c } = Pr = −V I sin θ

(4.23)

The relationship between the active power Pe , reactive power Pr , and apparent power Pa is as follows: Pa =

Pe2 + Pr2

(4.24)

4.1.2 Calculation Using the Z-Matrix Although we were able to derive the efficiency using the previous method, in situations where there are multiple coils (refer to Chaps. 9 and 10), it becomes difficult to calculate the efficiency using that method. Therefore, we will now use matrix calculations. It is also possible to use the matrix method to calculate the equations for the current for the case in which the relationship between the transmitter and receiver coils is 1-to-1. First, the self-inductance and mutual inductance can be represented by Eq. (4.25). Then, if we denote the impedances of each coil, excluding the load, as Z, then Eq. (4.26) applies. If we denote this impedance in terms of the coils, capacitors, internal resistances, and mutual inductances, the result is Eq. (4.27). Further, the components of the Z-matrix can be calculated using Eq. (4.28). However, it is more common to calculate the components of the Z-matrix (Eq. 4.27) based on Eq. (4.6) in practice.  [L] =  [Z] =

L1 Lm Lm L2 Z 11 Z 12 Z 21 Z 22

 (4.25)  (4.26)

98

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

 [Z] =

1 r1 + jωL 1 + jωC jωL m 1 jωL m r2 + jωL 2 + ⎧   ⎪ ⎪ Z 11 = VI11  ⎪ ⎪  I2=0 ⎪ ⎪ ⎪ ⎨ Z 12 = VI 1  2 I  1=0 V2  ⎪ Z = ⎪ 21 ⎪ I1  I ⎪  2=0 ⎪ ⎪ ⎪ ⎩ Z 22 = VI 2  2

 (4.27)

1 jωC2

(4.28)

I1=0

The relationship between Z 12 and Z 21 can be expressed as Z 12 = Z 21 . Therefore, the voltage V and current I are represented by Eq. (4.29). The relationship between the voltage, current, and impedance is shown in Eq. (4.30).  [V ] =

   I V1 , [I] = 1 V2 I2

(4.29)

[V ] = [Z][I]

(4.30)

Therefore, 

V1 0



⎡ =⎣

 r1 + j ωL 1 −

1 ωC1



⎤

jωL m  r2 + R L + j ωL 2 −

jωL m

1 ωC2

   ⎦ I1 I2

(4.31)

and during resonance, this becomes 

V1 0



 =

jω0 L m r1 jω0 L m r2 + R L



I1 I2

 (4.32)

where ω0 is the resonant angular frequency. Here, we redefine the voltage V  , impedance Z , and current I by Eq. (4.33), based on Eq. (4.31), which contains RL . Evidently, the current I is the same as in Eq. (4.30). Thus, the voltage V  and current I are given by Eq. (4.34).  

V





   V  = Z  [I] 

=

   I V1 , [I] = 1 0 I2

(4.33) (4.34)

Transforming Eq. (4.33), we obtain Eq. (4.35). Equation (4.35) can be expanded to obtain Eq. (4.36), and then Eqs. (4.37) and (4.38) can be derived. Here, we used the relationship shown in Eq. (4.39) for the calculation of the inverse matrix. And we can verify that the equations derived using matrices match Eqs. (4.9) and (4.10)

4.1 Derivation of Equivalent Circuit

99

derived earlier for the current. In cases in which a system of three or more coils is considered, it is necessary to perform the calculations using matrices with the Z parameters.  −1 [I] = Z  [V ] 

(4.35)



1      = 1 r1 + j ωL 1 − ωC1 r2 + R L + j ωL 2 − ωC1 2 − ωL m · ωL m ⎡   ⎤   r2 + R L + j ωL 2 − ωC1 2 − jωL m  ⎦ V1  ×⎣ (4.36) 0 − jωL m r1 + j ωL 1 − ωC1 1   r2 + R L + j ωL 2 − ωC1 2    V1   I1 =  r1 + j ωL 1 − ωC1 1 r2 + R L + j ωL 2 − ωC1 2 − ωL m · ωL m I1 I2

(4.37)  I2 = −  r1 + j ωL 1 − 

1 ωC1

ab cd

jωL m   r2 + R L + j ωL 2 − −1

1 ωC2

  1 d −b = ad − bc −c a



+ ω2 L 2m

V1

(4.38)

(4.39)

Using the equations above, we verify the frequency response characteristics for different values of the air gap distance and the load. We varied the air gap distance g over three distances and varied the load over the values 10, 50, and 100 . Figures 4.2, 4.3, and 4.4 show graphs for the cases in which g = 150 mm, 210 mm, and 300 mm, respectively. We define ηr1 to be the proportion of the loss that occurs at the internal resistance coil r 1 on the primary side, and ηr2 to be the proportion of the loss at the internal resistance r 2 on the secondary side. The parameters for the coil, internal resistance, and capacitor are L = 11.0 µH, r = 0.8 , and C = 12.5 pF, respectively. Figure 4.2 shows that when g = 150 mm, the graph for the power has two peaks. The peaks occur near the two locations at which the input impedance |Z in | becomes small. However, the efficiency always has only one peak. Figure 4.3 shows that when g = 210 mm, the optimal load at which the efficiency is maximized is 50 . In this case, there are two peaks in the graph for the power, with one on each side of the frequency at which the maximum efficiency is achieved. In other words, by aligning the frequency with these two peaks, power can be optimized while only sacrificing efficiency slightly. The graph for the case in which the load is 100  shows that when the impedance is too large, there is only one point at which the impedance becomes small, and the power only has one peak. Figure 4.4 shows that when g = 300 mm, the optimal load at which the efficiency is maximized is between 10 and 50 .

Fig. 4.2 g = 150 mm (10 , 50 , 100 )

(b) Load 50 Ω

(e) Load 50 Ω

(a) Load 10 Ω

(d) Load 10 Ω

(f) Load 100 Ω

(c) Load 100 Ω

100 4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

4.2 Equivalent Circuit at the Resonant Frequency

(a) Load 10 Ω

(b) Load 50 Ω

(d) Load 10 Ω

101

(c) Load 100 Ω

(e) Load 50 Ω

(f) Load 100 Ω

Fig. 4.3 g = 210 mm (10 , optimal load 50 , 100 )

(a) Load 10 Ω

(b) Load 50 Ω

(d) Load 10 Ω

(c) Load 100 Ω

(e) Load 50 Ω

(f) Load 100 Ω

Fig. 4.4 g = 300 mm (10 , 50 , 100 )

4.2 Equivalent Circuit at the Resonant Frequency 4.2.1 Derivation of the Equivalent Circuit at the Resonant Frequency, and Its Maximum Efficiency In this section, we assume that the circuit is operated under resonance (f = f 0 ). Under this condition, the reactance due to the self-inductances, L 1 and L 2 , and the reactance due to the capacitances, C 1 and C 2 , cancel each other out, as seen in Eq. (4.40). Additionally, because the resonant angular frequencies ω0 of the primary and secondary sides are equal, Eq. (4.41) results.

102

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)



jωL 1 + jωL 2 +

ω0 = √

1 jωC1 1 jωC2

=0 =0

1 1 =√ L 1 C1 L 2 C2

(4.40) (4.41)

Under those conditions, Eq. (4.6) for the voltage becomes Eq. (4.42). Simultaneously, the voltage across the loads connected to each coil can be represented by Eq. (4.43). Therefore, using Eqs. (4.42) and (4.43) results in Eq. (4.44). This equation can be represented in matrix form by Eq. (4.45). 



V1 = jω0 L m I2 + r1 I1 V2 = jω0 L m I1 + r2 I2  V1 = V1 V2 = −I2 R L

V1 = jω0 L m I2 + r1 I1 0 = jω0 L m I1 + r2 I2 + R L I2      jω0 L m I1 V1 r1 = 0 jω0 L m r2 + R L I2

(4.42) (4.43) (4.44) (4.45)

Based on Eq. (4.45), the following equations for the currents and voltage result: I1 =

r2 + R L V1 (ω0 L m ) + r1 (r2 + R L ) 2

jω0 L m V1 (ω0 L m ) + r1 (r2 + R L )

(4.47)

jω0 L m R L V1 (ω0 L m )2 + r1 (r2 + R L )

(4.48)

I2 = − V2 =

(4.46)

2

Based on these equations we obtain the input power P1 and the power consumed by the load, P2, as done for Eq. (4.13). Similarly, the power consumed by the internal resistance of each coil can be represented by Eq. (4.14). Based on the discussion above, the efficiency η at the load is P2 P2 R L (ω0 L m )2 = = 2 P1 Pr 1 + Pr 2 + P2 r1 (r2 + R L ) + r2 (ω0 L m )2 + R L (ω0 L m )2 R L (ω0 L m )2  = (4.49) (r2 + R L ) r1 (r2 + R L ) + (ω0 L m )2

η=

By using Eqs. (4.49) and (4.50), the condition for the optimal load that results in the maximum efficiency can be obtained by Eq. (4.51). Substituting this into

4.2 Equivalent Circuit at the Resonant Frequency

103

Eq. (4.49) gives Eq. (4.52) for the maximum efficiency [5]. ∂η =0 ∂ RL  r2 (ω0 L m )2 = r22 + r1

R Lopt

(ω0 L m )2 ηmax =  2  √ r1r2 + r1r2 + (ω0 L m )2

(4.50)

(4.51) (4.52)

Based on Eq. (4.46), the input impedance becomes Z in =

V1 (ω0 L m )2 = r1 + I1 r 2 + R2

(4.53)

Because this equation does not contain the symbol j, which represents the imaginary component, it can be concluded that the impedance is always a pure resistance, even when the load or the mutual inductance changes. Therefore, the power factor is always 1. This property is a characteristic of the resonant frequency f 0 in S–S type systems.

4.2.2 Derivation of Voltage Ratio and Current Ratio Knowing the ratios of the voltage and the current, between the inputs and the outputs, can be useful when designing a circuit. One property of S–S type circuits in particular is that the maximum efficiency is reached when the voltage ratio and current ratio are close to 1. We will let AV represent the voltage ratio and AI represent the current ratio. AV is the ratio between the voltage on the primary side and the secondary side, and AI is the ratio between the current on the primary side and the secondary side. These ratios can be expressed as AV =

V2 V1

(4.54)

AI =

I2 I1

(4.55)

Therefore, AV and AI , at f 0 , can be expressed as AV = j

ω0 L m R L r1 R L + r1r2 + (ω0 L m )2

(4.56)

104

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

AI = j

ω0 L m R L + r2

(4.57)

At the resonant frequency f 0 , AV , and AI have no real component and consist of an imaginary component only. Therefore, the phase difference between the input and the output for both the voltage and current is always 90°. Furthermore, the phase difference does not depend on the load or the mutual inductance. Next, we consider the transmission efficiency. The ratio between the power, that is, input to the primary side, and the power that is output from the secondary side is the efficiency. The transmission frequency is thus η = AP =

V2 · I2 V1 · I1

 =

V2 V1

 ·

I2 I1

= AV · A I

(4.58)

Because the power factor on both the primary side and the secondary side is 1, at f 0 , the efficiency η can be represented as the product of AV and only the complex conjugate of AI . Therefore, the efficiency is η=

(ω0 L m )2 R L  (R L + r2 ) r1 R L + r1r2 + (ω0 L m )2

(4.59)

By making the power transmission angular frequency equal to the resonant angular frequency ω0 , of the transmitter and receiver, it is possible to simplify the characteristic equations for the input and output. Figure 4.5 shows a graph of AV and AI with respect to the coupling coefficient for different load values. Although the plot of AV has a peak, considering the situation in which the circuit is operated near maximum efficiency, the actual operating point would lie to the right of the peak. Under this condition, the value of AV will increase as the air gap increases and the amount of coupling decreases, for all load values. Therefore, in the case in which the load is not adjusted, as the air gap increases the secondary-side voltage increases. However, AI decreases, so the secondary-side current becomes smaller than the primary-side current. Note that this relationship is referring to the relative value. If the secondary-side voltage is increasing, the current flowing in the secondary side is also increasing. In order to achieve maximum efficiency control, the load is adjusted to the optimal load RLopt , according to the coupling coefficient k. In that case, AV ≈ AI ≈ 1, except for the operating region in which k is small. Therefore, setting the voltage and current amplifications equal to 1 causes the circuit to operate at approximately the maximum efficiency condition. Furthermore, by setting V 1 = V 2 and I 1 = I 2 , it is possible to transmit power at that condition. Note that this only applies to cases in which k is large. In regions in which the air gap is large, such as in regions when k is less than 0.1, or with coils with large internal resistance and low Q values, there is some deviation from AV ≈ AI ≈ 1, and the conditions deviate from the maximum efficiency condition. Therefore, it is necessary to use caution when setting AV = AI = 1. However, in regions in which this equality can be used, even if there is misalignment or even when the air

4.2 Equivalent Circuit at the Resonant Frequency RL=5 RL=50 RL=10000

0.04

0.08

1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.02

0.06

k

k

(a) Av

(b) Av, Enlarged

RL=5 RL=50 RL=10000

RL=2 RL=20 RL=1000 2

RL=10 RL=100 Rlopt

RL=5 RL=50 RL=10000

RL=10 RL=100 Rlopt

0.04

0.08

0.1

1.5

Ai

300

Ai

RL=10 RL=100 Rlopt

0.5

RL=2 RL=20 RL=1000 400

200 100 0

RL=5 RL=50 RL=10000

1.5

30 20 10 0

RL=2 RL=20 RL=1000 2

RL=10 RL=100 Rlopt

Av

Av

RL=2 RL=20 RL=1000 60 50 40

105

1 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0

0.02

0.06

k

k

(c) Ai

(d) Ai, Enlarged

0.1

Fig. 4.5 Av and Ai versus coupling coefficient

gap becomes large, the condition for maximum efficiency is AV ≈ AI ≈ 1. Therefore, if the voltage of the power source is known beforehand, then by setting V 1 = V 2 , it is always possible to achieve power transmission at maximum efficiency.

4.2.3 Load Characteristics at the Resonant Frequency The characteristics of various quantities with respect to the load resistance at the resonant frequency, for different values of the air gap, are shown in Fig. 4.6. Regarding the voltage, the condition for maximum efficiency is AV ≈ 1, for cases in which the coupling is somewhat strong. Verifying that for V 2 shows that, because the input voltage is 100 V, the maximum efficiency happens when V 2 ≈ 100 V. Note that AV ≈ 1 is an approximation. Similarly, for the current, the condition for maximum efficiency is AI ≈ 1, for cases in which the coupling is somewhat strong. In the graphs for I 1 and I 2 , it is possible to identify the approximate value of the optimal load RLopt , at which the maximum efficiency is achieved, by considering the point at which I 1 ≈ I 2 for each value of air gap. Note that AI ≈ 1 is also just an approximation. Therefore, because the precise value of the optimal load RLopt , for each air gap distance g, is

106

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

g=200

g=250

g=300

g=350

g=450 100

g=500

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 6

g=500

g=550

g=600

5

40

Av

4

60

Av

η [%]

80

3 2

20 0

1 0

0

10 20 30 40 50 60 70 80 90 100

0

(a)η g=300

g=350

g=450 5

g=500

g=550

g=600

g=400

2

0

0

10 20 30 40 50 60 70 80 90 100

g=300

g=350

g=450 600

g=500

g=550

g=600

g=400

400 300 200

(d)AI

(e) V2

g=300

g=350

g=450 20

g=500

g=550

g=600

g=400

5

g=250

g=300

g=350

g=450 120 100

g=500

g=550

g=600

60 40

0

10 20 30 40 50 60 70 80 90 100

g=250

g=300

g=350

g=450 20

g=500

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 100,000

g=500

g=550

g=600

5 0

60,000 40,000 20,000 0

10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

RL [Ω]

(g) I1, Scale change

(h)I

(i) VL1

g=250

g=300

g=350

g=500

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 100,000

g=500

g=550

g=600

g=400

80,000

10,000 5,000 0 10 20 30 40 50 60 70 80 90 100

40,000 20,000 0

g=200

g=250

g=300

g=350

g=450 20,000

g=500

g=550

g=600

g=400

15,000

60,000

VC2 [V]

VC1 [V]

15,000

g=400

80,000

10

RL [Ω]

g=450 20,000

g=400

80

RL [Ω]

g=200

10000

(f) I1

g=200

0

10 20 30 40 50 60 70 80 90 100

8000

RL [Ω]

VL1 [V]

I2 [A]

10

6000

g=200

0

15

15

4000

20 0 10 20 30 40 50 60 70 80 90 100

RL [Ω]

g=250

0

2000

(c) AV, until 10,000 Ω

g=250

RL [Ω]

g=200

0

0

RL [Ω]

g=200

100 0

g=400

40 30 20 10

I1 [A]

V2 [V]

Ai

g=350 g=600

500

3 1

I1 [A]

g=300 g=550

(b) AV

g=250

4

VL2 [V]

g=250 g=500

RL [Ω]

g=200

0

g=200 g=450 60 50

10 20 30 40 50 60 70 80 90 100

RL [Ω]

0

g=400

0 10 20 30 40 50 60 70 80 90 100

10,000 5,000 0

0 10 20 30 40 50 60 70 80 90 100

RL [Ω]

RL [Ω]

RL [Ω]

(j) VL1

(k)VC1

(l) VC2

Fig. 4.6 Properties versus load resistance RL

shown in Fig. 4.6u, it is necessary to refer to Fig. 4.6u when verifying the location of RLopt in Fig. 4.6a–t. S–S type circuits have the property that the value of the current becomes constant on the secondary side (Fig. 4.6h). This implies that the value of the secondary-side current I 2 does not change, when the air gap is fixed, even when the load is changed. If the air gap becomes too large, then the property of having constant current is lost. However, if the air gap is less than the approximate diameter of the coil (g = 300 mm), which is a distance that would be considered large, a relatively constant current is maintained. As can be seen from the voltage and the current, as the value of the load deviates from the optimal load, large voltages and large currents arise. Therefore, if the designer did not specify the operating region during the design phase, or if the circuit

4.2 Equivalent Circuit at the Resonant Frequency g=300

g=350

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 500

g=500

g=550

g=600

400

6,000

300

P1 [W]

8,000 4,000 2,000

100 0

g=350 g=600

1,000 0

0 10 20 30 40 50 60 70 80 90 100

RL [Ω]

RL [Ω]

(m) P1

(n)P1, Scale change

(o) P2

g=250

g=300

g=350

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 8,000

g=500

g=550

g=600

400

g=400

100

4,000 2,000 0

0 10 20 30 40 50 60 70 80 90 100

g=250

g=300

g=350

g=500

g=550

g=600

Pr1 [W]

Pr1 [W]

200

g=200 g=450 50

g=400

40

6,000

300

g=400

2,000

RL [Ω]

g=500

30 20 10 0

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

RL [Ω]

RL [Ω]

RL [Ω]

(p) P2, Scale change

(q) Pr1

(r) Pr1, Scale change

g=200

g=250

g=300

g=350

g=450 300

g=500

g=550

g=600

g=400

g=200

g=250

g=300

g=350

g=450 50

g=500

g=550

g=600

250

g=400

40

Pr2 [W]

200 150 100

g [mm]

P2 [W]

g=300 g=550

0 10 20 30 40 50 60 70 80 90 100

g=200

Pr2 [W]

g=250 g=500

0 10 20 30 40 50 60 70 80 90 100

g=450 500

30 20

0 10 20 30 40 50 60 70 80 90 100

0

g=200

g=250

g=300

g=350

g=450 800

g=500

g=550

g=600

g=400

600 400 200

10

50 0

g=200 g=450 4,000 3,000

200

0

0

g=400

P2 [W]

g=250 g=500

P1 [W]

g=200 g=450 10,000

107

0 10 20 30 40 50 60 70 80 90 100

0

0

10 20 30 40 50 60 70 80 90 100

RL [Ω]

RL [Ω]

RLopt [Ω]

(s) Pr2

(t) Pr2, Scale change

(u) Optimal load and gap

Fig. 4.6 (continued)

is loaded far from the optimal load, it becomes necessary to use expensive parts with higher voltage and current ratings. Furthermore, the efficiency will decrease as well. It is particularly necessary to pay attention to the voltage and current applied to the capacitors. Furthermore, because the circuit is resonating, V L1 = V C1 and V L2 = V C2 .

4.2.4 kQ Representation In wireless power transmission, the kQ product is sometimes used as an index because it is intuitively easy to understand. This is because it is possible to calculate the maximum efficiency, at the resonant frequency, based on the coupling coefficient k and the Q value alone. As the air gap increases, the coupling coefficient k decreases and approaches 0. However, if a coil that can hold energy with a large Q value is used, then it is possible to transmit power with high efficiency. Although we derived the equation for the maximum efficiency in Sect. 4.2.1, here we present the equation for the maximum efficiency using the kQ product representation [6, 7]. First, the coupling coefficient k is given by Eq. (4.60). And the Q value is given by Eq. (4.61). Therefore, the kQ product, for a case in which the same coils are used

108

4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

for the transmitter and receiver, is given by Eq. (4.62). In the case in which the coils in the transmitter and receiver are different, then the product becomes k 2 Q1 Q2, as seen in Eq. (4.63). Lm Lm =√ L L1 L2

(4.60)

ωL 1 ωL 2 ωL , Q1 = , Q2 = r r1 r2

(4.61)

ωL m L m ωL = L r r

(4.62)

k= Q=

kQ = k 2 Q 1 Q 2 = √

L 2m L1 L2

2

ω2 L 1 L 2 ωL m =√ r1 r2 r1 r2

(4.63)

The kQ product is ultimately a relationship between ω, L m , and r. For example, as the frequency increases, the values of ω and Q increase. Note that, because r increases due to the skin effect, Q is determined by the relationship between ω and r in practice. In addition, in order to create a large value for L m , it is necessary to have a large value for L. Note that, because increasing the number of windings to increase L would also result in a larger value for r, this relationship also becomes a trade-off between L and r. It is possible to transform Eq. (4.62) into the form shown in Eq. (4.64). Substituting Eq. (4.64) into Eq. (4.65), the equation for the optimal load at which the maximum efficiency can be achieved, as shown in the next chapter, gives Eq. (4.66). Lm =  R Lopt =

k Qr ω

r22 +

r2 (ωL m )2 r1

R Lopt = r 1 + (k Q)2

(4.64)

(4.65) (4.66)

Because the equation for the efficiency of S–S type magnetic resonance coupling is as seen in Eq. (4.67), substituting the optimal load from Eq. (4.66) into RL gives a representation of the maximum efficiency ηmax , using the kQ product, as seen in Eq. (4.68). η=

(ωL m )2 R L (r2 + R L )2 r1 + (ωL m )2 r2 + (ωL m )2 R L k2 Q1 Q2 ηmax =  2  1 + 1 + k2 Q1 Q2

(4.67) (4.68)

4.2 Equivalent Circuit at the Resonant Frequency

109 100

Fig. 4.7 kQ and maximum efficiency

ηmax [%]

80 60 40 20 0

0 10 20 30 40 50 60 70 80 90 100

kQ

Note the graph in Fig. 4.7 and the values in Table 4.1. For example, a maximum efficiency of ηmax = 81.9% at kQ = 10 means that for a case, in which k = 0.1, this efficiency can be achieved using a coil with Q = 100, and for a case in which k = 0.01, this value can also be achieved using a coil with Q = 1000.

4.2.5 Maximum Efficiency Considering Coil Performance From the viewpoint of circuit topology, the maximum efficiency is achieved under the condition of magnetic resonance coupling. However, in order to achieve the maximum efficiency, considering the coil characteristics, it is necessary to consider the frequency response characteristics of the coil as well. In this subsection, we described a method for achieving ultimate maximum efficiency, by using a circuit topology that has magnetic resonance coupling that leverages the coil characteristics. Therefore, because the frequency that will be used is generally determined in advance, it is necessary to design the coil so that the values of Q and k are maximized at that frequency. When calculating the maximum efficiency, the results will be the same whether done using r, L m , and ω, or using k and Q. Here, we will provide an explanation using k and Q. Assuming that k is constant, in order to leverage the coil characteristics, the maximum efficiency is achieved when the circuit is used at the frequency at which Q is maximized. Obviously, because efficiency is determined by the product of k and Q, it also depends on k. However, as the frequency is changed, the change in r is dominant, and the changes in L and k are small in comparison, and these quantities can be considered almost constant. Therefore, the circuit will be used at the frequency at which Q is maximized. Figure 4.8 shows Q and the maximum efficiency ηmax . Here, we are using a different coil than the one used in the previous section. The Q value is strongly affected, by the frequency response characteristics of r, and has a peak. If the coupling coefficient does not change, then the maximum efficiency ηmax is achieved at the frequency at which Q has a peak, as seen in Fig. 4.8b. Therefore, it is desirable to draw out the maximum performance of the coil, and use it in the operating region where the peak is achieved. Note that there are limits on the coil size, however, and that sometimes it may be difficult to use the coil in the region

30

20

90.5

ηmax (%)

93.6

38.2

17.2

ηmax (%) kQ

2

1

kQ

Table 4.1 kQ and maximum efficiency ηmax

95.1

40

51.9

3

96.1

50

61.0

4

96.7

60

67.2

5

97.2

70

71.8

6

97.5

80

75.2

7

97.8

90

77.9

8

98.0

100

80.1

9

99.8

1000

81.9

10

110 4 Basic Circuit for Magnetic Resonance Coupling (S–S Type)

4.2 Equivalent Circuit at the Resonant Frequency

(a)Q and maximum efficiency ηmax

(b)Q and maximum

111

(c) L and r

efficiency ηmax (enlarged)

Fig. 4.8 Q and maximum efficiency ηmax

where the peak is achieved. However, it is still desirable to use the coil in the region where Q becomes high.

References 1. T. Imura, Y. Hori, Unified theory of electromagnetic induction and magnetic resonant coupling. Trans. Inst. Electr. Eng. Japan 135(6), 697–710 (2015) 2. T. Imura, T. Uchida, Y. Hori, Experimental analysis of high efficiency power transfer using resonance of magnetic antennas for the near field: geometry and fundamental characteristics. IEE Japan Ind. Appl. Soc. Conf. 2(2–62), 539–542 (2008) 3. T. Imura, H. Okabe, T. Uchida, Y. Hori, Study of magnetic and electric coupling for contactless power transfer using equivalent circuits: wireless power transfer via electromagnetic coupling at resonance. Trans. Inst. Electr. Eng. Japan D 130(1), 84–92 (2010) 4. M. Kato, T. Imura, T. Uchida, Y. Hori, Loss Reduction in antenna for wireless power transfer by magnetic resonant coupling. EVTeC’11, 20117264 (2011) 5. M. Kato, T. Imura, Y. Hori, New characteristics analysis changing transmission distance and load value in wireless power transfer via magnetic resonance coupling. in INTELEC 2012 34nd International Telecommunications Energy Conference (2012) 6. T. Imura, Electromagnetic resonance coupling. Handbook Power Electron. 1(11), 195–198 (2010) 7. T. Tohi, Y. Kaneko, S. Abe, Maximum efficiency of contactless power transfer systems using k and Q. Trans. Inst. Electr. Eng. Japan 132(1), 123–124 (2012)

Chapter 5

Comparison Between Electromagnetic Induction and Magnetic Resonance Coupling

Differences have been noted between electromagnetic induction and magnetic resonance coupling, as the latter was first introduced. However, when this technology was first announced, it was considered to operate at around 10 MHz, and the phenomenon was explained using a coupling mode theory that is not familiar to researchers of electromagnetic induction. It was considered a mysterious technique. After that, it was explained by the equivalent circuit theory, and it seemed to be solved for the time being. However, the five corresponding circuit mechanisms were not thoroughly compared considering the conditions for magnetic resonance coupling. Consequently, researchers had had no clear response to basic questions such as how these concepts differ, what they share in common, and why only magnetic resonance coupling, which possesses the same resonant frequencies on the primary and secondary sides, yields high efficiency and power. In this chapter, we compare the five circuits and show the transition from the conventional method of electromagnetic induction. This way, we analyze the mechanism of magnetic resonance coupling, which sets the same resonance frequencies on the primary and secondary sides and can exert high efficiency and power under large air gaps and misalignment. Moreover, we discuss how magnetic resonance coupling corresponds to electromagnetic induction with narrower conditions to unify the understanding of electromagnetic induction and magnetic resonance coupling [1]. We also discuss improvements in efficiency and power attributable to differences in circuit topology. Note that this discussion is not related to the performance improvements in the Q value of the coil itself, as those described in Sects. 4.2.4 and 4.2.5. Section 5.1 describes the circuits for comparison. Section 5.2 explains nonresonant circuits that do not use resonance, which is the most basic form of electromagnetic induction. Section 5.3 explains the efficiency improvement method driven by capacitor resonance on the secondary side based on electromagnetic induction. Section 5.4 explains the primary-side capacitor-based power factor correction method for electromagnetic induction. Sections 5.5 and 5.6 explain the S–S (series–series) and S–P (series–parallel) methods of magnetic resonance coupling. Section 5.7 reports the comparison of efficiency and power among the five circuits. As S–P © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_5

113

114

5 Comparison Between Electromagnetic Induction and Magnetic …

circuits are essentially equivalent to S–S circuits, we use four circuits excluding the S–P type in the next section. In addition, starting from Sect. 5.8, our comparisons are based on reactance using the X 1 and X 2 axes, and we comprehensively analyze the comparisons from the previous section. Section 5.9 focuses on magnetic flux distribution, and Sect. 5.10 discusses the role of the primary magnetic flux.

5.1 Five Basic Types of Circuits: N–N, N–S, S–N, S–S, S–P Figure 5.1 shows the five types of circuit topologies analyzed in this chapter: N–N (no resonance), N–S (secondary-side resonance), S–N (primary-side resonance), S– S, and S–P. The main objective through Sect. 5.7 is to compare their equations and basic characteristics. Therefore, we verify the coupling coefficient using only k = 0.10, the representative characteristic of a large air gap. Detailed examination will be done, including varying air gaps, from Sect. 5.8 onwards. Figure 5.2 shows the efficiency and power obtained from magnetic resonance coupling (S–S) and conventional electromagnetic induction (N–N). The efficiency is roughly the same with a narrow air gap, but it notably diverges when the air gap

(a) No resonance (N–N)

(c) Primary-side resonance (S–N)

(b) Secondary-side resonance (N–S)

(d) Magnetic resonance coupling (S–S)

(e) Magnetic resonance coupling (S–P) Fig. 5.1 Five types of circuit configurations for comparison

5.1 Five Basic Types of Circuits: N–N, N–S, S–N, S–S, S–P

100

S-S

N-N P2 [W]

η [%]

80 60 40 20 0 0

50

100

150

200

1200 1000 800 600 400 200 0 0

115

S-S

50

N-N

100

150

g [mm]

g [mm]

(a) Efficiency

(b) Power

200

Fig. 5.2 Comparison of magnetic resonance coupling (S–S) and conventional electromagnetic induction (N–N)

exceeds a radius of about 150 mm (i.e., large air gap). Furthermore, the S–S circuit greatly outperforms the N–N circuit in terms of power. Thus, magnetic resonance coupling (S–S) provides high efficiency and power even for large air gaps. We consider efficiency η as the ratio of effective power given by Eq. (5.1), where P1 is the input power, P2 is the power consumed by the load, Pr1 is the energy loss from the primary-side internal resistance, and Pr2 is the energy loss from the secondaryside internal resistance. In addition, r 1 , r 2 , and RL , respectively, represent primaryside internal resistance, secondary-side internal resistance, and load resistance, where the internal resistance includes the radiation resistance. η=

P2 P2 = P1 Pr 1 + Pr 2 + P2   P1 = Re I1 · V¯1

(5.1) (5.2)

  Pr 1 = Re I1 I¯1 · r1

(5.3)

  Pr 2 = Re I2 I¯2 · r2

(5.4)

  P2 = Re I2 I¯2 · R L

(5.5)

Table 5.1 lists the circuit parameters and coil dimensions considered in this chapter, which were obtained from experimental values. We employed smaller Q values than usual settings to make the phasor diagram more visible. However, depending on the circuit topology, the values of C 1 and C 2 may differ from those in Table 5.1 due to resonance. We display such instances in separate tables. Figure 5.3 shows the employed transmitter and receiver coils. For experiments, we considered coupling coefficient k = 0.10 at air gap g = 162 mm. Figure 5.4 and Table 5.2 show the air gap characteristics of the employed coil.

116

5 Comparison Between Electromagnetic Induction and Magnetic …

Table 5.1 Circuit and coil parameters Cal.

Exp.

f (kHz)

100.0

100

Outer radius (mm)

300

L 1 (μ H)

159.2

158.7

Inner radius (mm)

100

L 2 (μ H)

159.2

159.2

Turns

27.5

L m (μ H)

15.9

15.9

Wire diameter, a (mm)

2

Space between wires, s (mm)

2

k (−)

0.10

0.10

C1 (nF)

15.9

C1 (nF)

15.9

15.9

r1 ()

1.3

1.4

r2 ()

1.3

1.3

Q 1 (−)

75.6

72.6

Q 2 (−)

75.6

78.7

Exp.

15.9

Fig. 5.3 Transmitter and receiver coils

5.2 Non-resonant Circuit (N–N) In this section, we consider a non-resonant circuit (N–N) without capacitor (Fig. 5.1a) [1], being the most basic electromagnetic induction circuit and equivalent to a transformer circuit. As mentioned previously, the circuit does not operate at k = 1 when used for wireless power transmission, and thus we set k = 0.10.

5.2 Non-resonant Circuit (N–N)

117 100

0.8

80

Lm [uH]

1

k

0.6 0.4

40 20

0.2 0

60

0

50

100

150

200

0 0

50

g [mm]

100

150

200

g [mm]

Fig. 5.4 Coupling coefficient and mutual inductance versus air gap

5.2.1 Verification of Equivalent Circuit (N–N) To better understand the T-type equivalent circuit, we use an extended T-type equivalent circuit that separates the L 1 − L m components instead of an integrated circuit (Fig. 5.5). Figure 5.6 shows the diagram of this extended T-type equivalent circuit using converted secondary impedance and induced electromotive force. The postZ 2  impedance is converted into the primary-side impedance, and the secondary-side uses induced electromotive force V Lm2 . Figure 5.7 shows the converted secondary impedance circuit, Z 2  , and the overall circuit incorporating this converted circuit during resonance. Z 2  is sometimes referred to as reflected impedance. Equations (5.6) and (5.7), which represent the voltage on the primary and secondary sides of N–N, become Eqs. (5.8) and (5.9). Thus, we can calculate currents I 1 and I 2 using Eqs. (5.10) and (5.11). In these equations, ω is the angular frequency, V L1 = V L11 + V Lm1 and V L2 = V Lm2 - V L22 .

I1 =

V1 = VL11 + Vr 1 + VLm1

(5.6)

VLm2 = VL22 + Vr 2 + V2

(5.7)

V1 = jω L 1 I1 + I1r1 + jω L m I2

(5.8)

0 = jωL 2 I2 + I2 r2 + I2 R L + jωL m I1

(5.9)

r2 + R L + jωL 2 V1 (r1 + jω L 1 )(r2 + R L + jωL 2 ) + ω2 L 2m

I2 = −

jω L m V1 (r1 + jω L 1 )(r2 + R L + jωL 2 ) + ω2 L 2m

Additionally, the ratio of I 1 and I 2 can be expressed as

(5.10) (5.11)

30

0.54

85.6

130

0.14

22.7

g (mm)

k

L m (μH)

g (mm)

k

L m (μH)

20.0

0.13

140

72.8

0.46

40

17.7

0.11

150

63.8

0.40

50

16.1

0.10

160

63.9

0.35

60

Table 5.2 Coupling coefficient and mutual inductance versus air gap 70

15.9

0.10

162

54.9

0.31

80

15.4

0.10

165

48.5

0.26

90

14.4

0.09

170

42.0

0.23

100

12.9

0.08

180

36.5

0.21

110

11.5

0.07

190

28.6

0.18

120

10.4

0.07

200

25.1

0.16

118 5 Comparison Between Electromagnetic Induction and Magnetic …

5.2 Non-resonant Circuit (N–N)

119

Fig. 5.5 Extended T-type equivalent circuit (N–N)

Fig. 5.6 Equivalent circuit (N–N) of secondary-side impedance conversion and induced electromotive force

(a) Converted secondary impedance

(b) Overall circuit

Fig. 5.7 Circuit incorporating converted secondary impedance circuit (N–N)

I1 jω L 2 + r2 + R L = −I2 jωL m

(5.12)

120

5 Comparison Between Electromagnetic Induction and Magnetic …

Fig. 5.8 Relationship between I 1 and V Lm2 in a T-type equivalent circuit

The induced electromotive force, V Lm2 , excited on the secondary side is given by Eq. (5.13). Therefore, substituting Eq. (5.10) of current I 1 into the primary side results in Eq. (5.14). VLm2 = jω L m I1 VLm2 =

jωL m (r2 + R L + jωL 2 ) V1 (r1 + jω L 1 )(r2 + R L + jωL 2 ) + ω2 L 2m

(5.13) (5.14)

Figure 5.8 illustrates the formula in Eq. (5.13), which is frequently used. Focusing on Z Lm2 , I 1 , and I 2 , we can obtain the following equation to calculate Eq. (5.13) VLm2 = − jωL m I2 + jωL m (I2 + I1 ) = jωL m I1

(5.15)

Thus, based on either Eqs. (5.13) and (5.14) or (5.10) and (5.11), input power factor Z in2 on the secondary side becomes Z in2 =

VLm2 jωL m I1 = = r2 + R L + jωL 2 −I2 −I2

(5.16)

We can express Z 2  using Eq. (5.16) as Z 2 =

jωL m I2 (ωL m )2 = I1 Z in2

(5.17)

This is also referred to as immittance characteristics or K-inverter characteristics. Besides coefficient (ωL m )2 , the numerator and denominator of impedance Z in2 inverts to become a reciprocal. Therefore, Z in2 is given by Z in1 = r1 + jωL 1 + Z 2

(5.18)

Regarding input power factor Z in2 on the secondary side, Eq. (5.16) shows that input impedance Z in2 to the secondary side does not become purely resistive due to term jωL 2 . As explained below, Z in2 not being purely resistive reduces efficiency.

5.2 Non-resonant Circuit (N–N)

121

5.2.2 Efficiency and Power (N–N) Next, we consider efficiency. The power ratio can be expressed as follows based on Eqs. (5.3)–(5.5) and (5.10) and (5.11):   Pr 1 : Pr 2 : P2 = (r2 + R L )2 + (ωL 2 )2 r1 : (ωL m )2 r2 : (ωL m )2 R L

(5.19)

Thus, from Eqs. (5.1) and (5.19), efficiency is given by Eq. (5.20), and Eq. (5.21) derives in Eq. (5.22), which represents the load condition at maximum efficiency. Substituting Eq. (5.22) into Eq. (5.20) yields the maximum efficiency. (ωL m )2 R L  η=  (r2 + R L )2 + (ωL 2 )2 r1 + (ωL m )2 r2 + (ωL m )2 R L

(5.20)

∂η =0 ∂ RL

(5.21)

 R Lopt =

r22 +

r2 (ωL m )2 + (ωL 2 )2 r1

(5.22)

If we consider the input impedance on the primary side, Z in1 is given as follows based on Eqs. (5.16)–(5.18) or (5.10): Z in1 =

V1 (r1 + jωL 1 )(r2 + R L + jωL 2 ) + ω2 L 2m = I1 r2 + R L + jωL 2

(5.23)

Table 5.3 lists the calculation results for the non-resonant circuit, and Fig. 5.9 shows the phasor diagram and enlarged views of the circuit values. The phasor diagram uses V 1 as reference to draw the vectors. Figure 5.9 shows the vectors with scale 10 (i.e., 1 V = 10 A) due to the small current. As the non-resonant circuit has no capacitor on the secondary side, it becomes a series circuit comprising coil L 2 and resistance (r 2 + RL ) for the power generated on the secondary side, called induced electromotive force V Lm2 . As the two voltage vectors generated by the coil and resistance are purely resistive and inductive (i.e., value is an imaginary number), they are orthogonal and fall within a circular range with diameter of induced electromotive force V Lm2. Furthermore, the voltage of V Lm2 is used as load voltages V 2 and V r2 , as well as coil voltage V L22 , consequently undermining the voltage utilization rate. If the voltage comes from a normal source and not as induced electromotive force V Lm2 , the low voltage utilization rate only increases the voltage of the secondary-side coil without causing loss. Still, as shown in Eq. (5.13), the voltage on the secondary side is induced by electromotive force V Lm2 and derives from current I 1 on the primary side. A voltage that can only be generated by passing a current on the primary side as coil voltage without being a load is not useful, thus inducing loss (Eq. 5.16). The input power factor on the secondary

122

5 Comparison Between Electromagnetic Induction and Magnetic …

Table 5.3 Calculated values for non-resonant circuit N–N (P2 , Pr 1 , Pr 2 are purely resistive with the Im component being 0, and abs correspond to are Re values) Re.

Im.

ABS

θ (°)

I1 (A)

0.0

−1.0

1.0

271.0

I2 (A)

−0.1

0.0

0.1

136.6

VL11 (V)

100.5

1.8

100.5

1.0

0.0

−1.3

1.3

271.0

VL m 1 (V)

−0.5

−0.5

0.7

226.6

VL m 2 (V)

10.0

0.2

10.0

1.0

VL22 (V)

−4.8

−5.1

7.0

226.6

Vr 2 (V)

−0.1

0.1

0.1

136.6

5.1

−4.9

7.1

316.6

Vr 1 (V)

V2 (V) Z in1 ()

1.8

99.5

99.5

89.0

Z 2 ()

0.5

−0.5

0.7

315.5

101.8

100.0

142.7

44.5

100.5

Z in2 () P1 (W)

1.8

100.5

P2 (W)

0.5

θ I2 I1 (I2 /I1 )(◦ )

225.5

Pr 1 (W)

1.3

θV2 V1 (V2 /V1 )(◦ )

316.6

Pr 2 (W) Efficiency (%) R Lopt ()

0.0 27.1 100.5

89.0

)(◦ )

89.0

θ2 (V2 / − I2 )(◦ )   θ Zin2 VL m2 / − I2 (◦ )

44.5

θ1 (V1 /I1

0.0

(a) General view (N–N)

(b) Primary -side voltage, enlarged

(c) Secondary-side voltage, enlarged

Fig. 5.9 Phasor diagram of non-resonant circuit (N–N)

5.2 Non-resonant Circuit (N–N)

123

side of the non-resonant circuit is cosθ Zin2 = 0.71, but this can be perceived as an unrepresentative figure. As described above, the ratio of I 2 to I 1 is important regarding efficiency, but from Eq. (5.30), I 1 is 14.3 times larger than I 2 , thus reducing efficiency. A more detailed explanation is given for comparison with N–S in the next section. Overall, the N–N circuit is a low-efficiency topology with limited capacity to produce high power. However, as it does not use resonance, it can handle a wide range of frequency characteristics.

5.3 Secondary-Side Resonant Circuit (N–S) In this section, we examine the secondary-side resonant circuit (N–S) for electromagnetic induction with resonant capacitor on the secondary side [1, 2].

5.3.1 Verification of Equivalent Circuit (N–S) Figure 5.10 shows the extended T-type equivalent circuit, and Fig. 5.11 shows the equivalent circuit of secondary input impedance conversion and induced electromotive force. In addition, Fig. 5.12 shows the converted impedance circuit, Z 2  , on the secondary side during resonance and the overall circuit incorporating Z 2  during resonance. As shown in Fig. 5.1b, N–S is a circuit with resonance between L 2 and C 2 on the secondary side. Based on the equivalent circuits in the figure, Eqs. (5.24) and (5.25), which represent the voltages on the primary and secondary sides, become Eqs. (5.26) and (5.27). Then, currents I 1 and I 2 can be calculated using Eqs. (5.28) and (5.29). V1 = VL11 + Vr 1 + VLm1

(5.24)

VLm2 = VL22 + VC2 + Vr 2 + V2

(5.25)

V1 = jωL 1 I1 + I1r1 + jωL m I2

(5.26)

1 I2 + I2 r2 + I2 R L + jωL m I1 jωC2   r2 + R L + j ωL 2 − ωC1 2 

 V1 I1 = (r1 + jωL 1 ) r2 + R L + j ωL 2 − ωC1 2 + ω2 L 2m 0 = jωL 2 I2 +

I2 = −

jωL m  (r1 + jωL 1 ) r2 + R L + j ωL 2 −

1 ωC2



+ ω2 L 2m

V1

(5.27)

(5.28)

(5.29)

124

5 Comparison Between Electromagnetic Induction and Magnetic …

(a) General operation

(b) During resonance Fig. 5.10 Extended T-type equivalent circuit (N–S)

Fig. 5.11 Equivalent circuit (N–S) of secondary-side conversion and induced electromotive force

The ratio of I 1 and I 2 can be expressed as 1 jωL 2 + jωC + r2 + R L I1 2 = −I2 jωL m

The impedance of each component is given by

(5.30)

5.3 Secondary-Side Resonant Circuit (N–S)

125

(a) Converted secondary impedance

(b) Overall circuit

Fig. 5.12 Circuit incorporating Z 2  during resonance (N–S)

Z in2 =

VLm2 jωL m I1 1 = = jωL 2 + + r2 + R L −I2 −I2 jωC2

(5.31)

jωL m I2 (ω L m )2 = I1 Z in2

(5.32)

Z 2 =

Z in1 = r1 + jωL 1 + Z 2

(5.33)

Furthermore, Eq. (5.34) represents the secondary-side resonance conditions, as the voltage of the secondary-side coil is offset by the voltage of the secondary-side capacitor. Therefore, the resonant angular frequency can be expressed by Eq. (5.35). VL22 + VC2 = jωL 2 +  ω2 =

1 =0 jωC2

1 L 2 C2

(5.34)

(5.35)

To satisfy this equation, currents I 1 and I 2 can be, respectively, calculated as follows: I1 =

r2 + R L V1 (r1 + jω L 1 ) (r2 + R L ) + ω2 L 2m

I2 = −

jω L m V1 (r1 + jω L 1 ) (r2 + R L ) + ω2 L 2m

(5.36) (5.37)

The ratio of I 1 and I 2 can be expressed as I1 r2 + R L = −I2 jωL m

(5.38)

126

5 Comparison Between Electromagnetic Induction and Magnetic …

In addition, during this secondary-side resonance, Eq. (5.13) defines induced electromotive force V Lm2 on the secondary side. Therefore, substituting Eq. (5.36) for current I 1 on the primary side gives the following equation: VLm2 =

jω L m (r2 + R L ) V1 (r1 + jω L 1 ) (r2 + R L ) + ω2 L 2m

(5.39)

On the other hand, based on Eq. (5.34), Eq. (5.25) for VLm2 becomes the following, and the voltage on the secondary side is used entirely for resistance: VLm2 = Vr 2 + V2

(5.40)

In other words, unlike Eq. (5.16), which shows Z in2 in a non-resonant circuit, input impedance Z in2 on the secondary side becomes a pure resistance during secondaryside resonance. Hence, the input power factor on the secondary side becomes 1. This can be directly seen from the following equation, calculated using Eqs. (5.36) and (5.37): Z in2 =

VLm2 jωL m I1 = = r2 + R L −I2 −I2

(5.41)

5.3.2 Efficiency and Power (N–S) Next, we consider efficiency. From Eqs. (5.3)–(5.5) and (5.28) and (5.29), the power ratio can be expressed as 

Pr 1 : Pr 2 : P2 = (r2 + R L ) + ωL 2 − 2

1 ωL C

2  r1 : (ωL m )2 r2 : (ωL m )2 R L (5.42)

Thus, from Eqs. (5.1) and (5.42), efficiency can be expressed as η=   (r2 + R L )2 + ωL 2 −

(ωL m )2 R L 2  1 r1 + (ωL m )2 r2 + (ωL m )2 R L ωC2

(5.43)

Given the resonance conditions in Eq. (5.34), the power ratio given by Eq. (5.42) and efficiency given by Eq. (5.43), respectively, become the following equations at secondary-side resonance: Pr 1 : Pr 2 : P2 = (r2 + R L )2 r1 : (ωL m )2 r2 : (ωL m )2 R L

(5.44)

5.3 Secondary-Side Resonant Circuit (N–S)

η=

(ωL m )2 R L (r2 + R L ) r1 + (ωL m )2 r2 + (ωL m )2 R L 2

127

(5.45)

Comparing efficiency of the non-resonant circuit in Eq. (5.20), or pre-resonance efficiency of the secondary-side resonant circuit in Eq. (5.43), to the post-resonance efficiency in Eq. (5.45), we can see that canceling the L 2 component with C 2 decreases the denominator and improves efficiency. This reduces the loss ratio caused by primary internal resistance Pr1 , substantially improving efficiency. On the other hand, when there is no resonance by C 2 , the ratio of Pr1 increases because secondary-side inductance L 2 bears most of the voltage on this side. These situations can be explained as follows. In a non-resonant circuit, it is necessary that current I 1 flows on the primary side to induce electromotive force V Lm2 on the secondary side. With respect to V Lm2, by the limited voltage of the induced electromotive force, the ratio used in the secondary coil (V L22 ) becomes higher than that used in the load (V 2 ). Therefore, the proportion of current I 1 increases with respect to current I 2 , and the ratio of power consumption, Pr1 , on the primary side relatively increases due to the outflow of wasted primary current. This can be seen in Eqs. (5.12), (5.30), and (5.38), representing the ratio of I 1 and I 2 . Power in Eqs. (5.3)–(5.5) shows that, as the ratio of a current increases, segment increases power consumption. On the other hand, the efficiency of the secondary-side resonant circuit increases by canceling the L 2 component with C 2 to increase the ratio of I 2 with respect to current I 1 . Similar to Eq. (5.41), secondary input impedance Z in2 becomes a pure resistance, and the secondary input power factor, generated from the relationship between V Lm2 and I 2 , becomes 1. The secondary input power factor is cosθ Zin2 , and θ Zin2 = arg(V Lm2 /I 2 ) (°), being θ Zin2 = 0° by their same phase. For maximum efficiency, the conditional load in Eq. (5.47) is obtained from Eqs. (5.43) and (5.46). Substituting this equation into Eq. (5.43) gives the maximum efficiency. Furthermore, if the resonance conditions in Eq. (5.34) are satisfied, the load condition becomes Eq. (5.48). Equation (5.49) gives the maximum efficiency under resonance.

 R Lopt =

r22 +

R Lopt

∂η =0 ∂ RL

r2 (ωL m )2 1 2 + ωL 2 − r1 ωC2  r2 (ωL m )2 = r22 + r1

(ω0 L m )2 ηmax =  2  √ r1r2 + r1r2 + (ω0 L m )2

(5.46)

(5.47)

(5.48) (5.49)

128

5 Comparison Between Electromagnetic Induction and Magnetic …

Input impedance Z in1 on the primary side is relevant for power. However, based on Eqs. (5.31)–(5.33) or (5.36) Z in1 becomes

Z in1



 1 + ω2 L 2m + jω L + R + j ω L − r ) (r 1 1 2 L 2 ω C2 V1   = = I1 r2 + R L + j ωL 2 − ω 1C2

(5.50)

Under the secondary-side resonance conditions in Eq. (5.35) Z in1 becomes Z in1 =

V1 (r1 + jω L 1 )(r2 + R L ) + ω2 L 2m = I1 r2 + R L

(5.51)

Comparing Eqs. (5.50) and (5.51), we observe that C 2 causes changes in the power factor, but the improvements are not considerable due to the large impact of jωL 1 from the coil on the primary side. In other words, C 2 does not notably contribute to obtain high electric power. Next, we compare a non-resonant circuit without capacitor and a secondary-side resonant circuit with inserted C 2 . The calculations assume constant connection of optimal load RLopt that retrieves maximum efficiency and an input voltage of 100 V. The same input voltage is used for other circuits. Table 5.4 lists the calculation results for the secondary-side resonant circuit, and Fig. 5.13 shows the overall phasor diagram during resonance by L 2 and C 2 . The phasor diagram uses V 1 as reference to draw the vectors. Similar to Figs. 5.9, 5.13 shows the vectors with scale 10 (i.e., 1 V = 10 A) due to the small current. However, the scale is appropriately changed for the enlarged views of current. The two circuits have smaller currents compared to the primary-side resonant circuit and magnetic resonance coupling, which are described below. Accordingly, the coils in these circuits generate relatively small voltages, and the maximum voltage slightly exceeds (N–N) or does not exceed (N–S) the 100 V input voltage. Figure 5.14 shows a diagram of this transition to explain the effects of the C 2 insertion. We discuss the effects of C 2 insertion. In Fig. 5.14a, the non-resonant circuit (Fig. 5.9) has no capacitor on the secondary side, and thus voltage V Lm2 is used not only as V 2 and V r2 (load voltages) but also as V L22 (coil voltage). The low voltage utilization rate reduces the input power factor on the secondary side, thus reducing the circuit efficiency. Next, we insert a very small C 2 (Fig. 5.14b) such that it does not cause resonance. In this case, the voltage utilization ratio, or secondary input power factor, increases. However, the power loss persists as the voltage is still used as coil voltage V L22 . Figure 5.14c (Fig. 5.13) shows C 2 insertion to cause resonance on the secondary side. The coil voltage is canceled by the resonance capacitor, and induced electromotive force V Lm2 is entirely used as V 2 and V r2 . This is consistent with Eq. (5.36), and we can say that induced electromotive force V Lm2 generated by the primary side is completely used on the secondary side. Therefore, we obtain input power factor cosθ Zin2 = 1 on the secondary side, and the ratio of I 2 to I 1 is roughly 1.1 based on Eq. (5.38). Efficiency η of 27.1% (at optimal load) in a non-resonant

5.3 Secondary-Side Resonant Circuit (N–S)

129

Table 5.4 Calculated values for resonant circuit (N–S) on the secondary side (P2 , Pr1 , Pr2 are purely resistive with the Im component being 0, and abs values correspond to Re values) Re.

Im.

ABS

θ(◦ )

I1 (A)

0.1

−1.0

1.0

275.8

I2 (A)

−0.9

−0.1

0.9

185.8

VL11 (V)

99.0

10.0

99.5

5.8

Vr 1 (V)

0.1

−1.3

1.3

275.8

VL m 1 (V)

0.9

−8.7

8.7

275.8

VL m 2 (V)

9.9

1.0

9.9

5.8 275.8

8.8

−86.8

87.2

VC2 (V)

−8.8

86.8

87.2

95.8

Vr 2 (V)

−1.1

−0.1

1.2

185.8

VL22 (V)

V2 (V) Z in1 () Z 2 ()

8.8

0.9

8.8

5.8

10.0

100.0

100.5

84.3

8.8

0.0

8.8

0.0

Z in2 ()

11.4

0.0

11.4

0.0

P1 (W)

10.0

99.0

99.5

84.2

P2 (W)

7.7

θ I2 I1 (I2 /I1 )(◦ ) )(◦ )

Pr 1 (W)

1.3

θV2 V1 (V2 /V1

Pr 2 (W)

1.0

θ1 (V1 /I1 )(◦ )

Efficiency (%)

76.8

R Lopt ()

10.1

θ2 (V2 / − I2 )(◦ )   θ Zin2 VL m2 / − I2 (◦ )

270.0 5.8 84.2 0.0 0.0

circuit improves to 76.8% (at optimal load) in the secondary-side resonant circuit. Therefore, the mechanism described above allows to achieve high efficiency. On the primary side, induced electromotive voltage V Lm2 generated on the load side is small, at approximately 10 V, for an input voltage of 100 V during both nonresonance and secondary-side resonance. The primary input phase in a secondaryside resonant circuit is θ 1 = 84.2°, while that with no resonance is θ 1 = 89.0°. Although this phase slightly improves, the effect is small. As the power factor on the primary side is low, the primary-side coil uses most of the source voltage, resulting in an extremely small current I 1 and low power transmission. The load power improves to P2 = 7.7 W in a secondary-side resonant circuit from P2 = 0.5 W in a non-resonant circuit. However, the amount of power is still small for an input of 100 V. Despite achieving high efficiency under large air gaps and misalignment, high power is not obtained. In other words, the secondary-side resonant circuit N–S has high efficiency but low power. Figure 5.15 shows the efficiency of N–N and N–S according to RL . N–N is dominated by the loss in Pr1 , whereas N–S shows two areas, one with Pr2 dominance and the other with Pr1 dominance, and the maximum efficiency is obtained at their intersection.

130

5 Comparison Between Electromagnetic Induction and Magnetic …

(a) General view (N–S)

(b) Primary-side voltage, enlarged

(c) Secondary-side voltage, enlarged

Fig. 5.13 Phasor diagram of secondary-side resonant circuit (N–S)

(a) No C2 (non-resonant)

(b) Added C2 (non-resonant)

(c) Added C2 (resonant) Fig. 5.14 Transition from non-resonant circuit (N–N) to secondary-side resonant circuit (N–S)

131 100%

90%

90%

80%

80%

70% 60% 50%

PRL

40%

Pr2

30%

Pr1

Power ratio [%]

100%

70% 60% 50%

PRL

40%

Pr2

30%

Pr1

20%

20%

10%

10%

0%

0%

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Power ratio [%]

5.4 Primary-Side Resonant Circuit (S–N)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

RL [Ω]

RL [Ω]

(a) N–N

(b) N–S

Fig. 5.15 Load optimization and loss ratio (N–N vs. N–S)

5.4 Primary-Side Resonant Circuit (S–N) 5.4.1 Verification of Equivalent Circuit (S–N) Figure 5.1c shows the primary-side resonant circuit (S–N), an electromagnetic induction approach in which resonant capacitor C 1 is inserted on the primary side [1]. Figure 5.16 shows the extended T-type equivalent circuit. Figure 5.17 shows the equivalent circuit of secondary impedance conversion and induced electromotive force. In addition, Fig. 5.18 shows the converted secondary impedance circuit, Z 2  , and the overall circuit, in which L 1 and C 1 incorporates Z 2  during resonance. Based on the equivalent circuit shown in Fig. 5.1c, Eqs. (5.52) and (5.53) representing the primary-side and secondary-side voltages become Eqs. (5.54) and (5.55), respectively. Thus, currents I 1 and I 2 can be calculated using Eqs. (5.56) and (5.57). V1 = VL11 + VC1 + Vr 1 + VLm1

(5.52)

VLm2 = VL22 + Vr 2 + V2

(5.53)

V1 = jωL 1 I1 +

1 I1 + I1r1 + jωL m I2 jωC1

0 = jωL 2 I2 + I2 r2 + I2 R L + jωL m I1  I1 = r1 + j ω L 1 −

r2 + R L + jωL 2 

V1 1 (r2 + R L + jωL 2 ) + ω2 L 2m ω C1

 I2 = − r1 + j ω L 1 −

1 ω C1

jω L m 

V1 (r2 + R L + jωL 2 ) + ω2 L 2m

(5.54) (5.55) (5.56)

(5.57)

132

5 Comparison Between Electromagnetic Induction and Magnetic …

(a) General operation

(b) During resonance Fig. 5.16 Extended T-type equivalent circuit (S–N)

Fig. 5.17 Equivalent circuit (S–N) of secondary impedance conversion and induced electromotive force

Additionally, the ratio of I 1 and I 2 can be expressed as the following equation, which is the same as Eq. (5.12) in a non-resonant circuit: jω L 2 + r2 + R L I1 = −I2 jωL m

(5.58)

Based on Eqs. (5.56) and (5.57), input impedance Z in2 on the secondary side becomes

5.4 Primary-Side Resonant Circuit (S–N)

(a) Converted secondary impedance

133

(b) Overall circuit

Fig. 5.18 Circuit incorporating Z 2  by L 1 and C 1 during resonance (S–N)

Z in2 =

VLm2 jωL m I1 = = r2 + R L + jωL 2 −I2 −I2

(5.59)

5.4.2 Efficiency and Power (S–N) Input impedance Z in2 on the secondary side is not purely resistive due to term jωL 2 . Equation (5.59) is the same as Eq. (5.16) of a non-resonant circuit. Hence, the secondary-side input power factor is low, not reaching 1. Therefore, low efficiency occurs like in the non-resonant circuit. The efficiency formula verifies the inability to improve efficiency and shows no impact from C 1 . In addition, from Eqs. (5.3)–(5.5) and (5.56) and (5.57), C 1 impacts the denominator of the current, changing the magnitude of I 1 and I 2 , but not affecting efficiency. The respective power ratios can be calculated from Eqs. (5.3)–(5.5) and (5.56) and (5.57), which do not fit the conditions of Eq. (5.65), as follows:   Pr 1 : Pr 2 : P2 = (r2 + R L )2 + (ωL 2 )2 r1 : (ωL m )2 r2 : (ωL m )2 R L

(5.60)

Based on Eqs. (5.1) and (5.60), efficiency is given by η= 

(ωL m )2 R L  (r2 + R L )2 + (ωL 2 )2 r1 + (ωL m )2 r2 + (ωL m )2 R L

(5.61)

C 1 not being included in this equation also reveals the lack of impact from the primary-side capacitor to efficiency. In other words, the efficiency of the primaryside resonant circuit S–N corresponds to that in Eq. (5.20) of the non-resonant circuit N–N. The optimal load and maximum efficiency formulas at maximum efficiency also correspond to those in a non-resonant circuit.

134

5 Comparison Between Electromagnetic Induction and Magnetic …

On the other hand, converted secondary impedance Z 2  and primary-side input impedance Z in1 can be respectively expressed as 

Z2 =

jωL m I2 (ω L m )2 = I1 Z in2

Z in1 = r1 + jωL 1 +

1  + Z2 jωC1

(5.62) (5.63)

Therefore, based on Eqs. (5.59), (5.62) and (5.63), or Eq. (5.56), the primary-side input impedance is given by Eq. (5.64) and thus differs from that of the N–N circuit.

Z in1



 r1 + j ωL 1 − ωC1 1 (r2 + R L + jωL 2 ) + ω2 L 2m V1 = = I1 r2 + R L + jωL 2

(5.64)

5.4.3 Design Using Primary-Side Resonance Conditions (S–N) Magnetic resonance coupling, which is discussed in the next section, achieves resonance using primary-side L 1 and C 1 , and then using secondary-side L 2 and C 2 . Thus, the circuits are coupled using a magnetic field. However, inserting a resonant capacitor, C 1 , on the primary side without one, C 2 , on the secondary side produces an imaginary component on the secondary side. Consequently, this case cannot be regarded simply as resonance between L 1 and C 1 , and we set cosθ Zin1 = 1 as resonance condition to complement the power factor and operate the primary-side resonant circuit. We consider the conditions of Eqs. (5.71) and (5.72) as described in Sect. 5.4.4. On the other hand, Eq. (5.65) can be derived by using the value of C 1 from magnetic resonance coupling (S–S) described in 5.5, along with L 1 and C 1 to create the resonance conditions only on the primary side. The angular frequency is represented by Eq. (5.66). We refer to these conditions as primary-side resonance conditions. VL1 + VC1 = jωL 1 +  ω1 =

1 =0 jωC1

1 L 1 C1

(5.65)

(5.66)

When these equations are satisfied, currents I 1 and I 2 can be determined as I1 =

r2 + R L + jωL 2 V1 r1 (r2 + R L + jωL 2 ) + ω2 L 2m

(5.67)

5.4 Primary-Side Resonant Circuit (S–N)

I2 = −

135

jω L m V1 r1 (r2 + R L + jωL 2 ) + ω2 L 2m

(5.68)

In this case, Eq. (5.13) defines induced electromotive force V Lm2 on the secondary side, and the following equation can be obtained by substituting current I 1 on the primary side (Eq. 5.67): VLm2 =

jω L m (r2 + R L + jωL 2 ) V1 r1 (r2 + R L + jωL 2 ) + ω2 L 2m

(5.69)

Inserting C 1 on the primary side does not affect the impedance on the secondary side, and thus the secondary-side input power factor, Z in2 , remains given by Eq. (5.59). Based on Eqs. (5.59) and (5.62), the primary-side input impedance under primaryside resonance conditions becomes Z in1 = r1 + Z 2 = r1 +

(ωL m )2 r2 + R L + jωL 2

(5.70)

However, under these conditions, the power factor does not become 1, because Z2  contains jωL 2.

5.4.4 Design with Overall Resonance Condition (Power Factor = 1) The condition under which primary-side input power factor cosθ Zin1 = 1 occurs when the imaginary component in Eq. (5.64) is 0. Thus, calculating Im{Z in1 } = 0

(5.71)

derives in the following conditional formula (5.72) for C 1 : C1 =

(r2 + R L )2 + (ω L 2 )2     ω ω L 1 (r2 + R L )2 + (ω L 2 )2 − ω L 2 (ω L m )2

(5.72)

In this text, we refer to this condition as overall resonance condition. The calculation requires substitution with the formula of C 1 . However, as the equation becomes complicated, we verify the calculations by substituting the values of this formula and with power factor = 1 in Sect. 5.4.5.

136

5 Comparison Between Electromagnetic Induction and Magnetic …

5.4.5 Comparison Between Primary-Side and Overall Resonance Conditions (Power Factor = 1) We use calculation results to compare the primary-side and overall resonance conditions (power factor = 1) in a primary-side resonant circuit with resonance capacitor C 1 on the primary side. We also evaluate a non-resonant circuit without a capacitor. A graph is created based on the optimal load values to assess the calculation conditions. Table 5.5a lists the calculation results for primary-side resonance conditions, and Table 5.5b shows the calculation results for the overall resonance condition (power factor = 1). In addition, Fig. 5.19 shows the calculations for primary-side resonance conditions during resonance by L 1 and C 1 , and Fig. 5.20 shows the phasor diagram of the calculations for the overall resonance condition at primary input power factor cosθ Zin1 = 1. Note that V L11 and V C1 are drawn at scale 1/6, and each phasor diagram uses V 1 as reference to draw the vectors. The current is drawn at scale 1 V = 1 A. First, the efficiency becomes equal under both primary-side and overall resonance conditions, as well as in non-resonant circuits. This is consistent with Eqs. (5.61) and (5.20). Efficiency is low, at 27.1%, whereas θ I2I1 , the phase differences of I 2 and I 1 , are 225.5° and remain constant even under varying C 1 . Based on Eq. (5.58), the amplitudes of I 2 and I 1 are not related to C 1 . Therefore, the triangles formed by V Lm2 , (V 2 + V r2 ), and V L22 have similar relationships in Figs. 5.19 and 5.20, and efficiency does not change. Next, we analyze the improvements in power factor that enable high power. Primary input phase θ 1 improves to θ 1 = 344.9° = –15.1° under primary-side resonance conditions, compared to θ 1 = 89.0° in a non-resonant circuit. Naturally, under the overall resonance condition, the primary input phase improves to θ 1 = 0°. Even compared to primary input phase θ 1 = 84.2° of the secondary-side resonant circuit (N–S), both are substantial improvements. This can also be verified from I 1 and V 1 in the phasor diagram. Under the overall resonance condition, V L11 = V C1 but I 1 // V 1 , as shown in Fig. 5.20, and the power factor becomes 1. On the other hand, under primary-side resonance conditions, I 1 and V 1 do not face the same direction, as shown in Fig. 5.19. As there is resonance in L 1 and C 1 on the primary side, V L11 = V C1 (Fig. 5.19). The action of resonant capacitor C 1 cancels the induction component, but no notable design advantages are observed. The large value of P1 reveals the achievement of high electric power, but it is not necessary to resonate in L 1 and C 1 . However, if relatively high power is desired instead of maximum power, Eq. (5.66) allows the simple calculation of C 1 , instead of the resonance condition in Eq. (5.72), which is represented under the overall resonance condition and a primary input power factor of 1. Another benefit is that the power factor is not very low. Therefore, the primary-side resonant circuit S–N can realize high power under any condition, but similar to the N–N circuit, efficiency is low.

5.4 Primary-Side Resonant Circuit (S–N)

137

Table 5.5 Calculated values for primary-side resonant circuit (S–N) (P2, Pr1, Pr2, are purely resistive with the Im component being 0, and abs values correspond to Re values) (a) Primary-side resonance conditions Re.

Im.

ABS

θ(◦ )

I1 (A)

51.1

13.8

53.0

15.1

I2 (A)

−1.8

−3.2

3.7

240.6

−1377.5

5114.9

5297.2

105.1

VC1 (V)

1377.5

−5114.9

5297.2

285.1

Vr 1 (V)

67.7

18.2

70.1

15.1

VL m 1 (V)

32.3

−18.2

37.1

330.6

VL m 2 (V)

−137.8

511.5

529.7

105.1

VL22 (V)

323.3

−182.2

371.2

330.6 240.6

VL11 (V)

Vr 2 (V)

−2.4

−4.3

4.9

V2 (V)

183.2

325.0

373.0

60.6

1.8

−0.5

1.9

341.8 315.4

Z in1 () Z 2 ()

0.5

−0.5

0.7

Z in2 ()

101.8

100.0

142.7

44.6

P1 (W)

5114.9

1377.5

5297.2

15.1

P2 (W)

1384.5

θ I2 I1 (I2 /I1 )(◦ )

Pr 1 (W)

3712.1

θV2 V1 (V2 /V1 )(◦ )

Pr 2 (W)

18.2

θ1 (V1 /I1 )(◦ )

Efficiency (%)

27.1

θ2 (V2 / − I2 )(◦ )   θ Zin2 VL m2 / − I2 (◦ )

R Lopt ()

100.5

225.5 60.6 344.9 0.0 44.5

(b) Overall resonance condition (power factor = 1) Re.

Im.

ABS

θ(◦ )

I1 (A)

54.9

0.0

54.9

0.0

I2 (A)

−2.7

−2.7

3.8

225.5

0.0

5485.9

5485.9

90.0

VC1 (V)

0.0

−5459.0

5459.0

270.0

Vr 1 (V)

72.6

0.0

72.6

0.0

VL m 1 (V)

27.4

−26.9

38.4

315.5

VL11 (V)

VL m 2 (V)

0.0

548.6

548.6

90.0

VL22 (V)

274.3

−269.3

384.4

315.5

Vr 2 (V)

−3.6

−3.6

5.1

225.5

V2 (V)

45.5

270.7

275.6

386.3

Z in1 ()

1.8

0.0

1.8

0.0

Z 2 ()

0.5

−0.5

0.7

315.5

101.8

100.0

142.7

44.5

Z in2 ()

(continued)

138

5 Comparison Between Electromagnetic Induction and Magnetic …

Table 5.5 (continued) (b) Overall resonance condition (power factor = 1) Re.

Im.

ABS

θ(◦ )

P1 (W)

5485.9

0.0

P2 (W)

1485.0

θ I2 I1 (I2 /I1 )(◦ )

225.5

Pr 1 (W)

3981.4

θV2 V1 (I2 /I1 )(◦ )

45.5

5485.9

)(◦ )

0.0 0.0

Pr 2 (W)

19.5

θ1 (V1 /I1

Efficiency (%)

27.1

θ2 (V2 / − I2 )(◦ )   θ Zin2 VL m2 / − I2 (◦ )

R Lopt () C1 (nF)

100.5

0.0

44.5

16.0

(a) General view (S–N, primary-side resonance conditions)

(b) Primary-side voltage, enlarged

(c) Secondary-side voltage, enlarged

Fig. 5.19 Phasor diagram of primary-side resonant circuit (S–N, primary-side resonance conditions)

5.5 Circuit of Magnetic Resonance Coupling (S–S)

139

(a) General view (S–N, overall resonance conditions, power factor = 1)

(b) Primary-side voltage, enlarged

(c) Secondary-side voltage, enlarged

Fig. 5.20 Phasor diagram of primary-side resonant circuit (S–N, overall resonance condition, power factor = 1)

5.5 Circuit of Magnetic Resonance Coupling (S–S) In this section, we examine the S–S circuit for magnetic resonance coupling, which is shown in Fig. 5.1d [1].

5.5.1 Verification of Equivalent Circuit (S–S) Figure 5.21 shows the extended T-type equivalent circuit, and Fig. 5.22 shows the equivalent circuit of secondary-side impedance conversion and induced electromotive force. In addition, Fig. 5.23 shows the converted secondary impedance circuit, Z 2  , during resonance, and the overall circuit incorporating Z 2  during resonance. The impedance of each component, Z in2 , Z 2  , and Z in1 , is expressed by the following

140

5 Comparison Between Electromagnetic Induction and Magnetic …

(a) General operation

(b) During resonance Fig. 5.21 Extended T-type equivalent circuit (S–S)

Fig. 5.22 Equivalent circuit (S–S) of secondary conversion and induced electromotive force

equations. As shown in Eq. (5.74), Z 2  remains purely resistive if the load is a pure resistance. Z in2 = jω L 2 + 

Z2 =

1 + r2 + R L jω C2

jωL m I2 (ω L m )2 = I1 Z in2

(5.73) (5.74)

5.5 Circuit of Magnetic Resonance Coupling (S–S)

141

(b) Overall circuit

(a) Converted secondary impedance circuit

Fig. 5.23 Circuit incorporating Z 2  during resonance (S–S)

Z in1 = r1 + jωL 1 +

1  + Z2 jω C1

(5.75)

Applying the equivalent circuit in Fig. 5.1d to Eqs. (5.76) and (5.77), which represent the primary-side and secondary-side voltages, results in Eqs. (5.78) and (5.79). Thus, currents I 1 and I 2 can be calculated from Eqs. (5.80) and (5.81). V1 = VL11 + VC1 + Vr 1 + VLm1

(5.76)

VLm2 = VL22 + VC2 + Vr 2 + V2

(5.77)

V1 = jωL 1 I1 +

1 I1 + I1r1 + jωL m I2 jωC1

(5.78)

1 I2 + I2 r2 + I2 R L + jωL m I1 jωC2   r2 + R L + j ωL 2 − ωC1 2  

  V1 I1 = r1 + j ωL 1 − ωC1 1 r2 + R L + j ωL 2 − ωC1 2 + ω2 L 2m 0 = jωL 2 I2 +

 I2 = − r1 + j ωL 1 −

1 ωC1

jωL m   r2 + R L + j ωL 2 −

1 ωC2



+ ω2 L 2m

V1

(5.79)

(5.80)

(5.81)

Additionally, the ratio of I 1 and I 2 can be expressed as I1 = −I2

 r2 + R L + j ωL 2 − jωL m

1 ωC2

 (5.82)

142

5 Comparison Between Electromagnetic Induction and Magnetic …

Equation (5.83) shows the resonance conditions of the secondary side, while Eq. (5.84) shows the resonant frequency of the primary side. When the resonant frequency is equal on the primary and secondary sides, the resonant angular frequency is given by Eq. (5.85). VL22 + VC2 = jωL 2 +

1 =0 jωC2

1 =0 jωC1   1 1 ω0 = ω1 = = ω2 = L 1 C1 L 2 C2 VL11 + VC1 = jωL 1 +

(5.83) (5.84)

(5.85)

If these equations of resonance conditions are satisfied, currents I 1 and I 2 can be calculated as I1 =

r2 + R L V1 r1 (r2 + R L ) + ω2 L 2m

I2 = −

jω L m V1 r1 (r2 + R L ) + ω2 L 2m

(5.86) (5.87)

The ratio of I 1 and I 2 can be expressed by Eq. (5.88). During resonance, this equation is the same as Eq. (5.38) from a resonant circuit on the secondary side (N–S). I1 r2 + R L = −I2 jω L m

(5.88)

In this case, Eq. (5.13) defines induced electromotive force V Lm2 on the secondary side. Substituting Eq. (5.86) for the primary-side current I 1 we obtain VLm2 =

jω L m (r2 + R L ) V1 r1 (r2 + R L ) + ω2 L 2m

(5.89)

On the other hand, V Lm2 is given by the following equation based on Eq. (5.77), showing the purely resistive second-side voltage: VLm2 = Vr 2 + V2

(5.90)

In other words, during resonance, input impedance Z in2 on the secondary side becomes a pure resistance, and thus the input power factor of the secondary side becomes 1. This can be seen directly from the following equation, calculated from Eqs. (5.86) and (5.87):

5.5 Circuit of Magnetic Resonance Coupling (S–S)

Z in2 =

143

VLm2 jωL m I1 = = r2 + R L −I2 −I2

(5.91)

The value of V Lm2 differs from the resonant circuit on the secondary side. However, the values of Z in2 are the same, and as with Eq. (5.91), the secondary side is purely resistive, and input power factor cosθ Zin2 on the secondary side also becomes 1.

5.5.2 Efficiency and Power (S–S) Next, we consider efficiency. First, from Eqs. (5.3)–(5.5) and Eqs. (5.80) and (5.81), which do not use resonance conditions, we express the equations for power ratio and efficiency as 

 1 2 Pr 1 : Pr 2 : P2 = (r2 + R L ) + ωL 2 − r1 : (ωL m )2 r2 : (ωL m )2 R L ωC2 (5.92) 2

η=   (r2 + R L )2 + ω L 2 −

(ωL m )2 R L 2  1 r1 + (ωL m )2 r2 + (ωL m )2 R L ωC2

(5.93)

Equation (5.93) shows that the capacitor on the primary side has no effect on efficiency, as it does not contain C 1 . Instead, the value of C 2 impacts efficiency. Furthermore, considering the resonance conditions of Eqs. (5.83)–(5.85), the equations for power ratio and efficiency become Pr 1 : Pr 2 : P2 = (r2 + R L )2 r1 : (ωL m )2 r2 : (ωL m )2 R L η=

(ωL m )2 R L (r2 + R L )2 r1 + (ωL m )2 r2 + (ωL m )2 R L

(5.94) (5.95)

In other words, the efficiency of magnetic resonance coupling (S–S) matches efficiency in Eq. (5.45) of the secondary-side resonant circuit (N–S). The maximum efficiency for the optimal load is the same as that in N–S. Therefore, the following equations for S–S during resonance, that is, optimum load in Eq. (5.96) and maximum efficiency in Eq. (5.97) are also equal to those for N–S during resonance.  R Lopt =

r22 +

r2 (ωL m )2 r1

(ω0 L m )2 ηmax =  2  √ r1r2 + r1r2 + (ω0 L m )2

(5.96) (5.97)

144

5 Comparison Between Electromagnetic Induction and Magnetic …

Next, we consider electrical power. From Eq. (5.80), the input impedance of the primary side can be expressed as

Z in1 =

V1 = I1

 r1 + j ωL 1 −

  r2 + R L + j ωL 2 −   r2 + R L + j ωL 2 − ωC1 2

1 ωC1

1 ωC2



+ ω2 L 2m

(5.98) Furthermore, from the resonance conditions in Eqs. (5.83)–(5.86), we derive Eq. (5.99), which can also be calculated using Eqs. (5.73)–(5.75) after applying the resonance conditions. Z in1 =

V1 r1 (r2 + R L ) + ω2 L 2m = I1 r2 + R L

(5.99)

This way, there is a pure resistance when viewed from the primary-side input, making the primary input power factor to reach 1. Normally, a resonance by L 1 and C 1 only on the primary side (e.g., Eq. 5.84) reveals inductance L 2 on the secondary side. In other words, imaginary numbers appear, and Eq. (5.84) cannot improve the overall power factor. Consequently, complex resonance conditions like C 1 in the primaryside resonant circuit (S–N) of Eq. (5.72) becomes necessary, and resonance in L 1 and C 1 on the primary side is not feasible. However, under the conditions of magnetic resonance coupling, the secondary-side resonates, that is, its input impedance becomes purely resistive. Equation (5.74) shows that, if the load is a pure resistance, then Z 2  remains purely resistive. Hence, the phase does not change despite a K-inverter (see Sect. 9.2) that expresses the coupling involving L m , and thus Z 2  remains purely resistive. Therefore, if primary-side resonance occurs under these conditions, whereby the action of L 1 is canceled by that of C 1 , the power factor becomes 1 for the entire circuit, thus enabling operation at high power. The above mentioned mechanism allows magnetic resonance coupling to operate at both high efficiency and high power even under large air gaps and shifts in positioning. In other words, such mechanism can realize high efficiency and power during large air gaps or displacement if the resonant frequencies on the primary and secondary sides are matched and coupled with a magnetic field. Thus, we conclude that magnetic resonance coupling is electromagnetic induction with narrower conditions.

5.5.3 Calculations for Magnetic Resonance Coupling (S–S Type Circuit) In this section, we use calculation results to examine the mechanism of magnetic resonance coupling and create a graph based on optimal load values to assess the

5.5 Circuit of Magnetic Resonance Coupling (S–S)

145

Table 5.6 Calculated values for magnetic resonance coupling S–S (P2, Pr1, Pr2 are purely resistive with the Im component being 0, and abs correspond to Re values) Re.

Im.

ABS

θ(◦ )

I1 (A)

9.9

0.0

9.9

0.0

I2 (A)

0.0

−8.7

8.7

270.0

VL1 (V)

0.0

991.4

991.4

90.0

VC1 (V)

0.0

−991.4

991.4

270.0

Vr 1 (V)

13.1

0.0

13.1

0.0

VL m 1 (V)

86.9

0.0

86.9

0.0 90.0

VL m 2 (V)

0.0

99.1

99.1

VL2 (V)

868.8

0.0

868.8

0.0

VC2 (V)

−868.8

0.0

868.8

180.0

Vr 2 (V)

0.0

−11.5

11.5

270.0

V2 (V)

0.0

87.6

87.6

90.0

10.1

0.0

10.1

0.0 0.0

Z in1 () Z 2 ()

8.8

0.0

8.8

Z in2 ()

11.4

0.0

11.4

0.0

P1 (W)

991.4

0.0

991.4

0.0

P2 (W)

761.5

θ I2 I1 (I2 /I1 )(◦ )

Pr 1 (W)

130.0

θV2 V1 (V2 /V1 )(◦ )

Pr 2 (W)

99.9

Efficiency (%)

76.8

R Lopt ()

10.1

θ1 (V1 /I1 )(◦ ) )(◦ )

θ2 (V2 / − I2   θ Zin2 VL m2 / − I2 (◦ )

270.0 90.0 0.0 0.0 0.0

calculation conditions. Table 5.6 list the calculations for magnetic resonance coupling. Figure 5.24 shows the phasor diagram of the calculations. The diagram uses V 1 as reference to draw the vectors, which have scale 10 (i.e., 1 V = 10 A) due to the small current compared to voltage. The efficiency notably improves compared to the non-resonant circuit (N–N), achieving ηmax of 76.8% and being equivalent to the secondary-side resonant circuit (N–S). The optimal load is also the same, RLopt = 10.1 . The high efficiency is caused by the resonance of L 2 and C 2 on the secondary side, making the input power factor on the secondary side, cosθ Zin2 , to reach 1. In other words, Z in2 becomes purely resistive, as shown in the phasor diagram, V Lm2 // I 2. The input power factor on the primary side, cosθ 1 , also becomes 1, indicating a large current flow on this side to achieve high power. This can also be verified from I 1 //V 1 in the phasor diagram. Hence, the resonance conditions of L 1 and C 1 exactly match the input power factor of 1 on the primary side. This differs from the primary-side resonant circuit (S–N), as the converted secondary impedance Z 2  is purely resistive, and the primary-side input efficiency becomes 1 simply by canceling primary-side inductance L 1 with

146

5 Comparison Between Electromagnetic Induction and Magnetic …

(a) General view (S–S)

(b) Primary-side voltage, enlarged (c) Secondary-side voltage, enlarged Fig. 5.24 Phasor diagram of magnetic resonance coupling (S–S)

primary-side resonant capacitance C 1 . In this case, the ratio of I 1 and I 2 obtained from Eq. (5.82) is approximately 1.1. After confirming the mechanism of magnetic resonance coupling, we determine the conditions to achieve maximum efficiency. Figure 5.25 shows the load optimization and loss ratio, with the values being equal to those shown in Fig. 5.15b from the secondary-side resonant circuit. Varying the load only affects efficiency, and thus only the amplitude but not the phase varies in the phasor diagram. In other words, the length of the arrow changes but its direction remains fixed. Equation (5.94) shows that if the load is very small, the power ratio becomes dominated by loss Pr2 caused by the internal resistance on the secondary side, at ratio r 2 :RL . On the other hand, a very large load increases the ratio of loss Pr1 caused by the internal resistance on the primary side. The largest ratio of power consumed by the load (P2 ), and thus the maximum efficiency, occurs at optimal load RLopt = 10.1 . Moreover, θ V 2V 1 = 90° during S–S magnetic resonance coupling. As described in Sect. 4.2.3, a constant current characterizes the S–S circuit. As this phenomenon is difficult to verify without a given level of Q, when we draw a

5.5 Circuit of Magnetic Resonance Coupling (S–S)

147

100% 90%

Power ratio [%]

80% 70% 60% 50%

PRL

40%

Pr2

30%

Pr1

20% 10% 0%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

RL [Ω]

Fig. 5.25 Load optimization and loss ratio (S–S)

I2 [A]

Fig. 5.26 Constant current characteristic of S–S circuit 12 10 8 6 4 2 0

k=0.1

k=0.2

k=0.4

k=0.5

0

50

k=0.3

100 150 200

RL [Ω]

graph of I 2 assuming r 1 = r 2 = 0.1 , we can verify that I 2 is almost constant near the optimal load, as shown in Fig. 5.26, where the diamond markers indicate the optimal load per coupling coefficient. However, if the region has weak coupling or large internal resistance, the constant current is not maintained.

5.6 Circuit of Magnetic Resonance Coupling (S–P) In this section, we examine the S–P circuit for magnetic resonance coupling, which is shown in Fig. 5.1e [1, 3–6].

148

5 Comparison Between Electromagnetic Induction and Magnetic …

5.6.1 Verification of Equivalent Circuit (S–P) Figure 5.27 shows the extended T-type equivalent circuit, and Fig. 5.28 shows the equivalent circuit of the secondary impedance conversion and induced electromotive force. In addition, Fig. 5.29 shows the secondary-side capacitor and load of Z in2 as a series conversion, and Fig. 5.30 shows the converted secondary impedance circuit, Z 2  , during resonance, and the overall circuit incorporating Z 2  during resonance. Equation (5.100) represents impedance Z CR comprising the secondary-side resonant capacitor and load. Its real part is denoted RCR and given by Eq. (5.101), and

(a) General operation

(b) During resonance Fig. 5.27 Extended T-type equivalent circuit (S–P)

Fig. 5.28 Equivalent circuit (S–P) of secondary-side conversion and induced electromotive force

5.6 Circuit of Magnetic Resonance Coupling (S–P)

149

Fig. 5.29 Serial conversion (S–P) of secondary-side capacitor and load in equivalent circuit of induced electromotive force

(b) Overall circuit

(a) Converted secondary impedance

Fig. 5.30 Circuit incorporating Z 2  during resonance (S–P)

the imaginary part is denoted as −jX CR and given by Eq. (5.102). Both RCR and RL are almost inversely proportional. Z in2 is then expressed as Eq. (5.103), Z 2  as Eq. (5.104), and Z in1 as Eq. (5.105). From Eq. (5.104), Z 2  remains purely resistive if the load is a pure resistance. Z C R = R L //

R L · jω1C2 1 R L − jω C2 R L2 = = = RC R − j X C R 1 jω C2 1 + ω2 C22 R L2 R L + jω C2

(5.100)

RL 1 + ω2 C22 R L2

(5.101)

RC R ≡

− j XC R ≡

− jω C2 R L2 1 + ω2 C22 R L2

Z in2 = jω L 2 + r2 + RC R − j X C R 

Z2 =

jωL m I2 (ω L m )2 = I1 Z in2

(5.102) (5.103) (5.104)

150

5 Comparison Between Electromagnetic Induction and Magnetic …

Z in1 = r1 + jωL 1 +

1  + Z2 jω C2

(5.105)

Based on the equivalent circuit in Fig. 5.1e, Eqs. (5.106) and (5.107), which represent the voltages on the primary and secondary sides, become Eqs. (5.108) and (5.109). Thus, currents I 1 and I 2 can be calculated using Eqs. (5.110) and (5.111). V1 = VL11 + VC1 + Vr 1 + VLm1

(5.106)

VLm2 = VL22 + Vr 2 + V2

(5.107)

V1 = jωL 1 I1 +

1 I1 + I1r1 + jω L m I2 jωC1

0 = jωL 2 I2 + I2 r2 + I2 Z C R + jω L m I1  I1 = r1 + j ωL 1 −

r2 + R L + j(ωL 2 − X C R ) 

V1 1 {r2 + R L + j(ωL 2 − X C R )} + ω2 L 2m ωC1

 I2 = − r1 + j ωL 1 −

1 ωC1

jωL m 

V1 {r2 + R L + j(ωL 2 − X C R )} + ω2 L 2m

(5.108) (5.109) (5.110)

(5.111)

The ratio of I 1 and I 2 can be expressed as I1 r2 + R L + j(ωL 2 − X C R ) = −I2 jωL m

(5.112)

Let us set the resonance conditions to be similar to those of the S–S circuit: jωL 2 +

1 =0 jωC2

1 =0 jωC1   1 1 ω0 = ω1 = = ω2 = L 1 C1 L 2 C2 VL11 + VC1 = jωL 1 +

(5.113) (5.114)

(5.115)

Then, there is a mismatch of the resonance conditions on the primary and secondary sides. The problem arises from the inability of resonance on the secondary side in Eq. (5.113). Like the S–S circuit, to operate the S–P circuit under magnetic resonance coupling, resonance conditions must be established on the secondary side. Let us consider a conditional equation for this purpose. As shown in Fig. 5.29 and the conversion of

5.6 Circuit of Magnetic Resonance Coupling (S–P)

151

parallel-to-serial connection in Eq. (5.100), voltage V 2 comprises RCR and −jX CR , as shown in Eq. (5.116). Thus, Eq. (5.117) can be derived from Eq. (5.107). Hence, as only the imaginary component should be 0, Eq. (5.118) becomes the resonance condition. From Eqs. (5.102), (5.116), and (5.118), the resonant angular frequency on the secondary side is obtained from Eq. (5.119). As described below, Eq. (5.114) becomes the resonance condition on the primary side. In the S–P circuit, the resonance frequencies match on the primary and secondary sides, but the conditional equations for the resonance frequency are collectively expressed as Eq. (5.120). V2 = VRC R + VXC R = I2 RC R − I2 j X C R

(5.116)

−VLm2 = VL22 + Vr 2 + VRC R + VXC R

(5.117)

VL22 + VXC R = 0

(5.118)





2 1 C2 R L  

2 1 1 1 = ω2 = − ω0 = ω1 = L 1 C1 L 2 C2 C2 R L ω2 =

1 − L 2 C2

(5.119)

(5.120)

When these resonance conditions are satisfied, currents I 1 and I 2 can be, respectively, calculated as I1 =

r 2 + RC R V1 r1 (r2 + RC R ) + ω2 L 2m

I2 = −

jωL m V1 r1 (r2 + RC R ) + ω2 L 2m

(5.121) (5.122)

Given that the ratio of I 1 and I 2 can be expressed by Eq. (5.112), it becomes the following equation during resonance, resembling Eq. (5.88) for the S–S circuit: r 2 + RC R I1 = −I2 jωL m

(5.123)

Hence, Eq. (5.13) defines induced electromotive force V Lm2 on the secondary side. Therefore, substituting Eq. (5.121) for current I 1 on the primary side gives the following equation: VLm2 =

jωL m (r2 + RC R ) V1 r1 (r2 + RC R ) + ω2 L 2m

Based on Eqs. (5.116) and (5.107), V Lm2 is expressed as

(5.124)

152

5 Comparison Between Electromagnetic Induction and Magnetic …

VLm2 = Vr 2 + VRC R

(5.125)

In other words, similar to the S–S circuit, input impedance Z in2 on the secondary side becomes a pure resistance during resonance. In this case, the input power factor on the secondary side is 1. This can be seen from the following equation derived from Eqs. (5.121) and (5.122): Z in2 =

VLm2 jωL m I1 = = r 2 + RC R −I2 −I2

(5.126)

The value of V Lm2 differs from that in the S–S circuit, but the secondary side in Eq. (5.126) remains a pure resistance, and the input power factor on the secondary side is still 1.

5.6.2 Efficiency and Power (S–P) Next, we consider efficiency. From Eqs. (5.3)–(5.5), (5.109), and (5.110), which do not use resonance conditions, the power ratio and efficiency can be, respectively, expressed as   Pr 1 : Pr 2 : P2 = (r2 + R L )2 + (ω L 2 − X C R )2 r1 : (ωL m )2 r2 : (ωL m )2 RC R (5.127) (ωL m )2 RC R  η=  (r2 + R L )2 + (ωL 2 − X C R )2 r1 + (ωL m )2 r2 + (ωL m )2 RC R

(5.128)

Under resonance condition of Eq. (5.120), the power ratio and efficiency are expressed as Pr 1 : Pr 2 : P2 = (r2 + RC R )2 r1 : (ωL m )2 r2 : (ωL m )2 RC R η=

(ωL m )2 RC R (r2 + RC R ) r1 + (ωL m )2 r2 + (ωL m )2 RC R 2

(5.129) (5.130)

In other words, the efficiency of magnetic resonance coupling in an S–P circuit matches the efficiency in Eq. (5.95) of an S–S circuit, where RL is replaced by RCR . Therefore, if the resonance frequency conditions in Eq. (5.119) are satisfied, we can consider the same case as the S–S circuit. The condition in Eq. (5.132) of the optimal load for maximum efficiency can be obtained from Eqs. (5.130) and (5.131). Substituting this back into Eq. (5.130) gives the maximum efficiency. Furthermore, we obtain RL and C 2 from the resonance condition in Eq. (5.120) and the expression of RCR in Eq. (5.101). When RCR satisfies the optimal load condition for maximum

5.6 Circuit of Magnetic Resonance Coupling (S–P)

153

efficiency, RL and C 2 , respectively, become the conditions in Eq. (5.133) for optimal load RLopt and Eq. (5.134) for the optimal secondary-side resonant capacitor, C 2opt .

RC Ropt

∂η =0 ∂ RC R  r2 (ωL m )2 = r22 + r1

R Lopt = C2opt =

(5.131)

(5.132)

RC2 R + (ωL 2 )2 RC R

(5.133)

L2 + (ωL 2 )2

(5.134)

RC2 R

Next, we consider power. From Eq. (5.110), the input impedance on the primary side becomes 

 r1 + j ωL 1 − ωC1 1 {r2 + R L + j(ωL 2 − X C R )} + ω2 L 2m V1 Z in1 = = I1 r2 + R L + j(ωL 2 − X C R ) (5.135) This impedance based on the resonance condition in Eq. (5.120) or (5.121) is given by Z in1 =

V1 r1 (r2 + RC R ) + ω2 L 2m = I1 r 2 + RC R

(5.136)

Hence, this is a pure resistance when viewed from the primary side, that is, the input power factor on the primary side becomes 1. Like in the S–S circuit, the input impedance on the secondary side becomes purely resistive in the S–P circuit, and the input power factor on the secondary side is 1 during resonance on this side. Therefore, as the conditions are the same as those in the S–S circuit, the same considerations can be used. From Eq. (5.104), if the load is purely resistive, Z 2  remains a pure resistance. In other words, the phase does not change even through a K-inverter (see Sect. 9.2) that expresses the coupling involving L m , and thus Z 2  remains a pure resistance. Therefore, if primary-side resonance is achieved under these conditions, the power factor becomes 1 for the entire circuit. As the overall power factor is 1, the circuit can be operated with high power. Thus, the S–P circuit, like the S–S circuit, enables high-efficiency and high-power operation under magnetic resonance coupling even with large air gaps and shifts in positioning.

154

5 Comparison Between Electromagnetic Induction and Magnetic …

5.6.3 Calculations for Magnetic Resonance Coupling (S–P Circuit) In this section, we use calculation results to analyze magnetic resonance coupling and create a graph based on the optimal load values to assess the calculation conditions. Table 5.7 lists the calculation results for the primary-side resonance conditions. Figure 5.31 shows the phasor diagram of the calculations, where V 1 is the reference to draw the vectors, and a scale of 10 (i.e., 1 V = 10 A) is used due to the small current. First, efficiency reaches the same value as in the S–S circuit. However, the optimal load is approximately 100 times, and roughly (1/k)2 times, that of the S–S circuit, Table 5.7 Calculated values for magnetic resonance coupling S–P (P2, Pr1, Pr2 are purely resistive with the Im component being 0, and abs correspond to Re values) Re.

Im.

ABS

θ(◦ )

I1 (A)

9.9

0.0

9.9

0.0

I2 (A)

0.0

−8.7

8.7

270.0

IC (A)

−0.9

8.6

8.6

95.8

I R (A)

0.9

0.1

0.9

5.8

VL1 (V)

0.0

991.4

991.4

90.0

VC1 (V)

0.0

−991.4

991.4

270.0

Vr 1 (V)

13.1

0.0

13.1

0.0

VL m 1 (V)

86.9

0.0

86.9

0.0 90.0

VL m 2 (V)

0.0

99.1

99.1

VL2 (V)

868.8

0.0

868.8

0.0

Vr 2 (V)

0.0

−11.5

11.5

270.0

V RC R (V)

0.0

87.6

87.6

90.0

V XC R (V)

868.8

0.0

868.8

0.0

V2 (V)

868.8

87.6

873.3

5.8

10.1

0.0

10.1

0.0

8.8

0.0

8.8

0.0 0.0

Z in1 () Z 2 () Z in2 ()

11.4

0.0

11.4

P1 (W)

991.4

0.0

991.4

P2 (W)

761.5

θ I2 I1 (I2 /I1 )(◦ )

Pr 1 (W)

130.0

θV2 V1 (V2 /V1 )(◦ )

Pr 2 (W)

99.9

θ1 (V1 /I1 )(◦ )

Efficiency (%)

76.8

θ2 (V2 / − I2 )(◦ )   θ Zin2 VL m2 / − I2 (◦ )

R Lopt ()

1001.4

RC R ()

10.1

C2 (nF)

15.8

X C R ()

0.0 270.0 5.8 0.0 275.8 0.0 −100

5.6 Circuit of Magnetic Resonance Coupling (S–P)

155

(a) General view (S–P)

(b) Primary-side voltage, enlarged (c) Secondary-side voltage, enlarged Fig. 5.31 Phasor diagram of secondary-side resonant circuit (S–P)

being RLopt = 1001.4 . The high efficiency is caused by the resonance of −X CR , which includes L 2 , C 2 , and RL on the secondary side, making input power factor cosθ Zin2 on the secondary side equal to 1. In other words, Z in2 becomes a pure resistance, as verified from V Lm2 //I 2 in the phasor diagram. Comparing the currents I 2 and I 1 flowing through equivalent series resistance RCR , we confirm that I 1 is approximately 1.1 I 2 by using Eq. (5.123). We also confirm that equivalent series resistance RCR is equal to optimal load RLopt in the S–S circuit during magnetic resonance coupling. Input power factor cosθ 1 on the primary side also becomes 1, indicating a large current flow on this side to achieve high power. This can be verified from I 1 //V 1 in the phasor diagram. Hence, the resonance conditions of L 1 and C 1 exactly match the input power factor of 1 on the primary side. As the converted secondary impedance, Z 2  , is a pure resistance, primary-side inductance L 1 simply should be canceled by primary-side resonant capacitance C 1 . In this case, the primary side operates as in the

156

5 Comparison Between Electromagnetic Induction and Magnetic …

Fig. 5.32 Load optimization and loss ratio (S–P)

S–S circuit. Therefore, we confirm the mechanism of magnetic resonance coupling in the S–P circuit. Next, we determine the conditions to achieve maximum efficiency. Figure 5.32a shows the load optimization and loss ratio. Like the S–S circuit, load variations only affect efficiency, as the amplitude but not the phase varies in the phasor diagram. Thus, except for V 2 , the length of the arrow changes but its direction remains fixed. Equation (5.129) shows that if RCR is very small, the power ratio becomes dominated by loss Pr2 caused by the internal resistance on the secondary side, at ratio r 2 :RCR . On the other hand, if RCR is very large, the ratio of loss Pr1 increases by the internal resistance on the primary side. At optimal load RLopt = 1001.4 , RCR = 10.1 , resulting in the highest ratio of power P2 consumed by the load and thus maximum efficiency. In the S–P circuit, V RCR has a 90° phase difference with respect to V 1 . It holds for both the S–S and S–P circuits that, rather than ω1 = √ L1 C = ω2 = √ 1 , L 2 C2

1

1

cosθ Zin1 = cosθ Zin2 = 1, and f 0 = f 1 = f 2 . Hence, the following is true for

S–P:



 ω0 = ω1 =

1 = ω2 = L 1 C1

1 − L 2 C2



1 C2 R L

2 (5.137)

Although we have stated that considering cosθ Zin2 = 1 is correct for the S–P circuit in magnetic resonance coupling, as shown in Fig. 5.32b, ω2 = √ L1 C does not 2

2

produce a notable difference, and thus ω1 = √ L1 C = ω2 = √ L1 C does not affect 1 1 2 2 the analysis in practice. Moreover, this consideration provides advantages such as ease in design, because ω2 does not depend on RL . Figure 5.25 shows the load and efficiency characteristics of the S–S circuit, and Fig. 5.32 shows those characteristics of the S–P circuit. Comparing the figures, the magnitude of loss characteristics with respect to load resistance is reversed. As the S–P circuit can be considered as a modification of the S–S circuit, RCR corresponds to RL , that is, a large RL in the S–S circuit corresponds to a large RCR in the S– P circuit. On the other hand, from Eq. (5.101) and Fig. 5.33, if we consider the

5.6 Circuit of Magnetic Resonance Coupling (S–P)

157

Fig. 5.33 Relationship between RCR and RL

k=0.05 k=0.1 k=0.3

Fig. 5.34 Constant voltage from S–P circuit

k=0.08 k=0.2

2500

V2 [V]

2000 1500 1000 500 0 200

1200

RL [Ω]

2200

inversely proportional relationship between RCR and RL in the S–P circuit, a large RCR corresponds to a small RL , or the actual load of the S–P circuit. Therefore, a large RCR reduces Pr2 and increases Pr1 . Likewise, a small RCR corresponds to a large RL , the actual load of the S–P circuit, and Pr2 increases while Pr1 decreases. Therefore, the magnitude of loss characteristics for load resistance is the inverse when comparing the S–S and S–P circuits. Although the S–P circuit provides a constant voltage, this is difficult to verify without a large Q value. Thus, when we assume r 1 = r 2 = 0.1  and determine V 2 , we can verify that voltage V 2 is almost constant even near the optimal load, as shown in Fig. 5.34, where each diamond marker indicates the optimal load per coupling coefficient.

5.7 Comparison Summary of Five Types of Circuits For 100 V input and resonance frequency f = f 0 , Fig. 5.35 shows the efficiency, and Fig. 5.36 shows the input power and power consumed by load of the five evaluated circuits. We verify that N–N and S–N have low efficiency, whereas N–S, S–S, and S–P have high efficiency. In addition, S–N, S–S, and S–P achieve high power, but only the S–S and S–P circuits using magnetic resonance coupling achieve both high efficiency and power.

5 Comparison Between Electromagnetic Induction and Magnetic …

Efficiency [%]

158

90 80 70 60 50 40 30 20 10 0

N-N

N-S

S-N

S-S

S-P

Circuit topology Fig. 5.35 Efficiency of five evaluated circuits

P1

P2

6000

10

5000

8

4000

P2 [W]

P1 [W]

12

6 4 2

P1

P2

3000 2000 1000

0

0 N-N N-S Circuit topology

S-N S-S S-P Circuit topology

Fig. 5.36 P1 (input power) and P2 (power consumed by load) of five evaluated circuits

5.8 Evaluation and Transition Across Four Types of Circuits Along X 1 and X 2 Axes The basic comparison in Sect. 5.7 of the N–N, S–N, N–S, S–S, and S–P circuits allows to omit the S–P circuit, which is simply a modification of the S–S circuit. Throughout the sections above, we derived equations and thoroughly evaluated the N–N, S–N, N–S, and S–S circuits. In this section, we verify the transition from the N–N to the S–S circuit. We determine the primary and secondary reactances along the X 1 and X 2 axes and verify the transition of the 3-D map drawn over these axes [7]. Table 5.8 summarizes the results from the previous sections in terms of efficiency and power of the four types of circuits (excluding S–P) under a large air gap (i.e., weak coupling), and Fig. 5.37 shows the changes across the circuits, emphasizing resonant capacitor C 1 inserted on the primary side and resonant capacitor C 2 inserted on the secondary side. Power increases with the insertion of C 1 , and the increase is small with the insertion of C 2 , whose main effect is increasing the efficiency. Figure 5.38

5.8 Evaluation and Transition Across Four Types …

159

Table 5.8 Efficiency and power of four types of circuit (excluding S–P) according to resonance η

P2

C1

C2

N–N

Small

Low

-

-

N–S

Max

Low

-

✔

S–N

Small

High

✔

-

S–S

Max

High

✔

✔

Fig. 5.37 Role of primary- and secondary-side resonant capacitors

Fig. 5.38 T-type equivalent circuit for S–S

shows the T-type equivalent circuit for an S–S circuit to summarize the transition from the N–N to the S–S circuit. We further verify the transition in Fig. 5.40, but the concept is easy to visualize from the S–S circuit. From Eqs. (5.139) and (5.11), the imaginary component of the primary-side impedance in the S–S circuit, that is,

160

5 Comparison Between Electromagnetic Induction and Magnetic …

the reactance is X 1 . Similarly, the reactance on the secondary side is X 2 , as shown in Eq. (5.140), and the reactance values comprise either L 1 and C 1 or L 2 and C 2 . Assuming ωL 1 = ωL 2 = 100  (i.e., the same coil as in the previous sections), we consider the resonant capacitor. When C 1 is set to infinity (i.e., no capacitor exists on the primary side), it becomes conductive, and Eq. (5.139) becomes X 1 = ωL 1 . Similarly, when C 2 is set to infinity, Eq. (5.140) becomes X 2 = ωL 2 . In other words, if both resonant capacitors are set to infinity, an N–N circuit is obtained. Thus, the transition from the N–N to the S–S circuit occurs by reducing the values of the C 1 and C 2 resonant capacitors toward an open circuit. For example, in the transition from the N–N to the N–S circuit, C 2 and C 1 should be inserted such that X 2 = 0 and X 1 = 0, respectively, in the S–S circuit. X 1 = ωL 1 −

1 ωC1

(5.138)

X 2 = ωL 2 −

1 ωC2

(5.139)

Equation (5.93), which represents the S–S circuit without resonance, is suited as general equation for the transition from N–N to S–S. We reprint this equation as Eq. (5.140). η= 

 (r2 + R L )2 + ωL 2 −

(ωL m )2 R L 2  1 r1 + (ωL m )2 r2 + (ωL m )2 R L ωC2

(5.140)

Moreover, Eq. (5.47) is the conditional equation of the optimal load for maximum efficiency. We reprint this equation as Eq. (5.141), which is adjusted to obtain the maximum efficiency under the conditions of each circuit topology.  R Lopt =

r22 +

r2 (ωL m )2 1 2 + ωL 2 − r1 ωC2

(5.141)

These equations were derived in Sects. 5.3 and 5.5, and power equations can be derived analogously, but we omit them in this section. As these equations lack C 1 , the primary capacitor does not impact efficiency, which is only affected by C 2 . By varying C 1 and C 2 , we can calculate not only the parameters for N–N, N–S, S–N, and S–S, but also their transitions. Regions with a large air gap often dip below k = 0.10, but for explanation, we first consider k = 0.50 and show coupling coefficient k according to air gap g in Fig. 5.39. Figure 5.40 illustrates the transition from the N–N to the S–S circuit for transmitted power P1 , received power P2 , power factor cosθ Zin1 on the primary side, efficiency η, and power factor cosθ Zin2 on the secondary side for k = 0.5. The right side of the graphs, where X 1 = 100  corresponds to the absence of C 1 , and their upper side, where X 2 = 100  corresponds to the absence of C 2 .

5.8 Evaluation and Transition Across Four Types …

161

1

Fig. 5.39 Coupling coefficient k versus air gap g

0.8

k

0.6 0.4 0.2 0

0

50

100

150

200

g [mm]

The Z-axis in each figure corresponds to P1 , P2 , cosθ Zin1 , η, and cosθ Zin2 (Fig. 5.40a–e, respectively). The load is optimal, RLopt , yielding the maximum efficiency, which depends on the X 2 axis, that is, C 2 . A large amount of power is transmitted when power factor cosθ Zin1 on the primary side is 1. For example, if X 2 = 100 , high power does not necessarily occur at X 1 = 0 . Instead, high power is located away from origin of the X 1 axis, where power factor cosθ Zin1 is 1, as described above. Moreover, the maximum efficiency is achieved when power factor cosθ Zin2 on the secondary side is 1. The power factor on the secondary side, which is the condition that determines efficiency, is 1 in the N–S and S–S circuits. However, in the N–S circuit, the power factor on the primary side is far from 1, indicating low power. Therefore, the S–S circuit during magnetic resonance coupling provides higher efficiency and power. Furthermore, the power factor becomes 1 on the X 1 -axis only when X 2 = 0 . In other words, X 1 = X 2 = 0 , which corresponds to the S–S circuit. Efficiency and power substantially change depending on resonance, as well as on other air gaps but with different scales. Figures 5.41 and 5.42, respectively, show the results at k = 0.02 and k = 0.1. Although this chapter mainly focused on large air gaps, the conditions change with very strong coupling (k ≈ 1). First, k ≈ 1 is not achieved by simple contact but by devices such as transformers designed to maximally reduce magnetic flux leakage, in which no air gaps exist. Hence, there is high efficiency in all regions as shown in Fig. 5.43. Therefore, the design concept is different from a system with large air gaps. As efficiency is high, the secondary-side capacitor is unnecessary. Next, it is desirable to have a circuit topology that yields high power with few additional components. The topology that retrieves the highest power is S–N under overall resonance conditions, followed by S–S, N–N, and N–S. The S–S circuit is not preferable in this case as it contains several components, being inferior to the S–N circuit in this sense. Similarly, topology N–S is not desirable because it provides less power than N–N. Therefore, under no air gaps in circuits with close contact and very strong coupling, the S–N circuit can be adopted to transmit high power, whereas the N–N circuit minimizes the number of components.

162

5 Comparison Between Electromagnetic Induction and Magnetic …

Fig. 5.40 Parameters from N–N, N–S, S–N, and S–S circuits for k = 0.5

(a) Transmitted power P1

(b) Received power P2

(c) Primary-side power factor cosθZin1

(d) Efficiency η

(e) Secondary-side power factor cosθZin2

5.8 Evaluation and Transition Across Four Types …

163

Fig. 5.41 Transition from N–N to S–S circuit along X 1 - and X 2 -axes for k = 0.02

(a) Transmitted power P1

(b) Received power P2

(c) Primary-side power factor cosθZin1

(d) Efficiency η

(e) Secondary-side power factor cosθZin2

164

5 Comparison Between Electromagnetic Induction and Magnetic …

Fig. 5.42 Parameters from N–N, N–S, S–N, and S–S circuits for k = 0.1

(a) Transmitted power P1

(b) Received power P2

(c) Primary-side power factor cosθZin1

(d) Efficiency η

(e) Secondary-side power factor cosθZin2

5.8 Evaluation and Transition Across Four Types …

165

Fig. 5.43 Transition from N–N to S–S circuit along X 1 - and X 2 -axes for k = 1.0

(a) Transmitted power P1

(b) Received power P2

(c) Primary-side power factor cosθZin1

(d) Efficiency η

(e) Secondary-side power factor cosθZin2

166

5 Comparison Between Electromagnetic Induction and Magnetic …

A deeper understanding of the N–N, S–N, N–S, and S–S circuits requires comparisons along the k- and RL-axes, as well as the recognition of characteristics along the f-axis.

5.9 Comparison of Four Types of Circuits During Magnetic Flux Distribution In this section, we show the magnetic flux distribution in the four types of circuits. The distribution takes different shapes, as the efficiency is equivalent in the N–N and S–N circuits, and in the N–S and S–S circuits [8]. Figure 5.44 shows the basic magnetic flux distribution in the S–S circuit with the resonant capacitor included on both the primary and secondary sides. Note that the removal of any resonant capacitor converts this circuit into other topologies. Equations (2.29)–(2.39) in Sect. 2.2.4 correspond to the magnetic flux of the N– N circuit. They are equal except for the resonant capacitor. The following must be considered for the resonant capacitors added to the primary and secondary sides:

Fig. 5.44 Overview of magnetic flux distribution (S–S)

vC1 =

1 I1 = −v L11 jωC1

(5.142)

vC2 =

1 I2 = −v L22 jωC2

(5.143)

5.9 Comparison of Four Types of Circuits During Magnetic Flux …

167

Hence, the self-induced voltage can be canceled by the voltage of the resonant capacitor. In addition, we consider the magnetic flux distribution. From Eqs. (5.35)–(5.38) in Sect. 2.2.4 and given Φ = L I , the relationship between each magnetic flux, inductance, and current can be expressed as Φ11 + Φ21 = L 1 I1

(5.144)

Φ12 = L m I2

(5.145)

Φ22 + Φ12 = L 2 I2

(5.146)

Φ21 = L m I1

(5.147)

Therefore, Φ11 and Φ22 are given by Φ11 = L 1 I1 − L m I1

(5.148)

Φ22 = L 2 I2 − L m I2

(5.149)

Equation (5.150) represents the relationship between primary magnetic flux Φm and components Φ21 and Φ12 , where Φ21 (Φ12 ) is the magnetic flux produced by the current on the primary (secondary) side and links to the secondary (primary) side. Φm = Φ21 + Φ12

(5.150)

The ratio of currents on the primary and secondary sides is given by Eq. (5.151) for the S–S to the N–S circuit, and by Eq. (5.152) for the S–N to the N–N circuit. r2 + R L I1 = I2 jωL m

(5.151)

I1 r2 + R L + jωL 2 = I2 jωL m

(5.152)

As above, we calculate the relationships among the magnetic flux generated in the coils of each circuit topology, current, and voltage, whose results are listed in Table 5.9 at optimal load R Lopt for maximum efficiency per topology. The relationship among the magnetic flux generated in the N–N, S–N, N–S, and S–S circuits with current and voltage are explained as follows. In the N–N circuit, I1 , I2 , VL1 , and VL2 are small, and the magnetic fields produced by the primary and secondary coils are also small. In addition, both the current and magnetic flux are larger on the primary side as the current ratio in Eq. (5.152) is

168

5 Comparison Between Electromagnetic Induction and Magnetic …

Table 5.9 Calculated parameters for N–N, S–N, N–S, and S–S circuits (P2 , Pr 1 , Pr 2 are purely resistive with the Im component being 0, and abs correspond to Re values) (a) N–N Re.

Im.

ABS

θ(◦ )

I1 (A)

0.0

−1.0

1.0

271.0

I2 (A)

−0.1

0.0

0.1

136.6

VL11 (V)

100.5

1.8

100.5

1.0

0.0

−1.3

1.3

271.0

−0.5

−0.5

0.7

226.6

Vr 1 (V) VL m 1 (V) VL m 2 (V)

10.0

0.2

10.0

1.0

VL22 (V)

−4.8

−5.1

7.0

226.6

Vr 2 (V)

−0.1

0.1

0.1

136.6

V2 (V)

5.1

−4.9

7.1

316.6

Φ11 (μW b)

2.6

−143.9

143.9

271.0

Φ21 (μW b)

0.3

−16.0

16.0

271.0

Φ22 (μW b)

−7.3

6.9

10.1

136.6

Φ12 (μW b)

−0.8

0.8

1.1

136.6

Φm (μW b)

−0.5

−15.2

15.2

268.0

100.5

100.5

89.0

P1 (W)

1.8

P2 (W)

0.5

η(%)

Pr 1 (W)

1.3

R Lopt ()

Pr 2 (W)

0.0

27.1 100.5

(b) S–N Re. I1 (A)

Im. 51.1

ABS 13.8

53.0

θ(◦ ) 15.1

−1.8

−3.2

3.7

240.6

−1377.5

5114.9

5297.2

105.1

VC1 (V)

1377.5

−5114.9

5297.2

285.1

Vr 1 (V)

67.7

18.2

70.1

15.1

VL m 1 (V)

32.3

−18.2

37.1

330.6

VL m 2 (V)

−137.8

511.5

529.7

105.1

VL22 (V)

323.3

−182.2

371.2

330.6

Vr 2 (V)

−2.4

−4.3

4.9

240.6

V2 (V)

183.2

325.0

373.0

60.6

Φ11 (μW b)

7326.6

1973.2

7587.6

15.1

Φ21 (μW b)

814.1

219.2

843.1

15.1

Φ22 (μW b)

−261.0

−463.1

531.6

240.6

Φ12 (μW b)

−29.0

−51.5

59.1

240.6

I2 (A) VL11 (V)

(continued)

5.9 Comparison of Four Types of Circuits During Magnetic Flux …

169

Table 5.9 (continued) (b) S–N Re. Φm (μW b)

Im.

ABS

θ(◦ )

785.1

167.8

802.8

12.1

P1 (W)

5114.9

1377.5

5297.2

15.1

P2 (W)

1384.5

η(%)

Pr 1 (W)

3712.1

R Lopt ()

Pr 2 (W)

18.2

27.1 100.5

(c) N–S Re.

Im.

ABS

I1 (A)

0.1

−1.0

θ(◦ )

1.0

275.8 185.8

I2 (A)

−0.9

−0.1

0.9

VL11 (V)

99.0

10.0

99.5

5.8

Vr 1 (V)

0.1

−1.3

1.3

275.8

VL m 1 (V)

0.9

−8.7

8.7

275.8

VL m 2 (V)

9.9

1.0

9.9

5.8

VL22 (V)

8.8

−86.8

87.2

275.8

VC2

−8.8

86.8

87.2

95.8

Vr 2 (V)

−1.1

−0.1

1.2

185.5

8.8

0.9

8.8

5.8

Φ11 (μW b)

14.3

−141.8

142.5

275.8

Φ21 (μW b)

1.6

−15.8

15.8

275.8

Φ22 (μW b)

−124.3

−12.5

124.9

185.8

Φ12 (μW b)

−13.8

−1.4

13.9

185.8

Φm (μW b)

−12.2

−17.1

21.1

234.5

P1 (W)

10.0

99.0

99.5

84.2

P2 (W)

7.7

η(%)

76.8

Pr 1 (W)

1.3

R Lopt ()

10.1

Pr 2 (W)

1.0

V2 (V)

(d) S–S Re.

Im.

ABS

θ(◦ )

I1 (A)

9.9

0.0

9.9

0.0

I2 (A)

0.0

−8.7

8.7

270.0

VL11 (V)

0.0

991.4

991.4

90.0

VC1 (V)

0.0

−991.4

991.4

270.0

Vr 1 (V)

13.1

0.0

13.1

0.0 (continued)

170

5 Comparison Between Electromagnetic Induction and Magnetic …

Table 5.9 (continued) (d) S–S Re.

Im.

ABS

θ(◦ )

VL m 1 (V)

86.9

0.0

86.9

0.0

VL m 2 (V)

0.0

99.1

99.1

90.0

VL22 (V)

868.8

0.0

868.8

0.0

−868.8

0.0

868.8

180.0

Vr 2 (V)

0.0

−11.5

11.5

270.0

V2 (V)

0.0

87.6

87.6

90.0

Φ11 (μW b)

1420.0

0.0

1420.0

0.0

Φ21 (μW b)

157.8

0.0

157.8

0.0

Φ22 (μW b)

0.0

−1244.5

1244.5

270.0

VC2

Φ12 (μW b)

0.0

−138.3

138.3

270.0

Φm (μW b)

157.8

−138.3

209.8

318.8

P1 (W)

991.4

0.0

991.4

0.0

P2 (W)

761.5

η(%)

76.8

Pr 1 (W)

130.0

R Lopt ()

10.1

Pr 2 (W)

99.9

maintained. There is high power on the primary side, low power on the secondary side, and low efficiency. In the S–N circuit, I1 , I2 , VL1 , and VL2 increase, and the magnetic fields generated in the primary and secondary coils increase. However, as the current ratio in Eq. (5.152) is maintained, the current and magnetic flux are larger on the primary side, like in the N–N circuit. Therefore, the power on the primary side is large and that on the secondary side is small, with efficiency remaining low. Compared to N–N, the N–S circuit has higher current on the secondary side due to resonance, and I2 , VL2 are relatively larger than I1 , VL1 , respectively. However, the current ratio in Eq. (5.151) is maintained. At optimal load, I1 ≈ I2 , and hence the currents increase until the primary and secondary sides are nearly equivalent. VL2 also becomes almost equal to VL1 . Therefore, the magnetic flux of the secondary coil increases to nearly the same level as that of the primary coil. Despite the high efficiency, the received power is low, as there is almost no current flowing through the primary and secondary sides. Compared to N–N, the S–S circuit has resonance on the primary and secondary sides, and the currents that flow through these sides are larger. In addition, it maintains the current ratio in Eq. (5.151). Like the N–S circuit, I1 ≈ I2 at optimal load, and hence the currents increase on the primary and secondary sides until they are roughly equal. VL2 also becomes almost equal to VL1 . Therefore, the magnetic flux of the secondary coil increases to nearly the same level as that of the primary coil. The efficiency is high and the received power is high.

5.9 Comparison of Four Types of Circuits During Magnetic Flux …

171

Figure 5.45 shows the magnetic flux distribution, whose maximum strength is 8.3 A/m in the N–N, 844.5 A/m in the S–N, 8.2 A/m in the N–S circuit, and 96.6 A/m in the S–S circuit, with the maximum value being used to normalize each graph. The magnetic flux densities are as those described above. In other words, the magnetic flux on the primary side is large in the N–N and S–N circuits, but it is small in the former and large in the latter. In the N–S and S–S circuits, the magnetic flux on the primary and secondary sides is almost equal because I1 ≈ I2 at optimal load. However, N–S has a small magnetic flux, while S–S has a large magnetic flux. In each circuit topology, as the maximum strength of magnetic flux is normalized, the N–N and S–N circuits have the same magnetic flux distribution because they maintain the current ratio from Eq. (5.152). Likewise, the N–S and S–S circuits have the same magnetic flux distribution because they maintain the current ratio from Eq. (5.151).

(a) N-N

(b) S-N

(c) N-S

(d) S-S

Fig. 5.45 Magnetic flux distribution in four types of circuits

172

5 Comparison Between Electromagnetic Induction and Magnetic …

5.10 Role of Primary Magnetic Flux

1000

80

800

60

600

P2 [W]

100

40 20

200 0 70 80 90 100 110 120 130

Frequency [kHz]

Frequency [kHz]

(a) Efficiency

(b) Power

I1

[A]

400

0 70 80 90 100 110 120 130

I1 [deg]

I2

14 12 10 8 6 4 2 0 70 80 90 100 110 120 130

[deg]

Efficiency [%]

The relationship between circuit topology and magnetic flux has been established in the previous section. In this section, we further analyze the magnetic flux by discussing its role on the primary side for the S–S circuit [8]. In general, primary magnetic flux Φm is assumed to perform magnetic flux coupling, but a more accurate assessment is necessary. In wireless power transmission, magnetic flux Φ21 is an important component of the primary magnetic flux and the source of induced electromotive force. This magnetic flux is produced by passing the primary-side current. Naturally, the current flows through the load on the secondary side, and hence Φ12 causes a decline in voltage on the primary side during power transmission. Therefore, Φ12 is also important to correctly understand this relationship. We focus on two characteristic frequencies below. In magnetic resonance coupling, regions with strong coupling exhibit f m and f e ( f m ≤ f e ), two peak frequencies for electric power on either side of resonance frequency f o of a single coil. At each peak frequency, a magnetic wall and an electric wall are generated, which are, respectively, called even and odd modes. At frequencies lower than a single resonance frequency, the current approaches coordinated phase, and at higher frequencies, the current approaches antiphase (Fig. 5.46). The

I2 [deg]

180 135 90 45 0 -45 -90 -135 -180 70 80 90 100 110 120 130

Frequency [kHz]

Frequency [kHz]

(c) Current amplitude

(d) Current phase

Fig. 5.46 Efficiency, power, and current distribution at optimal load 10.1  (S–S)

5.10 Role of Primary Magnetic Flux

173

magnetic flux distribution is thus as follows. At low peak frequency f m , the distribution becomes similar to a magnetic wall as the current is similar in phase, and thus the magnetic flux gathers at the center and becomes perpendicular to the plane of symmetry. At high peak frequency f e , the distribution becomes similar to an electric wall as the magnetic flux gathers at the edges and becomes horizontal to the plane of symmetry (Fig. 5.47). A primary magnetic flux seems to exist in f m but not in f e . Hence, f e seems to impede power transmission, but this verification is difficult, because the magnetic flux distribution near the coil is almost entirely leaked, magnetic flux components Φ11 and Φ22 are dominant, and the proportion of primary magnetic flux Φm is small, as shown in Fig. 5.48a. In addition, primary magnetic flux Φm is larger in f m and smaller in f e . Still, Fig. 5.46 (efficiency and power) shows that the parameters are not proportional to the primary magnetic flux, but instead they are proportional to

(a) fm

(b) fe

Fig. 5.47 Magnetic flux distribution in f m and f e

Φ11

Φ22

Φm

Φ21 Φ12

1500

[uWb]

[uWb]

2000

1000 500 0 70 80 90 100 110 120 130

Φm

350 300 250 200 150 100 50 0 70 80 90 100 110 120 130

Frequency [kHz]

Frequency [kHz]

(a) Φ11, Φ22, Φm

(b) Φ21, Φ12, Φm

Fig. 5.48 Magnetic flux distribution (S–S)

174

5 Comparison Between Electromagnetic Induction and Magnetic …

Φ12 and Φ21 regarding power. In other words, the primary magnetic flux can be used to determine whether the phases of Φ12 and Φ21 match, but does not affect power transmission after Φ12 and Φ21 are merged. The same interpretation is obtained from Eqs. (5.145), (5.147), and (5.150). Therefore, the most important parameters for power transmission are Φ12 and Φ21 , which constitute primary magnetic flux Φm . As an extreme case to understand the role of Φ12 , I2 = 0 when load R L is not connected, and thus Φ12 = 0 from Eq. (5.145). When current I2 starts flowing, Φ12 is generated and power transmission occurs. Among them, magnetic flux Φ21 is especially important as it is generated by the induced electromotive force on the secondary side, where power transmission begins. This is because induced electromotive force VLm2 generated on the secondary side becomes the power source for this side.

References 1. T. Imura, Y. Hori, Unified theory of electromagnetic induction and magnetic resonant coupling. Trans. Inst. Electr. Eng. Japan 135(6), 697–710 (2015) 2. T. Matsuzaki, H. Matsuki, Transcutaneous energy-transmitting coils for FES. J. Mag. Soc. Japan 18(2), 663–666 (1994) 3. O.H. Stielau, G.A. Covic, Design of loosely coupled inductive power transfer systems. in Proceedings of the 2000 International Conference on Power System Technology, vol. 1, pp. 85–90 (2000) 4. Y. Nagatsuka, N. Ehara, Y. Kaneko, S. Abe, Efficiency of contactless power transfer systems using series resonant capacitors. in Proceedings of the Japan Industry Applications Society Conference, pp. 2–27 (2009) 5. T. Kai, K. Throngnumchai, A study on receiver circuit topology of non-contact charger for electric vehicle. Trans. Inst. Electr. Eng. Japan D 132(11), 1048–1054 (2012) 6. Y.H. Sohn, B.H. Choi, E.S. Lee, G.C. Lim, G. Cho, C.T. Rim, General unified analyses of two-capacitor inductive power transfer systems: equivalence of current-source SS and SP compensations. IEEE Trans. Power Electron. 30(11), 6030–6045 (2015) 7. T. Imura, Y. Hori, Superiority of Magnetic Resonant Coupling in Electromagnetic Induction. in IEICE Technical Report WPT2015-21, pp. 1–6 (2015) 8. T. Imura, Relation between magnetic flux and magnetic resonant coupling. in IEICE Technical Report WPT2016-9, pp. 1–4 (2016)

Chapter 6

Feature of P–S, P–P, LCL-LCL, and LCC-LCC

We previously discussed the differences between conventional electromagnetic induction and magnetic resonance coupling by comparing five circuits (N–N, N– S, S–N, S–S, and S–P). In this chapter, we describe S–S, S–P, P–S, and P–P, and LCL-S, LCL-P, S-LCL, P-LCL, LCL-LCL, and LCC-LCC, which have resonant capacitors on the transmission and receiving sides. √For a uniform comparison, we shall emphasize formula visibility and use ω = 1/ LC as a resonance condition.

6.1 S–S, S–P, P–S, and P–P First, we shall describe S–S, S–P, P–S, and P–P, which have one resonant capacitor on the transmission side and one on the receiving side (Fig. 6.1) [1, 2]. S indicates a series connection, and P indicates a parallel connection. The power supply for such a circuit is commonly designed using a constant voltage source or a constant current source. However, in a few design methods, the current flowing to the transmitting coil is converted into a constant current rather than using a power supply. Here, we shall mainly describe the instances employing a common constant voltage source. Furthermore, we will discuss a few properties when a constant current source is connected. It should be noted that the constant voltage and constant current mentioned here are those during load fluctuations and not those during gap fluctuations. When describing the load-side constant voltage and constant current during wireless power transmission, the abovementioned assumptions may hold unless otherwise noted. In addition, we shall ignore the internal resistance to simplify the calculation. Although it cannot be ignored when discussing efficiency, it is possible to ascertain tendencies when examining constant voltage and current properties even if they are ignored. In reality, a slight internal resistance does have a significant effect and may cause deviations in constant voltage and current properties, but it is still possible to ascertain the overall tendencies. © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_6

175

176

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

(a) S-S

(b) S-P

(c) P-S

(d) P-P Fig. 6.1 S–S, S–P, P–S, and P–P equivalent circuits

6.1 S–S, S–P, P–S, and P–P

177

Fig. 6.2 S–P circuit realizing an ideal transformer [3]

First, we shall describe S–S and S–P. The maximum efficiency is the same for both S–S and S–P. Additionally, S–S and S–P are characterized by the optimal load required for realizing a maximum efficiency that is low for S–S and high for S– P, which should be considered for each application. In addition, S–S and S–P can realize higher power. Therefore, they are considered to possess magnetic resonance coupling. A constant current appears on the load side of S–S when a constant voltage source is used. Based on the gyrator properties (K-inverter), a constant voltage source is converted into a constant current source because a gyrator exists only at the Tshaped part. This can be explained as a numerical formula as shown in formula (6.1). In addition, when we connect a constant current source, a constant voltage is obtained.      0 − jωL m V2 V1 = (6.1) 1 0 I1 I2 jωL m A constant voltage appears on the load side of S–P when the design is devised using a constant power source. Figure 6.2 shows the S–P circuit in reference literature [3]. Here, the use of formula (6.2) is mentioned. Thus, formula (6.3) is derived, and a constant voltage is obtained. 1 = x p = x0 + x2 ω0 C p        x0 0 V1 V2 V2 = ABCDE = x0 +x2 x0 +x2 0 I1 I2 I2 x0

(6.2)

(6.3)

It should be noted that the air gap is used as a variable for a robust evaluation [3]. Regarding the S–P circuit, when designed with limited conditions, a constant voltage appears. When designing by accounting for the gyrator properties while conscious only of LC vibrations, it behaves as in formula (6.4) even if a constant voltage source

178

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

Fig. 6.3 S–P circuit using LC vibrations

k=0.05 k=0.1 k=0.3

k=0.08 k=0.2

2500

V2 [V]

2000 1500 1000 500 0 200

1200

2200

RL [Ω]

is used. Furthermore, constant voltage does not appear on the load side in an orderly form. Hence, this needs to be studied using a graph. Even if the circuit is designed to be conscious only of LC vibrations, as shown in Fig. 6.3, if the conditions are set, constant voltage may be confirmed to such an extent that it may be considered as a quasiconstant voltage. If the conditions are such that the values of Q are high, then the load values are also in a high range, in the vicinity of the values at which maximum efficiency can be realized. In Fig. 6.3, the values are calculated at Q = 1000, and the diamonds represent optimal loads that can realize maximum efficiency. In this manner, when the constant voltage is demonstrated using set conditions, S–P is known to represent a constant voltage. However, this can be more easily understood in terms of the constant current. When supplementing the constant current source, as in formula (6.5), which is derived from formula (6.4), if a constant current source is connected, then a constant current emerges on the load side in an orderly form, even if it is conscious of only LC vibrations. Thus, C (P or LC) arranged in parallel is characterized as having quasigyrator properties. 

V1 I1



 =

I1 =

Lm L2

0

− jωL m L2 Lm



V2 I2

L2 Lm I2 ⇔ I2 = I1 Lm L2

 (6.4) (6.5)

Next, we confirm P–S and P–P. We shall describe the relationship of a squarewave drive and primary-side P when k = 0 in Sect. 6.4, and a sine-wave drive is considered to be a prerequisite. In this case, both P–S and P–P are naturally the same as S–S and S–P in maximum efficiency [4]. However, when power is designed with regular LC resonance, it exhibits a significantly higher reduction than S–S because the imaginary numbers remain. Hence, an additional L may be included to eliminate the remaining imaginary numbers. When an additional L is included in P–S on the power transmission side, the configuration is referred to as LCL-S [5, 6]. This may also be referred to as LCL-LC. However, from this point onward, things get more complicated. This is because a systematic explanation becomes more difficult owing to the differences in plans and

6.1 S–S, S–P, P–S, and P–P

179

the dependence of the resonance production on the naming and the circuit designer. Regarding the naming, S was previously used when resonant capacitors were in series, and P was used if they were in parallel. However, with an additional L, an expression with only S and P and mentioning a capacitor is no longer possible. On the contrary, if L and C are used, there is no information regarding the series or parallel configuration. Hence, this cannot be understood without reading an explanation. Based on experience, LCL can be imagined as indicating that C is in parallel. In addition, last L indicates inductance of transmitting coil when it is used on transmitting side. However, for LCC, a series or parallel arrangement cannot be understood without observing the circuit diagram. Moreover, the transmitting coil L is not included in the name for LCC. On the other hand, one views LCCL when observing the circuit diagram. In addition, in the case of the previous S–P, we mentioned that there were design methods that were conscious of constant voltage. However, even with the same S–P, if the design is conscious of LC resonance, then the imaginary number components remain on the secondary side, and thus maximum efficiency is not realized [7]. Thus, as the number of circuits increases, it becomes more difficult to comprehend the intention. Moreover, in wireless power transmission, L and C are occasionally added as filters to achieve an output resembling a sine wave because higher harmonics are undesired. Another reason is that the purpose of the circuit components is not clearly specified. Let us return to the discussion of P–S. This circuit is as shown in Formula (6.6). Orderly gyrator properties are not obtained with the P part. Hence, when only LC resonance is considered, a detailed analysis is required to determine if a constant voltage is obtained on the load side when there is a constant voltage source. On the contrary, when we use a constant current source, a constant current is obtained on the load side, as shown in Formula (6.7). 

V1 I1



 =

I1 =

L1 Lm

0

− jωL m Lm L1



V2 I2

Lm L1 I2 ⇔ I2 = I1 L1 Lm

 (6.6) (6.7)

For P–P, when considered in terms of the gyrator properties and only LC resonance is accounted for, it is not possible to achieve either a constant voltage or a constant current. On the contrary, when we add the assumption that the current flowing on the power-transmission-side coil is constant, the conditions are the same as with S–P. In this case, it is the same as the conclusion obtained from S–P when using a constant current source, that is, even if conscious of only the LC resonance, a constant current emerges in an orderly form on the load side. If control is provided such that the current flowing through a coil on the power transmission side is constant, then it is possible to assemble a system by using properties that cannot be grasped simply with S–P and P–P circuit topology.

180

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

Table 6.1 List of CV and CC properties of S–S, S–P, P–S, and P–P S–S

S–P

P–S

P–P

Constant voltage source

CC

–

–

–

Constant current source

CV

CC

CC

–

Table 6.1 shows a summary of S–S, S–P, P–S, and P–S properties discussed up to this point. The method in this table accounts only for LC resonance; hence, it does not include anything that realizes constant voltage (CV) or constant current (CC) using other conditions explained in the chapter.

6.2 LCL and LCC, etc. Here we explain circuits slightly expanded from those that have only one capacitor each on the transmission side and receiving side such as S–S, S–P, P–S, and P– P. LCL-S was briefly touched upon in a previous section, but here we explain it in a complete fashion. Specifically, we will talk about LCL-S, LCL-P, S-LCL, PLCL, LCL-LCL, and LCC-LCC. In general, if the number of passive components is increased, then both losses and costs are increased. Hence, when the above merits are achieved, these circuits are used. There are various ways to interpret LCL, but we describe the most typical example in terms of gyrator properties in which the properties emerging on the load side are either constant voltage or constant current when various circuits are connected to a constant voltage source. When a constant voltage source is used, S–S has constant current, and S–P has constant voltage. Based upon this, we shall describe an LCL circuit. In addition, we exclude LCC, and L 0 = L 1 = L 2 = L’0 . LCL uses a gyrator; hence, it is capable of converting ideal constant voltage to constant current or constant current to constant voltage. Hence, in a configuration such as that of LCL-S in Fig. 6.4, the load is constant voltage. As a stepwise explanation, once the current becomes constant at the LCL part, and as the T-shaped coupling part is also an ideal gyrator, the constant current becomes a constant voltage. In the last S, the properties do not change and become constant voltage. In other words, as

Fig. 6.4 LCL-S

6.2 LCL and LCC, etc.

181

Fig. 6.5 LCL-P

the number of passes through the gyrator is two, there is a return to the original constant voltage. As for formulas, two-stage gyrator properties are shown by Formula (6.8), and as a result, Formula (6.9) is obtained. Constant voltage can be expressed as in Formula (6.10). 

V1 I1



 =

   0 − jωL m V2 0 jωL 1 1 0 jωC1 0 I2 jωL m       L1 0 V1 V2 = Lm Lm I1 I2 0 L1

(6.8)

(6.9)

L1 Lm V2 ⇔ V2 = V1 Lm L1

V1 =

(6.10)

Next, in the case of LCL-P as shown in Fig. 6.5, current is constant when there is a constant voltage source. This is explained almost the same as LCL-S, but at the final P place, constant voltage changes into constant current. As in formula (6.11), there are two gyrator properties. Additionally, at P, there are quasigyrator properties, so it is three times worth. As a result, the gyrator properties are not orderly, but constant current appears as in formulas (6.12) and (6.13). 

V1 I1



 =

0 jωL 1 jωC1 0 

V1 I1



0 1 jωL m



 =

V1 = jω

− jωL m 0

0 jωC2 LLm1



0 jωL 2 jωC2 1   jω LL1 mL 2 V2 Lm I2 L1

L1 L2 Lm IL ⇔ IL = V1 Lm jωL 1 L 2



V2 I2

 (6.11)

(6.12) (6.13)

There is also S-LCL, as shown in Fig. 6.6. In this case, S has no effect, so the first stage has constant voltage. After constant current is reached in a T-shaped equivalent circuit, a constant voltage is found in LCL. As for the formulas, two-stage gyrator properties are shown in Formula (6.14), which results in Formula (6.15). Furthermore, constant voltage can be expressed as in Formula (6.16).

182

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

Fig. 6.6 S-LCL



V1 I1



 =

   − jωL m 0 jωL 0 VL 1 0 jωC2 0 IL jωL m       Lm 0 V1 VL L2 = L I1 IL 0 L m0 0

V1 =

(6.14)

(6.15)

Lm L2 VL ⇔ VL = V1 L2 Lm

(6.16)

Since P-LCL as shown in Fig. 6.7 is cumbersome, it is often not discussed. Ultimately, for the same reasons as P–S, the first P deviates from the gyrator properties. Therefore, if there is an earnest LC resonance, then the properties appearing on the load side will not achieve a constant current when a constant voltage source is used. On the contrary, when there is a constant current source, this P-LCL, by passing from the constant current source through three places (P, T-shaped part, and LCL), is characterized by the constant voltage being maintained. The numerical formulas are presented as formulas (6.17)–(6.19). 

V1 I1



 =

Fig. 6.7 P-LCL

1 jωL 1 jωC1 0



0 1 jωL m

− jωL m 0



0 jωL 0 jωC2 0



V2 I2

 (6.17)

6.2 LCL and LCC, etc.

183

Fig. 6.8 LCL-LCL



V1 I1



 =

I1 = jωC1

Lm L2

jωC1 LLm1

L2

jω L m1 0



V2 I2

 (6.18)

Lm L1 V2 ⇔ V2 = I1 L1 jωC1 L m

(6.19)

Figure 6.8 presents the LCL-LCL. The LCL-LCL is composed of an ideal threestage gyrator such that there is a constant current in the first LCL, constant voltage in the T-shaped equivalent circuit, and constant current in the last LCL. As the P part in P-LCL is replaced with a proper LCL, a constant current is achieved, as shown in formulas (6.20)–(6.22). In addition, if a constant current source is connected, then a constant voltage is achieved. 

V1 I1



 =

0 jωL 0 jωC1 0 

V1 I1



0 1 jωL m



 =

V1 = jω

− jωL m 0



L2

0 jω L m1 Lm jωC1 L 1 0

0 jωL 0 jωC2 0   V2 I2

L 21 Lm I2 ⇔ I2 = V1 Lm jωL 21



V2 I2

 (6.20)

(6.21) (6.22)

If orderly gyrator properties exist, then the power supply may be similarly considered even with a constant current source. For example, as described above, when connecting LCL-LCL to a constant current source, there is a constant voltage in the first LCL, constant current in the T-shaped equivalent circuit, and constant voltage in the last LCL. In addition, there is also a circuit called LCC-LCC [8–10]. When looking at the circuit, since LCC-LCC is symmetrically connected to the power transmission side and power-receiving side, it is also called a double LCC. In addition, a power transmission and receiving coil L is not included in the name. Therefore, on observing the circuit, it is actually LCCL-LCCL (Fig. 6.9). LCC emerges as a derivative of LCL. Inserting C 1S results in the effect of reducing L 0 in LCL and enlarging the power [8]. Since the design method is broadly the same

184

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

Fig. 6.9 Double LCC circuit

Table 6.2 List of LCL-related CV and CC properties LCL-S

LCL-P

S-LCL

P-LCL

LCL-LCL

Double LCC

Constant voltage source

CV

CC

CV

–

CC

CC

Constant current source

CC

–

CC

CV

CV

CV

as with LCL and constitutes a three-stage gyrator, the formula for LCC is formula (6.23). In this case, the resonance conditions are the same in concept. Thinking about the secondary side, since the resonance in three places of the T-shaped circuit of the gyrator is made the same, there is resonance in L’0 and C 2P , and resonance in C 2P and the “L 2 and C 2S series.” Hence, the resonance conditions are as in formula (6.24). The primary side may be considered similarly. Constant voltage and constant current are the same as with LCL, so they have been omitted. 

V1 I1



  0 − jωL m 0 jωL 0 = 1 0 jωC1P 0 jωL m    0 jωL 0 V2 jωC2P 0 I2 

ω0 = 

1 L 0 C2P

=

1 C2P L 2 CC2S2S+C 2P

(6.23) (6.24)

Table 6.2 shows a summary of the LCL and LCC seen to this point. Here, the method of classification considers the gyrator properties and accounts for the LC resonance only.

6.3 Relay Coil and Gyrator Properties Relay (repeater) coils can be explained using the K-inverter theory [11]. This Kinverter theory is the same as with a gyrator; hence, it may similarly be considered

6.3 Relay Coil and Gyrator Properties

185

(a) One relay coil

(b) Two relay coils

Fig. 6.10 Relay coil circuit

for constant voltage and constant current inversion. If a single relay coil is added, resulting in T-shaped equivalent circuits in two places, inversion occurs twice. Thus, when a constant voltage source is connected, a constant voltage is obtained (formula (6.25), (Fig. 6.10)). 

V1 I1



 =

0 1 jωL m

− jωL m 0



0 1 jωL m

− jωL m 0



V2 I2

 (6.25)

If two relay coils are added, making T-shaped equivalent circuits in three places, then inversion occurs three times. Thus, when a constant voltage source is connected, constant current results [Formula (6.26)]. 

V1 I1



 =

0 1 jωL m

− jωL m 0



0 1 jωL m

− jωL m 0



0 1 jωL m

− jωL m 0



V2 I2

 (6.26)

Below, the number of connection stages alternately changes between odd numbers and even numbers. Table 6.3 shows a summary of relay-coil CV and CC properties. Table 6.3 Relay-coil CV and CC properties

1 relay (odd number)

2 relays (even number)

Constant voltage source

CV

CC

Constant current source

CC

CV

186

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC

6.4 k = 0 Properties As in Sect. 6.2, it is possible to classify LCL circuits in terms of constant voltage and constant current properties. On the other hand, below we describe the merits of using LCL-LCL, in which the same properties as S–S are obtained even if the number of parts is increased. For example, when considering charging for moving EV (electric vehicle), the load suddenly appears and disappears in view of road power supply side. A similar phenomenon appears when placing a smartphone on a charging pad, even if this is not as fast as charging when driving. Therefore, we describe whether this is safe based upon electrical properties when not on the power-receiving side, that is, k = 0. We determine whether this is safe depending on whether large current is flowing on the primary side. As an aside, for S–S and P–S, the properties of the primary side when k = 0 are almost the same as when the secondary circuit is suddenly opened when there is a connection. The reason is that the self-inductance on the primary side changes slightly owing to the effects of the secondary side, but this is minor when the gap is large, so often it is negligible. On the contrary, other circuit systems have P in parallel on secondary side; hence, the properties are different when the circuit is open. Considering this, we describe k = 0. With S–S, S–P, and S-LCL, when there is no secondary side, only S remains, so the circuit is an LCR series resonant circuit (Fig. 6.11). As for inductance, the reactance is 0, so only the real component remains. The real component is only internal resistance r 1 , so the voltage is all applied to this load, and thus a large current flows. This is I = V /r 1 . Now, we describe P–S, P–P, and P-LCL. As only P remains, the circuit is an LC parallel resonant circuit (Fig. 6.12). Therefore, the inductance is ideally infinite, and the current is squeezed and may be considered to be safe. However, in an actual Fig. 6.11 When there is S on primary side

Fig. 6.12 If there is P on primary side

6.4 k = 0 Properties

187

Fig. 6.13 When there is LCL on primary side

circuit, when P–S and P–P are driven by a rectangular wave, a large current flows; hence, a sine-wave drive is required. As a result, L is ultimately added as a filter to the first stage, resulting in an increase in the number of components. When L is added to the DC side, a constant current source operation occurs. When L is added to the AC side, the circuit configuration cannot be differentiated from that of LCL. Nevertheless, if L is added as a filter and a near sine wave is achieved, then P–S and P–P driving are possible, during which it is possible to squeeze current at k = 0. Next, we describe LCL-S, LCL-P, and LCL-LCL. If there is no secondary side, then only LCL remains (Fig. 6.13). As anti-resonance is formed on the latter-stage CL, the input inductance is infinite and the same as P–S. In addition, the first-stage L sets resonance conditions such that LCL has gyrator properties. Nevertheless, there is ideally no current passing through the current circuit. Therefore, LCL is preferred. For example, when considering the charging during driving, when there is no primary-side control and no car, the current can be turned OFF. Additionally, when a car is present, it may turn ON of its own accord. This may be termed as a sensorless or passive system, that is, there is no need to add a sensor later, and the coil itself plays the role of a sensor. In addition, even if an inverter is just kept 50% duty is achieved, the current for powering will start to flow of its own accord; thus, the system may be considered as passive type. However, a constant current flows to L 1 as a result of the gyrator properties. Hence, there is the disadvantage that when it is completely passive, the standby power is consumed by the internal resistance of the coil. Now we will describe an S–S system. When turned ON with no consideration, a large current flows, as described previously, and the circuit is now broken. Therefore, when there is an S–S system, a sensorless and active control system is used. Although it is a sensorless system, there is a method to drive a pulse with the duty reduced as much as possible for detection. When a car arrives, the mode is switched to wireless power transfer (WPT) mode with 50% duty. We have explained the properties of k = 0 with a constant voltage source. Table 6.4 includes the properties when a constant current source is connected.

188 Table 6.4 k = 0 list

6 Feature of P–S, P–P, LCL-LCL, and LCC-LCC (a) S or P on primary side S

P

Constant voltage source

Large current

i =0

Constant current source

CC

Large voltage

(b) LCL or LCC on primary side LCL

LCC

Constant voltage source

i =0

i =0

Constant current source

Large voltage

Large voltage

(c) Relay coil on primary side 1 relay (odd number)

2 relays (even number)

Constant voltage source

i =0

Large current

Constant current source

Large voltage

Constant current

References 1. C. Wang, G.A. Covic, S. Member, O.H. Stielau, Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems. IEEE Trans. Industr. Electron. 51(1), 148–157 (2004) 2. C.Wang, O.H. Stielau, G.A. Covic, S. Member, Design considerations for a contactless electric vehicle battery charger. IEEE Trans. Ind. Electron. 52(5), 1308–1314 (2005) 3. T. Fujita, Y. Kaneko, S. Abe, Contactless power transfer systems using series and parallel resonant capacitors. Trans Inst. Electr. Eng. Japan D 127(2), 174–180 (2007) 4. Y.H. Sohn, B.H. Choi, E.S. Lee, G.C. Lim, G. Cho, C.T. Rim, General unified analyses of two-capacitor inductive power transfer systems: equivalence of current-source SS and SP compensations. IEEE Trans. Power Electron. 30(11), 6030–6045 (2015) 5. B. Wei et al., A study on constant current output control in wireless power transfer, pp. 2–5 (2017) 6. F. Liu, Y. Zhang, K. Chen, Z. Zhao, L. Yuan, A comparative study of load characteristics of resonance types in wireless transmission systems. Asia-Pacific Int. Symp. Electromagn. Compat. APEMC 2016, 203–206 (2016) 7. K. Takeda, T. Koseki, Analytical investigation on asymmetric LCC compensation circuit for trade-off between high efficiency and power. IPEC (2018) 8. T. Takeuchi, D. Kobayashi, T. Imura, Y. Hori, Fundamental experiment on dynamic wireless power transfer using double-LCC. IEICE Technical Report WPT2016-10, pp. 5–10 (2016) 9. T. Kan, T.-D. Nguyen, J.C. White, R.K. Malhan, C. Mi, A new integration method for an electric vehicle wireless charging system using LCC compensation topology: analysis and design. IEEE Trans. Power Electron. (2016) 10. T. Kan, S. Member, T. Nguyen, J.C. White, R.K. Malhan, C.C. Mi, A new integration method for an electric vehicle wireless charging system using LCC compensation topology: analysis and design. IEEE Trans. Power Electron. 32(2), 1638–1650 (2017) 11. K.K. Ean, B.T. Chuan, T. Imura, Y. Hori, impedance matching and power division using impedance inverter for wireless power transfer via magnetic resonant coupling. IEEE Trans. Ind. Appl. 50(3), 2061–2070 (2014)

Chapter 7

Open and Short-Circuit-Type Coils

There are two types of resonating magnetically coupled coils, namely, open-circuit or short-circuit type coils. While both could be operated as a magnetic resonance coupling, the open-circuit type can self-resonate without a resonant capacitor. On the other hand, the short-circuit type requires an external capacitor. Also, there is a difference in the resonant frequency. Open- and short-circuit type coils will be discussed in this chapter.

7.1 Overview of Open- and Short-Circuit Type Coils Spiral and helical coils will be used to explain open- and short-circuit type coils. There are two ways to achieve LC resonance, one is through an open-circuit type circuit, and the other is through a short-circuit type circuit. A single-element spiral coil is used to illustrate the difference between the two in Fig. 7.1. Here, with a single element, a spiral coil with an open-circuit type double-layer structure is used (Fig. 7.2). In addition, a helical coil is shown in Fig. 7.3. A power-transmitting coil and a power-receiving coil make up a single pair. The helical coil used in this chapter has a 150 mm radius, 5 turns, and a 5 mm pitch for both the open-circuit type and short-circuit type. When the conductive wires on the other side of the port—the connector portion of the power supply used for electrical input—are severed and thus in a disconnected condition, the circuit is referred to as an open-circuit type. On the other hand, if the wires on the other side of the port are connected, they are referred to as a short-circuit type. Most typical coils are not disconnected in the middle; rather, they are of the short-circuit type in which the wires are connected. In this chapter, the majority of the coils described are of the disconnected opencircuit type for the MHz band in which the coil ends are not connected to anything. The short-circuit type coil in which the ends of the coil are not open can be used as a magnetic resonance coupling as well; the only difference is how the LC resonance is made to occur. Generally, the open-circuit type is used more often with the MHz © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_7

189

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7 Open and Short-Circuit-Type Coils

Fig. 7.1 Open- and short-circuit type spiral coils (a) Open-circuit type

(b) Short-circuit type

Fig. 7.2 Spiral coil of double-layer structure (a) Full structure

(b) Top layer and bottom layer

Fig. 7.3 Open-circuit and short-circuit type helical coil

(a) Open-circuit type (end)

(b) Open-circuit type (power supply side)

(c) Short-circuit type (end)

(d) Short-circuittype (power supply side)

band. A schematic diagram is shown in Fig. 7.4. The open-circuit type uses the C (capacitance) of the stray capacity that is generated in the coil itself to generate LC resonance. That is, it is self-resonant, and its frequency is called the self-resonant frequency. For this reason, it is generally unnecessary to have an external capacitor, but if an external resonant capacitor were to be added, the open-circuit type would operate at a frequency higher than the self-resonant frequency. On the other hand, the short-circuit type is not able to resonate without a capacitor. Therefore, resonance is generated by adding an external capacitor C. While both types can operate as a magnetic resonance coupling, they differ in their operational range. The short-circuit type uses an external resonant capacitor and is capable of operating

7.1 Overview of Open- and Short-Circuit Type Coils

191

Fig. 7.4 Open-circuit and short-circuit type helical coils

(a) Open-circuit

(b) Short-circuit

at a lower frequency than the anti-resonance frequency, and such a frequency is lower than the self-resonant frequency of the open-circuit type. That is, the short-circuit type is better suited for operating at lower frequencies, and the open-circuit type is better suited for operating at higher frequencies. Since the open-circuit type requires no capacitors (capacitor-less) they are easy to fabricate because winding of the coils is all that is required. Also, they are more advantageous in terms of cost and space, and there is no need to be concerned about the issue of capacitor tolerance. Moreover, there is an added benefit that there is no increase in loss due to the internal resistance of the capacitor. On the other hand, since self-resonance is achieved through the capacitor component of the stray capacity, there is an inherent challenge that the precision of the coil is all that determines the precision of the resonant frequency. Additionally, since it uses the stray capacity, it has a greater tendency to be influenced by the surrounding environment than the short-circuit type; thus, the resonant frequency tends to shift easily. Furthermore, since the voltage becomes higher at the coil end if the coils are too close together it could cause a dielectric breakdown resulting in arcs. The dielectric breakdown of air occurs at approximately 3 kV/mm. This means that when 3 kV is generated at the end of a coil, and if there is only a 1 mm distance between the coils, arcing will occur. Therefore, it is necessary to ensure that the ends of the coil are not too close together. On the other hand, the short-circuit type coil’s L value can be measured using an LCR meter, so it is easier to design as it is only necessary to select the capacitor’s C so that it will resonate at the resonant frequency. For resonant capacitors, multi-layer ceramic capacitors (MLCC) and film capacitors are used. Because they are operable at lower frequencies than the self-resonant frequencies of the open-circuit type coils, they are used in a great number of applications. This is because the self-resonant frequency of an open-circuit type tends to be higher than what is preferable to use in the kHz band. However, short-circuit type coils must have a capacitor in addition to the coil itself; therefore, it is necessary to use a capacitor with minimal loss due to equivalent series resistance (ESR). In addition, since capacitors that are resonating have a voltage nearly equivalent to that of the coils applied, they need to have a high voltage capability. If one were to consider the most severe conditions, Q times more voltage than the power supply voltage could be generated. This occurs when decoupling occurs with the secondary side. LC resonance occurs only on the primary side, and the load becomes just the internal resistance. For example, if the Q factor of the coil is 100 (Q = 100) and the input voltage is 100 V (V in = 100 V), the capacitor voltage will be V c = 10 kV. If the voltage tolerance level of the capacitor

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7 Open and Short-Circuit-Type Coils

is insufficient, it will go into failure mode. Therefore, it is necessary to consider the air gaps and load based on the range of operation when designing a circuit. Under normal operation, in the case of the example in Chap. 5, even if the input is 100 V when k = 0.1, the capacitor voltage will be less than 1 kV (V c < 1 kV). Furthermore, depending on the accuracy of the capacitor’s specifications, there would indeed be an issue of variability. Typically, there is at least a ±5% margin of error for capacitors; in real applications, minor adjustments would be necessary. Thus, both the open-circuit types and short-circuit types have advantages and disadvantages; it is necessary to use them with a proper understanding of each of their characteristics.

7.2 An Intuitive Understanding of the Open-Circuit Type Through the Dipole Antenna It is fairly easy to grasp that, in the case of the short-circuit type, capacitor C is added onto the coil L; thus, this generates LC resonance. Although it was explained in the discussion of the stray capacity in the case of open-circuit types, for those who understand antenna theory, it is easier to imagine this by considering the half-wave dipole antenna [1]. Since the magnetic field type and the electric field type are also better understood in the same way, an explanation of the electric field type—the subject of Chap. 11—will also be provided in this section. A helical coil that has a magnetic field type coupling is shown in Fig. 7.5a, and a meander-line resonator that has an electric field type coupling is shown in Fig. 7.5b. In each case, power is supplied from the center of the line length, and they are opencircuit types with disconnected ends. The helical coil operates like a magnetic field type coil by turning a linear dipole antenna into a loop structure to concentrate its magnetic field. On the other hand, the meander-line resonator operates as an electric field type coil by generating and concentrating the electric field in the y-axis direction. Furthermore, the alternately meandering lines cancel out the magnetic field that is generated in the space of the coupling direction on the z-axis, which causes it to operate as an electric field type resonator. As shown in Fig. 7.6, the principle of resonance is the same as the resonance of a half-wave dipole antenna, and the line length is the primary determining factor of the

(a) Helicalcoil, magnetic-field type

(b) meander-line resonator, electric-field type

Fig. 7.5 Coupler for electromagnetic resonance coupling (coil and resonator)

7.2 An Intuitive Understanding of the Open-Circuit Type …

(a)

/4 line

(b) Half-wave dipole antenna

193

(c) Magnetic and electric resonance coupling resonators

Fig. 7.6 Relation to the half-wave dipole antenna

frequency. When the half-wave dipole antenna’s total length is half the wavelength, it will appear as though it is a quarter wavelength line (see Sect. 7.3); in this state, the reactance becomes zero and causes resonance. However, in the case of helical coils and meander-line resonators, they have a looped or meandering line structure. These structures impact the inductance and capacitance, resulting in resonance in a frequency shifted from the resonant frequency of the half-wave dipole antenna. When the coils are wound just a few times, the shift is minimal, so a relation between the wavelength and the line length is seen in the case of MHz coils, but there is still a shift from the half-wavelength. When the coils are wound a few dozen times, the shift becomes much greater, so in real applications, it is necessary to calculate the resonant frequency by determining the L value and C value as a concentrated constant. Furthermore, in the case of magnetic resonance coupling and electric resonance coupling coils and resonators, the total size of the coils is reduced in relation to the wavelength. Thus, the size is that of a small antenna. For this reason, just like a small antenna that has very little radiation resistance, such a coil is unable to match the impedance of the free space around it as a single element, and unlike a half-wave dipole antenna, it is difficult for electromagnetic waves to be emitted. Therefore, if one were to try to use a power-transmitting coil on its own, electrical energy would hardly be emitted from the power-transmitting coil. On the other hand, if a power-receiving coil is nearby, a coupling can be established, and it will transmit power. Thus, when this approach is used for power transmission, it takes advantage of the characteristic inability to produce power unless a receiver is nearby. Generally, components that have these characteristics are all called resonators by those in the filtering field.

7.3 Lumped-Element Model and Distributed Constant Circuit All the explanations and descriptions covered in this book are basically within the scope that can be covered and discussed within the lumped-element model. However, the details of open- or short-circuit type coils are quite difficult to explain without

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7 Open and Short-Circuit-Type Coils

the concept of a distribution constant. For this reason, the lumped-element model and the distributed constant circuit will be briefly explained. In brief, the difference between the above two is this: with the lumped-element model, the length of the wavelength can be ignored, and in general, it is more widely known. At a given time, the voltage on a line is constant at any of its locations. One the other hand, in the case, that the length of the wavelength cannot be ignored, it is necessary to consider the use of a distributed constant circuit. When the physical length of a wire or antenna of a circuit is close to or longer than the wavelength, the length of the wavelength cannot be ignored. In this case, at a given time, the voltage on a line would differ from one location to another; therefore, it becomes harder to design a circuit. A schematic diagram of the transmission line and the waveform are shown in Fig. 7.7. When the wavelength is short, as in Fig. 7.7b, it is necessary to consider it as a distributed constant since the voltage is not constant at any given location on the circuit. However, if the wavelength is long, as seen in Fig. 7.7c, at any location, the voltage would be the same; thus, it would be sufficient to consider the circuit under the lumped-element model. The schematic diagram of a transmission line of a balanced two-wire line is shown in Fig. 7.8. Figure 7.8b is in an open state, while Fig. 7.8c is in a short state. “Open” means that there is no load at the end of the transmission lines, and the circuit is in a disconnected state. In this state, the impedance at the load is ∞[]. “Short” means that the load of the transmission line has taken an unintended path, and the impedance at the load is 0[]. Following the convention of the transmission line theory, the location in which there is a load is defined as d = 0, and the distance traveled toward the power supply is defined as d. The equations related to the voltage, current, and impedance at location d on the transmission line are expressed in Eqs. (7.1) to (7.3). Here, Z L is the impedance of the load, Z 0 is the characteristic impedance of 50 , and β is the phase constant and β = 2π/λ.

(a) Transmission line

(b) Short wavelength

(c) Long wavelength

Fig. 7.7 Diagram of the transmission line and the waveforms

7.3 Lumped-Element Model and Distributed Constant Circuit

(a) Balanced two-wire line

(b) Open-type circuit

195

(c) Short-type circuit

Fig. 7.8 Schematic diagram of a transmission line

V (d) = V + (d) + V − (d) = I L {Z L cos(βd) + j Z 0 sin(βd)}

(7.1)

IL {Z 0 cos(βd) + j Z L sin(βd)} Z0

(7.2)

Z L cos(βd) + j Z 0 sin(βd) Z L + j Z 0 tan(βd) = Z0 Z 0 cos(βd) + j Z L sin(βd) Z 0 + j Z L tan(βd)

(7.3)

I (d) = I + (d) − I − (d) = Z (d) = Z 0

Generally, the coils used in kHz-band wireless power transmission is sufficiently small in relation to the wavelength, so there is no issue to consider it under the lumpedelement model. On the other hand, in the range of a MHz band, it is necessary to consider otherwise. At the GHz band, the wavelength cannot be ignored. For example, at 100 kHz, the wavelength is 3 km; this is more than sufficiently larger than the coil size. However, at 10 MHz the wavelength is 30 m; this is a cause for some concern. At 1 GHz, the wavelength is 30 cm and must be taken into consideration. Additionally, in the territory where it is no longer possible to ignore the length of the wavelength, it is necessary to consider the reflection phenomenon. That is, the stationary wave created by the traveling wave V + , I + and the reflecting wave (regressive wave) V − , I − appears in the line. Therefore, the voltage appears as the sum of the traveling wave and the reflecting wave. The traveling wave travels from the power supply to the load, and the reflecting wave is generated at the load and returns toward the power supply. Equations (7.1)–(7.3) take these matters into consideration. As an example, let us consider a 30 m line length with load Z L fixed at 100 . Here, the intention is to verify the difference in frequencies. For the 30 m line, the wavelength is 10 MHz. Therefore, the voltage distribution on this 30 m transmission line at 10 MHz—for which the wavelength cannot be ignored—is shown in Fig. 7.9a. For comparison, the voltage distribution on the same 30 m transmission line at 100 kHz— which is a frequency at which the wavelength would be considered long enough against a 30 m line—is shown in Fig. 7.9b. In this way, when the wavelength is close to the line length, the influence of reflection cannot be ignored. On the contrary, when the wavelength is long enough, the influence of reflection can be ignored. Besides, in the case of 50 Hz/60 Hz commercial frequencies, the wavelength is sufficiently long; therefore, there is no reason to consider the influence of reflection when used at home, so the circuit can be considered under the lumped-element model.

7 Open and Short-Circuit-Type Coils 200 100 0 -100 -200

Re{V(d)}

Im{V(d)}

|V(d)|

0

10

20

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d [m]

(a) f = 10 MHz, ZL = 100 Ω, VL = 100 V

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[V]

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Im{V(d)}

|V(d)|

100 50 0

0

10

d [m]

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30

(b) f = 100 kHz, ZL = 100 Ω, VL = 100 V

Fig. 7.9 Voltage distribution on a transmission line, current distribution, and impedance (wavelength comparison)

Next, Fig. 7.10 shows the voltage V (d), current I(d), and impedance Z in at 10 MHz; the distance from the load is defined as d. In this scenario, the impedance is varied. The horizontal axis is standardized at one wavelength λ. Figure 7.10a alone shows the load voltage as V L = V (0) = 1 V, and in the other parts of Fig. 7.10, it is set as V L = V (0) = 100 V. Figure 7.10a is 1  at the load, and it is nearly in a state of being short. In Fig. 7.10b it is 25 , in Fig. 7.10c it is 50 , in Fig. 7.10d it is 100 , Fig. 7.10e is 10000 , and it is nearly at a state of being open at the load. Figure 7.10a shows the voltage that would be required to achieve V L = V (0) = 1 V at the load if the power supply was set at V (d). In the same way, Fig. 7.10b–e shows the load and the voltage that would be necessary to achieve V L = V (0) = 100 V at the load if the power supply was set at V (d). For example, if the condition is that of Fig. 7.10a and V (0.25) = 50 V, it would only be possible to achieve V L = V (0) = 1 V at the load. Furthermore, the maximum current would be I L = 1 A at the load. That is, at the load, Z L = Z in (0) = 1 ; therefore, it is practically in a state of being short-circuited. However, with a 0.25 λ power source, the impedance becomes Z in (0.25) = 2500 , and it can be seen as an open circuit. It can also be observed that at 0.25 λ, V (0.25) = 50 V and I (0.25)  0 A, which means hardly any current flows, and it is in an open condition. In the case shown in Fig. 7.10b, while Z L = Z in (0) = 25  at load, if the power supply is set at 0.25 λ, the impedance would be Z in (0.25) = 100 . Thus, the load value is seen at the power supply, and the actual load value is not the same. Figure 7.10c, the load value is Z L = Z in (0) = 50 . The characteristic impedance and the load impedance match. In this case, reflecting waves are not generated. Since there are no reflections, the impedance at power supply and the impedance at the load where d = 0 are the same. Also, irrespective of 0.25 λ, no matter where the power supply is set, the impedance would be 50 . In the case shown in Fig. 7.10d, the impedance at load is 100 , but at Z in (0.25) = 25 . In the case shown in Fig. 7.10e at the power supply where d = 0.25 λ, even in the case where V (0.25) = 0.5 V, at load it increases to V L = V (0) = 100 V, and conversely, the load current becomes the minimal current, I L = 0.01 A. That is, because Z L = Z in (0) = 1000  at load, it is practically in a state of an open circuit. On the other hand, at the power supply of 0.25 λ, the impedance becomes Z in (0.25)

Im{V(d)}

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|Zin|

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3000 2000 1000 0 -1000

[Ω]

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[A]

[V]

7.3 Lumped-Element Model and Distributed Constant Circuit

0

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Im{V(d)}

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(a) ZL = 1Ω, VL = 1 V, equivalent to a short circuit

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[A]

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(b) ZL = 25 Ω, VL = 100 V

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[A]

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(c) ZL = 50 Ω, VL = 100 V

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[A]

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Re{I(d)}

4 2 0 -2 -4

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[Ω]

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(e) ZL = 10000 Ω, VL = 100 V, equivalent of an open circuit Fig. 7.10 Voltage distribution, current distribution, and impedance (Z L change) on a transmission line

= 0.25 , and it could practically be seen as a short-circuit. We can also observe from the fact that, at the setting of 0.25 λ, V (0.25) = 0.5 V and I (0.25) = 2 A, that while the voltage is, the current is relatively high, and it is practically in a state of a short-circuit. The equation for impedance at the quarter wavelength can be determined by substituting d = λ/4 in Eq. (7.3), which results in Eq. (7.4). Here, Z L is the impedance

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7 Open and Short-Circuit-Type Coils

of the load, Z 0 is the characteristic impedance, and the phase constant β = 2π/λ. The short circuit at the load may seem like an open circuit from the perspective of the power supply, while the open circuit at the load may look like a short circuit from the perspective of the power supply. Also, since the imaginary component is not included in Eq. (7.4), the length of the quarter wavelength is often the value used for impedance conversion. In particular, as the impedance inverts based on its characteristics, they are referred to as emittance characteristics or gyrator characteristics. What matters is not simply how great or small the impedance is in terms of pure resistance, but if Z L is inductive then Z becomes capacitive, and if Z L is capacitive, then Z becomes inductive.  λ  λ + j Z 0 sin 2π Z L cos 2π Z2 λ 4 λ 4 = 0  2π λ   2π (7.4) Z (λ/4) = Z 0 λ ZL Z 0 cos λ 4 + j Z L sin λ 4 The explanation provided in this example so far has assumed there is power supply at d = λ/4 = 0.25 λ, but if we look at the graph, we will understand the voltage on other activities no matter where the power supply is set. Again, since the effects of wavelengths can be ignored for topics other than this chapter, there is no need to be aware of the content introduced here.

7.4 Open-Circuit and Short-Circuit Type Coils from the Perspective of a Distributed Constant Circuit The previous section mentioned that in a quarter (1/4) wavelength, there would be a difference in characteristics depending on whether the end of the line is short or open. If the end of the line is open, then it will seem as though the line is short-circuited from the perspective of the input side, while if the end of the line is, in fact, short, then it will seem as though the line is open from the input side. This was demonstrated in Eq. (7.4), which had substituted d = λ/4 in Eq. (7.3) concerning the voltage, current, and impedance at location d on a transmission line. Furthermore, it was mentioned that it is often used for impedance conversion as one of the characteristics of a quarter (1/4) wavelength line. Thus, in the previous section, we fixed the frequency and discussed the matter from the perspective of the impedance variation over the length of a line. In this section, the line length will be fixed, and we will discuss the topic from the perspective of frequency variation. To start off this section, we will consider the line length d of a balanced two-wire line in which the resonant frequency of an open-circuit type coil is f 0 = 10.0 MHz. The wavelength is λ0 = c/f 0 = 30 m. The speed of light is set at c = 299,792,458 m/s  3.0 × 108 . Therefore, the line length is d = λ0 /4 = 7.5 m (Fig. 7.11). The phase constant β is the function of an operating frequency; therefore, its expression would be Eq. (7.5). If we substitute these into Eq. (7.3), this will become Eq. (7.6), and the

7.4 Open-Circuit and Short-Circuit Type Coils …

199

Fig. 7.11 Transmission line of a balanced two-wire line

graph in Fig. 7.12 is the result of these equations. Since pure resistance is taken into consideration, Z L = RL . The characteristic impedance Z 0 = 50 . 2π f ω 2π = = λ c c ω    Z L cos c d + j Z 0 sin ωc d     Z in ( f ) = Z 0 Z 0 cos ωc d + j Z L sin ωc d β=

(7.5) (7.6)

In Fig. 7.12, the load value is changed for the sake of reference, but the one that is equivalent to the open-circuit type coil is shown in Fig. 7.12e, and the one that is equivalent to the short-circuit type coil is shown in Fig. 7.12a. Here, we will discuss the subject with a focus on these two. Im{Zin}

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(a) RL = 1 Ω

60

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150 [Ω]

[Ω]

[Ω]

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10 20 30 40 Frequency [MHz]

(e) RL = 1000 Ω

Fig. 7.12 Input impedance when the line length is a quarter wavelength to 10 MHz

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7 Open and Short-Circuit-Type Coils

First, the coil equivalent to the open-circuit type will be discussed. As shown in Fig. 7.12e, when the load is practically open, at 10 MHz, the input impedance looks as though it is short-circuited. In other words, it is in a resonant state, and the resonant frequency is 10 MHz. At the second-harmonic wave of 20 MHz, the line seems to be open, and the waves appear as an anti-resonance frequency. At the third harmonic wave of 30 MHz, it appears that line is short-circuited. If we were to look at it on a distributed constant circuit, we would observe a repeating phenomenon as such. However, the original resonant frequency is generally used in power transmission, so this part is used as an equivalent circuit of a lumped-element model. In a more detailed equivalent circuit, it would be necessary to consider the waves beyond the second-harmonic waves. Next, the coil equivalent to the short-circuit type will be discussed. As shown in Fig. 7.12a, when the load is practically short-circuited, at 10 MHz, the input impedance appears as though it is in an open state. This is, in other words, an antiresonance frequency. While the resonant frequency is generally at the second harmonic of 20 MHz, this is not used for power transmission. Then, the question is, what frequency is used? For this, a resonant capacitor is used to create a resonance point below the 10-MHz anti-resonance frequency, and the power is transmitted there. Furthermore, one can see that the resonant frequency of an open-circuit type coil matches the anti-resonance frequency of a short-circuit type coil. However, in reality, unlike the balanced two-wire line model, the coil used for wireless power transmission has a large inductance, and it also has capacitance because it also generates stray capacity. For this reason, there will be a shift from the values, such as that of the resonant frequency, that would be indicated in a balanced two-wire line model. However, by considering this from the perspective of a distributed constant circuit, it is possible to understand the generation of resonance, anti-resonance, and harmonics. When the open-circuit type coil resonates at f 0 = 17.6 MHz, the wavelength is λ0 = c/f 0 = 17.05 m. L = 8.5 μH, C = 9.7 pF, L m = 0.71 μH, r = 1.46 . The air gap is 150 mm. Since the speed of light c = 299,792,458 m/s  3.0 × 108 , the line length is d = λ0 /4 = 4.26 m. If we were to create a graph using the same steps, it would look like Fig. 7.13. Figure 7.13a shows the input impedance at a state close to a short, while Fig. 7.13b shows the input impedance in a state close to open. The electromagnetic field analysis results of an actual helical coil are shown in Fig. 7.14. Unlike the transmission line model, we can see that the intervals between Re{Zin}

Im{Zin}

Re{Zin} 1500 1000 500 0 -500 -1000

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[Ω]

[Ω]

3000 2000 1000 0 -1000 -2000

0

20 40 60 80 Frequency [MHz]

(a) RL = 1 Ω, equivalent to a short circuit

0

20 40 60 80 Frequency [MHz]

(b) RL = 1000 Ω, equivalent to an open circuit

Fig. 7.13 Input impedance when the line length is quarter wavelength to 17.6 MHz

7.4 Open-Circuit and Short-Circuit Type Coils … im(Zin)

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im(Zin)

200

Im [Ω]

Re [Ω]

re(Zin)

201

-1000 0

20

40

60

80

-2000 100 120 140 160 180 200

Frequency [MHz]

(b) Open-circuit type

(a) Short-circuit type

Fig. 7.14 Resonance and anti-resonance in an actual helical coil (150 mm radius, 5 winds, 5 mm pitch)

the resonance and anti-resonance are not evenly spaced out, which is caused by the inductance and capacitance of the coil itself.

7.5 The Open-Circuit Type Coil A diagram of an open-circuit type coil is shown in Fig. 7.15 [2]. When considering power transmission, whether it is an open-circuit or short-circuit type coil, the equivalent circuit will be that shown in Fig. 7.16. A diagram of the open-circuit type transmission line and the input impedance are shown in Fig. 7.17. As discussed earlier, resonance (as marked by ◯) occurs when the frequency is at a quarter wavelength, and anti-resonance (as marked by an ×) occurs when the frequency is at a half-wavelength, and the rest repeats this pattern. For power transmission, the resonant frequency is the important part; if we consider just Fig. 7.15 Diagram of an open-circuit type coil

Fig. 7.16 Equivalent circuit

(a) Single element

(b) Power transmission and receiving coil

202

7 Open and Short-Circuit-Type Coils

Fig. 7.17 Diagram of an open-circuit type and transmission line and input impedance

(a) Open-circuit type

(b) Input impedance

(a) Lumped-element model of open-circuit type

(b) Imaginary component

Fig. 7.18 Diagram of an open-circuit type lumped-element model and imaginary component

this part as a lumped-element model, then as shown in Fig. 7.18a, we can consider the like as a series resonance. The input impedance of the imaginary component is shown in Fig. 7.18b. Since the real components in the distributed constant circuit are generally not simulated to take on frequency characteristics, the real components of the input impedance are treated as constant. We will now discuss the characteristics of open-circuit type coils when there are no external capacitors. An open-circuit type coil self resonates. The characteristics of a single element are shown in Fig. 7.19. As seen here, it will resonate at 17.6 MHz. An overview of the efficiency, power, and input impedance is shown in Fig. 7.20, and a magnified view is shown in Fig. 7.21. (1) The theory of Cs and Cp in an open-circuit type coil Generally, resonance capacitors are not necessary when there is self-resonance unless there is a need to adjust the frequency. There is a different behavior when connecting C s in a series (Fig. 7.22a) and when connecting C p in parallel (Fig. 7.22b). Here, C 1s is the stray capacity of a coil expressed as a connection in series. As shown in

5

im(Zin)

10 15 20 25 30 Frequency [MHz]

(a) Over view Fig. 7.19 Open-circuit type, no C (single element)

im(Zin)

200

2000

150

1000

100

0

50 0

Im [Ω]

Re [Ω]

re(Zin)

200000 150000 100000 50000 0 -50000 -100000 -150000 -200000

Im [Ω]

Re [Ω]

re(Zin) 400000 350000 300000 250000 200000 150000 100000 50000 0

-1000 5

10 15 20 25 30 Frequency [MHz]

-2000

(b) Magnified view

7.5 The Open-Circuit Type Coil

203

Power [W]

η[%]

80 60 40 20 0

250 200 150 100 50 0

Frequency [MHz]

P1

re(Zin) 5000 3000

P2

[Ω]

100

im(Zin)

abs(Zin)

1000

-1000 -3000 -5000 Frequency [MHz]

Frequency [MHz]

Fig. 7.20 Open-circuit type (RL = 50 , overview)

Power [W]

η[%]

80 60 40 20 0 Frequency [MHz]

250 200 150 100 50 0

P1

P2

[Ω]

100

re(Zin) 600 400 200 0 -200 -400 -600

Frequency [MHz]

im(Zin)

abs(Zin)

Frequency [MHz]

Fig. 7.21 Open-circuit type (RL = 50 , magnified view)

Fig. 7.22 Capacitors connected in series and in parallel on an open-circuit type coil

(a) Series, Cs

(b) Parallel, Cp

Fig. 7.23a, when capacitor C s are introduced in a series, there will be a shift from the resonant frequency to a higher frequency, and it is possible to make a shift until it reaches an anti-resonance frequency. In this case, the anti-resonance frequency does not shift. However, as shown in Fig. 7.23b if the capacitor C p is introduced Fig. 7.23 Effect of connecting a capacitor in series and in parallel on an open-circuit type coil

(a) Series, Cs

(b) Parallel, Cp

204

7 Open and Short-Circuit-Type Coils

Fig. 7.24 Capacitors connected in series and in parallel in an equivalent circuit of an open-circuit type coil

(a) Series,

(b) Parallel

in parallel, the anti-resonance frequency becomes lower, and it is possible to adjust the frequency until it reaches the resonant frequency. In such a case, the resonant frequency does not shift. To explain this behavior with an equation, it would be necessary to consider a more detailed open-circuit type equivalent circuit. Figure 7.24 shows an equivalent circuit of an open-circuit type coil where consideration is given to the stray capacity C 1p of the coil placed in parallel. First, we will consider Fig. 7.24a, in which C s are placed in series. In this case, Fig. 7.22a could also be used for the explanation, but at present, the circuit in Fig. 7.24a will be used. The equation for Z in in Fig. 7.24a is written as Z in =

1 + jωCs

1 jωC1 p 1 jωC1 p



+

1 jωC1s



+ jωL 1

1 jωC1s



+ jωL 1



(7.7)

The resonance condition is the condition in which Eq. (7.7) is expanded, and the numerator becomes 0. When this point is reached, the resonance angular frequency can be expressed as Eq. (7.8). When there are no C s it will result in Eq. (7.9). When we look at the ratio of Eqs. (7.8) and (7.9), as seen in Eq. (7.10), it will always be larger than 1, and we can see that the resonant frequency goes up by placing C s in series. While this can be verified by Fig. 7.22a as well, this is equal to the value that can be obtained as C 1p = 0 in Eq. (7.8). ω0

1 =√ C1s L 1



C1 p + Cs + C1s C1 p + Cs

1 ω0 = √ C1s L 1  C1s ω0 = 1+ >1 ω0 C1 p + Cs

(7.8) (7.9)

(7.10)

7.5 The Open-Circuit Type Coil

205

Next, let us consider Fig. 7.24b when C p is placed in parallel. In this situation, we are unable to use Fig. 7.22b to explain, so the equation for Z in in Fig. 7.24b will be shown in Eq. (7.11). Here, C p and C 1p are placed in parallel, so they are added together in the calculation.

Z in =

1 jω(C p +C1 p ) 1 jω(C p +C1 p )



+

1 jωC1s



+ jωL 1

1 jωC1s



+ jωL 1



(7.11)

On the other hand, the equation for the time before C p is placed in parallel, that is to say, without C p , will be the following:

Z in =

1 jωC1 p 1 jωC1 p



+

1 jωC1s



+ jωL 1

1 jωC1s



+ jωL 1



(7.12)

The anti-resonance frequency can be determined by expanding the equation under the condition that the denominator will become 0. The ratio of the anti-resonance  frequency ωa , which can be determined in Eq. (7.11), and the anti-resonance frequency ωa , which can be determined in Eq. (7.12), is expressed by Eq. (7.13). The result will always be less than 1, and the anti-resonance frequency will go lower.     2   C C + C + C C 1 p 1s p 1 p 1 p ωα   =  > C1 p , the next equation is used: 1 ω0 = √ Cs L 1

(7.16)

Since the anti-resonance frequency is a condition in which the denominator becomes 0, if we were to determine this, it would result in the following equation: 1 ωa =  C1 p L 1

(7.17)

Because C 1p is small, we can understand that ωa is higher than ω0 . If this is lower than the anti-resonance frequency, then either Eq. (7.15) or Eq. (7.16) can generate a resonant frequency by implementing C s in series. Next, let us consider Fig. 7.36b with C p implemented in parallel. The equation for Z in under this condition is expressed in Eq. (7.18). Because C p and C 1p are parallel, they are calculated after they are added together. Z in =

1 · jωL 1 jω(C p +C1 p ) 1 + jωL 1 jω(C p +C1 p )

(7.18)

The anti-resonance frequency is determined by expanding the equation under the condition that the denominator becomes 0, which results in the following equation: ωα =

1   L 1 C p + C1 p

(7.19)

7.6 The Short-Circuit Type Coil im(Zin)

re(Zin)

im(Zin)

200

2000

150

1000

150

1000

150

1000

100

0

100

0

100

0

50 0 0

-1000 5 10 15 20 Frequency [MHz]

-2000

(a) Cs = 10 pF

50

-1000

0 0

5 10 15 20 Frequency [MHz]

(b) Cs = 68 pF

-2000

Re [Ω]

2000

Im [Ω]

200

Re [Ω]

2000

50

Im [Ω]

re(Zin)

im(Zin)

200

Im [Ω]

Re [Ω]

re(Zin)

213

-1000

0 0

5 10 15 20 Frequency [MHz]

-2000

(c) Cs = 400 pF

Fig. 7.38 Short-circuit type + series C s (single element)

To determine the anti-resonance of the angular frequency before C p is placed, it is only necessary to remove C p from the equation. This would yield the following equation: ωα = 

1 L 1 C1 p

(7.20)

When we compare Eqs. (7.19) and (7.20), we can see that Eq. (7.19) is always smaller than Eq. (7.20). (2) The analysis results of Cs and Cp on a short-circuit type The abovementioned theory will be verified using an electromagnetic field analysis result. First, concerning the results of placing capacitor C s in series, the input impedance of a single element is shown in Fig. 7.38. An overview of the characteristics of two elements is shown in Fig. 7.39, and a magnified view is shown in Fig. 7.40. Based on Fig. 7.38, we can understand that resonant frequency is generated, and so long as it is between 0 Hz and the anti-resonance frequency, it is possible to adjust the frequency. Next, concerning the results of implementing the capacitor C p in parallel, the input impedance in the case of a single element is shown in Fig. 7.41. The characteristics when there are two elements are shown in Fig. 7.42. Compared to Fig. 7.41, the anti-resonance frequency is lowered, and as long as it is down to 0 Hz the antiresonance frequency can be adjusted. However, this is not a common usage method. The concept was introduced as a way for us to understand its behavior.

7.7 Summary of the Open-Circuit Type and Short-Circuit Type Coils Finally, if the intention is to operate such circuits for wireless power transmission, in the case of the open-circuit type coil, it is possible to operate it at a self-resonant frequency, or by implementing capacitors C s in series so it can be operated at a higher frequency. On the other hand, if one were to use a short-circuit type coil, then

214

7 Open and Short-Circuit-Type Coils 200 Power [W]

[%]

80 60 40 20

P1

P2

150 100

0

50 0

Frequency [MHz]

re(Zin) 5000 3000 1000 -1000 -3000 -5000

im(Zin)

abs(Zin)

[Ω]

100

Frequency [MHz]

Frequency [MHz]

(a) Cs = 10 pF

Power [W]

η[%]

80 60 40 20 0

300 250 200 150 100 50 0

Frequency [MHz]

P1

re(Zin) 5000

P2

im(Zin)

abs(Zin)

3000

[Ω]

100

1000

-1000 -3000 -5000

Frequency [MHz]

Frequency [MHz]

(b) Cs = 68 pF

Power [W]

η[%]

80 60 40 20 0 Frequency [MHz]

1600 1400 1200 1000 800 600 400 200 0

P1

re(Zin) 5000

P2

im(Zin)

abs(Zin)

3000

[Ω]

100

1000

-1000 -3000 -5000

Frequency [MHz]

Frequency [MHz]

(c) Cs = 400 pF Fig. 7.39 Short-circuit type + series C s (RL = 50 , overview)

capacitors C s could be implemented in series, and it could be operated as a power transmitter between 0 Hz and the anti-resonance frequency. However, a frequency that is too low will cause the Q value of the coil to decline, and the efficiency will go down. Additionally, the self-resonant frequency of an open-circuit type is equal to the anti-resonance frequency of a short-circuit type.

7.7 Summary of the Open-Circuit Type and Short-Circuit Type Coils 100

40 20

re(Zin) 600 400 200 0 -200 -400 -600

P2

150 100 50

0

im(Zin)

abs(Zin)

[Ω]

Power [W]

60

[%]

P1

200

80

215

0 Frequency [MHz]

Frequency [MHz]

Frequency [MHz]

(a) Cs = 10 pF

Power [W]

η[%]

80 60 40 20 0

P1

300 250 200 150 100 50 0

re(Zin) 200 150 100 50 0 -50 -100 -150 -200

P2

Frequency [MHz]

Frequency [MHz]

im(Zin)

abs(Zin)

[Ω]

100

Frequency [MHz]

(b) Cs = 68 pF

Power [W]

η[%]

80 60 40 20 0

P1

1600 1400 1200 1000 800 600 400 200 0

Frequency [MHz]

re(Zin) 200 150 100 50 0 -50 -100 -150 -200

P2

im(Zin)

abs(Zin)

[Ω]

100

Frequency [MHz]

Frequency [MHz]

(c) Cs = 400 pF Fig. 7.40 Short-circuit type + series C s (RL = 50 , magnified view) re(Zin)

im(Zin)

re(Zin)

im(Zin)

200

2000

150

1000

150

1000

150

1000

100

0

100

0

100

0

-1000

50 0 0

5 10 15 20 Frequency [MHz]

(a) Cp = 10 pF

-2000

-1000

50 0 0

5 10 15 20 Frequency [MHz]

-2000

(b) Cp = 100 pF

Fig. 7.41 Short-circuit type + parallel C p (single element)

Im [Ω]

2000

Re [Ω]

200

Re [Ω]

2000

-1000

50 0 0

5 10 15 20 Frequency [MHz]

(c) Cp = 1000 pF

-2000

Im [Ω]

im(Zin)

Im [Ω]

Re [Ω]

re(Zin) 200

216

7 Open and Short-Circuit-Type Coils 20

Power [W]

60 40 20 0

5

Power [W]

80 40 20

1000

-3000 -5000 Frequency [MHz]

(a) Cp = 10 pF 20

0

P1

re(Zin) 5000

P2

15 5

1000

-3000 -5000 Frequency [MHz]

Frequency [MHz]

(b) Cp = 100 pF 20

60 40 20 0

P1

15

im(Zin)

abs(Zin)

3000

10

1000

-1000

5

-3000

0 Frequency [MHz]

re(Zin) 5000

P2

[Ω]

Power [W]

η[%]

80

abs(Zin)

-1000

0

100

im(Zin)

3000

10

Frequency [MHz]

abs(Zin)

-1000

Frequency [MHz]

100

im(Zin)

3000

10 0

60

re(Zin) 5000

P2

15

Frequency [MHz]

η[%]

P1

[Ω]

η[%]

80

[Ω]

100

-5000 Frequency [MHz]

(c) Cp = 1000 pF

Frequency [MHz]

Fig. 7.42 Short-circuit type + parallel C p (RL = 50 , overview)

References 1. Takehiro Imura, Yoichi Hori, Wireless power transfer using electromagnetic resonant coupling. J. Inst. Electr. Eng. Jpn. 129(7), 414–417 (2009) 2. T. Imura, H. Okabe, T. Uchida, Y. Hori, Study on open and short end helical antennas with capacitor in series of wireless power transfer using magnetic resonant couplings, in IEEE Industrial Electronics Society Annual Conference (2009), pp. 3848–3853

Chapter 8

Magnetic Resonance Coupling Systems

A magnetic resonance coupling circuit is a circuit topology (circuit structure) that can achieve high efficiency and high power even in the presence of large air gaps. However, when considering the design of these systems, it is important to achieve both high efficiency and the desired power simultaneously. The desired power refers to the amount of power required by the load on the receiver side. For example, to achieve the maximum efficiency, it is necessary to set the load equal to the optimal load. Consequently, the optimal load determines the amount of power received. Therefore, the amount of power that arrives at the receiver side is not necessarily equal to the desired power. However, if the value of the load is adjusted so that the power becomes equal to the desired power, then the efficiency will not always be equal to the maximum efficiency. In this chapter, we verify these tradeoffs and describe methods to overcome them. In addition, all the circuits handled in this chapter are S–S type.

8.1 Overview of Wireless Power Transfer Systems The basic structure of a wireless charging system is shown in Fig. 8.1. DC power is generated by the DC power supply, and DC/AC conversion is performed by the inverter. Here, the system is made to generate high-frequency AC power with a frequency of 85 kHz or 6.78 MHz. To change the voltage, it is necessary to create a waveform with three levels using an inverter, or to add a DC/DC converter before and after the inverter (refer to Sect. 1.4). The transmitter coil and receiver coil are coupled through a magnetic field. This is the area where wireless power transfer occurs. The AC power sent to the receiver coil is converted back to DC power by a rectifier. By including a smoothing capacitor C DC , it is possible to stabilize the voltage in the DC link. The maximum efficiency is achieved by adjusting the impedance at the DC/DC converter. Finally, the power is sent to the load. If the load consists of a battery, then the load is a constant-voltage load. However, there are also many loads—including electronic devices—that operate as constant-power loads. Control © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_8

217

218

8 Magnetic Resonance Coupling Systems

(a) Overall system

(b) Actual circuit diagram Fig. 8.1 Basic system structure

is often performed through wireless communication. However, it is also possible to eliminate the necessity for wireless communication by estimating the quantities on the primary side from the secondary side, or vice versa.

8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries), and Constant-Power Loads (Motors and Electric Devices) There are many types of loads, including resistive loads, constant-voltage loads (secondary batteries), and constant-power loads (motors and electric devices). In wireless power transfer, it is necessary to take the load into consideration when designing the system. In this section, we explain the need for considering the load characteristics when designing the system. We illustrate our point using the examples of a resistive load, a constant-voltage load (secondary battery), and a constant-power load. In this section, we use the V 2 –V 1 axis efficiency map for explanation. The efficiency map obtained when L m is set to 12.5, 25, and 50 μH is shown in Fig. 8.2. In all cases, a high efficiency is achieved in the vicinity of V 1 = V 2 . In the following section, we fix L m = 25 μH and k = 0.05, except for some cases as noted and f =

8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries) …

219

Fig. 8.2 Efficiency map (for changes in mutual inductance)

Table 8.1 Coil parameters

Coils r1, r2

1



L1 , L2

500

μH

C1, C2

5070

pF

100 kHz and Q = 314. Furthermore, the parameters for the coils and capacitors used in this section are shown in Table 8.1.

8.2.1 Resistive Loads The resistive load is the most familiar load for everyone. However, systems that include this type of load are rare in practice. Nevertheless, this structure is useful for studying the fundamental properties of the system. As we will explain later, if a load with a resistance that makes the resistance of the equivalent circuit in front of the rectifier RLAC (RL ) equal to the optimal load is connected to the circuit, then it is possible to achieve power transmission with the maximum efficiency. In this section, we focus on the equivalent circuit resistance RL . In the stage after the rectifier, a smoothing capacitor C DC is placed in parallel with the actual resistance R L for suppressing the ripple (Fig. 8.3).

Fig. 8.3 Resistive load and smoothing capacitor

220

8 Magnetic Resonance Coupling Systems

The impedance after the rectifier can be considered to be equivalent to a pure resistance [1]. Therefore, the phases of V 2 and I 2 are the same and can be expressed as shown in Eq. (8.1). I 2 is given by Eq. (8.2). Therefore, V 2 can be expressed as shown in Eq. (8.3). Based on the above equations, the relationship among the primary side voltage, secondary side voltage, and efficiency can be obtained by deriving Eq. (8.4) for RL from Eqs. (8.1) and (8.2) and substituting the result into Eq. (8.5) for the efficiency. The relationship among the primary side voltage, secondary side voltage, and efficiency is shown in Eq. (8.6). .

RL =

V2 .

I2 . jωL m I2 = V1 r1 (r2 + R L ) + ω2 L 2m .

RL = η=

(8.1) (8.2)

V2 = j V2

(8.3)

ω2 L 2m + r1r2 V2 ωL m V1 − r1 V2

(8.4)

(ωL m )2 R L (r2 + R L ) r1 + (ωL m )2 r2 + (ωL m )2 R L

(8.5)

V2 (ωL m V1 − r1 V2 ) V1 (ωL m V2 + r2 V1 )

(8.6)

2

η=

Based on Eq. (8.4), we can obtain a plot of the equivalent circuit resistance RL versus voltage V 2 for different values of V 1 , as shown in Fig. 8.4. This plot shows the relationship among V 1 , V 2 , and RL. This figure is useful for understanding the relationship among the transmitted and received voltages and the resistance. The relationships between the efficiency and V 2 for V 1 = 100 V and for V 1 set to 50, 100, and 200 V are shown in Fig. 8.5. The maximum efficiency occurs when Fig. 8.4 Secondary side voltage V 2 and equivalent resistance RL

8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries) …

221

Fig. 8.5 Secondary side voltage V 2 and efficiency η

the voltage amplification is Av ≈ 1, or in other words, when V 1 ≈ V 2 . Therefore, in the relationship between the efficiency and V 2 , the efficiency depends on V 1 . On the other hand, in the vicinity of the optimal load, the change in the efficiency η with respect to a change in V 2 is extremely small. Therefore, we can say that the efficiency of wireless power transfer is robust (not easily affected by external influences and not susceptible to problems). For example, when V 1 = 100, as shown in Fig. 8.5a, the optimal receiver voltage, which achieves maximum efficiency, is V 2opt = 93.8 V, but even if V 2 is varied within a range of 50–150 V, the change in the efficiency is still very small. Next, we consider the efficiency η on the V 1 -axis and the V 2 -axis as viewed on a map. The relationship among the primary side voltage, secondary side voltage, and load is shown in Eq. (8.7), which can be transformed into the form shown in Eq. (8.8). V2 = I2 R L = V1 =

jωL m R L V1 r1 (r2 + R L ) + ω2 L 2m

r1 (r2 + R L ) + ω2 L 2m V2 jωL m R L

(8.7) (8.8)

A map of the efficiency η, when the value of the resistance is varied, is shown in Fig. 8.6a. The optimal load RLopt occurs when approximately V 1 ≈ V 2 . If the only goal is to approach the optimal load condition at which the maximum efficiency is achieved, then the only requirement is to continue to satisfy the relationship V 1 ≈ V 2 . If the primary side voltage is already known, then it is simple to design a system that satisfies this relationship. However, this is only true in the region in which the voltage amplification at the maximum efficiency Av ≈ 1 is satisfied. In regions in which the coupling is weak, it is not possible to satisfy Av ≈ 1. Therefore, control becomes necessary. Figure (b) shows a plot of the power and the resistance. It is possible to understand the relationship among the load RL , transmitted voltage V 1 , received voltage V 2 , and received power P2 from this single graph, which makes this graph a convenient one for reference. For example, the graph shows that, when V 1 = 100 V, it is not possible to obtain P2 = 1000 W while achieving the maximum efficiency. The graph also shows that, when V 1 = 100 V, it is possible to achieve

222

8 Magnetic Resonance Coupling Systems

(a) Resistance

(b) Resistance and power

Fig. 8.6 Efficiency map for specified values of resistance

only the maximum efficiency or only the desired power by changing V 2 . On the other hand, following the line for the optimal load RLopt = 15.7 , it is possible to find the location where it intersects the line for P2 = 1000 W. At this point, V 1 and V 2 are uniquely determined. At this condition, the primary side voltage V 1 and the secondary side voltage V 2 are controlled to be the same. The efficiency map can be used to confirm many phenomena visually as shown above and is very convenient. The derivation of these power lines will be shown in the section on constant-power loads. In addition, as there is a limit on the voltage amplification AV , in the region where V 2 is extremely large compared with V 1 , power transmission is not possible in the first place. Transforming Eq. (8.7), we obtain V2 =

jωL m r1 +

r1 r2 +ω2 L 2m RL

V1

(8.9)

Here, if we set RL = ∞[], then we obtain Eqs. (8.10) and (8.11). This becomes the boundary from which we determine whether power transmission is possible. The region beyond the boundary in which power transmission is not achievable is shown in white. jωL m V1 r1

(8.10)

V2 jωL m | R L =∞ = V1 r1

(8.11)

V2 | R L =∞ = A V | R L =∞ =

Furthermore, if we substitute Eq. (8.12) shown below for the optimal load RLopt into Eq. (8.7) for the relationship between V 2 and V 1 , then we obtain Eq. (8.13) for the receiver voltage V 2opt at which the maximum efficiency is achieved.

8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries) …

 R Lopt =  V2opt =

r22 +

r2 (ωL m )2 r1

r2 ωL m  V1 r1 r1r2 + (ωL m )2 + √r1r2

223

(8.12) (8.13)

8.2.2 Constant-Voltage Load (Secondary Battery) A common type of load is secondary battery charging (Fig. 8.7). In the case in which the load consists of a secondary battery, the voltage is determined by the secondary battery. In other words, the load consists of a constant-voltage load. Thus, this type of load is different from a resistive load. In practice, it is common to adjust the voltage using a DC–DC converter placed after the rectifier. Although it is possible to control the voltage in front of the rectifier, we consider a structure in which the load is connected directly to the rectifier in order to keep the discussion simple. The difference between a constant-power load and this type of load is important. Secondary batteries can absorb any amount of power, as long as the transmitted power satisfies the battery specifications. In this regard, secondary batteries are similar to resistances. Therefore, it is not necessary to adjust the power quickly, and it is not necessary to use advanced control. In other words, regardless of whether the system transmits 100 W or 1000 W, the battery can absorb the power without any problems. Therefore, it is sufficient to perform maximum efficiency control. Although it is necessary to switch the charging mode between the commonly used constant-current mode and constant-voltage mode, it is not necessary to control the system quickly as is required for constant-power loads in general. The efficiency map for a constant-voltage load is shown in Fig. 8.8. For example, when V 2 = 100 V, to conduct charging at maximum efficiency, it is possible to set the load to the optimal load and set the primary voltage to the value that achieves the optimal load. Therefore, V 1 should be set to V 1 = 106.6 V. Consequently, the received power becomes P2 = 635.3 W [Note that the value of V 2 for AC (the effective value) and the value of VL  for DC (the voltage for DC) are different. In this case, VL  is 111.1 V]

Fig. 8.7 Secondary battery

224

8 Magnetic Resonance Coupling Systems

(a) Constant voltage

(b) Constant voltage and power lines

(c) Constant voltage, power lines, and optimal load Fig. 8.8 Efficiency map for constant-power loads

In cases in which a particularly large amount of power is desired, it is necessary to increase V 1 . However, as the load side has constant voltage with a voltage of V 2 = 100 V, the value of the load begins to deviate from the optimal load. If a loss of a few percent is allowable, then the circuit can be used in this manner. In addition, although the system deviates from a constant-voltage load condition, if there is a DC–DC converter, and the system has a structure that makes it possible to make both V 1 and V 2 large, then it is possible to increase the power while tracking the optimal load.

8.2.3 Constant-Power Load Loads that are only capable of consuming a constant amount of power are referred to as constant-power loads (Fig. 8.9). Constant-power loads include electronic devices and motors. In a circuit with a secondary battery, which is a constant-voltage load, the wirelessly transmitted energy is stored in the secondary battery. In contrast, in the case of a constant-power load, the load is only capable of consuming a constant

8.2 Resistive Loads, Constant-Voltage Loads (Secondary Batteries) …

225

Fig. 8.9 Constant-power load

amount of power. It is not permissible to transmit an excess or insufficient amount of power. If the amount of energy transmitted is even slightly larger than the amount of power consumed, then the power, which has nowhere to go, will cause the voltage across the smoothing capacitor to rise, which may result in damage to the circuit. This phenomenon is unique to wireless power transfer [2]. For most devices, it is possible to cut off the supplied energy instantaneously in a relatively simple manner. However, the system is transmitting power wirelessly from the primary side to the secondary side, and both sides are operating in coordination while being independent. From the perspective of mitigating risk, it is not desirable to choose a structure in which rapid control is conducted on the primary side through a communication line, such as in the case of controlling voltage rise in the smoothing capacitor. Therefore, it is desirable to provide the system the ability to adjust power in the secondary side alone, including cutting off the power. Therefore, the short mode, which will be described in Sect. 8.8, is used. The characteristics of V 1 and V 2 for constant-power loads are shown in Fig. 8.10. This power curve can be desired based on Eqs. (8.14) and (8.15), which result in Eq. (8.16). V2 = I2 R L = P2 =

Fig. 8.10 Efficiency map for constant-power loads

jωL m R L V1 r1 (r2 + R L ) + ω2 L 2m V22 V2 ⇔ RL = 2 RL P2

(8.14) (8.15)

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8 Magnetic Resonance Coupling Systems

V1 =

 r1 r2 +

V22 P2



jωL m

+ ω2 L 2m V22 P2

V2

(8.16)

When the constant-power load is connected to the circuit, in the case in which the power consumption of the load is 100 W, the circuit can only operate along with the line corresponding to P2 = 100 W. The situation is similar for the cases in which the load is 500 and 1000 W. If the power transmitted to the secondary side is even slightly larger than the power along this line, then the voltage across the smoothing capacitor rises suddenly, and the circuit will be damaged. In contrast, if the power is even slightly lower than the power along this line, then the voltage will drop suddenly, and the amount of power transmitted will fall below the amount of power required by the device, which may cause the device to stop functioning. Therefore, it is necessary to handle these situations using the short mode, as will be explained in Sect. 8.8.

8.3 High Power Through Frequency Tracking Control A graph of the efficiency of magnetic resonance coupling versus power is shown in Fig. 8.11. Here, the optimal load at which the maximum efficiency is achieved for f = 100 kHz is connected to the circuit. When viewed over a wide range of frequencies,

(a) Efficiency

(c) Efficiency (magnified) Fig. 8.11 Power and efficiency (S–S)

(b) Power

(d) Power (magnified)

8.3 High Power Through Frequency Tracking Control

227

the efficiency appears to have a sharp peak. However, magnifying the graph shows that the efficiency is robust against variations in the frequency. However, the power is strongly affected by the frequency and is not robust against changes in the frequency. When aiming to achieve high efficiency, it is sufficient to make the circuit operate at the resonance frequency f 0 of the system with one element present; hence, it is not necessary to change the frequency. This is a characteristic of magnetic resonance. In cases in which the operating frequency and the resonance frequency f 0 deviate from each other, the operating frequency should be brought back to the resonance frequency f 0 . The resonance frequency f 0 of the system, with one element present, does not change depending on changes in the air gap. Therefore, it is sufficient to set the frequency only once. In contrast, even if the frequency deviates somewhat, the efficiency will still be high. It is sufficient to match the frequency only to a certain extent. Here, it is possible to design the system in a way that prioritizes power. In this case, it is necessary to perform control so that the system tracks the two peaks in the power, or in other words, to perform frequency tracking control. Specifically, it is necessary to change the frequency of the power source. As it is only possible to control the frequency on the primary side, this indicates that the control is applied to the primary side. The control simply adjusts the system to a frequency in which more current flows and aligns the frequency to one of the two peaks shown in Figure (d). Thus, it is simple to conduct tracking control such that high power is obtained. Note that it is not possible to conduct maximum efficiency control through frequency tracking alone, in theory. The optimal load for achieving maximum efficiency can only be achieved on the secondary side. Therefore, although it is possible to achieve high efficiency by setting the operating frequency to f 0 , whether the theoretical maximum efficiency is achieved depends on the value of the load, which is not possible to control actively. This is because, to achieve the theoretical maximum efficiency for magnetic resonance coupling, it is necessary to set the load to the optimal load while setting the operating frequency to the resonance frequency f 0 .

8.4 Overview of Achieving Maximum Efficiency Through Impedance Tracking Control Here, we explain a method for achieving maximum efficiency through impedance tracking control [1, 3–7]. First, we provide an overview of impedance tracking control. Impedance tracking control involves achieving maximum efficiency by constantly optimizing the impedance of the equivalent load resistance on the AC side (AC equivalent load resistance). Unlike the resonance frequency, the actual value of the load changes frequently. Hence, it is necessary to apply constant control to the circuit, such that the equivalent load resistance becomes equal to the optimal load. The process is shown below. The system is controlled to achieve maximum efficiency in steps (4) and (5).

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8 Magnetic Resonance Coupling Systems

(1) (2) (3) (4) (5)

Transmit power. Rectify power from AC to DC using a rectifier. Smooth the voltage using a smoothing capacitor. Control the impedance using an impedance converter DC/AC converter. As a result of (4), the equivalent load resistance RLAC (RL ) is controlled by the receiver resonator in front of the rectifier in an equivalent manner, and the maximum efficiency is achieved.

Thus, it is possible to make the circuit operate in such a way that the maximum efficiency is constantly achieved. Ultimately, the maximum efficiency is achieved by optimizing the equivalent load resistance RLAC (RL ) on the AC side. However, the control is applied to the DC side. In other words, even though the location at which the DC/DC converter directly performs impedance transformation is at R L and R DC , the change affects the AC side as well, and the circuit operates as if impedance transformation of R L and RLAC (RL ) was performed. We explain each of the above steps in the following paragraphs. A simple explanatory diagram is shown in Fig. 8.12, and a detailed diagram is shown in Fig. 8.13. In the previous chapters, the load side was represented using RL . However, in this chapter, as we are considering the entire system and are investigating the load side in detail, we represent the load on the AC side as RLAC . The important factor in wireless

Fig. 8.12 Simplified diagram of equivalent load resistance and impedance tracking control system

Fig. 8.13 Detailed diagram of impedance tracking control system

8.4 Overview of Achieving Maximum Efficiency Through …

229

power transfer is not the impedance R L of the load connected to the latter stage, but rather the equivalent load resistance RLAC (RL ) of the AC section. Naturally, the efficiency of the converter is important as well. However, as investigating the efficiency of the converter would involve examining the converter itself, we leave this matter for another book. First, to make the system operate as a magnetic resonance coupling system, we set the operating frequency to the resonance frequency f 0 for one element. Then, if we set the impedance of the load connected to the resonator on the second side (which is equivalent to the section of the circuit after the output of the resonator) equal to the equivalent load resistance RLAC (which was referred to as RL in the previous chapters), then by setting the equivalent load resistance equal to the optimal load RLACopt (which was referred to as RLopt in the previous chapters), we can achieve the maximum efficiency. In other words, regardless of the resistance RL  of the load actually connected to the circuit, the maximum efficiency ηmax can be achieved by transforming the impedance and setting the impedance to be equal to the optimal load RLACopt that achieves the maximum efficiency in the power transmitter (impedance optimization). As shown in Fig. 8.14, even if the air gap and the value of L m change, the maximum efficiency can still be obtained if the AC equivalent load resistance can achieve the maximum efficiency. For example, in Figure (b), the equivalent load resistance that achieves the maximum efficiency is 15.7 , and the efficiency is 88.1%. In other words, when performing maximum efficiency tracking control, the system should be controlled such that the AC equivalent load resistance that achieves the maximum efficiency is always set to the optimal load, corresponding to changes in the air gap and the load. Note that, for each value of air gap distance, in the case that the voltage across the primary side is fixed (the voltage is fixed in most cases), if the AC equivalent load resistance is uniquely determined (one value), then the received power is also uniquely determined. When the equivalent load resistance is 15.7 , the received power is 559.4 W. In Sects. 8.4, 8.6, and 8.7, we mainly focus on maximum efficiency tracking control. However, in some situations, obtaining the required power (desired power) is often prioritized, surpassing the maximum efficiency in reality. This situation will also be discussed. As shown in Fig. 8.14, as long as it is not possible to achieve the maximum efficiency and the maximum power simultaneously through control of the

Fig. 8.14 Equivalent load resistance and efficiency

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8 Magnetic Resonance Coupling Systems

impedance of the secondary side load alone, it is necessary to separate the strategy for system design into two cases, depending on the application. [Note: We will explain later that it is possible to achieve both maximum efficiency and maximum power through control of the secondary side alone by adding an additional degree of freedom consisting of the time axis.] The first method is to perform power control on the secondary side such that the electronic device that makes up the load receives the required amount of power, or in other words, such that the load power is equal to the desired power. However, in this case, the efficiency deviates from the maximum efficiency. This must be permitted. If there is no secondary side control, and the load receives an amount of power greater than the desired power, then the energy will have nowhere to go, and the voltage of the DC link will rise rapidly, so that the circuit will be damaged. Therefore, it is important to perform desired power control. Most electronic devices and motors can be considered to consist of constant-power loads (which only consume the required amount of power instantaneously). The load fluctuates in real time depending on the amount of power required by the electronic device. Therefore, if the system will perform power control on the primary side, then the system will require communication. However, due to the presence of communication delay, it is not simple to implement such a system. Furthermore, in situations where the communication is lost, it becomes dangerous to operate the system. Therefore, it is necessary to design the receiver power control system so that it can operate within the receiver side alone. In addition, it is necessary to have the ability to shut off the power on the secondary side. This functionality will be described in Sect. 8.8. The second method is to perform maximum efficiency control and to use the circuit for applications in which the load can absorb any amount of power that flows into it. For example, if the load consists of a battery, then maximum efficiency control can be performed. It is not a problem to have the power to be determined by the value of the load at any particular instant in time. As the energy has somewhere to go, regardless of how much energy flows into the load, there is no problem with performing control to stabilize the DC link voltage and performing control to set the voltage equal to the value that achieves the maximum efficiency. In other words, this strategy can be used for applications such as wireless charging. In addition, as wireless charging is often performed over a span of several hours, the impedance does not change as suddenly as in the case of constant-power loads. Furthermore, similar to the method of secondary side efficiency control and primary side power control, which will be described in Sect. 8.6, it is acceptable, even if the transmitter controller is slower than the receiver power controller. Therefore, it is not a problem to build the power control into the primary side to achieve the desired power. For example, if the amount of power that can be obtained for the optimal load that achieves maximum efficiency, or P2 = 559.4 W, is much smaller than the desired power, and charging would take a long time, then it is realistic to raise V 1 and increase P2 . If we use the method in which we add an additional degree of freedom, consisting of the time axis as described in Sect. 8.9, then it will be possible to achieve both the maximum efficiency and the desired power simultaneously through secondary

8.4 Overview of Achieving Maximum Efficiency Through Impedance …

231

side control alone. The efficiency and power are uniquely determined, as described above, as a result of the theory through adjusting the secondary side impedance.

8.5 Preliminary Knowledge for Discussing Efficiency Maximization Through Impedance Tracking Control 8.5.1 AC–DC Conversion Through Rectifiers In actual circuits, the DC side after the rectifier is involved in maximum efficiency tracking control. In this section, we discuss important points related to AC/DC conversion in the rectifier. First, we assume that the wireless power transfer is 85 kHz AC, and that in the S–S type, the secondary side has constant-current characteristics. Therefore, we can consider the system a circuit with a constant-current power source connected immediately before the rectifier. In wireless power transfer, once the AC power has been received by the secondary side coils, the power is first rectified and converted to DC. A circuit diagram of the rectifier and its waveforms is shown in Fig. 8.15. The circuits depicted in (a) and (b) are identical. The circuit is simply represented in both diamond form and rectangular form, to suit preferences. All the waveforms shown in this section refer to the waveforms in the system after the system has settled to steady state. As there is no smoothing capacitor present in Fig. 8.15, the waveforms have a large amount of ripple. However, the voltage and current are always positive and never become negative. In the case in which the load is a resistance, then without a smoothing capacitor, the waveforms of the voltage and current both consist of the absolute value of a sine wave. Note that the circuit is normally not used in this configuration without smoothing. In practice, the ripple is usually eliminated when converting the power to DC by using a smoothing capacitor, as shown in Fig. 8.16. The load is connected after the voltage and current have been stabilized to a constant DC waveform. The smoothing capacitor is simply mitigating the ripple. Hence, the relationship between the voltage and current at steady state is the same before and after the DC link, and RLDC = RDC  in Fig. 8.17. Therefore, this quantity is often represented simply as RLDC (Fig. 8.13). When the circuit is S–P type, it has constant-voltage characteristics. Similar to how a large voltage is generated when a coil is connected in series to a constantcurrent coil, a large current is generated when a capacitor is connected in parallel to a constant-voltage source. This structure is generally not preferred. However, it is still shown in Fig. 8.18 for comparison with the S–S type, which has the characteristics of a constant-current source. The figure shows that, in this structure, I AC is not sinusoidal. [Note: To prevent a large current from flowing through the resistor, we have increased the resistance compared with the case of a constant-current source. For the

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8 Magnetic Resonance Coupling Systems

(a) Full bridge rectifier (diamond)

(c) Input AC voltage and current

(b) Full bridge rectifier (rectangular)

(d) Output DC voltage and current

(e) Diode current Fig. 8.15 Rectifier diagrams (two representations) and voltage and current waveforms

case of a constant-voltage source, a choke coil is usually inserted in series with the source.]

8.5.2 AC/DC Voltage, Current, and Equivalent Load Resistance When analyzing a rectifier that converts AC–DC, it is necessary to be careful when calculating the root mean squared (RMS) value, waveform amplitude, and average value, or else there may be errors in the calculation of the voltage and amplitude. Therefore, we explain how to handle the AC/DC waveforms before and after the rectifier.

8.5 Preliminary Knowledge for Discussing Efficiency …

233

(a) Rectifier and smoothing capacitor

(b) Input voltage and current

(d) Current before and after smoothing capacitor

(c) Current after rectifier

(e) Output voltage and current

Fig. 8.16 Reduction of ripple through smoothing capacitors (constant-current source) Fig. 8.17 Impedance before and after the DC link

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8 Magnetic Resonance Coupling Systems

(a) Rectifier and smoothing capacitor

(b) Input voltage and current

(d) Current before and after smoothing capacitor

(c) Current after rectifier

(e) Output voltage and current

Fig. 8.18 Rectification of constant-current source and smoothing capacitor (constant-voltage source)

8.5.2.1

Fundamental Wave and 3rd Harmonic of a Square Wave

Here, we explain square waves, fundamental waves, and higher harmonics (3rd harmonic, 5th harmonic, etc.) First, we discuss the final waveforms in S–S type systems. When the primary side is driven by a square wave, the primary side current is a sinusoidal wave. Placing a smoothing capacitor in the load side makes the secondary side voltage become approximately equal to a square wave, and the current approximately equal to a sinusoidal wave. We focus our discussion on the square wave voltage on the primary side and the sinusoidal wave current on the secondary side. First, the square wave can be expressed as

8.5 Preliminary Knowledge for Discussing Efficiency …

235

  1 1 4 V1DC sin(ωt) + sin(3ωt) + sin(5ωt) + . . . π 3 5 ∞ 4 1 = V1DC sin{(2n − 1)}. π 2n − 1 n=1

V1DC =

(8.17)

As shown here, the wave can be synthesized from the fundamental wave and the odd higher harmonics (3rd harmonic, 5th harmonic, etc.). Notably, if the amplitude of the square wave is defined to be 1, then the amplitude of the fundamental wave is 4/π. Note that this value refers to the wave amplitude, not the RMS value. The amplitude of the 3rd harmonic is 1/3 of the amplitude of the fundamental, and the amplitude of the 5th harmonic is 1/5 of the amplitude of the fundamental. The amplitude decreases as the harmonic number increases. The wave synthesized from the fundamental wave, the 3rd harmonic wave, and the 5th harmonic wave, when V 1DC = 1 V, are all shown in Fig. 8.19. Figure 8.20 shows the wave synthesized from all harmonics up to the 19th harmonic and the wave synthesized from all harmonics up to the 99th harmonic. The synthesized wave approaches a square wave as the harmonic number increases. In other words, the voltage v1 of the fundamental wave included in a square wave with an amplitude of 1 on the primary side is given by the following equation (the voltage waveform is shown in Fig. 8.21).

(a) Fundamental and higher harmonics

(b) Synthesized wave

Fig. 8.19 Fundamental, 3rd harmonic, and 5th harmonic

(a) 19th harmonic Fig. 8.20 Higher harmonics

(b) 99th harmonic

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8 Magnetic Resonance Coupling Systems

(a) Primary side power source (square wave)

(b) Voltage waveform

Fig. 8.21 Power source (square wave) and voltage waveforms

v1 f =

4 V1DC sin(ωt) π

(8.18)

In other words, considering only the amplitude, the relationship between the amplitude V 1m of the fundamental wave and the amplitude V 1DC of the square wave is as follows. V1m =

4 V1DC π

(8.19)

As we will explain later, it is sufficient to consider just the fundamental wave when analyzing the power transmission efficiency. The RMS √ value of the fundamental wave can be calculated by dividing the amplitude by 2. Using the amplitude V 1DC of the square wave, the voltage v1f using the RMS value V 1f of the fundamental wave is given by the following equation. v1 f

√ 1 4 2 2 V1DC sin(ωt) = V1DC sin(ωt) = V1 f sin(ωt) =√ π 2π

(8.20)

In other words, extracting the RMS value V 1f yields the following equation. V1 f

√ 2 2 V1DC ≈ 0.90V1DC = π

(8.21)

In addition, the voltage is converted to a sine wave once during the process of being transferred to the secondary side, but as the voltage cannot exceed the voltage of the smoothing capacitor, the voltage on the secondary side becomes a square wave and becomes equal to the voltage V 2DC of the smoothing capacitor. The voltage can be considered in a similar way as the voltage on the primary side; therefore, if we let V 2DC represent the amplitude of the square wave, then the RMS value V 2f of the

8.5 Preliminary Knowledge for Discussing Efficiency …

237

fundamental component included in the square wave can be represented as shown in the following equation (Figs. 8.22 and 8.23a). v2 f

√ 2 2 V2DC sin(ωt) = V2 f sin(ωt) = π

Fig. 8.22 DC/AC transform before and after rectifier, RAC and RLDC

(a) Voltage waveform

(b) Current wave form before rectifier Fig. 8.23 Smoothing capacitor voltage and current waveforms

(c) Current waveform after rectifier

(8.22)

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8 Magnetic Resonance Coupling Systems

Collecting the terms for the voltage amplitude, the amplitude V 1m of the sine wave for the fundamental wave is π4 V1DC (amplification by a factor of 1.27), the amplitude of the square wave is V 1DC , and the RMS value V 1f of the fundamental of the sine √ 2 2 wave is π V1DC (amplification by a factor of 0.90). The voltage on the secondary side is similar. Next, we consider the current on the secondary side. The current waveforms before and after the rectifier are shown in Fig. 8.23b, c. During the conversion from AC to DC, there is ideally no loss. Hence, based on the law of conservation of energy, and the definition of the RMS value, we obtain Eq. (8.23). The value is not the product of the amplitudes (V 2m × I 2m ), but rather, the product of the RMS values (V 2f × I 2f ). [Note: The RMS value is defined as the value at which the average powers over one period as represented by the product of the DC current and voltage and the product of the AC current and voltage match]. V2 f · I2 f = V2DC · I2DC

(8.23)

Based on Eqs. (8.21) and (8.23), the relationship between the equivalent DC current (average current) I2DC and the RMS value I 2f of the sine wave is given by the following equation. π I2 f = √ I2DC ≈ 1.11I2DC 2 2

(8.24)

In other words, the numerator and denominator in the relationship between the AC and DC amplitudes for the current are switched with respect to the numerator and denominator in the relationship for the voltage. Representing this relationship in terms of the amplitude of the sine wave yields Eq. (8.25). I2m =

π I2DC 2

(8.25)

Collecting the terms for the current amplitude, the amplitude I 2m of the sine wave is π2 I2DC (amplification by a factor of 1.57), the equivalent DC current (average π I (amplification current) is I2DC , and the RMS value I 2f of the sine wave is 2√ 2 2DC by a factor of 1.11). Therefore, the equivalent load resistance RAC (RL ) for AC is different from the value of the load for DC. The value of the load for AC is given by the following equation. RL =

V2 f I2 f

(8.26)

RLDC on the DC side can be represented by the following equation using the AC equivalent load resistance RAC (RL ) for AC based on Eqs. (8.22), (8.24), and (8.26).

8.5 Preliminary Knowledge for Discussing Efficiency …

R LDC

V2DC = = I2DC

π √ V 2 2 2f √ 2 2 I π 2f

239

=

π2 RAC 8

(8.27)

To clarify the relationship further, we solve for RAC (RL ). RAC =

8 R LDC ≈ 0.81R LDC π2

(8.28)

RAC is 0.81 times RLDC. When setting the value for the optimal load that achieves maximum efficiency, the result will be incorrect if the difference between these values is overlooked; hence, it is necessary to note this point. In summary, from the perspective of the DC side, the load on the AC side becomes smaller; hence, the current I 2 is large, and the voltage V 2 is small. In other words, the resistance RAC (RL ) appears to be small. From the perspective of the AC side, the result is the reverse. The load RLDC on the DC side appears to be large. The voltage V 2DC becomes larger, and the current I2DC becomes smaller. In other words, the resistance RLDC appears to be larger.

8.5.2.2

Square Wave Drive + Resistance (S–S) and Band-Pass Filter Characteristics

The previous section showed how to handle square waves and sinusoidal waves in equations. Based on the previous discussion, we explain the voltage and current waveforms in the primary and secondary sides in detail. Immediately after the rectifier, the AC has been converted to DC. Both square waves and sinusoidal waves (sine waves) coexist in the circuit. First, we will verify the waveform for a case in which there is no rectifier (Figs. 8.24 and 8.25), although this structure is not often used in practice. Although the voltage on the primary side is a square wave, the current on the primary side is a sinusoidal wave. In contrast, for the secondary side, both the voltage and current are sine waves.

Fig. 8.24 Square wave drive (no rectifier)

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8 Magnetic Resonance Coupling Systems

(a) Primary side voltage and current

(b) Secondary side voltage and current

Fig. 8.25 Voltage and current in square wave drive (no rectifier)

Here, we verify the method for calculating the secondary side voltage from the voltage of the square wave on the primary side. In Fig. 8.24, the primary side input voltage V 1 is a square wave and is equal to the power supply voltage V s , and hence, V 1 = V s = V 1DC = 100 V. Based √ on Eq. (8.21), the RMS value of the fundamental component is V 1f = V1DC · 2 2/π = 90.0 V. The resistance RL  is directly visible to the voltage, and the resistance of the AC section is RL (RLAC ) = RL  = 15.7 . Therefore, substituting into Eq. (8.7) for V 2 yields the secondary voltage, and V 2f = 84.5 V. Next, we explain why the current on the primary side is a sinusoidal wave. First, the circuit as viewed from the primary side is equivalent to Fig. 8.26. Z  2 refers to the secondary side impedance including the coupled section (reflected impedance). As the impedance is high for all frequencies except the resonance frequency due to C 1 and L 1, the circuit acts as a band-pass filter (BPF) that lets only the resonance frequency through and cuts off all other frequencies. Calculating the frequency response from V 1 to I 1 , which is equivalent to the transfer function, yields Fig. 8.27a. [Note that we did not consider the secondary resonance, etc. to simplify our explanation of the basic principle. In practice, secondary resonance arises as seen in this chapter as described in the sections on open and short circuits. Hence, it is necessary to be more rigorous.] Fig. 8.26 Overall circuit as viewed from the primary side

8.5 Preliminary Knowledge for Discussing Efficiency …

(a) Transfer function from V1 to I1

241

(b) Imaginary component of input impedance as viewed from power source

Fig. 8.27 Band-pass filter characteristics (overall view)

Except for the frequency band around the resonance frequency, the impact of the secondary side is small, and the impact of Z 2  in the circuit in Fig. 8.26 can be ignored. The overall shape of the impedance is shown in Fig. 8.27b, and a rough estimate of the impedance is given by the sum of jωL 1 and 1/(jωC 1 ). Note that this discussion only applies to the characteristics as considered across a wide band range. Near the resonance frequency related to power transmission, the impedance does not match the rough estimate, as shown in Fig. 8.28b. Using the rough estimate will result in large errors. Hence, it is necessary to use a detailed calculation when analyzing power transmission, as explained in the previous chapters. In other words, the impedance is only low at the resonance frequency, and the current flows only at this frequency. Only the resonant frequency component passes through. As the impedance is high for the 3rd harmonic (300 kHz) and the 5th harmonic (500 kHz), the current does not flow for the higher harmonics (Fig. 8.29). Therefore, only the fundamental component remains, and a sine wave that consists of

(a) Transfer function from 1 to 1

(b) Imaginary component of input impedance as viewed from power source

Fig. 8.28 Band-pass filter characteristics (magnified view)

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8 Magnetic Resonance Coupling Systems

(a) Overall view

(b) Magnified view

Fig. 8.29 Input impedance

only the fundamental component appears as the current. On the secondary side, when the load is a resistance, and when the primary side current consists of a sinusoidal wave of only the fundamental component, the induced voltage is also a sine wave. Consequently, the secondary side voltage is a sine wave, and the secondary side current is also a sine wave. For example, this result is easier to understand if we decompose the wave into its separate frequencies, or f 1 = 100 kHz, f 3 = 300 kHz, and f 5 = 500 kHz. The current for each frequency is shown in Eqs. (8.29) through (8.31). I f 1 is large compared with both I f 3 and I f5. It is apparent that the fundamental wave constitutes most of the wave. If1 =

V 100 = = 6.36 Z f1 15.7

(8.29)

If3 =

V 100 = 0.12 = Z f3 835.3

(8.30)

If5 =

V 100 = 0.07 = Z f5 1503.8

(8.31)

The transfer function from the supply voltage V 1 on the primary side to the voltage V 2 and current I 2 on the secondary side is shown in Fig. 8.30. These figures show that the response is a band-pass filter for these quantities as well. The voltage V 2 on the secondary side as viewed from V 1 is equivalent to the power amplification AV .

8.5.2.3

Square Wave Drive + Rectifier + Resistance (S–S)

Next, we consider the case in which only the rectifier has been added (Fig. 8.31). This corresponds to the case in which there is no smoothing capacitor. Although this structure is not often used, we still present it here to verify its behavior. The voltage and current after the rectifier have large ripples and are not suitable for practical use (Fig. 8.32).

8.5 Preliminary Knowledge for Discussing Efficiency …

(a) Transfer function from 1 to 2

(b) Transfer function from 1 to 2

Fig. 8.30 Band-pass filter characteristics between the transmitter and receiver

Fig. 8.31 Square wave drive (no smoothing)

(a) Primary side voltage and current

(c) Diode current

(b) Secondary side voltage and current

(d) Load voltage and current

Fig. 8.32 Voltage and current in square wave drive (Rectifier and resistor; No smoothing)

243

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8 Magnetic Resonance Coupling Systems

Fig. 8.33 Rectifier + smoothing capacitor

8.5.2.4

Square Wave Drive + Rectifier + Smoothing Capacitor + Resistance (S–S)

Next, we consider the case in which there is a smoothing capacitor after the rectifier. We consider this structure, which is more practical (Figs. 8.33 and 8.34). Note that we exclude the DC/DC converter that performs impedance transformation. The smoothing capacitor functions to maintain the voltage at a constant level. Compared with the case in which there is no rectifier or smoothing capacitor, the voltage and current are more stable, and the waveforms are both DC waveforms without ripple when the load is a resistive load. On the primary side, the voltage is a square wave, and the current is a sine wave. Even with a smoothing capacitor, the behavior of the primary side is the same. In contrast, the voltage on the secondary side V 2 becomes a square wave when a smoothing capacitor is added. As mentioned above, V 2 took the form of a sine wave, but as there is a smoothing capacitor, in this case, the waveform is a smoothed square wave. Furthermore, the current is approximately a sine wave. In this section, we verify the method for calculating the secondary side voltage from the square wave voltage on the primary side. In Fig. 8.33, the input voltage V 1 on the primary side is a square wave, and V 1 is equal to the power supply voltage V. Based on Eq. (8.21), the RMS value of the V s . Hence, V 1 = V s = V 1DC = 100√ fundamental component is V 1f = 2 2/π = 90.0 V. If we set the resistance in the DC section to RL  (RLDC ) = 19.4 , then based on Eq. (8.28), the equivalent load resistance in the AC section becomes RL (RLAC ) = RL  (RLDC ) = 19.4 × 8/π2 = 15.7 . Substituting into Eq. (8.7) for V 2 , we obtain the fundamental component that appears in the secondary side voltage, which is V 2f = 84.5 V. Due to the effect of the smoothing capacitor, the voltage V 2 is a DC voltage. Based on the relationship π V 2f = 93.8 V. As the secondary side in Eq. (8.22), we obtain V 2 = V 2DC = 2√ 2 consists of a resistance, the calculation can be performed as shown above.

8.5.2.5

Square Wave Drive + Rectifier + Battery (S–S)

Finally, we consider a structure in which a constant-voltage load consisting of a secondary battery is connected after the rectifier (Fig. 8.35). Note that, as we have

8.5 Preliminary Knowledge for Discussing Efficiency …

(a) Primary side voltage and current

(c) Diode current

245

(b) Secondary side voltage and current

(d) Current before and after smoothing capacitor

(e) Load voltage and current Fig. 8.34 Voltage and current in square wave drive (Rectifier + smoothing capacitor + resistor)

Fig. 8.35 Rectifier + secondary battery

246

8 Magnetic Resonance Coupling Systems

(a) Primary side voltage and current

(b) Secondary side voltage and current

(c) Diode current

(d) Battery voltage and current

Fig. 8.36 Voltage and current in square wave drive (rectifier and battery, V 2 = 100 V)

excluded the smoothing capacitor and the DC/DC converter that performs impedance transformation, the current I L that flows into the battery has a large ripple (Fig. 8.36). Here, we verify the calculation method for the case in which a constant-voltage load, such as a battery, is present. In the case in which the load consisted of a resistance as discussed above, we presented the calculations for the secondary side voltage using the square wave voltage on the primary side. In the case of a battery, as the secondary side voltage is fixed, we show the calculation of the secondary side current from the square wave voltage on the primary side here. In Fig. 8.35, the input voltage V 1 on the primary side is a square wave, and V 1 is equal to the power supply voltage V. Based on Eq. (8.21), the RMS value of the V s . Hence, V 1 = V s = V 1DC = 100 √ 2 2 fundamental component is V 1f = π V 1 = 90.0 V. Next, due to the presence of the = 100 V. √Based battery, the secondary side voltage V 2 is also DC, and V 2 = V 2DC √ on the relationship in Eq. (8.21) or (8.22), we obtain V 2f = 2 π 2 V 2DC = 2 π 2 V 2 = 90.0 V. Based on the relationship in Eq. (8.4) among RL , V 1 , and V 2 , we obtain average RL (RLAC ) = 16.8 . As I 2 = V 2 /RL = 90.0/16.8 = 5.3 A, the equivalent √ current I2DC = I L that flows into the battery is given by I2DC = 2 π 2 I 2 = 4.8 A based on the relationship in Eq. (8.24). As a battery is present on the secondary side, the secondary side voltage is fixed, and it is possible to calculate the current in this manner.

8.5 Preliminary Knowledge for Discussing Efficiency …

8.5.2.6

247

Fundamental Component Power and Harmonic Component Power

In cases in which the current is a sine wave, the fundamental is the only component that affects power transmission, and it is possible to ignore the higher harmonics. This is a major characteristic of sine wave currents. Here, we explain this point using equations. Note that, for simplicity, we define the power factor of the fundamental component to be equal to 1. The voltage is a square wave and can be expressed as shown in Eq. (8.32). In cases in which the current is a sine wave, it can be expressed by Eq. (8.33). The power can be represented as the product of Eqs. (8.32) and (8.33), which results in Eq. (8.34).   1 1 4 V1DC sin(ωt) + sin(3ωt) + sin(5ωt) + . . . π 3 5 ∞ 4 1 V1DC sin{(2n − 1)ωt} = π 2n − 1 n=1

v1 (t) =

i 1 (t) = I1m sin(ωt)

(8.32) (8.33)

p1 (t) = v1 (t)i 1 (t) =

  1 1 4 V1DC I1m sin(ωt) · sin(ωt) + sin(ωt) · sin(3ωt) + sin(ωt) · sin(5ωt) + . . . π 3 5

=

∞ 4 1 V1DC I1m sin(ωt) · sin{(2n − 1)ωt} π 2n − 1

(8.34)

n=1

First, we consider the power Pf 1 over one period, which can be calculated from the product of the fundamental components of the voltage and current, as written in Eq. (8.34). This yields the result shown in Eq. (8.35). Note that the relationship between V 1DC and V 1m uses the relationship in Eq. (8.19) and assumes the relationship shown in Eq. (8.36). V 1m and I 1m both refer to amplitudes, and V 1f and I 1f both refer to RMS values. Pf 1 = = = = =

1 T 4 ∫ V1DC I1m sin(ωt) sin(ωt)dt T 0π ω 2π/ω ∫ V1m I1m sin(ωt) sin(ωt)dt 2π 0 ω 2π/ω 1 − cos(2ωt) ∫ dt V1m I1m 2π 0 2 V1m I1m ω 2π/ω ∫ {1 − cos(2ωt)}dt 2 2π 0 2π/ω

1 ω V1 f I1 f T− sin(2ωt) 2π 2ω 0

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8 Magnetic Resonance Coupling Systems

= V1 f I1 f

ω 2π = V1 f I1 f 2π ω

(8.35)

2π ω

(8.36)

ω = 2π f ⇔ T =

Next, we consider the term in the product of a specified harmonic of the voltage and the fundamental component of the current represented between the parentheses in Eq. (8.34). Here, we focus on the equation for the term that involves the 3rd harmonic, which is shown in Eq. (8.37). As we have excluded the coefficient, we define this value as P f 3 . T

P f 3 = ∫ sin(ωt) · sin(3ωt)dt

(8.37)

0

Using the assumption in Eq. (8.36), we obtain the following equation, which yields P f 3 = 0. This shows that the third harmonic does not transmit any power. T 1T P f 3 = ∫ sin(ωt) · sin(3ωt)dθ = − ∫{cos(4ωt) − cos(−2ωt)}dt 20 0

T

T 1 1 1 =− sin(4ωt) − sin(2ωt) =0 2 4ω 2ω 0 0

(8.38)

In Eq. (8.38) above, we used the following formula for the product-to-sum formula for trigonometric functions. 1 sin(α · β) = − {cos(α + β) + cos(α − β)} 2

(8.39)

Here, we focused on the 3rd harmonic, but the analysis for the 5th harmonic, 7th harmonic, etc., is the same. As shown above, power generated in the higher harmonics is not transmitted. In other words, in wireless power transfer, it is sufficient to consider only the fundamental component.

8.5.3 Concept of Impedance Transformation by Using a DC/DC Converter Next, we discuss impedance transformation using power electronics, which is the theme of Sect. 8.5. DC/DC converters are one example of a transformer that can optimize impedance (Fig. 8.37). Normally, DC/DC converters are used for transforming voltages. For example, DC/DC converters can be used to make a 5 V input to produce a 12 V output. As the converter is used to transform the voltage, there is no loss in the ideal state. As the law of conservation of energy applies, the input power and

8.5 Preliminary Knowledge for Discussing Efficiency …

249

Fig. 8.37 Impedance transformation through DC/DC converter

output power are equal. In an ideal DC/DC converter where loss can be ignored, we can have an input of 5 V × 4.8 A = 24 W, and an output of 12 V × 2 A = 24 W as an example. From the perspective of an impedance transformer, the converter can be considered as a transformer in which the output is RL  = 12 V/2 A = 6  whereas the input is R DC = 5 V/4.8 A ≈ 1 . As shown above, it is possible to use a DC/DC converter to transform the impedance. Although there are some circuits in which the transformation ratio is fixed, in many circuits, it is possible to change the transformation ratio for the impedance by changing the duty cycle D. Therefore, it is possible to achieve the maximum efficiency constantly by optimizing the impedance constantly by changing the duty cycle in real time. This method is referred to as impedance tracking control (maximum efficiency tracking control). The details are explained in Sect. 8.5.4.

8.5.4 Impedance Transform in a Step-Down Chopper, Step-up Chopper, and Step up/Down Chopper Using a DC/DC Converter There are three main types of DC/DC converters. Here, we describe bidirectional choppers. [Note: As this circuit contains a rectifier, there is no benefit in using a bidirectional chopper instead of a chopper that is not bidirectional. However, in cases in which the rectifier is a PWM converter, it is possible to use the chopper to return power to the primary side by making the PWM converter (AC–DC) operate as an inverter (DC–AC). In these systems, the chopper is very flexible. The PWM converter and inverter both have equivalent circuits.] The three types of choppers are step-down choppers, which can only be used to lower the voltage, step-up choppers, which can only be used to raise the voltage, and step-up/down choppers, which can be used to both raise and lower the voltage. All these types can be used as impedance transform circuits. In Fig. 8.38, we show a circuit that has the functionality to adjust the impedance on the secondary side. In the next section, we use a step-down chopper. In this section, we introduce step-down choppers and step-up choppers. First, we will describe step-down choppers. A circuit for a step-down chopper in a DC/DC converter is shown in Fig. 8.39. The diagrams of the circuit when the switch on top is ON and OFF (or in other words, for T on and T off ) are shown in Fig. 8.40. The operating waveforms are shown in Fig. 8.41. Here, as S–S type circuits have

250

8 Magnetic Resonance Coupling Systems

(a) Step-down chopper

(b) Step-up chopper

(c) Step-up/down chopper Fig. 8.38 Circuit diagram of DC–DC converter in impedance tracking control system Fig. 8.39 Impedance transformation through step-down chopper

(a) When

on

Fig. 8.40 Step-down chopper (for T on and T off )

(b) When

off

8.5 Preliminary Knowledge for Discussing Efficiency …

(b) Gate signal for upper switch

(a) Input and output voltage

(d) Output current

(e) Input power

251

(c) Input current

(d) Output power

Fig. 8.41 Step-down chopper operating waveform

constant-current characteristics, there is a constant-current source on the input side I 2DC = 1 A. As there is a battery connected to the output side, we then set the output voltage to V outDC = 10 V, and define the switching frequency to be 10 kHz (switching period 0.1 ms), the duty cycle to be 0.5, L s = 2 mH, r s = 0.1 , and C DC = 1 mF in our analyses. We define the voltage and current at the input to the DC/DC converter to be V inDC (V 2DC ) and I inDC , respectively. The voltage and current at the output are defined to be V outDC (V  L ) and I outDC (I  L ), respectively. Here, we consider an ideal DC/DC converter. Therefore, the input and output powers are equivalent to each other. The throughput rate D (duty cycle) shown in Eq. (8.40) represents the proportion of the cycle for which T on (0D1). D is based on the upper switch. Based on the law of conservation of energy, the average powers of the input and output are equal over one period. When T on, it implies I inDC = I L = I outDC, and when T off, it implies I inDC = 0, I L = I outDC, which yields Eq. (8.41). Using Eq. (8.40) and summarizing yields Eq. (8.42). D= VinDC IinDC

Ton Ton = T Ton + Toff

Ton Ton + Toff = VoutDC IoutDC T T

VinDC IinDC D = VoutDC IoutDC

(8.40) (8.41) (8.42)

252

8 Magnetic Resonance Coupling Systems

Furthermore, if we assume that the inductance of the smoothing impedance (smoothing coil) is sufficiently large so that the current flowing through the smoothing impedance is constant at steady-state and also assume that the capacitance of the smoothing capacitor is sufficiently large so that the current ripple can be ignored and that the current can be considered constant, then we obtain I inDC = I outDC in Eq. (8.41). The relationship between the input and output voltages is expressed by the following equation (Fig. 8.41a). VoutDC = DVinDC

(8.43)

As the time-averages of the input power and output power are equal, we obtain the following equation. VinDC I¯inDC = VoutDC IoutDC

(8.44)

Based on Eqs. (8.43) and (8.44), the following equation for the average input current can be obtained. I¯inDC = D I outDC

(8.45)

The relationship among the voltage, the actual load RL  across the output side of the DC/DC converter, and the current is given by the following equation. VoutDC = R L IoutDC

(8.46)

Based on Eqs. (8.43), (8.45), and (8.46) above, the impedance RL (RLDC ) virtually visible from the input side is given by the following equation. R LDC =

VinDC R = L2 D I¯inDC

(8.47)

As the range for the throughput ratio D is (0  D  1), the step-down chopper can make the virtually visible load RLDC to be larger than the actual load RL  . In other words, the range over which the load can be varied is given by the following equation. R L R LDC ∞

(8.48)

The step-up chopper can be considered in a similar manner. We show the circuit for a chopper in a DC/DC converter in Fig. 8.42, the circuit for T on and T off in Fig. 8.43, and the operating waveform in Fig. 8.44. The conditions are the same as those for the step-down chopper. As there is no loss in a DC/DC converter, the input and output powers are equal. Based on the law of conservation of energy, the average powers for the input and output over one period are equal. When T on , it implies that I inDC = I L = I outDC , and when T off , it implies that I inDC = I L , I outDC = 0. Therefore, we obtain

8.5 Preliminary Knowledge for Discussing Efficiency …

253

Fig. 8.42 Impedance transformation through step-up chopper

(b) When

(a) When Fig. 8.43 Step-up chopper (T on , T off )

(a) Input and output voltage

(d) Output current

(b) Gate signal for upper switch

(e) Input power

(c) Input current

(f) Output power

Fig. 8.44 Operating waveform of step-up chopper

Eq. (8.49). Using Eq. (8.40) and summarizing the results, we obtain Eq. (8.50). VinDC IinDC

Ton + Toff Ton = VoutDC IoutDC T T

(8.49)

254

8 Magnetic Resonance Coupling Systems

VinDC IinDC = VoutDC IoutDC D

(8.50)

In addition, if we assume that the inductance of the smoothing coil is sufficiently large, we can ignore the current ripple and treat the current as a constant value. Then, in Eq. (8.49), we obtain I inDC = I outDC and obtain the following equation for the relationship between the input and output voltages (Fig. 8.44a). VoutDC =

VinDC D

(8.51)

As the time-averages of the input power and output power are equal, we obtain the following equation. VinDC IinDC = VoutDC IoutDC

(8.52)

Based on Eqs. (8.51) and (8.52), the equation for the average input current can be obtained as follows. IinDC =

IoutDC . D

(8.53)

The relationship among the voltage, the actual load RL  on the output side of the DC/DC converter, and the current is as follows. VoutDC = R L IoutDC

(8.54)

Based on Eqs. (8.51), (8.53), and (8.54), the impedance RL (RDC ) virtually visible from the input side is shown in the following equation. R LDC =

VinDC = D 2 R L IinDC

(8.55)

As the range for the throughput ratio D is (0D1), the step-up chopper can make the virtually visible load RLDC be smaller than the actual load RL  . In other words, the range over which the load can be varied is given by Eq. (8.56). If r s is small enough to be ignored as well, then we obtain Eq. (8.57). rs ≤ R L DC R L

(8.56)

0 ≤ R L DC R L

(8.57)

Summarizing the equations above yields the results in Table 8.2.

8.6 Realization of Maximum Efficiency Tracking Control …

255

Table 8.2 Chopper operation Method

Input/output: voltage

Variable impedance value

Variation range

Step-down chopper

VoutDC = DVinDC

RLDC = R L /D2

R L R LDC ∞

Step-up chopper

Vout DC = Vin DC /D

RLDC = D2 /R L

0R LDC R L

8.6 Realization of Maximum Efficiency Tracking Control Through Impedance Optimization Using our established knowledge thus far, we will now study the actual method used for achieving the maximum efficiency. Various technical explanations are provided in this section; however, the process itself, in general, remains consistent. The optimal load that determines maximum efficiency depends on the equivalent load resistance RAC (RL ) on the AC side before the rectifier. Therefore, we shall apply the optimum load RACopt (RLopt ) to obtain the maximum efficiency (ηmax ).

8.6.1 Simple Design for Maximum Efficiency Control is necessary for actually achieving the maximum efficiency. Therefore, it is necessary to assemble a formula from a control viewpoint. Although this part does require knowledge of the modern control theory, we have attempted to explain it to the extent that it can be understood without prior knowledge. In this book, only the feedback control is adopted. There are various control methods. A well-designed control is described in the next section. Here, we shall study a control method that maximizes secondary side efficiency in a very simple manner. Owing to the simplicity of this method, the design of the PID controller has to be determined through trial and error which undermines the stability of the system; however, it can still be operated if the parameters are selected carefully. It works adequately despite imperfections while making it easier for us to understand the whole picture. As shown in Fig. 8.45, in this method, the DC/DC converter, i.e., the

Fig. 8.45 Circuit diagram for achieving maximum efficiency through a step-down chopper control

256

8 Magnetic Resonance Coupling Systems

control target (plant) which is shown in a black box, does not need to be modeled. The primary side input voltage and the mutual inductance are fixed and known. Let us now examine the input/ output voltage of the plant (DC/DC converter). We have already obtained the voltage condition expression (v2dc ) that provides maximum efficiency; therefore, we only need to determine the duty ratio D to obtain a relationship between v2dc and the battery voltage (E) of the load to be finally charged. However, it should be noted that voltage varies between AC and DC. The optimal load RACopt (RLopt ) that provides maximum efficiency is obtained using the optimal load equation which is expressed as follows.

RACopt

   (ω0 L m )2 = r2 + r2 . r1

(8.58)

Substituting this equation in the equation for the effective voltage V 2 of the secondary side resonator results in V2∗

 =

r2 ω0 L m  V1 . r1 r1r2 + (ω0 L m )2 + √r1r2

(8.59)

The obtained secondary side voltage can achieve maximum efficiency if controlled in a way that its effective value is first determined as the command value (target value) and then following that command value instead. The asterisk * at the upper right of V 2 indicates that it is a command value. Although the secondary side voltage and the primary side voltage are both square waves, we intend to use the fundamental wave component V 1 in the formula. Reformulating Eq. (8.22) as Eq. (8.60), and then including Eq. (8.59) in it, results in Eq. (8.61). It should be noted that the primary side voltage shows a similar relationship as well. Equation (8.62) shows the relationship between V 1DC and the effective voltage V 1 , thus V*2DC can be reformulated as shown in Eq. (8.63). √ 2 2 ∗ π ∗ V = ⇔ V2DC = √ V2∗ . π 2DC 2 2  r2 ω0 L m π ∗  V2DC = √ V1 . √ r 2 2 1 r1r2 + (ω0 L m )2 + r1r2 π V1DC = √ V1 . 2 2  r2 ω0 L m ∗  V2DC = V1DC. r1 r1r2 + (ω0 L m )2 + √r1r2 V2∗

(8.60) (8.61) (8.62) (8.63)

Furthermore, if the DC/DC converter works ideally, the duty ratio D is given by eq. (8.64); therefore, the relationship between the input and output voltages can be expressed as in eq. (8.65). Moreover, by including Eq. (8.63), the point of equilibrium

8.6 Realization of Maximum Efficiency Tracking Control …

257

Fig. 8.46 Block diagram of secondary voltage control

D can be obtained. D=

Ton . Ton + Toff

∗ ⇔D= E = DV2DC

(8.64) E

∗ V2DC

.

(8.65)

The maximum efficiency (ηmax ) can be achieved theoretically if the point of equilibrium D is set such that the optimum load is obtained; however, ηmax cannot be achieved in practice. Therefore, to obtain RLopt , feedback control is applied. The corresponding block diagram is shown in Fig. 8.46. It depicts the part of the DC/DC converter, or the black box, where the PID controller is working on a trial and error basis. The v2DC on the right side is a value that is being output and measured. It is also called the control amount as it is the amount being controlled. This amount is fed back and compared with the command value, and the difference (error) thus obtained is input to the controller. The controller then determines the duty ratio d, which is the deviation from the duty ratio D, such that the amount of error (deviation) that has been input to the controller becomes 0. Here, we make use of a commonly used PID controller. Each letter stands for the following. P: proportionality, I: integration, D: differentiation. The ideal PID controller is represented by Eq. (8.66). CPID (s) = K p +

KI + K D s. s

(8.66)

P is multiplied by the amount that has been input to represent the proportional gain K P . Similarly, K I is the integral gain and K D is the differential gain. Additionally, integration has the effect of reducing the deviation to zero, whereas differentiation has the effect of suppressing the vibrations. The major difference in the method described in the next section is that the PID controller here needs to determine the parameters K P , K I , and K D on a trial and error basis, which means that the system stability is not guaranteed. This is because the plant model for the DC/DC converter, the black box, could not be modeled using mathematical formulas, forcing us to design the PID controller on a purely empirical basis. In other words, the controller for the part shown in the frame in Fig. 8.46 is not based on a theoretical model because the DC/DC converter could not be modeled.

258

8 Magnetic Resonance Coupling Systems

Therefore, there is no guarantee for its stability; however, the system can still be operated.

8.6.2 Strict Design for Maximum Efficiency Control In the previous section, because the plant had not been modeled, the PID gain of the controller had to be determined empirically. Therefore, in this section, we will study a method for modeling the DC/DC converter, i.e., the plant, using mathematical equations. Furthermore, we will be designing the system based on a plant model expressed through these mathematical equations [6, 7]. As a result, the entire system can be designed strictly, thus enabling anyone to design a stable controller that does not rely on trial and error. In short, the controller design relies on pole placement based on a modeling that combines the modern as well as the classical control theory. First, in order to get a grasp on the general flow, the block diagram for realizing the maximum efficiency through the secondary side voltage control is shown in Fig. 8.47. With regard to the plant (i.e., the control target or the DC/DC converter), a smallsignal model ΔPv (s), using minute fluctuations, is expressed using a mathematical expression. In other words, first, the transfer function ΔPv (s) between Δd(s) and Δv2DC is obtained, which is a necessary step prior to designing the controller. Next, the controller’s PID gain setting is determined through the pole placement. Prior to this, however, it is necessary to linearize the nonlinear model so that it can be calculated and controlled.

8.6.2.1

Preparations for the Linearization of the Model (Transforming It into a Computable Expression)

The first problem is the switching of the circuit by turning it ON and OFF. At the time of switching, the system, in terms of its mathematical description, becomes discontinuous making its calculation very difficult. Therefore, we counter this problem by modeling the average state of the circuit; in other words, by devising a model using a state space averaging method. Next, even after the circuit is modeled, it still behaves nonlinearly; therefore, it is linearized using a small-signal model. This procedure makes the calculations much easier. Moreover, the primary side input voltage and mutual inductance are fixed and known quantities. The state space averaging method is a method of adding a circuit equation when the switch is ON and OFF, Fig. 8.47 Block diagram of secondary voltage control

8.6 Realization of Maximum Efficiency Tracking Control …

259

and distributing the sum at a ratio of operating time. Therefore, it is also named the averaging method. In this case, as shown in Fig. 8.48, a step-down chopper is used, and the load is a battery. The circuits when the switch SW1 is turned ON and OFF are shown in Fig. 8.49. Moreover, when SW1 is turned ON, SW2 is turned OFF, and vice versa. Furthermore, the current in the step-down chopper at this time is shown in Fig. 8.50. The current iL flowing through the smoothing reactor L is never allowed to be 0 A and is used in a continuous range (continuous current mode). i2DC is treated as an average value. [Note: Analysis becomes difficult when there is no current flowing, that is, when the current is in a discontinuous mode including zero amperes.] When SW1 is turned ON and SW1 is turned OFF, the following Eqs. (8.67) and (8.68) are obtained:

 r 1 

 1 

 d −L L −L 0 i L (t) E i L (t) = + , − C1 0 0 C1 v2DC (t) i 2DC (t) dt v2DC (t)

 r 

 1 

 d −L 0 E i L (t) −L 0 i L (t) = + . 0 C1 0 0 v2DC (t) i 2DC (t) dt v2DC (t)

(8.67) (8.68)

These equations are called state equations in the field of modern control theory, and are expressed using a variable and its derivative. They belong to the basic expression

Fig. 8.48 Maximum efficiency control through step-down chopper control

Fig. 8.49 Circuit diagram when SW1 is turned ON and OFF

260

8 Magnetic Resonance Coupling Systems

Fig. 8.50 Step-down chopper current waveform

methods in modern control and allow for the control to be handled mathematically. State equations have many advantages such as being able to take into account changes in variables that do not appear in the output of classical control transfer functions, and being able to deal with multiple inputs and outputs. Thus, they constitute a significant section in modern control theory. Equations (8.67) and (8.68) are averaged with the duty ratio, that is, the ratio of conductivity d(t)(0 ≤ d(t) ≤ 1). This is according to the state space averaging method. The value of d(t) is updated in real time, and is a function of time t. For the time frame in which SW1 is ON, the following equation can be obtained when weighting is performed with the conductivity d(t); specifically, Eq. (8.67) is multiplied by d and Eq. (8.68) by (1 − d) and then both these equations are added together.

 r d(t) 

 1 

 d −L L −L 0 i L (t) E i L (t) = + − d(t) 0 0 C1 v2DC (t) i 2DC (t) dt v2DC (t) C

8.6.2.2

(8.69)

Formulation Using the State Space Expression (Modern Control Theory)

First, general and basic state space equations in modern control are provided as Eqs. (8.70) and (8.71). If we summarize the Eq. (8.69) and express it in one state space equation, we obtain the following Eqs. (8.72)–(8.75), where, x(t) is the state variable vector, A the system state matrix, B the input matrix, and c the output matrix (in this case, it may be an output vector because there is only one output). Equations (8.70) and (8.72) are called state equations whereas Eqs. (8.71) and (8.73) are called output equations. Equation (8.74), which provides the contents of x(t) shows that it is a variable, and referred to as a state variable because it represents a state. A is a matrix containing the coefficients which represent the variable change and is hence called the coefficient matrix. B is related to the input, and c is related to the output. O is a

8.6 Realization of Maximum Efficiency Tracking Control …

261

zero vector and is ignored because it is only used for the transformation. d x(t) = Ax(t) + bu(t). dt

(8.70)

y(t) = cx(t).

(8.71)

 d E x(t) = A(d(t))x(t) + B . i 2DC (t) dt

(8.72)

v2DC (t) = cx(t).

(8.73)



 i L (t) x(t) = . v2DC (t) ⎡ r d(t) 1 ⎤

 −L L −L 0 A B ⎣ := − d(t) 0 0 C1 ⎦ C c 0 0 1 0

(8.74)

(8.75)

At this point, it should be noted that the output is v2DC as provided in Eq. (8.73), and the input vector u(t) consists of E and i2DC (t) as shown in eq. (8.72).

8.6.2.3

Linearization Using Minute Fluctuations Around the Equilibrium Point

In this case, the transfer function ΔPv (s) is obtained by linearizing Eq. (8.69) using minute fluctuations around the equilibrium point. This is shown in the thick frame in Fig. 8.51. Since Eq. (8.69) is nonlinear, the transfer function cannot be calculated directly. Therefore, the linearization is performed using an arbitrary equilibrium point and the minute fluctuations around it, as shown in Fig. 8.52. The equilibrium point is defined at the time of linearization, and so located that it is not necessary to consider its change with time. Moreover, since the equilibrium point does not vary with time, the temporal change instead needs to be taken into account with minute fluctuations. Therefore, as shown in Eq. (8.76), the equilibrium points of iL (t), v2DC (t), i2DC (t), and d(t) are represented by I L , V 2DC , I 2DC , and D, respectively, and their respective minute fluctuations are given by iL (t), v2DC (t), i2DC (t), and d(t). Using these Fig. 8.51 Block diagram of secondary side voltage control

262

8 Magnetic Resonance Coupling Systems

Fig. 8.52 Linearization at equilibrium point (small-signal model)

representations, the state equation (Eq. 8.77) around the equilibrium point is obtained. Hereafter, we shall consider the equilibrium points and the corresponding minute fluctuations around them.  i 2DC (t) = I2DC (t) + i 2DC (t) i L (t) = I L (t) + i L (t), (8.76) v2DC (t) = V2DC (t) + v2DC (t), d(t) = D(t) + d(t)

 r D 

 V2DC 

 d 0 −L L d(t) i L (t) i L (t) C = + (8.77) − CD 0 v2DC (t) i 2DC (t) − ICL C1 dt v2DC (t) If summarized in state space expressions, the following Eqs. (8.78)–(8.80) are obtained. The breakdown formula is as given in Eq. (8.81). Since the output is v2DC (t), Eq. (8.79) can be obtained from the general output equation (Eq. 8.71). d x(t) = A x(t) + B u(t). dt

(8.78)

v2DC (t) = c x(t).

(8.79)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎡

⎤ − Lr DL V2DC 0 L A B := ⎣ − CD 0 − ICL C1 ⎦ c O 0 1 O

 i L (t) x(t) = X + x(t), x(t) = v2DC (t) 

d L (t) . u(t) = U + u(t), u(t) = i 2DC (t) 



D IL ,U = X= V2DC I2DC



(8.80)

(8.81)

The next step is to obtain a relational equation at the equilibrium point. V 2DC , which can achieve maximum transmission efficiency, is obtained from Eq. (8.63), which makes it possible to obtain I 2DC . Equation (8.82) shows AV , which represents the relationship between the effective values V 1 and V 2 . When this equation is modified using the equivalent load resistance RL given by Eq. (8.83), the result is Eq. (8.84) which gives the effective value I 2 of the secondary side current.

8.6 Realization of Maximum Efficiency Tracking Control …

AV =

263

V2 ω0 L m R L = . V1 r1r2 + r1 R L + (ω0 L m )2

(8.82)

V2 . I2

(8.83)

ω0 L m V1 − r1 V2 . r1r2 + (ω0 L m )2

(8.84)

RL = I2 =

The relationship between the peak values (V 1DC , V 2DC ) and effective values (V 1 , V 2 ) of the square wave on the primary and secondary sides is √ √ 2 2 2 2 V1DC , V2 = V2DC , V1 = π π

(8.85)

which results in the following equation if substituted in Eq. (8.84): √ 2 2 ω0 L m V1DC − r1 V2DC I2 = . π r1r2 + (ω0 L m )2

(8.86)

The relationship between the AC and DC current on the secondary side is given by π I2 = √ I2DC 2 2

(8.87)

Substituting this in Eq. (8.86), we obtain I2DC =

8 ω0 L m V1DC − r1 V2DC π 2 r1r2 + (ω0 L m )2

(8.88)

The circuit configuration of the DC/DC converter is shown in Fig. 8.53. In addition to the aforementioned properties, the equilibrium points I L, V 2DC, I 2DC , and D also satisfy the following equations: Fig. 8.53 DC/DC converter circuit configuration

264

8 Magnetic Resonance Coupling Systems

IL =

I2DC , D

(8.89)

DV2DC = −r I L + E,

(8.90)

E D − r I2DC . D2

(8.91)

V2DC =

If the duty ratio D and the ratio of I L to I 2DC are known, we can derive Eq. (8.89) from the average current provided in Eq. (8.45). Furthermore, when considering the voltage generated in SW2, it becomes DV 2DC , and Eq. (8.91) can be obtained from Eqs. (8.89) and (8.90). When shown collectively, the equilibrium points V 2DC , I 2DC , D, and I L are represented by Eqs. (8.92)–(8.95), respectively. It can be seen that Eq. (8.94) is obtained from Eq. (8.91).  V2DC =

r2 ω0 L m  V1DC . r1 r1r2 + (ω0 L m )2 + √r1r2

8 ω0 L m V1DC − r1 V2DC . π 2 r1r2 + (ω0 L m )2  E + E 2 − 4r V2DC I2DC D= . 2V2DC

I2DC =

IL =

I2DC . D

(8.92) (8.93) (8.94) (8.95)

Next, the model is simplified using the obtained I 2DC in Eq. (8.93). As for the procedure, the linearization is performed using Eq. (8.96), which gives the average value i2DC of the secondary side direct current, which flows into the smoothing capacitor after rectification. Here, the secondary side voltage and current are time-varying; therefore, lower case v2DC and i2DC , respectively, are used for their representation. i 2DC =

8 ω0 L m V1DC − r1 v2DC . π 2 r1r2 + (ω0 L m )2

(8.96)

When this i2DC is linearized around the equilibrium point, the following equation is obtained. i 2DC = −

r1 8 v2DC 2 π r1r2 + (ω0 L m )2

(8.97)

Let us now describe the process for obtaining Eq. (8.97). First, the i2DC in Eq. (8.96) is a function of v2DC ; hence, it is expressed as i2DC (v2DC ) as shown in Eq. (8.98). Furthermore, when v2DC, which is equal to V 2DC at the equilibrium point, is linearly approximated using the Taylor’s expansion up to the first order, the following equation

8.6 Realization of Maximum Efficiency Tracking Control …

265

is obtained.  di 2DC (v2DC )  i 2DC (v2DC ) ≈ i 2DC (V2DC ) + (v2DC − V2DC ) dv 2DC v2DC =v2DC  di 2DC (v2DC )  v2DC = I2DC + dv 2DC v2DC =v2DC   dv2DC  1DC  ω0 L m dV − r  1 dv2DC v =v dv2DC v =V 8 2DC 2DC 2DC 2DC = I2DC + 2 v2DC π r1r2 + (ω0 L m )2   r1 8 v2DC = I2DC + − 2 π r1r2 + (ω0 L m )2 = I2DC + i 2DC . (8.98) Therefore, Δi2DC is expressed by Eq. (8.97) as discussed earlier. In this case, V 1DC is independent of v2DC , i.e., dV 1DC /dv2DC = 0. Further, to obtain the difference from the equilibrium point, the following Eq. (8.99) is used, which is a part of Eq. (8.76). v2DC = v2DC − V2DC

(8.99)

The equilibrium point V 2DC is given to i2DC , as a result, the relationship given by the following equation is also used. I2DC = i 2DC (V2DC )

(8.100)

The following Eq. (8.101) is obtained by substituting Eq. (8.97) in either Eq. (8.77) or (8.78). The model can thus be simplified, and the input can be unified using one d. Further, as previously confirmed from the output equation (Eq. 8.79), the output is Δv2DC (t).

  r −L d i L (t) = − CD − C1 dt v2DC (t)

D L



r1 8 π 2 r1 r2 +(ω0 L m )2

 V2DC  i L (t) C d(t). + v2DC (t) − ICL (8.101)

This can be summarized using Eqs. (8.102)–(8.104) expressed as follows: d x(t) = A x(t) + B u(t), dt v2DC (t) = c x(t),

A B c O



⎡

− Lr ⎢−D − 8 := ⎣ C π 2 C 0

D L

(8.102) V2DC L − ICL

⎤

r1 ⎥ ⎦ (r1 r2 +(ω0 L m )2 ) 1 O

(8.103)

266

8 Magnetic Resonance Coupling Systems

⎧

 ⎪ i L (t) ⎪ ⎪ x(t) = X + x(t), x(t) = ⎪ ⎪ v2DC (t) ⎨ u(t) = D + d(t), u(t) = d(t)

 ⎪ ⎪ ⎪ IL ⎪ ⎪ X = ,U = D ⎩ V2DC

(8.104)

As a result, we were able to formulate the state space equations. We can now derive the transfer function between Δd and Δv2DC by using a conversion formula from the state space equation to the transfer function equation. The conversion formula is shown in Eq. (8.105), where I is an identity matrix, U(s) is the input, Y (s) is the output, and G(s) is the resulting transfer function. In this case, U(s) is Δd, Y (s) is Δv2DC and G(s) is ΔPv (s). Therefore, the transfer function ΔPv (s) between Δd(s) and Δv2DC can be obtained using Eqs. (8.102) and (8.103), and by modifying the Eq. (8.106). Finally, upon expansion, the required transfer function ΔPv (s) can be obtained as given in Eq. (8.107). G(s) =

Y (s) = c(s I − A)−1 b. U (s)

 −1       D   V2DC − Lr v2DC 1 0 L C Pv (s) = . = 0 1 s − r1 − CD − C1 π82 d − ICL 0 1 r1 r2 +(ω0 L m )2 −1      s+ r V2DC −D L L C Pv (s) = 0 1 . r1 D s + C1 π82 − ICL C r1 r2 +(ω0 L m )2  r1 D   s + C1 π82 1 L r1 r2 +(ω0 L m )2   = 0 1 ! ! D "! D " " D r1 r 1 8 − s + − − s + L s + C π2 C L C r1 r2 +(ω0 L m )2   V2DC C

− ICL

.

 r L

(8.106) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

1s +b0 Pv (s) = s 2b+a 1s +a0 r1 a1 = Lr + π82 C r r +(ω ( 1 2 0 L m )2 )  .  rr1 1 ⎪ a0 = LC D 2 + π82 r r +(ω ⎪ 2 ⎪ 1 2 0 Lm ) ⎪ ⎩ 1 b1 = ICL , b0 = LC (r I L − DV2DC )

8.6.2.4

(8.105)

(8.107)

Controller Design (Classical Control Theory and Pole Placement Method)

We have obtained the transfer function model for the DC/DC converter; therefore, we now enter the area of classical control. The next step is to obtain the controller marked through the thick frame in Fig. 8.54. As discussed earlier, one method to theoretically find a value that allows for a stable operation, rather than attempting its solution empirically or by trial and error, is the pole placement method, which we

8.6 Realization of Maximum Efficiency Tracking Control …

267

Fig. 8.54 Block diagram of secondary side voltage control

adopted in this book. Using the pole placement method, the response speed and the control performance can be determined as they depend on the pole placement. The slower the poles, the more stable the operation; however, this also results in a slower response. Therefore, it is important to completely identify this trade-off relationship. First, in order to perform feedback control, the transfer function v2DC / v2DC * of the closed-loop system is obtained. This transfer function is the ratio of actual output to the target value (command value). The command value v2DC * enters from the left, takes the difference from the output, which enters the controller and then enters the plant, whose output, as a result, is v2DC . This can be expressed as Eq. (8.108) and further be rearranged as Eq. (8.109). The numerator and denominator of Eq. (8.109) are Bc1 (s) and Ac1 (s), respectively, where cl stands for closed loop. " ! ∗ − v2DC v2DC = Pv (s)C P I D (s) v2DC

(8.108)

Bcl (s) C P I D (s) Pv (s) v2DC = = ∗ v2DC 1 + C P I D (s) Pv (s) Acl (s)

(8.109)

The plant model is quadratic; therefore, using the PID controller allows for arbitrary pole placement. The controller employs a PID controller, therefore, the following Eq. (8.110) is used, where, K P is the proportional gain, K I is the integral gain, and K D is the differential gain. CPID (s) = K P +

K DS KI + s TD S + 1

(8.110)

In order to use a practical differentiator, noise countermeasures are necessary. Differentiating a noisy signal results in an excessive amplification. For this reason, (τ D s + 1) is inserted into the denominator to remove the harmonic noise generated during differentiation. That is, a first-order low-pass filter (first-order lag filter) is included. τ D (where D stands for differential) is the time constant of the first-order lag filter. A low-pass filter is a filter that only passes low frequencies and attenuates high frequencies. In a first-order case, there is only one s in the denominator; however, in a second-order case, s2 appears. The order is used to change the rate of attenuation with respect to the frequency of the filter; the higher the order, the sharper the cut. In this case, instead of determining each gain through trial and error, the pole placement method is implemented, using which a back calculation is made from a system that is stabilized by the appropriate placement of poles. It is known that the poles are determined only by the denominator and the system is stable when it is a negative real number. However, the poles diverge when it becomes a positive real

268

8 Magnetic Resonance Coupling Systems

number. For example, if a is the only pole (in a first-order case), the inverse Laplace transform becomes eat . If a is positive, it diverges to infinity, and if it is negative, it converges to 0. Furthermore, if it has an inappropriately large imaginary component, it will eventually stabilize; however, the vibrations immediately after the signal input will be intense. Therefore, control is designed to only have negative real number poles that are stable, i.e., without vibration. Pole placement is thus performed in such a way that the closed-loop pole has s = −1000 rad/ s (quadruple root) as given in the following Eq. (8.111). Acl (s) = (s + 1000)4

(8.111)

At the pole position, the vibration would converge faster if set to an even higher negative value; however, if set too high, it could diverge. First, the system´s model is one where ON/OFF switching is performed at a speed of 10 kHz with a state space averaging method; hence, it appears to be averaged and linearized only in slower ranges. Therefore, the state space averaging method can be applied in this case. On the other hand, speeding up the poles of this system would mean that the system is approaching an un-averaged state. Therefore, generally, the pole speed is preferred to be about 1/10th of that value, about 1 kHz in this case. Considering this, it was set to −1000 rad/s = −159 Hz (since ω = 2π f ). By comparing Eqs. (8.107), (8.109), (8.110), and (8.111), the values for K P , K I , K D , and τ bc can be obtained, at which point, we can say that the PID controller has been completely designed. This completes the small-signal modeling and the design of the controller, leaving us only its execution. Due to the control being actually performed using a PC program, it is necessary to convert the signals into digital signals for mounting. In other words, it is necessary to design a discrete controller from a continuous controller. While there are various methods for discretization, here, the discrete controller is designed according to the Tustin´s method, utilizing the following equation: s=

2(z − 1) . T (z + 1)

(8.112)

With this discretized controller C PID (z), the system is complete. The block diagram for the completed secondary side voltage control is shown in Fig. 8.55. Here, z is used because it is the block diagram after discretization, but the diagram itself is the same as the one in Fig. 8.46. The noticeable difference between this system and the one discussed in Sect. 8.6.1 is whether the area around the plant is modeled, and whether Fig. 8.55 Block diagram of secondary side voltage control

8.6 Realization of Maximum Efficiency Tracking Control …

269

the controller design is based on trial and error or on mathematical expressions that guarantee a stable operation. Table 8.3 shows the coil and circuit parameters. Considering the power supply, the input voltage is a square wave input of 100 V. The frequency of the power transmission unit is 100 kHz. Moreover, the carrier frequency in the DC/DC converter is 10 kHz. The results of performing control with (w/: with) and without (w/o: without) the maximum efficiency for varying mutual inductance are shown in Fig. 8.56. The battery voltage in this case is considered 30 V. Therefore, by optimizing the impedance of the equivalent load resistance, that is, by optimizing the secondary side voltage, the maximum efficiency tracking control can be achieved; moreover, maximum efficiency can be realized in any air gap. The results of performing maximum efficiency control for a varying battery voltage are shown in Fig. 8.57. In this case, the mutual inductance is considered to be L m = 25 μH. These results too appropriately display the effect of the maximum efficiency tracking control. [Column] The reason for adopting the plant model order and a PID controller: We will further explain why we opted for a PID controller instead of a PI or other controller. (1) When using a PI controller The expression for the plant and the controller are given by Eqs. (8.113) and (8.114), respectively. We will define both of these by considering their numerators (n: numerator) and denominators (d: denominator), separately. The plant is quadratic and the controller is first order due to the presence of the integrator, therefore, the resulting closed-loop transfer function is of third order. This becomes apparent in Eq. (8.116) because the transfer function contains s3 in the denominator. Table 8.3 DC–DC converter circuit parameters

Fig. 8.56 Mutual inductance maximum efficiency control

DC–DC converter r

0.1



L

1000

μH

C

1000

μF

fc

10

kHz

p

−1000

rad/s

270

8 Magnetic Resonance Coupling Systems

Fig. 8.57 Results for battery voltage maximum efficiency control

ΔPv (s) =

ΔPvn (s) b1 s + b0 ≡ s 2 + a1 s + a0 ΔPvd (s)

C(s) = K P +

sKP + KI Cn (s) KI = ≡ s s Cd (s)

(8.113) (8.114)

The transfer function of the closed loop (cl: closed loop) is expressed in the following Eq. (8.115). Here, we shall consider the numerator and denominator separately. Cn (s) ΔPvn (s)

Bcl (s) v2DC C(s)ΔPv (s) Cn (s)ΔPvn (s) Cd (s) ΔPvd (s) ≡ ≡ = = ∗ n (s) ΔPvn (s) v2DC 1 + C(s)ΔPv (s) Cd (s)ΔPvd (s) + Cn (s)ΔPvn (s) Acl (s) 1+ C Cd (s) ΔPvd (s)

(8.115)

If we only look at the denominator, the following Eq. (8.116) is obtained. Acl (s) = Cd (s)ΔPvd (s) + Cn (s)ΔPvn (s) = s 3 + (K P b1 + a1 )s 2 + (K 1 b + K P b0 + a0 )s + K 1 b0

(8.116)

Therefore, in order for it to become a triple root design, the system is designed such that Acl becomes Eq. (8.117). Acl (s) = (s + ω)3 = s 3 + 3ωs 2 + 3ω2 s + ω3

(8.117)

Furthermore, comparing the coefficients associated with s in Eqs. (8.116) and (8.117), the following three equations, given by Eq. (8.118), can be formulated. The result is two unknown numbers and three equations. For this reason, the controller design parameters K P and K I cannot be determined. ⎧ ⎨

3ω = K P b1 + a1 3ω = K 1 b + K P b0 + a0 ⎩ ω3 = K I b0 2

(8.118)

(2) When using a PID controller (with pseudo-differentiation) The PID control without pseudo-differentiation suffers from harmonic noise, which is why it is not used for mounting. Therefore, a PID controller with a low-pass filter

8.6 Realization of Maximum Efficiency Tracking Control …

271

is essential. When pseudo-differentiation is added to the PID, the design parameters become K P , K I , K D , and τ D . To remove harmonic noise at the time of differentiation, a simple low-pass filter called first-order lag filter is added. Owing to this low-pass filter, differentiation without a harmonic noise becomes possible. The expression for the plant and the controller is shown in Eqs. (8.119) and (8.120), respectively. We will define both of these by considering their numerators and denominators separately. ac2 , bc2 , etc., are representations for each coefficient. ΔPv (s) =

ΔPvn (s) b1 s + b0 , ≡ s 2 + a1 s + a0 ΔPvd (s)

(8.119)

K Ds s(τ D s + 1)K P + (τ D s + 1)K I + K D s 2 KI + = s τD s + 1 s(τ D s + 1) 2 (τ D K P + K D )s + (K P + τ D K I )s + K I = τD s2 + s τ D K P +K D 2 K P +τ D K I s + s + τ1D K I Cn (s) bc2 s 2 + bc1 s + bc0 τD τD = ≡ . = 1 2 2 s + ac1 s Cd (s) s + τD s (8.120)

ΔC(s) = K P +

The feedback of the closed loop (cl: closed loop) is given by Eq. (8.121), which too, we shall define by considering the numerator and denominator separately. Cn (s) ΔPvn (s)

Bcl (s) v2DC C(s)ΔPv (s) Cn (s)ΔPvn (s) Cd (s) ΔPvd (s) ≡ ≡ . = = ∗ ΔPvn (s) v2DC 1 + C(s)ΔPv (s) Cd (s)ΔPvd (s) + Cn (s)ΔPvn (s) Acl (s) 1 + CCdn (s) (s) ΔPvd (s)

(8.121)

Considering only the denominator, Eq. (8.122) is obtained. It can be understood from Eqs. (8.119) and (8.120), this equation contains s4 in the denominator, meaning it is of fourth order, as shown below. Acl (s) = Cd (s)ΔPvd (s) + Cn (s)ΔPvn (s) = s 4 + (bc2 b1 + ac1 + a1 )s 3 + (bc2 b0 + bc1 b1 + ac1 a1 + a0 )s 2 + (bc1 b0 + bc0 b1 + ac1 a0 )s + bc0 b0 . (8.122) Therefore, for this system to become a quadruple root design, it is designed such that Acl becomes Acl (s) = (s + ω)4 = s 4 + 4ωs 3 + 6ω2 s 2 + 4ω3 s + ω4 .

(8.123)

Comparing the coefficients associated with s in Eqs. (8.122) and (8.123), the following four equations, given by Eq. (8.133), can be formulated. The results are four unknown numbers and four equations; as a result, the controller design parameters K P , K I , K D , and τ D can be calculated, and the controller parameters can be determined through pole placement.

272

8 Magnetic Resonance Coupling Systems

⎧ ⎪ ⎪ ⎨

4ω = bc2 b1 + ac1 + a1 6ω2 = bc2 b0 + bc1 b1 + ac1 a1 + a0 ⎪ 4ω3 = bc1 b0 + bc0 b1 + ac1 a0 ⎪ ⎩ ω4 = bc0 b0

(8.124)

8.7 Realization of the Maximum Efficiency and the Desired Power When the wireless power transfer is performed, the positions of the power transmission coil and power reception coil are fixed. This means that the air gap between them is often fixed at a certain value. In this section, the air gap is fixed at L m = 25 μH and k = 0.05. It has been described before that it is necessary to operate the system at the optimum load RLopt in order to achieve maximum efficiency under these conditions. At other load values, the relationship between efficiency and power is shown in Fig. 8.58. The load value at which the efficiency is highest is different from the load value at which the power is highest. Furthermore, once the load value is determined, the efficiency (η) and the received power (P2 ) are determined independently. In other words, when the value of R is determined as the optimum load value RLopt for maximum efficiency, the received power P2 is also determined arbitrarily, and it does not match the desired power on the secondary side. Therefore, it is impossible to achieve maximum efficiency and desired power by simply changing the load resistance value on the secondary side. However, it can be accomplished by tinkering with the circuit and utilizing the flexibility of the time axis. The details are presented in Sect. 8.9. Fig. 8.58 Relationship between load efficiency and power

8.7 Realization of the Maximum Efficiency and the Desired Power

273

8.7.1 Secondary Side Maximum Efficiency and Primary Side Desired Power Even if it is impossible to simultaneously achieve maximum efficiency and desired power on the secondary side alone, the desired power can also be obtained by controlling the primary side. Thus, it is possible to achieve both maximum efficiency and desired power. Therefore, our goal now is to obtain both of these at the same time. This section covers the secondary side maximum efficiency and primary side desired power. (1) Optimal load First, the optimum load RLopt that maximizes efficiency on the secondary side is selected. By adjusting the primary side voltage V 1 to obtain the desired power, both the maximum efficiency and the desired power can then be achieved as shown in Fig. 8.59. The efficiency can be changed by adjusting the load value RL on the secondary side. However, the received power varies simultaneously. On the other hand, changing the primary side voltage does not affect the efficiency, whereas the power alone can be adjusted accordingly as shown in Fig. 8.59. The position of the power peak with

(a) Overall image, log display

(b) Linear display,up to 100 Ω

(c) Detailed display near maximum efficiency Fig. 8.59 Relationship between efficiency and power with respect to load

274

8 Magnetic Resonance Coupling Systems

respect to the load does not fluctuate, as can be seen in Fig. 8.59a. In this figure, the input voltage is varied as 50, 100, and 200 V. For example, when setting the secondary side to the optimum load value RLopt = 15.7  for maximum efficiency, V 1 may simply be adjusted to obtain the desired power. For instance, if the desired value is 567 W, V 1 can be adjusted to 100 V after setting the optimal load RLopt to 15.7 . Furthermore, as the power is proportional to the square of the voltage, doubling the voltage V 1 quadruples power P2 ; and if V 1 is quadrupled (4V 1 ), P2 changes to 16P2 . Thus, the power can be controlled on the primary side. To summarize, the efficiency control is performed on the secondary side, whereas the desired power control is performed on the primary side. The parameters that can be changed for wireless power transfers are the mutual inductance L m , and the load RL related to the primary side voltage V 1 and the air gap. However, the coil position is generally fixed; hence, the mutual inductance L m is already determined. Therefore, only the primary side voltage V 1 and the secondary side load RL can be varied (controlled). The optimum load for maximum efficiency is given by the Eq. (8.125). Substituting that value in the efficiency equation gives us the maximum efficiency as Eq. (8.126).  R Lopt = η=

r22 +

r2 (ωL m )2 . r1

(8.125)

(ωL m )2 R L . (r2 + R L )2 r1 + (ωL m )2 r2 + (ωL m )2 R L

(8.126)

The last remaining parameter is the primary side voltage V 1 . The secondary side current is given by Eq. (8.127), therefore, considering the relationship of P = I 2 R, the received power P2 of the secondary side load can be given by Eq. (8.128). I2 = − R L (ωL m )2

P2 = # r1 (r2 + R L ) + ω2 L 2m

jωL m V1 . r1 (r2 + R L ) + ω2 L 2m

2 $2 V1

r1 (r2 + R L ) + ω2 L 2m ⇔ V1 = ωL m

(8.127)  P2 . (8.128) RL

In other words, only the primary side voltage V 1 can be changed (controlled). Therefore, by adjusting the voltage on the primary side, it is possible to control the secondary side to achieve the desired power. This premise only holds under the condition that the mutual inductance L m is known. Moreover, the mutual inductance can be estimated from both the primary and secondary side; therefore, it can be regarded as a known quantity (see Sect. 8.10). In conclusion, the maximum efficiency and desired power on the secondary side can be realized by controlling the secondary side maximum efficiency and primary side desired power.

8.7 Realization of the Maximum Efficiency and the Desired Power

275

(2) Optimal secondary side voltage The above explanation is focused on the optimal load; however, there is another approach possible for achieving the secondary side maximum efficiency, which is introduced here. By applying the load value RL to the secondary side current in Eq. (8.129), the secondary side voltage given by Eq. (8.129) can be obtained. By using this equation and substituting the optimum load value in the secondary side voltage equation, the optimum secondary side voltage that can achieve maximum efficiency can be obtained as shown in Eq. (8.130). V2 =  V2opt =

jωL m R L V1 . r1 (r2 + R L ) + ω2 L 2m

r2 ωL m  V1 . r1 r1r2 + (ωL m )2 + √r1r2

(8.129) (8.130)

Although Eq. (8.130) had already appeared in Eq. (8.59), its meaning was not explained in detail earlier. Although they are the same in the sense that the maximum efficiency can be realized, Eq. (8.130) is used to adjust the voltage V 2 on the secondary side, instead of adjusting the load value RL , as the control target. To realize the maximum efficiency by optimizing the load value, the secondary side voltage V 2 and secondary side current I 2 are first measured to calculate the load value RL ; however, for calculating only the optimized secondary side voltage V 2opt , as in Eq. (8.130), the secondary side current I 2 becomes unnecessary. In Eq. (8.125), for calculating the optimum load RLopt , the primary side voltage V 1 is unnecessary; however, it is clear from Eq. (8.130) that for optimizing the secondary side voltage, the primary side voltage V 1 is required. In other words, this method is different such that in the previous case, V 2 is controlled using I 2 to obtain RLopt , whereas in this case, V 2 is controlled using information about V 1 to obtain the optimal secondary side voltage V 2opt . (3) Map operation of primary desired power control and secondary side efficiency control Figure 8.60 helps with the explanation of a specific example. Let us consider the desired power to be P2 = 500 W. If the input voltage is V 1 = 50.0 V and control is conducted to realize maximum efficiency on the secondary side, i.e., at point ➀, the secondary side voltage is V 2 = 46.9 V, and the power is 140.0 W, which does not reach the desired power. If desired power control is gradually conducted on the primary side, V 1 gradually increases and moves to the upper right along the curve of maximum efficiency (RLopt = 15.7 ). Since the maximum efficiency control is performed on the secondary side, there is no deviation from the straight line, and V 1 increases until it moves to position ➁, where the desired power is achieved. Eventually, V 1 = 94.5 V and V 2 = 88.7 V, at which the desired power of 500 W is obtained. The efficiency is controlled to ensure that it is the maximum efficiency; therefore, it matches the optimum load value RLopt = 15.7 , and is found to be 88.1%.

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8 Magnetic Resonance Coupling Systems

Fig. 8.60 The desired power control on the primary side and the efficiency control on the secondary side

8.7.2 The Primary Side Maximum Efficiency and the Secondary Side Desired Power In the previous section, after the maximum efficiency was achieved on the secondary side, voltage control was performed on the primary side such that the power received on the secondary side would match the desired power. Here, a method will be described in which, the received power on the secondary side is first controlled to match desired power, and the voltage control is then performed on the primary side such that the optimum load and maximum efficiency on the secondary side are achieved. In other words, a method for primary side maximum efficiency control and secondary side desired power control will be described [8]. Considering the equivalent circuit formula, it would seem logical to adjust the load value on the secondary side to achieve optimum load; however, with a different perspective, it becomes possible to realize both the primary side maximum efficiency and the secondary side desired power. First, the desired power P2 can be obtained by adjusting the value of the load on the secondary side. The received power P2 is given by Eq. (8.131). This means that the voltage on the primary side at this time can be expressed as Eq. (8.132). R L (ωL m )2 2 P2 = # $2 V1 , 2 2 r1 (r2 + R L ) + ω L m # $2 r1 (r2 + R L ) + ω2 L 2m V1 = P2 . R L (ωL m )2

(8.131)

(8.132)

The optimum load RLopt for achieving maximum efficiency is given by Eq. (8.133). When this value of RLopt is substituted in Eq. (8.131), the received power P2 , when the maximum efficiency is obtained, is obtained as Eq. (8.134). In other words, the

8.7 Realization of the Maximum Efficiency and the Desired Power

277

power when maximum efficiency is achieved is given by Eq. (8.131). Therefore, the corresponding voltage on the primary side is given by Eq. (8.135). Furthermore, P2 can be obtained using calculation or estimation. 

r2 (ωL m )2 , r1 % # $ r2 r1r2 + (ωL m )2 (ωL m )2 R Lopt =

P2 =

r22 +

2 % '2 V1 , # $ √ & 2 2 2 r1 r1r2 r1r2 + (ωL m ) + r1r2 + ω L m ( ) % '2 # $ )√ & ) r1 r1r2 r1r2 + (ωL m )2 + r1r2 + ω2 L 2m % # V1 = ) P2 . * $ r2 r1r2 + (ωL m )2 (ωL m )2

(8.133)

(8.134)

(8.135)

In other words, when the power control is performed such that the received power P2 becomes the desired power on the secondary side, the value of the load on the secondary side will eventually become the optimum load when the value of V 1 , which achieves maximum efficiency on the primary side, is realized. As can be seen from Eq. (8.136), the secondary side power control is performed by simply measuring the voltage and current and by controlling the secondary side to obtain the desired power. P2 =

V22 = V2 I2 . RL

(8.136)

This can be further explained by a specific example described using Fig. 8.61. Let us consider the desired power to be P2 = 100 W, and then perform the desired power Fig. 8.61 Desired power control on secondary side and the efficiency control on the primary side

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8 Magnetic Resonance Coupling Systems

control on the secondary side. Assuming that the input voltage is V 1 = 20.0 V and power control is performed such that P2 = 100 W on the secondary side only, i.e., at point ➀. The secondary voltage is V 2 = 150.0 V in this case, and the efficiency is as low as 52.0%. From ➀, if maximum efficiency control is gradually performed on the primary side, V 1 gradually increases and moves leftwards along the curve P2 = 100 W. The power control is performed on the secondary side, as a result, there is no deviation from this curve, and V 1 increases until it moves to point ➁, where the maximum efficiency is achieved. Eventually, V 1 = 42.3 V and V 2 = 39.7 V, which coincides with the optimum load value RLopt = 15.7 , with a maximum efficiency of 88.1%. Of course, the received power is still controlled to maintain the desired power, i.e., P2 = 100 W. In this case, the power control is performed on the power receiving side; as a result, there are several merits to this method such as quicker control of power and more stable operation.

8.8 Secondary-Side Power on–off Mechanism (Correspondence to Short Mode and Constant Power Load) In general circuits, power can be interrupted easily, but in wireless power transfer, energy flow control for instant termination of the primary-side power is challenging because of the separation of the primary and secondary sides. Although this control is possible with wireless communication, delays can occur and risk loss of communication with the primary side, which could damage the device in critical cases. As the system must be designed with consideration for the worst-case scenario, it is important to obtain the desired power only by controlling the secondary side. For this reason, the short mode is required. Power is normally adjusted by primary-side voltage control. The received power can be controlled by adjusting the secondary-side power load, but the power cannot be turned off. There are, however, two modes wherein the power can be turned off in a pseudo manner or completely; these are the short mode and the open mode shown in Fig. 8.62. The short mode is a realistic and effective means to achieve this requirement. As discussed in Chap. 5, the secondary impedance at resonance Z in2 , the secondary impedance referred to the primary side Z  2 , and the overall circuit impedance referred to the power supply Z in1 are shown in Fig. 8.63 and are defined by the following equations. Z in2 = r2 + R L

(8.137)

(ωL m )2 Z in2

(8.138)

Z 2 =

8.8 Secondary-Side Power on–off Mechanism …

(a) Short mode

279

(b) Open mode

Fig. 8.62 Short mode (left) and open mode (right)

Fig. 8.63 Secondary impedance referred to the primary side Z 2 

Z in1 = r1 + Z 2

(8.139)

In the short mode, power control is enabled on the secondary side using a short circuit by setting the load value to zero. When RL = 0 , substituting Eq. (8.137) into Eq. (8.138) gives Eq. (8.140). Next, by substituting this into Eq. (8.139), Eq. (8.141) is obtained. When the secondary side is short-circuited with the zero load value, the secondary impedance referred to the primary side Z 2  becomes very large, thereby increasing the primary impedance Z in1 . Z 2 = Z in1 = r1 +

(ωL m )2 r2

(ωL m )2 (ωL m )2 = r1 + Z in2 r2

(8.140) (8.141)

This implies that the primary-side current is reduced by turningoff the power in a pseudo manner. Because a small current still flows on the primary side, a slight power loss occurs. On the secondary side, the current flows normally owing to the constant current characteristic. However, as the secondary-side internal resistance is low similar to that of the primary side, the secondary-side power consumption is minimal despite the larger current. Consequently, the power can be turned off on the secondary side. This turning off mechanism on the secondary side enables

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8 Magnetic Resonance Coupling Systems

independent power control based on decisions on the secondary side, which allows many advantages. On the other hand, in the open mode, there is no secondary-side current flow; hence, there is no power loss on the secondary side, making this mode superior to the short mode. Nevertheless, when the secondary side is open, the secondary impedance referred to the primary side Z 2  becomes very small, and a large current flows on the primary side, causing large power losses. Such a large current is problematic and not recommended for S–S. This will be further described with mathematical expressions as below. In Eq. (8.137), when RL = ∞ , substitution into Eq. (8.138) gives Eq. (8.142). Further substituting this into Eq. (8.139) gives Eq. (8.143). This shows that the overall circuit impedance becomes very small, resulting in a large current. Z 2 =

(ωL m )2 =0 ∞

Z in1 = r1

(8.142) (8.143)

The short mode is therefore adopted and will be discussed further. Such a technology that can turn off the power whenever unnecessary, based on independent decisions on the secondary side without communication with the primary side, has many advantages and is essential for a variety of circuits such as those with constant power loads. Without this technology, the energy losses in the device will be critical and the device will be damaged. The specific requirements for the constant power load are described next. First, the short mode mechanism is indispensable to a circuit without an energy buffer on the secondary side. For instance, if an electronic device or a motor has a constant power load on the secondary side in the absence of a secondary battery or electrical double-layer capacitor to store the extra energy, damages occur when the extra energy is transmitted to the device because of non-dissipation. For example, in the case of the secondary side having an inverter and a motor, the voltage of the smoothing capacitor suddenly increases and destroys the capacitor (Fig. 8.64). On the other hand, if even a small amount of extra energy is consumed

(a) Voltage rise due to power overload

(b) Voltage drop due to power shortage

Fig. 8.64 Energy overload and energy shortage in a smoothing capacitor

8.8 Secondary-Side Power on–off Mechanism …

281

with respect to the received power on the secondary side, the voltage of the smoothing capacitor will decrease drastically and stop the motor. For this reason, it is ideal to make fine adjustments on the secondary side allowing for some margin in the system. This type of load, which strictly requires an exact amount of necessary power, is called a constant power load. It is also the most difficult type of load to handle wireless power transfer. In another example, even if a constant power load has a secondary battery that stores energy and requires no additional energy, when energy is supplied from the power receiving coil, the circuit will be damaged. Thus, the short mode is necessary in such cases.

8.8.1 Half Active Rectifier (HAR) While the concept of constructing the short mode is simple, many problems arise in the design of the actual circuit. For instance, it is not possible to realize the short mode using a step-down chopper (Fig. 8.65). If the upper and lower MOSFET switches SW1 and SW2 are turned on in an attempt to engage the short mode, the charge stored in the smoothing capacitor is released at once, damaging the element. To overcome this issue, another circuit design will be considered. Figure 8.66 illustrates the half active rectifier (HAR) that enables the short mode using two diodes and two MOSFETs (switches) [9–11]. The short mode can also be realized in another HAR design using a PWM converter module, as shown in Fig. 8.67. When using the PWM converter, either the upper or the lower switch is not used. Either

Fig. 8.65 Rectifier DC/DC converter circuits (step-down) (short mode is not possible)

Fig. 8.66 Half active rectifier (HAR)

282

8 Magnetic Resonance Coupling Systems

Fig. 8.67 HAR design with a PWM converter

one can be selected, but in this example, the upper side is always off and operates as a diode. Although the PWM converter has an identical circuit configuration as an inverter, it is referred as a PWM converter because it generates a direct current from alternating current. This not only enables rectification using a body diode and synchronous rectification but also offers many advantages. For example, a PWM converter can be a power source when operated as an inverter to transmit power from the receiving side to the transmission side, thereby enabling bidirectional wireless power transfer. With HAR, the original overall system design shown in Fig. 8.1a is updated as shown in Fig. 8.68. The secondary control is performed by HAR. While wireless communication can be used for efficiency control in the primary side, it is not the main subject here; thus, the focus will be on the secondary-side HAR. HAR is unique in its operation. Normally, the upper and lower sides are not turned on at the same time, but when operating as HAR, the lower MOSFET (switch) is turned on simultaneously. Then, owing to the upper diode, the current flows to the lower MOSFET and the short mode is enabled to cut off power. When the lower MOSFET is off, the diode is capable of full-wave rectification, thereby entering the normal operation of rectification mode. In summary, the short mode is active when the lower MOSFET is on and the rectification mode is active when it is off. It should be noted that the power can be transmitted when the switch is off and not on.

Fig. 8.68 Overall system diagram

8.8 Secondary-Side Power on–off Mechanism …

283

Fig. 8.69 On–off operation modes for HAR

(a) Rectification mode

(b) Short mode

Fig. 8.70 Voltage waveform for HAR operation

When the HAR is on, it operates in the short mode by cutting the power, and when it is turned off, it operates in the rectification mode to transmit power. In the current section, however, the time of the rectification mode with current flow is expressed as T r , and the time for the short mode is given as T s . The operation modes and the voltage waveforms during operation are presented in Fig. 8.69 and Fig. 8.70, respectively. The HAR is operated with a period of a few milliseconds, which is sufficiently fast. Herein, the upper and lower limits are fixed as defined in Eq. (8.144), and V DC is maintained within this range. It is also possible to operate with a constant ratio of T r and T s instead of the upper and lower limits. 

∗ + V Vhigh = V2DC ∗ Vlow = V2DC − V

(8.144)

8.8.2 Maximum Efficiency Control with HAR Thus far, HAR was discussed from the viewpoint of received power control. In this section, maximum efficiency control is discussed from another viewpoint as possible with HAR [12].

284

8 Magnetic Resonance Coupling Systems

With HAR, the voltage V 2DC is determined by the voltage accumulated in the smoothing capacitor. In a simple rectifier, V 2DC cannot be controlled. In contrast, HAR can control V 2DC and thereby V 2 as well. This means that the efficiency control on the power receiving side is possible by voltage control of the smoothing capacitor. It is shown already in Sect. 8.7.1 (2) that the optimum load RLopt for maximum efficiency can be described using V 2opt . Therefore, for maximum efficiency, the short mode can be used to achieve V 2opt . It should be noted that, because of losses in the short mode, it is recommended to operate within the range where this loss is negligible.

8.9 Maximum Efficiency and Desired Power Simultaneously by Secondary Side Control Alone In principle, maximum efficiency and desired power cannot be achieved by secondary side control alone, as described in Sect. 8.7. Therefore, these objectives were achieved by controlling both the primary and secondary sides. However, using the time axis wisely, maximum efficiency and desired power can be achieved simultaneously and solely by secondary-side control regardless of changes in the air gap or load [13, 14]. Specifically, this method is used to control power on the secondary side via the short mode while achieving maximum efficiency control with impedance optimization. However, wireless communication is not necessary with only secondary-side control, i.e., maximum efficiency and desired power are achieved by independent secondaryside control as well. The corresponding overall system is illustrated in Fig. 8.71. Only the value of the mutual inductance L m is required, which can be obtained by the estimation described in Sect. 8.10. Here, this value will be described using the circuit diagram (Fig. 8.72). First, maximum efficiency tracking control is performed by impedance optimization using a secondary-side DC/DC converter. Then, the relationship between the

Fig. 8.71 Overall system diagram

8.9 Maximum Efficiency and Desired Power Simultaneously …

285

Fig. 8.72 Independent secondary-side control diagram

optimum load, efficiency, and received power, which is summarized in Eqs. (8.145)– (8.147), is reintroduced. These values are uniquely determined with a specific RL . However, thus far, the received power is determined by the load value alone. In this case, if the primary-side voltage does not equal the desired voltage, the desired power cannot be obtained on the secondary side.  R Lopt = η=

r22 +

r2 (ωL m )2 r1

(ωL m )2 R L (r2 + R L )2 r1 + (ωL m )2 r2 + (ωL m )2 R L R L (ωL m )2 2 P2 = # $2 V1 2 2 r1 (r2 + R L ) + ω L m

(8.145) (8.146) (8.147)

To address this issue, the HAR is inserted on the secondary side to create the short mode in front of the DC/DC converter. The HAR is able to turn the power on and off; it cuts off power in the short mode and transmits power in the rectification mode. In other words, the received power can be adjusted depending upon the on–off duty ratio of the HAR. The time of the rectification mode with current flow is expressed as T r , and the time of the short mode is expressed as T s . The duty ratio D is defined as follows: D=

Tr Tr + Ts

(8.148)

With the received power at T r given by P2r and that at T s given as P2s , P2s = 0; thus, the average power on the receiving side P2 is expressed as P¯2 =

Tr P2r = D P2r = PL Tr + Ts

(8.149)

This average power corresponds to the power sent to the battery in the load PL . Based on the assumption of no primary-side control, the transmission power P1s in the short mode is substantially reduced, but also slightly lost (P1r  P1s ). Only P1r

286

8 Magnetic Resonance Coupling Systems

can be transmitted, and this leads to the strictly defined average efficiency given by Eq. (8.150). Here, since P1r  P1s , when P1s is ignored, Eq. (8.151) is obtained. Incidentally, while it is out of the scope of the independent secondary-side control, if the primary side detects the short mode on the secondary side and automatically reduces the power output, Eq. (8.151) can be developed without approximation (Fig. 8.71). η=

P2r P1r + P1s

(8.150)

P2r P1r

(8.151)

η= (1) Explanation by time axis

The waveforms for maximum efficiency and power expected from the independent secondary-side control are shown in Fig. 8.73. It is assumed that desired power and maximum efficiency control are possible regardless of changes in power or load. For instance, here we assume the case where the desired power increases during operation. Owing to space limitations, the time axis is compressed to present the situation under two desired power conditions. The power is transmitted to the secondary side at T r , when the maximum efficiency is achieved with the optimal load RLopt . The power is transmitted to the secondary side at T r and is cut off in the short mode at T s . The desired power P2 can therefore be obtained by weighting the power with the on–off time ratio, which is the duty ratio D. As seen in the waveform, the desired power PL is small at first, and thereby the first T r ; the time for the rectification mode is short. When P2 is averaged over time including T s , it gives P2 = PL , and the desired power PL is obtained at the load without interruption. At T s , the secondary impedance is 0 owing to the short mode.

Fig. 8.73 Conceptual diagram for independent secondary-side control

8.9 Maximum Efficiency and Desired Power Simultaneously …

287

Here, we assume that the desired power PL has increased before the second cycle. The second T r , the time for the rectification mode, thus becomes longer. Similar to the first cycle, P2 can be averaged over time including T s , and then P2 = PL , which provides the increased desired power PL at the load without interruption. Since maximum efficiency control is always performed during this process, the maximum efficiency and desired power are achieved solely by secondary-side control. In summary, the power fluctuations in the conventional methods directly affect the load value, thereby decreasing the efficiency. In contrast, the independent secondaryside control for maximum efficiency and desired power enables changing only the desired power while maintaining the optimal load and achieving maximum efficiency. (2) Explanation by R-axis This mechanism will be explained herein from another viewpoint, using a graph with R-axis (Fig. 8.74) instead of the time axis. In Sect. 8.7.1, there was no degree of freedom in time, and as shown in the figure, once the load value was determined, both efficiency and power were uniquely determined. In the example, with the optimal load for achieving maximum efficiency, a maximum efficiency of 88.1% was obtained and the received power was uniquely determined as P2 = 559 W. We will explain the case of independent secondary-side control for maximum efficiency and desired power next using the degree of freedom in time. Figure 8.75 presents the relationship between the R-axis of resistance, efficiency, and power. The entire profile is shown in (a) and the enlarged view is given in (b). The primary Fig. 8.74 Efficiency and received power on the R-axis (with no degree of freedom in time)

(a) Entire profile

(b) Enlarged view

Fig. 8.75 Efficiency and received power on the R-axis with secondary side only control for efficiency and power

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8 Magnetic Resonance Coupling Systems

Fig. 8.76 Operating point on the efficiency map with V 2 –V 1 axes

voltage is fixed at V 1 = 100 V. To achieve maximum efficiency, the optimal load RLopt = 15.7  and maximum efficiency is 88.1% with a fixed secondary voltage V 2 = 93.8 V. In the conventional method, the desired power is also uniquely fixed at P2 = 559 W. However, using the degree of freedom in time in this case, maximum efficiency and desired power are achieved solely by secondary-side control. Using this control method, P2 = 559 W can be obtained as the maximum power, and the power in the range of P2 = 0–559 W can be obtained freely only by secondary control while maintaining maximum efficiency. Strictly speaking, the loss during the short mode needs to be considered, but it is ignored in this consideration as the actual operation is under the condition of low loss. As a reference, an efficiency map using V 2 and V 1 is presented in Fig. 8.76. As the voltages are constant both on the primary and secondary sides, there is only one operating point on the map; P2 = 559 W is the possible maximum power.

8.10 Estimation of Mutual Inductance In general, wireless power transfer achieves communication with a different system as power transmission, but it can also be communicationless. This is done via estimation [15–19]. By estimating the information required from the primary or secondary side, the maximum efficiency or desired power can be obtained without communication. This section describes the method of estimation from the secondary side for achieving maximum efficiency. For achieving maximum efficiency by the secondary side, it is necessary to estimate the mutual inductance L m . As shown below, substituting Eq. (8.152) for optimal load into Eq. (8.153) for efficiency gives the maximum efficiency, but these equations have L m . In addition, the estimation of L m is necessary for power control on the secondary side by the primary voltage V 1 , as L m can be seen from Eq. (8.154) for the received power on the secondary side. In this section, we focus on the estimation

8.10 Estimation of Mutual Inductance

289

of L m and assume that all other parameters are known.  R Lopt = η=

r22 +

r2 (ωL m )2 r1

(8.152)

(ωL m )2 R L (r2 + R L )2 r1 + (ωL m )2 r2 + (ωL m )2 R L R L (ωL m )2 2 P2 = # $2 V1 2 2 r1 (r2 + R L ) + ω L m

(8.153) (8.154)

The estimation must obviously be performed using measurable parameters. Here, we describe the simplest method of calculating from equations of measurable parameters. Specifically, the secondary voltage V 2 will be used. In the following Eq. (8.155) for the secondary current, multiplying by the load value RL gives Eq. (8.156) for the secondary voltage. Substituting Eq. (8.152) for the optimal load value into Eq. (8.156) for the secondary voltage gives Eq. (8.157), which describes the secondary voltage V 2opt for achieving maximum efficiency, enabling maximum efficiency control on the secondary side. However, only L m remains unknown. Here, L m will be estimated. I2 =

jωL m V1 r1 (r2 + R L ) + ω2 L 2m

(8.155)

V2 =

jωL m R L V1 r1 (r2 + R L ) + ω2 L 2m

(8.156)

 V2opt =

r2 ωL m  V1 r1 r1r2 + (ωL m )2 + √r1r2

(8.157)

On the secondary side, the secondary voltage V 2 and secondary current I 2 can be estimated. From Kirchhoff’s voltage law, the following equation is obtained. This is the equation obtained from the Z matrix in Chap. 4.

V1 V2



=

jω0 L m r1 jω0 L m r2



I1 I2

 (8.158)

From Eq. (8.158), the estimated mutual inductance Lˆ m is described in the following equation. Lˆ m =

V1 ±

%

V12 − 4r1 I2 (V2 + r2 I2 ) 2I2 ω0

(8.159)

The estimated values are indicated using the caret symbol (ˆ). Since the sign in Eq. (8.159) may change during calculations, care must be taken when using it. For the

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8 Magnetic Resonance Coupling Systems

Fig. 8.77 Load–efficiency relationship

secondary-side control, by substituting the estimated value from this equation into Eq. (8.157), which describes the secondary voltage V 2opt , maximum efficiency ηmax can be achieved. When actually using the estimation equation, the original form is vulnerable to noise, and the estimated value has some errors. It is therefore necessary to minimize the error using a noise-reducing filter such as a recursive least-squares (RLS) estimator. As discussed above, maximum efficiency control is possible without communication via estimations. The example shown for the maximum efficiency on the secondary side can be used for this purpose. Moreover, even if RL is set to 1  before control and the efficiency is low at first, maximum efficiency can be achieved by changing it to the optimal load of 15.7  (Fig. 8.77).

References 1. K. Takuzaki, N. Hoshi, Consideration of operating condition of secondary-side converter of inductive power transfer system for obtaining high resonant circuit efficiency. The Trans. Inst. Electr. Eng. Jpn. D, A Publ. Ind. Appl. Soc. 132(10), 966–975 (2012) 2. D. Gunji, T. Imura, H. Fujimoto, Stability analysis of secondary load voltage on wireless power transfer using magnetic resonance coupling for constant power load, IEE Jpn. Ind. Appl. Soc. Conf. 2–15, 139–142 (2014) 3. Y. Moriwaki, T. Imura, Y. Hori, A study on reduction of reflected power using DC/DC converter in wireless power transfer system. IEEJ, JIASC 2, II-403–II-406 (2011) 4. T.C. Beh, M. Kato, T. Imura, S. Oh, Y. Hori, Automated impedance matching system for robust wireless power transfer via magnetic resonance coupling. IEEE Trans. Ind. Electron. 60(9), 3689–3698 (2013) 5. M. Kato, T. Imura, Y. Hori, Study on maximize efficiency by secondary side control using DC–DC converter in wireless power transfer via magnetic resonant coupling, in EVS27 (IEEE, 2013), pp. 1–5 6. K. Hata, T. Imura, Y. Hori, Maximum efficiency control of wireless power transfer using secondary side DC–DC converter for moving EV in long distance transmission. IEICE Technical Report WPT2014-33 (2014), pp. 51–56 7. D. Kobayashi, T. Imura, Y. Hori, Real-time maximum efficiency control in dynamic wireless power transfer system. Trans. Inst. Electr. Eng. Jpn. D Publ. Ind. Appl. Soc. 136(6), 425–432 (2016) 8. T. Hiramatsu, X. Huang, M. Kato, T. Imura, Y. Hori, Independent control of maximum transmission efficiency by the transmitter side and power by the receiver side for wireless power transfer. Trans. Inst. Electr. Eng. Jpn. D Publ. Ind. Appl. Soc. 135(8), 847–854 (2015)

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9. D. Gunji, M. Sato, T. Imura, H. Fujimoto, Experimental validation of load voltage control method using secondary converter for wireless power transfer by magnetic resonance coupling, in IEICE Society Conference 2014, BI-8-4 (2014), pp. 61–62 10. D. Gunji, T. Imura, H. Fujimoto, Fundamental research of power conversion circuit control for wireless in-wheel motor using magnetic resonance coupling, in 40th Annual Conference of the IEEE Industrial Electronics Society (2014), pp. 3004–3009 11. D. Gunji, T. Imura, H. Fujimoto, Fundamental research on control method for power conversion circuit of wireless in-wheel motor using magnetic resonance coupling. Trans. Inst. Electr. Eng. Jpn. D Publ. Ind. Appl. Soc. 135(3), 182–191 (2015) 12. G. Yamamoto, D. Gunji, T. Imura, H. Fujimoto, Basic Study on maximizing power transfer efficiency of wireless in-wheel motor by primary and load-side voltage control. Trans. Inst. Electr. Eng. Jpn. D Publ. Ind. Appl. Soc. 136(2), 118–125 (2016) 13. G. Lovison, M. Sato, T. Imura, Y. Hori, Secondary-side-only control for maximum efficiency and desired power in wireless power transfer system, in 41st Annual Conference of the IEEE Industrial Electronics Society (2015), pp. 4825–4829 14. K. Hata, T. Imura, Y. Hori, Dynamic wireless power transfer system for electric vehicles to simplify ground facilities—power control and efficiency maximization on the secondary side, in The 31st Applied Power Electronics Conference and Exposition (2016), pp. 1731–1736 15. V. Jiwariyavej, T. Imura, Y. Hori, Coupling coefficients estimation of wireless power transfer system via magnetic resonance coupling using information from either side of the system, in The 2012 International Conference on Broadband and Biomedical Communications (2012) 16. M. Tsuboka, J. Vissuta, T. Imura, H. Fujimoto, Y. Hori, Secondary parameter estimation for wireless power transfer system using magnetic resonance coupling. IEEJ Tech. Meet. Industr. Instrum. Control, IIC-12-063, 77–80 (2012) 17. D. Kobayashi, T. Imura, Y. Hori, Coupling coefficient estimation using impedance inverter coil in dynamic wireless power transfer system for electric vehicles. IEICE Technical Report WPT2014-54, pp. 21–26 (2014) 18. K. Hata, T. Imura, Y. Hori, Fundamental study on dynamic wireless power transfer system for electric vehicle to simplify ground facilities: primary voltage estimation based on secondary side information. IEICE Technical Report WPT2014-53, pp. 17–20 (2014) 19. V. Jiwariyavej, T. Imura, Y. Hori, Coupling coefficients estimation of wireless power transfer system via magnetic resonance coupling using information from either side of the system. IEEE J. Emerg. Sel. Top. Power Electron. 3(1), 191–200 (2015)

Chapter 9

Repeating Coil and Multiple Power Supply (Basic)

A repeating coil (repeater) is a coil inserted into the air gap formed between a transmitting coil and a receiving coil that operates at the same resonance frequency as both coils. The purpose of a repeating coil is to further extend this air gap such that the power transmission may exceed the limitations of the gap. In addition, a power transmission involving multiple transmitting and receiving coils is referred to as a multiple power supply. Transmitting coils are coils connected to a power source, and receiving coils are coils with connected loads. The repeating coils and multiple power supply use the multiple coils together. Chapters 9 and 10 have many common mathematical formulas, so the formulas common will be described in this chapter. Figure 9.1 shows an example of a repeating coil and a multiple power supply. Figure 9.1a shows the repeating coil, where No. 1 indicates a transmitting coil, No. 2 is a repeating coil, and No. 3 represents a receiving coil. The repeating coil has no load and is used for interrupting the power. When this occurs, the distance between the transmitting coil and receiving coil closes, a cross-coupling results, and k 13 = 0 holds. However, often when the coils are arranged linearly, cross-coupling may be ignored, and k 13 = 0 may be regarded as true. In the case shown in Fig. 9.1a, a repeating coil is placed between the transmitting coil and receiving coil, meaning a coupling is produced between them, which jumps over the repeating coil. In addition, there are

(a) Repeating coil

(b) Multiple loads

Fig. 9.1 Conceptual diagram of repeating coil and multiple power supply © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_9

293

294

9 Repeating Coil and Multiple Power Supply (Basic)

also areas with an unintended coupling, which is referred to as cross-coupling. For example, Fig. 9.1b shows a diagram of a power supply for multiple loads, with an essentially unnecessary cross-coupling between receiving coil No. 2 and receiving coil No. 3.

9.1 Linear Arrangement of Repeating Coil In this section, we discuss the linear arrangement of a repeating coil [1]. In addition, to verify the basic properties of a repeating coil, we examine under the condition that an equivalent circuit does not produce a cross-coupling effect. The parameters of the transmitting and receiving coils and the repeating coils described in this chapter are self-inductance L = 11.0 µH, capacitance C = 12.5 pF, and internal resistance r = 0.77 , with a consistent load of 50 .

9.1.1 Linear Arrangements of Three Repeating Coils Figure 9.2 shows the results from distance sa = 10 mm between the coils in a basic configuration with only transmitting and receiving coils. Here, η is the efficiency, ηr1 is the percentage loss from the internal resistance of the primary coil, and ηr2 is the percentage loss from the internal resistance of the secondary coil. The resonance frequency is 13.56 MHz, and the inter-coil distance is small, and therefore the efficiency is high. If the inter-coil distance is widened to sa = 320 mm, the efficiency drastically worsens, as shown in Fig. 9.3. Therefore, by inserting a repeating coil between the transmitting and receiving coils, as indicated in Fig. 9.4, the efficiency is improved. In Fig. 9.4, the mutual inductance is L 12 = L 23 = 0.542 µH, and the coupling coefficients are k 12 = L 12 /L = 0.049 and k 23 = L 23 /L = 0.049. Figure 9.4 shows the calculation results for the equivalent circuit. When the cross-coupling is η[%]

ηr1 [%]

ηr2 [%]

Efficiency [%]

100 80 60 40 20 0

11

12

13

14

15

Frequency [MHz]

(a ) Transmitting and receiving coils Fig. 9.2 Transmitting and receiving coils (sa = 10 mm)

(b) Efficiency

16

9.1 Linear Arrangement of Repeating Coil

295 η[%]

ηr1 [%]

ηr2 [%]

Efficiency [%]

100 80 60 40 20 0

11

12

13 14 15 Frequency [MHz]

16

(b) Efficiency

(a)Transmitting and receiving coils

Fig. 9.3 Effects when air gap between transmitting and receiving coils is widened (sa = 320 mm)

η[%]

ηr1 [%]

ηr3 [%]

Efficiency [%]

100 80 60 40 20 0

(a) Repeating coil

11

12

13 14 15 Frequency [MHz]

16

(b) Efficiency

Fig. 9.4 Repeating coil and efficiency (sa = 320 mm, sp = 10 mm)

largely ignorable, the equivalent circuit for the repeating coil may appear as a T-type equivalent circuit, as shown in Fig. 9.5. The two values L 2 /2 in the repeating coil are written based on the coupling on both sides, and thus represent the total L 2 for the coils.

(a) Repeating coil equivalent circuit

(b) Repeating coil T-type equivalent circuit

Fig. 9.5 Repeating coils equivalent circuits (3 circuits) when no cross-coupling occurs

296

9 Repeating Coil and Multiple Power Supply (Basic)

9.1.2 Linear Arrangement of Repeating Coil (N Coils) In the previous section, we verified the conditions that occur when one repeating coil is applied; however, the use of two or more repeating coils may also be handled by applying an equivalent circuit. Figure 9.6 shows the configuration when increasing the number of repeating coils to five or ten, the results of which are shown in Fig. 9.7. Figure 9.8 shows an equivalent circuit. At the far left is a transmitting coil, followed by five or ten interposed repeating coils, with a receiving coil at the far right. The equivalent circuit is shown for a case in which the number of repeating coils is n. The mutual inductance is 0.542 µH, the coefficient of coupling is 0.049, and Rn = 50 .

(a)Five repeating coils

(b)Ten repeating coils

Fig. 9.6 Multiple repeating coils (sp = 10 mm)

η[%]

ηr1 [%]

ηr5 [%]

Pin

80 Power [W]

Efficiency [%]

100

60 40 20 0

11

12

13 14 15 Frequency [MHz]

350 300 250 200 150 100 50 0

16

11

12

P RL

P r1

13 14 15 Frequency [MHz]

P r5

16

(a) Five coils η[%]

ηr1 [%]

ηr10 [%]

Pin

80 Power [W]

Efficiency [%]

100

60 40 20 0

11

12

13 14 15 Frequency [MHz]

16

350 300 250 200 150 100 50 0 11

(b) Ten coils Fig. 9.7 Efficiency and power (sp = 10 mm)

12

P RL

P r1

13 14 15 Frequency [MHz]

P r10

16

9.2 K-Inverter Theory (Gyrator Theory)

297

(a) Equivalent circuit

(b)T-type equivalent circuit. Fig. 9.8 Repeating coil (n) and equivalent circuit when no cross-coupling occurs

9.2 K-Inverter Theory (Gyrator Theory) A K-inverter is a circuit that converts the impedance Z connected to the load side into a reciprocal number and has a function proportional to its value. Admittance Y = Z −1 is obtained using the reciprocal value of the impedance. Because the inverter is intended to convert the circuit impedance, it should be pointed out that it differs from the inverter for the DC/AC conversion described in the previous chapter. In addition, a K-inverter is also referred to as a gyrator. A typical K-inverter form appears during magnetic resonance coupling, and thus it is important to know the properties of the K-inverter [2–4]. Figure 9.9 shows a specific circuit. In addition, the power impedance Z 1 is shown through the following formula: Z 1 = − jωL m +

Fig. 9.9 Typical example of K-inverter

jωL m (− jωL m + Z 2 ) K (ωL m )2 = = jωL m + (− jωL m + Z 2 ) z2 Z2

(9.1)

298

9 Repeating Coil and Multiple Power Supply (Basic)

This is very clear when (ωL m ) 2 is K. As can be seen from formula (9.1), when looking at load Z 2 through a K-inverter, it is possible to show Z 1 in a form proportional to K, as well as the reciprocal number of Z 2 , or in other words, the admittance. [Note: K-inverter converts the impedance into the admittance, although there is also a J-inverter that converts the admittance into the impedance. In addition, because this is an impedance and admittance converter, it may be referred to simply as an immitance converter for short. A late conversion using a quarter-wave line also occurs, which is described using a constant distributed circuit with a resonancetype immitance converter [5] and transmission line, which can convert only specific resonance frequencies.]

9.2.1 Calculation Method Using K-Inverter When using a K-inverter, it is easy to make calculations for a linear coil [Note: Capable of being applied to multiple coils]. Here, as shown in Fig. 9.10, we examine a situation involving three coils. When discussing the properties of a K-inverter, the impedance of Z in3 , Z in2 , and Z in1 is shown in formulas (9.2)–(9.4). Z in3 = r3 + R3

(9.2)

Z in2 = r2 +

(ωL m )2 Z in3

(9.3)

Z in1 = r1 +

(ωL m )2 Z in2

(9.4)

Accordingly, Z in2 and Z in1 are shown in formulas (9.5) and (9.6). The impedance of Z in2 is inverted once, but the impedance of Z in1 is inverted again, and thus the properties of Z in1 and Z in3 become closer. For example, when r 1 = r 2 = r 3 = 0 holds, the properties of Z in1 and Z in3 may be understood as being closer from Z in1 = Z in3 = R3 .

Fig. 9.10 Linear coil T-type equivalent circuit

9.2 K-Inverter Theory (Gyrator Theory)

299

Z in2 = r2 + Z in1 = r1 +

(ω L m )2 r2 +

(ω L m ) Z in3

2

(ω L m )2 r 3 + R3

= r1 +

(9.5)

(ω L m )2 r2 +

(ω L m )2 r3 +R3

≈ R3

(9.6)

In addition, the relationship between the induced electromotive force and the current is shown in formula (9.7) for coils 1 and 2, and formula (9.8) for coils 2 and late 3. VLm2 = jωL m I1

(9.7)

VLm3 = jωL m I2

(9.8)

The currents I 1 , I 2 , and I 3 in each coil are shown in formulas (9.9)–(9.11). I1 =

V1 Z in1

(9.9)

I2 =

VLm2 Z in2

(9.10)

I3 =

VLm3 Z in3

(9.11)

The voltage applied to the load is shown in formula (9.12). V3 = I3 R3

(9.12)

The input power P1 is based on formula (9.13), and the power consumption of the load is based on formula (9.14), and thus the efficiency is as indicated in formula (9.15)   P1 = Re V1 I1

(9.13)

  P3 = Re V3 I3

(9.14)

η=

P3 P1

(9.15)

In this way, it is possible to calculate the power and efficiency by setting the numerical formula on an axis of the K-inverter properties. In addition, although three coils were examined here, the above may be similarly expanded to n number of coils.

300

9 Repeating Coil and Multiple Power Supply (Basic)

9.2.2 Dead Zone Here, we confirm the improved prospects from utilizing the K-inverter properties using the dead zone phenomenon [6]. A dead zone is an intermittent phenomenon in which power is no longer transmitted when attempting to do so using a repeating coil. We confirm that when a repeating coil is disposed on the ground, the power is transmitted to a receiving coil above it. First, as shown in Fig. 9.11, if a coil is placed at the end, and regardless of whether there is an even or odd number of ground coils, the power is transmittable. The impedance inverts the same number of times as the number of coils for the input impedance Z in1 from the power source. If the mutual inductance is equal, and no internal resistance occurs, then formula (9.16) holds. When the total number of coils (number of ground coils plus one receiving coil) is an even number (odd number on the ground), the impedance is inverted, and as shown in formula (9.16), all of the coils return to their original state as indicated in formula (9.6), which is equal to RL . [Note: Configurations that finally have a coil at the end were specifically indicated before, focusing on the fact they are included in an “Odd & Odd” or “Even & Even” framework described later, but with a load at the end.] Z in1 = r1 +

(ω L 12 )2 r2 +

(ω L 23 ) ω L 34 )2 r3 + (r +R 2

4

≈

(ω L 12 )2 RL

(9.16)

L

Next, we look at when the receiving coil is not at the end but is in the middle. First, we consider when the ground coils are odd in number. When this happens, a receiving coil is placed on the odd numbered coil, which we marked as Odd & Odd, as indicated in Fig. 9.12. If there is no internal resistance, formula (9.17) holds

Fig. 9.11 When a load coil is placed at the end

9.2 K-Inverter Theory (Gyrator Theory)

301

Fig. 9.12 When there is an odd number on the ground and a receiving coil is placed on an odd numbered coil (Odd & Odd)

for the receiving coil. The third ground coil is a repeating coil, and thus R3 = 0. In addition, when this happens, Z  2 is as indicated in formula (9.18). In other words, the impedance is inverted as indicated in formula (9.19) only for the input impedance Z in1 from the power source. This also includes configurations in which there is a receiving coil at the end under an odd number of ground coils. Z 4 =

(ωL 14 )2 (ωL 14 )2 ≈ r4 + R L RL

Z 2 =

(ωL 12 )2 r2 +

(ωL 23 )2 r3 +R3

Z in1 = r1 + Z 4 + Z 2 ≈

(9.17)

≈0

(9.18)

(ωL 14 )2 RL

(9.19)

Subsequently, we considered when the ground coils are odd in number. However, now a receiving coil is placed on an even numbered coil (Odd & Even). When this occurs, the situation is as indicated in Fig. 9.13. If there is no internal resistance, formula (9.20) holds for the receiving coil. The third ground coil is a repeating coil, and thus R3 = 0. In addition, when this occurs, Z  3 is as shown in formula (9.21). In other words, the state is an open one. In contrast, the input impedance Z in1 from the power source is in a shorted state, as indicated in formula (9.22). In other words,

302

9 Repeating Coil and Multiple Power Supply (Basic)

Fig. 9.13 When there is an odd number on the ground and the receiving coil is placed on an even numbered coil (Odd & Even)

power is not transmitted to the load, and a large current only flows in the transmitting coil. Z 4 =

(ωL 14 )2 (ωL 14 )2 ≈ r4 + R L RL

(9.20)

(ωL 23 )2 ≈∞ r 3 + R3

(9.21)

Z 3 =

Z in1 = r1 +

(ωL 12 )2 ≈ r1 r2 + Z 4 + Z 3

(9.22)

Next, we consider when the ground coils are even in number. When this occurs, a receiving coil is placed on an even numbered coil (Even & Even), as shown in

9.2 K-Inverter Theory (Gyrator Theory)

303

Fig. 9.14. If there is no internal resistance, formula (9.23) holds for the receiving coil. The fourth ground coil is a repeating coil, and thus R4 = 0. In addition, when this happens, Z  3 is as indicated in formula (9.24). In other words, if L 14 and L 25 are equal for the input impedance Z in1 from the power source, as indicated in formula (9.25), RL still holds, and power is transmitted. This also includes configurations in which there is a receiving coil at the end when there is an even number of ground coils. Z 5 =

(ωL 25 )2 (ωL 25 )2 ≈ r5 + R L RL

Z 3 =

(ωL 23 )2 r3 +

(ωL 34 )2 r4 +R4

≈0

(9.23) (9.24)

Fig. 9.14 When there is an even number on the ground and the receiving coil is placed on an even numbered coil (Even & Even)

304

9 Repeating Coil and Multiple Power Supply (Basic)

Z in1 = r1 +

(ωL 12 )2 (ωL 14 )2 RL ≈ RL   ≈ Z5 + Z3 (ωL 25 )2

(9.25)

We will continue with our consideration of cases in which there is an even number of ground coils. When this occurs, a receiving coil is placed on an odd numbered coil (Even & Odd), as shown in Fig. 9.15. If there is no internal resistance, formula (9.26) holds for the receiving coil. The fourth ground coil is a repeating coil, and thus R4 = 0. In addition, when this occurs, Z  4 is as indicated in formula (9.27). In other words, the state is an open one. In contrast, the input impedance Z in1 from the power source is in an open state as indicated in formula (9.28), and no power is transmitted. Z 5 =

(ωL 35 )2 (ωL 35 )2 ≈ r5 + R L RL

(9.26)

Fig. 9.15 When there is an even number on the ground and the receiving coil is placed on an odd numbered coil (Even & Odd)

9.2 K-Inverter Theory (Gyrator Theory)

Z 4 = Z in1 = r1 +

305

(ωL 34 )2 ≈∞ r 4 + R4

(ωL 12 )2 r2 +

(ωL 23 ) r3 +Z 5 +Z 4 2

≈

 (ωL 12 )2   Z 5 + Z 4 ≈ ∞ 2 (ωL 23 )

(9.27) (9.28)

Even with Odd & Even, as indicated in Fig. 9.13, or Even & Odd, as indicated in Fig. 9.15, in both cases dead zones occur in which power is not sent to a load. However, although the current flowing in the very first transmitting coil very dangerously shorts with Odd & Even, it is open with Even & Odd, and thus the difference in safety is significant. Thus, if the ground coils are even in number, it is desirable to arrange the coils linearly. Therefore, a K-inverter is an extremely convenient method when calculating the impedance. [Note: Naturally, because the impedance is determined at different positions, the current flowing in each coil is determined by dividing the impedance-induced electromotive force, which also makes it possible to calculate the power and efficiency.]

9.3 Calculation Using Z-Matrix Accounting for Cross-Coupling (Three Coils) We consider the relationship between three coils, including cross-coupling [1]. A repeating coil is placed under conditions in which the distance between the transmitting and receiving coils is large, and thus the coupling coefficient between the transmitting and receiving coil is almost zero. Thus, any effect may be ignored without a problem, and an equivalent circuit may be arranged in a ladder-shaped T-type equivalent circuit, as shown in Figs. 9.5 and 9.8. However, if we consider cross-coupling between the transmitting and receiving coils, then a T-type equivalent circuit is difficult to present. Multiple cross-coupling will occur during multiple loads. Therefore, in this section, we examine an equivalent circuit using a Z-matrix accounting for the cross-coupling. In addition, the repeating coil is one in which the load value is zero. Thus, if we consider a repeating coil, then coil 1 is a transmitting coil, coil 2 is a repeating coil, and coil 3 is a receiving coil (Fig. 9.16a). Thus, if we consider the power supply to multiple loads, we can consider coil 1 as a transmitting coil, coil 2 as a receiving coil, and coil 3 as another receiving coil (Fig. 9.16b). If there are multiple transmitting coils, they may similarly be considered by accounting for the input voltage. Self-inductance and mutual inductance are each defined by formula (9.29). Here, L 1 is transmitting coil self-inductance, L 2 is self-inductance of the repeating coil or receiving coil, and L 3 is self-inductance of the receiving coil.

306

9 Repeating Coil and Multiple Power Supply (Basic)

(a) Repeatingcoil

(b) Multipleloads

Fig. 9.16 Equivalent circuit diagram accounting for cross-coupling of repeating coil and multiple power supply

⎤ L 1 L 12 L 13 [L] = ⎣ L 21 L 2 L 23 ⎦ L 31 L 32 L 3 ⎡

(9.29)

Here, L 12 = L 21 , L 13 = L 31 , and L 23 = L 32 , and thus formula (9.30) holds. If there is no cross-coupling between the transmitting and receiving coils, then L 13 = 0, and formula (9.31) holds. ⎡

L1 [L] = ⎣ L 12 L 13 ⎡ L1 ⎣ [L] = L 12 0

⎤ L 12 L 13 L 2 L 23 ⎦ L 23 L 3 ⎤ L 12 0 L 2 L 23 ⎦ L 23 L 3

(9.30)

(9.31)

The Z parameters in the circuit are as shown in formula (9.32). ⎡

⎤ Z 11 Z 12 Z 13 [Z] = ⎣ Z 21 Z 22 Z 23 ⎦ Z 31 Z 32 Z 33

(9.32)

The voltage and current may be as shown through formula (9.33). ⎤ ⎡ ⎤ I1 V1 [V ] = ⎣ V2 ⎦ , [I] = ⎣ I2 ⎦ V3 I3 ⎡

(9.33)

9.3 Calculation Using Z-Matrix Accounting for Cross …

307

The relationship between the voltage, current, and impedance is as shown in formula (9.34). [V ] = [Z] [I]

(9.34)

The voltage in the circuit is as indicated in formulas (9.35)–(9.39). Here, r 1 is the internal resistance of coil 1, and r 2 and r 3 are the internal resistances of coils 1 and 2, respectively. ⎧ ⎪ ⎨ V1 = VL1 + VC1 + Vr 1 V2 = VL2 + VC2 + Vr 2 ⎪ ⎩ V3 = VL3 + VC3 + Vr 3

(9.35)

⎧ ⎨ VL1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 V = jωL 12 I1 + jωL 2 I2 + jωL 23 I3 ⎩ L2 VL3 = jωL 13 I1 + jωL 23 I2 + jωL 3 I3 ⎧ 1 ⎪ I1 ⎪ VC1 = ⎪ ⎪ jωC 1 ⎪ ⎪ ⎨ 1 VC2 = I2 ⎪ jωC2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ VC3 = I3 jωC3 ⎧ ⎪ ⎨ Vr 1 = r1 I1 Vr 2 = r2 I2 ⎪ ⎩ Vr 3 = r3 I3

(9.36)

(9.37)

(9.38)

⎧ 1 ⎪ V1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 + I1 + r 1 I1 ⎪ ⎪ ⎪ jωC1 ⎪ ⎪ ⎨ 1 V2 = jωL 12 I1 + jωL 2 I2 + jωL 23 I3 + I2 + r 2 I2 ⎪ jωC2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ V3 = jωL 13 I1 + jωL 23 I2 + jωL 3 I3 + I3 + r 3 I3 jωC3

(9.39)

In other words, formula (9.34) can be written as formula (9.40). ⎡

⎤ ⎡ V1 r1 + jωL 1 + ⎢ ⎥ ⎢ jωL 12 ⎣ V2 ⎦ = ⎣ jωL 13 V3

1 jωC1

jωL 12 r2 + jωL 2 + jωL 23

1 jωC2

jωL 13 jωL 23 r3 + jωL 3 +

⎤⎡ ⎤ I1 ⎥⎢ ⎥ ⎦⎣ I2 ⎦ 1 jωC3

I3 (9.40)

In contrast, the voltage for the load is as indicated in formula (9.41). If coil 2 is a repeating coil, because there is no load resistance, R2 = 0 should be satisfied. In

308

9 Repeating Coil and Multiple Power Supply (Basic)

addition, R2 is the load resistance of coil 2, and R3 is the load resistance of coil 3. ⎧ ⎨ V1 : const. V = −I2 R2 ⎩ 2 V3 = −I3 R3

(9.41)

Formula (9.42) is obtained from formulas (9.39) and (9.41). ⎤ ⎡ V1 r1 + jωL 1 + ⎢ ⎥ ⎢ jωL 12 ⎣ 0⎦ = ⎣ jωL 13 0 ⎡

1 jωC1

⎤⎡ ⎤ I1 jωL 12 jωL 13 ⎥⎢ ⎥ 1 r2 + jωL 2 + jωC2 + R2 jωL 23 ⎦⎣ I2 ⎦ 1 jωL 23 r3 + jωL 3 + jωC3 + R3 I3 (9.42)

Because the Z parameters included the load, the formula corresponding to formula (9.42) is re-defined as formulas (9.43)–(9.45).     V = Z [I]

(9.43)

 −1 [I] = Z  [V ]

(9.44)

⎡ ⎤ ⎡ ⎤ V1 I1   V = ⎣ 0 ⎦, [I] = ⎣ I2 ⎦ I3 0

(9.45)

The power relationship is obtained from formula (9.46).   ⎧ ⎨ P1 = Re V1 I¯1   P2 = PR2 = Re V2 (− I¯2 ) ⎩ P3 = PR3 = Re V3 (− I¯3 )

(9.46)

The efficiency of each coil is indicated in formula (9.47). ⎧ P2 ⎪ ⎪ ⎨ η21 = P1 P ⎪ ⎪ ⎩ η31 = 3 P1

(9.47)

The total efficiency is indicated in formula (9.48). η = η21 + η31 The resonance conditions are shown in formula (9.49).

(9.48)

9.3 Calculation Using Z-Matrix Accounting for Cross …

jωL n +

1 =0 jωCn

309

(9.49)

Thus, it is possible to rewrite the formulas through formula (9.50) under the resonance conditions. ⎧ ⎪ ⎨ V1 = jωL 12 I2 + jωL 13 I3 + r1 I1 V2 = jωL 12 I1 + jωL 23 I3 + r2 I2 (9.50) ⎪ ⎩ V3 = jωL 13 I1 + jωL 23 I2 + r3 I3 This is as indicated in formula (9.51) when in matrix form. ⎡

⎤

⎤⎡ I ⎤ 1 r1 jωL 12 jωL 13 ⎢ ⎥ ⎣ ⎢ ⎥ ⎦ 0 I = jωL 12 r2 + R2 jωL 23 ⎣ 2 ⎦ ⎣ ⎦ jωL 13 jωL 23 r3 + R3 0 I3 V1

⎡

(9.51)

It is possible to derive answers for both multiple power supply and repeating coils from the above formulas. However, it should be particularly noted that, if there is no cross-coupling between the transmitting and receiving coils, then L 13 = 0. Then if the above is used for a repeating coil, the coil load resistance value is set to zero. For example, when coil 2 is operating as a repeating coil, then R2 = 0 should be satisfied, in which case, formula (9.51) becomes formula (9.52). ⎤

⎤⎡ I ⎤ 1 r1 jωL 12 jωL 13 ⎢ ⎥ ⎣ ⎢ ⎥ ⎦ 0 I = jωL 12 r2 jωL 23 ⎣ 2 ⎦ ⎣ ⎦ jωL 13 jωL 23 r3 + R3 0 I3 ⎡

V1

⎡

(9.52)

9.4 Positive and Negative Mutual Inductance If a repeating coil is not used, then the power transmission occurs with two elements: a transmitting coil and a receiving coil. Therefore, regardless of whether the mutual inductance is positive or negative, the power transmission efficiency has the same frequency characteristics, thus no consideration of the sign of the mutual inductance is needed. Figure 9.17 shows the results when the mutual inductance of an equivalent circuit is positive and negative. The coefficient of coupling k is 0.092, and the mutual inductance L m = 1.019 µH or L m = −1.019 µH. In this section, we used the twolayer open-type spiral coil shown in Fig. 6.2. Figure 9.17a shows a diagram of this configuration from a side view. From these results, we can see that, when considering the efficiency and power, the sign of the mutual inductance may be ignored if only transmitting and receiving coils are applied.

310

9 Repeating Coil and Multiple Power Supply (Basic)

(a) Coil arrangement η[%]

η[%]

ηr2 [%]

ηr1 [%]

Efficiency [%]

Efficiency [%]

ηr2 [%]

ηr1 [%]

100

100 80 60 40 20 0 11

12 13 14 Frequency [MHz]

15

80 60 40 20 0 11

16

12 13 14 Frequency [MHz]

15

16

(b) Efficiency, +Lm (c) Efficiency, -Lm P RL

P r1

P r2

Pin

250

250

200

200 Power [W]

Power [W]

Pin

150 100 50 0

11

12

13 14 15 Frequency [ MHz ]

16

P RL

P r1

P r2

150 100 50 0

11

12

13 14 15 Frequency [ MHz ]

16

(d) Power, +Lm (e) power, -Lm Fig. 9.17 Influence of sign of mutual inductance if only transmitting and receiving coils are applied (ga = 150 mm)

However, if a repeating coil is used, the sign cannot be ignored [1]. Therefore, we verify the behavior of the magnetic field within the vicinity of the coil. If the transmitting and receiving coil parameters are as indicated in Fig. 9.18, then the power transmission efficiency distribution in the vertical and horizontal directions is as shown in Fig. 9.19. However, the efficiency distribution in Fig. 9.19 is the value at which the load is consistently 50 , and the operating frequency is properly adjusted. In the case of (A) in Fig. 9.19, in which the receiving coil is disposed in the vertical direction, and in the case of (B) in Fig. 9.19, in which it is disposed in the horizontal

9.4 Positive and Negative Mutual Inductance

311

Fig. 9.18 Transmitting and receiving coil parameters

Fig. 9.19 Efficiency distribution on d and g axes

direction, the power transmission has a high frequency; however, in (C) in Fig. 9.19, the receiving coil position moves from the vertical to horizontal direction, which is a null direction in which the efficiency rapidly worsens. Because the direction in which the magnetic fields penetrate is geometrically determined, when discussing the behavior of a magnetic field, an outline drawing as shown in Fig. 9.20 is used. In certain cases, as indicated in Fig. 9.20a, magnetic

(a) Same direction

(b) Opposite direction (c) Null direction

Fig. 9.20 Three types of magnetic flux interlinking

312

9 Repeating Coil and Multiple Power Supply (Basic)

flux H g penetrates like a loop C g in the same direction from the bottom side of the transmitting and receiving coils, and in certain cases, as shown in Fig. 9.20b, there is a loop C s in which the magnetic flux H s penetrates from the bottom of the transmitting coil, and the magnetic flux Hs penetrates from the top of the receiving coil. The state in Fig. 9.20a corresponds to (A) in Fig. 9.19, and the state in Fig. 9.20b corresponds to (B) in Fig. 9.19, and thus the power transmission in each has high efficiency. In contrast, in the null direction, as indicated in Fig. 9.20c, the amount of magnetic field penetration is equivalent to the mutual cancellation of C g in the same direction, and C s in the opposite direction, and the efficiency becomes lower as indicated in (C) in Fig. 9.19. In the case of the transmitting and receiving coils, the direction of the magnetic flux becomes the same, and thus even if the mutual inductance is always thought of as positive, it poses no obstacle; however, if a repeating coil is interposed, then the direction of the magnetic flux is occasionally in the opposite direction. Therefore, in the next section, when dealing with an equivalent circuit, we show the influence of the direction of the magnetic flux and the sign of the mutual inductance L m concerning the arrangement in the vertical direction and the arrangement in the planar direction.

9.4.1 Vertical Direction and +Lm Here, we verify the repeating coil equivalent circuit in the vertical direction. First, to verify the results for a repeating coil in the vertical direction, we show the power transmission efficiency with and without a repeating coil in Fig. 9.21. If the distance between the transmitting and receiving coils is ga = 610 mm, then as indicated in Fig. 9.21a–c, the power transmission efficiency is extremely poor, whereas if a repeating coil is disposed between the transmitting and receiving coils, and the distance between the transmitting and repeating coils, or the distance between the receiving and repeating coils, is gp = 300 mm, then as indicated in Fig. 9.21d–f, the power transmission distance is extended, and the effects of the repeating coil in the vertical direction are verified. For the calculations here consider the occurrence of cross-coupling. Next, to evaluate the influence of the cross-coupling of the transmitting and receiving coils, we examine reducing the air gap to a distance at which the influence of the transmitting and receiving coils is likely to appear, that is, gp = 150 mm and ga = 310 mm, as shown in Fig. 9.22. Coupling between the transmitting and receiving coils is L 13 = 0.271 µH and k 13 = 0.025 when ga = 310 mm. Coupling between the transmitting coil and repeating coil, and coupling between the repeating coil and receiving coil, is L 12 = L 23 = 1.019 µH and k 12 = k 23 = 0.092, when gp = 150 mm. Figure 9.23 shows the calculation results determined from an equivalent circuit accounting for the cross-coupling. Here, to discuss the sign of the mutual inductance, Fig. 9.23a shows a case in which the mutual inductance is positive, and Fig. 9.23b shows when the mutual inductance is negative. If vertically arranged, the direction

9.4 Positive and Negative Mutual Inductance η[%]

313

ηr1 [%]

ηr2 [%]

Pin

Efficiency [%]

100 80

P r2

6000

40

4000

20

2000 11

12

13

14

15

16

0 11

12

Frequency [MHz]

(a)Without repeating coil

(b) Efficiency η[%]

ηr1 [%]

ηr3 [%]

80 60 40 20 0

11

12

13

14

15

16

1200 1000 800 600 400 200 0 11

Frequency [MHz]

(d)With repeating coil

13 14 [ MHz ]

15

16

(c)Power Pin

100 Efficiency [%]

P r1

8000

60

0

P RL

10000

12

P RL

P r1

13 14 [ MHz ]

P r3

15

16

(f)Power

(e) Efficiency

Fig. 9.21 Repeating coil effects (ga = 610 mm and gp = 300 mm)

(a) Coil arrangement

(b) Magnetic flux

Fig. 9.22 Vertical direction (gp = 150 mm, ga = 310 mm)

of the magnetic flux created by the induced electromotive force generated by the interlinking of the magnetic flux created by a transmitting coil with a receiving coil, and the direction of the magnetic flux created by the induced electromotive force generated by the interlinking of the magnetic flux created by a transmitting coil with a repeating coil, matches in both directions, and thus the mutual inductance is positive. That is, Fig. 9.23a is the correct graph. The results of inverting the sign describe a curve inverting the resonance frequency around the center. In addition, Fig. 9.24 shows the results found from an equivalent circuit when ignoring the cross-coupling. When doing so, k 13 = 0. When there is a negligible degree of coupling between the transmitting and receiving coils, no matching occurs if the cross-coupling is ignored.

314

9 Repeating Coil and Multiple Power Supply (Basic) η[%]

η[%]

ηr3 [%]

ηr1 [%]

80

80

Efficiency [%]

100

Efficiency [%]

100

60 40 20 0 11

12

13 14 15 Frequency [MHz]

60 40 20 0

16

11

P r1

P r3

12

13 14 [ MHz ]

15

13 14 15 Frequency [MHz]

Pin

[W]

[W]

300 250 200 150 100 50 0 11

P RL

12

16

(b) Efficiency, -L23

(a) Efficiency, +L23 Pin

ηr3 [%]

ηr1 [%]

16

300 250 200 150 100 50 0

11

12

(c) Power, +L23

P RL

P r1

P r3

13 14 [ MHz ]

15

16

(d) Power, -L23

Fig. 9.23 Vertical direction and repeating coil when accounting for cross-coupling (gp = 150 mm, ga = 310 mm) ηr1 [%]

ηr3 [%]

Pin

250

80

200

60

150

[W]

Efficiency [%]

η[%] 100

40

100

20

50

0 11

12 13 14 Frequency [MHz]

(a) Efficiency

15

16

0

11

12

P RL

P r3

P r1

13 14 [ MHz ]

15

16

(b) Power

Fig. 9.24 Vertical repeating coil arrangement when cross-coupling is ignored (gp = 150 mm, ga = 310 mm)

9.4 Positive and Negative Mutual Inductance

315

9.4.2 Horizontal Direction and −Lm We verify a repeating coil equivalent circuit in the horizontal direction. To verify the influence of cross-coupling between the transmitting and receiving coils, verification is conducted using a reduced transmission distance. First, Fig. 9.25 shows a configuration diagram for sa = 10 mm and sp = 10 mm, and the state of the magnetic flux. The lower left shows the transmitting coil, the lower right shows the receiving coil, and the repeating coil is at the top. The parameters used for coupling with the configuration shown in Fig. 9.25, which accounts for the cross-coupling, are k 12 = k 13 = k 23 = 0.049, L 12 = L 13 , and −L 23 = −0.542 µH. Figure 9.26 shows the results of accounting for the cross-coupling under each parameter. Here, to discuss the sign of the mutual inductance, Fig. 9.26a shows a positive mutual inductance, and Fig. 9.26b shows a negative mutual inductance. If horizontally arranged, after the magnetic flux generated by the transmitting coil is sent to the repeating and receiving coils, the direction in which the magnetic field generated by the coils becomes reversed between the repeating and receiving coils, and thus the direction of the magnetic flux created by the inducted electromotive

(a) Coil arrangement

(b) Magnetic flux

Fig. 9.25 Repeating coil horizontal arrangement (sa = 10 mm, sp = 10 mm)

η[%]

ηr1 [%]

ηr3 [%]

η[%]

80

80

Efficiency[%]

100

Efficiency[%]

100

60 40 20 0 11

12 13 14 Frequency [MHz]

(a) Efficiency, +L23

15

16

ηr1 [%]

ηr3 [%]

60 40 20 0 11

12

13 14 15 Frequency [MHz]

16

(b) Efficiency, -L23

Fig. 9.26 Repeating coil in the horizontal direction accounting for cross-coupling (sa = 10 mm, sp = 10 mm)

Fig. 9.27 Horizontally arranged repeating coil ignoring cross-coupling (sa = 10 mm, sp = 10 mm)

9 Repeating Coil and Multiple Power Supply (Basic)

η[%]

ηr1 [%]

ηr3 [%]

100

Efficiency [%]

316

80 60 40 20 0 11

12

13

14

15

16

Frequency [MHz]

force induced by each coil is also reversed, resulting in negative mutual inductance; hence, Fig. 9.26b is correct. Figure 9.27 shows the results of determinations made from an equivalent circuit when ignoring the cross-coupling between the transmitting and receiving coils. When doing so, k 13 = L 13 = 0. In this type of configuration, with a strong coupling between the transmitting and receiving coils, no matching occurs when ignoring the crosscoupling with an equivalent circuit.

9.4.3 Combination of Vertical and Horizontal Directions Here, we verify a mixed vertical and horizontal arrangement. Figure 9.28a, b shows the coil arrangement, coupling, and power transmission efficiency when transmitting and repeating coils are in a planar arrangement with a distance of sp = 10 mm, and repeating and receiving coils are in a vertical arrangement with a distance of gp = 200 mm. The magnetic flux between the repeating and receiving coils is reversed from the direction of the magnetic flux of the repeating and receiving coils created by the magnetic flux generated by the transmitting coil, and thus the mutual inductance is negative. Figure 9.28c, d shows the calculation results for an equivalent circuit with k 12 = 0.049, k 13 = 0.0044, k 23 = 0.058, L 12 = 0.542 µH, L 13 = 0.480 µH, and −L 23 = −0.643 µH. Figure 9.28c shows the results of negative mutual inductance between the repeating coil and receiving coil, and Fig. 9.28d shows the results of inverting all of the mutual inductance signs. In this way, it is possible to make similar calculations even if mixing a horizontal and vertical arrangement by determining the mutual inductance in the direction of the magnetic flux. Figure 9.29 shows the results when the number of repeating coils is increased by 1 for large changes in wavelength under the proposed coil parameters because they are small with one repeating coil. Because the total number of coils is increased to four, the calculations are not in a square matrix on the order of 3, but are expanded to a square matrix on the order of 4. When this occurs, L 12 = 0.542 µH, L 13 = 0.306

317

(a) Coil arrangement

(b) Magnetic flux

η[%]

η[%]

ηr1 [%]

ηr3 [%]

100

100

80

80

Efficiency [%]

Efficiency [%]

9.4 Positive and Negative Mutual Inductance

60 40 20 0 11

12

13

14

Frequency [MHz]

(c) Correct sign

15

16

ηr1 [%]

ηr3 [%]

60 40 20 0

11

12

13

14

15

16

Frequency [MHz]

(d)Sign inversion

Fig. 9.28 Vertical and horizontal arrangements (sp = 10 mm, gp = 200 mm)

µH, L 14 = 0.147 µH, −L 23 = −0.711 µH, −L 24 = −0.223 µH, and L 34 = 1.019 µH.

9.4.4 Multiple Power Supply Equivalent Circuit (Three Coils) In previous sections, we dealt with a repeating coil, and in this section we examine multiple coils. In addition, as shown in Sect. 9.3, a repeating coil and multiple coils can be handled within the same equivalent circuit framework. Because power is supplied to multiple loads, we imagine a load is connected to each receiving coil. The coil arrangement, parameters, and direction of the magnetic flux are the same as in Fig. 9.22. However, No. 2 is not a repeating coil, but a load-connected coil. The load resistance of coil No. 2 is R2 , and the load resistance of coil No. 3 is R3 . Figure 9.30 shows the results when varying the load. We show the parameters of 50  for both coils, as well as R2 = 200  and R3 = 50 , and R2 = 50  and R3 = 200 . In this way, it is possible to calculate multiple loads using the formulas shown in Sect. 9.3.

318

9 Repeating Coil and Multiple Power Supply (Basic)

(a) Coil arrangement

(b) Magnetic flux

(a) Correct sign

(b) Sign inversion

Fig. 9.29 When there are two repeating coils in a vertical and horizontal arrangement (sa = 10 mm, d p = 200 mm, gp = 150 mm) η2

ηr1

12

13

14

Frequency[MHz]

(a)

15

16

100 90 80 70 60 50 40 30 20 10 0 11

η3

ηr1

η2

Efficiency [%]

η3

Efficiency [%]

Efficiency [%]

η2 100 90 80 70 60 50 40 30 20 10 0 11

12

13

14

Frequency[MHz]

(b)

15

16

100 90 80 70 60 50 40 30 20 10 0

11

12

η3

13

ηr1

14

15

16

Frequency [MHz]

(c)

Fig. 9.30 Results for a multiple coil equivalent circuit in the vertical direction with gp = 150 mm, ga = 310 mm. a R2 = 50 , R3 = 50 , b R2 = 200 , R3 = 50 , and c R2 = 50 , R3 = 200 

9.5 Calculation with Z-Matrix Accounting for Cross …

319

9.5 Calculation with Z-Matrix Accounting for Cross-Coupling (N Coils) The previous discussion involved only three coils. However, it was similarly applied even when there are n or more coils, or when there are coexistent multiple repeating coils and multiple loads. In this section, we show the general formula for n coils using a Z-matrix. An equivalent circuit with multiple loads is shown in Fig. 9.31. However, a case of n = 6 is shown. In addition, self-inductance and mutual inductance accounting for cross-coupling in all places with this setup are as indicated in formula (9.53). The subscript n = 1 represents the transmitting coil, and n = 2, 3, 4… represent the receiving coils. ⎡

L 1 L 12 L 13 . . . ⎢ ⎢ L 12 L 2 L 23 . . . ⎢ ⎢ [L] = ⎢ L 13 L 23 L 3 . . . ⎢ ⎢ . . . . ⎣ .. .. . . . . L 1n L 2n L 3n . . .

L 1n

⎤

⎥ L 2n ⎥ ⎥ ⎥ L 3n ⎥ ⎥ .. ⎥ . ⎦

(9.53)

Ln

The impedance in each place when the coil loads are removed is represented as Z. The voltage V and current I are based on formula (9.54) and formula (9.55),

Fig. 9.31 Multiple coil equivalent circuit accounting for cross-coupling (with six coils)

320

9 Repeating Coil and Multiple Power Supply (Basic)

respectively, the relationship of which is expressed through formula (9.56). When this is done, each element of voltage V is as indicated in formula (9.57), and the relationship among V n , V Ln , V Cn , and V rn is indicated in formulas (9.58)–(9.60). Here, ω is the angular frequency. ⎡

Z 11 Z 12 Z 13 . . . ⎢ ⎢ Z 21 Z 22 Z 23 . . . ⎢ ⎢ [Z] = ⎢ Z 31 Z 32 Z 33 . . . ⎢ ⎢ . .. . . . . ⎣ .. . . .

Z 1n

⎤

⎥ Z 2n ⎥ ⎥ ⎥ Z 3n ⎥ ⎥ .. ⎥ . ⎦

Z n1 Z n2 Z n3 . . . Z nn ⎡ ⎤ ⎡ ⎤ V1 I1 ⎢ V2 ⎥ ⎢ I2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [V ] = ⎢ V3 ⎥, [I] = ⎢ I3 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ Vn In [V ] = [Z][I] 

V1 = VL1 + VC1 + Vr 1 Vn = VLn + VCn + Vr n

⎧ VL1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 + . . . + jωL 1n In ⎪ ⎪ ⎪ ⎨ VL2 = jωL 12 I1 + jωL 2 I2 + jωL 23 I3 + . . . + jωL 2n In .. ⎪ ⎪ . ⎪ ⎩ VLn = jωL 1n I1 + jωL 2n I2 + jωL 3n I3 + . . . + jωL n In ⎧ 1 ⎪ ⎪ ⎨ VC1 = jω C I1 1

⎪ ⎪ ⎩ VCn = 

1 In jω Cn

Vr 1 = r1 I1 Vr n = rn In

(9.54)

(9.55)

(9.56)

(9.57)

(9.58)

(9.59)

(9.60)

When formulas (9.58)–(9.60) are substituted into formula (9.57), formula (9.61) is obtained.

9.5 Calculation with Z-Matrix Accounting for Cross …

⎧ V1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 + · · · + jωL 1n In + ⎪ ⎪ ⎪ ⎪ ⎨ V2 = jωL 12 I1 + jωL 2 I2 + jωL 23 I3 + · · · + jωL 2n In + .. ⎪ ⎪ . ⎪ ⎪ ⎩ V = jωL I + jωL I + jωL I + · · · + jωL I + n 1n 1 2n 2 3n 3 n n

321 1 I jωC1 1 1 I jωC2 2

+ r 1 I1 + r 2 I2

1 I jωCn n

+ r n In (9.61)

Because this study aims to achieve an operation under the resonance conditions, the reactance through self-inductance L n and the reactance through capacitance C n cancel each other out, as shown in formula (9.62) and they can be aggregated into formula (9.63). jω L n +

1 =0 jω Cn

⎧ V1 = 0 + jωL 12 I2 + jωL 13 I3 + · · · + jωL 1n In + r1 I1 ⎪ ⎪ ⎪ ⎨ V2 = jωL 12 I1 + 0 + jωL 23 I3 + · · · + jωL 2n In + r2 I2 .. ⎪ ⎪ . ⎪ ⎩ Vn = jωL 1n I1 + jωL 2n I2 + jωL 3n I3 + · · · + 0 + rn In

(9.62)

(9.63)

In contrast, the voltage of loads connected to each coil is expressed through formula (9.64). Accordingly, formula (9.65) is obtained from formulas (9.62) and (9.64). 

V1 = V1 Vn = −In Rn

⎧ V1 = 0 + jωL 12 I2 + jωL 13 I3 + · · · + jωL 1n In + r1 I1 ⎪ ⎪ ⎪ ⎨ 0 = jωL 12 I1 + 0 + jωL 23 I3 + · · · + jωL 2n In + r2 I2 + R2 I2 .. ⎪ ⎪ . ⎪ ⎩ 0 = jωL 1n I1 + jωL 2n I2 + jωL 3n I3 + · · · + 0 + rn In + Rn In

(9.64)

(9.65)

Here, when the formula (9.65) voltage V  , impedance Z , and current I are redefined as in formula (9.66), they can be collectively expressed through formula (9.67). Naturally, current I does not change from formula (9.56). 

   V  = Z  [I]

(9.66)

322

9 Repeating Coil and Multiple Power Supply (Basic)

⎡ ⎤ r1 V1 ⎢ ⎢ 0 ⎥ ⎢ jωL 12 ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢ ⎢ ⎥ = ⎢ jωL 13 ⎢ . ⎥ ⎢ ⎣ .. ⎦ ⎢ .. ⎣ . 0 jωL 1n ⎡

⎤ jωL 12 jωL 13 . . . jωL 1n ⎡ ⎤ ⎥ I1 .. ⎢ ⎥ . jωL 2n ⎥ r2 + R2 jωL 23 ⎥⎢ I2 ⎥ ⎢ I3 ⎥ ⎥ . ⎢ ⎥ jωL 23 r3 + R3 . . jωL 3n ⎥ ⎥⎢ . ⎥ ⎥⎣ .. ⎦ .. .. .. .. ⎦ . . . . In jωL 2n jωL 3n . . . rn + Rn

(9.67)

Voltage V  and current I are based on formula (9.68). The current for each load can be determined from formula (9.69). ⎡ ⎤ ⎤ I1 V1 ⎢ I2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥   ⎢ ⎢ ⎥ ⎢ ⎥ V = ⎢ 0 ⎥, [I] = ⎢ I3 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ⎡

(9.68)

In

0  −1 [I] = Z  [V ]

(9.69)

The input power P1 and power Pn consumed at each load are determined through formula (9.70). The power consumed with the internal resistance of each coil is expressed using formula (9.71). From the above, the efficiency ηn1 at each load is as indicated in formula (9.72), and the total efficiency η is based on the formula (9.73). 

  P1 = Re V1 I¯1   Pn = PRn = Re Vn (− I¯n )    Pr 1 = Re  Vr 1 I¯1  Pr n = Re Vr n (− I¯n ) ⎧ P2 ⎪ ⎨ η21 = P1 .. . ⎪ ⎩ ηn1 = PPn1

η = η21 + . . . + ηn1 =

n 

ηm1

(9.70) (9.71)

(9.72)

(9.73)

m=2

Here, we show an example in which six coils are used when changing the number of repeating coils, the number of multiple loads, and the values of the loads. Figure 9.32 shows a configuration in which the vertical and horizontal directions are combined. In this case, the value of mutual inductance accounting for the direction of the magnetic flux is L 12 = −L 34 = −L 56 = 0.542 µH, L 13 = −L 24 = L 35 = L 46 = 1.019 µH, L 14 = −L 23 = L 36 = L 45 = 0.011 µH, L 15 = −L 26 = 0.271 µH, and L 16 = −L 25 = 0.062 µH. Figure 9.33 shows the equivalent circuit calculation results

9.5 Calculation with Z-Matrix Accounting for Cross …

323

Fig. 9.32 Multiple and repeating coils in the vertical direction with sa = 10 mm, ga = 150 mm

Fig. 9.33 Calculation results with multiple coils and repeating coils in the vertical direction, with a R2 = R3 = ··· = R6 = 50 , b R5 = R6 = 50 , R2 = R3 = R4 = 0 , and c R5 = 200 , R6 = 300 , R2 = R3 = R4 = 0 

when all impedance values are set to 50 , when R5 = R6 = 50 , R2 = R3 = R4 = 0 , and when R5 = 200 , R6 = 300 , and R2 = R3 = R4 = 0 . Impedance 0  refers to a repeating coil. In this way, it is possible to make calculations without problems occurring even when multiple coils are used.

324

9 Repeating Coil and Multiple Power Supply (Basic)

References 1. T. Imura, Equivalent circuits of repeater antennas for wireless power transfer via magnetic resonant coupling. Trans. Inst. Electr. Eng. Japan D 131(12), 1373–1382 (2011) 2. Y. Narusue, Y. Kawahara, T. Asami, Variable-hop impedance matching method for magnetic resonance based wireless power transmission relays. in IEICE Technical Report WPT2012-11, pp. 9–14 (2012) 3. K.-E. Koh, T. Imura, Y. Hori, Impedance inverter based analysis of wireless power transfer consists of repeaters via magnetic resonant coupling. in IEICE Technical Report WPT2012-38, pp. 41–45 (2012) 4. K.E. Koh, T.C. Beh, T. Imura, Y. Hori, Impedance matching and power division using impedance inverter for wireless power transfer via magnetic resonant coupling. IEEE Trans. Ind. Appl. 50, 2061–2070 (2014) 5. H. Irie, H. Yamana, Immittance converter suitable for power electronics. Electr. Eng. Japan 124(4), 53–62 (1998) 6. K.E. Koh, T. Imura, Y. Hori, Analysis of dead zone in wireless power transfer via magnetic resonant coupling for charging moving electric vehicles. Int. J. Intell. Transp. Syst. Res. (2015)

Chapter 10

Applications of Multiple Power Supplies

The previous chapter discussed the fundamentals of repeating coils and multiple power supplies. This chapter will focus on the phenomena involved when multiple coils are increased as well as actual scenarios. The first half of this chapter will discuss the phenomenon of improved efficiency that is generated when multiple power supplies are used as a result of multiple receiving coils. The latter half will discuss the effects of cross-coupling that are generated when the distance between receiving coils is reduced, as well as the cross-coupling canceling method which cancels out cross-coupling.

10.1 Improved Efficiency When Using Multiple Power Supplies To implement high-efficiency power transmission, a coil with a high Q value is required; however, there are also limitations to an approach involving optimization via coils with a high Q value. In contrast, when considering power supply for multiple loads, there are other methods that improve total efficiency without relying on the Q value. This is achieved by increasing the number of receiving coils. This section will discuss the phenomenon that improves total efficiency by supplying power for multiple loads, as well as discuss the method thereof [1]. As shown in Fig. 10.1, we will discuss the case in which a large coil is used to collectively supply power to multiple small receiving coils and their loads. The total efficiency discussed in this section is a ratio of the total power consumed by the load of the receiving coil to the transmission power. P1 represents transmission power, Pr1 represents power lost due to internal resistance at the transmission side, Pn represents power consumed by the resistance transferred to the nth receiving coil, and Prn represents the power lost due to the internal resistance; that is, the subscript n = 1 represents the transmitting coil and subscripts n = 2, 3, 4, etc., represent the receiving coils. Moreover, when the total number of coils is n, and there is one © Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_10

325

326

10 Applications of Multiple Power Supplies

Fig. 10.1 Transmitting coil and eight receiving coils

transmitting coil and (n−1) receiving coils, the total efficiency is defined by formula (10.1). For example, if there are two transmitting coils, total efficiency is then defined by formula (10.2). n η= η=

m=2

P1

Pm

n =

Pr 1 +

n m=2

Pm  Pr m + nm=2 Pm

m=2

P2 + P3 P2 + P3 = P1 Pr 1 + Pr 2 + Pr 3 + P2 + P3

(10.1) (10.2)

10.1.1 The Model Considered in This Chapter Figure 10.2 shows a one-to-one case of a transmitting/receiving coil and an enlarged view of a transmitting coil. Here, the transmitting coil is centrally positioned so as to show the size differences between the two. The resonance frequency is operated by setting it to 200 kHz for both transmitting and receiving. The transmitting coil is in the shape of a cube with dimensions of 500 × 500 × 500 mm has 20 turns whose wires are rectangular and made of aluminum, and each wire has a width of 15.0 mm and a thickness of 1.0 mm, and its coils are of a short type which resonate with an external capacitor. The receiving coil is in the shape of a cube one-tenth in size of the Fig. 10.2 Cube-shaped coils

(a) Transmitting coil and receiving coil

(b) Receiving coil (magnified)

10.1 Improved Efficiency When Using Multiple Power Supplies Table 10.1 Parameters of the transmitting and receiving coils

327

Transmitting coil

Receiving coil

lx × ly × lz (mm)

500 × 500 × 500

lx × ly × lz (mm)

50 × 50 × 50

Turns

20

Turns

20

ah (mm)

1.0

a (mm)

1.0

aw (mm)

15.0

ph (mm)

2.6

ph (mm)

10.5

L n (uH)

17.57

L 1 (uH)

168.85

C n (nF)

36.05

C 1 (nF)

3.75

r n ()

0.52

Qn

42.12

r 1 ()

2.13

Q1

99.46

former cube, having dimensions of 50 × 50 × 50 mm and is of a short type, has 20 turns whose wires have a thickness radius of 1 mm and are made of copper. Detailed parameters of the transmitting and receiving coils are shown in Table 10.1. The pitch ph is the distance between the centers of the wires. The parameter of ah is thickness of wire, aw is width of wire, and a is radius of wire. L 1 , C 1 , r 1 , and Q1 represent the transmitting coils’ self-inductance, external capacitor, internal resistance, and Q value, respectively. L n , C n , r n , and Qn represent the receiving coils’ self-inductance, external capacitor, internal resistance, and Q value, respectively. Once again, we will consider a structure that has eight receiving coils. Figure 10.3a shows an explanatory illustration of the arrangement of the receiving coils. Considering the points A, B, C, etc., from the bottom, the central locations of the receiving coils are set 400 mm apart. The outermost frame (broken-lined cube) is the position of the transmitting coil. Figure 10.3b shows a model in which the receiving coils are arranged when the magnetic flux within the coils is consistent. Although the magnetic fields inside the coils are not uniform, if the relative positions with respect to the center were symmetrical, the linking fluxes would be uniform. In the model of Fig. 10.3b, as the linking fluxes are identical, the coupling coefficients of the transmitting coil and the receiving coil would both be identical, and k = 0.032 in the equivalent circuit calculation. Since the coupling coefficient of the coupling between the receiving coils and the coefficient of the cross-coupling in this configuration are 0.000, when considering up to three decimal places, the cross-coupling can be regarded as absent. The relationship between the transmitting coil L 1 and the mutual inductance L m of the receiving coil L n is defined by the following formula. k=√

Lm L1 Ln

(10.3)

The values of the loads connected to each receiving coil are equal. Figure 10.3c shows photos of the coils used. The material of the transmitting coil is aluminum,

328

10 Applications of Multiple Power Supplies

(a) Arrangement of the receiving coil

(b) Placement locations of the receiving coils (the coupling coefficient has the same positions)

1) Transmitting coil 2) Receiving coils (c) Actual cube-shaped coils Fig. 10.3 Arrangement of the transmitting coils

which is light, and weighs 1.6 kg. This section discusses these models as specific examples.

10.1.2 Deriving the Formula for Total Efficiency and Optimal Load Value As shown in Sect. 9.5, the equivalent circuit in multiple loads is the same as Fig. 9.31 in Chap. 9, as well as formula (10.4); however, we will now consider an area where cross-coupling between the receiving coils can be ignored, and thus, the equivalent circuit is expressed in Fig. 10.4 and formula (10.5). ⎡

L 1 L 12 L 13 . . . ⎢ ⎢ L 12 L 2 L 23 . . . ⎢ ⎢ [L] = ⎢ L 13 L 23 L 3 . . . ⎢ ⎢ . . . . ⎣ .. .. . . . . L 1n L 2n L 3n . . .

L 1n

⎤

⎥ L 2n ⎥ ⎥ ⎥ L 3n ⎥ ⎥ .. ⎥ . ⎦ Ln

(10.4)

10.1 Improved Efficiency When Using Multiple Power Supplies

329

Fig. 10.4 Equivalent circuit for multiple loads without cross-coupling

⎡

L 1 L 12 L 13 . . . ⎢ ⎢ L 12 L 2 0 . . . ⎢ ⎢ [L] = ⎢ L 13 0 L 3 . . . ⎢ ⎢ . . . . ⎣ .. .. . . . . L 1n 0 0 . . .

L 1n

⎤

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ .. ⎥ . ⎦

(10.5)

Ln

Z represents the impedance of the locations with the load of each coil removed. The voltage V and current I are at that time based on formula (10.6) and formula (10.7), respectively, and their relationship is expressed in formula (10.8). At this time, the voltage V of each element is indicated in formula (10.9), and the relationships among V n , V Ln , V Cn , and V rn are expressed in formula (10.10) to formula (10.12). ω is the angular frequency.

330

10 Applications of Multiple Power Supplies

⎤ Z 11 Z 12 Z 13 . . . Z 1n ⎥ ⎢ ⎢ Z 21 Z 22 Z 23 . . . Z 2n ⎥ ⎥ ⎢ ⎥ ⎢ [Z] = ⎢ Z 31 Z 32 Z 33 . . . Z 3n ⎥ ⎥ ⎢ ⎢ . .. . . . . .. ⎥ ⎣ .. . . . ⎦ . Z n1 Z n2 Z n3 . . . Z nn ⎡ ⎤ ⎡ ⎤ V1 I1 ⎢ V2 ⎥ ⎢ I2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [V ] = ⎢ V3 ⎥, [I] = ⎢ I3 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ Vn In ⎡



(10.6)

(10.7)

[V ] = [Z][I]

(10.8)

V1 = VL1 + VC1 + Vr 1 Vn = VLn + VCn + Vr n

(10.9)

VL1 = jωL 1 I1 + jωL 12 I2 + · · · + jωL 1n In VLn = jωL 1n I1 + jωL n In

1 I1 VC1 = jωC 1 1 VCn = jωC In n Vr 1 = r1 I1 Vr n = rn In

(10.10)

(10.11)

(10.12)

The following formulas are obtained when substituting formula (10.10) to formula (10.12) into formula (10.9).

V1 = jωL 1 I1 + jωL 12 I2 + · · · + jωL 1n In + 1 Vn = jωL 1n I1 + jωL n In + jωC In + r n In n

1 I jωC1 1

+ r 1 I1

(10.13)

Owing to the fact that this study aims to achieve the operation under resonance conditions, the reactance due to self-inductance L n and the reactance due to capacitance C n cancel each other out, as shown in formula (10.14), and they can be aggregated into formula (10.15). jωL n +

1 =0 jωCn

V1 = 0 + jωL 12 I2 + · · · + jωL 1n In + r1 I1 Vn = jωL 1n I1 + 0 + rn In

(10.14) (10.15)

10.1 Improved Efficiency When Using Multiple Power Supplies

331

Meanwhile, the voltage of the load connected to each coil is expressed through formula (10.16). Accordingly, formula (10.17) is obtained via formula (10.14) and formula (10.16).

V1 = V1 Vn = −In Rn

(10.16)

V1 = 0 + jωL 12 I2 + · · · + jωL 1n In + r1 I1 0 = jωL 1n I1 + 0 + rn In + Rn In

(10.17)

Here, when the voltage V  , inductance Z , and current I in formula (10.17) are redefined as in formula (10.18), they can be collectively expressed through formula (10.19). Of course, the current I does not change in formula (10.8).  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

⎡

r1

V1 ⎢ ⎢ 0 ⎥ ⎥ ⎢ jωL 12 ⎥ 0 ⎥=⎢ ⎢ jωL 13 ⎢ .. ⎥ ⎢ . ⎦ . ⎣ .. 0 jωL 1n

 V  = Z  [I]

(10.18)

⎤ jωL 12 jωL 13 . . . jωL 1n ⎡ ⎤ ⎥ I1 .. ⎢ ⎥ . r 2 + R2 0 0 ⎥ ⎥⎢ I2 ⎥ ⎢ I3 ⎥ ⎥ . ⎢ ⎥ 0 ⎥ 0 r 3 + R3 . . ⎥⎢ . ⎥ ⎥⎣ .. ⎦ .. .. .. .. ⎦ . . . . In 0 0 . . . r n + Rn

(10.19)

The voltage V  and current I are in the form of formula (10.20). The current for each load can be determined from formula (10.21). ⎡

⎡ ⎤ ⎤ V1 I1 ⎢ 0 ⎥ ⎢ I2 ⎥ ⎢ ⎥ ⎥   ⎢ ⎢ ⎥ ⎢ ⎥ V = ⎢ 0 ⎥, [I] = ⎢ I3 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ 0

(10.20)

In

 −1 [I] = Z  [V ]

(10.21)

The input power P1 and the power consumed at each load Pn are determined through formula (10.22). The power consumed by the internal resistance of each coil is expressed using formula (10.23). Based on the above, the efficiency ηn1 at each load is as indicated in formula (10.24), and the total efficiency η is based on formula (10.25).



P1 = Re V1 I 1   Pn = PRn = Re Vn −I n

(10.22)

332

10 Applications of Multiple Power Supplies





Pr 1 = Re Vr 1I 1  Pr n = Re Vm −I n ⎧ P2 ⎪ ⎨ η21 = P1 .. . ⎪ ⎩ ηn1 = PPn1

η = η21 + · · · + ηn1 =

n 

(10.23)

(10.24)

ηm1

(10.25)

m=2

10.1.3 When Mutual Inductance Lm Varies First, we will consider the conditions L 12 = L 13 = ··· = L 1n = L m assuming that the mutual inductance varies. This is only a general condition, and its formula would be slightly more complex; however, formularization is possible. Based on formula (10.18) and formula (10.19), the currents in formula (10.26) are obtained as follows, and the voltages in formula (10.27) are obtained by substitution into formula (10.16):

r +R  V n r1 R+ω2 ( i=2 L 21i )+r1 r 1 − jωL 1n n V r1 R+ω2 ( i=2 L 21i )+r1 r 1

(10.26)

V1 : const. R n1n 2 V Vn = r R+ω2 jωL L 1i )+r1 r 1 ( i=2 1

(10.27)

I1 = In =



Based on formula (10.22), the input power and the received power at each load are determined through the following formula: ⎧ r +R ⎪ V2 n ⎪ P1 = r1 R+ω2 (i=2 L 21i )+r1 r 1 ⎨ R(ωL 1n )2 2 Pn = n 2V r1 R+ω2 ( i=2 L 21i )+r1 r } 1 ⎪ { ⎪ ⎩ PRall = P2 + · · · + Pn

(10.28)

Based on formula (10.23), the power consumed by the internal resistances of the transmitting coil and the receiving coil is determined through the following formula: ⎧ r1 (r +R)2 2 ⎪ 2 V n ⎪ Pr 1 = {r1 R+ω2 (i=2 L 21i )+r1 r } 1 ⎨ r (ωL 1n )2 2 Pr n = n 2 V1 2 2 ⎪ r R+ω { ( 1 i=2 L 1i )+r1 r } ⎪ ⎩ Prall = Pr 2 + · · · + Pr n

(10.29)

10.1 Improved Efficiency When Using Multiple Power Supplies

333

The total efficiency formula is obtained based on formulas (10.23), (10.24), and (10.25) as follows: n  2 i=2 L 1i n  

η= L 21i + r1r (r + R) r1 R + ω2 i=2 Rω2

(10.30)

Next, the optimum load Ropt will be determined. The maximum value is determined through formula (10.31), and the formula for optimum load Ropt can be calculated through formula (10.32). This is the optimum load when there are (n−1) receiving coils, and the maximum efficiency ηmax during this load is obtained as shown in formula (10.33): n  n   L 21i r1r 2 + r ω2 i=2 L 21i − r1 R 2 ω2 i=2 ∂η = =0 (10.31)

n  2 ∂R (r + R)2 r1 R + ω2 i=2 L 21i + r1r  n  r ω2 i=2 L 21i + r2 (10.32) Ropt = r1 n n  r ω2 (i=2 L 21i )+r1 r 2 ω2 i=2 L 21i r1 ηmax =  2 n 2   

n  r ω ( i=2 L 1i )+r1 r 2 2 n 2 2 2 ω i=2 L 1i + 2r 1 r + 2r ω i=2 L 1i + r 1 r r1 (10.33)

10.1.4 When Mutual Inductance Lm Is Constant Next, we will consider the condition in which mutual inductance is constant and L 12 = L 13 = ··· = L 1n = L m . This is the condition of the model in Sect. 10.1.1 Since the formula is simplified, it is easy to understand the phenomenon when the receiving coils are increased. The formulas for the currents (10.34) are obtained based on formula (10.18) and formula (10.19), and the formulas for the voltages are obtained by substituting into formula (10.16).



r +R V r1 R+(n−1)ω2 L 2m +r1 r 1 − jωL m V r1 R+(n−1)ω2 L 2m +r1 r 1

(10.34)

V1 : const. jωL m R Vn = r1 R+(n−1)ω 2 L 2 +r r V1 1

(10.35)

I1 = In =

m

Based on formula (10.22) and formula (10.23), the input power P1 , the received power at each load Pn , and the power consumed at the internal resistance of the coil Prn are determined through the following formulas, respectively:

334

10 Applications of Multiple Power Supplies

⎧ r +R 2 ⎪ ⎪ P1 = r1 R+(n−1)ω2 L 2m +r1 r V1 ⎨ 2 R(ωL m ) 2 Pn = 2 V r1 R+(n−1)ω2 L 2m +r1 r } 1 { ⎪ ⎪ ⎩ P = P + ··· + P Rall 2 n ⎧ 2 r1 (r +R) ⎪ P = V2 ⎪ ⎨ r1 {r1 R+(n−1)ω2 L 2m +r1 r }2 1 r (ωL m )2 2 Pr n = 2V ⎪ r1 R+(n−1)ω2 L 2m +r1 r } 1 { ⎪ ⎩ Prall = Pr 2 + · · · + Pr n

(10.36)

(10.37)

Formula (10.38) expresses the ratios of the loss at the internal resistance of the transmitting side, the loss at the internal resistance of the receiving side, and the consumed power at the load of the receiving side. Formula (10.39) for the total efficiency η is obtained based on formulas (10.23), (10.24), and (10.25).  Pr 1 : Prall : PRall = r1 : r (n − 1) η=

ωL m r+R



2 : R(n − 1)

ωL m r+R

(n − 1)Rω2 L 2m 

(r + R) r1 R + (n − 1)ω2 L 2m + r1r )

2 (10.38) (10.39)

To investigate the case in which the number of coils is increased, the following formula (10.40) is used when η is differentiated with respect to n, which shows that the increase is monotonous. Since n is the number of coils, n ≥ 2. r1 Rω2 L 2m ∂η = 2 > 0 ∂n r1 R + (n − 1)ω2 L 2m + r1r

(10.40)

Next, the optimum load Ropt will be determined. The maximum value is determined through formula (10.41), and the formula for optimum load Ropt can be calculated through formula (10.42). This is the optimum load when there are (n−1) receiving coils, and the maximum efficiency ηmax during this load is obtained as shown in formula(10.43).

 (n − 1)ω2 L 2m r1r 2 + (n − 1)r ω2 L 2m − r1 R 2 ∂η = =0 (10.41)

2 ∂R (r + R)2 r1 R + (n − 1)ω2 L 2m + r1r  (n − 1)r ω2 L 2m + r2 (10.42) Ropt = r1  (n−1)r ω2 L 2m + r 2 ω2 L 2m (n − 1) r   1  ηmax =   2 2 (n−1)r ω2 L m (n−1)r ω2 L m 2 2 + (n − 1)ω2 L 2 + r r r+ r + r + r 1 1 m r1 r1 (10.43)

10.1 Improved Efficiency When Using Multiple Power Supplies

335

In this section, calculations were performed by revisiting formula (10.16); however, calculations were also determined by substituting L 12 = L 13 = ··· = L 1n = L m into formulas determined in the previous Sect. 10.1.3.

10.1.5 Graph of Multiple Power Supply Efficiency Increase As shown in the model in Sect. 10.1.1, it is easy to understand the effect when mutual inductance is constant and the number of receiving coils is increased. Accordingly, this section will investigate the theory in which inductance L m is assumed to be constant. By using the formula(s) previously obtained in Sect. 10.1.4, we will confirm the effects of efficiency, power, voltage, and current via the number of transmitting and receiving coils n and the load resistance R. Since the configuration is symmetrical, the mutual inductances L m are identical in the eight corners. When there are nine or more corners, the positions where the mutual inductances are identical are no longer easily identifiable; however, this theoretical investigation will not focus on the positions and will increase the number of coils n by using the value of L m used in the model. Figure 10.5 shows a 3-D plot of the relationship between the number of receiving coils, the load resistance value, and the efficiency when L m is constant. As shown in the figure, even if the load value deviates from the optimum value or is the optimum value, the total efficiency increases by merely increasing the number of coils. This is shown in formula (10.40), which expresses that the total efficiency is a monotonous increase with respect to the number of coils, and a graph for this is shown in Fig. 10.6. Based on this, it can be understood that merely increasing the number of coils increases the efficiency. The phenomenon by which efficiency is increased by merely increasing the number of receiving coils offers great advantages; for example, in supplying wireless power to sensors. Since the size of sensors is generally small in one-to-one transmitting–receiving coils, the coils are also small; that is, the coupling coefficient is often small, and thus, achieving high efficiency is difficult; however, since efficiency Fig. 10.5 Efficiency versus load resistance and receiving coil

336

10 Applications of Multiple Power Supplies

Fig. 10.6 Result of differentiating efficiency with number of receiving coils

can be improved by increasing the number of sensors, it can be said to be compatible with spatial simultaneous power supply systems that employ many sensors. The same graph is shown as a 2-D plot in Fig. 10.7 which includes a range of 2 ≤ n ≤100. Figure 10.7a shows the total efficiency when R = 50 , and Fig. 10.7b shows the maximum total efficiency value when R is at an optimum load. Figure 10.8 shows the optimum load value corresponding to the number of receiving coils and a fixed value when R = 50 . As shown in Fig. 10.5, as the number of coils increases, the resistance value of the optimum load increases. Even if the load resistance deviates from the optimum value, the efficiency will increase; however, by applying the optimum 100 Efficiency [%]

Efficiency [%]

100 80 60 40 20 0

0

20

40

60

80

100

80 60 40 20 0

0

20

40

60

80

100

Number of coils [n]

Number of coils [n]

(a) R = 50 Ω

(b) Optimum load, Ropt

Fig. 10.7 Relationship between efficiency and number of receiving coils

Ropt

Resistance [Ω]

Fig. 10.8 Relationship between load resistance and number of receiving coils

60 50 40 30 20 10 0

0

R = 50 [ohm]

10 20 30 40 50 60 70 80 90 100

Number of coils [n]

10.1 Improved Efficiency When Using Multiple Power Supplies

337

load corresponding to the number of receiving coils, the maximum total efficiency can be achieved. Finally, the efficiency when the number of receiving coils n is increased can be determined via formula (10.39). As expressed in formula (10.44), the final efficiencies at the internal resistance and the load resistance of the load side are determined. Furthermore, even if there is increased mutual inductance and resonance frequency, the same result is obtained as expressed in formula (10.45) and formula (10.46). lim η =

R r+R

(10.44)

lim η =

R r+R

(10.45)

lim η =

R r+R

(10.46)

n→∞

L m →∞

ω→∞

If we consider a realistic scenario, the number of sensors to be added in a limited space is also limited; therefore, there is a limit to the increase of n. Additionally, when the mutual inductance is large, the self-inductance will need to be larger than the mutual inductance. Due to the fact that a large self-inductance or a large mutual inductance indicates that the size of the receiving coil will be large and the increase in L m will also be limited. Furthermore, when the resonance frequency is raised, it would eventually reach self-resonance, and the resistance value would increase drastically, which would not allow ω to become excessively large. Based on the above, they will not necessarily reach the values shown in formula (10.44), formula (10.45), and formula (10.46); however, knowing the final value would allow us to determine a plan for the objective. In the system in Sect. 10.1.1, receiving coils that are one-tenth the size of a transmitting coil were used. According to Fig. 10.7b, in the case of in-coil power supply, the one-to-one efficiency, when the load is optimized, is known to be 39.6%; however, in some cases, the size is expected to be smaller than one-tenth. In this case, the internal resistance value and mutual inductance would decrease. For example, if only the mutual inductance is assumed to be 1/2 or 1/4, the one-to-one efficiency would be extremely small regardless of whether the load value was to be optimized, as shown in Fig. 10.9. Even so, when the number of receiving coils is increased, the increase in total efficiency can be implied from the case in which n = 20 (Fig. 10.9) and n = 100 (Fig. 10.10). Next, we will confirm the relationship between voltage, current, and power. The input voltage is 10 V. The primary input current I 1 is shown as a 3-D plot in Fig. 10.11. Figure 10.12 shows 2-D plots of the states of I 1 and load current I n for both R = 50  and at the optimum load. Figure 10.13 shows a 3-D plot of the load voltage V n . Figure 10.14 shows the respective voltage states for both R = 50  and at the optimum load. Figure 10.15 shows individual 3-D plots of the received power Pn and consumed power Pr1 at the transmitting side’s internal resistance. Figure 10.16 shows the states

338

10 Applications of Multiple Power Supplies

(a) Lm/2

(b) Lm/4

Fig. 10.9 Efficiency versus load resistance and receiving coil (20 coils)

(a) Lm/2

(b) Lm/4

Fig. 10.10 Efficiency versus load resistance and receiving coil (100 coils)

Fig. 10.11 I 1 Primary current I 1 versus load resistance and receiving coil

of power for both R = 50  and at the optimum load. The relationship between the load consumed power PRn and load internal consumed power Prn is r/R based on formula (10.38), and thus, when R = 50 , Prn becomes almost 0. The graph of the sum of the values, and not individual values, is shown below. Figure 10.17a, b show 3-D plots of the coil number, load resistance value, and total received power PRall ; and the coil number, load resistance value, and total

10.1 Improved Efficiency When Using Multiple Power Supplies In

I1

In

3

Current [A]

Current [A]

I1

5 4 3 2 1 0

339

2

4

6

8 10 12 14 16 18 20

2 1 0

2

4

6

Number of coils [n]

8 10 12 14 16 18 20

Number of coils [n] (b)

(a) R = 50 Ω

Optimum load Ropt

Fig. 10.12 Relationship between primary current I 1 , load current I n , and number of coils

Fig. 10.13 Load voltage V n versus load resistance and receiving coil

Vn

Vr1

10 5 0

V1

Vrn

Voltage [V]

Voltage [V]

V1

15

2

4 6

8 10 12 14 16 18 20

Number of coils [n]

(a) R = 50 Ω

Vn

Vr1

Vrn

15 10 5 0

2

4

6

8 10 12 14 16 18 20

Number of coils [n]

(b) Optimum load Ropt

Fig. 10.14 Various voltages versus load resistance and receiving coil (V 1 , V r1 , V rn , and V n )

consumed power at the receiving side’s internal resistance Prall , respectively. Additionally, Fig. 10.18 shows the voltage state when R = 50  and at the optimum load. If the optimum load value is not used, not only does the efficiency decrease, but a large amount of unneeded current will also flow to the transmitting side as shown in Fig. 10.12a), and more voltage than necessary will be applied to the load as shown in Fig. 10.14a; that is, applying the optimum load amount will not only increase efficiency, but can also control the voltage and current values. Furthermore, since it

340

10 Applications of Multiple Power Supplies

(b) Pr1

(a) Pn

Fig. 10.15 Receiving load Pn versus load resistance and receiving coil, and consumed power at internal resistance of transmitting coil Pr1 Pr1

2

4

6

8 10 12 14 16 18 20

Number of coils [n]

(a) R = 50 Ω

Pr1

Pn

Prn

Power [W]

Power [W]

Pn

50 40 30 20 10 0

10 8 6 4 2 0

2

4

6

Prn

8 10 12 14 16 18 20

Number of coils [n]

(b) Optimum load Ropt

Fig. 10.16 Relationship between each voltage (individual) and number of coils

(a) PRall

(b) Prall

Fig. 10.17 Total received power PRall versus load resistance and receiving coil, and loss at internal resistance of receiving coil Prall

10.1 Improved Efficiency When Using Multiple Power Supplies Pr1

PRall

Prall

Power [W]

Power [W]

PRall

50 40 30 20 10 0

2

4

6 8 10 12 14 16 18 20 Number of coils [n]

(a) R = 50 Ω

341

10 8 6 4 2 0

2

4

Pr1

Prall

6 8 10 12 14 16 18 20 Number of coils [n]

(b) Optimum load Ropt

Fig. 10.18 Relationship between (sum of) each voltage and number of coils

is known that a large amount of current flows to the transmitting side, the ratio of the power consumed by the internal resistance at the transmitting side increases, causing loss, which is seen in Figs. 10.16 and 10.18. Next, although calculated through formula (10.36), we will consider the distribution of power when the number of coils is increased. For example, we will consider when the input power of the model in Sect. 10.1.1 is 10 V. When n = 9, that is, when there are eight receiving coils, in a load of R = 50 , each load is 1.13 W, the total load is 9.04 W, the consumed power due to the internal resistance of the transmitting coil is 25.32 W, the total consumed power due to the internal resistance of the receiving coil is 0.01, and the efficiency is 26.2%. In contrast, when using the optimum load, which is the optimum maximum efficiency, of Ropt = 3.13 , the following improvements were observed: each load is 0.70 W, the total load is 5.59 W, the consumed power due to the internal resistance of the transmitting coil is 1.31 W, the total consumed power due to the internal resistance of the receiving coil is 0.94 W, and the efficiency is 71.3%. Important indicators include not only efficiency, but also whether necessary power can be sufficiently transmitted to the loads. Although simply raising the voltage allows necessary power to be transmitted, if the load value is not at its optimum, it may lead to the flow of a large amount of current, and a large voltage is applied. Therefore, a suitable system design is necessary. Finally, the frequency characteristics will be shown. The resistance is the optimum load value. The order of increasing coils is A → B→C → … as observed from the dots in Fig. 10.3. Figure 10.19 shows the frequency characteristics of the model when n = 2, 4, and 9. As observed in the frequency axis, efficiency increases as the number of coils increase. The efficiency indicates unimodality that peaks at the resonance frequency. This section discusses the optimal load and the number of multiple receiving coils within a large transmitting coil. As receiving coils are increased, the total efficiency is increased monotonically; that is, the total efficiency improves, and the value is eventually determined by the internal resistance at the load side of the load value. At the same time, formularization of the optimum load corresponding to the number

342

10 Applications of Multiple Power Supplies

Efficiency [%]

Fig. 10.19 Efficiency versus frequency (n = 2, 4, 9)

80 70 60 50 40 30 20 10 0

2 4 9

140

160

180 200 220 Frequency [kHz]

240

260

of multiple devices is known to allow power transmission to be implemented at maximum efficiency. Through this, if we assume, for example, supplying power to many small sensors in a large space, not only would we approach the situation by largely improving the Q value of the small coils to improve the individual efficiencies, but it is also possible to approach the situation by using multiple sensors to improve the total efficiency.

10.2 Cross-Coupling Canceling Method (CCC Method) The aim of wireless power transfer includes achieving not only a one-to-one transmitting–receiving coil, but also a one-to-many. In this case, cross-coupling may be generated between the relay coil and the multiple receiving coils. The phenomenon of cross-coupling affects the characteristics of wireless power transmission and improves efficiency; thus, it is the method for improving the efficiency by canceling the effects of cross-coupling [2]. Cross-coupling occurs in various scenarios. One example is illustrated in Fig. 10.20. Tx represents the transmitting coil, and Rx represents the receiving coil. Cross-coupling is referred to as the coupling generated between receiving coils; not the main coupling generated between the receiving coil and the transmitting coil. L m represents the mutual inductance between the transmitting and receiving coils. L c represents the mutual inductance due to cross-coupling between the receiving coils.

Fig. 10.20 Cross-coupling generated in various places

10.2 Cross-Coupling Canceling Method (CCC Method)

(a) Transmitting coil and two receiving coils

343

(b) Transmitting coil

(d) Photo of a transmitting coil and a receiving coil

(c) Receiving coil

(e) Photo of receiving coils

Fig. 10.21 Smallest configuration for cross-coupling with transmitting coil and two receiving coils

To validate this phenomenon, we will discuss the smallest configuration for generating cross-coupling which includes one receiving coil and two transmitting coils, as shown in Fig. 10.21. Cross-coupling is generated in various scenarios. However, Fig. 10.21 depicts a setup in which a transmitting coil is configured under a desk to supply wireless power to two nearby mobile devices on the desk. Here, even if there are multiple receiving coils, the mutual inductance L m between the transmitting and receiving power is constant. Furthermore, even if the receiving coils were increased, the cross-coupling L c between the receiving coils would still be constant. Figure 10.21 shows transmitting and receiving coils. The resonance frequency is 200 kHz for both receiving and transmitting coils, and operates at resonance frequency. The transmitting coil in Fig. 10.21a is 1200 × 250 mm, has five turns of copper wire with a radius of 1.0 mm, and is a short type that resonates with an external capacitor. For the two receiving resonators, the coil used is of the same shape and the radius is 50 mm. They have 10 turns of copper wire with a 0.5 mm radius, is a two-layer configuration, and has a distance of ph = 10.0 mm between the layers. These are also short type that are resonated with an external capacitor. Table 10.2 tabulates the detailed parameters of the transmitting coil and the receiving coil; n = Table 10.2 Coil parameters

Transmitting

Resonator 1

Receiving resonator

lx × ty (mm)

1200 × 250

Radius (mm)

50

Turns

5

Turns

10

a (mm)

1.0

a (mm)

p (mm)

4.0

p (mm)

L 1 (µH)

54.94

0.5 2.0

ph (mm)

10.0 36.28

C 1 (nF)

11.53

L n (µH)

r 1 ()

0.41

C n (nF)

17.45

Q1

168.15

r n ()

0.29

Qn

158.60

344

10 Applications of Multiple Power Supplies

Fig. 10.22 Position and parameters of coils

1 is the parameter concerning the transmitting coil, and n = 2, 3 are the parameters of the receiving coil. The receiving coils have the same configuration/structure; and therefore, there is no difference between n = 2 and n = 3. Figure 10.22 illustrates positions of the coils. The parameter s represents the distance between the receiving coils. The transmitting side coil’s self-inductance, external capacitor, internal resistance, and Q value are represented by L 1 , C 1 , r 1 , and Q1 , respectively. The receiving side coil’s self-inductance, external capacitor, internal resistance, and Q value are represented by L n , C n , r n , and Qn , respectively. Table 10.2 tabulates the positional parameters of the transmitting coil. When the position of the receiving coil is represented by x, the central position of the receiving coil is shown. The relationship between the coupling coefficient k, transmitting coil L 1 , and the mutual inductance L m of the receiving coil L n is expressed in the following formula: k=√

Lm L1 Ln

(10.47)

The optimum load value is used by taking the value of the load connected to each receiving coil.

10.2.1 Formulas for Multiple Loads and Cross-Coupling Here, a formula for the efficiency of power transmission for multiple loads that takes into account cross-coupling is derived. Although fundamentally the same as Sect. 9.3, the location(s) where cross-coupling is generated is/are described by L c . First, Fig. 10.23 shows the equivalent circuit for multiple loads. The self-inductance and mutual inductance that take into account the cross-coupling in all locations during this time is expressed in formula (10.48). Here, since the mutual inductance L 23 between the receiving coils corresponds to the cross-coupling, L 23 is reassigned as L c . ⎡

⎤ L 1 L 12 L 13 [L] = ⎣ L 12 L 2 L c ⎦ L 13 L c L 3

(10.48)

10.2 Cross-Coupling Canceling Method (CCC Method)

345

Fig. 10.23 Equivalent circuit with cross-coupling taken into account

Z represents the inductance at the locations with the load of each of the coils removed. The voltage V and the current I at this time are expressed in formula (10.49) and formula (10.50), respectively. Their relationship is expressed in formula (10.51). Each of the elements’ voltages V at this time is also expressed in formula (10.52). ⎤ Z 11 Z 12 Z 13 [Z] = ⎣ Z 21 Z 22 Z 23 ⎦ Z 31 Z 32 Z 33 ⎡ ⎤ ⎡ ⎤ I1 V1 [V ] = ⎣ V2 ⎦, [I] = ⎣ I2 ⎦ V3 I3 ⎡

(10.49)

(10.50)

[V ] = [Z][I]

(10.51)

⎧ ⎨ V1 = VL1 + VC1 + Vr 1 V = VL2 + VC2 + Vr 2 ⎩ 2 V3 = VL3 + VC3 + Vr 3

(10.52)

The relationships of the voltages generated in the coil, capacitor, and internal resistor are expressed in formulas (10.53) to (10.55). When formulas (10.53) to

346

10 Applications of Multiple Power Supplies

(10.55) are substituted into formula (10.52), formula (10.56) is obtained. ⎧ ⎨ VL1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 V = jωL 12 I1 + jωL 2 I2 + jωL c I3 ⎩ L2 VL3 = jωL 13 I1 + jωL c I2 + jωL 3 I3 ⎧ 1 ⎪ ⎨ VC1 = jωC1 I1 1 VC2 = jωC I2 2 ⎪ ⎩V = 1 I C3 jωC3 3 ⎧ ⎨ Vr 1 = r1 I1 V = r 2 I2 ⎩ r2 Vr 3 = r3 I3

⎧ 1 ⎪ ⎨ V1 = jωL 1 I1 + jωL 12 I2 + jωL 13 I3 + jωC1 I1 + r1 I1 1 V2 = jωL 12 I1 + jωL 2 I2 + jωL c I3 + jωC I2 + r 2 I2 2 ⎪ ⎩ V = jωL I + jωL I + jωL I + 1 I + r I 3 13 1 c 2 3 3 3 3 jωC3 3

(10.53)

(10.54)

(10.55)

(10.56)

In contrast, the voltages of the loads connected to each coil are expressed in formula (10.57). Accordingly, they can be collectively expressed as in formula (10.58). The input power P1 and the powers P2 and P3 consumed at each load are calculated through formula (10.59). Based on the above, the efficiencies η21 and η31 at each load are derived in formula (10.60), and the total efficiency η is derived in formula (10.61).

⎤ ⎡ r1 + jωL 1 + V1 ⎣ 0 ⎦ =⎢ jωL 12 ⎣ 0 jωL 13 ⎡ ⎤ I1 ⎣ I2 ⎦ ⎡

I3

⎧ ⎨ V1 : constant V = −I2 R2 ⎩ 2 V3 = −I3 R3

1 jωC1

(10.57)

⎤ jωL 12 jωL 13 ⎥ 1 r2 + jωL 2 + jωC + R2 jωL c ⎦ 2 1 jωL c r3 + jωL 3 + jωC3 + R3 (10.58)



⎧ ⎨ P1 = Re V1 I 1   P = PR2 = Re V2 −I 2  ⎩ 2 P3 = PR3 = Re V3 −I 3

η21 = PP21 η31 = PP31 η = η21 + η31

(10.59)

(10.60) (10.61)

10.2 Cross-Coupling Canceling Method (CCC Method)

347

Moreover, when aiming to operate at a separately excited resonance frequency f = f 0 , the reactance due to self-inductance L n and the reactance due to capacitance C n cancel each other out, as shown in formula (10.62); thus, they can be aggregated into formula (10.63). In this study, the resonance frequencies produced by self-inductance and the capacitance from the capacitor prior to being affected by cross-coupling, etc., shall be termed as separately excited resonance frequencies. Accordingly, they can be collectively expressed as in formula (10.64). The next section will discuss these formulas. jωL n +

1 =0 jωCn

⎧ ⎨ V1 = jωL 12 I2 + jωL 13 I3 + r1 I1 V = jωL 12 I1 + jωL c I3 + r2 I2 ⎩ 2 V3 = jωL 13 I1 + jωL c I2 + r3 I3 ⎡ ⎤ ⎡ ⎤⎡ ⎤ V1 jωL 12 jωL 13 I1 r1 ⎣ 0 ⎦ = ⎣ jωL 12 r2 + R2 jωL c ⎦⎣ I2 ⎦ jωL 13 jωL c r3 + R3 I3 0

(10.62)

(10.63)

(10.64)

10.2.2 Verification of the Effects of Cross-Coupling and Simple Frequency Tracking Method (Method A) To confirm the effects of cross-coupling, frequency-efficiency curves determined through numerical calculation for when cross-coupling is not present (L c = 0) and for when cross-coupling is present (L c = 1593.2 nH, sw = 3 mm) are shown in Fig. 10.24a, b. The value of the load uses the optimized value Ropt = 2.55  to achieve the maximum efficiency for when cross-coupling is not present and at the separately excited resonance frequency f 0 = 200 kHz. Ropt is determined using formula (10.66). Figure 10.24a shows the case in which cross-coupling is negative and Fig. 10.24b shows the case in which cross-coupling is positive. In either case, frequency is known to shift for cases in which cross-coupling is not present. Because cross-coupling is negative in this configuration, frequency shifts toward the higher values. The measured results of the experiment are shown in Fig. 10.24. When s is small and coupling is stronger, the frequency shift is larger. In contrast, if cross-coupling is positive, the frequency shifts toward the lower values, as shown in Fig. 10.24b. When considering the separately excited resonance frequency f = f 0 , the efficiency would decrease due to the shift in frequency characteristics. When s = 3 mm, the calculated value in graph (a) decreases by 4.9% in efficiency. When wireless power transmission is performed, in many cases, it is not possible to easily change the frequency in conjunction with the Radio Law, and thus, decreased efficiency at

Fig. 10.24 Effects of cross-coupling

10 Applications of Multiple Power Supplies

Efficiency [%]

348

50 40 30 20 10 0 160

Lc = 0 -Lc

180 200 220 Frequency [kHz]

240

Efficiency [%]

(a) When mutual inductance is negative (current configuration) 50 40 30 20 10 0 160

Lc = 0 +Lc

180 200 220 Frequency [kHz]

240

(b) When mutual inductance is positive

Efficiency [%]

50

sw=3mm

40

sw=20mm

30

sw=300m

20 10 0 160

180

200 220 Frequency [kHz]

240

(c) Test measurement results

the separately excited resonance frequency f 0 becomes an issue; that is, the frequency shift is significant in the degeneration of the cross-coupling efficiency. The efficiency at f 0 = 200.00 kHz during the absence of cross-coupling is 44.121%; the efficiency at peak frequency f p = 204.91 kHz when cross-coupling is negative is 44.919%; the efficiency at peak frequency f p = 196.00 kHz when crosscoupling is positive is 43.402%. The reason why efficiency is higher for negative cross-coupling compared to that of positive cross-coupling, at their respective peaks, is due to the increase in the Q value as frequency increases since the internal resistance was kept at a constant in this study when frequency increases, resulting in improved efficiency. Since the internal resistance also increases as an actual phenomenon, we

10.2 Cross-Coupling Canceling Method (CCC Method)

349

cannot necessarily state whether the efficiency improves or deteriorates; therefore, this difference can be ignored considering the nature of this section. From this point on, unless otherwise noted, the cross-coupling will be negative in accordance with the proposed configuration. As previously discussed, the reduction in efficiency at the separately excited resonance frequency f 0 due to cross-coupling is validated. However, the significance of cross-coupling is its shift in frequency characteristics. Accordingly, transmitting power at the peak frequency f p at which the maximum efficiency occurs through frequency tracking is able to achieve higher efficiency more easily than at the separately excited resonance frequency f 0 . This is referred to as method A. However, frequency tracking alone deviates from the optimum load that should be the maximum efficiency at the peak frequency, though this will be allowed. The conditions here can be summarized as the frequency tracking method at the load optimized at f 0 during the point in time in which cross-coupling is absent. Here, this can be simply referred to as the frequency tracking method, as well as method A. Method A is only for changing the power frequency.

10.2.3 Optimization for Load Resistance Only (Method B) and Limits Thereof We found in the previous section that the significance of the cross-coupling effect is its shift in frequency characteristics. We also found that using a simple frequency tracking method such as method A can allow a high-efficiency power transmission; however, in power transmission, the operating frequency is often fixed. As such, method B and method C aim to achieve maximum efficiency in this situation. Method B examines the conventional method for improving efficiency by optimizing only the real component of the load impedance, that is, the resistance value. This method is referred to as method B. Conventional cross-coupling without the generation of crosscoupling may achieve maximum efficiency through the simple method of merely optimizing the resistance value. If maximum efficiency was achievable, the merits would then be great; however, the limits thereof will be demonstrated here. First, the conditions will be summarized. The formulas in this section are examined at the separately excited resonance frequency f 0 , and only optimize the resistance value. In this configuration, the mutual inductance L m is substantially the same at the same height except at the ends. Investigating the central area allows for verifying the effect on efficiency based solely on the presence/absence of cross-coupling L c . Formula (10.4) and formula (10.5) are the times during which the cross-coupling between receiving coils can be ignored, and the maximum efficiency calculated with reference to this is expressed by ηa . Here, L 12 = L 13 = L m ; this is because the distance between the transmitting and receiving power is the same for all receiving coils. Local maximum is determined by differentiating formula (10.25) with a load, and formula (10.66) is obtained when the optimum load condition formula is determined.

350

10 Applications of Multiple Power Supplies

Substituting formula (10.66) into the efficiency ηa formula (10.67) determines the maximum efficiency during which cross-coupling can be ignored. ⎤ L 1 L 12 L 13 [L] = ⎣ L 12 L 2 0 ⎦ L 13 0 L 3  2r ω2 L 2m Ropt = + r2 r1 ⎡

ηa =

2Rω2 L 2m   (r + R) r1 R + 2ω2 L 2m + r1r

(10.65)

(10.66) (10.67)

On the contrary, the maximum efficiency when cross-coupling is considered, is expressed by ηb . Since cross-coupling is taken into account, formula (10.48) is obtained when formula (10.4) is used. Similarly, the local maximum is determined by differentiating formula (10.25) with a load, and formula (10.68) is obtained when the optimum load condition formula is determined. Substituting formula (10.68) into the efficiency ηb formula (10.69) determines the maximum. The ratio of the crosscoupling L c between the receiving coils to the mutual inductance L m between the transmitting and receiving coils is expressed as α, as shown in formula (10.70). By calculating formula (10.69) of the efficiency using α, the effects of cross-coupling can be confirmed.  2r ω2 L 2m Ropt = ω2 L 2c + + r2 (10.68) r1 ηb =

2Rω2 L 2m   (r + R) r1 R + 2ω2 L 2m + r1r + r1 ω2 L 2c    Lc    α= Lm 

(10.69) (10.70)

The difference δη between the efficiency when cross-coupling is absent and the efficiency when cross-coupling is present is expressed in formula (10.71); that is, δη is the efficiency that is lost due to the effects of cross-coupling. As seen in formula (10.66) and formula (10.68), the optimum load value increased slightly due to the effects of cross-coupling from the influence of ω2 L 2c . On the contrary, as seen in formula (10.67) and formula (10.69), the maximum efficiency is known to slightly deteriorate when cross-coupling is present due to the leftover effects of r 1 ω2 L 2c . δη = ηa − ηb

(10.71)

10.2 Cross-Coupling Canceling Method (CCC Method)

351

The following discusses the effects when the mutual inductance L m between a transmitting coil and a receiving coil is kept at a constant and when cross-coupling L c is changed. An air gap is fixed (g = 50 mm) to change a gap s between receiving coils. The change in cross-coupling L c can be confirmed by fixing the mutual inductance L m between the transmitting coil and the receiving coil. Figure 10.25a illustrates a strong coupling when s = 3 mm, and Fig. 10.25b illustrates a negligible coupling when s = 150 mm (x = 125 mm). Because these mutual inductance values are not able to be easily calculated from the theoretical formulas, the results from electromagnetic field analysis will be compared to the results from the experiment, and the uniformity thereof will be verified, upon which the theoretical calculations of the equivalent circuit will be made using the values of the experimental results. As such, the experimental results of the electromagnetic field analysis findings are shown in Fig. 10.26. Figure 10.26a shows the cross-coupling L c and the distance between receiving coils, and Fig. 10.26b shows the coupling coefficient and the distance between receiving coils. The uniformity

(a) s = 3 mm

(b) s = 300 mm

1200 1000 800 600 400 200 0

Sim. Exp.

0

100

200

sw [mm]

(a) Mutual inductance Lc

300

Coupling coefficient k

Mutual Inductance [nH]

Fig. 10.25 Coil positions a s = 3 mm, cross-coupling is present b s = 300 mm, cross-coupling is absent

0.04 Sim. Exp.

0.03 0.02 0.01 0.00

0

100

200

sw [mm]

(b) Coupling coefficient k

Fig. 10.26 Mutual inductance generated by cross-coupling between receiving coils

300

352

10 Applications of Multiple Power Supplies

between the results of the experiment and the results of the electromagnetic field analysis can be confirmed. The electromagnetic field analysis was performed with the method of moments. From here, all calculations of the equivalent circuit utilize the findings of experimental results. Furthermore, in this section, the height is fixed and g = 50 mm. Owing to the fact that the mutual inductance L m is constant at the fixed height except at the ends, this study employs a mutual inductance L m = 1498.3 nH at the center of the transmitting coil, determined by the experiment. Next, the efficiency will be verified. In method B, formula (10.68) is always considered to be satisfied; that is, the optimum load based on the resistance value with respect to the change in the distance between coils s is always applied. Figure 10.27 depicts the effect of mutual inductance L c of the cross-coupling between the receiving coils with respect to the mutual inductance L m of the transmitting coil and receiving coil, where α of formula (10.70) is represented on the x-axis and δη is represented on the y-axis. This was determined via theoretical formula (10.70) and formula (10.71). When cross-coupling is not present, that is, when α = 0, efficiency is maximum, and as the cross-coupling ratio increases, the efficiency deteriorates. For example, when L m = L c , that is, when α = 1, the efficiency deteriorates by 3.92%. In this manner, when cross-coupling becomes stronger, method B shows that the efficiency cannot be improved using α as an index; however, as a specific example that is much easier to understand, Fig. 10.28 shows the results of the theoretical formula when the distance between the receiving coils s is changed. Figure 10.29

Fig. 10.28 Effects of cross-coupling and distance of receiving coils s, g = 50 mm

12 10 8 6 4 2 0 0.0

0.5

η [%]

w/o CC 45 44 43 42 41 40 39 0

20

1.0 α

1.5

w CC

40 60 sw [mm]

80

2.0

δη 6 5 4 3 2 1 0 100

δη [%]

δη [%]

Fig. 10.27 Cross-coupling and efficiency

10.2 Cross-Coupling Canceling Method (CCC Method)

1.2 1 0.8 0.6 0.4 0.2 0

α

Fig. 10.29 Relationship between α and coil distance sw, g = 50 mm

0

353

20

40 60 sw [mm]

80

100

shows the difference of the efficiencies at f 0 when calculated without accounting for cross-coupling and at f 0 when accounting for cross-coupling. Through this, efficiency is known to deteriorate due to the effects of cross-coupling when the distance between the receiving coils is small. For example, when g = 50 mm and s = 3 mm, L m = 1498.3 nH and L c = −1593.2 nH, and thus, α = 1.06 and δη = 4.3%. On the contrary, as the distance between coils becomes larger, the effects of cross-coupling become less, and thus, the deteriorated efficiency due to the effects of cross-coupling is improved. For example, when g = 50 mm and s = 300 mm, in which cross-coupling is set to be negligible, L m = 1498.3 nH and L c = −17.2 nH, and thus, α = 0.012 and δη = 6.4 × 10−4 % . Moreover, similar results determined via theoretical formulas are shown as references in Fig. 10.30, in which the x-axis is the mutual inductance due to cross-coupling, and in Fig. 10.31, where the x-axis is the coupling coefficient k c due to cross-coupling. k c is expressed by the following formula. kc =

Lc Ln

(10.72)

Though not covered here, the condition for α increasing occurs not only when the distance between the receiving coils increases and L c decreases, but also when used at a position slightly separated from the transmitting coil. Owing to the fact that the coupling of the transmitting coil and the receiving coil would weaken and the mutual

w CC

w/o CC

η [%]

Fig. 10.30 When distance between receiving coils that generate cross-coupling is expressed via mutual inductance, (g = 50 mm)

45 44 43 42 41 40 39

0

500

1000 c

[nH]

1500

2000

354

10 Applications of Multiple Power Supplies

w CC

w/o CC

η [%]

Fig. 10.31 When distance between receiving coils that generate cross-coupling is expressed via coupling coefficient, (g = 50 mm)

45 44 43 42 41 40 39

0

0.01 0.02 0.03 0.04 0.05

kc inductance L m would decrease, L c would experience a relative increase, resulting in α becoming larger and the effects of cross-coupling becoming enhanced. As discussed, through method B, which is a simple method that merely optimizes the resistance value, there are limits to improving efficiency when subjected to the effects of cross-coupling.

10.2.4 Cross-Coupling Canceling Method (Method C) Based on the effects of the shift in frequency characteristics due to cross-coupling, efficiency deterioration at the separately excited resonance frequency f 0 and optimization limitations based on only conventional resistance values were confirmed in the previous section. As such, this section will introduce the cross-coupling canceling method (CCC). The method will be referred to as method C. Furthermore, this section will continue from the previous section to investigate the maximum efficiency at the separately excited resonance frequency f 0 . According to method B, it is assumed that the efficiency will be improved if the mutual inductance L c of the cross-coupling in the voltage-current relational expression (10.19) at the resonance frequency is canceled and the efficiency returns to the state without the cross-coupling.

10.2.4.1

−Lc and Canceling Coil Lcan

In this configuration, cross-coupling is negative. Therefore, a coil for canceling crosscoupling L can is inserted into the receiving end. The formula for this is expressed in formula (10.73). For formula (10.73), a coil L Tx for confirming the effects at both the receiving and transmitting ends is also inserted; however, as discussed below, there is no impact on efficiency improvement. ⎤⎡ ⎤ ⎤ ⎡ I1 V1 r1 + jωL T x jωL m jωL m ⎦⎣ I2 ⎦ (10.73) ⎣ 0 ⎦ = ⎣ jωL m r2 + jωL can + R2 − jωL c jωL m − jωL c r3 + jωL can + R3 I3 0 ⎡

10.2 Cross-Coupling Canceling Method (CCC Method)

355

The following formula (10.74) for efficiency is calculated based on formula (10.73). η=

2L 2 ω2 R  m  (r + R) r1 R + 2ω2 L 2m + r1r + r1 ω2 L 2c − 2r1 ω2 L c L can + r1 ω2 L 2can (10.74)

We will determine the value of the optimum load for implementing maximum efficiency. Differentiating the efficiency via the load and determining the optimum load from the extreme value gives the following formulas.  Ropt = ω2 L 2c + 

2r ω2 L 2m + r 2 − 2ω2 L c L can + ω2 L 2can r1

= ω2 (−L c + L can )2 +

2r ω2 L 2m + r2 r1

(10.75)

First, L Tx is not involved in these formulas; that is, L Tx does not affect the efficiency. Second, when formula (10.75) is compared with both formula (10.66) (during which cross-coupling is absent) and formula (10.68) (which is affected by cross-coupling), if L c is canceled out by L can , the effects of cross-coupling from the formula of the optimum load would be canceled as in formula (10.76); that is, it is only necessary to add a canceling coil while maintaining the optimum load value with respect to the resonance frequency f 0 during which cross-coupling is absent. The inductance value of the canceling coil becomes identical to the value generated due to cross-coupling. Furthermore, to cancel out -L c , the coil is required instead of the capacitor. −L c + L can = 0

(10.76)

Verification is accomplished through numerical calculation using the value s = 3 mm, at which the cross-coupling is strong. Figure 10.32 shows the improved efficiencies when cross-coupling is absent (L c = 0), when cross-coupling occurs (L c− ),

Efficiency [%]

Fig. 10.32 CCC method, for −L c (using L can )

50

Lc = 0

40

-Lc

30

prop. C

20 10 0

160

180

200

220

Frequency [kHz]

240

356

10 Applications of Multiple Power Supplies

and at the separately excited resonance frequency f 0 due to the cross-coupling canceling applied in method C. These results were determined using theoretical calculations. When calculating the maximum efficiency at f 0 under the conditions of method C, the results for the coil used for canceling L can , the optimum load Ropt , and the total efficiency η are 1.593 uH, 2.554 , and 44.12%, respectively. The values of Ropt and the total efficiency are completely uniform when cross-coupling is absent. When the effects of the cross-coupling due to coil L can , which is inserted in the receiving side, are canceled, the maximum efficiency is achievable. At this moment, the conditions of (10.76) are satisfied. The shift in frequency characteristics returns to its original, and its original waveform also resonated at 200 kHz. The measured results of the experiment are shown in Fig. 10.33. Here, the experiment was conducted in which the canceling coil was inserted to remove the shift in frequency characteristics after the distance between coils was reduced to allow cross-coupling to be generated when s = 3, 5, 10, and 20 mm. Naturally, the closer the distance between the coils, the larger the shift in frequency characteristics. By inserting the canceling coil, the frequency returned to the resonance phenomenon at 200 kHz.

40 30 20 10 0 180

190

200

210

sw=5mm

40 30 20 10 0 180

220

190

200

210

Frequency [kHz]

(a) s = 3 mm

(b) s = 5 mm

sw=10mm

sw=20mm (Lcan)

Efficiency [%]

50 40 30 20 10 0 180

sw=5mm (Lcan) 50

Frequency [kHz]

sw=10mm (Lcan)

Efficiency [%]

sw=3mm

Efficiency [%]

Efficiency [%]

sw=3mm (Lcan) 50

190

200

210

220

220

sw=20mm

50 40 30 20 10 0 180

190

200

210

Frequency [kHz]

Frequency [kHz]

(c) s = 10 mm

(d) s = 20 mm

Fig. 10.33 Measured values of CCC method, for −L c (using L can g = 50 mm)

220

10.2 Cross-Coupling Canceling Method (CCC Method)

10.2.4.2

357

+Lc and Canceling Capacitor C can

Although cross-coupling is negative in this configuration, analysis was conducted while assuming that the cross-coupling was positive. Accordingly, the capacitor C can for canceling the cross-coupling is inserted into the receiving side. The formula for this is expressed in formula (10.77). Similar to the previous analysis, formula (10.77) comprises a capacitor C Tx at both the receiving and the transmitting ends so as to confirm the results; however, as discussed below, there is no impact on efficiency improvement. ⎤⎡ ⎤ ⎤ ⎡ jωL m jωL m r1 + jωC1 T x V1 I1 ⎥⎣ ⎦ 1 ⎣ 0 ⎦=⎢ + jωL c ⎦ I2 ⎣ jωL m r2 + jωCcan + R2 I3 0 jωL m + jωL c r3 + jωC1 can + R3 ⎡

(10.77)

The following formula (10.78) for efficiency is determined based on formula (10.77). η=

2L 2m ω2 R   (r + R) r1 R + 2ω2 L 2m + r1r + r1 ω2 L 2c −

2r1 L c Ccan

+

r1 2 ω2 Ccan

(10.78)

We will determine the value of the optimum load for implementing maximum efficiency. Differentiating the efficiency via the load and determining the optimum load from the extreme value gives the following formulas: 

2r ω2 L 2m 2L c 1 + r2 − + 2 2 r1 Ccan ω Ccan  2 1 2r ω2 L 2m ωL c − = + + r2 ωCcan r1

Ropt = ω2 L 2c +

(10.79)

First, C Tx is not involved in these formulas; that is, C Tx does not affect the efficiency. Second, when formula (10.79) is compared to both formula (10.66) (during which cross-coupling is absent) and formula (10.68) (which is affected by crosscoupling), if L c is canceled out by C can , the effects of cross-coupling from the formula of the optimum load would be canceled (formula 10.80). This implies that it is only necessary to add a canceling capacitor while maintaining the optimum load value with respect to the resonance frequency f 0 during which cross-coupling is absent. Based on formula (10.80), the capacitance value C can of the canceling capacitor can be considered canceled out by resonating with the mutual inductance L c and f 0 that are generated by the cross-coupling. Furthermore, to cancel out +L c , the coil is required instead of the capacitor. ωL c −

1 =0 ωCcan

(10.80)

358

10 Applications of Multiple Power Supplies

Verification is performed through numerical calculation using the value s = 3 mm, at which the cross-coupling is strong. As previously described, +L c does not occur in this configuration. Accordingly, values other than that of the mutual inductance L c generated via the cross-coupling are used as reference values applied in 10.2.4.1. The results of the theoretical calculations are shown in Fig. 10.34. This depicts the improved efficiencies when cross-coupling is absent (L c = 0), when cross-coupling occurs (+L c ), and at the resonance frequency f 0 , due to the cross-coupling canceling applied in method C. When calculating the maximum efficiency at f 0 under the conditions of method C, the results for the capacitor used for canceling C can , the optimum load Ropt , and the total efficiency η are 397.5 nF, 2.554 , and 44.12%, respectively. The values of Ropt and the total efficiency are completely uniform when cross-coupling is absent. When the effects of the cross-coupling due to capacitor C can , which is inserted in the receiving side, are canceled, the maximum efficiency can be achieved. At this moment, the conditions of (10.80) are satisfied. Each method is summarized in the Table. Table 10.3 tabulates the comparisons between method B and method C (when coupling is canceled out) during which f = f 0 (original state) and when the occurrence of cross-coupling causes deterioration. In regard to the method B and method C, both of which aim for improved efficiency at f = f 0 , method B showed that there were limits to optimization at only the resistance component, while the cross-coupling canceling method of method C revealed that, by removing the effects of cross-coupling, it is possible to improve efficiency to the original efficiency of which cross-coupling is absent.

Efficiency [%]

Fig. 10.34 CCC method, for +L c (using C can )

Lc = 0

50 40

+Lc

30

prop. C

20 10 0

160

180

200

220

240

Frequency [kHz]

Table 10.3 Presence/absence of cross-coupling and comparisons between method B and method C (f = f 0 ) f 0 (kHz)

Efficiency at f0 (%)

Ropt

Frequency where R is optimized (kHz)

Canceling L can or Ccan

w/o CC

200.0

44.12

2.55

200.00

N/A

Prop. B

200.0

39.79

3.24

200.00

N/A

Prop. C (−Lc)

200.0

44.12

2.55

200.00

1.593 (µH)

Prop. C (+Lc)

200.0

44.12

2.55

200.00

397.5 (nF)

References

359

References 1. T. Imura, Simultaneous wireless power transfer via magnetic resonant coupling to multiple receiving coils. Trans. Inst. Electr. Eng. Jpn D, A Publ. Ind. Appl. Soc. 134(6), 625–633 (2014) 2. T. Imura, Cross-coupling canceling method for wireless power transfer via magnetic resonance coupling. Trans. Inst. Electr. Eng. Japan D 134(5), 564–574 (2014)

Chapter 11

Basic Characteristics of Electric Field Resonance

Electric resonance coupling was briefly introduced in Sect. 11.3, but will be described in detail in this chapter and part of Chap. 12. The relationship between electromagnetic induction (IPT: Inductive power transfer) and magnetic resonance coupling is equal to the relationship between electric field coupling (CPT: Capacitive power transfer) and electric resonance coupling, and forms a dual relationship. By replacing the magnetic field with an electric field, it is easy to understand electric resonance coupling.

11.1 Capacitors as Electromagnetics and the True Nature of Displacement Current At the time of magnetic resonance coupling, it was sufficient to understand the capacitor as a component of the circuit. However, in order to understand electric field coupling, it is still necessary to understand the capacitor electromagnetically. Therefore, the relationship between the capacitor and the displacement current will be described. The principle of a capacitor is shown in Fig. 11.1, which is an object in which two planar metals (plate) are arranged in parallel. Since the plate shape is wide Fig. 11.1 Displacement current

© Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_11

361

362

11 Basic Characteristics of Electric Field Resonance

and takes a place, it is usually the type that is rolled up to be compact and cylindrical, but there are many plate-like objects when electric field coupling is performed. The area between the plates of the capacitors is a space. In the first place, the capacitor transfers energy over the space. What is flowing here is the displacement current. [Note: Capacitance C is small with air alone, so a common capacitor holds a dielectric between the plates to increase C.] What flows in the space between the capacitors is a displacement current. The displacement current is dD/dt and was discovered by Maxwell. The electric flux density is D, and the amount of change is the displacement current dD/dt. The change in electric flux density dD/dt, that is, the change in electric field dE/dt (time derivative of electric field) is important as well as the change in magnetic field at the time of electromagnetic induction (Eq. 11.1). Equation (11.2) shows the relationship between the electric flux density D and the electric field E. The dielectric constant is ε. When the permittivity in vacuum is ε0 = 8.854 × 10−12 (F/m) and the relative permittivity is εr , the relationship between the permittivity and the relative permittivity is expressed by Eq. (11.3). The relative permittivity of air is 1.0006, and water is about 80. The displacement current dD/dt is a change in the electric flux density D after all, and can be said to be a change in the electric field E. Its dimension is equal to the current I (Eq. 11.4). Since the change of the electric flux density and an electric field displacement current, the identity of the power transmission in electric field coupling is displacement current shown in Fig. 11.1. Here, S is an area where displacement current flows.

I =

dE dD =ε dt dt

(11.1)

D = εE

(11.2)

ε = ε0 εr

(11.3)

dE dD S=ε S dt dt

(11.4)

As in the Eqs. (11.2) and (11.5), in the relationship between the electric field and the magnetic field, the electric field E corresponds to the magnetic field strength H, and the electric flux density D corresponds to the magnetic flux density B. B = μH

(11.5)

Hereafter, it is confirmed here that the current and the displacement current multiplied by the area are the same unit system by a simple method. In power transmission, it is sufficient to understand that a change in the electric flux density D, that is, a change in the electric field E has an effect. However, in order to understand the basic

11.1 Capacitors as Electromagnetics and the True …

363

equation around the capacitor, it is shown that the current and the electric flux density are the same unit system. The capacitance is C and is given by Eq. (11.6). The distance is d. The relationship between the charge Q, the voltage V, and the capacitance C is represented by expression (11.7). The capacitance C is a proportional coefficient indicating how much charge is accumulated when a voltage is applied. εS d

(11.6)

Q = CV

(11.7)

C=

Since the amount of change in the electric charge is the current, the current is expressed by Eq. (11.8). I =

dQ dt

(11.8)

Therefore, expression (11.9) of the current, which is the amount of change in the voltage, is obtained. I =C

dV dt

(11.9)

Therefore, integrating both sides of Eq. (11.9) results in a voltage equation, which is an integral form of the current. V =

1 C

 I dt

(11.10)

In addition, since the potential is a physical quantity obtained by integrating the electric field with the distance, the potential is represented by expression (11.11). Here, for simplicity, assuming that the electric field is constant, the potential is the product of the electric field and the distance d and therefore, Eq. (11.12) is obtained.  V =

E dr

V = Ed

(11.11) (11.12)

The purpose here is to confirm by a simple method that the current I and the displacement current dD/dt are in the same unit system, which is as follows. The results are shown in Eq. (11.13). First, the current and voltage relational expressions in expression (11.9) are started. After expressing the electric field, expressing the electric flux density, taking out the constant, using the definition Eq. (11.6) of the capacitor, the last can be expressed as the product of the area and the displacement

364

11 Basic Characteristics of Electric Field Resonance

current. As described above, the unit system obtained by multiplying the area by the displacement current matches the unit system of the current. From this fact, it can be seen that the same type as the current is moving in the space between the capacitors. That is the displacement current.   ∂ Dε d ∂V ∂(Ed) Cd ∂ D ∂D I =C =C =C = =S ∂t ∂t ∂t ε ∂t ∂t

(11.13)

The relationship between the charge Q, the electric flux density D, and the area S is given by expression (11.14) by integrating Eqs. (11.8) and (11.13). Incidentally, the relationship between the magnetic flux Φ, the magnetic flux density B, and the area S is represented by expression (11.15). Strictly speaking, the magnetic flux Φ corresponds to the electric flux Φ e , but since it is rarely used except in a textbook on electromagnetism, the electric charge Q of the source of the electric flux Φ e is used. Eventually, both have the same value. Q = DS

(11.14)

 = BS

(11.15)

11.2 Electric Field Coupling The electric field coupling will be described based on the above knowledge. After all, a change in the electric field, as well as a change in the magnetic field, is the essence of energy transmission. If it is an electric field, it is simply named displacement current dD/dt. Figure 11.2 illustrates the electric field coupling. When energy is sent from the AC power supply, plus and minus of the secondary side are induced according to plus and minus of the electric field of the primary side. Therefore, the equivalent circuit is shown in Fig. 11.2. Since the coupling here is an electric field, the mutual capacitance C m indicates the coupling strength. Figure 11.2b is a c in which the coupling portion in the electric field is written in an easily understandable manner. Figure 11.2c is obtained by modifying the schematic diagram so that the connection can be easily understood. Thus, a part of the electric field is transmitted to the secondary side. The equivalent circuit is represented by the π-type, and Fig. 11.2d, e are used. The transformation from Fig. 11.2d, e is shown in Fig. 11.2f. Z C//R is the sum of the parallel impedances made by C 2 and RL . When C m1 = C m2 , C m = C m1 /2. However, since resonance is not used in this state, high-efficiency power transmission and high-power power transmission cannot be realized. This is because the secondary side is capacitive, and the capacitor uses the voltage induced on the secondary side rather than the load.

11.2 Electric Field Coupling

365

(a) Equivalent circuit

(b) Schematic diagram

(c) Schematic diagram

(d) π-type equivalent circuit, Cm1 and Cm2

(e) π-type equivalent circuit, Cm

(f) Circuit deformation

Fig. 11.2 Principle of electric field coupling

366

11 Basic Characteristics of Electric Field Resonance

11.3 Introduction of Electric Resonance Coupling After the electric power transmission by the electric field itself becomes possible, the goal is to achieve high efficiency and high power. Regarding electric field coupling, as shown in Fig. 11.3, depending on the presence or absence of resonance, electric field coupling of non-resonant (N–N), primary-side resonance (S–N), secondary-side resonance (N–S), and both-side resonance (S–S), that is, electric resonance coupling is classified. When the same conditions as the magnetic field resonance are satisfied, the electric field resonance also has high efficiency and high power, so that the primaryside resonance frequency and the secondary-side resonance frequency are eventually equalized (Fig. 11.3d). What satisfies the condition is electric resonance coupling. Figure 11.4 shows an equivalent circuit of π-type electric resonance coupling. By adding a resonance coil to the power transmission side and the power reception side, it operates as electric resonance coupling. Details are given in Sect. 11.7. Here, a self-resonant electric field resonance will be described as an example. The electric field type resonator used in this chapter is a meander line resonator shown

(a) Non-resonant (N-N)

(b) Primary-side resonance (S-N)

(c) Secondary-side resonance (N-S)

(d) Electric resonance coupling (S-S)

Fig. 11.3 Electric field coupling and electric resonance coupling Fig. 11.4 π-type electric resonance coupling, S–S

11.3 Introduction of Electric Resonance Coupling

367

Fig. 11.5 Meander line resonator parameters [1–3]

in Fig. 11.5 [1–3]. It is assumed that lx = 500 mm, ly = 495 mm, w = s = 5 mm, and the number of steps n = 49. The number of portions elongated in a strip shape in the lx direction is one stage. This structure is referred to as a meander line resonator because it is a meander line structure. This is an open-type resonator that does not require an external resonance coil and can self-resonate by itself. Figure 11.6 shows a simple principle diagram. Figure 11.6a shows an electric field and current distribution in the case of one element, and Fig. 11.6b shows a side view. Since the positive charges move in the direction in which the current flows, an electric field is generated in the same direction as the current (+y-axis direction). On the other hand, as a space, an electric field is generated from +E to −E, so that it is generated in a direction opposite to the direction of the current (−y-axis direction). In other words, the meander line resonator looks like a single resonator structure, but has two plates. Next, Fig. 11.6c shows two elements. When a power receiving resonator is arranged, the polarity (positive and negative) opposite to the electric charge (electric field) induced on the power transmission side is shown. This is entirely the nature of the capacitor. This enables power transmission via the electric field. Electric power cannot be transmitted in an electric field E that does not fluctuate over time like a magnetic field. The fluctuating E, that is, dE/dt enables power transmission. Figure 11.6c is a schematic diagram viewed from right beside during power transmission.

Fig. 11.6 Operating principle of meander line resonator

368

11 Basic Characteristics of Electric Field Resonance

The meander line resonator is an electric field type resonator. The voltage increases at the end of the resonator, and the two resonators are coupled by an electric field. Like the helical resonator, the proposed meander line resonator is small in size with respect to wavelength, and the radiation resistance of one element is small and cannot be matched. It can be used only as a power transmission resonator. Furthermore, the meander line resonator has a characteristic that, due to its meandering shape, the current flows in the opposite direction, so that the magnetic field is neglected neatly. Although the meander line resonator has been introduced here, the electric field resonance can be similarly realized by a combination of a plate and a resonance coil.

11.4 Air Gap Characteristics Figure 11.7 shows the air gap characteristics of the electric field resonance. Here, for simplicity, it is assumed that the resonance condition is Eq. (11.16) and the load resistance is RL = 50 . f0 =

1 1 = √ √ 2π L 1 C1 2π L 2 C2

(11.16)

When the air gap is close, the number of resonance frequencies at which the received power peaks is two, and when the air gap is large, the peak is one. This phenomenon is similar to magnetic resonance coupling. However, in the electric field coupling, the relation between the resonance frequency f e  and the f m  , which are two peaks, is defined as f e  < f m  . The frequency corresponding to the center valley 100

100

40

50

20

0

0

25

30

20

Frequency [MHz]

30

P2 [W]

40 20

25

30

20

25

Frequency [MHz]

Frequency [MHz]

(c) g

400mm

40 20

20

25

(b) g

200mm

60

0

60

0

30

20

Frequency [MHz]

80

η [%]

P2 [W]

0

100

20

50

Frequency [MHz]

(a) g 250 200 150 100 50 0

25

100

30

500 400 300 200 100 0

25

30

Frequency [MHz]

300mm 100 80

η [%]

20

80

150

60

P2 [W]

η [%]

P2 [W]

150

100

200

80

η [%]

200

60 40 20

20

25

30

Frequency [MHz]

(d) g

0

20

25

30

Frequency [MHz]

500mm

Fig. 11.7 Air gap characteristics. (lx = 500 mm, ly = 495 mm, g = 150 mm, s = w = 5 mm, n = 49 steps)

11.4 Air Gap Characteristics

369

Fig. 11.8 Air gap and coupling coefficient

of the two peak frequencies shown in Fig. 11.7a, b, or the resonance frequency when one peak in Fig. 11.7d is almost equal to the resonance frequency when the resonator is one element. This phenomenon is also similar to magnetic resonance coupling. The case when the air gap is changed will be described. When the air gap changes, the positional deviation dx = dy = 0. The air gap and the coupling coefficient are shown in Fig. 11.8. The coupling coefficient k e represents the ratio of electric field coupling between the transmitting and receiving resonators, and is determined from two resonance frequencies when the load value is set to zero as much as possible. ω2 − ωe2 ke = m2 ωm + ωe2    ∵ ωm = 2π f m , ωe = 2π f e

(11.17)

The electric field coupling in the meander line resonator has the same tendency as the magnetic field coupling as a whole. When the gap is small, the coupling is strengthened, the resonance frequency is divided into two, and high-efficiency power transmission becomes possible. On the other hand, when the gap becomes large, the coupling becomes weak, the frequency peak becomes one, and when the gap becomes further large and ke ≈ 0, power transmission becomes impossible.

11.5 Misalignment Characteristics Figure 11.9 shows the relationship between efficiency and displacement. When the displacement is changed, g is set to 200 mm. The load is unified to 50 . When Fig. 11.9 Efficiency when misaligned

dy

dx

η[%]

100

50

0

0

100

200

300 [mm]

400

500

600

370

11 Basic Characteristics of Electric Field Resonance

shifting dx, dy = 0 mm, and when shifting dy, dx = 0 mm. As the positions of both dx and dy shift, the coupling becomes weaker. Since the shift in the dx direction keeps the symmetry of the electric field and only increases the distance, the efficiency gradually decreases. On the other hand, with respect to the shift in the dy direction, the efficiency is low because the direction of coupling of the electric field is switched once around dy = 300 mm where the resonator is shifted by about half.

11.6 Near-Field Electric Field The behavior of the electromagnetic field near the resonator is shown. The electric field vectors are shown in Figs. 11.10 and 11.11. Figure 11.12 shows the electric and magnetic field distributions. Figure 11.13 shows the electric power density and the magnetic power density. Figure 11.13 is normalized by the maximum value. In the electric field coupling as well as the magnetic field coupling, a very characteristic distribution is shown at two resonance frequencies f e  and f m  . This appears in the state of the electric field in the plane of symmetry between the transmitting resonator and the receiving resonator. In f e  , an electric field is

(a) Electric wall fe’

(b) Magnetic wall fm’

Fig. 11.10 Electric field distribution in near field (vector)

Fig. 11.11 Schematic diagram of electric field and current

(a) Electric field, fe’

(b) Electric field, fm’

11.6 Near-Field Electric Field

(a) Electric field, fe’

(c) Magnetic field, fe’

371

(b) Electric field, fm’

(d) Magnetic field, fm’

Fig. 11.12 Electric and magnetic field distribution

(a) fe’, x 0mm

(b) fm’, x 0mm

Fig. 11.13 Power density in symmetry plane of magnetic and electric fields

distributed perpendicular to the symmetry plane to form an electric wall, and in f m  , an electric field is distributed horizontally to the symmetry plane to form a magnetic wall. The distribution of the electric wall and the magnetic wall is confirmed in the same way as the magnetic field coupling. However, considering the resonance frequency generated by the magnetic wall and the electric wall, in the magnetic field coupling, f m  < f e  . In the electric field coupling, f e  < f m  . From Fig. 11.12, it can be seen that the magnetic field is neglected neatly. This is because the magnetic field is canceled by the current flowing in the opposite direction due to the meandering line meandering shape. Therefore, on the symmetry plane, the ratio of the magnetic power density to the electric power density is less than 0.001%.

372

11 Basic Characteristics of Electric Field Resonance

11.7 Comparison of Electric Field Coupling and Electric Resonance Coupling (N–N, N–S, S–N, S–S, S–P) This section describes a comparison of N–N, N–S, S–N, S–S, and S–P. The equivalent circuit can be obtained in the same manner as the magnetic field coupling. However, in the case of electric field coupling, mutual capacitance is used instead of mutual inductance because it is not magnetic field coupling [4]. Figure 11.14 shows the equivalent circuit. Also, the basic mathematical expression is shown in expression (11.18). The method of obtaining the efficiency and the like is the same as that of the magnetic field resonance. Since it is expressed by π-type coupling, there are three current loops. Therefore, calculation is performed using a matrix of three rows and three columns. ⎤ ⎡ r1 + jωL 1 + jω(C11−Cm ) V1 ⎢ ⎥ ⎢ − jω(C11−Cm ) ⎣ 0 ⎦=⎢ ⎣ 0 0 ⎡

−

1 jω(C1 −Cm ) 1 1 1 jω(C1 −Cm ) + jωCm + jω(C2 −Cm ) − jω(C21−Cm )

0 −

1 jω(C2 −Cm )

r2 + R L + jωL 2 +

1 jω(C2 −Cm )

⎤⎡ ⎤ I ⎥⎢ l1 ⎥ ⎥⎣ Im ⎦ ⎦ Il2

(11.18)

11.7.1 Electric Field Coupling Type Coupler Structure and Capacitance Figure 11.15 shows the transmitting and receiving couplers for electric field coupling. The electric field coupling is composed of four metal plates, and the power

(a) Equivalent circuit

(b) π-type equivalent circuit Fig. 11.14 Equivalent circuit of electric resonance coupling (S–S)

11.7 Comparison of Electric Field Coupling and …

373

Fig. 11.15 CPT coupler with vertical structure

Fig. 11.16 Equivalent circuit diagram of CPT coupler

supply side is called a power transmission coupler and the load side is called a power reception coupler. Figure 11.16 shows the equivalent circuit of the power transmission/reception coupler. At this time, the self- capacitances C 1 and C 2 and the mutual capacitance C m can be expressed by (11.19) to (11.21), respectively. C1 = C12 +

(C13 + C14 )(C23 + C24 ) C13 + C14 + C23 + C24

(11.19)

C2 = C34 +

(C13 + C14 )(C23 + C24 ) C13 + C14 + C23 + C24

(11.20)

Cm =

C24 C13 − C14 C23 C13 + C14 + C23 + C24

(11.21)

11.7.2 Each Topology of Electric Field Coupling There are five circuit topologies: non-resonant circuit (NN), primary-side resonance circuit (SN), secondary-side resonance circuit (NS), electric field resonance coupling circuit (SS), and electric field resonance coupling circuit (SP) topology. Figure 11.17 shows each topology. The power transmission efficiency η is expressed by Eq. (11.22). Further, Pr 1 , Pr 2 and P2 are represented by Eqs. (11.23)–(11.25). Note that Pr 1 , Pr 2 , and P2 are power consumption at r1 , r2 , and R L , respectively.

374

11 Basic Characteristics of Electric Field Resonance

(a) Non resonance (N-N)

(b) Primary side resonance (S-N)

(c) Secondary side resonance(N-S)

(d) Electric resonant coupling(S-S)

(e) Electric resonant coupling(S-P) Fig. 11.17 Five circuit topologies

η=

P2 Pr 1 + Pr 2 + P2

(11.22)

Pr 1 = r1 |I1 |2

(11.23)

Pr2 = r2 |I2 |2

(11.24)

P2 = R L |I2 |2

(11.25)

Note that Pr2 and P2 of the electric field resonance coupling circuit (S–P) are represented by (11.26) and (11.27).

11.7 Comparison of Electric Field Coupling and …

375

 2 Pr2 = r2  Ir2 

(11.26)

 2 P2 = R L  I R L 

(11.27)

11.7.3 Derivation of Theoretical Formula In this subsection, we derive the resonance condition, efficiency, output power, and optimal load in the target topology shown in Sect. 11.7.2. First, the derivation process of the equation will be described using an electric resonance coupling circuit (S–S) as an example. Figure 11.18 shows an extended π-type circuit with a separate capacitance to study the electric resonance coupling circuit (S–S) shown in Fig. 11.17d. The values of each impedance are shown in Eqs. (11.28)–(11.30). Z in2 =

1 1 R L +r2 + jωL 2

Z2 = Z in1 =

+ jωC2

1 ω2 Cm2 Z in2

1 jωC1 +

1 Z2

+ jωL 1 + r1

(11.28) (11.29) (11.30)

Since the condition that becomes the secondary-side resonance condition is when the imaginary part of expression (11.28) becomes 0, condition (11.31) of ω2 is obtained.

Fig. 11.18 Expanded π-type equivalent circuit (S–S)

376

11 Basic Characteristics of Electric Field Resonance

Fig. 11.19 Loop current circuit (S–S)

ω2 =

1 − L 2 C2

R L + r2 L2

2 (11.31)

Also, the condition for the primary-side resonance condition is when the imaginary part of the expression (11.3) becomes 0, and the condition (11.32) is obtained.     C1 (R L + r2 )2 − ω12 {C2 A − L 2 } C1 L 2 − C1 C2 − Cm2 A L1 =    2 ω14 C1 L 2 − C1 C2 − Cm2 A + ω12 C12 (R L + r2 )2

(11.32)

Next, we consider efficiency. I l1 , I m , and I l2 shown in Fig. 11.19 are loop currents, I 1 and I 2 are branch currents, and have the following relationship. Il1 = I1

(11.33)

Il2 = I2

(11.34)

From the circuit of Fig. 11.19 and Eq. (11.18), the ratio of |I1 |2 and |I2 |2 can be shown as Eq. (11.35). Further, from the expressions (11.23) to (11.25), the ratio of the power consumption at each resistor can be expressed as the expression (11.36). Note that the following parameters A and B are used to avoid complicated expressions.   |I1 |2 : |I2 |2 = ω2 C1 C2 A + C12 1 − ω2 L 2 C2 : Cm2 Pr 1 : Pr2 : PR L = r1 |I1 |2 : r2 |I2 |2 : R L |I2 |2

(11.35)

     = r1 ω2 C1 C2 − Cm2 C1 A − L 2 C1 − Cm2 B + C12 : r2 Cm2 : R L Cm2

(11.36)



A = (R L + r2 )2 + (ωL 2 )2 B = r22 + (ωL 2 )2

From Eqs. (11.22) and (11.36), the equation for efficiency is shown in Eq. (11.37). In addition, when the resonance condition (11.31) on the secondary side, which affects the efficiency, is taken into consideration, the efficiency is expressed by

11.7 Comparison of Electric Field Coupling and …

377

Table 11.1 Resonance condition Primary-side resonance condition N–N

–

S–N

L1 =

N–S

–

Secondary-side resonance condition –

  2 C1 +ω12 C2 (r2 +R L )2 C1 C2 −Cm   2 2 2 2 2 ω1 ω (r2 +R L ) (C1 C2 −Cm ) +C12

– ω2 =  1 L 2 C2

S–S

S–P

L1 =

    2 A C1 (R L +r2 )2 −ω12 {C2 A−L 2 } C1 L 2 − C1 C2 −Cm 2 ) A} +ω2 C 2 (R +r )2 ω14 {C1 L 2 −(C1 C2 −Cm 2 L 1 1 2

L 1 =

    2 R 2 B(L −C B) (R L r2 +B)2 +ω12 R 2L (L 2 −C2 B)2 C1 (R L r2 +B)2 +ω12 R 2L (L 2 −C2 B)2 +ω12 Cm 2 2 L     4 R 2 B 2 (R r +B)2 + ω C (R r +B)2 +ω2 R 2 (L −C B)2 +ω3 C 2 R 2 B(L −C B) 2 ω14 Cm 1 1 2 2 2 2 L 2 L 2 1 L 1 m L L

−

ω2 =  1 L 2 C2

−

ω2 =  1 C2 L 2

−







R L +r2 L2

R L +r2 L2

r2 L2

2

2

2

the following Eq. (11.38). Further, the optimum load Eq. (11.40) is obtained from Eq. (11.39). η(ω) =

η(ω2 ) =

R L Cm2  2     r1 C1 + ω2 C1 C2 − Cm2 C1 C2 − Cm2 A − 2L 2 C1 + (r2 + R L )Cm2 (11.37) R L L 2 C22 Cm2   2 2   (11.38) 2 r1 C2 (R L + r2 ) C1 C2 − Cm4 + L 2 Cm4 + (r2 + R L )Cm2 C22 L 2

R Lopt

dη =0 d RL   L 2 Cm2 r1 Cm2 + r2 C2 2   = r2 + r1 C2 C12 C22 − Cm4

(11.39)

(11.40)

Next, output power is derived. Since the equation of the output power is complicated, only the derivation method is shown. If a circuit after the input impedance Z in1 is represented by a real number R and an imaginary number X, it can be expressed as in Eq. (11.41). From Eq. (11.41), the output power can be expressed by Eq. (11.42). Z in1 = R + j X P1 =

R V2 R2 + X 2 1

(11.41) (11.42)

Similarly, for the N–N, S–N, N–S, and S–P topologies other than S–S, the resonance condition was derived if there was a resonance condition. Then, efficiency, optimum load, and output power were also derived. Table 11.1 shows the equations for the secondary resonance conditions.

378

11 Basic Characteristics of Electric Field Resonance

The secondary resonance condition is a condition for maximizing the efficiency, and the primary resonance condition is a condition for maximizing the output power under the condition. Therefore, in order to realize the maximum efficiency and the large power, it is necessary to set the optimum load at the time of the secondary resonance and to perform the power transmission in a state where the primary resonance is satisfied. The primary resonance condition can be satisfied by adjusting the primary coil L 1 , and the secondary resonance condition can be satisfied by adjusting the secondary coil L 2 . The derived equations are summarized in the table. Table 11.2 shows the efficiency equation η(ω) at an arbitrary frequency. Table 11.3 shows the efficiency equation η(ω2 ) at the secondary resonance, and Table 11.4 shows the optimal load equation at the resonance. The process of deriving the values of the optimum load R L opt , the primary resonance condition L 1 , and the secondary resonance condition L 2 is performed as follows. First, by combining the Eqs. (11.31) and (11.40), the optimum load R L opt in efficiency and the secondary coil L 2 at that time are obtained. After that, the primary coil L 1 is obtained by substituting each value into the Eq. (11.32). Specific calculation examples are shown in Sect. 11.7.6. Table 11.2 Efficiency (at any frequency) η(ω) N–N

2 R L Cm   2 )2 +(r +R )C 2 r1 C12 +ω2 (r2 +R L )2 (C1 C2 −Cm 2 L m

S–N

2 R L Cm   2 )2 +(r +R )C 2 r1 C12 +ω2 (r2 +R L )2 (C1 C2 −Cm 2 L m

N–S

2 R L Cm   2 ){(C C −C 2 ) A−2L C } +(r +R )C 2 r1 C12 +ω2 (C1 C2 −Cm 1 2 2 1 2 L m m

S–S

2 R L Cm   2 ){(C C −C 2 ) A−2L C } +(r +R )C 2 r1 C12 +ω2 (C1 C2 −Cm 1 2 2 1 2 L m m

S–P

2 R L BCm   2 ){ B (C C −C 2 )−2C L }+C 2 A +(B+r R )R C 2 r1 ω2 R 2L (C1 C2 −Cm 1 2 1 2 2 L L m m 1

Table 11.3 Efficiency (at secondary-side resonance) η(ω2 ) N–S

2 R L L 2 C22 Cm     4 +L C 4 +(r +R )C 2 C 2 L r1 C2 (R L +r2 )2 C12 C22 −Cm 2 m 2 L m 2 2

S–S

2 R L L 2 C22 Cm     4 +L C 4 +(r +R )C 2 C 2 L r1 C2 (R L +r2 )2 C12 C22 −Cm 2 m 2 L m 2 2

S–P

2 R L L 22 C2 Cm     4 +(L +C r R )R C 2 C L r1 C12 C2 (C2 r2 R L +L 2 )2 + L 2 −C2 r22 R 2L Cm 2 2 2 L L m 2 2

11.7 Comparison of Electric Field Coupling and … Table 11.4 Optimum load at secondary-side resonance N–N S–N

379

R L opt  r22 +  r22 +

N–S

 r22 +

S–S

 r22 +

S–P

 L 2 C1

2 r1 C12 +r2 Cm

2) ω2 r1 (C1 C2 −Cm 2 r1 C12 +r2 Cm

2) ω2 r1 (C1 C2 −Cm

2

2

  2 r C 2 +r C 2 L 2 Cm  1 m 2 2 4 r1 C2 C12 C22 −Cm   2 r C 2 +r C 2 L 2 Cm  1 m 2 2 4 r1 C2 C12 C22 −Cm r1 C2     4 +C 2 C 2 r L r1 C12 C23 r22 + L 2 −C2 r22 Cm 2 m 2 2

11.7.4 Validation of Theoretical Formula and Analysis In this subsection, we confirm the validity of the derived efficiency and output power by confirming that the graph (Cal.) of the equation matches the graph (Sim.) analyzed using a circuit simulator. Use the efficiency and load relationships for each topology shown in Table 11.2. Figure 11.20 shows the relationship between efficiency and frequency, and Fig. 11.21 shows the relationship between output power and frequency. Table 11.5 shows the values of each element. As can be seen from Figs. 11.20 and 11.21, the theoretical equation and the graph of the analysis agree, indicating the usefulness of the theoretical equation.

11.7.5 Efficiency Comparison The secondary resonance frequency is related to the efficiency value, and the efficiency becomes the highest value when the secondary resonance condition is satisfied. In this subsection, the efficiency of the secondary-side resonance (Table 11.3) was compared, and the transmission characteristics in each topology were compared. Figure 11.22 shows graphs of N–N and S–N efficiencies, N–S and S–S efficiencies, and S–P efficiencies. The resonance frequency is 450 kHz. Except for the variables L 2 and R L , the values in Table 11.5 were used for the values of each element. From Fig. 11.22, it can be seen that the efficiency is maximum when the load is around 900  for N–N and S–N, around 80  for N–S and S–S, and around 10 k for S–P.

380

11 Basic Characteristics of Electric Field Resonance

(a) N-N

(b) S-N

(c) N-S

(d) S-S

(e) S-P Fig. 11.20 Efficiency

11.7 Comparison of Electric Field Coupling and …

381

(a) N-N

(b) S-N

(c) N-S

(d) S-S

(e) S-P Fig. 11.21 Power

11.7.6 Conditions for High Efficiency and High-Power Transfer This subsection describes the conditions for high-efficiency and high-power transmission in each topology. As described in Sect. 11.7.3 for N–S, S–S, and S–P, the optimum load in efficiency and the value of the inductance of the secondary coil are first determined. The resonance frequency is 450 kHz and the coupling coefficient k = 0.1. From the formula of the optimal load at resonance derived in Sect. 11.7.3, the optimal load of N–N and S–N is R Lopt = 874.8 , the optimal load of N-S and

382

11 Basic Characteristics of Electric Field Resonance

Table 11.5 The value of each element N–N

S–N

N–S

S–S

S–P

f (Hz)

450 k

450 k

450 k

450 k

450 k

C1 (F)

390 p

390 p

390 p

390 p

390 p

C2 (F)

410 p

410 p

410 p

410 p

410 p

Cm (F)

40 p

40 p

40 p

40 p

40 p

L 1 (H)

–

320 μ

–

320 μ

320 μ

L 2 (H)

–

–

300 μ

300 μ

300 μ

r1 ()

2.67

2.67

2.67

2.67

2.67

r2 ()

2

2

2

2

2

V1 (V)

100

100

100

100

100

RL

100

100

100

100

100 k

k

0.1

0.1

0.1

(a) N-N and S-N

0.1

(b) N-S and S-S

(c) S-P Fig. 11.22 Efficiency (secondary-side resonance)

0.1

11.7 Comparison of Electric Field Coupling and …

383

S-S is R Lopt = 76.8 , and the optimal load of S–P is R Lopt = 9653.4 . This can be seen from the graph of Fig. 11.22. The value of the secondary coil at the optimal load is L 2 = 302.5 μH for N–S and S–S, and L 2 = 305.1 μH for S–P from the secondary resonance condition formula (Table 11.1). Next, the resonance condition L 1 is obtained. From the equation of the primaryside resonance condition (Table 11.1), L 1 = 322.4 μH for S–N, L 1 = 317.0 μH for S–S, and L 1 = 320.7 μH for S–P. Based on the above results, graphs of efficiency and load power consumption P2 when power is transmitted with the optimal load are shown in Figs. 11.23 and 11.24. The power supply voltage V1 is set to 100 V. From Fig. 11.23, under the optimum load, N–S, S–S, and S–P can transmit with nearly 95% efficiency. From Fig. 11.24, it is possible to transmit power of 92 W in S–S and about 95 W in S–P. From this, it can be seen that S–S and S–P can transmit power of about 100 W with high efficiency, and S–S and S–P are superior to other topologies. S–N can transmit power of about 860 W, but the maximum efficiency is as low as about 63%, so S–S and S–P are considered to be better. As described above, a total of five circuit topologies of the non-resonant circuit (N–N), the primary-side resonant circuit (S–N), the secondary-side resonant circuit (N–S), the electric resonance coupling circuit (S–S), and the electric resonance coupling circuit (S–P) was examined. The efficiency of each topology was compared by deriving the resonance condition, efficiency, optimal load, and output power in the five topologies. As a result, S–S and S–P are superior to other topologies because they can transmit both efficiency and power at high values.

Fig. 11.23 Efficiency of five different circuits

Fig. 11.24 Consumed power of five different circuits

384

11 Basic Characteristics of Electric Field Resonance

References 1. T. Imura, T. Uchida, Y. Hori, A proposal of meander line antenna for contactless power transfer. in IEICE Society Conference (2008) 2. T. Imura, H. Okabe, T. Uchida, Y. Hori, Wireless power transfer during displacement using electromagnetic coupling in resonance: magnetic-versus Electric-type antennas. Trans. Inst. Electr. Eng. Jpn D, A Publ. Ind. Appl. Soc. 130(1), 76–83 (2010) 3. T. Imura, H. Okabe, T. Uchida, Y. Hori, Study of magnetic and electric coupling for contactless power transfer using equivalent circuits: wireless power transfer via electromagnetic coupling at resonance. Trans. Inst. Electr. Eng. Japan D 130(1), 84–92 (2010) 4. S. Kuroda, T. Imura, Derivation and comparison of efficiency and power in non-resonant and resonant circuit of capacitive power transfer. Asian Wireless Power Transf. (2019)

Chapter 12

Unified Theory of Magnetic Field Coupling and Electric Field Coupling

In this chapter, magnetic field coupling and electric field coupling are compared under unified conditions. Comparing the circuit topology under the same conditions, it shows that the efficiency and the power equation are the same, and shows that magnetic field coupling (IPT: Inductive power transfer) and electric field coupling (CPT: Capacitive power transfer) can be interpreted in a unified manner.

12.1 Unified Interpretation of Magnetic Field Coupling and Electric Field Coupling and Preparation for Comparison In this section, the structure of magnetic field coupling and electric field coupling, and the unified analysis method applied to the resonance coupling method are mentioned [1]. For comparison under unified conditions, the resonance conditions are described again. For the convenience of description, WPT by magnetic field coupling is represented by IPT, and WPT by electric field coupling is represented by CPT. First, the power transmission and reception coupler (coil/plate) structures of both power transmission methods, their equivalent circuits, and the power transmission distance characteristics are shown. Next, the compensation topology of the resonance coupling method common to both power transmission methods is described. The following conditions are adopted as unified design conditions. We adopt optimal load conditions for maximum efficiency transmission. A reactance compensating element design method that can obtain CUPE (Constant Unity Power Factor) characteristics and CC (Constant Current) characteristics or CV (Constant Voltage) characteristics is adopted. These are based on gyrator characteristics and ideal transformer characteristics.

© Springer Nature Singapore Pte Ltd. 2020 T. Imura, Wireless Power Transfer, https://doi.org/10.1007/978-981-15-4580-1_12

385

386

12 Unified Theory of Magnetic Field Coupling and Electric Field …

12.1.1 Magnetic Field Coupling (IPT) The magnetic field coupling will be described. As presented in Sect. 12.2 circuit analysis, k and Q are important design parameters that contribute to efficiency and output power. k=√ Q1 =

ωL 1 , R1

Lm L1 L2 Q2 =

(12.1) ωL 2 R2

(12.2)

Here, the coils used in the study are described (Figs. 12.1 and 12.2). We use a symmetrical power transmission and reception coil with an inner radius r1 = 124 mm, an outer radius r2 = 310 mm, and a magnetic flux linkage cross section S = 0.302 mm. Distance characteristics were measured with an impedance analyzer Fig. 12.1 IPT coupler

Fig. 12.2 Experimental equipment of IPT

12.1 Unified Interpretation of Magnetic Field Coupling …

387

Fig. 12.3 Equivalent circuit diagram of IPT coupler

(a) Circuitstructure

(b) T-type circuit

(KeySight E4990A). The winding is a KIV wire with a diameter of 1.63 mm (2 Sq) and has an air core structure of 54 turns. The following relational expression based on the equivalent circuit structure in Fig. 12.3b was used to measure the coil parameters. Note that L i,op and Ri,op are the power transmission side (power receiving side) inductance and ESR when the power receiving side (power transmitting side) coupler end is open. L i,sh is a power transmission side (power reception side) inductance when the power reception side (power transmission side) coupler end is in a shortcircuit state. The average value of the values calculated on the power transmission and reception sides is adopted as the coupling coefficient.

ki =

L i = L i,op (i = 1, 2)

(12.3)

Ri = Ri,op (i = 1, 2)

(12.4)

 1 + L i,sh /L i,op (i = 1, 2) k = (k1 + k2 )/2

(12.5) (12.6)

In addition, the mutual inductance L m is calculated by substituting the parameters of the above equation into (1). As the distance between the transmitting and receiving coils decreases, the coil parameters change due to the interaction. Therefore, all coil parameters are obtained for each transmission distance. Figure 12.4 shows

388

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Fig. 12.4 Transmission distance characteristics for coils

the measurement results at 85 kHz for the change characteristics of the values of the self-inductance, mutual inductance, and coupling coefficient with respect to the transmission distance, and the change characteristics of the ESR. As the transmission distance increases, all coil parameter values decrease due to the decrease in the interaction between the transmission and reception coils. When the power transmission and reception coils were measured individually, the coil parameters were measured as L 1 = 1.43 mH, L 2 = 1.46 mH, R1 = 2.33 Ω, and R2 = 2.42 Ω. Considering the ESR constant regardless of the transmission distance may cause a difference with the assumed transmission characteristics. Fortunately, since the self-inductance hardly depends on the transmission distance, it is thought that resonance deviation due to the self-inductance change is unlikely to occur except when the distance is very close.

12.1.2 Electric Field Coupling (CPT) The electric field coupling will be described [2–8]. A typical electric-coupled power transmission/reception coupler is such that the metal plates Plate 1–Plate 4 are arranged horizontally as shown in Fig. 12.5 (horizontal coupler structure) or vertically as shown in Fig. 12.6 (vertical coupler structure). Plate 1 and Plate 2 are located on the power transmitting side with power supply, and Plate 3 and Plate 4 are located on the power receiving side with load. The circuit diagram of this structure is represented by capacitances C12 to C34 between each metal plate as shown in Fig. 12.7a. The relational expression between the self-capacitances C1 and C2 and the mutual capacitance Cm is represented by Expressions (12.7)–(12.9). Considering electric resonance coupling discussed in this section, the internal resistance of the compensation inductor used is sufficiently larger

12.1 Unified Interpretation of Magnetic Field Coupling …

389

Fig. 12.5 CPT coupler with horizontal structure

Fig. 12.6 CPT coupler with vertical structure

Fig. 12.7 Equivalent circuit diagram of CPT coupler

(a) Structure circuit

(b)

-type circuit

390

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Fig. 12.8 Experimental equipment in CPT

than the internal resistance of the metal plate. Therefore, the analysis is performed ignoring the internal resistance of the metal plate. The circuit diagram using these capacitors is equivalently converted to a π-type circuit as shown in Fig. 12.7b. C1 = C12 +

(C13 + C14 )(C23 + C24 ) C13 + C14 + C23 + C24

(12.7)

C2 = C34 +

(C13 + C14 )(C23 + C24 ) C13 + C14 + C23 + C24

(12.8)

Cm =

C24 C13 − C14 C23 C13 + C14 + C23 + C24

(12.9)

In addition, a coupling coefficient k representing the strength of coupling between the power transmission and reception sides is defined as the following equation. The coupling coefficient k is a design parameter that contributes to efficiency and output power. Cm k=√ C1 C2

(12.10)

Here, the plate used in the study is explained. The distance characteristics of the plate parameters were measured with an impedance analyzer (KeySight E4990A). Experimental setup is shown in Fig. 12.8. Figure 12.5 shows a horizontal arrangement of copper plates with short side l1 = 250 mm, long side l2 = 600 mm, plate thickness t = 0.4 mm, area S = 0.150 m2 (Table 12.1), and Fig. 12.6 Plate 1 and 3: short side lout1 = 500 mm, long side lout2 = 600 mm, plate thickness t = 0.8 mm, area Table 12.1 Plate parameters in Fig. 12.8 of CPT (horizontal arrangement)   l1 (mm) l2 (mm) S m2 t(mm) w(mm) 250

600

0.150

0.400

70.0

g(mm) 7.00

12.1 Unified Interpretation of Magnetic Field Coupling …

391

S = 0.300 m2 , Plate 2 and Plate 4: short side lin1 = 400 mm, long side lin2 = 480 mm, plate thickness t = 0.8 The copper plate with mm and area S = 0.192 m2 is arranged vertically. In this structure, the total area occupied by the metal plates on the power transmission and reception sides on the same plane is unified to S = 0.300 m2 , and the cross-sectional area of the interlinking magnetic flux of the transmission and reception coils in Sect. 12.1.1 is also unified. No special dielectric is inserted, and the transmitting and receiving couplers are installed in the air. The following relational expression based on the equivalent circuit structure in Fig. 12.7b is used to measure the plate parameters. Ci,op is a power transmission side (power reception side) capacitance when the power reception side (power transmission side) coupler end is in an open state. Ci,sh is a power transmission side (power reception side) capacitance when the power reception side (power transmission side) coupler end is in a short-circuit state. The average value of the values calculated on the power transmission and reception sides is adopted as the coupling coefficient. Ci = Ci,sh (i = 1, 2) ki =



1 + Ci,op /Ci,sh

k = (k1 + k2 )/2

(12.11) (12.12) (12.13)

In addition, the mutual capacitance Cm is calculated by substituting the parameters of the above equations into Eq. (12.10). As with the magnetic field coupling, all plate parameters are obtained for each transmission distance. Figure 12.9 shows the measurement results at 400 kHz of the change characteristics of the self-capacitance, mutual capacitance, and coupling coefficient with respect to the transmission distance. As the transmission distance increases, all plate parameter values decrease. Furthermore, there is a remarkable difference that a relatively large coupling coefficient and a small self-capacitance can be obtained with the horizontal coupler structure, while a relatively small coupling coefficient and a large self-capacitance can be obtained with the vertical coupler structure.

12.1.3 Unified Design Method for Resonance Conditions In this section, power transmission characteristics are evaluated fairly by applying one unified resonance design method to magnetic resonance coupling and electric resonance coupling. In the magnetic resonance coupling and the electric resonance

392

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Fig. 12.9 Transmission distance characteristics for plates

(a) Horizontal structure

(b) Vertical structure

coupling, there are various methods such as (i) a design method based on series–parallel LC resonance, (ii) a design method based on input/output power factor compensation, (iii) a design based on gyrator characteristics and ideal transformer characteristics. In this section, “(iii) Design method based on gyrator characteristics and ideal transformer characteristics” will be adopted.

12.1 Unified Interpretation of Magnetic Field Coupling …

12.1.3.1

393

Design Method Based on Series–Parallel LC Resonance (i)

The design method based on the series–parallel LC resonance is a method to match the resonance frequency ω1 of the power transmitting side resonator, the resonance frequency ω2 of the power receiving side resonator, and the operating frequency ω of the power supply as follows. ω1 = √

1 1 , ω2 = √ L 1 C1 L 2 C2

(12.14)

This conditional expression is frequently adopted in the SS topology of the magnetic resonance coupling. It achieves CUPF (Constant Unity Power Factor) characteristics and CC (Constant Current) characteristics in addition to high efficiency and high power transmission. However, this conditional expression deviates from the resonance condition of the circuit in the compensation topology except the SS topology of the magnetic resonance coupling and the PP topology of the electric resonance coupling.

12.1.3.2

Design Method Based on Input Power Factor Compensation (ii)

The design method based on input/output power factor compensation will be described using an equivalent circuit of the resonance coupling method shown in Fig. 12.10. This design method is a compensating element design method to cancel out the reactance between the input impedance on the transmitting side and the input impedance on the receiving side as shown in Eq. (12.15). Unlike the series–parallel LC resonance design that considers only the resonance of the resonator, this design is a resonance design considering the entire equivalent circuit and secondary-side resonance. ω1 → Im(Z in1 ) = 0, ω2 → Im(Z in2 ) = 0

(12.15)

Only when the optimal load condition is satisfied by using this design and the optimal load design together, an input/output power factor of 1 can be achieved in all the compensation topologies of the magnetic resonance coupling and the electric resonance coupling. However, the compensating element conditions according to this design are complicated conditions including the parameters of L 1 , L 2 , C1 , C2 , Q 1 , Q 2 , ω, k, and R L .

394

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Fig. 12.10 Circuit diagram for power factor compensation design

(a) IPT

(b) CPT

12.1.3.3

Design Method Based on Gyrator Characteristics and Ideal Transformer Characteristics (iii)

The design method based on gyrator characteristics and ideal transformer is a design method that can achieve CC characteristics or CV characteristics in addition to CUPF characteristics by appropriate resonance conditions. By achieving the CC and CV characteristics, it becomes possible to supply power to the motor whose equivalent load resistance fluctuates, and to supply power stably even during the charging process of the secondary battery. In addition, it has been reported that in the magnetic resonance coupling, each compensation topology can be understood graphically by treating the transmission circuit as a combination of a network called a gyrator. In the following, the transmission characteristics obtained by the gyrator and its combination are mentioned. In a two-port circuit of an electric circuit, the network is represented as a circuit equation with the following F parameters. 

V˙1 I˙1





A˙ B˙ = ˙ ˙ C D



V˙2 I˙2

 (12.16)

12.1 Unified Interpretation of Magnetic Field Coupling …

395

Fig. 12.11 T-type and -type gyrators

Gyrator characteristics A network in which A˙ = D˙ = 0 in the circuit equation shown in Eq. (12.17) is called a gyrator. 

V˙1 I˙1





0 ±jZ = ± Zj 0



V˙2 I˙2

 (12.17)

Equation (12.18) corresponds to Fig. 12.11a, c, and Eq. (12.19) corresponds to Fig. 12.11b, d. 

  0 jZ V˙2 = = 1 − jZ 0 I˙2          0 −jZ V˙1 V˙2 0 −jZ V˙2 = = 1 0 I˙1 I˙2 I˙2 − Zj 0 jZ 

V˙1 I˙1





0 jZ j 0 Z



V˙2 I˙2



(12.18)

(12.19)

Figure 12.12a shows the Eq. (12.19) in Fig. 12.11b where Z = ωL m . Figure 12.12b shows the Eq. (12.18) in Fig. 12.11c with Z = 1/ωCm . In particular, the network in Fig. 12.11a is called the K inverter, and the network in Fig. 12.11b is also called the J inverter. When the network satisfies A˙ = D˙ = 0, the input impedance Z in is expressed by Eq. (12.20). It can be seen that the value is a real number when there is no ESR in the circuit. In other words, the input voltage and the input current have the same phase regardless of the load value, and the input power factor cos θin = 1 is achieved. Here, R L is the resistance connected to the load side.

396

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Fig. 12.12 Example circuit diagram of gyrator

Z in =

V˙1 Z 2 I˙2 Z2 = = RL I˙1 V˙2

(cos θin = 1)

(12.20)

The output voltage V˙2 at this time can be expressed as follows. V˙2 = ±R L /Z V˙1

(12.21)

The output current I˙2 at this time can be expressed as follows. I˙2 = ±1/Z V˙1

(12.22)

Equations (12.21) and (12.22) show that the output current does not depend on the load for an output voltage whose value is determined in proportion to the load value. The CUPF characteristics and CC characteristics obtained when the network satisfies A˙ = D˙ = 0 are collectively called gyrator characteristics. Although not dealt with in this document, CUPF characteristics and CV characteristics can be obtained when a constant current source is used as the power supply. Ideal transformer characteristics By cascading an even number of gyrators, the network in Eq. (12.17) satisfies B˙ = C˙ = 0.      ± j Z 0 V˙2 V˙1 = (12.23) j I˙1 I˙2 0 ± Z If the network satisfies B˙ = C˙ = 0, the input impedance Z in is expressed by (12.24). If there is no ESR in the circuit, the value is a real number. In other words, the input voltage and the input current have the same phase regardless of the load value, and the input power factor cos θin = 1 is achieved. Z in =

V˙1 Z 2 V˙2 = = Z 2 R L (cos θin = 1) I˙1 I˙2

The output voltage V˙2 at this time can be expressed as follows.

(12.24)

12.1 Unified Interpretation of Magnetic Field Coupling …

1 V˙2 = ±  Z V˙1

397

(12.25)

The output current I˙2 at this time can be expressed as follows. I˙2 = ±1/Z  R L V˙1

(12.26)

Equations (12.25) and (12.26) show that the output voltage does not depend on the load for the output current whose value is determined in inverse proportion to the load value. Focusing on the characteristic that the output voltage depends only on the input voltage and the transmission circuit, the CUPF characteristic and the CV characteristic obtained when the circuit network satisfies B˙ = C˙ = 0 are called the ideal transformer characteristic. Although not dealt with in this document, CUPF characteristics and CC characteristics can be obtained when a constant current source is adopted as the power supply. From the above, in the evaluation comparison in this chapter, the advantages of the CUPF characteristics, CC and CV characteristics are expected, so “(iii) Design method of resonance conditions based on gyrator characteristics and ideal transformer characteristics” is adopted. In the circuit analysis at the beginning of Sects. 12.2 and 12.3, the resonance condition that satisfies either A˙ = D˙ = 0 or B˙ = C˙ = 0 under the condition of R1 = R2 = 0 is derived for each compensation topology based on this design method. The following parameters are used to simplify the analysis. ⎧ ⎨

a1 = 1 − ω2 L 1 C1 , a2 = 1 − ω2 L 2 C2   a3 = 1 − ω2 L 1 C1 1 − k 2 , a4 = 1 − ω2 L 2 C2 1 − k 2 ⎩ a5 = 1 + ω4 L 1 L 2 C1 C2 1 − k 2 − ω2 (L 1 C1 + L 2 C2 )

(12.27)

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP) In this section, the circuit analysis of magnetic resonance coupled wireless power transmission is performed, and the transmission characteristics of each compensation topology are evaluated based on various formulas and various characteristic calculations. First, we derive the compensation capacitor condition of each compensation topology to obtain the gyrator characteristic or the ideal transformer characteristic under the condition of R1 = R2 = 0. Next, we derive exact and approximated formulas for each transmission characteristic when the compensation capacitor condition is applied under the condition of R1 = R2 = 0, and evaluate the transmission characteristics of each compensation topology.

398

12 Unified Theory of Magnetic Field Coupling and Electric Field …

12.2.1 Derivation of Compensation Capacitor Condition (R1 = R2 = 0) In this section, under the condition of R1 = R2 = 0, we derive the compensation capacitor condition of the two-element compensation type topology by following the derivation process in the SS topology. In a two-port pair circuit with SS topology, the F parameter A˙ to D˙ of the circuit equation can be expressed as follows. A˙ = −

a1 ω2 L

m C1

,

B˙ =

ω3 L

ja5 1 , C˙ = , jωL m m C1 C2

D˙ = −a2 /ω2 L m C2 (12.28)

In the SS topology, it is possible to satisfy A˙ = D˙ = 0 by appropriate design of the compensation capacitor. At that time, C1 and C2 are as follows. C1 =

1 ω2 L

, C2 = 1

1 ω2 L

(12.29) 2

When the compensation capacitor condition is satisfied, the F parameter of the network is rewritten as follows.      ˙1 1 0 − jωL V˙2 V m A˙ = 0, B˙ = − jωL m , C˙ = , D˙ = 0 ˙ = 1 0 I1 I˙2 jωL m jωL m (12.30) Equation (12.31) shows the impedance and the input power factor at this input time. Also, the output voltage V˙2 and the output current I˙2 when using the constant voltage source are shown in Expressions (12.32) and (12.33). Z in =

ω2 L 2m , ∴ cos θin = 1 RL

(12.31)

V˙2 =

RL ωL m V˙1

(12.32)

I˙2 =

1 ωL m V˙1

(12.33)

It can be seen that cos θin = 1 and constant I˙2 can be obtained without depending on R L . Therefore, it was shown that the SS topology can obtain the gyrator characteristic by satisfying the compensation capacitor condition. Other compensation topologies are analyzed in the same way, and the results are shown in Tables 12.2, 12.3 and 12.4. If different compensation capacitor conditions are satisfied for each compensation topology, it is possible to obtain gyrator characteristics satisfying A˙ = D˙ = 0

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)

399

Table 12.2 F parameters in IPT (R1 = R2 = 0) SS topology

SP topology

PS topology

PP topology

A˙ B˙

−a1 /ω2 L m C1

−a5 /ω2 L m C1

L 1 /L m

ja5 /ω3 L m C1 C2

a3 L 2 /jωL m C1

a4 L 1 /jωL m C2

a4 L 1 /L m   jωL 1 L 2 1 − k 2 /L m

C˙ D˙

1/jωL m

a2 /jωL m

a1 /jωL m

a5 /jωL m

−a2 /ω2 L m C2

L 2 /L m

−a5 /ω2 L m C2

a3 L 2 /L m

Table 12.3 Compensation capacitor in IPT SS topology

SP topology

PS topology

PP topology

C1

1 ω2 L 1

1 ω2 L 1 (1−k 2 )

1 ω2 L 1

1 ω2 L 1 (1−k 2 )

C2

1 ω2 L 2

1 ω2 L 2

1 ω2 L 2 (1−k 2 )

1 ω2 L 2 (1−k 2 )

Table 12.4 Gyrator and transformer characteristics in IPT (R1 = R2 = 0) SS topology

SP topology

PS topology

PP topology 0

A˙ B˙ C˙

0

L m /L 2

L 1 /L m

− jωL m

0

0

1/jωL m

0

0

  jωL 1 L 2 1 − k 2 /L m   −k 2 /jωL m 1 − k 2

D˙

0

L 2 /L m

L m /L 1

0

Z in V˙2 I˙2

ω2 L 2m /R L R L /ωL m V˙1 1/ωL m V˙1

L 2m R L /L 22 L 2 /L m V˙1 L 2 /L m R L V˙1

L 21 R L /L 2m L m /L 1 V˙1 L m /L 1 R L V˙1

Characteristics

Gyrator

Ideal transformer

Ideal transformer

 2 ω2 L 2m 1 − k 2 /k 4 R L   k 2 R L /ωL m 1 − k 2 V˙1   k 2 /ωL m 1 − k 2 V˙1 Gyrator

for SS and PP topologies, and ideal transformer characteristics satisfying B˙ = C˙ = 0 for SP and PS topologies. SP, PS, and PP are given by Eqs. (12.34), (12.35), and (12.36), respectively.  

V˙1 I˙1



 = 



V˙2 I˙2



  0 L 1 /L m V˙2 I˙2 0 L m /L 1        V˙1 0 jωL 1 L 2 1 − k 2 /L m V˙2  = 2 2 ˙I1 0 −k /jωL m 1 − k I˙2 V˙1 I˙1



0 L m /L 2 0 L 2 /L m

=

(12.34) (12.35) (12.36)

400

12 Unified Theory of Magnetic Field Coupling and Electric Field …

From the next subsection, we evaluate the power transmission characteristics of each compensation topology when these compensation capacitor conditions are applied, taking into account the resistance components of the circuit.

12.2.2 Circuit Analysis and Characterization (R1  = R2  = 0) In this subsection, we derive the transmission characteristics formulas when the compensation capacitor condition is applied under the condition of R1 = R2 = 0. Then, by simplifying it as an approximate expression, we evaluate the transmission characteristics of each compensation topology based on clear formulas and compare them fairly. The transmission characteristics to be evaluated are of six types: transmission efficiency, optimal load, output power, CUPF characteristics, CC characteristics, and CV characteristics. In the circuit analysis of CC characteristics and CV characteristics, the analysis process is divided into the derivation of the output current formula of the SS topology for the CC characteristics and the derivation of the output voltage formula of the SP topology for the CV characteristics. For simplification of the formulas, the parameters of Eq. (12.37) are used in addition to the coupling coefficient k of Eq. (12.1) and the Q value of the transmitting and receiving coil in Eq. (12.2). Note that these parameters are different from those used for circuit analysis of electric field coupling. ⎧      2 Q 1d = Q 1 + k 2 Q 2 , Q 2d = k 2 Q 1 + Q 2 , R2d = R2 1 + k 2 , R Ld = R L 1 + k 2 / 1 − k 2 ⎪ ⎪  2  ⎪ ⎪ 2 2 ⎪ ⎨ A1 = R2 Q 2 − k R L Q 1 , A2 = R2 k Q 1 + Q 2 + k R L Q 1 A3 = R2 Q 2 1 − k 2 − k 2 R L (Q 1 + Q 2 ) ⎪   ⎪ 2 2 ⎪ B ⎪ ⎪ 1 = R L + R2 Q 2 , B2 = R L + R2 Q 2 1 − k (Q 1 + Q 2 ) ⎩ C = (1 − k 2 ){R L + R2 Q 22 + (R2 + R L )(1 + k 2 Q 1 Q 2 )}, D = Q 1 Q 2 (1 − k 2 ) − 1

(12.37)

The characteristics of each of the four groups (a) efficiency and optimal load, (b) output power, (c) input impedance and input power factor, and (d) output current and output voltage are described below.

12.2.2.1

Efficiency and Output Power

Tables 12.5 and 12.6 show the exact and approximate solutions for efficiency and optimal load. The equations for efficiency and maximum efficiency are also given. Since the coil Q value often has a value on the order of several hundreds, approximation of Eq. (12.38) is effective. Q = Q1 = Q2, 1  Q2

(12.38)

PP topology

PS topology

2  k 2 R2 R L Q 1 Q 3 1−k 2  2  2 2 C R2 Q 2 2 +R L 1+k Q 1d Q 2 

k2 R R Q4    2 L R2d Q 2 +R L 1+k 2 Q 2

R Ld +R2 Q 2

k2 R R Q4    2 L R2d Q 2 +R L 1+k 2 Q 2

k 2 R2 R L Q 2   R2 +R L +k 2 R2 Q 2

R L +R2 Q 2

R2d +R L



k 2 R2 R L Q 1 Q 2    R2 +R L 2 +k 2 R2 Q 2 R2 Q 1d +R L Q 1

SP topology





k 2 R2 R L Q 1 Q 3 2   A1 R2 Q 2 +B1 A2 Q 2 +R L

SS topology

k 2 R2 R L Q 2    R2 +R L R2 +R L +k 2 R2 Q 2

  η Q = Q1 = Q2 , 1  Q2

k 2 R2 R L Q 1 Q 2    R2 +R L R2 +R L +k 2 R2 Q 1 Q 2

η(Q 1  = Q 2 )

Table 12.5 Formulas of transmission efficiency in IPT



k2 Q2 2  1+k 2 + 1+k 2 Q 2

k2 Q2 2  1+k 2 + 1+k 2 Q 2

k2 Q2 2  1+k 2 + 1+k 2 Q 2 1−k 2





k2 Q2 2   1+ 1+k 2 Q 2

ηmax

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP) 401

402

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.6 Formulas of optimal load in IPT R Lopt (Q 1 = Q 2 )  R2 1 + k 2 Q 1 Q 2

  R Lopt Q = Q 1 = Q 2 , 1  Q 2  R2 1 + k 2 Q 2

PS topology

 R2 1 + k 2 Q 1d Q 2

R2

PP topology

 √ R2 Q 2 1−k 2 1+Q 2 Q 2d

SS topology SP topology

√ R2 Q 2 1+Q 2 Q 2d √ 1+k 2 Q 1 Q 2

√

1+k 2 Q

√ 2 1+k 2 R√ 2Q 1+k 2 Q 2

   1 + k2 1 + k2 Q2

  R2 Q 2 1−k 2

√

1+k 2 Q 2

1d Q 2

In addition, in order to unify the equations of SS, SP, PS, and PP, the number of parameters is increased as well as deleted within the effective range of approximation. For example, in the optimal load equation of the PS topology, the approxi    mation of R Lopt = R2 1 + k 2 Q 1d Q 2  R2 1 + k 2 Q 2 1 + k 2 is approximated as    R2 1 + k 2 1 + k 2 Q 2 . When k < 0.3, the approximation of 1 + k 2 ≈ 1 − k 2 ≈ 1 is possible when the coupling coefficient is small. Under such weak coupling conditions, the maximum efficiencies of SS, SP, PS, and PP can be approximated by Eq. (12.39) and have the same value. all = ηmax

k2 Q2 2  1 + 1 + k2 Q2

(12.39)

Similarly, under such weak coupling conditions, the optimum load is divided into two, and SS and PS can be approximated by Eq. (12.40), and SP and PP can be approximated by Eq. (12.41).  SS,PS R Lopt = R2 1 + k 2 Q 2 SP,PP R Lopt =

R2 Q 2 1 + k2 Q2

(12.40) (12.41)

The relationship between Eqs. (12.40) and (12.41) is Eq. (12.42). Considering k < 1, it can be seen that SP and PP have larger optimum load values than SS and PS. SS,PS SP,PP : R Lopt = 1 + k2 Q2 : Q2  k2 : 1 R Lopt

(12.42)

Table 12.7 shows the exact and approximate solutions for the output power. The output power at maximum efficiency is also shown. When the approximation of 1 + k 2 ≈ 1 − k 2 ≈ 1 is performed under the weak coupling condition, the output power at the maximum efficiency is divided into two, SS and SP are expressed by Eq. (12.43), and PS and PP can be approximated by Eq. (12.44).

PP topology

PS topology

SP topology

SS topology

2 2 V˙1

2 k 2 R2 R L Q 1 Q 2     V˙1 R2 +R L 2 + R2 Q 1d +R L Q 1 2



  2 2 k 2 R2 R L Q 1 Q 3 2 2 1−k   2  V˙1 R1 B22 + R L Q 1d +D R2 Q 2 1−k 2

R1



2 k 2 R2 R L Q 1 Q 3 2   2  V˙1 R1 A 2 1 + A2 Q 2 +R L

R1 R2 +R L +k 2 R2 Q 1 Q 2



k 2 R2 R L Q 1 Q 2

Pout (Q 1  = Q 2 )

Table 12.7 Formulas of output power in IPT 2 2 V˙1

2 k 2 R2 R L Q 2 2 V˙1 2   R Ld +R2 Q 2 R1 1−k 2

2 k 2 R2 R L   V˙1 R1 R2d +R L 2



2 k 2 R2 R L Q 4  2 V˙1 R1 R2d Q 2 +R L 1+k 2 Q 2

R1 R2 +R L +k 2 R2 Q 2



k 2 R2 R L Q 2

  Pout Q = Q 1 = Q 2 , 1  Q 2







2 ηmax   V˙1 1+k 2 Q 2 1+k 2

 ηmax 1+k 2 Q 2 2 V˙1  R1 Q 2 1+k 2  ηmax 1+k 2 Q 2 2   V˙1 R1 Q 2 1−k 2

R1

V˙1 2 ηmax R1 1+k 2 Q 2

Pout,ηmax

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP) 403

404

12 Unified Theory of Magnetic Field Coupling and Electric Field …

2 ηmax V˙1  R1 1 + k 2 Q 2  ηmax 1 + k 2 Q 2 ˙ 2 V1 = R1 Q 2

SS,SP Pout,η = max

PS,PP Pout,η max

(12.43) (12.44)

The relationship between Eqs. (12.43) and (12.44) is Eq. (12.45). Considering k < 1, it can be seen that the output power of SS and SP is larger than that of PS and PP. SS,SP PS,PP : Pout,η = Q2 : 1 + k2 Q2  1 : k2 Pout,η max max

12.2.2.2

(12.45)

Input Impedance and Input Power Factor

Based on the circuit analysis, the input impedance Z in when the SS topology resonance condition was applied was derived as follows.   R1 R2 + R L + k 2 R2 Q 1 Q 2 Z in = R2 + R L

(12.46)

Here, the input impedance formula of the SS topology is expressed only by the real component, and it is clear that the CUPF characteristic independent of the load is achieved, but the input impedance formula of other compensation topologies is expressed by complex numbers. It is necessary to derive the input power factor equation for exact CUPF characteristic evaluation, but it is derived as a complicated equation. Therefore, the change of the input power factor at the time of load change is presented as a graph, and the result is used to judge whether the CUPF characteristic can be achieved or not, and to apply it to the simplification of the input impedance formula. The input power factor was calculated as shown in Fig. 12.13 in the load range of R L = 1 − 100 M, with the design values of the inductance and resonance conditions uniquely determined under the condition of Q 1 = Q 2 = 300, R1 = R2 = 1 , f = 85 kHz. Here, two types of coupling coefficients, k = 0.1 and 0.3, are assumed. Figure 12.13 shows the input power factor fluctuation slightly with the load fluctuation, but hardly fluctuates. From the graph, the input power factor is almost 1 regardless of the magnitude of the coupling coefficient, and it can be said that all the compensation topologies can achieve the CUPF characteristics. Therefore, the input impedance formula in all compensation topologies is approximated only by real components. In addition, the approximation of the Q value in (12.38) simplifies the input impedance equation as follows. Z in 

  R1 R2 + R L + k 2 R2 Q 2 R2 + R L

(12.47)

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)

(a)

= 0.1

(b)

= 0.3

Fig. 12.13 Calculation results of load characteristics for input power factor in IPT

405

406

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.8 Formulas of input impedance in IPT Z in (Q 1  = Q 2 ) 

SS topology SP topology PS topology PP topology

+k 2 R

R1 R2 +R L 2 Q1 Q2 R2 +R L

  Z in Q = Q 1 = Q 2 , 1  Q 2



  R1 R2 +R L +k 2 R2 Q 2 R2 +R L

R1 {A1 + j(A2 Q 2 +R L )} R2 Q 2 + j B1

   R1 R2d Q 2 +R L 1+k 2 Q 2 R L +R2 Q 2

R1 Q 1 {R2 +R L + j(R2 Q 1d +R L Q 1 )} −k 2 R2 Q 2 + j ( R2 +R L +k 2 R2 Q 1 Q 2 )

R1 Q 2 (R2d +R L ) R2 +R L +k 2 R2 Q 2

     R1 Q 1 1−k 2 R L Q 1d +D R2 Q 2 1−k 2 − j B2 R L +A2 Q 2 (1−k 2 )− j A3

 2   R1 Q 2 1−k 2 R Ld +R2 Q 2 R2d Q 2 +R L (1+k 2 Q 2 )

Z in,ηmax  R1 1 + k 2 Q 2 R1

   1 + k2 1 + k2 Q2

√ 2 1+k 2 R√ 1Q 1+k 2 Q 2   R1 Q 2 1−k 2

√

1+k 2 Q 2

From Eqs. (12.40) and (12.47), the input impedance Z in,ηmax when both the resonance condition and the optimal load condition are satisfied can be obtained as follows.  (12.48) Z in,ηmax  R1 1 + k 2 Q 2 Input impedance equations for other compensation topologies are analyzed in the same way, and the results are shown in Table 12.8. As described above, even when the resonance conditions are applied to the SP, PS, and PP topologies, the input impedance formula is a complex number, and perfect CUPF characteristics cannot be obtained. However, by using high Q transmitting and receiving coil, it is possible to achieve almost CUPF characteristics. In addition, Z in,ηmax in each compensation topology is included in Pout,ηmax in Table 12.7, and it can be seen that it is an important parameter for increasing output power.

12.2.2.3

Output Current and Output Voltage

Output current (CC characteristics)

Based on the circuit analysis, the output current I˙2 when the compensating inductor condition of SS topology was applied was derived as follows. I˙2 = √

√ k R2 Q 1 Q 2   V˙1 2 R1 R L + R2 + k R2 Q 1 Q 2

(12.49)

Unlike the circuit analysis that disregards the internal resistance, I˙2 is expressed as a function of R L , so the complete CC characteristics cannot be obtained. However, in the load range close to R L = 0, the CC characteristics are obtained along with the maximum current I˙2 max . In this section, the load range where the damping ratio to the maximum current is 1/2 or less is defined as the CC load range. I˙2 = √

√ k Q1 Q2   V˙1 R1 R2 1 + k 2 Q 1 Q 2

(12.50)

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)

407

  CC range : R L  R2 1 + k 2 Q 1 Q 2

(12.51)

If the design value R Lopt satisfies the above load range, the SS topology obtains CC characteristics within that range. The output current equation in the PP topology where CC characteristics were expected in Sect. 12.2.1 was also analyzed in the same way, and the results are shown in Table 12.9. In the PP topology, as in the SS topology, the output current includes R_L in the equation, and the value changes depending on the load fluctuation. Therefore, in order to obtain CC characteristics in SS and PP topologies, it is necessary to confirm that, in addition to the optimal load being in the CC load range, the load fluctuation expected in the application is in the CC load range. Output voltage (CV characteristic)

Based on the circuit analysis, the output voltage V˙2 when the compensation inductor condition of the SP topology was applied was derived as follows. V˙2 = 

√ k R L Q 2 R2 Q 1 Q 2 V˙1  2  2 R1 A1 + (A2 Q 2 + R L )

(12.52)

Unlike the circuit analysis ignoring the internal resistance, V˙2 is expressed as a function of R L , so the complete CV characteristics cannot be obtained. However, in the load range close to R L = ∞, CV characteristics can be obtained together with the maximum voltage V˙2 max . Here, this chapter defines the load range where the damping ratio to the maximum voltage is 1/2 or less as the CV load range. V˙2 max = 

√ k Q 2 R2 Q 1 Q 2 V˙1     2 R1 k 4 Q 21 + 1 + k 2 Q 1 Q 2

(12.53)

Table 12.9 Formulas of output current and output voltage in IPT I˙2

SS topology PP topology

I˙2

max

√ k R2 Q 1 Q 2 V˙1 √ R1 ( R2 +R L +k 2 R2 Q 1 Q 2 ) 



 √ k Q 2 1−k 2 R2 Q 1 Q 2 V˙1

R1 B22 +{ R L Q 1d +D R2 Q 2 (1−k 2 )}

V˙2 SP topology

√ k R Q R2 Q 1 Q 2 V˙1  L 2  R1 A21 +(A2 Q 2 +R L )2

PS topology



√ k R L R2 Q 1 Q 2 V˙1   R1 (R2 +R L )2 +(R2 Q 1d +R L Q 1 )2

2



√ k Q 1 Q 2 V˙1 √ R1 R2 (1+k 2 Q 1 Q 2 )  √ k 1−k 2 Q 1 Q 2 V˙1    2 R1 R2 Q 22 +D 2 (1−k 2 )

V˙2

max

√ k Q R2 Q 1 Q 2 V˙1   2  2 R1 k 4 Q 21 +(1+k 2 Q 1 Q 2 ) √ k R2 Q 1 Q 2 V˙1    R1 1+Q 21

CC range

  R L  R2 1 + k 2 Q 1 Q 2 RL 

√

+Q 2 )2 +D 2  (Q 1 1+Q 21d /R2 Q 2 (1−k 2 )

CV range

 R2 Q 2 1+Q 22d  2 k 4 Q 21 +((1+k 2 Q 1 Q 2 )  R2 1+Q 21d  1+Q 21

 RL

 RL

408

12 Unified Theory of Magnetic Field Coupling and Electric Field …

 R2 Q 2 1 + Q 22d CV range :  2  R L  k 4 Q 21 + 1 + k 2 Q 1 Q 2 If the design value R Lopt satisfies the above load range, the SP topology obtains CV characteristics within that range. The output voltage equation in the PS topology where the CV characteristic was expected in Sect. 12.2.1 was also analyzed in the same way, and the results are shown in Table 12.9. In the PS topology, as in the SP topology, the output voltage includes R L in the equation, and the value changes depending on the load fluctuation. Therefore, in order to obtain CV characteristics in the SP and PS topologies, it is necessary to confirm that the load fluctuation expected in the application exists in the CV load range in addition to the fact that the optimum load exists in the CV load range.

12.2.3 Power Transmission Characteristics Evaluation During Load Fluctuation In the above, we mainly evaluated the optimal load value and simplified equations for the transmission efficiency and output power at the optimal load. Therefore, in this section, we confirm whether or not the simplified evaluation is effective by rigorous calculation of the transmission characteristics under optimal load.

12.2.3.1

Prerequisites and Notes

Table 12.10 shows the circuit parameters that are commonly used for the calculation of k = 0.1 and k = 0.3 when calculating the characteristics. The self-inductances L 1 and L 2 are assumed to be equivalent values assuming a symmetric structure. The resonance frequency is 85 kHz based on standards and ministerial ordinances. The input voltage is given a constant value of 100 V to evaluate the characteristics of the power supply with the same output. Tables 12.11 and 12.12 show the circuit parameters used for each coupling coefficient. Table 12.10 Calculation conditions for all topologies with k = 0.1 and k = 0.3 L 1 /L 2 (mH)

Q 1 /Q 2 (−)

R1 /R2 (−)

f 0 (kHz)

V˙1 (V)

1.40/1.40

300/300

2.49/2.49

85.0

100

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)

409

Table 12.11 Calculation conditions with k = 0.1 in IPT SS topology k(−)

0.100

SP topology

PS topology

PP topology

L m (μH)

140

C1 /C2 (nF)

2.50/2.50

2.53/2.50

2.50/2.53

2.53/2.53

R Lopt ()

74.8

7510

75.2

7398

SP topology

PS topology

PP topology

Table 12.12 Calculation conditions with k = 0.3 in IPT SS topology k(−)

0.300

L m (μH)

420

C1 /C2 (nF)

2.50/2.50

2.75/2.50

2.50/2.75

2.75/2.75

R Lopt ()

224

2602

234.2

2268

12.2.3.2

Transmission Characteristics at Optimal Load (Comparison Between Exact and Approximate Solutions)

The result of the exact calculation without approximation will be described. Tables 12.13 and 12.14 show the transmission characteristics and the CC and CV load ranges at the optimal load for each coupling coefficient. By substituting the calculation conditions into the simplified transmission efficiency Eq. (12.39), the Table 12.13 Calculation results with k = 0.1 in IPT (Exact solution) SS topology

SP topology

PS topology

PP topology

ηmax (%) Pout,ηmax (W)

93.6

93.5

93.5

93.4

125

124

1.25

1.26

cosin,ηmax (−)

1.00

1.00

1.00

1.00

CC range ()

R L  2.24 k

–

–

R L  218 k

CV range ()

–

251  R L

2.51  R L

–

Table 12.14 Calculation results with k = 0.3 in IPT (Exact solution) SS topology

SP topology

PS topology

PP topology

ηmax (%) Pout,ηmax (W)

97.8

97.7

97.7

97.4

43.5

41.7

3.76

4.29

cosin,ηmax (−)

1.00

1.00

1.00

1.00

CC range ()

R L  20.2 k

–

–

R L  170 k

CV range ()

–

30.2  R L

2.71  R L

–

410

12 Unified Theory of Magnetic Field Coupling and Electric Field …

transmission efficiency for k = 0.1 was calculated to be 93.6% for all compensation topologies and 97.8% for k = 0.3. By substituting the calculation conditions into the simplified output power Eqs. (12.43) and (12.44), in case of k = 0.1, the output power of the SS and SP topologies are 125.2 W and PS and PP topologies are 1.25 W, and in the case of k = 0.3, they were calculated as 43.6 and 3.93 W. Compared to the exact solution result in Tables 12.13 and 12.14, it can be seen that there is little difference between the exact calculation and the simplified formulas. The difference of output power between SS and SP topologies and PS and PP topologies was about k 2 times. Furthermore, it is clear that the CUPF characteristics can be obtained in all compensation topologies, but the input power factor was 1 at the optimal load. For the CC and CV load ranges, it can be seen that the optimal load values designed for each compensation topology exist within the load ranges. Furthermore, it can be seen that the CC load range of the SS topology at k = 0.3 is narrower than that at k = 0.1, whereas the CV load range of the SP topology is wide. It can be seen that in the PS and PP topologies, the width of the load range does not depend on the magnitude of the coupling coefficient. From the above, the following features can be said: Simplified formula evaluation using approximation has validity. The SS and SP topologies are excellent from the viewpoint of output power. There is a trade-off between power transmission efficiency and output power depending on the magnitude of the coupling coefficient. All compensation topologies achieve CUPF characteristics. The SS and PP topologies obtain CC characteristics near the optimum load. The SP and PS topologies obtain CV characteristics.

12.2.3.3

Power Transmission Characteristics During Load Fluctuation

The transmission characteristics during a load change for each magnitude of the coupling coefficient were calculated as shown in Figs. 12.14 and 12.15. In order to explain the characteristics of output power, output current, and output voltage in detail, the optimal load values for each compensation topology are described in these graphs. First, similarities that do not depend on the magnitude of the coupling coefficient are described. In Figs. 12.14a and 12.15a, all compensation topologies achieve almost the same maximum efficiency at the designed optimal load value. Also, as is clear from the similarity of the transmission efficiency formulas, the load fluctuation characteristics of SS and PS topologies and SP and PP topologies are almost the same. In Figs. 12.14b and 12.15b, it can be seen that the output power values of the SS and SP topologies are larger than the PS and PP topologies over the entire load range. Figures 12.14c and 12.15c are the same as the above graphs, and although the input power factor fluctuates with the load fluctuation, the value is negligible. It shows that CUPF characteristics can be achieved with all compensation topologies. Figures 12.14d and 12.15d show that all compensation topologies can achieve CV

12.2 Magnetic Field Coupling, IPT (SS, SP, PS, and PP)

Fig. 12.14 Calculation results of load characteristics with k = 0.1 in IPT (Exact solution)

Fig. 12.15 Calculation results of load characteristics with k = 0.3 in IPT (Exact solution)

411

412

12 Unified Theory of Magnetic Field Coupling and Electric Field …

characteristics from a constant load value to a large load range. However, considering the added optimal load position, it can be said that only SP and PS topologies achieve CV characteristics with load fluctuations near the optimal load. Similarly, in Figs. 12.14e and 12.15e, all compensation topologies can achieve CC characteristics from a fixed load value to a small load range. It can be said that only SS and SP topologies achieve CC characteristics with load fluctuations near the optimum load. Next, we describe the characteristics of k = 0.3 compared to k = 0.1, that is, the characteristic differences that can be confirmed as the coupling coefficient increases. As for the transmission efficiency, it can be seen that the larger the coupling coefficient, the wider the load range where the transmission efficiency close to the maximum efficiency can be obtained, and the maximum efficiency itself has a large value. As for the output power, the maximum power value of the PS and PP topologies increases as the coupling coefficient increases, whereas the maximum power value does not change in the SS and SP topologies. Instead, as the coupling coefficient increases, the maximum power load increases in the SS topology and decreases in the SP topology. It is thought that this causes the output power to increase or decrease according to the magnitude of the coupling coefficient as shown in Tables 12.13 and 12.14. Regarding the output voltage, there is no difference in the load range where the CV characteristic is obtained in the PP topology, but it can be seen that the larger the coupling coefficient in the SP topology, the wider the CV load range. On the other hand, for the output current, there is no difference in the load range where the CC characteristic can be obtained in the PS topology, but it can be seen that the larger the coupling coefficient in the SS topology, the wider the CC load range. Also, as described above, it can be seen that the larger the coupling coefficient, the narrower the CC load range of the SS topology, while the wider the CV load range of the SP topology. It can be seen that in the PS and PP topologies, the width of the load range does not depend on the magnitude of the coupling coefficient. From the above characteristic investigations, three items to consider in the design are the compensation topology, coupling coefficient, and load. The SS and SP topologies should be selected for the compensation topology, because all compensation topologies achieve the same maximum efficiency and the output power is superior under any conditions.

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP) In this section, we analyze the circuit of electric resonance coupled wireless power transmission and evaluate the power transmission characteristics of each compensation topology based on various formulas and characteristic calculations. First, we derive the compensation inductor conditions for each compensation topology to obtain the gyrator characteristics or ideal transformer characteristics under the condition of R1 = R2 = 0. Next, under the condition of R1 = R2 = 0, the exact and approximated formulas of each transmission characteristic when the compensation

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

413

inductor condition is applied are derived, and the transmission characteristics of each compensation topology are evaluated.

12.3.1 Derivation of Compensation Inductor Condition (R1 = R2 = 0) In this section, under the condition of R1 = R2 = 0, we derive the compensation capacitor condition of the two-element compensation topology by following the derivation process in the SS topology. In a two-port pair circuit with SS topology, the F parameter A˙ to D˙ of the circuit equation can be expressed as follows. a3 C 2 , A˙ = Cm

  jωCm 1 − k 2 a 5 B˙ = , C˙ = , jωCm k2

a4 C 1 D˙ = Cm

(12.54)

In the SS topology, it is possible to satisfy A˙ = D˙ = 0 by proper design of the compensation inductor, and L 1 and L 2 at that time are as follows. L1 =

ω2 C

1  , 2 1 1−k

L2 =

ω2 C

1   2 2 1−k

(12.55)

When the compensation inductor condition is satisfied, the F parameter of the network is rewritten as follows.   2 jωCm 1 − k 2 −k   , C˙ = , A˙ = 0, B˙ = k2 jωCm 1 − k 2      −k 2 0 ˙1 2 V˙2 V jωC 1−k ) m( (12.56) D˙ = 0 ˙ = jωC (1−k 2 ) m I1 I˙2 0 2 k The input impedance at this time are shown in (12.57). factor and the input power The output voltage V˙2 and output current I˙2 when using a constant voltage source are shown in Eqs. (12.58) and (12.59). Z in =

k4  2 , ∴ cos θin = 1 ω2 Cm2 R L 1 − k 2   ωCm 1 − k 2 R L V˙2 = k 2 V˙1   ωCm 1 − k 2 I˙2 = k 2 V˙1

(12.57)

(12.58) (12.59)

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12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.15 F parameters in CPT (R1 = R2 = 0) A˙ B˙ C˙ D˙

SS topology

SP topology

PS topology

PP topology

a3 C2 /Cm

−a5 /ω2 L 2 Cm

C2 /Cm

−a2 /ω2 L 2 Cm

a5 /jωCm   jωCm 1 − k 2 /k 2

a1 /jωCm

a2 /jωCm

1/jωCm

a4 C1 /jωL 2 Cm

a3 C2 /jωL 1 Cm

−a5 /jω3 L 1 L 2 Cm

a4 C1 /Cm

C1 /Cm

−a5 /ω2 L 1 Cm

−a1 /ω2 L 1 Cm

It can be seen that cos θin = 1 and constant I˙2 can be obtained without depending on R L . Therefore, it was shown that the SS topology can obtain the gyrator characteristics by satisfying the compensation inductor condition. Other compensation topologies are analyzed in the same way, and the results are shown in Tables 12.15, 12.16 and 12.17. If different compensation inductor conditions are satisfied for each compensation topology, SS and PP topologies can obtain gyrator characteristics satisfying A˙ = D˙ = 0, and SP and PS topologies can obtain ideal transformer characteristics satisfying B˙ = C˙ = 0. SP, PS, and PP are Eqs. (12.60), (12.61), and (12.62), respectively. 

V˙1 I˙1



 =

Cm C1

0

0



C1 Cm

V˙2 I˙2

 (12.60)

Table 12.16 Compensation inductor conditions in CPT SS topology

SP topology

PS topology

PP topology

L1

1 ω2 C1 (1−k 2 )

1 ω2 C 1

1 ω2 C1 (1−k 2 )

1 ω2 C 1

L2

1 ω2 C2 (1−k 2 )

1 ω2 C2 (1−k 2 )

1 ω2 C 2

1 ω2 C 2

Table 12.17 Gyrator and ideal transformer characteristics in CPT (R1 = R2 = 0) A˙ B˙ C˙ D˙

SS topology

SP topology

PS topology

0

Cm /C1

C2 /Cm

0

0

0

1/jωCm

  −k 2 /jωCm 1 − k 2   jωCm 1 − k 2 /k 2 0

Z in V˙2 I˙2

 2 k 4 /ω2 Cm2 R L 1 − k 2   ωCm R L 1 − k 2 /k 2 V˙1   ωCm 1 − k 2 /k 2 V˙1

Characteristics

Gyrator

PP topology

0

0

− jωCm

C1 /Cm

Cm /C2

0

Cm2 R L /C12 C1 /Cm V˙1 C1 /Cm R L V˙1

C22 R L /Cm2 Cm /C2 V˙1 Cm /C2 R L V˙1

ω2 Cm2 /R L ωCm R L V˙1 ωCm V˙1

Ideal transformer

Ideal transformer

Gyrator

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)







 V˙2 I˙2 0 CCm2      1 0 V˙1 V˙2 jωC m ˙I1 = − jωCm 0 I˙2 V˙1 I˙1



415

=

C2 Cm

0

(12.61)

(12.62)

From the next subsection, we evaluate the power transmission characteristics of each compensation topology when these compensation inductor conditions are applied, taking into account the resistance components of the circuit.

12.3.2 Circuit Analysis and Characterization (R1  = R2  = 0) In this subsection, we derive formulas for transmission characteristics when the compensation inductor condition is applied under the condition R1 = R2 = 0. Furthermore, by simplifying it as an approximate expression, we evaluate the transmission characteristics of each compensation topology based on various formulas and compare fairly. The transmission characteristics to be evaluated are of six types: transmission efficiency, optimal load, output power, CUPF characteristics, CC characteristics, and CV characteristics. In the circuit analysis of CC and CV characteristics, the analysis is divided into the derivation of the output current equation of the SS topology for the CC characteristic and the output voltage equation of the SP topology for the CV characteristic due to the difference in the derivation process. In addition, to simplify the formulas, the following parameters are used in addition to the coupling coefficient k in Eq. (12.10) and the Q value of the compensation inductor in Eq. (12.2). ⎧ 1+Q 2 1+Q 2 k2 ⎪ ⎨ Q 1δ = Q 1 1 , Q 2δ = Q 2 2 , δk = 1−k 2 2 A = R Q + k R Q , A = R Q + δk R L Q 1 , k 2 2 L 1 δ 2 2 k ⎪ ⎩ B = R L + R2 Q 22

Ak Q 1δ = R2 Q 2 + k 2 R L Q 1δ (12.63)

The characteristics of each of the four groups (a) efficiency and optimal load, (b) output power, (c) input impedance and input power factor, and (d) output current and output voltage are described below.

12.3.2.1

Efficiency and Output Power

Tables 12.18 and 12.19 show the exact and approximate solutions for efficiency and optimal load. The equations for efficiency and maximum efficiency are also given. Since the Q value of the coil often has a value of about several hundreds,

416

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.18 Formulas of transmission efficiency in CPT   η(Q 1 = Q 2 ) η Q = Q1 = Q2, 1  Q2 SS topology

k 2 R2 R L Q 1 Q 2 (R2 +R L )( R2 +R L +k 2 R2 Q 1 Q 2 )

k 2 R2 R L Q 2 (R2 +R L )( R2 +R L +k 2 R2 Q 2 )

SP topology

δk R2 R L Q 1 Q 22 Q 2δ   Aδk R2 Q 2 +B Aδk Q 2 +R L

( R L +R2 Q 2 )( R L +R2 Q 2 +δk R L Q 2 )

PS topology PP topology

δk R2 R L Q 4

δk R2 R L Q 1δ Q 2 (R2 +R L )(R2 +R L +δk R2 Q 1δ Q 2 )

δk R2 R L Q 2 (R2 +R L )( R2 +R L +δk R2 Q 2 ) k 2 R2 R L Q 4

k 2 R2 R L Q 1δ Q 22 Q 2δ   Ak Q 1δ R2 Q 2 +B Ak Q 1δ Q 2 +R L

( R L +R2 Q 2 )( R L +R2 Q 2 +k 2 R L Q 2 )

Table 12.19 Formulas of optimal load in CPT SS topology SP topology PS topology PP topology

ηmax k2 Q2

 1+

√

1+k 2 Q 2

δ Q2

 1+

√k

1+δk Q 2

δ Q2

 1+

√k

1+δk Q 2

k2 Q2

 1+

√

1+k 2 Q 2

2

2

2

2

  R Lopt Q = Q 1 = Q 2 , 1  Q 2  R2 1 + k 2 Q 2

R Lopt (Q 1 = Q 2 )  R2 1 + k 2 Q 1 Q 2 √ R√2 Q 2 Q 2 Q 2δ 1+δk Q 1 Q 2

√ R2 Q

2

1+δk Q 2

√ R2 1 + δk Q 1δ Q 2

 R 2 1 + δk Q 2

R2 Q 2 √

√ R2 Q

√

Q 2 Q 2δ 1+k 2 Q 1δ Q 2

2

1+k 2 Q 2

approximation of the Eq. (12.64) is effective. Q = Q1 = Q2, 1  Q2

(12.64)

When k < 0.3, the approximation of δk ≈ k 2 is possible when the coupling coefficient is small. Under such weak coupling conditions, the maximum efficiency of SS, SP, PS, and PP can be approximated by Eq. (12.65) and has the same value. all = ηmax

k2 Q2 2  1 + 1 + k2 Q2

(12.65)

Similarly, under such weak coupling conditions, the optimum load is divided into two, and SS and PS can be approximated by Eq. (12.66), and SP and PP can be approximated by Eq. (12.67).  SS,PS R Lopt = R2 1 + k 2 Q 2 SP,PP R Lopt =

R2 Q 2 1 + k2 Q2

(12.66) (12.67)

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

417

Table 12.20 Formulas of output power in CPT Pout (Q 1  = Q 2 )





SS topology

k 2 R2 R L Q 1 Q 2 ˙ 2 2 V1 R1 ( R2 +R L +k 2 R2 Q 1 Q 2 )

SP topology

2 δ R R Q Q2 Q  k 2 L 1 2 2δ   V˙1 2 R1 A2δ + Aδk Q 2 +R L

PS topology PP topology

k





δk Q 2 R2 R L ˙ 2 V1 R1 Q 1 (R2 +R L )2

  Pout Q = Q 1 = Q 2 , 1  Q 2 2 k 2 R2 R L Q 2 ˙ 2 V1 R1 ( R2 +R L +k 2 R2 Q 2 ) 2 δk R2 R L Q 4 ˙ 2 V1 R1 ( R L +R2 Q 2 +δk R L Q 2 ) δk R2 R L ˙ 2 2 V1 R1 (R2 +R L )



k 2 R2 R L Q 22 Q 2δ ˙ 2   V1 R1 Q 1 B 2 +R22 Q 22



k 2 R 2 R L Q 4 ˙ 2 2 V1 R1 ( R L +R2 Q 2 )

Pout,ηmax R1

R1

ηmax √

2 V˙1

ηmax √

2 V˙1

1+k 2 Q 2

ηmax

1+δk Q 2

√

1+δk Q 2 ˙ 2 V1 R1 Q 2

√

ηmax 1+k 2 Q 2 ˙ 2 V1 R1 Q 2

The relationship between these two equations is Eq. (12.68). Considering k < 1, it can be seen that SP and PP have larger optimum load values than SS and PS. SS,PS SP,PP : R Lopt = 1 + k2 Q2 : Q2  k2 : 1 R Lopt

(12.68)

Table 12.20 shows the exact and approximate solutions for the output power. The output power at maximum efficiency is also shown. Under the weak coupling condition, when approximating δk ≈ k 2 , the output power at the maximum efficiency is divided into two, and SS and SP can be approximated by Eq. (12.69), and PS and PP can be approximated by Eq. (12.70). 2 ηmax V˙1  2 2 R1 1 + k Q  ηmax 1 + k 2 Q 2 ˙ 2 V1 = R1 Q 2

SS,SP Pout,η = max

PS,PP Pout,η max

(12.69) (12.70)

The relationship between Expressions (12.69) and (12.70) is Expression (12.71). Considering k < 1, it can be seen that the output power of SS and SP is larger than that of PS and PP. SS,SP PS,PP : Pout,η = Q2 : 1 + k2 Q2  1 : k2 Pout,η max max

12.3.2.2

(12.71)

Input Power Factor and Input Current

Based on the circuit analysis, the input impedance Z in when the SS topology resonance condition is applied is derived as follows.   R1 R2 + R L + k 2 R2 Q 1 Q 2 Z in = R2 + R L

(12.72)

418

12 Unified Theory of Magnetic Field Coupling and Electric Field …

The input impedance formula of the SS topology is expressed only by the real component, and it is clear that the CUPF characteristic independent of the load is achieved. On the other hand, the input impedance formulas of other compensation topologies are represented by complex numbers. It is necessary to derive the input power factor equation for exact CUPF characteristic evaluation, but it is derived as a complicated equation. Therefore, the change of input power factor at the time of load change is presented as a graph, and based on the result, it is judged whether the CUPF characteristic can be achieved or not and applied to simplify the input impedance equation. The input power factor was calculated as shown in Fig. 12.16 in the load range of R L = 1 − 100 M, with the design values of the inductance and resonance conditions uniquely determined under the condition of Q 1 = Q 2 = 300, R1 = R2 = 1 , f = 400 kHz. The coupling coefficient was assumed to be k = 0.1 and 0.3. Figure 12.16 shows some input power factor fluctuations due to load fluctuations. From the graph, the input power factor is almost 1 irrespective of the magnitude of the coupling coefficient, and it can be said that all the compensation topologies can achieve the CUPF characteristics. Therefore, the input impedance formula in all compensation topologies is approximated only by real components. In addition, the approximation of the Q value in (12.64) simplifies the input impedance equation as follows.   R1 R2 + R L k 2 R2 Q 2 Z in  R2 + R L

(12.73)

From Eqs. (12.66) and (12.73), the input impedance Z in,ηmax when both the compensation inductor condition and the optimal load condition are satisfied is obtained as follows.  (12.74) Z in,ηmax R1 a + k 2 Q 2 The input impedance equations for other compensation topologies are analyzed in the same way, and the results are shown in Table 12.21. As mentioned above, even when the resonance condition is applied to the SP, PS, and PP topologies, the input impedance equation becomes a complex number and perfect CUPF characteristics cannot be obtained. However, as shown in Fig. 12.16, a compensation inductor with a high Q value must be used. By using it, it is possible to achieve almost CUPF characteristics. In addition, Z in,ηmax in each compensation topology is included in Pout,ηmax in Table 12.20, and it can be seen that it is an important parameter for increasing output power.

12.3.2.3

Output Current and Output Voltage

Output current (CC characteristics) Based on the circuit analysis, the output current I˙2 when the SS topology resonance condition was applied was derived as follows.

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

(a)

= 0.1

(b)

= 0.3

Fig. 12.16 Calculation results of load characteristics for input power factor in CPT

419

420

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.21 Formulas of input impedance in CPT Z in (Q 1 = Q 2 ) 

SS topology SP topology PS topology PP topology

R1 R2 +R L +k 2 R2 Q 1 Q 2 R2 +R L

  Z in Q = Q 1 = Q 2 , 1Q 2



  R1 R2 +R L +k 2 R2 Q 2 R2 +R L

Z in,ηmax  R1 1 + k 2 Q 2  R 1 1 + δk Q 2

   R1 Aδk + j Aδk Q 2 +R L R2 Q 2 + j B

  R1 R L +R2 Q 2 +δk R L Q 2 R L +R2 Q 2

R1 Q 1 (R2 +R L )(1+ j Q 1 ) δk R2 Q 2 + j(R2 +R L +δk R2 Q 1 Q 2 )

R1 Q 2 (R2 +R L ) R2 +R L +δk R2 Q 2

√ R1 Q

2

− j R1 Q 1 (R2 Q 2 + j B)(1+ j Q 1 ) Ak +k 2 R L Q 2 + j ( Ak Q 2 +R L (1−k 2 ))

  R1 Q 2 R L +R2 Q 2 R L +R2 Q 2 +k 2 R L Q 2

√ R1 Q

2

I˙2 = √

1+δk Q 2 1+k 2 Q 2

√ k R2 Q 1 Q 2  V˙1 2 R1 R L + R2 + k R2 Q 1 Q 2 

(12.75)

Unlike the circuit analysis that disregards the internal resistance, I˙2 is expressed as a function of R L , so the complete CC characteristics cannot be obtained. However, in the load range close to R L = 0, the CC characteristics are obtained along with the maximum current I˙2 max . In this section, the load range where the damping ratio to the maximum current is 1/2 or less is defined as the CC load range. √ k Q1 Q2 ˙ I˙2 max = √ R R 1 + k 2 Q Q  V1 1 2 1 2   CC range : R L  R2 1 + k 2 Q 1 Q 2

(12.76) (12.77)

If the design value R Lopt satisfies the above load range, the SS topology obtains CC characteristics within that range. The output current equation in the PP topology where CC characteristics were expected in Sect. 12.3.1 was also analyzed in the same way. The results are shown in Table 12.22. In the PP topology, as in the SS topology, the output current includes R L in the equation, and the value changes depending on the load fluctuation. Therefore, it is necessary to confirm that the optimum load exists in the CC load range in order to obtain CC characteristics in SS and PP topologies. Table 12.22 Formulas of output current and output voltage in CPT I˙2

SS topology PP topology

√ k R2 Q 1 Q 2 V˙ √ R1 ( R2 +R L +k 2 R2 Q 1 Q 2 ) 1 √ k Q R Q 2 2 2δ    V˙1 R1 Q 1 B 2 +R22 Q 22

V˙2 SP topology

√

 RL Q 2 δk R2 Q 1 Q 2δ 2  R1 A2δ + Aδk Q 2 +R L k

PS topology

√ √ R L δk R2 Q 2 V˙1 R1 Q 1 (R2 +R L )

 V˙1

I˙2

max√ k Q1 Q2 V˙ R1 R2 (1+k 2 Q 1 Q 2 ) 1 k ˙ √ V1 R1 R2 Q 1 Q 2 √

V˙2 



max √ Q 2 δk R2 Q 1 Q 2δ  2 2  V˙1 R1 δk Q 1 +(1+δk Q 1 Q 2 )2 δk R2 Q 2 R1 Q 1

V˙1

CC range

  R L  R2 1 + k 2 Q 1 Q 2 √ R L  R2 Q 2 Q 2 Q 2δ

CV range

√  R2 Q 2 Q 2 Q 2δ δk2 Q 21 +(1+δk Q 1 Q 2 )2

R2  R L

 RL

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

421

Also, it is necessary to confirm that the load fluctuation expected in the application exists within the CC load range. Output voltage (CV characteristic)

Based on the circuit analysis, the output voltage V˙2 when the resonance condition of the SP topology is applied is derived as follows. V˙2 = 

√ R L Q 2 δk R2 Q 1 Q 2δ ˙   V1   2 2 R1 Aδk + Aδk Q 2 + R L

(12.78)

Unlike the circuit analysis ignoring the internal resistance, V˙2 is expressed as a function of R L , so the complete CV characteristics cannot be obtained. However, CV characteristics are obtained with the maximum voltage V˙2 max in the load range close to R L = ∞. The load range where the attenuation ratio to the maximum voltage is less than 1/2 is defined as the CV load range. V˙2 max = 

√ Q 2 δk R2 Q 1 Q 2δ

V˙1  2 2  2 R1 δk Q 1 + (1 + δk Q 1 Q 2 ) √ R2 Q 2 Q 2 Q 2δ CV range :   RL δk2 Q 21 + (1 + δk Q 1 Q 2 )2

(12.79)

(12.80)

When the design value R Lopt satisfies the above load range, the SP topology obtains CV characteristics within that range. The output voltage equation in the PS topology for which CV characteristics were expected in Sect. 12.3.1 was also analyzed in the same way, and the results are shown in Table 12.22. In the PS topology, as in the SP topology, the output voltage includes R L in the equation and its value changes depending on the load fluctuation. Therefore, it is necessary to confirm that the optimum load exists in the CV load range in order to obtain CV characteristics in the SP and PS topologies. In addition, it is necessary to confirm that the load fluctuation expected in the application exists within the CV load range.

12.3.3 Power Transmission Characteristics Evaluation During Load Fluctuation In the above, we mainly evaluated the optimal load value and simplified equations for the transmission efficiency and output power at the optimal load. Therefore, in this section, we confirm whether or not the simplified evaluation is effective by rigorous calculation of the transmission characteristics under optimal load.

422

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Table 12.23 Calculation conditions for all topologies with k = 0.1 and k = 0.3 V˙1 (V) C1 /C2 (pF) Q 1 /Q 2 (−) f 0 (kHz) 500/500

300/300

400

100

Table 12.24 Calculation conditions with k = 0.1 SS topology

SP topology

PS topology

PP topology

k(−)

0.100

Cm (pF)

50.0

L 1 /L 2 (μH)

320/320

317/320

320/317

317/317

R1 /R2 ()

2.67/2.67

2.65/2.67

2.67/2.65

2.65/2.65

R Lopt ()

80.4

7.99 k

80.0

7.95 k

SP topology

PS topology

PP topology

Table 12.25 Calculation conditions with k = 0.3 SS topology k[−]

0.300

Cm (pF)

150

L 1 /L 2 (μH)

348/348

317/348

348/317

317/317

R1 /R2 ()

2.91/2.91

2.65/2.91

2.91/2.65

2.65/2.65

R Lopt ()

262

2.78 k

250

2.65 k

12.3.3.1

Prerequisites and Notes

Table 12.23 shows the circuit parameters commonly used for the calculation of k = 0.1 and k = 0.3. The self-capacitances C 1 and C 2 are equivalent values assuming a symmetric structure. The resonance frequency is 400 kHz based on standards and ministerial ordinances. The input voltage is given a constant value of 100 V to evaluate the characteristics of the power supply with the same output. Table 12.24 and Table 12.25 show the circuit parameters used for each coupling coefficient.

12.3.3.2

Transmission Characteristics Under Optimal Load (Comparison Between Exact and Approximate Solutions)

Tables 12.26 and 12.27 show the power transmission characteristics and the CC and CV load ranges at the optimal load for each magnitude of the coupling coefficient, based on the results of rigorous calculations without approximation. Here, by substituting the calculation conditions into the simplified transmission efficiency to be compared, Eq. (12.65), the transmission efficiency is 93.6% for k = 0.1 and 97.8% for k = 0.3 for all compensation topologies. By substituting the calculation conditions into Eqs. (12.69) and (12.70) for the simplified output power, when k = 0.1, the

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

423

Table 12.26 Calculation results with k = 0.1 in CPT (Exact solution) SS topology

SP topology

PS topology

PP topology

ηmax (%) Pout,ηmax (W)

93.6

93.6

93.6

93.6

116

117

1.17

1.18

cos θin,ηmax (−)

1.00

1.00

1.00

1.00

CC range ()

R L = 2.41 k

–

–

R L = 239 k

CV range ()

–

264  R L

2.65  R L

–

Table 12.27 Calculation results with k = 0.3 in CPT (Exact solution) SS topology

SP topology

PS topology

PP topology

ηmax π (%) Pout,ηmax (W)

97.8

97.9

97.9

97.8

37.2

39.1

3.53

3.68

cos θin,ηmax (−)

1.00

1.00

1.00

1.00

CC range ()

R L = 23.6 k

–

–

R L = 239 k

CV range ()

–

29.4  R L

2.65  R L

–

output power is 117 W and 118 W for the SS and SP topologies, and 1.17 W for the PS and PP topologies. When k = 0.3, it was calculated as 1.18 W, 37.3 W, 41.0 W, 3.36 W, and 3.69 W, respectively. Exact solutions Tables 12.26 and 12.27 show that there are some differences, but that the exact results and the simplified formulas are almost the same. The difference in output power between the SS and SP topologies and the PS and PP topologies was about k 2 times. Furthermore, it is clear that the CUPF characteristics can be obtained in all compensation topologies, but the input power factor was 1 at the optimal load. For the CC and CV load ranges, it can be seen that the optimal load values designed for each compensation topology exist within the load ranges. Further, it can be seen that the CC load range of the SS topology at k = 0.3 is narrower than that at k = 0.1, whereas the CV load range of the SP topology is wide. In addition, it can be seen that in the PS and PP topologies, the width of the load range does not depend on the magnitude of the coupling coefficient. From the above, the following features can be said: Simplified formula evaluation using approximation has validity. The SS and SP topologies are excellent from the viewpoint of output power. There is a trade-off between power transmission efficiency and output power depending on the magnitude of the coupling coefficient. All compensation topologies achieve CUPF characteristics. The SS and PP topologies obtain CC characteristics near the optimum load. SP and PS topologies obtain CV characteristics.

424

12.3.3.3

12 Unified Theory of Magnetic Field Coupling and Electric Field …

Power Transmission Characteristics During Load Fluctuation

The transmission characteristics when the load fluctuates for each magnitude of the coupling coefficient are shown in Figs. 12.17 and 12.18. In order to explain the characteristics of output power, output current and output voltage in detail, the optimal load values for each compensation topology are described in these graphs. First, similarities that do not depend on the magnitude of the coupling coefficient are described. In Figs. 12.17a and 12.18a, all compensation topologies achieve almost the same maximum efficiency at the designed optimum load value. Also, as it is clear from the similarity of the transmission efficiency formulas, the load fluctuation characteristics of the SS and PS topologies and the SP and PP topologies are almost the same. Figures 12.17b and 12.18b show that the output power values of the SS and SP topologies are larger than the PS and PP topologies over the entire load range. In addition, comparing the power transmission efficiency and the load fluctuation characteristics of the output power, the maximum efficiency load (optimum load) and the maximum power load do not match, so it is necessary to select the load value to be used according to the purpose. Figures 12.17c and 12.18c are the same as the graphs shown above. Although the input power factor fluctuates with the load fluctuation, its value is negligible, indicating that the CUPF characteristics can be achieved in all compensation topologies. Figures 12.17d and 12.18d show that all compensation topologies can achieve CV characteristics from a constant load value to a large load range. However, considering the added optimal load position, it can be said that

Fig. 12.17 Calculation results of load characteristics with k = 0.1 in CPT (Exact solution)

12.3 Electric Field Coupling, CPT (SS, SP, PS, and PP)

425

Fig. 12.18 Calculation results of load characteristics with k = 0.3 in CPT (Exact solution)

only SP and PS topologies achieve CV characteristics with load fluctuations near the optimal load. Similarly, in Figs. 12.17e and 12.18e, all compensation topologies can achieve CC characteristics from a constant load value to a small load range. It can be said that only the SS and SP topologies achieve CC characteristics with load fluctuations near the optimum load. Next, the features of k = 0.1 and k = 0.3 are compared. In other words, the characteristic differences that can be confirmed as the coupling coefficient increases are described. As for the power transmission efficiency, it is understood that the load range in which the power transmission efficiency close to the maximum efficiency can be obtained is wider as the coupling coefficient is larger. Also, it can be seen that the maximum efficiency itself has a large value. As for the output power, the maximum power value of the PS and PP topologies increases as the coupling coefficient increases, whereas the maximum power value does not change in the SS and SP topologies. Also, as described above, it can be seen that as the coupling coefficient increases, the CC load range of the SS topology decreases, whereas the CV load range of the SP topology increases. It can be seen that in the PS and PP topologies, the width of the load range does not depend on the magnitude of the coupling coefficient. From the above characteristic investigations, three items to consider in the design are the compensation topology, coupling coefficient, and load. While all compensation topologies achieve the same maximum efficiency, the SS and SP topologies should be selected for the compensation topology, since the output power is superior under any conditions.

426

12 Unified Theory of Magnetic Field Coupling and Electric Field …

12.4 Comparison and Unified Theory of Magnetic Resonance Coupling and Electric Resonance Coupling 12.4.1 Unified Interpretation We summarize and compare IPT and CPT [1]. For magnetic field coupling and electric field coupling, the reactance compensation element conditions that satisfy the gyrator characteristics or ideal transformer characteristics are shown again in Table 12.28 for each magnetic field and electric field coupling method. Here, the resonance condition using a compensation capacitor is shown for IPT, and the compensation inductor condition is shown for CPT. It is similar that C = 1/ω2 L for magnetic field coupling and L = 1/ω2 C for electric field coupling, but the presence or absence of 1 − k 2 for each compensation topology has a difference. This is because the transmitting and receiving coupler with magnetic field coupling is represented by a T-type equivalent circuit, and the transmitting and receiving coupler with electric field coupling is represented by a π-type equivalent circuit. For example, if an equivalent circuit expression with a light physical meaning is used, the magnetic field coupling can be expressed as a π-type equivalent circuit, and the electric field coupling can be expressed as a T-type equivalent circuit. The inclusion of 1−k 2 is switched symmetrically. However, since it is an equivalent transformation, the design result does not change, only the viewpoint changes. In the evaluations up to the previous chapter, it was concluded that both the IPT and CPT have excellent SS and SP topologies in terms of output power. If we want to adopt SS topology, CPT includes 1 − k 2 in the reactance compensation condition. On the other hand, when using the SP topology, it is necessary to pay attention to the fact that both IPT and CPT include 1 − k 2 in the resonance condition. Regarding IPT and CPT, the transmission efficiency, optimal load, and output power equations are shown in Table 12.29 when the resonance condition and the optimal load condition are satisfied and the approximation of Q = Q 1 = Q 2 , 1  Q 2 , k 2  1 is assumed. In all the compensation topologies (SS, SP, PS, and PP), the formulas expressing the transmission characteristics of both the magnetic field Table 12.28 Compensation conditions for reactance in IPT and CPT (R1 = R2 = 0) In IPT

SS topology

SP topology

PS topology

PP topology

C1

1 ω2 L 1

1 ω2 L 1 (1−k 2 )

1 ω2 L 1

1 ω2 L 1 (1−k 2 )

C2

1 ω2 L 2

1 ω2 L 2

1 ω2 L 2 (1−k 2 )

1 ω2 L 2 (1−k 2 )

In CPT

SS topology

SP topology

PS topology

PP topology

L1

1 ω2 C1 (1−k 2 )

1 ω2 C 1

1 ω2 C1 (1−k 2 )

1 ω2 C 1

L2

1 ω2 C2 (1−k 2 )

1 ω2 C2 (1−k 2 )

1 ω2 C 2

1 ω2 C 2

12.4 Comparison and Unified Theory of Magnetic Resonance Coupling …

427

Table 12.29 Formulas in IPT and CPT In IPT and CPT SS topology SP topology

ηmax k2 Q2

 √ 2 1+ 1+k 2 Q 2 k2 Q2  √ 2 1+ 1+k 2 Q 2

PS topology

k2 Q2  √ 2 1+ 1+k 2 Q 2

PP topology

k2 Q2  √ 2 1+ 1+k 2 Q 2

R Lopt  R2 1 + k 2 Q 2

Pout,ηmax R1

√ R2 Q

2

1+k 2 Q 2

 R2 1 + k 2 Q 2 √ R2 Q

2

1+k 2 Q 2

R1

ηmax √

2 V˙1

ηmax √

2 V˙1

1+k 2 Q 2

1+k 2 Q 2

√

ηmax 1+k 2 Q 2 ˙ 2 V1 R1 Q 2

√

ηmax 1+k 2 Q 2 ˙ 2 V1 R1 Q 2

coupling and the electric field coupling were all the same. Thus, since the equation of efficiency, the equation of optimal load, and the equation of power are the same, IPT and CPT can be treated unifiedly.

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