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AL-FARABI KAZAKH NATIONAL UNIVERSITY

THEORY OF ELECTRIC CIRCUITS Educational manual

Almaty «Qazaq University» 2020

UDС 621.3 LBC 31.2 T 11 Recommended for publication by the Academic Council of the Faculty of Mechanics and Mathematics and Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol No.4 dated 19.06.2020)

Reviewers: KazNPU named after Abay senior lecturer of the Department of Physics, PhD D.M. Nasirova and senior lecturer, Department of Solid State Physics and Nonlinear Physics, PhD B.Zh. Medetov

T 11

Theory оf Electric Circuits: educational manual / N.Sh. Alimgazinova, S.M. Manakov, M.T. Kyzgarina, A.B. Manapbayeva. – Almaty: Qazaq University, 2020. – 216 p. ISBN 978-601-04-4741-7 The educational manual contains lectures on the main sections of the discipline «Theory of Electric Circuits». It depicts physical phenomena and processes that occur in various electric circuits and devices. In addition, the textbook provides basic methods of calculation in DC and AC linear circuits. The course of lectures is intended for undergraduate students majoring in «5B071900 – Radio Engineering, Electronics and Telecommunications», «5B070300 – Information Systems» and «5B070400 – Computer Engineering and Software».

UDС 621.3 LBC 31.2 ISBN 978-601-04-4741-7

© Alimgazinova N.Sh., Manakov S.M., Kyzgarina M.T., Manapbayeva A.B., 2020 © Al-Farabi KazNU, 2020

INTRODUCTION

This manual was created on the basis of lectures, read by the author over the past ten years for the 2nd year students of the Faculty of Physics and Technology and the Faculty of Mathematics and Mechanics at Al-Farabi Kazakh National University. The course «Theory of electric circuits» («TEC») is included in the list of compulsory subjects for Bachelor’s program of specialities «5B071900 – Radio engineering, electronics and telecommunications», «5B070300 – Information Systems» and «5B070400 – Computers and Software». The contents of manuals and sequence of presentation are generally in line with the standard program for «TEC» discipline. The aim of the course «TEC» is the study of the fundamental laws, properties, and methods of calculation of DC and AC electric circuits. The main objective of the course is to teach students the methods of theoretical analysis and experimental study of the electric circuits of various types. As a result of studying of the course «TEC», presented in this manual, students will have an understanding of the basic concepts and definitions used in the «TEC», basic laws, methods of analysis of electric circuits in steady-state and transient conditions; will be able to make equation of the state of the circuit describing its operation, to calculate the transients in circuits with single and multiple energy storage units, to make spectral analysis with non-sinusoidal effects, independently analyze the physical processes occurring in the electric circuits. In preparing this lecture course well-known textbooks, collections of problems and manuals have been used [1…20].

3

Lecture 1

ELECTRIC CIRCUITS

Plan 1.1 Electric circuit. Concepts 1.2 Elements of electric circuits. Classification 1.3 Classification of electric circuits

1.1 Electric circuit. Concepts Electric energy is the most common form of energy used by mankind in our time. Electricity has become the basis for the development of all branches of engineering, transport, telecommunications, agriculture, and it is an integral part of our everyday life. Electric energy, being a part of the electromagnetic energy, is widespread due to the possibility of its transmission at large distances and transformation into other forms of energy: thermal, mechanical, chemical, and others. Almost all of the existing electric, radio and electronic devices are electromagnetic systems, in which the main processes are subject to the basic laws of electricity and magnetism. Accurate analysis of the processes described by Maxwell's equations in partial derivatives, even in the simplest case is a difficult task. Therefore, for engineering calculations and design of a variety of devices it is necessary to make quantitative assessment. In this regard, there is a need in the approximate methods of analysis that allow a certain degree of accuracy to solve a wide range of tasks. Such methods are given by the theory of electric circuits (TEC). However, not all electromagnetic processes can be analyzed by the TEC. For example, investigation of processes at high frequencies, determination of parameters of circuit elements, etc. should be performed using the methods of the electromagnetic field theory. Thus, depending on the conditions provided by the problem, there are two 4

ways to describe electrical and magnetic phenomena, using the basic concepts of the electromagnetic field theory (EMFT) and the theory of electric circuits (TEC) (Figure 1.1). EMFT studies the changes of the electric and magnetic quantities from point to point in space and time, it explores the electric and magnetic fields and uses them to study such phenomena as the emission of electromagnetic energy, the distribution of the space charge, current density, and others. TEC is based on replacing the real electric devices by an ideal equivalent circuit and allows us, with sufficient accuracy for engineering practice, to determine the direct voltage between the ends of the circuit section without calculating it between intermediate points. The currents are then determined without calculation of their density at various points in the conductor cross section.

Methods for describing EM phenomena

TEC

EMFT

Figure 1.1. Methods of describing electromagnetic phenomena

The subject of the course TEC is the study of quantitative and qualitative aspects of the electromagnetic processes in electric circuits. The main objective of TEC is to study the methods of analysis and synthesis of electric circuits. The problem of analysis reduces to the calculation of electric values for a given circuit. The problem of synthesis is the creation of an electric circuit with the desired properties. Electric circuit (EC) is a collection of devices for transmission, distribution and mutual conversion of electric energy and information, if the processes occurring in the devices can be described using the concepts of electromotive force (EMF), current and voltage. 5

The electric current is an orderly movement of charged particles in a conductive medium by an electric field. In metals and vacuum, this is motion of electrons, while in electrolytes and gases – motion of positive (+) cations and negative (-) ions. Numerically, the electric current is defined as the limit of the ratio of the amount of electricity transferred by charged particles through the conductor cross-section to the amount of time tending to zero:

i(t )  lim

t 0

q dq  , t dt

(1.1)

q is the total amount of electricity of positive and negative charges, which have moved in opposite directions over the time t . where

In the International System of Units (SI), the current is measured in amperes (A); charge in Coulombs (C) or ampere-seconds (A ∙ s); time in seconds (s). Ampere is the value of a constant current, which passing along two parallel rectangular conductors of infinite length and negligible cross-section, located at a distance of 1 meter apart in vacuum, would produce between these conductors a force equal to 2·10-7 N per meter of length. There is direct and alternating current. If the rate of motion of charged particles in time is constant, the current is called direct, if the current values change over time, it is called alternating. Electric current is attributed the direction, which coincides with the direction of movement of the positive charges and is opposite to the movement of negative charges. The current direction is characterized by the sign. The positive direction of the current is selected at random and is denoted by an arrow. Voltage is defined as the work done for moving a unit charge between specified points 1 and 2 of the space, which is defined by the expression: 2

u12   Edl , 1

6

(1.2)

or

u12 (t )  lim

q 0

W dW  , q dq

(1.3)

where E is the electric field strength, which is determined by charges in the region under consideration l , W is the energy required to move a unit charge q. Along with the above definitions the electric voltage is called potential difference. Electrical potential difference is the voltage in the irrotational electric field, which is characterized by the independent choice of integration path:

u12  1  2 ,

(1.4)

and the potential of the point is the amount of energy needed to move a unit charge from the selected point to the point at infinity ∞ in space, the potential of which is assumed to be zero   0 . In the International System of Units (SI), the voltage is measured in volts (V), the energy in joules (J). Volt is the electric voltage, which produces a constant current of 1A in the circuit at a power of 1W. There is direct and alternating voltage. The positive direction of the reference voltage corresponds to the direction of movement of the positively charged particles from the higher potential point 1 to a lower potential point 2. The electromotive force (EMF) is generated by external forces, which are understood as non-electrostatic forces, whose action on the conduction electrons in a conductor causes their orderly movement and supports the current in the circuit. External forces, unlike Coulomb forces do not connect opposite charges, but cause their separation and a potential difference is maintained at the conductor ends. External forces create a non-electrostatic electric field Eout that ensures orderly movement of electric charges. The total effect of the electric field strength along the outside of the loop L is characterized by EMF 7

e   Eout dl ,

(1.5)

L

which is crated by electric energy sources (electrochemical cells, electric generators, etc.). In the International System of Units (SI), the EMF is measured in volts (V). There are direct and alternating EMFs.

1

EC 2 Figure 1.2. Direction of current and voltage in the EC

1.2 Elements of EC. Classification EC consists of parts that perform certain functions, which are called elements. In the general case, the EC elements are divided into sources of electric energy (EES), receivers (load) (RE) and intermediates (IM) (wires, devices, measuring instruments, switches) which connect the sources with receivers (Figure 1.3). The sources of electric energy are electrical devices that produce electricity by converting chemical, molecular-kinetic, thermal, mechanical and other types of energy. These are chemical sources, thermocouples, batteries, generators and other devices. The receivers of electricity are electrical devices that consume electric energy by converting it into light, heat, mechanical or other types of energy. They are electric lamps, electric heaters, electric motors and other devices. The elements in the TEC are not the components of the electric devices that physically exist, but their idealized models that approximately reproduce phenomena in real devices. 8

Structure of EC

Energy sources

Intermediates Receivers of energy

Figure 1.3. Components of the EC

The elements are distinguished by the number of terminals (poles), by their intended purpose, by the type of equations, etc. (Figure 1.4). two-terminal Е l e m e n t s

the number of terminals (poles)

by intended purpose

multiterminal active lumped

passive

distributed

parameters by

linear

characteristics equation

non-linear

Figure 1.4. Classification of EC elements

Each circuit element has a certain number of terminals (poles), by which it is connected with other elements. If the element has one pair of terminals, it is called a two-pole network. Examples are power sources (with the exception of controlled and multiphase), resistors, inductors, capacitors. If there are terminals in a number of three or more, the element is referred to as a multi-terminal element. These include transistors, transformers, amplifiers, etc. 9

By the purpose of usage, all the circuit elements can be divided into active and passive. The active element includes in its structure the source of electric energy. The passive elements include those in which the energy is scattered (resistors) or stored (inductor and capacitor). The main characteristics of circuit elements include their current-voltage, Weber-voltage and coulomb-volt characteristics (IVC, WbAC, CVC), described by differential or algebraic equations. If the elements are described by linear differential or algebraic equations, they are called linear, otherwise they belong to the class of non-linear. In a rigorous examination, all the elements are nonlinear. When there is a possibility of considering them as linear, it substantially simplifies the mathematical description and analysis of the processes defined by the boundaries of change characterizing their variables and their frequencies. The coefficients related to the variables, their derivatives and integrals in these equations are called the parameters of the element. There are elements with lumped and distributed parameters. An element is called a lumped element if its parameters are not functions of spatial coordinates defining its geometrical dimensions. In turn, the lumped elements may be constant or variable. If the element is described by equations, which include spatial variables, it belongs to a class of elements with distributed parameters (for example, an electricity transmission line (a long line)).

1.3 Classification of electric circuits Like most systems, electric circuits can be distinguished and classified according to various criteria. The basic division of electric circuits is made depending on the type of the processed signals. Signals can be analog, discrete, and digital (Figure 1.5). An analog signal is a signal, continuous in time. A discrete signal is a signal, discrete in time and continuous in state. A digital signal is a signal that is used to represent data as a sequence of discrete values; at any given time it can only take one of a finite number of values. 10

Therefore we distinguish analog, digital, and digital circuits.

а) analog signal

b) discrete signal

c) digital signal Figure 1.5. Examples of electromagnetic waves

The TEC mainly considers analogue ECs. In turn, analog ECs are further classified by the following features: ‒ by the type of current: DC, AC sine wave, non-sinusoidal. ‒ by the number of phases: single-phase, three-phase. ‒ by the characteristics of elements: linear (in which all the elements are linear), nonlinear (contain at least one non-linear element); electric circuits with lumped and distributed parameters. 11

‒ by the method of connection with consumers: branched circuit, unbranched circuit. To produce quantitative calculations it is necessary to present the EC in the form of a mathematical model, called an electric scheme (ES). Electric scheme is a graphical design document, which using symbolic notation (SN) shows electric components of the EC and connection between them. All elements of EC in SN and ES are performed in accordance with generally accepted standards. ESs are classified according to four main groups (Figure 1.8).

by the type of signal being processed

by the type of current

EC

by the characteristics of elements by the number of phases by the method of connection with consumers

Figure 1.6. Classification of Ecs

Figure 1.7. Classification of analog circuits

12

Figure 1.8. Classification of electric schemes

Schemes of I group are for general guidance on electric components of the EC and study of the general principles of their work and relationships. It includes: structural and functional ES. These schemes are being developed in the design to pre-development schemes of other groups. Block diagram defines the components of the ECs, their functions and interactions. Functional diagram clarifies certain processes in certain functional parts (element, device or functional group having in the EC a well-defined functionality) or in the EC as a whole. Group II schemes are designed to determine the total composition and detailed study of the principles of work of the EC, as well as for its calculation. These schemes are the basis for the development of other design documents, in particular drawings, and diagrams of groups III and IV.They are used when setting up, adjusting, monitoring, maintenance and repair of electric devices. There are basic and equivalent circuits. The basic circuit defines the complete composition of elements and relations between them and gives a detailed presentation of the principles of the EC. All elements are represented with the real characteristics. Equivalent circuit is designed for the analysis and calculation of the parameters (characteristics) of the EC or its functional parts. 13

Moreover, this circuit consists of the elements and is called an idealized equivalent circuit. The real elements are replaced by close in functionality ideal elements. In the theory of electric circuits, we deal with schemes of group I and II. Figure 1.9 shows examples of the structural (a), the principal (b) and the equivalent circuit diagram (c). Rr Iin

Rout Iout

Еr Uin

Е

Rin

Rout

Uout

а)

RК С2

R1

С1 Rr Еr

R2

R

С

Z Uout

b)

С1

С2

transistor

h21I Uin

R

Сin rin

rout Сout RК СМ Z

Uout

c) Figure 1.9. Scheme of the RC – amplifier stage

Group III schemes are intended to provide information on the electrical connections or components of the EC circuit as a whole. They are divided into: the wiring diagram; circuit connections and 14

the general scheme. These circuits are used in the development of design documents, in particular drawings, determine the gasket and methods of attachment of wires, harnesses, cables in place, as well as for connections and commissioning, monitoring, maintenance facilities. Wiring diagram shows the electrical connections of individual components of the circuit and determines wire harnesses and cables for these connections, as well as place of their connection and input (terminals, connectors, bushings, etc.). Wiring diagram shows the connection of external circuits. General scheme determines the complex components and electrical connections between them on site. Group IV circuits are designed to determine the relative position of the EC and its constituent parts. The main types of this group are the electric circuit and the circuit of power supply and communication. They are used in the development of other design documents, as well as in the manufacture and operation. Electric wiring diagram on the plans determines the relative positions of components of the EC in buildings. The scheme of power supply and communication determines the relative positions of components of the EC on the ground. When constructing electric circuits the following concepts are used: node, branch and loop. A branch is part of electric circuit comprising one or more series-connected receivers and sources of electric energy, the current in which is the same. The branches containing sources of energy are called active. The branches that contain receivers of energy are called passive. The number of branches in the schemes is represented by n. node branch h

a

R2

R7

I2

E2

I

I7 II

R3

I3

I1

b

E3

Е1

R

I4 I6

R III

IV

loop

I5

R

R

c

Figure 1.10. Example of a circuit

15

Node is the connection of three or more branches. Consequently, the branch is a circuit part between two nodes, through which only one current flows. The number of nodes in the schemes is represented by m. Loop is any closed path formed by the branches of the circuit. The circuit may comprise one loop, as well as many loops. There are main and independent loops in the schemes. The main loop is the loop, which does not contain other loops. The loop can be independent from the current source, while the number of such loops can be determined as follows: k  n  m  ncs  1 , where ncs is the number of branches comprising current sources. Control questions 1. Define an electric circuit, branch, node and loop. 2. Explain the physical meaning of physical quantities in the theory of electric circuits. 3. Tell us about the elements of the electric circuits. 4. Tell us about the various schemes of electric circuits.

16

Lecture 2

ELECTRIC COMPONENTS

Plan 2.1 Idealized elements of the electric circuit 2.2 Real elements of the electric circuit

2.1 Idealized elements of the EC All elements of the electric circuits can be represented as the ideal (idealized) elements. The ideal (idealized) elements take into account the phenomena occurring in the actual electric circuit.

2.1.1 Sources of electric energy There are two types of ideal sources of electric energy: the ideal voltage source (EMF) and the ideal current source (Figure 2.1).

Figure 2.1. The difference between energy sources

17

An ideal voltage source (VS) is a two-terminal element – a source of energy, the voltage at the terminals of which is not dependent on the electric current passing through it (Figure 2.2, a). The ideal internal resistance of VS is infinitely small, so the voltage at its terminals, when the load changes, does not change, only the current changes. An ideal current source (CS) is a a two-terminal element – an energy source, the current in which is independent of the voltage at its terminals (Figure 2.2, b). The ideal internal resistance of CS is infinite, so the change in load does not change the CS current, only changes the voltage at its terminals.

а)

b)

Figure 2.2. Circuit shematic symbols of ideal sources

а)

b)

Figure 2.3. Circuit shematic symbols of real sources

The values of the internal resistances are taken into account in their symbols: shorted in a perfect VS circle and break – in an ideal CS. However, there are no ideal devices. Each source has a finite resistance value, it is little for VS, and large for CS (Figure 2.3). 18

IVCs of ideal and the real source of voltage (EMF) and current are shown in Figure 2.4. From Figure 2.4 (a), it is evident that the smaller the internal resistance Ri of the source, the greater the short circuit current and the more the VS power (EMF source). From Figure 2.4 (b), it can be seen that by increasing the resistance of the circuit connected to an ideal current source, the voltage at its terminals and, accordingly, force developed by them, also increases indefinitely. Hence, the greater the power, the greater the voltage of open terminals and the greater the capacity of the power source. IVC of real sources intersects both axes and these intersection points correspond to zero current through the source and zero voltage drop. The mode with zero current and non-zero voltage drop is called open circuit, and the mode with zero voltage drop and a non-zero output current – short circuit.

а)

b) Figure 2.4. IVC of sources

If the voltage at the terminals of energy sources and the setting current are only determined by the intrinsic properties of the sources and do not depend on external influences, such sources are called independent sources. A dependent VS or CS is a four-pole element with two pairs of terminals: two inputs (1 – 1 ') and two outputs (2 – 2'). In this case the control quantities are input current I1 and U1 voltage. There are four types of dependent sources: a voltage controlled voltage source (VCVS);a current controlled voltage source (CCVS); a voltage controlled current source (VCCS); a current controlled current source (CCCS). 19

Figure 2.5 shows a VCVS scheme, its input impedance is infinite, and the input current I1 = 0. The output voltage is connected with the input voltage by the equation U2 = КUU1, where КU is the voltage transmission coefficient. This scheme is an ideal voltage amplifier. Figure 2.6 shows a CCVS circuit, its input current I1 controls the output voltage U2, input conductivity is infinite: U1 = 0, U2 = rI1, where r is the coefficient of proportionality, which has the dimension of resistance.

Figure 2.5. VCVS scheme

Figure 2.6. CCVS scheme

Figure 2.7 (a) shows a VCCS diagram, its output current I2 is controlled by the input voltage U1, and I1 = 0 and the current I2 is connected with U1 by the equation I2 = gU1, where g is the coefficient having conductivity dimension. CCCS diagram is shown in Figure 2.7 (b), there is a controlling input current I1, and controllable output current I2, U1 = 0, I2 = βI1, where β is a dimensionless coefficient of current transfer. This scheme is an ideal current amplifier.

а)

b)

Figure 2.7. Schemes:VCCS (а), CCCS (b)

20

2.1.2 Electric energy receivers There are three idealized receiving elements in the EC: a resistive element, an inductive element and a capacitive element (Figure 2.8). R а)

L b)

C c)

Figure 2.8. Circuit shematic symbol of idealized passive elements

The resistive element is an idealized two-pole element of EC, characterizing the conversion of electromagnetic energy into any other form of energy, or it has only a property of irreversible energy dissipation (Figure 2.8, а). Features of the element: 1) The current flowing through the element is directed from high to low potential. 2) As a result of the current flow, a voltage drop (reduction in the potential) is observed. The voltage drop is positive when it is directed from high to low potential. 3) It has a resistance R, which is determined by the resistivity  of the material, length l and cross-sectionS of the conductor:

R

l . S

(2.1)

It is measured in Ohms (Ω), there may be values in kOhm (10 Ohm), МОhm (106Оhm). Graphically, the resistance is determined by the current-voltage characteristic (Figure 2.10), the slope of the current change at voltage change: 3

tg 

I 1.  U R

(2.2)

The inverse value of resistance is called conductivity: G

1. R

(2.3) 21

It is measured in Siemens (S). 4) The resistive element is a delayless element, since a change in the current and voltage on the element occurs without delay. This element is used as a load in active devices, in filtering circuits, as an additional element of timer (setting time constant), for redistributing of a potential level in the schemes, etc. The actual resistive element is called a resistor.

Figure 2.10. IVC of a linear resistive element

Figure 2.9

Inductive element is an idealized two-pole EC element, characterizing the magnetic field energy stored in the circuit (Figure 2.8, b). Features of the element: 1) According to the physical nature of the element it is dynamic. The voltage drop across it occurs only when the current flowing through it changes: di  0  uL  0 . In the static mode, when a

dt

DC is applied, there will be no voltage drop at the element. 2) The element is inertial, because when the voltage is applied, the current through it cannot change step-like: di    uL   .

dt

Figure 2.11

3) The numerical parameter characterizing the element is inductance, which determines the amount of magnetic flux generated by the current flowing: 22

L

, i

(2.4)

where   Bds , measured in Weber (Wb), B is magnetic induc S

tion in tesla (T), s is the square. Inductance is measured in Henry (H), may be in mH (10-3 H), μH (10-6 H). 4) The rate of change of magnetic flux determines the amount of voltage across the inductor

uL 

d , dt

uL  L

di . dt

(2.5)

Consequently, the inductance is a measure of the inertia of the element relative to the changes in the flowing current. Inductive elements are of limited use and are used in the filter elements, as well as elements of the high-frequency «decoupling». A real inductive element is the inductance coil. The capacitive element is an idealized two-pole EC element characterizing the energy of the electric field accumulated in the circuit (Figure 2.8,c). Features of the element: 1) The element is accumulator of electric energy. When the potential difference U  2  1 is applied to two electrodes, equal in magnitude and opposite in sign charges +q are accumulated on them. An electric field is created between separated charges. The magnitude of the flux  E of electric induction D through a closed surface S is defined by the charge enclosed within the volume bounded by the surface of the  E   Dds  q . 2) The numerical parameter characterizing the element is the electric capacity, which determines the amount of electric induction flux generated by this potential difference:

C

E q .  U U

(2.6)

23

Capacity is measured in farads (F), a possible value is mF (10F), µF (10-6 F), nF (10-9 F), pF (10-12 F). 3)The capacitive element by its physical nature is dynamic, because it responds to changes in the applied voltage. The current in the capacitive element is a consequence of changes in the electric charge on it: 3

iC 

dq du C . dt dt

(2.7)

The current through the element appears only when the applied voltage du  0  iC  0 . In the static mode, at a constant voltage,

dt

the current is absent.

Figure 2.12

4) The element is inertial since the voltage across it cannot be changed very rapidly: du    iC   .

dt

The real capacitive element is called a capacitor.

2.2 Real elements of the electric circuit Consider the real passive components of the EC. Resistor in reality, in addition to its main parameter – resistance, also has parasitic characteristics – capacitance and inductance. Capacitance arises between parts of adjacent resistors and other circuit elements and the inductance of the resistor is its ability to store energy of a magnetic field, for example, in its terminals or pins. Classification of resistors is shown in Figure 2.13. 24

Figure 2.13. Classification of resistors

Figure 2.14

The main parameters of resistors: 1) Nominal resistance is indicated in accordance with the standard series of resistors - En : E 6, E12, E 24, E 48, E 96, E192. Series E corresponds to a geometric progression with qn  n 10 . The value of nominal resistance is indicated by multiplication of the value of the number in the series by the ordinal value 10m , where m = 0, 1, 2, ... . 25

If E 6  q6  6 10  1,47 , then the series consists of values: 1,0; 1,5; 2,2; 3,3; 4,7; 6,8 etc. The magnitude of the error, or allowed deviation of the resistor from the nominal value is equal to:  R % . Possible error is: 2%, R 5%, 10%, 20%. 3) Rated power dissipation, the parameter defines the size of the resistor, is the maximum power that the resistor can dissipate without changing its settings. 4) Temperature coefficient of resistance, determining the change in the resistance of the resistor when the temperature changes by 1 degree, equals TCR 

R % .  100 R0 t grad

(2.8)

The technical documentation setting the nominal resistor values certainly contains its basic parameters, such as 0.25W – 100 kOhm + 2%. Typically, on the resistor body coded or color-coded designation is used. Colour coding is applied to the small-size resistors in several color stripes (Figure 2.15). A coded designation of the nominal resistance value contains numbers and letters. The letter corresponds to the decimal multiplier, which is multiplied by еру digital signage (Table 2.1). After the nominal there is the letter indicating the deviation in % (Table 2.2). Table 2.1 Letter Multiplier

R 1

K 103

M 106

G 109

T 1012 Table 2.2

Deviation Letter Deviation Letter

26

+ 0,001 + 0,002 + 0,005 + 0,01 E L R P + 0,1 + 0,2 + 0,5 + 1 + 2 B C D F G

+ 0,02 + 0,05 U X +5 + 10 + 20 I K M

+ 30 N

Figure 2.15. Colour code of resistors

Capacitor is a real capacitive element, which also has parasitic quantities: inductance and resistance. Classification of capacitors is shown in Figure 2.16.

Figure 2.16. Classification of Capacitors

27

The main parameters of the capacitors: 1) The nominal capacity of the capacitor, similar to the resistor may have standard values. 2) The permissible deviation from the nominal value: C  C0 100%. C0

2) Temperature coefficient of capacitance TCC 

C % 100 . C0 t град

(2.9)

3) Rated operating voltage is the maximum voltage at which the capacitor preserves the settings for a specified operating time. There are a number of additional parameters: insulation resistance and dissipation factor. Insulation resistance is the resistance of the capacitor to DC. The angle of dielectric loss or dissipation factor (Q-factor) is the angle complementary to the phase angle between current and voltage, during the flow of alternating current through a capacitor. Nominal capacity is marked with numbers and a letter, which defines a decimal factor (Table 2.3). When setting the nominal capacitance, capacity, tolerance and TCC must be specified, for example 100pF + 10%М. Table 2.3 Dimension

Farad, F

Letter Multiplier

F 1

Millifarad, mF m 10-3

Microfarad, µF µ 10-6

Nanofarad, nF n 109-9

Picofarad, pF p 10-12

Nominal voltage and coded designations are given in Table 2.4. Table 2.4 Nominal voltage, V 1 1,0 1,6 2,5

28

Code 2 I P M

Nominal voltage, V 3 20 32 40

Code 4 F H S

Nominal voltage, V 5 160 200 250

Code 6 Q Z W

1 3,2 4,0 6,3 10 16

2 A C B D E

3 50 63 80 100 125

4 J K L N P

5 315 350 400 450 500

6 X T Y U V

The technical documentation setting the nominal capacitor values must contain its basic parameters, on the case coded symbol or color-coding is used (Figure 2.17).

Figure 2.17. Color codes of capacitors

29

The designation of capacitors, which consists of letters and numbers, can be abbreviated or full. The first element is a letter or a combination of letters denoting a subclass of the condenser: C – constant capacitance, CT – Trimmers, CV – variable capacity The second element designates the capacitor group depending on the dielectric material. The third element is written with a hyphen and designates the registration number of a specific type of capacitor (the following letters can be used: DC – disc capacitors, SC – solid ceramic, CAS – ceramic alloy, sectional, etc.). Parameters and characteristics included in the full symbol, are indicated in the following sequence: 1. Description of embodiment. 2. Rated voltage. 3. Nominal capacity. 4. Tolerance capacity. 5. The group and class on temperature stability of capacitor. 6. Nominal reactive power. 7. Other required additional features. Control questions 1. Describe the main characteristics of the idealized sources of energy of the electric circuit. 2. Describe the main characteristics of idealized and real passive elements of an electric circuit.

30

Lecture 3

DC LINEAR ELECTRIC CIRCUITS. BASIC LAWS

Plan 3.1 Ohm's Law 3.2 Kirchhoff's Laws. Stages of application 3.3 Balance of power 3.4 Equivalent network reduction

3.1 Ohm's Law In 1820-1825, the German physicist Georg Simons Ohm, studying the phenomenon of conduction in materials, formulated one of the basic laws in the TEC, later named in his honor Ohm's law. This law shows the dependence of the current on the voltage both in the section of the circuit by the known potential difference, and in the full circuit through the known value of the EMF. The geometric interpretation of Ohm's law is a current-voltage characteristic of the conductor of electric current. To write the law, we assume that the positive direction of voltage U ab coincides with the positive direction of the current. Then Ohm's law for subcircuit (Figure 3.1, a):

I ab 

U ab a  b ,  R R

(3.1)

where  a , b are potentials of the nodes a and b. In general terms, the denominator is the sum of all resistances in the considered circuit path. Thus, Ohm's law states that the current in the subcircuit is directly proportional to the voltage drop across this area and inversely proportional to the resistance of the subcircuit. 31

For the full circuit (Figure 3.1, b), containing a power source, Ohm's law is written as follows:

I ab 

U ab  E a  b  E ,  Rr Rr

(3.2)

where E is the EMF source, r is its internal resistance. In general (3.2) has the form:

I ab 

U ab   E

 R  r 



a  b   E .

 R  r 

(3.3)

The sign «  » before EMF depends on the direction of EMF with respect to the current.

а)

b) Figure 3.1

3.2 Kirchhoff's Laws. Stages of application In 1845, Gustav Robert Kirchhoff, being a student, on the basis of his experiments wrote a scientific paper, in the notes to which he formulated the two laws, which became the main fundamental laws of electric circuits. There are two laws: Kirchhoff's law #1 describes the current balance in the branchings, Kirchhoff's law #2 – voltage balance in the closed parts of the circuit. Processes in all electrical circuits, both with constant and alternating currents and voltages, obey these laws. Kirchhoff's law #1 (Kirchhoff's current law): 1 interpretation: the algebraic sum of all currents that converge at the node is equal to zero: 32

I

k

 0.

(3.4)

k

Current signs are chosen as follows: «  » – if the current is directed to the node, and «  » – if the current is directed in the opposite direction (from node). 2 interpretation: the algebraic sum of the currents flowing to a node is equal to the algebraic sum of the currents flowing from the node). According to Kirchhoff's law #1, m  1 equations are written, where m is the number of nodes in the EC. Kirchhoff's law #2 (Kirchhoff's voltage law): The algebraic sum of all the voltages around a closed circuit equals zero. 1 interpretation: in any closed loop in a circuit, the algebraic sum of the voltage drops (i.e. the product of current and resistance) taken arround the loop is equal to the algebraic sum of all the EMFs acting in that loop):

I R  E k

l

k ,l

m

.

(3.5)

m

In order to make the equation of this law properly it is necessary to set the direction of bypass circuit. Accordingly, the marks are recorded by summing the voltage drop across elements: «  » – if the direction of the current flowing through the element, coincides with the direction of the bypass circuit, and «  » – if the current is directed in the opposite direction. It is similar with the signs of EMF: «  » for coinciding directions and «  » for opposite directions. 2 interpretation: the algebraic sum of the potential difference (voltage) of any closed part of the circuit (loop) is zero:

U

k

0,

(3.6)

k

where U  2  1 . According to Kirchhoff's law #2 the number of equations is equal to the number of independent loops in the EC: k  n  m  ncs  1 , where n is the number of branches, m is the number of nodes, ncs is the number of branches containing current sources. 33

3.3 Balance of power Considering energy in linear electric circuits it is neсessary to mention that the energy of electric current flowing through the resistances is converted into heat. According to the law of conservation and transformation of energy: energy generated by sources is completely consumed by receivers. On this basis, you can create the so called balance of power for each circuit: the algebraic sum of the source powers must be equal to the arithmetic sum of the powers of the loads: n

m

l

k 1

k 1

k 1

 I k2 Rk   Ek I k  U ab J k ,

(3.7)

where U ab J k is the power delivered to the circuit by a current source ( a is the node, to which the current flows J k , b is the node from which this current flows). If the direction of the source of EMF and the direction of the current flowing to the same branch are directed opposite to each other, the power component of this source in the power balance equation is taken with the «minus» sign, otherwise the sign «plus» is used.

3.4 The equivalent conversion of circuits In solving the problems in TEC it is often necessary to simplify the electrical circuits, i.e., there is a need to replace the complex parts of the EC with simpler equivalent parts. This equivalent conversion of electrical circuits is based on the principle of equivalence, which can be expressed in the following definitions: ‒ The currents in a converted part of the circuit remain unchanged; ‒ The potentials of nodes in which the converted part of the circuit is connected to untransformable part, remain unchanged; ‒ Power of transformed circuit must be equal to the power of the circuit before its transformation. 34

3.4.1 Converting passive elements of circuit All elements in EC circuits may be interconnected in series, parallel and mixed. In a series connection of several elements, the end of the first element is connected to the beginning of the second, and the end of the second – to the beginning of the third, etc. In this case, the current flowing through all elements is the same. The voltage on this part of the circuit will be the sum of voltages on each of the series-connected elements. When changing the parameters of one of them, at the same time entails a change in the voltage at the other connected elements. When you turn off or break the EC in one of the elements, the current stops in all other elements. Because of this dependence, this method of connecting elements is used rarely, with the exception of the case where the voltage of the electrical power source is greater than the nominal voltage for which the load is designed. In a series connection of resistors (Figure 3.2), the equivalent resistance is determined as follows: N

Req   Rn .

(3.8)

n

As the resistance is connected with conduction by the ratio: 1 , the (3.8) takes the form R G N 1 1 . (3.9)  Geq n Gn Thus it is possible to determine the equivalent conductance of the circuit.

Figure 3.2

35

In a series connection of inductors (Figure 3.3) equivalent inductance is determined as follows: N

Leq   Ln

(3.10)

n

Figure 3.3

In a series connection of capacitors, (Figure 3.3) equivalent capacitance is determined as follows: N 1 1  Ceq n Cn

(3.11)

Figure 3.4

In parallel connection of several elements, they are included between two nodes of EC forming parallel branches. The current in the unbranched part of the EC in accordance with the first Kirchhoff's law is equal to the algebraic sum of the currents in branched EC. When writing equations, take into account the directions of the currents in the branches. All elements are at the same voltage, and the operation of each is independent of the other elements. The current flowing through any of the branches do not have a significant influence onthe other elements in the other parallel branches. Any shutdown or failure of any element, does not affect other elements. Therefore, parallel connec36

tion, having substantial advantages over a serial connection, is widespread. For example, the devices designed to work at a certain (nominal) voltage are only connected in parallel. When resistors are connected in parallel (Figure 3.5), the equivalent resistance is determined as follows: N 1 1 .  Req n Rn

(3.12)

In the case of finding the equivalent conductivity of the elements, formula (3.12) takes the form N

Geq   Gn .

(3.13)

n

Figure 3.5

In parallel connection of inductors (Figure 3.6), equivalent inductance is determined as follows: N 1 1  Leq n Ln

(3.14)

Figure 3.6

37

In parallel connection of capacitors (Figure 3.7) equivalent capacitance is determined as follows: N

Ceq   Cn

(3.15)

n

Figure 3.7

Mixed connection is such type of connection, in which some of the elements are connected in series, and another part – in parallel. The equivalent circuit parameters for this kind of connections are usually determined by conversion method in which a complex EC by successive transformations is replaced by a simpler one. For example, for EC shown in Figure 3.8 the equivalent resistance will be equal to:

  R4  R5    R3  R6   R2    R  R5  . Req  R1    4   R4  R5    R3  R6   R2      R4  R5  

Figure 3.8

In complex EC the types of connection such as the «triangle» and «star» are often used. 38

а)

b)

Figure 3.9

Connection «star» is the connection of the three branches having a common node (Figure 3.9 a). Connection «triangle» is a combination of three branches, which has the form of a triangle whose sides are the branches and vertices – nodes (Figure 3.9, b). When there are such connections in diagrams, most often, to simplify calculations, it is required to convert one type of connection into another. With such a transformation, the potentials of the nodes and the power in these circuits must remain unchanged. When you convert a scheme of «star» into «triangle»: ‒ The resistance of the side of the «triangle» is equal to the sum of the resistances of the adjacent rays of the «star» and their product, divided by the resistance of the third ray 39

R1R2 ; R3 RR R23  R2  R3  2 3 ; R1 R3 R1 R31  R3  R1  . R2

R12  R1  R2 

(3.16)

‒ for conductivities

G1G2 ; G1  G2  G3 G2G3 G23  ; G1  G2  G3 G3G1 G31  . G1  G2  G3

G12 

(3.17)

When you convert a scheme from the «triangle» into the «star»: ‒ the resistance of the «star» ray is equal to the product of the resistances of the adjacent sides of the «triangle» divided by the sum of resistances of the three sides of the «triangle

R12 R31 ; R12  R23  R31 R23 R12 R2  ; R12  R23  R31 R31R23 R3  . R12  R23  R31

R1 

(3.18)

‒ For conductivities

G12G31 ; G23 G G G2  G12  G23  12 23 ; G31 G23G31 G3  G23  G31  . G12

G1  G12  G31 

40

(3.19)

In the particular case, when R1  R2  R3  RY и R12  R23  R31  R , equations (3.16) and (3.18) are written down in the form:

1 RY  R , 3

(3.20)

R  3RY .

(3.21)

3.4.2 Conversion of circuit active elements Consider the circuit shown in Figure 3.10.

J

R

Е

U

а)

U

b) Figure 3.10

For the series circuit formed by the voltage source and resistor the equality U  IR  E , s valid. Then current

I 

U E U     J eq . R R R

This equation corresponds to the circuit formed by the parallel connection of the current source J eq  E and resistor R (Figure R 3.10, b). Because currents and voltages at the terminals of the external circuits are the same, they are equivalent. A reverse transforma41

tion of a voltage source into an equivalent current source, i.e. Eeq  JR. is also possible. Control questions 1. Formulate Ohm's law for different segments of the circuit. 2. Formulate the laws of Kirchhoff and stages of their use in the calculation of electrical circuits. 3. Formulate the balance of power in the electrical circuits. 4. Explain the principle of equivalent conversion of circuits.

42

Lecture 4

THE TOPOLOGY OF ELECTRIC CIRCUITS

Plan 4.1 Basic concepts of circuit topology 4.2 Subgraphs 4.3 Topological matrices

4.1 Basic concepts of circuit topology The mathematical description of processes in electrical circuits (EC) is based on the equations of two types: component and topological. The component equations (or equations of branches) establish the relationship between current and voltage in each branch. Since the topology shows the geometrical structure of EC, the topological equations reflect the circuit properties, which are defined only by its topology and do not depend on any electrical components included in the branches. For writing the component equations Ohm's law is used, and for topological equations – Kirchhoff's laws. The main objectives of the topology of circuits are:  Determination of the number of independent nodes and loops;  Allocation of systems of corresponding nodes and loops. A system of independent nodes and a system of independent loops are a collection of nodes and loops, for which it is possiible to make a system of linearly independent equations using Kirchhoff's laws. The basic concept in the topology of electric circuits is a graph. Graph is a graphical representation of EC, in which all branches are replaced by lines. In other words, the graph is a geometric structure of the circuit, which is represented by a collection of segments of any length and shape, called branches (ribs) and their connection points called nodes (vertices) (Figure 4.1) 43

Figure 4.1. Electric circuit and its corresponding graph

Graphs are divided into two types: topological and directed graph of the signals. The topological graph is a simplified electrical circuit model reflecting only its topological (structural) properties. Directed graph of the signals or the signal graph is a graphical representation of the system of equations describing the processes in the electrical circuit. We consider only topological graphs. The topological graph (hereinafter referred to as the «graph») is build on its equivalent circuit. If in constructing graphs, the arrow is used to show the direction of each branch, such graphs are called directed or focused. To show the directions of branches the real direction of currents in these branches is usually choosen. Graph properties do not depend on the shape and length of the branches, as well as on the relative position of the graph nodes on the plane, and are defined only by the number of branches n, number of nodes m and the type of connection of branches with each other. Graphs, having the same number of nodes and branches interconnected in the same way are called isomorphic (Figure 4.2).

Figure 4.2. Isomorphic graphs

44

By varying the length and shape of the branches, as well as the relative position of nodes in the graph on the plane, it is possible to obtain an infinite number of graphs which are isomorphic to the original one. Such transformations are called isomorphic graph transformations. Each of the options of the graph image obtained by such transformations is called its geometric realization If a node i is the end or the beginning of a branch j, it is said that they are (i and j) incident. Thus, any branch is incident to two nodes. The degree of a node is the number of branches of the graph incident to a given node. The graphs isomorphic up to a second-degree nodes, are called homeomorphic. After removal from homeomorphic graphs nodes of the second degree, and the unification of branches incident to these vertices, homeomorphic graphs become isomorphic. Graphs corresponding to expanded and contracted topological description of the EC are called homeomorphic. When depicted, graphs can be planar and volumetric, in accordance with this, they are classified into planar and non-planar. Planar graph is a graph, which as a result of isomorphic transformations can be depicted on a plane without crossing branches. All graphs containing no more than 4 nodes are planar graphs. Nonplanar graph is a graph which cannot be represented on a plane without crossing branches. Examples of such graphs are shown in Figure 4.3. An arbitrary graph is planar if it contains no parts that are homeomorphic to one of the Pontryagin-Kuratowski graphs. Planar graph divides the plane in which it is depicted into the outer and inner regions. Inner regions, limited by the branches of the graph, are called cells or graph windows. The outer part of the plane, in relation to the graph, is called the basal cell. Graph properties do not depend on the shape and length of the branches, as well as the mutual arrangement of nodes on the graph, and are only determined by the number of branches n, number of nodes m and by the method of connection of branches. 45

a) Full pentagon

b) A bipartite graph

Figure 4.3. Pontryagin-Kuratowski’s graphs

4.2 Subgraphs Any part of the graph, the elements of which are elements of the original graph is called a subgraph. Subgraph can be obtained by removing the separate branches of the original graph. One branch or a node in the graph, as well as any set of branches and nodes contained in this graph can be a subgraph. Such subgraphs as the path, the loop, tree, branch connection and cross section of the graph are very important in the theory of electrical circuits. Path is an ordered sequence of branches wherein each two neighboring branches have a common node, each branch and each node on this path meet only once. The nodes connected by a graph are called terminal. A closed path in which the initial and the end terminal nodes are the same is called a loop. Two branches of the loop are incident to each of the circuit nodes. If any two nodes of the graph are connected by a path, the graph is called connected (linked). Otherwise, the graph is called unconnected. A connected subgraph containing all the nodes of the graph, but not containing any loops, is called the tree of the graph. The branches of the graph included in the tree are called the branches of the tree. The branches not included in the tree are called the branches of communication (bonds) or chords of that tree. The branches of the tree complete the connection of the full graph. If m is the number of nodes, n is the number of branches, the number of branches of the 46

tree is   m  1, and the number of connection branches is c  n  (m  1)  n  m  1 . The branches of the tree are drawn by the bold lines and the branches of communication – by thin lines. For each graph, you can form a few trees, differing in composition of tree branches (Figure 4.4). 1

2

3 6

5

4 7

8

Figure 4.4. Subgraphs of electrical circuit: A tree (lines) and a communication branch (dashed line)

To build a tree it is necessary to identify nodes of the graph. Then, starting with a node, called the root, plot tree branches connecting the other nodes of the graph, while avoiding the formation of loops. Addition of any branch to the tree graph forms a loop connection. Loops, formed by alternate addition of its communication branches to the tree graph, are called the main loops. The number of the main loops is equal to the number of communication branches. Numbers of the main loops are denoted by Roman numerals. For the direction of the loop around the branch take the direction of the communication loop. The cross-section of the graph is a system or a set of branches, removal of which splits the graph into two disjoint subgraphs, each of which is connected. In a particular case, one of these subgraphs may be an isolated node. The cross-section can be represented in the form of a closed surface of a path, which covers part of the cicuit with one or more nodes. None of the branches of the graph is intersected twice. The main cross-section of the graph is a cross-section, which includes only one branch of the selected tree. The remaining bran47

ches included in the main section, are branches of communication. The number of the main sections is the same as the number of branches of the tree. For a positive direction of the main section take the direction of tree branches, included in this section and point it by the arrow. Figure 4.5 shows the main cross-sections S1, S2, S3 and S4 of the graph. 1

S1S3 2

3

4

1

4 2

6

7

5

S2

3

8

S4 0

Figure 4.5. The cross sections of the graph

4.3 Topological matrices For the analytical description of the circuit structure, its graph and the main current distribution laws, topological matrices are used. Analytical representation of the graph is needed for the formation of of equations for a complex circuit using mathematical computer programs. In accordance with the type of the Kirchhoff equations there are three topological matrices: a matrix of connections (nodal matrix) [А], loop matrix (main loops) [В] and a matrix of the main cross sections [Q]. Full description of the structure of a directed graph is given by m*  n the matrix connections of (nodal matrix) [А], where m*  m  1 ( m is the number of nodes). m* – rows of the matrix are the serial numbers of nodes, and n – columns are numbers of branches. It is a table of the coefficients composed by the first Kirchhoff's law for the circuit nodes. 48

Each element of the matrix can take the values 0, +1 and -1:  matrix element aij, located at the intersection of the i-th row and the j-th column, is equal to +1, if branch j of the graph is connected with node i and directed from the node.  matrix elementaij = –1, if branch j of the graph is connected with node i and directed to the node.  matrix element aij = 0, if branch j of the graph is not connected to node i. Matrix of the main sections [Q] is a table-factor coefficients composed by the first Kirchhoff's law for the main sections. Rows of matrix [Q] correspond to sections ( s  n  m  1 ), columns – to branches ( n ), i.e., the dimension of the matrix is s  n . Each element can have values 0, +1 and -1:  matrix element qij = +1, if branch j is contained in crosssection i and its orientation coincides with the orientation of the section, i.e. with the orientation of the corresponding branch of the tree with respect to the section line.  matrix element qij = –1, if branch j is contained in crosssection i and it is directed opposite to the direction of the section.  matrix element qij = 0, if branch j is not contained in crosssection i. The main loop matrix (contour matrix) [B] is a table of coefficients of equations written by the second Kirchhoff’s law. The rows of the matrix [B] correspond to the main loops ( k  n  m  1 ), columns – branches ( n ), i.e., the dimension of the matrix is k  n. Each element can have values 0, +1 and -1:  matrix element bij = +1, if branch j is contained in cross-section i and direction of branches coincides with the direction of the loop tracing.  matrix element bij = –1, if branch j is contained in cross-section i and its direction is opposite to the direction the loop tracing.  matrix element bij = 0, if branch j is not contained in loop i. Control questions 1. Identify the basic concepts of topology of circuits. 2. Describe the classification of the graphs. 3. Describe the different types of subgraphs. 4. Explain the principle of the topological matrices.

49

Lecture 5

METHODS OF LINEAR ELECTRIC CIRCUIT ANALYSIS

Plan 5.1 The method of direct application of Ohm's and Kirchhoff's laws 5.2 The mesh current method 5.3 The node voltage method. Method of two nodes 5.4 Kirchhoff's laws 5.5 The method of equivalent generator 5.6 Principles of compensation and reciprocity

The objects of calculation and analysis of electric circuits are very multifarious. The most common tasks can be divided into three categories: 1) Determination of current, voltage, power in various circuit elements for given parameters of these elements. 2) Determination of parameters of the components that provide desired currents, power, voltage. 3) Determination of the changing nature of different values or relations between them when the circuit parameters change.

5.1 The method of direct application of Ohm's and Kirchhoff's laws Ohm's Law is used to calculate the modes of individual sections of the circuit consisting of one or more resistors and EMF sources. However, in combination with equivalent transformations it may be used for more complex tasks. In particular, it can be used for determining the current in any branches of the two contour circuit or on a separate voltage element. Stages of application of Ohm's law: 1. Determine the number of nodes (m) and circuit branches (n). 50

2. Make the equivalent transformation of the scheme (taking into account the conditions of the problem). Give an arbitrary conditionally positive direction of the real currents in each branch, denote nodes. 3. Determine the equivalent circuit parameters. 4. Using Ohm's law calculate unknown quantities. Kirchhoff's laws are used in the calculation of any electric circuit of any complexity. Stages of application of Kirchhoff's laws: 1. Determine the number of nodes – m , branches – n and branches containing current sources – ncs . Using this data determine the number of independent circuits by the formula: k  n  m  ncs  1 . 2. Name each node (a, b, c,...) and loop ( I , II ,...). On the circuit diagram indicate an arbitrary conditionally positive direction of the real currents in all branches (I1, I2, ..., In) and the direction of loop tracing. 3. For nodes (m  1) write the equation of the first law of Kirchhoff. 4. For k loops write the equation of the second law of Kirchhoff. 5. Solve the equations of the system. We find the unknown true currents in the branches. There is a wide variety of complex electric circuits, which in principle can be calculated using Ohm's and Kirchhoff's laws. However, the solution in this case can be too bulky and time consuming. For this reason, for the analysis of complex electrical circuits more rational methods of calculation developed on the basis of Ohm's and Kirchhoff's laws are used. Which method to apply for the calculation of the considered circuit is determined by the given conditions and the structure of the circuit calculated.

5.2 The mesh current method The mesh current method (MCM ) is based on application of the Kirchhoff's voltage law. This method is most efficiently used in applications where there is a small number of independent loops. 51

Any complex circuit consists of several adjacent loops, each of which has non-adjacent branches belonging to this loop and adjacent branches belonging to the neighboring loops. According to the MCM it is assumed that in each of the circuit loops the loop current is the same for all elements. Loop currents act as unknown quantities to be determined, so for them equations are written using the second law of Kirchhoff. MCM allows us to reduce the problem of calculation of electrical circuit to solving a system of equations of a lower order. This simplifies the calculation and makes this method preferred over the method of direct application of Kirchhoff’s laws. Stages of MCM application: 1. Determine the number of nodes – m , branches – n and branches containing current sources – ncs . Using this data determine the number of independent circuits by the formula: k  n  m  ncs  1 . 2. Define a conditional positive direction of the real currents in each branch. Designate independent loop. 3. We assume that the same current flows through all the elements of a predetermined loop. Loop currents are denoted by I11 , I 22 , ..., I kk , where k corresponds to the loop number. Set an arbitrary conditionally positive direction of loop currents in the inde-pendent circuits. Loops are chosen so that they do not include branches with current sources. Branches with current sources J form their own loops with predetermined currents. 4. Make k  n  m  ncs  1 equations of the second Kirchhoff's laws for the selected loops with contour currents I11 , I 22 , ..., I kk . The equations take into account the voltage drop on loop elements, both due to its own loop current, and due to adjacent loop currents. 5. Solve the system of equations for the loop, determine the unknown loop currents I11 , I 22 , ..., I kk . 6. True currents in branches are determined as the algebraic sum of the loop currents flowing in the branches. 7. If necessary, one can determine the voltage on the individual elements, a source power and power of energy receivers. 52

5.3 The node potential method. Method of two nodes Method of node potentials. (MNP) is based on the application of the Kirchhoff's current law. This method is most efficiently used in applications where the number of nodes is less than the number of contours. In this method, a circuit node is defined as a reference (or base), whose potential is taken as equal to zero, and the potentials of the remaining ( m  1 ) nodes are considered unknowns to be determined. For circuits containing multiple branches with both active and passive elements of the reference node can be selected depending on the application conditions, or at random. For circuits containing several branches only with ideal sources of EMF (without passive elements) having a common node, the common node is taken as the reference node. Then, the potentials of nodes connected by these ideal sources of EMF without passive elements to the reference node, are ideal sources of EMF (+E, if an ideal source of EMF is directed away from the reference node and –E otherwise). MNP allows us to reduce the problem electric circuit calculation to the solution of a system of simple equations of lower degree. Stages of application of MNP: 1. Determine the number of nodes – m , branches – n . 2. Set an arbitrary conditional positive direction of the real currents in each branch. Designate the nodes. 3. One of the nodes is grounded, i.e., define it as a reference. Its potential is accepted as zero. 4. For the rest ( m  1 ) nodes write the equation of the Kirchhoff's current laws. 5. According to Ohm's law, write the equations for all the currents in the potentials nodes. 6. Substitute the equations obtained in paragraph 5 in the equation in paragraph 4. Solve the resulting system of equations for the potentials. 7. True branch currents are determined by the equations of paragraph 5 and paragraph 4. 8. If necessary, the voltage can be determined on the individual elements, a power source and power energy receivers. 53

In some cases it is convenient to combine paragraph 5 and paragraph 4 after pre-calculating nodal conductivity. In the calculations of branched circuits, there are cases when the analyzed circuit is formed by parallel connection of several branches, i.e., it is a diagram of two nodes. Method of two nodes (MTN) is a special case of nodal analysis. Analysis of these circuits is simplified, if we first determine the voltage between the nodes a and b, using the conversion of the voltage sources into equivalent current sources. Stages of application of MTN: 1. Determine the number of nodes – m , branches – n . 2. Set an arbitrary conditional positive direction of the real currents in each branch. Designate the nodes. 3. Determine the voltage between the nodes а and b:

 E j g j   Ji U ab 

j

 gn

i

,

n

1 where g  is branch conductivity ( g j is conductivity of branches R containing sources of EMF E j , а g n is conductivity of all branches of the scheme). In calculating the voltage U ab those terms that correspond to the sources, directed to the node а are written with a positive sign.

5.4 Superposition method The basis of this method (SM) is the principle of superposition. Superposition Principle (Theorem) states that the current in any branch of a complex circuit containing multiple sources is equal to the algebraic sum of the partial currents (generated by only one source of energy) arising in the branch (for this element) from the independent action of each source individually. Superposition principle can also be interpreted for voltage. 54

The principle of superposition is performed only for those physical quantities, which are described by linear equations, such as voltages and currents in linear circuits. Superposition principle does not hold for power, which is connected with the current by a non-linear equation P = I2R. The essence of the SM is that in a complex scheme with several sources, sequentially partial currents from each source are calculated individually. Calculation of partial currents is performed, usually by transformation of the scheme. Actual currents are determined by algebraic addition of partial currents, taking into account their directions. Stages of application of SM: 1. Determine the number of nodes – m , branches – n , the number of EMF sources nE and the number of current sources n J . Then determine the total amount of energy nE , J  nE  nJ . 2. Set an arbitrary conditional positive direction of the real currents in each branch. Number the energy sources. 3. Make nE , J subcircuits, each of which retains only one source of energy, whereas instead of other sources of energy only their internal resistances remain (in an ideal source of EMF internal resistance tends to zero – do not put anything, and an internal resistance of an ideal current source approaches infinity – branch is not taken into account, the break in the circuit). 4. Determine partial current in each subcircuit. 5. The current branch is the algebraic sum of the partial currents in subcircuits. We use plus for partial currents, which coincide with the direction of the desired current branch, and a minus, if they do not coincide. 6. If necessary, we can determine the voltage on the individual elements, a power source and power energy receivers. 5.5 Method of equivalent generator The method of current calculation in selected branches of complex circuitry based on the application of the theorem of equivalent generator, is called the method of equivalent voltage (current) generator (MEG) or open-circuit and short-circuit method. 55

The equivalent generator theorem is formulated as follows: with respect to the terminals of the selected branches or a single element, the other part of the complex scheme can be replaced by a) an equivalent voltage generator EMF ЕE , equal to the open circuit voltage at the terminals of the selected branch or element (ЕE = Uoc) and the internal resistance R0, equal to the input resistance of the circuit from the side of the selected branch or element (R0 = Rin); b) equivalent to a current generator with JE, equal to the short-circuit current at the terminals of the selected branch or element (JE = Isc), and an internal conductivity G0, equal to the input conductivity of the circuit from the side of the selected branch or element (G0 = Gin). Stages of MEG application: 1. Determine the number of nodes – m, branches – n. 2. Set an arbitrary conditional positive direction of the real currents in each branch. Designate the nodes. 3. Remove the selected branch from the complex scheme, carry out calculation of the remaining parts of a complex scheme by any method and determine the open circuit voltage between the points of connection of the selected branch. 4. Remove the selected branch from the complex scheme, shorten the circuit in the points of connection of the selected branch, carry out calculation of the remaining part of the complex scheme by any method and determine the short-circuit current in the short-circuited part between the connection points of the selected branch. 5. Remove the selected branch from the complex scheme, in the remaining part of the circuit remove all sources (sources of EMF E are shortened and current sources with J are removed from the circuit), the conversion is performed by convolution of the passive circuit with respect to the connection points of the selected branch and thus the input resistance is determined. 6. Compile the equivalent circuit with a voltage generator or current generator. 7. Perform the calculation of the equivalent circuit and find the desired current (using Ohm's law, or the method of two nodes). 56

5.6 Principles of compensation and reciprocity Principle of compensation (PC) is needed to simplify calculations in electrical circuits. There are PC voltage (PCV) and PC current (PCC). PCV states, that currents in all branches of the circuit will not change, if any part of the circuit is replaced by the source of EMF, the value of which is equal to the voltage at this part with oposside direction to this voltage. PCC states, that currents in all branches of the circuit will not change, if any branch of the electrical circuit is replaced by the current source, the value of which is equal to the current flowing in the branch, and coincides with its direction. Note that the EMF or current source that replaces subcircuit of the circuit depends on the subcircuit current. If you change the parameters of the other elements of the circuit, the current in subcircuit, in general, is changed, and so this source is a dependent (controlled) source of EMF, or current. The principle of reciprocity (PR) determines the relationship between the currents and voltages in the two branches of passive circuit when there are sources of different nature in these branches. PR states, that if the source of EMF E , included in the branch m, creates in branch n partial current I , the same source of EMF E, included in the branch m, causes in branch m the same partial current I .

EC

EC

Figure 5.1

In practical use of PR it is important to keep in mind mutual correspondence of directions of currents and EMF sources. Circuits for which the PR is fulfilled are called reversible, and if not fulfilled – irreversible (for example, nonlinear circuits). 57

Control questions 1. Explain the methodology of solving problems with the use of Ohm's and Kirchhoff's laws. 2. Explain the stages of the mesh current method. 3. Explain the stages of the node potential method. 4. Explain the stages of the method of two nodes. 5. Explain the stages of applying the superposition method. 6. Explain the stages of application the method of the equivalent generator. 7. Formulate the principles of compensation and reciprocity.

58

Lecture 6

SINGLE-PHASE ELECTRIC CIRCUITS OF SINUSOIDAL CURRENT. BASIC CONCEPTS AND LAWS

Plan 6.1 Sinusoidal current. Basic characteristics 6.2 Methods of presentation of sinusoidal quantities 6.3 Ohm's and Kirchhoff's laws in the complex form

6.1 Sinusoidal current. Basic characteristics A quantity that changes over time is called a variable. If the variables: voltages and currents of all branches of EC are periodic functions of time or remain constant, then EC mode is called a steady-state mode of the circuit, otherwise – an unsteady-state mode of the circuit. A waveform is said to be sinusoidal if it is a sine function. Any sinusoidal periodic function is described by three characteristics: instantaneous, average and effective values. Values of the function at any given moment are called instantaneous, they are defined in terms of three parameters: the amplitude (the maximum value of the function), the angular frequency (the number of complete oscillations per one cycle) and the initial phase (sine value shift from the origin). The instantaneous value of the sinusoidal current is defined by:

i  I m sin(t   i ) ,

(6.1)

where I m is the amplitude,   2f is the angular frequency (rad/s),

f is the frequency (Hz). The frequency is inversely proportional to 59

the period f  1 . The period is defined as the time interval per one T complete oscillation (s). The argument of the sine, i.e. (t  i ) is called the full phase,  i is the initial phase. The phase characterizes the oscillation condition (a numerical value) at a given moment. Instant values of variables and their parameters separately do not give information about the work done by the source of electrical energy or power dissipated or converted in its elements. They do not give an idea about the power circuit parameters. It requires a value comprising an evaluation parameter, i.e. the time. In the DC circuits such values are not required, as there EMF, voltages and currents are constant over time. When several sine values (e, u, i) are considered together, it is necessary to study the difference of their phase angles. The phase shift is the difference of the initial phases of the sinusoidal current and voltage of the circuit. It is determined by subtracting the initial phase of current from the initial phase of voltage    u  i . The phase angle  is an algebraic quantity. If voltage u leads current i or vice-versa the phase shift φ can be either positive or negative. We define the average value of a sinusoidal current as the area under the waveform over the time interval:

2 I cp  av T

TT/ 22

2  i (t ) dt  T 0

TT/2 2

 I m sin(  t ) dt

0



2Im



,

(6.2)

we find the average value of a sinusoidal current from the peak of 2  0.638 .



The effective value of a sinusoidal current is equal to the value of DC current, which will produce the same heating or electrodynamic effect over the time qual to the perod of sinusoidal current. The effective value is called the root mean square (rms) value and is given by:

I 60

1T 2 1T 2 I i ( t ) dt  I m sin 2 (t ) dt  m  0.707  I m , (6.3)   T0 T0 2

We find that effective value of a sinusoidal current is equal to 0.707. Im. The concept of effective value is very widely used in circuits of alternating current. Most measuring instruments are calibrated to display the effective value of the current. Technical data of electrotechnical devices are specified in effective values. The effective values are represented by capital letters without index, emphasizing similarity of these concepts to analogues on direct current.

6.2 Methods of presentation of sinusoidal values Several methods of representation of a sinusoidal quantity are known:  trigonometric functions;  time diagrams;  rotating vectors;  complex numbers. In general case a sinusoidal quantity is defined by the following instantaneous values: a  Am sin(t   a ) . However, even the equation describing a simple electromagnetic circuit may have such a complicated form, that it is impossible to describe generally the parameter which we are interested in. Therefore the analysis of circuits of variable current represents these functions as a type of temporal diagrams (Figure 6.1).

Figure 6.1. Timing diagram of current

According to temporal diagrams it is possible to see visually high-quality changes of quantity over time. However, no one of the 61

above described methods gives an opportunity to visually receive representation of the quantitative and phase ratios of the main quantities. In this case we use a vector, that shows transformation from trigonometric so algebraic expressions and quantitatively evaluates characteristics of sinusoidal quantity. In the third case arbitrary sinusoidal function of time a (t )  Am sin(t   a ) (Figure 6.2, b) corresponds to a projection to vector axis 0Y with the module equal Am , rotating on the plane X0Y under constant angular speed ω making an angle  a with an axis 0X (Figure 6.2, а). If we use this method to figure several vectors on the plane corresponding to the different sinusoidal functions with the same frequency, they will rotate jointly, without changing mutual position, which is defined only by an initial phase of these functions. Therefore in the analysis of circuits all functions have identical frequencies and are restricted only by amplitude and an initial phase. In this case the vectors figuring sinusoidal functions will be fixed (Figure 6.2, c).

а)

b)

c)

Figure 6.2. Rotating a vector

A set of different sine quantities varying over the time interval with the same frequency can be represented visually on one diagram as rotating vectors. This method of summing and subtraction of the vectors is much simpler than those of trigonometric functions, the method is very widespread. Let's consider the fourth case. We may replace geometrical operations on vectors by algebraic operations on complex numbers since it significantly improves the accuracy of the obtained results. 62

Any vector can be represented as a set of two coordinates on the plane: two projections on axes of the Cartesian system of coordinates, or the magnitude of amplitude and an angle with the axis taken from the beginning of a reference frame in polar system of coordinates. Both coordinates in both cases can be integrated in the form of complex numbers or, it is possible to construct the vector figuring sinusoidal function on the plane of complex numbers. In order to record complex numbers, algebraic, trigonometric and demonstrative forms are used. Usually complex values are designated by a «point» on top or the lower underlining. Rectagular (Cartesian) form of record. Any point on the complex plane or the vector, which is taken from the beginning of the origin of coordinates to this point, corresponds to a complex number

A m  p  jq ,

(6.4)

where p is a vector coordinate on an axis of real numbers, and q is a vector coordinate on an axis of imaginary numbers. Trigonometric form of record. Representing real and imaginary parts of a vector by its length and an angle with an axis of real numbers, we write:

A m  Am cos a  jAm sin  a . e

j a

(6.5)

Exponential form of record. Using Euler's formula  cos a  j sin a it is possible to transform trigonometric form:

A m  Am (cos a  j sin a )  Ame j . a

(6.6)

where an amplitude of the sinusoidal function is the module of the complex number, and the initial phase is an argument of a sine. In calculations the rectagular and exponential forms of recording for complex numbers are used: the first for execution of operations of summing, and the second is for multiplication, division and exponentation. Transition from the algebraic form to the exponential form: 63

Am  p2  q2 ,

(6.7)

q  p

(6.8)

 a  arctg   .

A multiplier of e j  cos  j sin is called an operator of turn and it is itself a unit vector and,  is an angle of turn between it and an real axis. The name of the operator is related to the fact that its multiplication by of any vector leads to a turn of that vector to an angle  . We consider the real and imaginary numbers of 1, j, –1, –j as operators of turn 1  e j 0 ;   j  (6.9) 2;  j e   j  1  e ;    j  e  j 2 .  A complex number A m , whose magnitude is equal to the amplitude of a sinusoidal function, is called a complex amplitude. An amplitude and an effective value of the function are related through a 1 constant  0.707 , therefore we calculate effective values using 2 A complex numbers with the magnitude of A  m . This number is 2 called a complex effective value or just a phasor.

6.3 Ohm's and Kirchhoff's laws in the complex form The law of Ohm for different sections of a circuit a) Let's consider a circuit section which contains a resistive element with a resistance R through which a sinusoidal current flows 64

iR  I mR sin( t   i ) . R

According to the law of Ohm iR  element is

uR , the voltage across an R

u R  I mR R sin(t   i ) . R

It mmeans that in case of the sinusoidal current the voltage across a resistive element changes under the sinusoidal law too. Let's compare this expression with u R  U mR sin(t   u ) , then we obR

tain a relationship between the amplitude values

U mR  I mR R and initial phases

u i . R

(6.10)

R

It means that if the time-variyng current flows through a resistive element, there is no phase shift between current and voltage. Let's write the current and the voltage in a complex form: j j , IR  I mRe , U R  U mRe uR

iR

then Ohm’s law for a circuit section:

U R  IR  R .

(6.11)

This expression defines falling of voltage on a resistive element. b) Let's consider a circuit section which contains a capacitive element with the capacitance c. It is used as an energy source and voltage across it varies by the following law

uC  U mC sin(t   u ) . C

65

The current passing through a capacitive element iC  q  uC C . Then

dq , where dt

iC  U mCC cos(t   u ) . C

That is, at a sinusoidal voltage current on a capacitive element also changes by the sinusoidal law. If we transform expression using the function of a sine, we can write



iC  U mCC sin(t   u  ) . 2 C

Let's compare this expression with iC  I mC sin(t   i ) , then C

we obtain a ratio of initial phases

i u  C

C



(6.12)

2

and connection between amplitude values I mC  U mC C ,

1 is the capacitive resistance. C From this it is possible to make a conclusion that in the section of the circuit containing a capacitive element, when an alternating current flows, there will be a phase shift between current and voltage. Voltage on the capacitive element lags behind a phase current on an angle equal  . 2 Let's write all values in a complex form: where X C 

j , U C  U mC e uC

1 1 X C  j   jX C , C jC 66

(6.13)

then the law of Ohm for a circuit section: IC  I mC e j

U C  IC  X C   jX C  IC .

iC

(6.14)

This expression defines a voltage drop on a capacitive element. c) Let's consider a circuit section containing the inductive element with inductivity through which the current flows

i L  I mL sin(t   i ) . L

The voltage on the element is equal to

u L  eL , where eL   L

diL is self-induction EMF. Then dt u L  I mLL cos( t   i ) . L

i.e. at a sinusoidal voltage the voltage on the inductive element also changes by the sinusoidal law. If we transform the expression through the function of a sine, we obtain



uL  I mLL sin(t   i  ) . 2 L

Let's compare this expression with u L  U mL sin( t   u ) , L

then we obtain a ratio of initial phases

 u  i  L

L

 2

(6.15)

and connection between amplitude values

U mL  I mLL , where X L  L is the inductive resistance. 67

It is possible to make a conclusion from this that on the circuit section containing the inductive element in case of alternating current, there will be a phase shift between current and voltage. Voltage on the inductive element is ahead of the phase current by an angle equal

. 2

Let's write all values in a complex form: U L  U mLe j , IL  I mLe j , X L  j L  jX L , iL

uL

(6.16)

then the law of Ohm for a circuit section: U L  IL  X L  jX L  IL .

(6.17)

This expression defines a voltage drop on the inductive element. Kirchhoff's laws The sense of laws of Kirchhoff doesn't depend on how the quantities (current, voltage, EMF) depend on time: constants or variables. The difference is that all variable values must be presented in a complex form. І Kirchhoff's law (Kirchhoff's current law): The algebraic sum of all currents converging in the nodep is equal to zero:

 Ik  0

(6.18)

k

ІІ Kirchhoff's law (Kirchhoff's current law): For any closed path in a network the algebraic sum of the voltages drops on all passive elements is equal to the algebraic sum of all voltage sosurces in this loop:

U k   Em k

68

m

(6.19)

Control questions 1. Describe the basic characteristics of sinusoidal current. 2. Describe methods of representation of a sinusoidal waveform. 3. Formulate Ohm’s law in the complex form for a section of the circuit containing the inductive element. 4. Formulate Ohm’s law in the complex form for a section of the circuit containing the active resistance. 5. Formulate Ohm’s law in the complex form for a section of the circuit containing the capacitive element. 6. Formulate Kirchhoff's laws in a complex form.

69

Lecture 7

CURRENT, VOLTAGE, RESISTANCE AND POWER IN SINUSOIDAL CURRENT CIRCUITS

Plan 7.1 Impedance and conductivity 7.2 The active and reactive component of voltage and current 7.3 Energy and power in sinusoidal AC circuit

7.1 Impedance and conductivity of a circuit

Consider a series resonant circuit shown in Figure 7.1.

Figure 7.1

The source of energy produces the variable EMF e  Em sin(t  e ) , which creates current in the circuit. According to Kirchhoff’s voltage law: for the instantaneous values

e  u R  u L  uC , for complex numbers

E  U R  U L  U C ,

70

(7.1)

where

U R  IR  R , U L  jX L  IL , U C   jX C  IC are the voltages across the circuit elements. Then equation (7.1) takes the form E  I  ( R  j ( X L  X C )) .

Assume that x  X L  X C , where x is a circuit reactance, and R is the active resistance. Then

Z  R  jx

(7.2)

is circuit impedance. The active resistance is always positive, and reactance can be either positive or negative. If we show components of impedance on the complex plane, then the active resistance, reactance and impedance form the rectangular triangle called a triangle of impedance. Components of this triangle are related as x Z  R 2  x 2 ,   arctg . R

(7.3)

Consequently, phase shift between current and voltage on a section of a circuit is defined by a ratio of reactance and active resistance. In case of absence of the active component we find the phase shift to be: − +90° in case of the inductive reactance; − -90° in case of capacitive reactance. Existence of the active component defines for phase shift of sector: − 0 < φ < 90° in more inductive circuit (Figure 7.2, a); − 0 > φ > -90° in more capacitive circuit (Figure 7.2, b). 71

In case of absence of a reactive component or if XL – XC = 0 phase shift between current and voltage is absent, i.e.   0 .

а)

b)

Figure 7.2. The impedance triangles

The current is defined as E I  . Z

If

E  E m e j , Z  Z m e j , e

where Em , Z m are amplitudes,  e ,  are initial phases of EMF and impedance. Then

E e j E I  m j  m e j (  ) , Z me Zm e

e

(7.4)

in I  I m e j , get i

Im 

Em ,      . i e Zm

Therefore, the initial phase of impedance is numerically equal to the phase shift between voltage and current in the circuit:

   e  i . 72

(7.5)

Consider a parallel RLC circuit, shown in Figure 7.3. In parallel circuits we usually use conductivity instead of impedance to simplify calculations.

Figure 7.3

The admittance is the inverse of impedance Z , it is denoted by ܻሶ and is measured in siemens (S): 1 Y  . Z

(7.6)

The admittance is also a complex number: it contains the active g and reactive b components Y  g  jb.

(7.7)

The real part of the complex is the conductance g, and the imaginary part is called the susceptance b. The susceptance is the difference between the capacitive susceptance and inductive susceptance, i.e. b = bC – bL. Vector of complex admittance and its components form on the complex plane the rectangular triangle called by an admittance triangle (Figure 7.4). Components of this triangle ratios are related as: b Y  g 2  b 2 ,   arctg g

(7.8)

If the circuit consist of resistance R and inductance L, Z  R jXL, then 73

Y 

1 R  jX L R X  2  2  j L2  g  jbL , 2 R  jX L R  X L Z Z

(7.9)

where the conductance is g

R , 2 Z

(7.10)

bL 

XL . 2 Z

(7.11)

and inductive susceptance is

а)

b)

Figure 7.4. The admittance triangles

If the circuit consists of resistance R and capacitance C,  Z  R  jXC , then

Y 

1 R  jX C R X  2  2  j C2  g  jbC , 2 R  jX C R  X L Z Z

(7.12)

where the conductance is defined by formula (7.10), and capacitive susceptance is

bC 

XC . 2 Z

(7.13)

Thus, the imaginary part of total complex admittance is positive for a capacitive circuit, and negative for the inductive circuit. 74

7.2 The active and reactive components of voltage and current

Let's write Ohm’s law in the complex form for the circuit containing various passive elements U  I  Z  I  R  jx   I  R  I  jx  U A  U P ,

where U A , U P are complex voltages. Complex active voltage U A  IR

(7.14)

has the same direction as the current vector. Complex reactive voltage is located perpendicular to the current vector (7.15) U P  I  jx . The active voltage corresponds to the voltage across the element with active resistance, and reactive across the element with reactive resistance. Relations are obvious for components of complex voltage: U  U A  j UP ;

(7.16)

U A  U cos  ; U P  U sin  ;

(7.17)

U  U A2  U P2 ;   arctg

UP UA

(7.18)

the active voltage can be only the positive, and the sign of reactive voltage is defined by a sign of phase shift  . Voltage vector together with the active and reactive components on the complex plane form the rectangular triangle called a voltage triangle (Figure 7.5). 75

U R  IR, U L  jLI, U C 

1  1  I j I C j C

Figure 7.5. Voltage triangle for series circuits

Let's write Ohm’s law through complex admittance

I  U  Y  U  ( g  jb)  IA  IP , where IA , IP are complex current components. Complex active current IA  Ug

(7.19)

has the same direction as the voltage vector. Complex reactive current is located perpendicular to the current vector (7.20) IP   jb  U . The active current corresponds to the current passing through an element with the active conductivity, and reactive – with susceptance. Relations are obvious for components of complex current: I  I A  j I P ;

(7.21)

I A  I cos  ; I P  I sin  ; I  I A2  I P2 ;   arctg

IP , IA

(7.22) (7.23)

and the active current component can be only positive, and the sign of reactive current component is defined by the sign of phase shift  . 76

Current vector together with the active and reactive components on the complex plane form the rectangular triangle called by a triangle of currents.

U U IP  IL  IC    jCU  YU R j L Figure 7.6. Current triangle for parallel RLC circuit

7.3 Energy and power in AC circuit

Energy transmission W by an electrical circuit (for example, by the line of an electricity transmission), energy dispersion (transformation of electromagnetic energy into heat), and also other types of conversion of energy, are characterized by intensity with which the process proceeds, i.e. how much energy is transmitted by the line per unit time and how much energy dissipates per unit time. Intensity of transmission or conversion of energy is called power p, i.e. p

dW . dt

In electrical circuits the instantaneous power is expressed by current and voltage (7.24) p  u i . Let's consider a linear two-terminal network where there are no energy sources. Voltage and current on clamps of the two-terminal network change under the harmonic law

u  U m sin(t   u ) ,

(7.25) 77

i  I m sin(t   i ) .

(7.26)

Let's write (7.25) and (7.26) through effective values:

u  U 2 sin(t   u ) ,

(7.27)

i  I 2 sin(t   i ) .

(7.28)

By adding (7.27) and (7.28) in (7.26) we obtain

p  UI cos( u   i )  UI cos( 2t   u   i ) .

(7.29)

Consider    u   i , where  is phase shift between voltage and current. Then the average value of the instantaneous power for the period

pav 

1T p  UI cos( ) . T 0

(7.30)

(7.30) characterizing the speed of energy supply, average for the period, in the two-pole network is an active power:

P  UI cos( ) .

(7.31)

The active power is also known as the average power, which is the product of the RMS voltage, RMS current and cos(φ) in an AC circuit. It is actually the average power dissipation on the resistive load, i.e. the average power within one period of time (one full cycle) for a sinusoidal power waveform in an AC circuit. The active power is also called true or real power since the power is really dissipated by the load resistor, and it can be converted to useful energy. Instantaneous power always varies with time and it is difficult to measure it, so it is not very practical in use. Since it is the actual power dissipated in the load, the average or active power P is used more often in AC sinusoidal circuits. 78

When the active power P  0 , the element absorbs power; when the active power P  0 , the element releases power. SI unit of power is watt [W]. It is the power released on the active resistance R :

P  I 2R .

(7.32)

As the effect of charging/discharging in a capacitor C and storing/releasing energy from an inductor L is that energy is only exchanged or transferred back and forth between the source and the component, it will not do any real work for the load. So the average power dissipated on the load is zero. The reactive power Q can describe the maximum velocity of energy transferring between the source and the storage element L or C. Q  UI sin( ) .

(7.33)

If Q  0 – energy is accumulated in a magnetic field of a circuit; Q  0 – energy is accumulated in an electrical field of a circuit; Q  0 – in a circuit there is no exchange of energy with a source. SI unit of reactive power is [VAr]. Energies of magnetic and electrical fields are defined by WM  WE 

2 LI m , 2 2 CU m

2

(7.34)

,

where L is inductivity of the coil, C is capacitance of the capacitor. Then reactive powers in the inductive and capacity elements are defined by expressions QL  I 2 X L  I 2  L ,

QC  I 2 X C  I 2 

1 , C

(7.35) (7.36) 79

where ω is the angle frequency of alternating current. For a complete circuit the reactive power is equal to

Q  QL  QC  I 2 ( X L  X C )  I 2  x .

(7.37)

Total power is the maximum value of the active power in case of

 0

S  UI .

(7.38)

Total power is the value, equal to the work of operating values of current and voltage on circuit clamps. SI unit of S measurement is [VA]. Total, active and reactive capacities are connected by a ratio S  P 2  Q2 .

(7.39)

In a complex form the total power is defined by expression

S  P  jQ.

(7.40)

The rectangular triangle formed by a vector of the total power together with the active and reactive components on the complex plane is called a power triangle. The important characteristic of electrotechnical devices is the power factor P P (7.41) cos()   . UI S Control questions 1. Write formulas of impedance and admittance in a circuit. 2. Describe the active and reactive components of voltages and currents. 3. Formulate concepts of energy and power in AC circuit.

80

Lecture 8

METHODS OF AC CIRCUIT ANALYSIS

Plan 8.1 Vector diagrams 8.2 Symbolical method of AC circuit analysis

8.17 Vector diagrams

Application of laws of Ohm and Kirchhoff assumes use of a concept of direction: the direction of flowing current, the direction of EMF, the direction by the relation to a node and others. Since in electrical circuits of alternating current all values (EMF, voltage and currents) change the directions twice per a time period, for them we use a concept the positive direction, i.e. direction appropriate to the positive instantaneous values of the defined value. When we choose another direction an initial phase of sinusoidal quantity changes to an angle of π. Thus, all complex values of quantities can be defined only with the account for a choice of the positive direction. For a passive element the positive direction is selected randomly only for one of quantities: current or voltage. The direction of the second quantity must match the direction of the first, then the phase ratios between them, which follow from physical processes of conversion of energy, won't be violated. The positive direction of EMF is given and it is specified by an arrow in the conditional designation. With respect to this direction the initial phase of EMF is defined. For the analysis of the quantitative and phase ratios of values of alternating current on the complex plane we build the vectors corresponding to the mode of operation of an electrical circuit. This set of vectors is called a vector diagram. Let's construct vector diagrams and complex diagrams for the subcircuit of an electrical circuit containing different passive elements. 81

8.1.1 Section of an electrical circuit with a resistive element

Let's consider a part of an electrical circuit containing a resistive element with a resistance R  Z R , through which the sinusoidal current expressed by a complex effective value flows:

IR  Ie j , i

then according to Ohm’s law a complex voltage is:

U R  IR e j  Ue j . i

u

Complex current and voltage across a resistor have identical arguments and differ in magnitude in R times. On the complex plane IR and U R are drawn by vectors which match in the direction and differ only in scale (Figure 8.1, a). +j

+j

0

0

+1 а)

+1 b)

+j

0

+1 c)

Figure 8.1. Vector diagrams for current and voltage (a), the complex resistance (b) and complex conductance (c) of the active resistance

82

On the complex plane Z R is shown by a vector directed along the abscissa axis (Figure 8.1, b). Complex conductivity of the active resistance Y  1  1 is also shown by a vector with positive R Z R R direction with respect to the abscissa axis. The complex equivalent circuit diagram is shown in Figure 8.2, a.

а)

b)

c)

Figure 8.2. Complex equivalent circuit diagrams containing resistance (a), capacitance (b) and inductance (c)

8.1.2 Section of an electrical circuit with a capacitive element

Let's consider a section of an electrical circuit containing a capacitive element with a capacity C. Since capacitor voltage lags j ( current by  , complex current IC  Ie j  CUe i



u

 ) 2

and capa-

2 city voltage U C  Ue j are depicted as two vectors located in such way that a vector IC leads a vector U C by angle  (Figure 8.3). 2 On the complex plane complex capacity resistance ZC  X C   j 1 C 1 and complex admittance YC    jC  jbC are depicted by the XC u

vectors directed along the negative and positive imaginary axis (Figure 8.3, b, c). The complex equivalent circuit is shown in Figure 8.2, b. 83

+j +j  0

+1

+1

0 а)

b)

+j

+1

0 c)

Figure 8.3. Vector diagrams of current and voltage (a), complex impedance (b) and complex susceptance (c) of capacity

8.1.3 Section of an electrical circuit with an inductive element

Let's consider a section of an electrical circuit containing an inductive element with inductance L. As inductivity current iL lags inductivity voltage u L by angle  , complex current IL  Ie j and 2 i



j ( i  )

2 are shown on the plane as complex voltage U L  Ue j  LIe a couple of vectors, whose lengths are equal in a certain scale to effective values of voltage and current on inductance, and a vector U L leads a vector IL by angle  (Figure 8.4). 2 u

84

+j

+j

0

+1

+1

0 а)

b)

+j

0

+1 c)

Figure 8.4. Vector diagrams of current and voltage (a), complex impedance (c) and complex admittance (b) of inductance

On the complex plane complex impedance Z L  X L  jL and 1 1 complex admittance YL  j  jbL are figured by the vectors X L L directed correspondingly along the negative and positive ordinate axis (Figure 8.4, b, c). The complex equivalent circuit is shown in Figure 8.2, c. 8.1.4 Series RL circuit

Let's consider an idealized series electrical circuit consisting of a resistor R and an inductor L (Figure 8.5, а). Let voltage attached to external clamps of an electrical circuit change by the harmonical law u  U 2 sin (t   u ) , 85

where U ,  ,  u are the given values. Let's replace an electrical circuit by a complex equivalent sheme (Figure 8.5, b).

а)

b)

c)

Figure 8.5. Equivalent series RL-circuit

Vector diagrams for current and voltage in RL circuit are shown in Figure 8.6. As voltage on resistance is in phase with current, a vector U R has the same direction as vector I , vector U L leads vector by angle

 . Independent of the initial phase of voltage  vector I legs vecu

2 tor U  U R  U L by angle φ, i.e., current lags voltage by an angle, equal to an argument of complex input impedance of a circuit.

а)

b)

Figure 8.6. Vector diagrams of series RL-circuit

86

8.1.5 Series RC circuit

Let's consider an idealized series electrical circuit consisting of a resistor R and a capacitor C (Figure 8.7, а). The complex equivalent circuit is shown in Figure 8.7, b.

а)

b)

c)

Figure 8.7. Equivalent series RC-circuit

Vector diagrams of current and voltage for series RC circuit are shown in Figure 8.8. Vector U C is turned clockwise about vector I by angle  . Independent of the initial phase of current  i vector U 2 legs I by angle  , i.e. voltage lags behind a phase current by angle  , equal to an argument of complex input impedance of a circuit.

а)

b)

Figure 8.8. Vector diagrams of series RC circuit

87

8.1.6 Series RLC circuit

Let's consider a series RLC – circuit with harmonic voltage u (Figure 8.9).

b)

а)

c) Figure 8.9. Equivalent series RLC-circuit

The character of input resistance of a circuit depends on the ratio between imaginary components of complex capacitive susceptance 1 and inductive susceptance X L  jL : X C   j C  (Figure  when X L  XC the circuit is more inductive 0    8.10, а);  when X L  X C the circuit is more capacitive 

2

 2

   0 (Fi-

gure 8.10, b);  when X L  X C imaginary components of input resistance are mutually compensated and the circuit will look like a purely resistive circuit,   0 and input impedance Z = R (Figure 8.10, c). 88

а)

b)

c) Figure 8.10. Vector diagrams for resistance of series RLC-circuit

Vector diagrams for current and voltage of this circuit are given in Figure 8.11.

а)

b)

c) Figure 8.11. Vector diagrams for voltage and current in series RLC-circuit

89

Vector of voltage drop on resistance U R  IR has the same direction as vector I ; vector U  jL  I leads I by 90°; vector L

1 UC   j  I is directed opposite to vector U L . When X L  X C C vector U L  U C has the same direction as vector U L (Figure 8.11, а),

current in the circuit lags voltage by phase (   0 ). When X L  X C vector U L  U C has the same direction as vectorU C (Figure 8.11, b), current in the circuit leads voltage by phase (   0 ). When X L  X C vector U L  U C  0, voltage on circuit clamps U is equal to voltage on resistance U R (Figure 8.11, c), current in the circuit is in phase with voltage (   0 ).

8.1.7 Parallel RLC circuit

Let's consider a parallel RLC circuit with harmonic voltage (Figure 8.12).

а)

b) Figure 8.12. Equivalent parallel

c)

RLC

circuit

The character of input admittance of a circuit depends on the ratio between reactive components of input admittence: capacitive susceptance bC  C and inductive susceptance bL   1 :

L

90

 when bC  bL

the circuit is more inductive (argument of



   0 ) (Figure 8.13, а); 2  when bC  bL – the circuit is more capacitive (Figure 8.13, b);

complex admittence –

 when bC  bL reactive components of input admittance are

mutually compensated and input admittance of the circuit has purely resistive character (Figure 8.13, c). Vector diagrams for current and voltage of the circuit are given in Figure 8.14.

а)

b)

c) Figure 8.13. Admittance vector diagrams in parallel RLC – circuit

91

b)

а)

c) Figure 8.14. Vector diagrams for currents and voltage in parallel RLC – circuit at bC  bL (а), bC  bL (b), bC  bL (c)

8.2 Symbolical method for AC circuit calculation

In circuits of alternating current with several branches and elements it is almost impossible to execute the analysis of the mode of operation if the main values are provided by sinusoidal functions since difficult trigonometric equations turn out. In the case of representation of functions and parameters of a circuit by complex numbers, the mathematical description reduces to the linear algebraic equa92

tions. In this case, initial sinusoidal functions of time (the area of the real variable – t ) are originals, and complex numbers and vectors are their images or characters (the area of the imaginary argument – j ). Therefore the method is called symbolical. Stages of application of the symbolical method: 1. We determine topological parameters of the circuit: number of nodes, branches, circuits. 2. We represent all values and parameters of the circuit by complex numbers. 3. We make a complex equivalent of the electrical circuit on which all the data are specified in the complex form. We specify the positive direction of currents in circuit branches. 2. We determine the required values by any method of calculation known from the theory of DC circuit. 3. We transform the obtained complex values to the form of representation by their sinusoidal functions of time. Control questions 1. Describe the principle of creation of vector diagrams for currents and voltage in different parts of a circuit. 2. Describe the principle of creation of vector diagrams for complex impedance and admittance in different parts of a circuit. 3. Explain stages of application of the symbolical method for calculation of electrical circuits.

93

Lecture 9

RESONANCE IN ELECTRIC CIRCUITS

Plan 9.1 Series resonance 9.2 Parallel resonance 9.3 Resonance in a composite oscillating circuit

An oscillating circuit is the electrical circuit which contains an inductor, a capacitor and a resistor and where free oscillations of current and voltage are possible. In the theory of electrical circuits oscillating circuits play an important role as they possess resonating characteristics. A resonance is a mode of an oscillating circuit where the input reactance of a circuit or its input susceptance is equal to zero. Therefore in case of a resonance, voltage and current at the input of a circuit are in phase. In electrical circuits there are two types of a resonance: a series (voltage) resonance and a parallel (current) resonance. The voltage resonance is observed in a series oscillating circuit, and a current resonance is observed in a parallel oscillating circuit. In a complicated oscillating circuit both types of a resonance can take place. The phenomenon of a resonance is used in a wide range of applications in communication systems, such as filters, tuners, etc. The purpose of resonant circuits are the same – to select a specific frequency (resonant frequency fr) and reject all others, or select signals over a specific frequency range that is between the cutoff frequencies f1 and f2. The key circuit of a communication system is a tuned amplifier (tuning circuit). Simplified radio tuning circuit consists of inductance and capacitance. The combination of a practical parallel resonant circuit and an amplifier can select the appropriate signal to be amplified. The input signals in the radio tuner circuit have a wide frequency range, because there are many different radio signals from different radio stations. When adjusting the capacitance of the variable capacitor in the practical parallel resonant circuit (i.e. adjust94

ting the switch of the radio channel), the circuit resonant frequency fr will consequently change. Once fr matches the desired input signal frequency with the highest input impedance, the desired input signal will be passed, and this is the only signal that will be amplified. After it is amplified by the amplifier in the circuit, this signal of the corresponding station can be clearly heard. The resonant phenomena can also play a negative role. In electrotechnical installations the resonance is often dangerous by the unwanted phenomenon as can lead to accidents owing to overheating of elements of an electrical circuit or insulation breakdown in case of overvoltage. 9.1. Voltage resonance 9.1.1 Basic characteristics

Circuit operation mode with series connection of R, L, C – elements, when the capacitor reactance XC is equal to the inductor reactance XL, and the equivalent impedance of the series RLC circuit will be active and the lowest (Z = R) is called a voltage resonance.

Figure 9.1

Power source in series oscillating circuit (Figure 9.1) is the source of alternating voltage

u  U m sin t   U . Total input impedance of a circuit 1   j Z  R  j L    R  jx  Z e . C  

(9.1) 95

where X L  L is the inductive reactance and XC 

1 is the capaC

citive reactance. The circuit current is equal to U U I   e  j . Z Z

(9.2)

The sign of the current complex argument  depends on a ratio between the inductive and capacitive reactance of a circuit:  When the frequency of the circuit is above the resonant frequency fr, X L  X C , and the circuit is more inductive, the phase difference is between zero and positive 90° ( 0   

 ), and voltage 2

leads current (Figure 9.1, a) .  When the frequency of the circuit is below the resonant frequency fr, X L  X C , the circuit is more capacitive, X L  X C , and the phase difference is between zero and negative 90° ( 



2

   0 ).

And the voltage lags current (Figure 9.1, b).  When the frequency of the circuit is equal to the resonant frequency fr, X L  X C , the circuit is active and voltage and current are in phase, and the phase difference is zero ,   0 (Figure 9.1, c). Hence, at resonance mode reactance is zero: x  L 

1  0. C

(9.3)

In this case total complex impedance of a circuit

Z  R ,

(9.4)

and current will have the maximum value I0  96

U. R

(9.5)

a)

b)

c)

Figure 9.2. Phasor diagrams

It is seen from the diagram that the voltage across an inductor U L and a capacitor voltage U C compensate each other, and input voltage in a circuit U and current are in phase:

 u  i .

(9.6)

Therefore it is a voltage resonance. If frequency, at which the power source is connected to a circuit, doesn't change, the resonance mode is reached through changes of parameters of a circuit (L or C). L and C values must simultaneously correspond to the condition (9.3). If parameters of a circuit remain invariable, the resonance mode arises at a frequency  0 , which is called the resonance frequency of a circuit. In case of connection to an oscillating circuit of a power source with a frequency ω0, the ratio is satisfied:

X L  X C , 0 L  1 , 0C

0 

1 . LC

(9.7)

(9.8)

At a resonance the inductive and capacitive reactance are equal, therefore this resistance is called characteristic resistance or wave resistance of a circuit 97

  0 L 

1

0 C



L. C

(9.9)

The value characterizing resonating characteristics of a circuit at resonance is called quality factor of a circuit Q

 R



U L UC  . U U

(9.10)

In the electrical circuits used in radio engineering devices, good quality factor has value of an order 50-300. If the oscillating circuit is set up in a resonance, the voltage of the inductive and capacity elements many times exceeds the input voltage. Generally, the quality factor of a circuit is given by: Q  0 

where

W

max

Wmax , P

is the sum of the maximum values of the energy ac-

cumulated in reactive elements at resonance, P  I 02 R is the active power in a circuit at a resonance. Reciprocal value of the quality factor of a circuit is called attenuation U 1 R U (9.11) d    . Q  U L UC 9.1.2 Energy processes at series resonance

Total energy of an electromagnetic field in an oscillating circuit is equal to (9.12) W  WM  WE , where WM 

Li 2 is the magnetic field energy, WE 2

electric field energy. 98



CuC2 is the 2

If the instantaneous value of current at a resonance is

i  I m sin0 t . then the capacitive voltage lags current by

(9.13)

 and is equal to 2

  uC  U Cm sin 0 t    U Cm cos0 t . 2 

(9.14)

Taking into account (9.13) and (9.14), equation (9.12) will be W 

L C 2  I m sin  0 t   2 2



    U Cm sin   0 t  2  

2

   

L 2 C 2  I m sin 2  0 t    U Cm cos 2  0 t  , 2 2

As

U Cm 

Im L  I m  0 L    I m   Im , C 0 C

(9.15)

the total energy of an electromagnetic field will be equal

W

W

 

 

L 2 2 C L  I m sin 0 t   I m2 cos2 0 t . 2 2 C

  

 

2 LI m2 LI 2 CU Cm sin 2 0 t  cos2 0 t  m  2 2 2

(9.16)

As amplitude values of current and voltage are constants, we have W  const . Thus, during a resonance, the total energy of an electromagnetic field of a series oscillating circuit doesn't change: reduction of electric field energy is followed by an increase in magnetic field energy 99

and vice versa. Reactive elements of a circuit continuously exchange energy, and the external source doesn't participate in this exchange. If the circuit had no losses (R = 0), oscillatory process could be set without an external power source. When the charged capacity is closed on inductivity, an oscillatory process with a frequency  0 arises in the circuit. The energy accumulated in an electric field will pass into a magnetic field, then there will be the reverse transition of energy, etc. Such a process will last infinitely long. If the oscillating circuit has the active resistance, losses of energy take place and the oscillatory process has the damping character. The external power source is necessary for compensating the losses in the active resistance. The energy coming to a circuit from an external source at any moment entirely turns into heat. It is obvious that compensating of losses will happen only in that case when energy comes from a source in-phase with oscillations of the circuit. Thus, the frequency of a source must be equal to the resonance frequency of the circuit. 9.1.3 The frequency responses and resonance curves

Let's consider an electrical circuit to which alternating voltage is applied u  U m sin t   U  with a constant amplitude, but a variable frequency (from 0 to ∞). In case of frequency change there is a change of parameters of a circuit, its reactance and impedance, and also its phase shift. The dependence on frequency of parameters of a circuit is called the frequency response (Figure 9.3). The active resistance of the circuit R doesn't depend on frequency (a direct line in Figure 9.3). The direct line at some angle corresponds to the diagram of the inductive reacnance X L  L , hyperbola – to the capacitive reactance XC 

1

. C The total reactance of a circuit x is defined by the difference of ordinates of diagrams X L and X C :  in case of resonance frequency  0 reactances X L and X C are equal and x is equal zero;

100

 if frequencies   0 the nature of reactance of a circuit is more inductive (Figure 9.2, a);  if frequencies   0 the reactance of a circuit is more capacitive (Figure 9.2, b). The impedance of a circuit is defined by the formula 2

1  .  Z  R 2   L   C  

(9.17)

Dependences of effective or amplitude values of current and voltage on frequency are called resonance curves. Resonance curves can be ploted on the basis of the frequency responses (Figure 9.4). Current of a circuit is determined by Ohm’s law U . I  Z

(9.18)

As voltage is constant, diagrams I   and Z   are mutually reverse. In expression for voltage on inductivity U L  I X L both

multiplicands depend on frequency. In case   0 and X L  0 , therefore U L  0 . In case of increase in frequency to 0 , both I , and

X L increase, thus U L increases.

Figure 9.3. Frequency responses

101

Figure 9.4. Resonance curves

In case of    0 , current decreases, but X L grows, therefore U L at the beginning continues to grow, but at some frequency

   L this growth U L stops. In of case    L , U L decreases.

In expression U C  I X C both multiplicands are also functions of frequency. In case of   0 (direct current) the condenser is charged by source, therefore U C  U . In case of increase in  , the reactance X C decreases, but I at the beginning grows stronger. Therefore at the beginning U C grows, but at a frequency   C  0 the recession of this diagram begins. Grafs U C and U L are crossed in case if    0 .

The values of frequencies  L and  C , at which the curves U L  L  and U C C  reach a maximum, can be determined by studying the conditions of maximum of these functions. These frequencies can be determined as follows. Let's define  C from the equation

UC  Im  X C 

Um  XC  Z



102

Um  1   R 2   C L  C C   Um

R  C  C2 LC  1 2

2 C

2

2

.

2



1

C C



We find the derivative dU C , and then, equating the result to d zero, we will obtain a formula for frequency  C , at which the voltage on a capacity element U C reaches the maximum value

C 

C 2 L  1 1  1 R2 C   1  1  R  0 2LC 2 2 2 L LC

2  R2

 0 1 

1 2 d2   . 0 2 2Q2

(9.19)

Similarly it is possible to determine  L  0

2 . 2d2

(9.20)

Substitution of expressions for  L and  C in calculation formulas for voltage shows that the maximum values of functions U L and U C are identical U Lm  U Cm 

2U

.

(9.21)

d 4d2

The level of shifting of frequencies  L and  C from the resonance frequency  0 depends on the quality factor of a circuit. Fre-

quencies  L and C noticeably differ from each other only in case of high quality factor. 9.1.4 Frequency mismatch in series oscillating circuits

During operation of installations, deviations of oscillating circuits from a resonance take place. These deviations can be caused both by frequency change of a source, and change of parameters of circuits. 103

The circuit mode deviation caused by frequency change is evaluated by absolute or relative frequency mismatch. Absolute frequency mismatch  is called a difference in frequency of a source and resonance frequency of a circuit:     0 . As frequency  can be both more and less than resonance frequency 0 , absolute frequency mismatch  can be both positive and negative. The relative frequency mismatch is determined by relation:



0



f . f0

(9.22)

Except these two concepts of the theory of oscillating circuits the concept of the generalized frequency mismatch  is widely applied. The generalized frequency mismatch considers all reasons which can cause a circuit mode deviation from a resonance. It is equal of a reactance devided by active resistance: 

X  R



 1  1  0 L   0  0C  0 C   R R

L 

  0     0     Q   Q .   R  0    0  

The value of  

(9.23)

 0  is called the relative frequency mis0 

match. 9.1.5 The bandwidth of a series resonance

When an RLC series circuit is in resonance, its impedance will reach the minimum value and the current will reach the maximum value. The curve of the current versus frequency of the series resonant circuit is illustrated in Figure 9.5. 104

As displayed in the diagram, the current reaches the maximum value Imax as the frequency closes in on the resonant frequency fr, which is located at the center of the curve. The characteristic of the resonant circuit can be expressed in terms of its bandwidth (BW) or pass-band. The bandwidth of the resonant circuit is the difference between two frequency points f2 and f1.

BW  f 2  f1 ,

(9.24)

where f2 and f1 are called critical, cutoff or half-power frequencies.

Figure 9.5. Bandwidth of a series resonant circuit

As shown in Figure 9.5, the bandwidth of the resonant circuit is a frequency range between f2 and f1 when current I is equivalent to 0.707 or 2 of its maximum value Imax, or 70.7 per cent of the maximum value of the curve. The reason to define the term ‘half-power’ frequency can be derived from the following mathematical process. The power delivered by the source at the points f1 and f2 can be determined from the power formula:

Pf1  Pf 2 





2

2 I max R  0.5Pmax .

(9.25) 105

Therefore, at both points f2 and f1, the circuit power is only onehalf of the maximum power that is produced by the source at resonance frequency fr, where f2 is the upper critical frequency, and f1 is the lower critical frequency. The curve in Figure 9.5 is also called the selectivity curve of the series resonant circuit. The selectivity is the capability of a series resonant circuit to choose the maximum current that is closer to the resonant frequency fr. The steeper the selectivity curve, the faster the signal attenuation (reducing), the higher the maximum current value, and the better the circuit selectivity. In the devices containing oscillating circuits not only frequency ω0, but also a number of close to it frequencies called «side» frequencies are used. In order that these side frequencies passed through an oscillating circuit, it must have the appropriate bandpass range. Bandpass range of a series circuit is the area of frequencies within which the current decreases in comparison with resonant no more than in 2 times. The absolute bandpass range SA is a difference of boundary frequencies: f (9.26) S A  f 2  f1  0 . Q

The relative bandpass range:

SR 

SA 1   d. f0 Q

(9.27)

Therefore, the band pass filter becomes more selective (small BW) as Q increases.

9.2 Current resonance 9.2.1 Basic characteristics

The operation mode of RLC-circuit with parallel branches in case of which input reacance of a circuit is equal to zero is called a current resonance. 106

As admittance of a circuit

1 Y  g  jb  . Z

(9.28)

Transforming (9.26) we will receive Y 

1 R x ,  2 j 2 2 R  jx R  x R  x2

g

R , R  x2

(9.29)

Then b

2

As

x . R  x2 2

b  bL  bC , bL  

X XL , bC  2 C 2 . 2 R  XC R  XL 2

(9.30) (9.31) (9.32)

Figure 9.6. The parallel resonant circuit

Complex admittance of the first and second branches of a circuit are respectively equal Y1  g1  jb1 

R12

R1 X j 2 L 2, 2  XL R1  X L

(9.33) 107

Y2  g 2  jb2 

R22

X R2 j 2 C 2 . 2  XC R2  X C

(9.34)

Input conductance of a circuit Y  Y1  Y2   g1  g 2   j b1  b2  .

(9.35)

Resonance condition of currents b1  b2 ,

(9.36)

Then 

XL R12



X L2



XC R22

 X C2

.

(9.37)

From the condition of currents resonance it follows that reactive components of currents of branches in the resonance mode by the absolute value are equal: I P1  Ub1 ;

I P 2  Ub2 ;

I P1  I P 2 .

Equality I P1  I P 2 leads to the fact that the input current of a circuit has no reactive component and current is in phase with voltage. Therefore this type of a resonance is called a current resonance. The vector diagram is ploted in Figure 9.7.  

  Figure 9.7

108

Let's define resonance frequency  p pL R12



 pL

1



2

 pC



2

,

   1   C p   1 1 ,     R12 2  1    p L  p CR2    C   p    pL



R22



  R12  ,    p L   p CR22   1     L C p  p   



R12

as  

L



  2p L   2p CR22  1 , C

L C p 

1 LC

 2  R12 .  2  R22

(9.38)

Let's consider different cases:  if R1  R2   , then the current resonance is possible at any frequency;  if R1  R2   , then the current resonance will occur at a voltage resonance frequency  p  0 ;  if R1  R2 and R1 , R2   or R1 , R2   , then the current resonance will be at one frequency  p ;  if R1   и R2   , then the current resonance isn't possible. Total impedance of a parallel oscillating circuit is equal

Z Z Z  1 2 , Z1  Z 2

(9.39) 109

At resonance condition impedance is active, i.e. Z  R p , but its value, unlike in a series circuit, depends not only on the resistances of branches R1 and R2 , but also on reactive elements. According to a resonance condition of currents, input conductance of an oscillating circuit Y  g  g1  g 2 

R1 R12



 pL

1  1   R22     pC   



2



2



R2  1   R22     pC     p LC ,

2

,

(9.40)

R12   p L 2

expression (9.38) will take a form R1   2p  L  C  R2 , g R12   p L 2





then impedance of a circuit Z  Rp 

R1 R2   2 . R1  R2

(9.41)

Quality facor of a ciucuit, Q-factor Q

 R1  R2

.

(9.42)

Let's consider different cases:  if R1  R2   , and values are commensurable, the current resonance in case of energy losses is possible;  if R1  R2  0 , that the oscillating circuit has no losses;  if R1  R2    , then the oscillating circuit has small losses. 110

9.2.2 The frequency characteristics

Let's consider a parallel circuit without losses, then the frequency response of the inductive susceptance bL  1 is a hyperbola, and L for capacitive susceptance bC  C is a straight line.

Figure 9.8. Frequency characteristics

Frequency dependence of input susceptance b  can be obtained from subtraction of ordinates of diagrams bL and bC . If frequencies 0    0 input impedance of a circuit has the inductive character, if   0 – a capacitive character. At parallel resonance input impedance is infinitely large. The resonance curve of an input current of an ideal circuit has the same form, as the diagram b  f  , only with frequencies   0 it represents the mirror image of this dependence. It is connected to the fact that effective value of current is the positive value. 111

9.3 Resonance in a complex oscillating circuit

Let's consider a complex parallel circuit which contains inductivity and capacity in each branch. Let the circuit have a high quality factor, i.e. it has no losses R1  X 1  L1 

1 1 . , R2  X 2  L2  C2 C1

(9.43)

Figure 9.9

In this case, input impedance of a circuit

R1  jX 1 R2  jX 2   Z  Z Z  1 2    Z1  Z 2 R1  jX 1   R2  jX 2  

R1 R2  X 1  X 2  j R2  X 1  R1  X 2  , R  jX

where R  R1  R2 , X  X 1  X 2 , and R1  R2  X 1  X 2 . The term j R2  X 1  R1  X 2  can be neglected as it has a small value (for frequencies close to current resonance, frequencies X 1 and X 2 have opposite signs). 112

Input impedance of a circuit

X X Z   1 2  RE  jX E , R  jX where

RE 

R  X1  X 2 2

R X

2

, XE 

X  X1  X 2 R2  X 2

(9.44)

.

(9.45)

By considering a circuit without losses, i.e. in case when small resistances R1 and R2 are neglected, we get the admittance in case of a resonance (9.46) Y  Y1  Y2  0. then Y1  Y2 , X 1   X 2 , therefore  p  L1 

1 1    p  L2 .  p C1  p C2

If we assume that a full inductance is equal to L0  L1  L2 and aggregate capacitance – to C0  C1  C2 , the resonance frequency is  p  0 

1 . L0C0

(9.47)

In case of a resonance, circuit resistance is Rp 

X 12 X 22 .  R R

(9.48)

Transforming it, we get 2

    p  L1  1    p  C1   p  L0  Rp   R R



2  L1   L0 

2



  L   p 0 2  R L0   p  C1  1

2  L1  C0  2 . L  0

C1 

113

Let's use the designation pL 

C L1 , pC  0 , L0 C1

as the quality of this complex circuit is Q

we get

 p  L0 R1  R2



 p  L0 R



1

 p  C0  R

R p  Q 2  R  p L  pC 2 .

,

(9.49)

(9.50)

From this expression it is seen that in case of p L  1, pC  0 or in case of pL  0, pC  1 resistance will have the maximum value. In this case all inductivity L0 will be concentrated in one branch, and C0 – in the other branch, i.e. the complex parallel circuit will be transformed into a simple circuit. Control questions 1. Describe the resonance phenomenon. 2. Write the main characteristics of a circuit used in the series resonance. 3. Describe energy processes in the series resonance. 4. Explain origin of detuning of a series oscillating circuit. 5. Write formulas for determination of bandwidth in a series circuit. 6. Write the main characteristics of a circuit used in case of a current resonance. 7. Draw and explain the frequency characteristics and resonance curves in series resonance and in parallel resonance. 8. Explain a resonance in a complex oscillating circuit.

114

Lecture 10

COUPLED CIRCUITS

Plan 10.1 Coupled circuits. Basic concepts 10.2 Inductive coupling. Series and parallel connection of inductive coupled elements of a circuit 10. Circuit with mutual inductance: transformer coupling

10.1 Coupled circuits. Basic concepts

Electrical circuits are named coupled if the change of parameters and physical quantities in one circuit initiate a change of physical quantities in the other circuit. Coupled circuits may have internal and external connection (Figure 10.1).

a)

b)

Figure 10.1. Electrical circuits with internal connection (a) and external connection (b)

Let's consider two connected circuits in Figure 10.1, and here Z1 is a complex impedance of the first circuit, Z 2 is a complex impedance of the second circuit, Z12 is a complex coupled impedance. Let the first circuit be characterized by a connection level k1 , and the second – k 2 . Then coupling coefficient 115

k  k1k2 .

(10.1)

Depending on which elements of circuits connect circuits, different types of coupling are distinguished:  magnetic (the inductive element);  electrical (capacitive element);  combined (the inductive and capacitive elements)  galvanic (resistive element). Magnetic coupling. Two or more inductive coils will be coupled if changes in current in one of the coils causes appearance of EMF in the other coils. For example, for two connected coils u2  e2  M

di1 , dt

(10.2)

where i1 is the current of the first coil, u 2 is the induced voltage of the second inductive coil, e2 is the EMF of the second coil, M is the constant of proportionality or mutual inductance between coils. Mutual inductance is the ability of a coil to produce an induced voltage due to changing of the current in another coil nearby. Mutual inductance is said to exist between two circuits when a changing current in one induces, by electromagnetic induction, an EMF in the other. The unit of M is H (Henry). The phenomenon of induction of EMF in any coil when changing current in the other coil is called mutual induction, and the induced EMF is mutual induction EMF. In calculations voltage compensating this EMF is more often used. The polarity of the induced voltage across the mutually coupled coils can be determined by the dot convention method. This method can be used to indicate whether the induced voltage in the second coil is in phase or out of phase with the voltage in the first coil. The dot convention method places two small phase dots (•) or asterisks (*), one on the coil L1 and the other on the coil L2, to indicate that polarities of the induced voltage v1 in the coil L1 and v2 in the adjacent coil L2 are the same at these points, as shown in Figure 10.2(a). This means that the dotted terminals of coils have the same voltage 116

polarity and the coils are termed cumulatively coupled. In coils in Figure 10.2 (b) mutually induced voltage has opposite sign with a self-inductive voltage.

a)

b)

Figure 10.2. Coils connected in series

In Figure 10.1 the inductance of the first circuit is L1 , and the inductance of the second circuit is L2 . Then coupling coefficient k1 of the first circuit with the second is the ratio of the voltage in the second coil to the voltage in the first coil

k1 

U L 2 OC U L1

.

(10.3)

.

(10.4)

Similar for the second circuit

k2 

U L1 OC U L2

In case of magnetic coupling, coefficient k  k1k2 shows what part of magnetic flux of one coil is linked to the other coil. Let's consider extreme cases:  If the coils are spaced apart, only a part of the flux links with the second, and the coils are termed loosely-coupled. When there is no magnetic coupling, k  0 , there is no coupling between the coils, in this case the magnetic flux of one coil doesn't cross winds of the second coil.  If the magnetic coupling is perfect, i.e., all the flux produced in the primary links with the secondary, then k  1 . Coupling coefficient is used in engineering communications to denote the degree of coupling between two coils. If the coils are close together, most of the flux produced by current in one coil passes through the other, and the coils are termed tightly coupled. 117

There is a concept «leakage flux», it is defined as

 1 k2.

(10.5)

In case of tight coupling, the leakage flux is   0 , and when there is no coupling the leakage flux is   1 . Magnetic coupling can be theinductive (transformer) or conductive (auto-transformer) (Figure 10.3).

a)

b)

Figure 10.3. Magnetic coupling: inductive (transformer) and conductive (auto-transformer)

For the circuits with mutual inductance coupling the levels of coupling of the first and second circuits are equal k1 

M ; L1

k2 

M . L2

(10.6)

Then the coupling coefficient

k  k1k 2 

M . L1L2

(10.7)

For conductive magnetic coupling the levels of coupling of the first and second circuits are equal

k1  118

L12 ; L1  L12

k2 

L12 . L2  L12

(10.8)

Then the coupling coefficient is

k  k1k 2 

L12

L1  L12 L2  L12 

.

(10.9)

Electrical coupling. This type of coupling is provided through the capacitive element and circuits can have two types of coupling: capacitive internal (Figure 10.4) and capacitive external (Figure 10.5).

Figure 10.4. Electrical coupling: capacitive internal

Figure 10.5. Electrical coupling: capacitive external

For capacitive internal coupling the levels of coupling of the first and second circuits are equal k1 

C1 ; C1  C12

k2 

C2 . C2  C12

(10.10)

Then the coupling coefficient

k  k1k2 

C1C2 . C1  C12 C2  C12 

(10.11)

119

If we use the substitution

C11 

C1C12 , C1  C12 k

C22 

C2C12 , C2  C12

C11C22 . C12

(10.12)

(10.13)

For capacitive external coupling the levels of coupling of the first and second circuits are equal k1 

C12 . C2  C12

(10.14)

C12 . C1  C12 C2  C12 

(10.15)

C12 ; C1  C12

k2 

Then the coupling coefficient is k  k1k2 

Combination coupling. This type of coupling is provided through capacitive and the inductive elements and circuits can have two types of connection: the reactance-capacitive and conductive- capacitive (Figure 10.6).

a)

b)

Figure 10.6. Combination coupling: capacitive-inductive and capacitive-external

120

Galvanic coupling. This type of connection is provided out through a resistive element (Figure 10.7).

Figure 10.7. Galvanic coupling

10.2 Magnetic coupling. Series and parallel connection of inductive coupled elements of a circuit

Let's consider electrical circuits in which two coils are connected with each other, in the first case it is the serial connection, and in the second case it is the parallel connection.

M

Figure 10.8. The series circuit with mutural inductance

1. Voltage on a section of the circuit (Figure 10.8) is defined as

u  u R1  u L1  u12  u21  u L 2  u R 2 ,

(10.16)

where

u R1  iR1, u R 2  iR2 ,

di , dt di u L 2  L2 , dt u L1  L1

di , dt di u21  M 21 . dt

u12  M12

(10.17)

Mutual inductance in the linear circuits doesn't depend on the directions and values of currents, and depends only on the construction of the coils and their relative orientation. Therefore, for identical coils, the following expression is valid: 121

M12  M 21  M . Adding (10.17) to (10.16) we will obtain

u  iR1  R2  

di L1  L2  2M  . dt

(10.18)

Then the equivalent values

RE  R1  R2 , LE  L1  L2  2M .

(10.19)

In case of commulative coupling of coils in the second equation (10.16) the third term of equation will have a sign «+», and in case of differential coupling will have a sign «–». If applied voltage changes under some harmonic law, then the equations (10.16) and (10.17) will be transformed into a complex form U  U R1  U L1  U12  U 21  U L 2  U R 2 ,

U R1  IR1 , U R 2  IR2 ,

U L1  I  jL1 , U L 2  I  jL2 ,

U12  I  jM , U 21  I  jM .

(10.20) (10.21)

here

X L1  jL1 , X L 2  jL2 , Z M  jM .

(10.22)

Adding (10.21) to (10.20) we will obtain

U  IR1  R2   j L1  L2  2 M  ,

(10.23)

Then Z  R1  R2   j L1  L2  2 M  

 RE  jLE  RE  jx E .

122

(10.24)

Mutual inductance can be determined by the experimental method. For this purpose it is necessary to measure the equivalent inductivity in case of commulative coupling and differential coupling of coils, then using formulas Lcom  L1  L2  2M , Ldif  L1  L2  2M

determine mutual inductance as

M

Lcom  Ldif 4

.

(10.25)

2. Using Kirchhoff's laws we can write the system of equations, describing an electromagnetic status of a circuit drawn in Figure 10.9  I  I1  I2 ,    U  I1Z1  I2 Z M , U  I Z  I Z . 2 2 1 M 

(10.26)

Here

Z1  R1  jX L1  R1  jL1 , Z 2  R2  jX L 2  R2  jL2 , Z M  jM . Currents in parallel branches of a circuit are U Z 2  Z M  , I1  2 Z1Z 2  Z M

U Z1  Z M  , I2  2 Z1Z 2  Z M

(10.27)

then total current is

I 

U Z1  Z 2  2Z M  . 2 Z1Z 2  Z M

(10.28)

123

From here the complex impedance of a circuit is equal Z 

2 Z1Z 2  Z M . Z1  Z 2  2 Z M

(10.29)

Figure 10.9. The parallel circuit with mutual inductance

For determination of the equivalent inductivity we will accept that in a circuit there are no energy losses, i.e. R1  R2  0 , then adding formulas for complex resistance we will obtain LE 

L1L2  M 2 . L1  L2  2M

(10.30)

In case of commulative coupling of the coils we use the sign «+» in equation (10.30), and if we have differential coupling of coils we use the sign «-».

10.3 Circuit with mutual inductance: transformer coupling

A transformer is an electrical device formed by two coils that are wound on a common core. You may have seen transformers on top of utility poles. A transformer uses the principle of mutual inductance to convert AC electrical energy from input to output. Recall that 124

mutual inductance is the ability of a coil to produce induced voltage due to changing current in another coil nearby (Figure 10.10). The transformer is a static electromagnetic device which consists of two or more connected coils and is intended for voltage transformation of an alternating current without frequency change from one coils to the other coils by means of an electromagnetic induction.

Figure 10.10. The air transformer

For determination of currents in the transformer we use the following methods of calculation: 1. using Kirchhoff's laws; 2. introduction of entered resistance and inductance; 3. by means of equivalent circuits. 1. For inductively-coupled circuits (Figure 10.10) the equations of the second Kirchhoff’s law give:

U1  I1Z11  I2 Z M ,  0  I2 Z 22  I1Z M .

(10.31)

here / Z11  Z1  jL1  Z1  jx1  R11  jx11  jx1  R11  jx11 ,

Z 22  Z 2  jL2  Z H  Z1  jx 2  Z H  /  R22  jx22  jx2  RH  jx H  R22  jx22 ,

125

Z M  jM  jxM .

From a system of equations (10.26) it is possible to determine currents in transformer circuits I1 

U1Z 22 , 2 Z11Z 22  Z M

I2 

U1Z11 . 2    Z11Z 22  Z M

(10.32)

This method of calculation of currents is convenient only for the analysis of transformers, simple in structure, for example, for air transformers. For more complex magnetic devices the system of equations made under Kirchhoff's laws has more complex form. 2. It is possible to calculate operation of the transformer by means of introduction of the entered values. Such a method is realized by conversion of the two-circuit diagram into a single-circuit (Figure 10.11). In this case two coils are considered in the singlecircuit diagram as uncoupled.

Figure 10.11. The single-circuit diagram of the transformer

For the single-circuit diagram the equation is fair





* , U1  I1 Z11  Z 22

(10.33)

* where Z 22  Z 2*  jL*2  R2*  jx2* is the entered complex impedance. For determination of the entered values we will transform the first equation (10.32) to the following form

I1 

126

U1 . 2 Z11  Z M  Z 22

(10.34)

From the equation (10.33) we will obtain I1 

U 1

.

(10.35)

* Z11  Z 22

By comparison of (10.34) and (10.35) we will determine the entered complex impedance Z 2 * Z 22  M  . Z 22

(10.36)

By conversion of equation (10.36) R2* 

x2*  

 2M 2 2 2  x22 R22

 2M 2 2 R22



2 x22

R22 

 2M 2

x22  

2 Z 22

R22 ;

 2M 2 2 Z 22

x22 .

(10.37)

(10.38)

Also formula (10.37) can be derived from the condition of equality of capacities in the second circuit. Adding these equations in (10.35) it is possible to determine current in the single-circuit diagram of the transformer. From the physical point of view the entered resistance is a resistance that is switched in series with the primary winding and that allows us to take into account the influence of current of loading I2 on current I1 . 3. For the transformer shown in Figure 10.10 (without loading impedance), a corresponding equivalent circuit is shown in Figure 10.12

Figure 10.12. T-type equivalent circuit of the air transformer

127

In case of compilation of a system of equations under Kirchhoff's laws for this T-type diagram it is possible to obtain a system of equations (10.31). This equivalent circuit allows us to calculate currents in transformer circuits. Control questions 1. Give definition of the coupled circuits. 2. Describe the main types of coupling. 3. Write formulas of the equivalent values in case of a series and parallel circuits with mutual inductance. 4. Describe a transformer type of coupling.

128

Lecture 11

THREE-PHASE CIRCUITS

Plan 11.1 Multiphase system. Basic concepts 11.2 The symmetric operation mode of a three-phase circuit 11.3 The asymmetrical operation mode of a three-phase circuit 11.4 Power of a three-phase circuit

11.1 Multiphase system. Basic concepts Combining in one circuit of several circuits of a sinusoidal current of the same frequency, similar in structure, with independent power sources creates a multiphase system. A multiphase system is a means of distributing alternating-current electrical power where the power transfer is constant during each electrical cycle. Polyphase systems have three or more energized electrical conductors carrying alternating currents with a defined phase angle between the voltage waves in each conductor; for three-phase voltage, the phase angle is 120° or 2 radians. Polyphase systems are particularly useful for 3

transmitting power to electric motors that use alternating current to rotate. The most common example is the three-phase power system used for industrial applications and for power transmission. Compared to a single-phase two-wire system, a three-phase three-wire system transmits three times as much power for the same conductor size and voltage. The phase is part of a circuit relating to the appropriate winding of the generator or transformer, the line and load. A three-phase generator was first created in 1891 by M.O. Dolivo-Dobrovolsky. The system consisting of three connected circuits was widely adopted worldwide. Practically all electrical energy is worked out on power stations by three-phase generators, it is trans129

ferred to places of consuming by three-phase transmission lines and its main share is used in three-phase receivers. In devices of rectifycation six – and twelve-phase systems are applied, and in devices of automatic equipment and telemechanics two-phase systems are used. Advantages of the three-phase system:  For a given amount of power transmitted through a system, the three-phase system requires conductors with a smaller cross-sectional area. This means saving of copper (or aluminium) and thus the original installation costs are less.  Two voltages are available.  Three-phase motors are very robust, relatively cheap, usually smaller, have self-starting properties, provide a steadier output and require little maintenance compared with single-phase motors. On the stator 1 of generator the winding 2 consisting of three parts or phases is placed (it is conditionally shown as one winding). The beginnings of phases are denoted by letters A, B and C, and ends are denoted by letters X, Y, Z. The rotor 3 represents the electromagnet excited by a direct current of the drive winding 4 located on the rotor. Figure 11.1 shows cross-section of a three-phase generator in which Figure 11.1. Three-phase the stator has three electrically isolagenerator ted phase windings. If the generator rotor has two poles, the axes of the phase windings of the generator are rotated in the space at 120° with respect to each other. When the rotor rotates, phase sinusoidal EMFs are induced in the phase windings. Due to the symmetry, the amplitude E m and effective values E are equal in all phases. However, the magnetic field lines of the rotating rotor intersect the phase windings not simultaneously, so the EMFs of the windings are shifted in phase relative to each other T  2 . If the generator rotor is multi-pole, ach pair of its poles 3 3 corresponds to 3 isolated from each other three-phase coil windings 130

in the stator. Placed along the circumference of the stator, the individual coils, whose number is equal to the number of pairs of poles of each phase winding, are connected in series or in parallel. The phases are indicated by initial capital letters of the alphabet: A, B, C . Designation of the phase sequence is not accidental. They are chosen so that the EMF of phase A reached its maximum value at 1 period before the EMF of phase В and 2 period before the EMF 3 3 of phase С. This sequence is normal or straight (Figure 11.2, а). The direction of rotation of three-phase motors depends on the phase sequence. If you change the direction of rotation of the rotor of the generator, the phase sequence changes (Figure 11.2, b) and will be called reverse. In direct sequence the instantaneous values of voltage in the phases are e A  Em sin t  ,

(11.1)

2  ,  eB  Em sin  t   3  

(11.2)

4  2    eC  Em sin  t    Em sin  t  . 3  3   

a)

(11.3)

b)

Figure 11.2. Vector diagram of a three-phase circuit

131

б) a) Figure 11.3. Conditional graphical notation of the phase of power supply

Figure 11.4. Graphs of changes in instantaneous EMF values in a three-phase circuit over time

We represent the instantaneous values in a complex form E A  Eme j  Em , A

E B  Em e j  Em e B

E C  Em e j  Em e C

132

2   j A   3  

2   j A   3  

 A  0,

(11.4)

 Em e  j120 ,  B   0

 Em e j120 , C  0

2 (11.5) , 3

2 . 3

(11.6)

j

Here e

2 3

 a is a phase factor, then E A  Em ,

(11.7)

E B  Em a 2 ,

(11.8)

E C  Em a.

(11.9)

Multiplication of a complex value by a corresponds to the rotation of the imaging vector at an angle 2  120 in the positive direc3

tion, i.e. clockwise. In calculating the three-phase actual measurements the real axes are directed vertically upwards, so in diagram 11.2 the vector ĖA is directed vertically. According to the second law of Kirchhoff and vector diagrams of Figure 11.2, the geometric vector sum of EMF of all phases is zero for the three-phase system: E A  E B  E C  0 .

(11.10)

To prove (11.10), we use the expressions (11.4) – (11.6) Em  Em e  j120  Eme j120  0

0





 Em 1  cos(120°0 )  j sin( 120 0°)  cos(120°0 )  j sin(120°0 )   1 3 1 3   0.  Em 1   j  j 2 2 2   2

If the three-phase windings shown in Figure 11.3 are kept independent, six wires are needed to connect a supply source (such as a generator) to a load (such as a motor). To reduce the number of wires it is usual to interconnect the three phases. There are two ways in which this can be done, these are: (a) a star connection or wye (Y), and (b) a triangle connection or a delta (Δ) configuration. 133

A «star» connection is a connection in which the ends of the generator windings or consumer ends of windings are connected in one point, which is called a neutral or zero point. A «triangle» connection is a configuration in which the beginning of one phase is connected to the end of the other phase. There are five simple ways of connecting a threephasegenerator, presented in Figures 11.5 – 11.6. The mode of operation of threephase circuits depends on the amplitude values of EMF sources and complex impedances of each phase of the load, so there are different balanced and unbalanced modes.

a) Star-star, four-wire connection

b) Star-star, three-wire (without neutral conductor) connection Figure 11.5. Connection of a three-phase circuit

134

c) Triangle-Star connection

d) Triangle-delta connection

e) Star-delta connection Figure 11.5. Connection of a three-phase circuit

135

Conditions of operation modes of a three-phase circuit

symmetric

asymmetric

Z p  0;

Z p  0;

Z A  Z B  Z C EmA  EmB  EmC

Z A  Z B  ZC EmA  EmB  EmC

Figure 11.6. Modes of operation of three-phase circuits

Loads that are included in the three-phase circuit can be singleor three-phase. An example of a single-phase load can be electric incandescent lamps, lighting fixtures, single-phase motors, appliances, etc. Three-phase loads are induction furnaces and three-phase asynchronous motors. If the complex impedances of three-phase loads are equal, then they are called symmetric Z A  Z B  Z C  Ze j .

(11.11)

If the condition (11.11) is not satisfied, the receiver is called asymmetric.

11.2 The symmetric operation mode of a three-phase circuit Symmetrical mode in a three-phase circuit is with a symmetrical system of voltages and symmetrical load Z A  Z B  ZC ,

i.e., RA  RB  RC  R ph and x A  xB  xC  x ph . Then the phase currents and phase angles are the same 136

I A  I B  I C  I ph 

U ph Z ph

 A   B  C    arctan

,

x ph

(11.12)

.

(11.13)

R ph

Vector diagram, corresponding to the three-phase symmetric system, EMF, is shown in Figure 11.2, a. The «star- star» connection If the ends of the phase windings of the generator X , Y , Z are combined in one common point N, called the neutral point (or neutral), and the ends of the phase detectors are connected to a single point n, such a configuration is called «Star-Star» connection (Figure 11.7).

Figure 11.7. The three-phase circuit: «four-wire star-star» connection

The wires that connect the beginning of the phase generator and load are called linear: A − a, B − b and C − c. The wire that connects the point N of the generator with the point n of the load is called a neutral wire: N − n . Connection of a three-phase circuit with a neutral conductor is called a four- wire star, without a neutral wire it is called a three-wire star. In the case of a symmetrical load, current in the neutral wire IN is equal to 0. It follows from the current vector diagram shown in 137

Figure 11.8, because the geometrical sum of the three current vectors is zero: IA  IB  IC  0 . Therefore, the neutral conductor can be excluded.

Figure 11.8. Vector diagram of currents for symmetric receiver

The three-phase circuits there are two types of voltages and currents:  line – Лl  U AB , U BC , U CA , UU II  I , I , I  ; Лl

AB

BC

CA

 phase –   U ,U ,U , UU ph Ф A B C

   . IIph Ф  I A , I B , I C 

Phase voltage is the voltage between any line conductor and the neutral conductor or the voltage applied to the source coil or to the consumer. If the wiring resistance can be ignored, the phase voltage at the receiver is considered the same as the source voltage. The direction from the beginning to the end of the phase is conditionally taken as the positive phase voltage direction. Phase current is a current flowing through the source coil or through the load. 138

Line voltage is the voltage between line conductors or the voltage between any two phases. Conditionally positive direction of line voltages is the direction from the points corresponding to the first index to the points corresponding to the second index. Line current is a current flowing through the line conductor. If you neglect the resistance of phase wires and resistance of the neutral wire, the relationship between line and phase values will be as follows. When a connection is «three-wire star-star» for symmetrical load phase currents and line currents are equal

Phase voltages are equal

I ph  Il .

(11.14)

U A  E A , U B ,  E B ,

(11.15)

U C  E C .

From Figure 11.7 linear voltages are

U AB  E A  E B , U BC  E B  E C ,

(11.16)

U CA  E C  E A . Substituting in the first equation (11.16) formula (11.15) followed by (11.4) (11.6), we obtain





 1 3 U AB  E A  E B  Em  Em cos(120 0°)  j sin( 120 0°)  Em 1   j   2   2 jarctg  3  Em 1.5  j   Eme 2  

3 1  2 1.5

°

0

 Eme j 30  Em 3.

So, U AB  Em 3, the same connection will be for the other line voltages. As E A  Em , then 139

U l  3U ph .

(11.17)

Equations (11.14) and (11.17) show the relationship between line and phase values (current, voltage) in a symmetrical three-phase circuit with «three-wire star-star» connection. «Triangle-triangle» connection If the end of each phase winding of the generator is connected to the beginning of the next phase and connected by the three line conductors to the load, connected in a similar manner,this connection is called the «triangle-triangle» (Figure 11.9).

Figure 11.9. The three-phase connection: a «triangle-triangle» connection

In a symmetrical three-phase circuit connected in such a way, the phase voltages and the line voltages are equal U A  U AB , U B  U BC , U C  U CA

(11.18)

U ph  U l .

(11.19)

so

According to Kirchhoff’s voltage law line currents and phase load currents for nodes a, b, с are connected by relations IA  IAB  ICA , IB  IBC  IAB , IC  ICA  IBC . 140

(11.20)

where U U U IAB  AB , IBC  BC , ICA  CA . Z AB Z BC Z CA

(11.21)

Substituting in the first equation (11.20) formula (11.21) and (11.18), and taking into account that the load is symmetric, i.e.

Z AB  Z BC  Z CA  Z we get U U E U U E IA  AB  CA  A  C  A  C  Z AB Z CA Z Z Z Z 

Em  Em a Em  1 3 1   j   Z Z  2 2  

3 1 

  E  3  Em jarctg   2  1.5     m 1.5  j e  Z  2  Z E  j 30  j 30 E0 1 I  m e  m  E m Л1 .  Il .  meE   Z Z Z 3 Z 33 3 0

(11.22)

Then I I ph  l 3

or

Il  3I ph .

(11.23)

Equations (11.18) and (11.23) show the relationship between line and phase values (current, voltage) in a symmetrical three-phase circuit when it is connected in the «triangle-triangle».

11.3 Unbalanced mode of a three-phase circuit Consider various options for unbalanced load in a three-phase «star star» connection: 1. The general case – Z A  Z B  ZC and Z N  0 , and Z N  0 . 2. The case when Z  0,  , a Z B  ZC . A

141

3. The case where each phase load consists of various elements of different physical nature: Z A  C , Z B  L , ZC  R . 1. In a symmetric system of voltages and asymmetrical load when Z A  Z B  ZC и  A   B  C the currents in the phases of the consumer are different and will be determined by Ohm's law

U U IA  A , IB  B , Z A Z B

U IC  C . Z C

(11.24)

Consider the unbalanced mode three-phase circuit for «four-wire star-star» connection. The current in the neutral conductor IN is equal to the vector sum of the phase currents

IN  IA  IB  IC .

(11.25)

The difference in potentials between neutral terminal of the source N and the load neutral n, can be determined by the method of two nodes

Y U  Y U  Y U U nN  A A B B C C , YA  YB  YC  YN

(11.26)

where Y  1 , Y  1 , Y  1 , Y  1 – complex admitB C N A Z B ZC Z N Z A tance of phase loads and neutral wire. The bias voltage of the neutral with symmetrical load is absent, i.e., U nN  0 , and at asymmetrical load the neutral voltage bias is U nN  0 . Phase voltages in the loads are different from each other, and according the Kirchhoff’s voltage law U A/  U A  U nN ,

U B/  U B  U nN , U C/ 142

 U C  U nN .

(11.27)

phase currents U / IA  A  U A/  YA , Z A U / IB  B  U B/  YB , Z

(11.28)

B

U / IC  C  U C/  YC . ZC Vectors of power supply phase voltages and U nN are shown in Figure 11.10. If you change the value (or character) of phase load, values of U nN also change. In this case the neutral point of load n may take different positions, and the phase voltages U A/ , U B/ , U C/ will be very different from each other.

Figure 11.10. The vector diagram of voltages and currents

For a symmetrical load the neutral wire is not taken into account, as it does not affect the phase voltage of the load. In case of an asymmetrical load and no neutral wire, the load phase voltage is not connected rigidly to the phase voltage as the load affects only linear generator voltage. The unbalanced load in such conditions causes 143

asymmetry of the phase voltages U A/ , U B/ , U C/ and shifts its neutral point n from the center of the triangle voltage (neutral displacement). Neutral shift direction depends on the sequence of phases of the system and the nature of the load. Therefore, the neutral wire is needed in order to:  align the phase voltage of the receiver at asymmetrical load;  connect single-phase receivers with the rated voltage 3 times smaller than the nominal line voltage to the three-phase circuit; If Z N  0 , thanks to the neutral wire, voltages on the windings of the generator in each phase are equal to the voltages on the loads of the respective phases (Figure 11.11). Hence, the neutral wire ensures the symmetry of the phase voltage of the asymmetrical load. Therefore, in the four-wire network single-phase unbalanced loads (for example, an electric incandescent lamp) are included. The operating mode of each phase of the threephase load, under constant phase voltage, will not depend on the operating mode of the other phases.

Figure 11.11. The three-phase circuit: «three-wire star-star» connection

2. A symmetric system of voltages and asymmetrical load when there is a single variable impedance in one phase Z A  0,  , and Z B  ZC (Figure 11.12). We assume that all the phase loads are resistive in nature 144

YA  g A 

1 1 , YB  YC  g  . R RA

neutral bias voltage takes the form g U  gU B  gU C g A Em  gEm a 2  gEm a U nN  A A   g A  2g g A  2g 

(11.29)

 1 Em  3 1 3   Em  g A  g    g A  g    j  j .  g A  2g  2 2 2   g A  2g  2

We introduce the notation m  g A , then (11.29) takes the form g

m 1 U nN  Em  . m2

(11.30)

А

Figure 11.12

Consider the following cases: 1) If R A  0 , then g A   and m  0 . Neutral voltage shift will have a certain value, i.e. U nN  Em . In this case, phase voltage: 145

U A  0, U B  U AB , U C  U CA . Phase currents can be determined from the above formulas (11.28). 2) If R A   , then g A  0 and m  0 . Neutral voltage shift will be negative, i.e., U nN   Em , in this case the phase voltages and 2 phase currents are determined by the formulas (11.27) and (11.28). 3. In a symmetric system of voltages and asymmetrical load when phase a has a capacitive load, phase b – inductive, and phase c – resistive load: Z A  C , Z B  L , ZC  R (Figure 11.13). А

Figure 11.13

Complex admittances of phase loads are 1 1 1 YA   jbC , YB     jbL , YC    g .  ZC ZB ZA

(11.31)

Shift voltage of neutrals will take a form

E  jbC  Em a 2   jbL   Em a  g . U nN  m g  jbC  jbL

(11.32)

If we accept that reactive conductivities are identical, i.e. bC  bL  b , then 146





 





jb  Em 1  a 2  Em a  g Em U nN   jb 1  a 2  ga . g g

The phase voltage and currents are also determined by formulas (11.27) and (11.28). We are interested in current in phase С

 



 E IC  E C  U nN YC   Em a  m jb 1  a 2  ga g 



 









 g  

 Em ag  Em jb 1  a  ga   Em  jb 1  a . 2

2

From this formula we see that current in phase С doesn't depend on the active admittance in case of equality of conductivities in other phases. Similarly, using the above formulas it is possible to consider connection of the «triangle» load. An important feature of such a connection of phases of the load is that in case of change of resistance of one of phases the operation mode of other phases remains invariable as the linear voltages of the generator are constants. Only current in this phase and the line currents in the wires of the line connected to this phase will change. Therefore the connection by a triangle is widely used for switching on of the asymmetrical loading.

11.4 Power in a three-phase circuit In three-phase circuits as well as in single-phase, concepts of active, reactive and apparent powers are used. Let's consider «star» connection of customers of electrical energy. Generally, for an asymmetrical load the active power of the three-phase loads is equal to the sum of the active powers of separate phases (11.33) P  PA  PB  PC , where PA  U A I A cos A , (11.34) PB  U B I B cos B , PC  U C I C cos C , 147

U A ,U B ,U C are phase voltages, IA , IB , IC are phase currents,  A ,  B , C are angles of phase shift between voltage and current. Reactive power is respectively equal to the algebraic sum of reactive powers of separate phases Q  Q A  QB  QC ,

where

Q A  U A I A sin  A , QB  U B I B sin  B ,

(11.35) (11.36)

QC  U C I C sin  C ,

Apparent power of separate phases S A  U AI A, SB  U B I B ,

(11.37)

SC  U C I C .

Apparent power of the three-phase load S  P2  Q2 .

In case of the symmetric system of voltage

U A  U B  U C  U ph and the symmetrical load IA  IB  IC  Iph ,  A  B  C  

phase powers are equal PA  PB  PC  Pph  U ph I ph cos  , QA  QB  QC  Q ph  U ph I ph sin  . 148

(11.38)

The active and reactive powers in symmetric three-phase load are equal, respectively P  3Pph  3U ph I ph cos  , Q  3Q ph  3U ph I ph sin  ,

then the apparent power

S  3S ph  3U phI ph.

(11.39)

Therefore, in case of the symmetric system of voltage and the symmetrical load it is possible to measure the power only of one phase in a three-phase circuit, and then to treble the result. In case of «triangle» connection of customers of electrical energy with the asymmetrical loads, the active power of three-phase load is equal to the sum of active powers of separate phases

P  PAB  PBC  PCA , where

PAB  U AB I AB cos AB , PBC  U BC I BC cos BC ,

(11.40)

(11.41)

PCA  U CA ICA cosCA ,

U AB ,U BC ,U CA are the linear voltages, which are equal to the phase voltage of the loads, IAB , IBC , ICA are the line currents in each phase of the loads,  AB , BC ,CA are phase shifts between voltage and current. Reactive power is respectively equal to the algebraic sum of reactive powers of separate phases Q  QAB  QBC  QCA ,

where

QAB  U AB I AB sin  AB , QBC  U BC I BC sin  BC ,

(11.42)

(11.43)

QCA  U CA ICA sin CA , 149

Apparent power of separate phases S AB  U AB I AB , S BC  U BC I BC ,

(11.44)

SCA  U CA ICA.

Similar to the star connection in case of connection by «traingle» in case of the symmetric system of voltage

U AB  U BC  U CA  U ph and for symmetrical load IAB  IBC  ICA  I ph ,

 AB   BC  CA   phase powers are equal PAB  PBC  PCA  Pph  U ph I ph cos  , QAB  QBC  QCA  Q ph  U ph I ph sin  .

The active and reactive powers of the symmetric three-phase receiver are equal, respectively

P  3Pph , Q  3Qph. then apparent power

S  3S ph  3U phI ph.

(11.45)

As usual apply the line voltages and currents, it is more convenient to express the power through the line values U l and I l . In case of «star» connection of symmetric phase loads:

U ph  150

Ul , I ph  Il , 3

in case of «triangle» connection I U ph  U l , I ph  l , 3

Irrespective of mode of connection of consumers, phases for active and reactive powers in case of the symmetrical load are defined by P  3Pl  3U l Il cos  , Q  3Ql  3U l Il sin  ,

(11.46)

where U l and I l are the line voltage and current. Let's remove indexes, then compare the above formulas and we obtain the relation between the active powers in case of «Δ» – triangle connections and star «Υ» connection (11.47) P  3P . Thus, in case of not changed line voltage, after switching the receiver from a star to a triangle, its power increases three times. Measurement of the active power in three-phase circuits is performed by means of three, two or one wattmeter, using different diagrams of their switching on. The diagram of switching on of active wattmeters for measurement of the active power is defined by the diagram of the network (three – or four-wire), the connection of phases of the receiver («star» or «triangle»), character of loading (symmetric or asymmetrical), accessibility of a neutral point. Control questions 1. Give definition to a multiphase system and the used basic concepts. 2. Describe the symmetric operation mode of a three-phase circuit in case of connection by «star». 3. Describe the symmetric operation mode of a three-phase circuit in case of connection by «triangle». 4. Describe different cases of the asymmetrical operation mode of a threephase circuit. 5. Derive formulas for different powers of a three-phase circuit under different conditions.

151

Lecture 12

NON-LINEAR ELECTRICAL CIRCUITS

Plan 12.1 Basic concepts 12.2 Graphic method of calculation 12.3 Analytical method of calculation

12.1 Basic concepts Non-linear electrical circuits contain non-linear elements whose key parameters depend on the current flowing through them or voltage applied to them. All non-linear elements (NLE) are subdivided into two groups: uncontrollable and controlled. The glow lamp, a baretter, the diode, the gasotron belong to uncontrollable non-linear elements. Сontrolled non-linear elements are three – and more electrode vacuum valves, transistors, thyristors. The relationships between current I and voltage V for a non-linear elements, called the I-V characteristics, are curves. I-V characteristics of non-linear elements can be the symmetric I(U) = -I(-U) (Figure 12.1, a) and asymmetrical I(U)  -I(-U) (Figure 12.1, b, c)

a)

b)

c)

Figure 12.1. I-V characteristics of non-linear elements

152

In non-linear elements with the symmetric I-V characteristic the resistance doesn't depend on the direction of current and voltage, and in non-linear elements with the asymmetrical I-V characteristic the resistance depends on the direction of current and voltage. The type of I-V characteristic determines the application area of nonlinear element (NE). I-V characteristic of non-linear resistive elements can be monotonic (Figure 12.1, a) and non-monotone (Figure 12.1, c). For elements with the monotonic I-V characteristic growth of current occurs with voltage increase and vice versa, voltage on an element increases with an increase in current. In this case the relation between current and voltage will be a one-valued dependence, and derivatives of I-V characteristic exist in all points of the curve, and have the positive values (or 0)

dU  0, dI

dI  0. dU

(12.1)

I-V characteristic of NE will be non-monotone if increase in voltage on clamps of an element leads to lowering of current (in the range of change of currents and voltage), or on the contrary, increase in current leads to reduction of voltage on element clamps. In general, all I-V characteristics of NE, in particular two-terminal networks, are classified into six main types:  «Saturation of current» (Figure 12.2, a);  «Saturation of voltage» (Figure 12.2, b);  « S-type uncertainty of current» (Figure 12.2, c);  « N-type ucertainty of voltage» (Figure 12.2, d);  «Hysteresis type» (Figure 12.2, e);  «λ-type» (Figure 12.2, f). I-V characteristic can have non-sensitive zones, i.e. only in case of certain values of current or voltage it is possible to observe voltampere dependence (Figure 12.3, and, b). The type I-V characteristic can also depend on another value which is directly not connected to currents or voltage of a circuit in which this element is included. For example, for the non-linear resis153

tive two-terminal network its I-V characteristic depends on temperature, illuminance, pressure, and other parameters. Such elements refer to not electrically controlled two-terminal networks.

a)

b)

d)

c)

e)

f)

Figure 12.2. The main types of I-V characteristic of non-linear two-terminal networks

a)

b)

Figure 12.3. I-V characteristic with ambiguity on voltage (a) and on current

Variable resistance of a non-linear element is set by means of IV characteristic or dependences of static and differential resistance on current or voltage. Static resistance Rst characterizes a non-linear element in the invariable mode: 154

Rst 

U . I

(12.2)

Static resistance can be determined by a tangent of angle α between the appropriate axis of coordinates and the straight line connecting the operation point with zero (Figure 12.4). Differential (dynamic) resistance Rdif is equal to the relation of an infinitesimal increment of voltage on a non-linear element to the appropriate infinitesimal increment of current: Rdif 

U . I

(12.3)

Differential resistance can be determined by a tangent of angle β of a tangent line inclination to I-V characteristic in the operation point (Figure 12.4). Differential resistance can be negative if there is a section of I-V characteristic where current decreases when voltage increases, or in case of current increase with voltage reduction.

a)

b)

Figure 12.4. Determination of characteristics of a non-linear element

The electrical status of non-linear circuits is described by Kirchhoff's laws. The general methods of calculation of non-linear circuits don't exist as all known receptions and methods have different opportunities and scopes. However, in most cases the analysis of a nonlinear circuit can be made by graphic or analytical methods. 12.2 Graphic method of calculation In this method calculation of operation of a circuit is solved by graphic plotted on the planes. At the same time characteristics of all 155

branches of a non-linear circuit will be recorded as functions of one general argument, therefore the system of equations made according to Kirchhoff's laws will be reduced to one non-linear equation with one unknown parameter. In case of application of a graphic method of calculation for different types of connections of non-linear electrical circuits use the techniques described below.

12.2.1 Circuits with series connection of resistive elements In case of series connection of non-linear resistors as the general argument the current passing through connected in series elements is always accepted. Sequence of calculation is as follows: 1. By the given I-V characteristic U i I  of separate resistors in the system of orthogonal Cartesian coordinates U  I the resultant dependence U I   U i I  is plotted. 2. On the voltage axis the point, which corresponds in the selected scale to the given input value of voltage of the circuit, is marked and from it the perpendicular up to intersection with the dependence U I  is drawn. 3. From the point of intersection of the perpendicular with the curve U I  the orthogonal is drawn to the axis of currents – the obtained point corresponds to the required circuit current. 4. By the found value of required current using the dependences U i I  , voltages U i on separate non-linear resistive elements are determined. Application of this technique is shown in Figure 12.5. Graphic solution for a circuit with a series connection of resistive elements can also be obtained by the method of intersections. In this case one of non-linear resistors, for example, U1 I  , Figure 12.4.a, is considered internal resistance of a source with the EMF E, and another – loading. Then, as shown in Figure 12.6, based on the ratio E  U1 I   U 2 I  point «a» of intersection of curves I E  U1  156

and U 2 I  defines the circuit operation mode. Curve I E  U1  of I-V characteristic is plotted by subtraction of abscissa U1 I  from EMF for different values of current.

a)

b)

Figure 12.5. Illustration of a graphic method of calculation of a circuit with a series connection of resistive elements

Use of the method of intersections is most rational in case of series connection of the linear and non-linear resistors. In this case the linear resistor is accepted to be the internal resistance of a source, and its linear I-V characteristic is plotted on two points. 12.2.2 Circuits with a parallel connection of resistive elements In case of a parallel connection of non-linear resistors voltage applied to the elements connected in parallel is accepted as the common argument.

Figure 12.6. Illustration of the method of intersections

Algorithm of calculation is as follows: 1. By the given I-V characteristic U i I  of separate resistors in the system of Cartesian coordinates U  I the resultant dependence I U    I i U  is plotted. 157

2. On the voltage axis the point, which corresponds in the selected scale to the given input value of voltage of the circuit, is marked and from it the perpendicular up to intersection with the dependence U I  is drawn. 3. From the point of intersection of the perpendicular with the curve U I  the orthogonal is drawn to the axis of voltages – the obtained point corresponds to the required voltage on non-linear resistors. 4. By the found value of required voltage using the dependences U i I  , currents I i in branches with separate resistive elements are determined. Application of this technique is shown in Figure 12.7. 12.2.3 Circuits with series-to-parallel (mixed) connection of resistive elements The sequence of calculation of circuits with the mixed connection is as follows: 1. The initial diagram is reduced to a circuit with a series connection of resistors for which the resulting I-V characteristic of parallel connected elements is plotted. 2. Calculation of the obtained diagram with series connection of resistive elements is carried out, based on which currents in the initial parallel branches are determined.

a)

b)

Figure 12.7. Illustration of a graphic method of calculation of a circuit with resistive elements connected in parallel

158

12.3 Analytical method of calculation The cornerstone of the method is reduction of the task on finding a periodic solution of the non-linear equations to the determination of the periodic solution of a system of linear equations. The main stages of application of the method are: 1. Replace the main characteristic (volt-ampere, Weber – ampere, coulomb-volt) of a non-linear element for the instantaneous values by segments of direct lines. 2. Substitute in non-linear differential equations – the equations of direct lines in 1. Each non-linear equation will correspond to as many linear equations as many segments of straight lines replace the characteristic of a non-linear element; 3. Solve the system of linear differential equations. Each line section of the characteristic of a non-linear element will correspond a solution with the integration constants corresponding to it; 4. Determine constants of integrations by coordination of the solution on one line section with the solution on other line sections. Control questions 1. Give definition of non-linear circuits and elements. 2. Describe classification of non-linear elements based on I-V characteristic. 3. Explain the principle of application of a graphic method for calculation. 4. Explain the principle of application of an analytical method for calculation.

159

Lecture 13

ELECTRICAL CIRCUITS OF NONSINUSOIDAL CURRENT

Plan 13.1 Basic concepts 13.2 Fourier series representation of non-sinusoidal functions 13.3 Characteristics of non-sinusoidal quantities 13.4 Properties of the periodic curves of non-sinusoidal current having symmetry 13.5 Power in a circuit of non-sinusoidal current 13.6 Calculation of non-sinusoidal current in electric circuit

13.1 Basic concepts In the previous lectures we considered the linear electrical circuits with sinusoidal currents and voltages. In reality these quantities may be more or less non-sinusoidal. First of all, it is connected to the fact that real generators of current and voltage not always provide signals of strictly sinusoidal form, and also existence of elements with nonlinear characteristics gives distortion of the form of currents even in case of ideal sine wave voltages or EMF (Figure 13.1). In practice the non-sinusoidal voltage and currents can be considered from two points of view:  In the first case it is a negative side: in power industry nonsinusoidal currents cause additional losses of power, a moment pulsation on a shaft of engines, cause noises in communication lines; therefore here «all methods» of maintenance of the sinusoidal modes are necessary;  In the second case it is a positive side: in circuits of automatic equipment and communication where non-sinusoidal currents and voltage are the cornerstone of the principle of operation of electrotechnical devices, the task on the contrary consists in their gain and transmission with the smallest distortions. 160

Causes of nonsinusoidality

Dependence on time of non-sinusoidal currents and voltage can be periodic, almost periodic and non-cycle. In the lecture we consider periodic non-sinusoidal currents and voltage, which in turn can have the property of symmetry on time diagrams (Figure 13.2). Periodic non-sinusoidal currents and voltage are called the currents and voltage changing over time under the periodic non-sinusoidal law.

Imperfection (nonideality) of sources of sinusoidal voltages and currents

Presence of nonlinear elements

Presence of voltage generators and currents of special shape (rectangular, sawtooth, etc.)

Figure 13.1. Origins of non-sinusoidal currents and voltage in ECs

The following main operation modes of ECs may cause nonsinusoidal currents. 1. ECs operation mode when the EMF source (or a current source) gives non-sinusoidal EMF (or current), and all ECs elements: a resistor, an inductor and a capacitor are linear, i.e. don't depend on current. 2. ECs operation mode when the EMF source (or a current source) gives sinusoidal EMF (or current), but one or several ECs elements (a resistor, an inductor and a capacitor) are non-linear. 3. ECs operation mode when the EMF source (or a current source) gives non-sinusoidal EMF (or current), and one or several ECs elements (a resistor, an inductor and a capacitor) are non-linear. 4. ECs operation mode when the EMF source (or a current source) gives constant or sinusoidal EMF (or current), but one or several ECs elements (a resistor, an inductor and a capacitor) periodically change over time.

161

a)

b)

c)

d)

Figure 13.2. Periodic non-sinusoidal values (current or voltage): a, b, c – symmetric; d – asymmetrical quantity

Figure 13.2. An operation mode which corresponds to point 2: electric circuit (a), temporal implementation and I-V characteristic of an element of a circuit

13.2 Fourier series representation of a periodic non-sinusoidal function Any non-sinusoidal periodic function f ( x) , which has a finite number of maxima, minima and ruptures of the first kind for the complete period can be decomposed in Fourier series. 2 t , where T If variable x is connected to time t by ratio x  t  T is the period, then the function period on a variable x is equal to 2 , and the period of the same function on time t is equal to T. 162

f ( x)  A0  A1/ sin( x)  A2/ sin(2 x)  A3/ sin(3x)  ... ...  A1// cos( x)  A2// cos(2 x)  A3// cos(3 x)  ...,

(13.1)

where A0 is a constant component or zero harmonic, A1/ is the amplitude of sine (changing by the law of «sine») component of the first harmonic, A1// is the amplitude of cosine (changing by the law of «cosine») component of the first harmonic. Then 2 1 (13.2) A0  f ( x)dx ; 2 0 A1/ 

A1// 

1



1

 2



2



/ f ( x)sin( x)dx , … Ak 

0

f ( x)cos( x)dx , … Ak// 

0

1

2

 1



 f ( x)sin(kx)dx ;

(13.3)

0 2



f ( x)cos(kx)dx . (13.4)

0

Ak/ and Ak// are also called Fourier series coefficients. As Ak/ sin(kx)  Ak// cos(kx)  Ak sin(kx   k ) ,

(13.5)

where

A  A  / 2 k

Ak 

tg ( k ) 

// 2 k

,

Ak// . Ak/

(13.6) (13.7)

A set of amplitudes Ak and initial phases  k is called respectively amplitude and phase frequency spectra, and a set of coefficients Ak/ и Ak// is the frequency spectra of function. Then the Fourier series (13.1) can be written as: f ( x)  A0  A1 sin( x   1 )  A2 sin(2 x   2 )  ... 

f ( x)  A0   Ak sin(kx   k ) ,

(13.8)

k 1

163

where Ak is the amplitude of the k-th harmonic of a Fourier series. In case of k=1 the component of a Fourier series has a frequency equal to the function frequency f ( x) , which is also called the fundamental frequency or the first harmonic. If k  2n, n  1,2,3... is even number harmonics are called even and if k  2n  1, n  0,1,2... is odd number, harmonics are called odd.

13.3 Characteristics of non-sinusoidal values The main characteristics of non-sinusoidal alternating currents and voltage are the following parameters: 1. Maximum value of non-sinusoidal current (or voltage) imax . 2. Effective value or root mean square (rms) of non-sinusoidal current (or voltage) T

I

1 2 . i dt T 0

(13.9)



If i(t )   I km sin(kt   k ) , then (13.9) takes a form k 0

I 2  I 02  Ik 

1  2 ,  I km 2 k 1

I km , I 2



I k 0

2 k

(13.10)

3. An average by the module value of non-sinusoidal current (or voltage) 1T (13.11) I av   i dt . T0 4. An average for the period value of non-sinusoidal current (or voltage) 164

T

1 I 0   idt . T0

(13.12)

5. Amplitude coefficient, that is the relation of the maximum value of non-sinusoidal current (or voltage) to its effective value

imax . I

KA 

(13.13)

6. Form factor of a complex waveform whose negative halfcycle is similar in shape to its positive half-cycle is defined as: Kf 

I .

(13.14)

I av

7. Coefficient of distortions that is quotient of effective value of the first harmonic to the effective value of a variable I Kd  1 . I

(13.15)

8. Coefficient of harmonics is the ratio of effective values of the higher harmonics to the effective value of the first harmonica

 I k2

Kh  k 2 I1

.

(13.16)

13.4 Properties of periodic curves of non-sinusoidal current having symmetry Fourier series coefficients for fixed functions can be taken from reference books or can be generally calculated by the above-stated formulas. However, in case of the curves having symmetry, the task becomes much simpler as the whole ranges of harmonics fall out of 165

their expansion. The curves of non-sinusoidal current obeying the periodic law can have the following properties of symmetry: 1. If a periodic non-sinusoidal current is presented on the time diagram in the form of a curve which has a property of symmetry along the abscissa axis, i.e. f ( x)   f ( x  T ) (Figure 13.3). 2

Figure 13.3. Examples of symmetric curves

Then in case of function expansion in a Fourier series in the equation a constant component A0 and even harmonics Ak , where k  2n, n  1,2,3... , are absent i.e. A0  A2  A4  A6  ...  0 2. If a periodic non-sinusoidal current is presented on the time diagram in the form of a curve which has property of symmetry along the ordinate axis, i.e. f (t )  f (t ) . Then in case of function expansion in a Fourier series in the equation there are no sine components Ak , where k  1,2,3... , i.e. A1/  A2/  A3/  ...  0 .

Figure 13.4. Examples of symmetric curves

3. If periodic non-sinusoidal current is presented on the time diagram in the form of a curve which has property of symmetry with respect to the origin of coordinates, i.e. f (t )   f (t ) . Then in case of function expansion in a Fourier series in the equation there will be no constant – A0 and cosine components Ak , where k  1,2,3... , i.e. A0  A1//  A2//  A3//  ...  0 . 166

Figure 13.5. Examples of symmetric curves

13.5 Power in a circuit of non-sinusoidal current The active (average) power of a periodic current of complex form can be defined as the average power for the period: T

P

1  u  i  dt T 0

(13.17)

If circuit current 

i(t )   I km sin(kt   I k ) k 0

and voltage 

u (t )  U km sin(kt   U k ) , k 0

then

1     . U sin( k  t   )  km U k     I km sin( kt   I k )  dt   T 0  k 0   k 0  T

P

As the average for the period value of the work of sine functions of different frequency will be equal to zero, the last expression takes a form T 1  P   U km I km sin(kt   U k )sin(kt   I k )dt → T 0 k 0

U km I km cos(k )   U k I k cos(k ) , (13.18) 2 k 1 k 0 

P  U0 I0  

167

where k   U  I . If Pk  U k I k cos(k ) , then k k 

P   Pk .

(13.19)

k 0

The average power of the non-sinusoidal current is equal to the sum of average powers of separate harmonics. We obtain similar expression for the reactive power if Qk  U k I k sin(k ) , then 

Q   Qk .

(13.20)

k 0

If one of the curves (voltage or current) contains harmonics, which are not present in the other, this does not affect the value of the active and reactive power, but it raises the effective value of the first function. Therefore, if the apparent power in this circuit is defined as the product of the effective values of voltage and current: S  UI 





k 0

k 0

U k2  I k2  P 2  Q 2  T 2  P 2  Q 2 , (13.21)

where T is called the distortion power. It characterizes the degree of difference between the shapes of the voltage curve u(t) and current curve i(t).

13.6 Calculation of electric circuits of non-sinusoidal current If in the linear ECs there are sources of non-sinusoidal currents and voltage (EMF), then for the correct calculation of operation of such circuit it is necessary to adhere to the following sequence: 1. It is necessary to decompose all non-sinusoidal currents and voltage in a Fourier series. Then select constant and sinusoidal components. 168

2. Apply the method of superposition and calculate current and voltage for each component separately. 3. Integrate all obtained results, to find required values. Let's consider ECs with a voltage source, its EMF instantaneous value changes under the non-sinusoidal law. First of all, we will expand this function in a Fourier series, having selected a constant and sine components:

e(t )  E0  E1m sin(1t   1 )  E2 m sin(2t   2 ) . Then we see that the action of a source of such EMF is similar to the action of three sequentially connected sources of EMF: e0 (t )  E0 ; e1 (t )  E1m sin(1t   1 ); e2 (t )  E2 m sin(2t   2 ). Figure 13.6. Representation of non-sinusoidal EMF

Applying a method of superposition we define the values of the partial currents from each EMF separately on each section of the circuit. Then the true current in the k -th branch is equal to the sum of the partial currents: ik  I 0 k  i1k  i2 k . When solving problems, it is necessary to consider that for different frequencies the inductive and capacitive reactances will be different. Relation for them is written as follows: X Lk  k L  kX L1 ,

(13.22)

1 X  C1 , kC k

(13.23)

X Ck 

i.e. the inductive reactance for the k-th harmonics is k times more, and capacitive reactance is k times less, than for the first (main) harmonic. It follows from this that in case of increase of inductive 169

reactance in the circuit having a source of non-sinusoidal voltage; the form of a curve of current will aim at the form «sine curves». Because of existence of the higher harmonic components the non-sinusoidal current will be in direct dependence on capacitance even if in a circuit of sine wave voltage. Control questions 1. Give definition of nonsinusoidal quantities. 2. Write nonsinusoidal quantities by means of Fourier series. 3. Formulate definition and write equations for characteristics of nonsinusoidal quantities. 4. Describe properties of the periodic curves for nonsinusoidal current having symmetry. 5. Write the equation of apparent power in a circuit of nonsinusoidal current. 6. Explain the principle for calculation of electric circuits of nonsinusoidal current.

170

Lecture 14

TRANSIENTS IN LINEAR ELECTRICAL CIRCUITS

Plan 14.1 Transients. Commutation laws 14.2 Methods of transient analysis 14.3 Classical method for calculation of transient phenomena 14.4 Operator method for calculation of transient phenomena

14.1 Transients. Commutation laws Up to now we have been considering the steady state in linear networks, i.e. conditions in which voltages and currents are either invariable in time (DC circuits) or are periodic time functions (AC circuits). As a rule, the steady state, which is different from the circuit's initial operating conditions, is preceded by a transient in which voltages and currents change nonperiodically. The transition from one set of operating conditions to another can be affected by switching circuit elements in or out, changing the circuit's parameters, or changing the circuit itself. Transient is processes of transition from one operation mode of ECs to another, something differing from the previous one, for example, in value of amplitude, a phase, the form or frequency operating in the diagram EMF, diagram parameter values and also owing to configuration change of a circuit. Periodic regimes are the modes of a sinusoidal and direct current, and also the mode of absence of current in circuit branches. Transient phenomena are switching in a circuit. Switching is that process of closing or breaking of switches.

a) a switch on

b) a switch off Figure 14.1. Switching

171

Physically transient represents processes of transition from the energy state corresponding of the state before commutation to the energy state after commutation. Transients are rapid, their duration is the tenth, 100-th and less fractions of a second, however, their study is very important since it enables us to determine how the form and amplitude of signals is deformed in case of their passing through amplifiers, filters and other devices, it allows us to reveal exceeding of voltage on separate sections of a circuit which can be dangerous for insulation of installation, increase in amplitudes of currents, which can exceed the amplitude of current of the set periodic process in tens of times, and also to determine duration of transients. There are two types of circuit states in RL, RC or RLC circuit, the transient and steady states. The transient state is the dynamic state that occurs by a sudden change of voltage, current, etc., in a circuit. This means the dynamic state of the circuit has been changed, such as the process of charging/discharging a capacitor or energy storing/releasing for an inductor as the result of operation of a switch. The steady-state is an equilibrium condition that occurs in a circuit when all transients have finished. It is the stable-circuit state when all the physical quantities in the circuit have stopped changing. For the process of charging/discharging a capacitor or energy storing/releasing for an inductor, it is the result of operation of a switch in the circuit after a certain time interval. The analysis of a transient by the classic method (by solving differential equations) is, in the end, reduced to the following:  compiling a differential equation to describe the process under examination from Kirchhoff's laws ;  finding a general solution of a homogeneous differential equation (i.e. with the excitation function equal to zero) which corresponds to the free state of the network (complementary solution or natural response);  finding the components of the forced state (particular solution, forced or steady-state response);  determining the integration constants contained in the general solution of a homogeneous differential equation in accordance with initial conditions ;  adding together the electrical quantities of the natural and forced responses. 172

Let's consider the diagram provided in Figure 14.2. We write Kirchhoff’s second law for this circuit: uL  iR  e, di L  iR  e, dt

(14.1)

Figure 14.2

The general solution (14.1) consists of the solution of a homogeneous differential equation and a particular solution. The result of particular solution is the equation i  E , where E is a constant

R

component of EMF. The homogeneous differential equation can be obtained if we equate the right part (14.1) to zero: L

di  iR  0 . dt

(14.2)

The solution of equation (14.2) is i  Ae pt , where A and p are some coefficients, not time-dependent. Let's suppose t  0 , this time corresponds to the switching moment, then A 

E, R p   and the common solution of the equaR L

tion (14.1) takes a form i

E E  RL t  e R R

(14.3)

The first term is the particular solution of the non-uniform equation, which is called the forced or steady-state response, the second 173

item is the common solution of a homogeneous equation, that is a free component of the quantity (current or voltage). Then

i  i for  i fr .

(14.4)

The forced component (current or voltage) physically represents the component changing with the same frequency, as the operating forced EMF in the diagram. If EMF changes under the harmonic law, then calculation of the forced component is produced by means of a symbolical method and if EMF has a constant we usually use the known methods for calculation of direct current circuits. The direct current doesn't flow through a capacitor in circuits with sources of constant EMF, therefore the forced current component iC , for  0 . In case of a direct current the forced voltage component across the inductor is uL, for  0 . The free component of the quantity (current or voltage) physically represents a component that is free from driving force. The free component is the solution of a homogeneous equation (14.2). In the linear electrical circuits the free components of currents and voltage fade in time by the exponential law e pt . Total current i is a current which actually flows in this or that branch of a circuit in case of transient. Total voltage is a voltage that is actually available between some points of an electrical circuit in case of transient. Values of the total current and voltage can be written and measured. The forced and free components of currents and voltage during transient play a supporting role. They are estimated components which in the sum give the valid values of the total current and voltage. In case of any transition and steady processes two basic rules are observed: the current passing through the inductive element and the voltage on a capacity element can't change abruptly. Let's consider the circuit shown in Figure 14.2, we will accept that current and EMF can accept finite values. Let's say that in the equation (14.1) current i can change abruptly. The step of the current 174

means that for an infinitesimal interval of time t  0 current will change by a finite value i , then i . t

If in the equation (14.1) instead of L di add ∞, then the left part

dt

won't be equal to the right and the second law of Kirchhoff won't be executed. Therefore, the assumption about a possibility of spasmodic change of the current flowing through the inductive element contradicts the second law of Kirchhoff. The current passing through the inductance can't change abruptly, but the voltage on this element equal L di can abruptly change.

dt

Let's consider the circuit shown in Figure 14.3.

Figure 14.3

According the second law of Kirchhoff

iR  uC  E here i  dq , q  uC C , therefore

dt

iC

duC , dt 175

then

RC

duC  uC  E . dt

(14.5)

If to assume that uC can change spasmodically, then

uC duC   t dt and the left member of equation (14.5) won't be equal to the right. Thus, it also contradicts the second law of Kirchhoff. Voltage on a capacity element can't change spasmodically, but du the current passing through this element equal C C can change by dt spets. Two laws of switching follow from the above. First commutation law: The current passing through the inductive element just before switching iL 0 _  is equal to the current passing through the same inductive element directly after switching iL 0  : iL 0 _   iL 0 

(14.6)

Time t  0 _ represents the time just before switching, t  0 – time at the time of switching, t  0 – time at the first moment after switching. Second commutation law: Voltage on a capacity element just before switching uC 0 _  is equal to voltage on the same capacity element just after switching uC 0  : (14.7) uC 0 _   uC 0  . The essence of the so-called commutation laws is that the current in the inductance and the voltage across the capacitance remain constant at the instant of commutation. 176

14.2 Methods of transient analysis For calculation of transient phenomena in electrical circuits five main methods of calculation are widely used: classical, operator, the frequency method, a method of calculation by means of Duhamel integral and a method of state variables. Classical method consists in direct integration the differential equations describing an electromagnetic status of a circuit. These equations are based on Kirchhoff's laws for nodes and loops. The operator method consists in the solution of system of algebraic equations in Laplace representations of required variables with the subsequent inverse Laplace transform from the found representations to originals. Originals are original functions of variables, representations are the functions of variables transformed to an operator mode. Laplace transforms and Kirchhoff's laws are used. The frequency method is based on Fourier transform. It finds broad application in case of the solution of problems of circuit synthesis. The method of calculation by means of Duhamel integral is used in case of irregular shape of a curve of the perturbing action. The method of state variables represents the ordered method of determination of an electromagnetic status of a circuit on the basis of solution of a system of the differential equations of first order written in a normal form (Cauchy form). General stages of application of methods: 1. Select the positive directions of currents in branches of an electrical circuit. Define also the number of nodes, branches and independent loops. 2. Define initial conditions: values of currents and voltage just before commutation. In electrical circuits for transient analysis define independent and dependent, null and nonzero, major and minor starting values of currents and voltage. Independent initial conditions are values of all currents passing through inductance and voltage on capacities in the diagram before commutation. 177

Dependent initial conditions – values of all remaining currents and voltage, in case of t  0 in the circuit after commutation which are determined by independent starting values from Kirchhoff's laws. Zero initial conditions – if by the beginning of transient just before switching all currents and all voltages on undriven elements of an electrical circuit are equal to zero. Nonzero initial conditions – if by the beginning of transient at least a part of currents and voltages in an electrical circuit are not equal to zero. For complex electrical circuits with many drives of energy there are major and minor independent initial values. The major independent starting values are currents in the inductive elements and voltage on capacitive elements which can be set irrespective of others. The minor independent initial values are the remaining. Values of currents and voltage during a time frame t  0 _ , i.e. just before switching, are initial values. During a time frame t  0 , i.e. directly after switching, the values of currents and voltages are called initial values after commutation. 3. A characteristic equation is written out and its roots are determined. Here the circuit after commutation is considered. For full currents and voltages under Kirchhoff's laws the equations are written, from which an electromagnetic status of the circuit is determined. Unknown values are determined. 4. Obtain expressions for required currents and voltages as functions of time. 14.3 Classical method of calculation of transient phenomena It is a method of calculation of transient phenomena in which the solution of a differential equation represents the sum of the forced and free components. Determination of the constants of integration entering the expression for the free current (voltage) is made by the joint solution of the system of linear algebraic equations for the known values of roots of a characteristic equation, and also for the known values of the free component of current (voltage) and its derivatives taken in case of t  0 , i.e. directly after switching. 178

Let's consider application of a classical method of calculation on the example of the electrical circuit presented in Figure 14.4. 1. This circuit contains the inductive and capacity elements, has m  2, n  3, ncs  0, k  n  m  ncs  1  2 . In branches of the circuit currents are iL , i, iC

Figure 14.4

2. Zero initial conditions are set, as during the period before switching all values of currents and voltage are equal to zero. As in the first branch the key is off iL (0 _)  0,

and according to the first law of Kirchhoff iL (0 _)  i  iC

and through capacity a direct current doesn't pass: iC (0 _)  0 , therefore, i(0 _)  0 . Voltage on the capacity is defined from the equation uC (0 _)  i (0 _)  R2 ,

then uC (0 _)  0 .

3. Now we will consider the circuit after commutation. According to Kirchhoff's laws we will work out the equations describing an electromagnetic status of a circuit 179

 iL  i  iC  0,  diL   iR2  E , iL R1  L dt  1  C  iC dt  iR2  0.

(14.8)

Let's write the first two equations (14.8) for the forced current components iL, for  i for  iC , for  0,  (14.9)  diL, for i R  L  i R  E .  L, for 1 for 2 dt  Considering that iC , for  0 , uL, for  L

diL, for dt

 iL , for  i for  iC , for  0,   iL , for R1  i for R2  E.

 0 , let's obtain

(14.10)

E . R1  R2 Thus, we defined the forced current components. For determination of the free component currents we will write (14.8) in the following way  iL, f  i f  iC , f  0,  diL, f  (14.11)  i f R2  0, iL, f R1  L dt  1  C  iC , f dt  i f R2  0 Then iL , for  i for 

Let's transform the system of equations (14.11) to the algebraic form. Let's take into account that i f  Ae pt , then 180

diL, f

   

d Ae pt  Lp Ae pt  LpiL, f , dt

(14.12)

1 1 1 1 iC , f dt   Ae pt dt  Ae pt  iC , f . C C Cp Cp

(14.13)

L

dt

L

System of equations according Kirchhoff's laws, written for the free component of currents  i  i  i  0, f C, f  L, f iL, f R1  LpiL, f  i f R2  0,   1 iC , f  i f R2  0  Cp

(14.14)

Let's make determinant of system (14.14) and having equated it to zero, we will obtain a characteristic equation 1   R1  Lp 0

1 R2  R2

1 1 1 0  1  R2   R2 R1  Lp   R1  Lp  0 Cp Cp 1 Cp

p 2 LCR2  p(CR2 R1  L)  R1  R2   0 .

(14.15)

Usually, the degree of the characteristic equation is equal to the number of basic independent initial values in the post-commutation circuit after its maximum simplification and does not depend on the type of EMF of EMF sources in the circuit

  nL  nC  mL  kC ,

(14.16)

nL is the number of inductances in the diagram, nC is the number of capacities. mL is the number of inductances in which currents can't 181

be set randomly (the number of nodes in which only the branches containing inductor coils meet), kC is the number of capacities across which voltage can't be set randomly (or the number of circuits of the diagram which branches contain only condensers).  If the characteristic equation of a circuit is the equation of the first power, then the parameter of attenuation has one value for all currents in branches of the diagram, i.e. all circuit has the same transient phenomenon and the equation for free current component is

i f  Ae pt .

(14.17)

 The constant of integration of A is determined by value of the free current in case of t  0

i f 0   A.

(14.18)

 If the characteristic equation is of the second power and its roots are real and aren't equal, then

i f  A1e p1t  A2e p2t . Let's differentiate the equation on time

if  p1A1e p1t  p2 A2e p2t . also we will write the equations in case of t  0 , then we will get

 i f 0   A1  A2 ,     i 0  p A  p A  f 1 1 2 2  here i f 0, if 0 , p1, p2 are known values. The joint solution of the last system of equations will give values of constants of integration

182

if 0   p2i f 0   ,  A1  p1  p2   A  i 0   A . 1  2 f

(14.19)

 If roots of a characteristic equation are complex conjugate, then

i f  Ae pt sin 0t   ,

where ω0 – cyclic frequency and attenuation coefficient δ are known from the solution of a characteristic equation. unknowns A and ν in this case are also determined by values i f 0 , if 0  . Differentiating the last equation we will get

i f   Aet sin 0t     A0et sin 0t    . At t  0

i f 0  A sin  ,  if 0   A sin    A0 sin  

The value of a constant of integration A and parameter of initial phase ν are determined. As the characteristic equation (14.15) is square, the parameter of attenuation has two values p1, p2 and values of constants of integration are determined by formulas (14.19). 4. Full currents in branches of the considered circuit iL t   iL, for  iL, f , it   i for  i f ,

(14.20)

iC t   iC , for  iC , f .

Stages of application of a classical method of calculation of transient phenomena: 1. Select the positive directions of currents in the branches of the electrical circuit. 183

2. Determine the initial conditions: the values of currents and voltages immediately before commutation. 3. Construct a system of equations according to the laws of Kirchhoff. 4. Determine the forced values of currents and voltages in the circuit. 5. Write a system of equations for the free components of the currents. Construct the characteristic equation and determine its roots. 6. Obtain expressions for the desired currents and voltages as a function of time in the form of a sum of forced and free components.

14.4 Operator method of calculation of transient phenomena The essence of the operator method is that the function of time f(t) (original), defined for all real numbers t ≥ 0, is transformed into function F(p), which is called representation (image). New variable p = a + jb is a complex variable. The image F(p) of the given function f(t) is defined according to direct Laplace transform: F ( p) 



 f (t )e

 pt

dp

.

(14.21)

0

In abbreviated form compliance between the image F(p) and original f(t) is designated as F  p   f t  .

The elementary operator ratios for some functions found in the analysis of transient regimes in electrical circuits are given as an example in Table 14.1. 184

Table 14.1 Original 1 t 1



Image

Original

Image

1 p 1

t

1 p

e

sin t

p2

1  e    t

1 p p   

cos t

 p2   2

p p2   2

Using properties of transformation, the operator equivalent circuit (Table 14.2) is formed. Transition to images allows us to replace the system of integraldifferential equations by an algebraic linear equation system, whose transformation gives an image of required value. Then by means of inverse transformation of Laplace its original is determined f (t ) 

1 a  j pt  F ( p)e dt . 2j a  j

(14.22)

Table 14.2 Original 1

Operation expression 2 E p 

Equivalent circuit 3

E p

U ( p)  R  I ( p)

UL ( p)  Lp  I ( p)  L  i(0)

185

1

2

UC ( p ) 

3

U c (0) I ( p)  p Cp

Ohm's and Kirchhoff's laws in the operator form Let's consider part of the circuit shown in Figure 14.5. When closing a key in an external circuit there will be the transition process, in this case initial conditions for current in branches and voltage on the capacitance generally will be nonzero. For the instantaneous values of voltage it is possible to write uab t   u R t   u L t   uC t   et  ,

adding the appropriate formulas we will obtain

uab t   Ri  L

di 1 t  it dt  uC 0  et  . dt C 0

(14.23)

Figure 14.5

Let's transform originals of voltage into operator images U ab  p   RI  p   LpI  p   Li0 

1 u 0 I  p   C  E  p  , (14.23) Cp p

here Li0 is the internal EMF caused by the energy of a magnetic field of the inductive coil L due to current i 0 passing through it just before switching, uC 0  is the internal EMF caused by a stored p 186

energy of an electric field of the capacitor due to presence of voltage on it uC 0 just before switching. Let's express current

I  p 

uC 0  E p p . 1 R  Lp  Cp

U ab  p   Li0 

(14.24)

This relation expresses the Ohm's law in an operator form. Then total operator impedance of a circuit Z  p   R  Lp 

1 . Cp

(14.25)

According to formula (14.24) in Figure 14.6, the operator equivalent circuit of the circuit in Figure 14.5 is shown. In that specific case, in case of zero initial conditions, when in the considered branch ab there is no EMF et  and by the time of switching i0  0 and uC 0  0 , the law of Ohm in the operator form takes a simpler form:

I  p 

U ab  p  . Z  p

(14.26)

Figure 14.6. The Operator equivalent circuit of a circuit

Kirchhoff's current law in operator form: the algebraic sum of operator images of the currents meeting in a node is equal to zero: 187

n

 I  p  0 .

(14.27)

k 1

Kirchhoff's current law in operator form: the algebraic sum of operator images of voltage on stationary elements around any closed loop is equal in a closed circuit to the algebraic sum of operator images of the EMF operating in this circuit: m

U k  p  

k 1

n

 Ek  p  .

(14.28)

k 1

Taking into account nonzero initial conditions the second law of Kirchhoff takes a form



m

n

k 1

k 1

 I k  p Z k  p     Ek  p   Lk ik 0 

uCk 0  . p 

(14.29)

All procedures and methods of equations for electrical circuits based on Kirchhoff's laws (mesh current method, nodal potentials method, superposition method, the equivalent method, etc.) can be applied also in case of compilation of the equations to images. Stages of application of the operator method of transient calculation: 1. Select the positive directions of currents from branches of the electrical circuit. 2. Define initial conditions: values of currents and voltage just before commutation. 3. Transform all given values from originals to their images by means of direct Laplace transform or tables. 4. Make the operator of equivalent circuit taking into account all initial conditions. 5. For the operator of equivalent circuit write a system of equations according Kirchhoff's laws. 6. Define unknown currents and voltage which are presented in the operator form. 7. Transform the found images to originals. 188

Transformation from the obtained operator image of required function to the original can be realized: 1. By means of Laplace inverse transform; 2. According to tables of correspondences between originals and operator images; 3. By means of the expansion formula. The first method is carried out by means of formula (14.22). In special literature there is a table of correspondences in which according to the image it is possible to find expression of the original. It is the second method. In the third method consider the obtained image in the form of the relation of two polynomials

F  p 

N  p  am p m  am 1 p m 1  ...  a1 p  a0 ,  M  p bn p n  bn 1 p n 1  ...  b1 p  b0

(14.30)

and the power of numerator is less than denominator power (m < n), and polynomial M(p) = 0 has no multiple roots. Considering properties of images, we will obtain expression for the original n Np  p t k (14.31) f t    e k ,    M p k 1 k here M  pk  is the first derivative in parameter p. Expression (14.31) represents the expansion formula which is widely used when calculating and is considered to be the basic formula for transformation from the image to function of time. Control questions 1. Give a definition of transient phenomenon. Explain commutation laws. 2. Tell about methods of transient analysis. 3. Explain stages of application of a classical method of calculation of transient phenomena. 4. Explain stages of application of an operator method of calculation of transient. 5. Write Ohm's and Kirchhoff's laws in an operator form.

189

Lecture 15

FOUR-TERMINAL NETWORKS

Plan 15.1 Basic concepts of the theory of four-terminal networks 15.2 Systems of four-terminal network equations 15.3 Input impedance of four-terminal network 15.4 Characteristic (secondary) parameters of four-terminal network 15.5 Gear functions of four-terminal Network

15.1 Basic concepts of the theory of four-terminal networks Any part of an electrical circuit with two pairs of terminals may be represented as a for-terminal network (Figure 15.1). Thus, a line, a filter, a transformer, an amplifier, an attenuator and any other device with two terminal pairs joining a source and a load, may be referred to as a four-terminal network. The terminals of a four-terminal network to which an energy source is connected, are called the input terminals and those to which the load is connected are called the output terminals. For brevity, the terms input and output are used.

Figure 15.1

Four-terminal networks can be classified by various features. According to the linearity of the elements which they contain, they are divided into linear and non-linear networks. Only linear networks are considered below. 190

There are active and passive four-terminal networks. A four-terminal network is called active if it contains non-compensated energy sources. Then, if these sources are independent, when the four-terminal network is disconnected from the rest of the circuit a voltage appears at its open terminals (from one end or from both). This type of active four-terminal network is called an autonomous four-terminal network. We distinguish between symmetrical and unsymmetrical fourterminal networks. A four-terminal network is symmetrical if the currents and voltages in the circuit to which it is connected do not vary when its input and output terminals are reversed. In the opposite case the network is unsymmetrical. A four-terminal networks is symmetric if symmetry to OY axis remains (Figure 15.2)

Figure 15.2

The four-terminal network is balanced if symmetry with respect to OY axis remains (a Figure 15.2). Otherwise a four-terminal networks will be unbalanced. A four-terminal network is called reciprocal or bilateral if it obeys the reciprocity theorem, i.e. the ratio of input voltage to the output current of the network or, the same thing, the transfer impedance remains unchanged if the input and output terminal pairs are interchanged. In the opposite case a four-terminal network is called non-reciprocal or unilateral. 15.2 Systems of four-terminal network equations The principal significance of four-terminal network theory is that, by using certain generalized network parameters, it is possible 191

to examine and relate analytically voltages and currents at its input and output without calculating currents and voltages in the interior of the network itself. In turn, a complex electrical network (e.g. a multistage amplifier or a communication channel) with input and output terminals can be represented in the form of a combination of component four-terminal networks connected in series or in any other manner. By knowing the analytic relationships between the electrical quantities at the input and output of the component networks, it is possible to obtain relationships between the voltages and currents at the input and output of the over-all network. The generalized parameters used in the theory of four-terminal networks thus make it possible to find currents and voltages at the input and output of complex electrical devices without resorting to an examination of the processes within the given network. The electrical quantities at the input and output of the network which are thus obtained make it possible to estimate the operating conditions for transmission as a whole. The use of generalized network parameters then makes it possible to compare and estimate correctly the transfer properties. The main objective of the theory of four-terminal networks is to establish ratios between voltage on the input and output and currents passing through input and output terminals . The option with currents I1 , I2 is called forward transmission, and I1, I2 – reverse. It is obvious that I1   I1, I2   I2 . Two of four values defining the mode of the four-terminal network can be considered as the given influences, two remained as responses to these influences. Thus, ratios between currents and voltage on the input and output of the four-terminal network can be written in the six possible forms of systems of equations. Y – form Input current and output current are expressed in terms of input voltage and output voltage:

  I1  Y11U1  Y12U 2 ,    I 2  Y21U1  Y22U 2 . 192

(15.1)

Coefficients Y11, Y12, Y21, Y22 are called Y  parameters. These factors are generally complex and dependent on frequency; they have the dimension of admittance and can be determined thus: I Y11  1 is the input admittance at the terminals 1 when the U1 U  0 2

output terminals 2 are short-circuited. I Y22  2 is the input admittance at the terminals 2 when the U 2 U 0 1

terminals 1 are short-circuited. I Y12  1 is the transfer admittance when the terminals 1 are U 2 U 0 1

short-circuited. I Y21  2 is the transfer admittance when the terminals 1 are U1 U  0 2

short-circuited. The Y  parameters are called short-circuit admittances. In the case of a reciprocal four-terminal network Y12  Y21 , i.e. only three factors in equations (15.1) are independent. If the network is symmetrical, the condition Y11  Y22 is obtained together with Y12  Y21 . In this case the number of independent parameters is two ( Y11,Y12 ). Z – form Input voltage and output voltage are expressed in terms of input and output currents: U1  Z11I1  Z12 I 2 ,    U 2  Z 21I1  Z 22 I 2 .

(15.2)

The factors Z11, Z12, Z 21 and Z 22 are generally complex and depend on frequency; they have the dimension of impedance and can be determined thus: 193

Z11 

U1 I1

is the input impedance at the terminals 1 with the I 2  0

terminals 2 open. U is the transfer impedance with the terminals 1 open. Z12  1 I 2 I  0 1

Z 21  Z 22 

U2 I1

I 2  0

U2 I 2

I1  0

is the transfer impedance with the terminals 2 open. is the input impedance at the terminals 2 with the

terminals 1 open. The Z – parameters are called open-circuit impedances. In the case of a reciprocal four-terminal network Z12  Z 21 , i.e. only three factors in equations (15.2) are independent. If the network is symmetrical, the condition Z11  Z 22 is obtained. In this case only two independent parameters remain: Z11, Z12 . A – form If a four-terminal network is an intermediate stage between the source of a signal and a load with given voltage and current on its output ( U 2 , I2 ), input voltage and current ( U1, I1 ) can be calculated by using A – form equations. In this form U1 and I1 are expressed in terms of U and I : 2

2

 U1  A11U 2  A12 I 2 ,    I1  A21U 21  A22 I 2 . A11 

U1 U2

is the relationship of the voltages for an open-cirI2  0

cuited output;

194

(15.3)

I1 U2

A21 

is the transfer admittance for an open-circuited outI2  0

put. A12 

U1 I2 U

A22 

I1 I2

is the transfer impedance for a short-circuited value. 2 0

is the relationship of the currents for a short-circuiU2 0

ted output. Factors A11, A12 , A21 and A22 are generally complex and depend on frequency; A11 and A22 are dimensionless; A12 has the dimension of impedance and A21 has the dimension of admittance. In the case of a reciprocal four-terminal network A  A11 A22  A12 A21  1, i.e. only three factors in equations (15.3) are independent; the fourth is related to the others by above expression. Thus, given any three of the parameters, the fourth is defined by the above equation.   A . If the network is symmetrical one can obtain A 11 22 Let's find the relation between A – and Y – parameters. From the second equation of system of Y – parameters it follows

 I1  Y11U1  Y12U 2 ,   I 2  Y21U1  Y22U 2 .

(15.4)

 I 2  I 2  Y21U1  Y22U 2 , Y 1 . U1   22 U 2  I2 Y21 Y21 By adding the last expression in the first equation of system of

Y – parameters, we will obtain: 195

I1  Y11 (

Y22 1 Y Y Y Y Y U2  I 2 )  Y12U 2   11 22 12 21 U 2  11 I 2 . Y21 Y21 Y21 Y21

And finally,

 Y22 1 I2 , U1   U 2  Y21 Y21    I   Y11Y22  Y12Y21 U  Y11 I . 2 2  1 Y21 Y21

(15.5)

Therefore,

A11  

A21  

1 , Y22 , A12   Y21 Y21

Y , Y11Y22  Y12Y21 Y ,  A22   11 Y21 Y21 Y21

(15.6)

where Y  Y11Y22  Y12Y21 is the determinant made from Y – parameters. The determinant made from A – parameters is equal:

A  A11 A22  A12 A21 

Y12 . Y21

For a reciprocal four-terminal network

Y12  Y21 and

A  A11 A22  A12 A21  1. 196

(15.7)

B – form For the analysis of signal transmission from terminals 2 to terminals 1 the system of equations for the reverse transmission is used:  U 2  B11U1  B12 I1,    I 2  B21U1  B22 I1.

(15.8)

Values of B – parameters are defined also from experiments with open circuit in an input circuit ( I1  0 ) and short circuit ( U1  0 ). H – form When complex amplitudes of current on an input I1 and voltage on an output U 2 are given, required values U1 and I2 can be found from a system of equations in H – parameters:  U1  H11I1  H12U 2 ,    I 2  H 21I1  H 22U 2 .

(15.9)

Values of each of H – parameters are defined from experiments with short circuit on the output ( U 2  0 ) and open circuit on the input ( I1  0 ). G – form In that case when values of U1 and I2 are set, current on the input I1 and voltage on the output U 2 are defined from the equations in G – parameters:

  I1  G11U1  G12 I 2 ,   U 2  G21U1  G22 I 2 .

(15.10)

197

G – parameters in this system of equations can be found from experiments with of an output open circuit ( I2  0 ) and input short circuit ( U1  0 ). As all six systems of parameters describe one four-terminal network, they are connected among themselves by the recast formulas given in reference tables. In the radio engineering for simplification of the analysis and calculation of the electronic circuits containing the active elements, equivalent circuits which are built based on systems of equations of four-terminal networks are used. In practice U-, T-, L- shaped and bridge circuits are widely used.

a) T-type four-terminal network

b) Π – type four-terminal network

c) L – type four-terminal network

Figure 15.3. The equivalent circuits of four-terminal networks

Composite four-terminal networks In order to obtain parameters of a four-terminal network composed of a certain combination of simpler four-terminal networks, the parameters of which are given, it is convenient to use a matrix transcription. 198

a) cascade connection: U1  U1a ,U 2 a  U1b ,

U 2  U 2b , I1  I1a , I2 a  I1b , I2  I2b

b) parallel connection: U1  U1a  U1b ,U 2  U 2 a  U 2b ,

I1  I1a  I1b , I2  I 2 a  I2b

c) series connection: U1  U1a  U1b ,U 2  U 2 a  U 2b ,

I1  I1a  I1b , I2  I 2 a  I2b Figure 15.4. Types of connections of composite four-terminal network

199

Depending on how the composite four-terminal network is connected, one or other form of transcribing the equations is used, namely:  the [A] form for cascade connection (Figure 15.4 a);  the [Y] form for parallel connection (Figure 15.4 b);  the [Z] form for series connection (Figure 15.4 c);  the [H] form for series-parallel connection (Figure 15.4 d);  the [G] form for the for parallel-series connection (Figure 15.4 e). In Figure 15.4 parameters of the resultant four-terminal network: voltage – U1,U 2 and currents – I1 , I2 .

d) serial-parallel connection: U1  U1a  U1b ,U 2  U 2 a  U 2b ,

I1  I1a  I1b , I2  I 2 a  I2b

e) parallel-series connection: U1  U1a  U1b ,U 2  U 2 a  U 2b ,

I1  I1a  I1b , I2  I 2 a  I2b Figure 15.4. Types of connections of composite four-terminal network

200

For the diagram of the cascade connection use the equations written in A – form, for the diagram of serial connection in Z – form, for the diagram of a parallel coupling – in Y – orm, for the diagram of parallel-serial connection – in G – form and for serial-to-parallel connection – in H – form.

15.3 Input impedance of the four-terminal network Influence of the four-terminal network on the mode of a circuit to which it is connected is estimated by input impedances: U Zin1  1 , I1

U Zin 2  Z out  2 . I2

(15.11)

(15.12)

If, during forward transmission (Figure 15.5, a), an unsymmetrical four-terminal network is loaded with an impedance Z2, its input impedance, i.e. the ratio of the input voltage to the input current, is determined from the expression

A Z  A12 U A U  A12 I2 Z in1  1  11 2  11 load , I1 A 21U 2  A 22 I2 A 21Z load  A 22

(15.13)

 where Z load  U 2 . I2 In case of change of the direction of signal transmission (Figure 15.5, b) we will use the following method. In a system of equations in A – parameters we replace currents I1 by I1 and I2 by I2 and

solve the equations relatively U 2 and I2 , thus we will get equations in system of B – parameters, expressed through A – coefficients. 201

a)

b) Figure 15.5

If, during reverse transmission, this network is loaded with an impedance Z1 , its input impedance from the right is determined from the expression

U A U  A I A Z  A Zin 2  2  22 1 12 1  22 1 12 , I2 A 21U1  A11I1 A 21Z1  A11

(15.14)

as Z1  U1 / I1 . Expressions for input impedances can be provided also in the other form. Really,

A Z in1  11 A 21

A12   Z load A11  Z1OC A 22   Z load A 21

Zin 2  Z 2OC

202

Z 2 SC  Z load , Z 2OC  Z load

Z1SC  Z1 . Z1OC  Z1

(15.15)

(15.16)

A A Z1OC  11 , Z1SC  12 are input impedances at output open cirA 22 A 21 A A cuit regime and output short circuit mode, Z 2OC  22 , Z 2 SC  12 A A 21

11

are impedances at input open circuit regime and output short circuit mode. Thus, the four-terminal network transforms load impedance to the new one, both depending on values of load, and on four-terminal network parameters.

15.4 Characteristic (secondary) parameters of the four-terminal network Along with primary parameters considered above (coefficients in the systems of equations) the four-terminal network, in case of solution of many problems characteristic (secondary) parameters of the four-terminal network are used. Such parameters are useful and important for determination of power transfer and various gain computations. In the four-terminal network theory the following secondary parameters are known: characteristic impedance, propagation constant, current gain and voltage gain. It is known that the internal impedance of the generator Z i gives maximum power to the load Z on condition of Z  Z . If i

load

load

between the generator and loading there is a four-terminal network, then for transmission of the maximum power from the generator in the four-terminal network it is necessary to match input impedance of the four-terminal network Z in1 with internal resistance of the generator i.e. to satisfy the condition Z  Z , and for transmission of i

in1

the maximum power from the four-terminal network in loading to match output impedance of the four-terminal network with loading impedance, i.e. to satisfy the condition Z in 2  Z load . Four-terminal network operation mode, when Z i  Z in1 and Z  Z is called in 2

load

the mode of the coordinated switching on. 203

It appears that for any four-terminal network there is such a pair of impedances for which the condition is satisfied:

A Z  A12   A Z  A Z in1  11 load  Z i , Z in 2  22 1 12  Z load , (15.17) A 21Z load  A 22 A 21Z1  A11 these resistances are called characteristic resistances of the four-terminal network and are designated as Z1C and Z 2C . In other words we will assume that there are two impedances Z and Z which 1C

2C

satisfy the following condition: for forward transmission the input impedance of a four-terminal network loaded with an impedance Z 2C is Z1C ; and for reverse transmission the input impedance of a four-terminal network loaded from the left with an impedance Z1C equals Z 2C .

Z1C 

A Z  A12 . A11Z 2C  A12 , Z 2C  22 1C A21Z1C  A11 A21Z 2C  A22

(15.18)

These two impedances are called the characteristic or image impedances of an unsymmetrical four-terminal network. The condition when a four-terminal network is loaded with the appropriate characteristic impedance is called matched loading. The joint solution of these equations with respect to the unknown quantities Z in1  Z i  Z1C and Z in 2  Z load  Z 2C gives Z1C 

A11 A12 , Z 2C  A21 A22

A22 A12 . A21 A11

(15.19)

as

A A A A Z1OC  11 , Z1SC  12 , Z 2OC  22 , Z 2 SC  12 A 21 A21 A 22 A11

(15.20)

characteristic impedances are written through parameters of open circuit and short circuit: 204

Z1C  Z1OC Z1SC , Z 2C  Z 2OC Z 2SC .

(15.21)

If the four-terminal network is matched with loading

U Z load  2  Z 2C  I 2

A 22 A12 , A A

(15.22)

21 11

then the equations in the A-system of parameters  U2 , U1  A11U 2  A12 Z 2C  I  A Z I  A I . 21 2 C 2 22 2 1  A12 A21 A11 ), U1  U 2 ( A11  A22   A12 A21 A22  ).  I1  I 2 ( A22  A11 

(15.23)

then U A22  A11 A22  A12 A21 ,  1 A11 U 2  A11  I1  I  A  A11 A22  A12 A21 . 22  2

The quantity A11  A22

Z1C  nT Z 2C

(15.24)

is called a four-terminal network transformation ratio. Input impedance of the coordinated four-terminal network

Zin1  Z1C  nT2 Z 2C  nT2 Zload ,

(15.25) 205

i.e. the coordinated four-terminal network transforms loading impe2

dance in nT times. Thus, , 1 U1 I  nT 1  A11 A22  A12 A21  e g nT U 2 I2

(15.26)

where g is called the network's propagation function (propagation constant). If the four-terminal network is symmetric A11  A 22 , Z1C  Z 2C ,

nT  1 , then g  ln

U1 I  ln 1  ln( A11 A22  A12 A21 ) , U2 I2

(15.27)

i.e. the constant of propagation is defined only by primary parameters of the four-terminal network. In case of the coordinated load

Z load  Z 2C , Z2C I 2  U 2 , chg  shg  e g ,  Z1C U 2e g , U1  Z 2C   Z 2C g   I1  Z I 2e . 1C 

(15.28)

Propagation constant is generally a complex quantity g  a  jb . Characteristic impedances are complex values Z1C  Z1C e j1C , Z 2C  Z 2C e j2 C . 206

(15.29)

Amplitudes or effective values of voltage and currents on an input and output of the four-terminal network are connected through characteristic impedances and a constant of propagation   Z  1 j (1C  2 C ) 1C a  jU 2 2  U e jU 1   U e e e e jb ,  1  Z 2C 2       Z  1  j ( 2 C 1C ) 2C I 2 e a  e j I 2 e 2 e jb .  I1 e j I 1     Z1C    

(15.30)

The real part of the propagation constant (b) is called the attenuation coefficient, and the imaginary part (a) the phase coefficient. The real part of the propagation constant characterizes the change of amplitude or effective value of current and voltage when passing a signal via the four-terminal network. Imaginary component b characterizes the phase shift between the input and output voltage or currents, 1 2 1  I 1   I 2  ( 2C  1C )  jb 2

U 1  U 2  (1C   2C )  jb

(15.31)

For the symmetric four-terminal network  U1 e jU 1  U 2 e a e jU 2 e jb ,   j a j jb  I1 e I 1  I 2 e e I 2 e .

(15.32)

b is the coefficient of a phase is measured in radians or degrees

b  U 1  U 2   I 1   I 2 .

(15.33)

The coefficient a (natural attenuation) is defined as a  ln

U1 U2

 ln

I1 I2

(Neppeр).

(15.34) 207

To attenuation a = 1 Np there corresponds reduction of amplitude or effective value of voltage or current in е = 2.718 times. In radio engineering attenuation is calculated in bels (B) and more often in decibels (dB) from the formulae

a( B)  lg

U1I1 S  lg 1 . U2I2 S2

(15.35)

So

S1 U12 I12   S2 U 22 I 22 a( B)  2 lg

U1 I  2 lg 1 . U2 I2

(15.36)

Unit bel is large enough, therefore use 0.1 bel is called decibel. a(dB)  20 lg

U1 I S  20 lg 1  10 lg 1 , U2 I2 S2

(15.37)

1dB ≈ 0.115 Np; 1 Np ≈ 8.7 dB Effective attenuation and insertion loss of a four-terminal Let's consider the four-terminal network which is switched on between a source with internal impedance Z i and load impedance

Z load (Figure 15.6).

Figure 15.6

By the effective attenuation of the network we understand a half of the natural logarithm or ten times the decimal logarithm of the 208

ratio of the apparent power S0 which the source would supply directly to impedance Zi (Figure 15.6), to the apparent power S2 at the output of a four-terminal network loaded with an impedance Zload: For an impact assessment of conditions of coordination of the four-terminal network with the generator and load of signal transmission working attenuation is used

aW 

1 S0 ln , 2 S2

(15.38)

where S 0 is the maximum apparent power which the generator gives to the coordinated load, S 2 is the apparent power absorbed in the loading connected to a four-terminal network output. The maximum of apparent power is absorbed on impedance, equal to internal impedance of the generator: 2

E E2 . S0  Z I   Zi  2Zi 4Zi 2 i 1

(15.39)

The apparent power absorbed in load:

U 22 S 2  U 2 I2  . Z

(15.40)

load

Effective attenuation in this case 1 E 2  Z load E 1 Z aW  ln  ln  ln load . 2 2 4Z i  U 2 Z i 2U 2 2

(15.41)

Driving voltage of the generator

E  U1  Zi I1  A11U 2  A12 I 2  Zi ( A21U 2  A22 I 2 ) .

(15.42) 209

Considering that I2 

U 2 , Z load

let's obtain

A Z A E  A11  12  Z i A 21  22 i . U 2 Z load Z load Let's replace A – parameters by characteristic A11 

Z1C chg , A12  Z1C Z 2C shg , A21  Z 2C

1 Z1C Z 2C

shg , A22 

Z 2C chg Z1C

and add in the formula for effective attenuation: aW  a  ln

where p1 

Z i  Z1C Z  Z 2C  ln load  ln 1  p1 p2e 2 g , (15.43)     2 Z i Z1C 2 Z load Z 2C

Z load  Z 2C Zi  Z1C p  are reflection coefficients , 2 Z load  Z 2C Zi  Z1C

on an input and output of the four-terminal network, respectively. Thus, effective attenuation contains four components. The first component is natural attenuation of the four-terminal network a. The second component characterizes a generator mismatch level with a four-terminal network input, the third component is a four-terminal network output mismatch level with load. The fourth component appears only when there is a mismatch both on input and output, i.e. when both reflection coefficients aren't equal to zero. In practice this component is usually small, and it can be neglected. It is necessary to note that in case of matching of an input of the four-terminal network with the generator ( Zi  Z1C ), the second component is equal to zero. If to provide matching of the four-ter210

minal network with load ( Z load  Z 2C ), that the third and fourth components will also be zero, and the effective attenuation is equal to natural attenuation of the four-terminal network. Instead of effective attenuation the other parameter is called the insertion loss and it is quite often applied. In this case the apparent power coming to the load is compared to the apparent power which the generator would give to the load in case of their direct connection, i.e. the insertion loss aIN 

1 S12 , ln 2 S2

(15.44)

here S12 

E 2 Z load Z  Z 2 i

load

is apparent power which the generator would give to the load in case of their direct connection. The insertion loss is connected with effective attenuation as

aIN 

1 S12S0 1 S0 1 S0 1 S ln  ln  ln  aW  ln 0 . 2 S 2 S0 2 S 2 2 S12 2 S12

(15.45)

In case of replacement Z  Z load . 1 S0 ln  ln i 2 S12 2 Z i Z load

The insertion loss is

aIN  aW  ln

Z i  Z load , 2 Z Z i

(15.46)

load

i.e. the attenuation caused by the mismatching of the generator with the load is excluded from the effective attenuation. 211

Let's consider different cases for effective attenuation:  If aW = 0, apparent powers on the input and output of the four-terminal network are equal;  If aW < 0, the four-terminal network is the signal amplifier.

15.5 Transfer functions of the four-terminal network Transfer function is the ratio of the complex amplitudes or complex effective values of the electrical quantities at its output to those at its input in given transfer conditions. The ratios of like electrical quantities (output-to-input voltage ratio and current ratio) KU 

U2 E

Y21 

I2 , E

(15.47)

where K U is complex voltage transfer function, Y21 is complex transfer admittance. Voltage transfer function is dimensionless quotient, usually complex and dependent on frequency. In accordance with amplifying devices, these ratios are called voltage amplification or current amplification. The word 'gain' is sometimes used as a synonym for amplification. If input influence represents current on a four-terminal network input, and a four-terminal network response to this influence is current or voltage on an output, then in this case complex transfer functions are: I2 I1 U Z 21  2 . I1 KI 

(15.48)

K I is complex current transfer function, Z 21 is complex transfer impedance. 212

Control questions 1. Define the basic concepts used in the four-terminal network theory. 2. Write down the six forms of two-equation system of four-terminal network. 3. Define the coefficients in the equations of four-terminal network in different forms of equations. 4. Determine the input impedance and the characteristic (secondary) parameters of four-terminal network. 5. Write down the formula of transfer functions of the four-terminal networks.

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LITERATURE

1. Atabekov G.I. Osnovy teorii cepej: uchebnik / G.I. Atabekov. – 2-e izd., ispr. – SPb.: Lan', 2006. – 432 s. 2. Prjanishnikov V.A. Teoreticheskie osnovy jelektrotehniki. – M.: KORONA-Vek, 2012. – 368 s. 3. Atabekov G.I. Nelinejnye jelektricheskie cepi. Teoreticheskie osnovy jelektrotehniki. Uchebnoe posobie. – SPb.: Piter, Lan', 2010. – 432 s. 4. Bessonov L.A. Jelektricheskie cepi. Teoreticheskie osnovy jelektrotehniki. – M.: Jurajt, 2016. – 701 s. 5. Popov V.P. Osnovy teorii cepej: Uchebnik dlja stud. vuzov spec. Radiotehnika. – M.: Vysshaja shkola, 2000. – 574 s. 6. Dobrotvorskij I.N. Teorija jelektricheskih cepej. – M.: Radio i svjaz', 1989. – 472 s. 7. Bakalov V.P., Zhuravleva O.B., Kruk B.I. Osnovy analiza cepej: uchebnoe posobie dlja vuzov. – Gorjachaja linija: Telekom, 2007. – 591 s. 8. Bychkov Ju.A., Zolotnickij V.M., Chernyshev Je.P. Sbornik zadach i praktikum po osnovam teorii jelektricheskih cepej. 2-e izd. – SPb.: Piter, 2007. – 302 s. 9. Zapasnyj A.I. Osnovy teorii cepej: Uchebnoe posobie. – M.: RIOR, 2006. – 336 s. 10. Kuznecov A.P., Batura M.P., Kurulev A.P. Teorija jelektricheskih cepej. Uchebnik dlja VUZov. Vysshaja shkola, 2007. – 606 s. 11. Ponomarenko V.K. Posobie k prakticheskim zanjatijam po teorii jelektricheskih cepej. Uchebnoe posobie dlja jelektrotehnicheskih special'nostej vuzov. Ozersk: OTIMIFI, 2001. – 200 s. 12. Shilin L.Ju., Kovalenko V.M., Ivanickaja N.A., Derjushev A.A. Teorija jelektricheskih cepej. –Minsk: BGUIR, 2010. – 83 s. 13. Beleckij A.S. Teorija linejnyh jelektricheskih cepej. – M.: Lan', 2009. – 544 c. 14. Nikulin V.I. Teorija jelektricheskih cepej. – M.: RIOR, 2016. – 240 s. 15. Popov V. Sbornik zadach po teorii cepej. – M.: Vysshaja shkola, 2009. – 270 s. 16. Alimgazinova N.Sh. Teorija jelektricheskih cepej. Uchebnoe posobie. – Almaty: Kazakh universiteti, 2017. – 228 s. 17. James W. Nilsson, Susan A. Riedel. Electric Circuit. Ninth Edition. Pearson Education, Inc., publishing as Prentice Hall, One Lake Street, Upper Saddle River, New Jersey, 2011. – 882 p. 18. Charles A., Sadiku M. Fundamentals of Electric Circuits. Science Engineering & Math; 5 edition, 2012. – 992 p. 19. Nahvi M., Edminister J.A. Electric Circuits. Seventh Edition. McGraw-Hill Education, 2018. – 507 p. 20. Bird J.O. Electrical Circuit Theory and Technology, 6th ed., Routledge Kegan & Paul, 2017. – 858 p.

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CONTENT

Introduction ............................................................................................... 3 Lecture 1. Electric circuits ......................................................................... 4 Lecture 2. Electric components .................................................................. 17 Lecture 3. DC linear electric circuits. Basic laws ...................................... 31 Lecture 4. The topology of electric circuits ............................................... 43 Lecture 5. Methods of linear electric circuit analysis ................................. 50 Lecture 6. Single-phase electric circuits of sinusoidal current. Basic concepts and laws............................................................................. 59 Lecture 7. Current, voltage, resistance and power in sinusoidal current circuits ........................................................................................... 70 Lecture 8. Methods of AC circuit analysis ................................................. 81 Lecture 9. Resonance in electric circuits .................................................... 94 Lecture 10. Coupled circuits ...................................................................... 115 Lecture 11. Three-phase circuits ................................................................ 129 Lecture 12. Non-linear circuits .................................................................. 152 Lecture 13. Electric circuits of non-sinusoidal current .............................. 160 Lecture 14.Transients in electric circuits ................................................... 171 Lecture 15. Four-terminal networks ........................................................... 190 Literature ................................................................................................... 214

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Еducational issue

Alimgazinova Nazgul Shakarimovna Manakov Sergey Mikhailovich Kyzgarina Meiramgul Tuleubekovna Manapbayeva Arailym Bekbolatkyzy THEORY OF ELECTRIC CIRCUITS Educational manual Editor L. Strautman Typesetting G. Kaliyeva Cover design B. Malayeva Cover photo «halejandropmartz» is from www.pixabay.com.

IB No.13827 Signed for publishing 17.09.2020. Format 60x84 1/16. Offset paper. Digital printing. Volume 13,5 printer’s sheet. 100 copies. Order No.11513. Publishing house «Qazaq University» Al-Farabi Kazakh National University KazNU, 71 Al-Farabi, 050040, Almaty Printed in the printing office of the «Qazaq University» Publishing House.

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