Compendium of Lecture Notes on Meteorological Instruments for Training Class III and Class IV Meteorological Personnel [2/3] 9263106223
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&-J t1D
61 (t, J.2
WORLD METEOROLOGICAL ORGANIZATION
COMPENDIUM OF LECTURE NOTES ON
METEOROLOGICAL INSTRUMENTS FOR
TRAINING CLASS Ill AND CLASS_ IV METEOROLOGICAL PERSONNEL Prepared by Dr. D.A Simidchiev
VOLUME II PART 3 -
Basic electronics for the meteorologist
I WMO - No. 622 Secretariat of the World Meteorological Organization- Geneva- Switzerland
1986
© 1986, World Meteorological Organization ISBN 92-63-10622-3
NOTE The designations employed and the presentation of material in this publication do not imply the expression of any opinion whatsoever on the part of the Secretariat of the World Meteorological Organization concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.
P A R T 3*
BASIC ELECTRONICS FOR THE METEOROLOGIST
* Parts 1 and 2 are published separately
C 0 N T E N T S
PART 3
INTRODUCTION
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BASIC ELECTRONICS FOR THE METEOROLOGIST
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365
CHAPTER 18 - REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM General
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367
The nature of electricity - units of charge, current, voltage - resistance-related laws •• . . . . . . . . ., • • • • • • • • • • • • • • • 111.
368
18.2.1
Units of electrical charge and current
368
Exercises
IIIIIIOIIIIIDOIIIIIIIIIIIIOIIOIIIIIIIIIIIIIIIOIIIIIIIIIIIIIIIGIIIIIIIIOIIIIIIIIIIIIIIIIIIIIIIIIOIIIIIIIIIOIIIIIIIIO
18.2.2
Units of voltage
18.2.3
Units of electrical resistance
•
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377 The laws of Ohm and Kirchoff - the law governing the conversion of d.c. power into heat - connexion of resistances and d.c. electrical sources in series and parallel-circuit elements ••••••••••••••••••••••••••••••••••• "
Exercises 18.2.5
Exercises 18.3 Exercises
372 373
Exercises 18.2.4
371
•••••••••••
11
•••
·-
378
11 • • • • • • • • • • • • • • • • •
382
D.c. electric sources: carbon-type dry cell and lead-acid storage battery- use and maintenance ••••••••••••••••••••o••
384
Solution of simple electrical circuits - principle of superposition theories of Thevenin and Norton •••••••••••••••
387
••••••••••••••"••••••••••••••-••••••••••-••••••••••••••••••••eo•
394
The nature of magnetism - magnetic quantities and units of measurement ogg••••••••ooe••••••e••••••••••••••••••••••••••gc
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18.3.1
The magnetic circuit
Exercises
............. , .., ........................................... , ...... .
410
18.3.2
Electromagnetic induction
412
Exercises
••••••••o•••-•••••••••••••••·•••••••••••••e•••••••·•·•••••••••••••
430
18.4
Conversion of electrical energy into mechanical energy and vice versa ••••••e•o•e•••••••o••••••••••eoooeooeo••••••••••••
430
18.4.1
Direct-current generator
18.4.2
Torque and power pf d.c. machines
18.4.3
Types of d.c. generator
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XIV
CONTENTS
18.4.3.1
D,c. generator having an independent (external) excitation
436
18.4,3,2
D.c. shunt generator
437
18.4.3,3
D.c. series generator
18.4,3.4
D.c. compound generator
439
18.4.4
Types of d.c. motor
18.4.4.1
The d.c. shunt motor
439 440
18.4.4.2
The d.c. series motor
18.4.4.3
Compound d.c. motor
18.4.4.4
Thermal regime of electric motors
442
18.4.5
Alternating-current generators
444
Exercise
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Electrical sine quantities
8
.............................. .
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442
446 446
18,5,1
R.m.s. (or effective) currents and voltages
.................
449
18.5,2
Electrical sine quantities in a.c. circuits and their determination using complex numbers •••••••••••••••••••••••oo
450
Exercises
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...................... , .. ,. ................. .
18.5,3
The phasor method
18.5.4
A.co circuits- trigonometric approach
18.5.5
A.c. power relationships
Exercises
Ill
454 455
••••••••••••••••••••••
461
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465
·~~~··········lit··················································
467
18.5.6
Three-phase alternating current
469
18. 5.7
Three-phase a.c, power relationships
472
Exercises
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18.6
Ae c. motors
18.6.1
Monophase asynchronous motor
18.6.2
The selsyn angular shaft position transmission
18.7
Transformers
Exercise
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484
A.c.-into-d.c. conversion (rectification) and vice versa
485
18.8.1
Half-wave rectifiers
486
18.8.2
Full-wave rectifiers
18.8,3
The voltage doubler
18.9
Conversion of direct current into alternating current
18.10
Non-sinusoidal waveforms
18.11
Measurement of the basic electrical and electrical-circuit parameters
•••••••••••••••••••••••••••••
488
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492 497
XV
CONTENTS
18.11.1
Measurement of electrical current
498
18.11.2
Measurement of voltage
501
18.11.3
Measurement of electrical resistance, capacitance and inductance
Exercises
•••·••••••••te.oooe••••••••••••••a•••••••
502
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505
CHAPTER 19 - ELECTRONICS 19.1
The resonance effect in electrical circuits
509
19 .1.1
Series RLC circuit resonance
510
19 .1.2
Parallel RLC circuit resonance
512
19 .l. 3
Q-factor of a resonant circuit
513
19 .l. 4
Bandwidth of a resonant circuit
515
19.1.5
Coupled tuned resonant circuits
516
Exercises
••••e•••••••••••••••••••••••••••••••e••••••••••·•••••••••••••••
518
19.1.6
Filter circuits
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519
19.2
Vacuum tubes
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522
19.2.1
The diode, triode and pentode tubes - electrical parameters and characteristics - the diode detector and electron tube amplifiers ······················••a•••····~···•••oeoooooe•••
522
19.2.2
The tube load-line
533
19.2.3
Multi-grid tubes
19.2.4
The electron tube AF and RF sine-wave oscillators - the e e o o o o o o e e o e o o o o o e • o o o o o o o electron-tube transmitter
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537 540 545
19.2.5
The superheterodyne receiver principle
19.2.6
The phase-shift oscillator
19 .2.7
Non-sinusoidal waveform tube oscillators
19.2.7.1
The blocking oscillator
19.2.7.2
The free-running multivibrator
19.2.7.3
The one-shot multivibrator
19.2.7.4
The flip-flop device
19.3
Semiconductor devices- p-n junctions
•••••••••.•.••••.••.••.
554
19.3.1
Semiconductor diodes - the junction diode - point-contact diode .............. ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
556
19.3,2
The Zener diode
557
19.3.3
The variable capacitance diode
19.3.4
The tunnel diode
1903.5
The semiconductor photodiode
19.3.6
The light emitting diode
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• • • • • • • • • • • • • • • e • • • •
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XVI
CONTENTS
19.3.7
The transistor
561
19.3.8
Transistor amplifier configurations
563
19.3.9
The transistor equivalent circuits
19.3.9.1
T...;network equivalent circuit
19.3.9.2
Impedance-parameter
19.3,9.3
Admittance-parameters equivalent circuit
19.3.9.4
Hybrid, h .. -parameters equivalent circuit
19.3.10
The transistor static characteristic curves
570
19.3.11
Transistor biasing
•••••••••••.••••••••••...•.•••..••••.•••••
572
19.3.12
Th~
••••o•••o•
575
19.3.13
Load-lines - CE configuration
576
19.3.14
Circuit examples of various transistor amplifiers
578
Exercises
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.. • • .. .. • .. . • . . . . . . . . .. • .. . . . . ..
equi~alent
circuits
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565 566 567 567
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effect of temperature on the collector current
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567
582 586
19.3.15
Transistor sinusoidal oscillators
19.3.15.1
The phase-shift sine-wave oscillator
19.3.15.2
The transistor resonant circuit oscillator
587
19.3.16
Non-sinusoidal wave generators
•••• • ••••. • . •. . . • . • . •. . . •. . •• •
590
19.3.16.1
The bistable multivibrators
o••·················
19.3.16.2
The monostable multivibrator
590 592
19.3.16.3
The astable multivibrator
593
19.3.16.4
The blocking oscillator
595
••••••••.••••.•••••.•..•• o.
········••o••··········· 586
(flip-flop)
596
Exercises
19.3.17
Semiconductor rectifier-voltage stabilizer
•..•••.•.•...•.••.
598
19.3.17.1
Approximate analysis of capacitor filters
•••••.•....•.••..••
601
19.3.17.2
The voltage stabilizer
o ••••••• o •••••••••••••••••••••••
603
Exercises
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605
19.3o18
The d.c.-into-a.c. converter
•••••••
606
19.3.19
Other semiconductor active elements
••••••••.••.•••••••••••••
607
19,3.19.1
The field-effect transistor
........................ ..
607
19.3.19.2
The insulated-gate
•••••••••.•••••..•.•••..••••
610
19.3.1903
The silicon controlled rectifier
19.3.20
Integrated circuits
FET
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(FET)
(IGFET)
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0
611 0
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fl
•
612
CONTENTS
XVII
Page
ANNEX
615
REFERENCES
622
NOTE CONCERNING FIGURES ACKNOWLEDGEMENTS
625 625
I NT R0 DUCT I 0 N
New electrical and electronic meteorological instrument• are being introduced for making routine meteorological observations at an increased rate. Electronic techniques in the research field have greatly improved the experimental potential of the atmospheric sciences. The use of these new techniques, and the need to appreciate their limitations and possibilities, as well as to know how to interpret the measurement results correctly call for a more profound knowledge of electrotechnical disci~ ~lines. The training of meteorologists in this respect is lagging behind the accelerated introduction of electronic instrumentation into the operational as well as theoretical fields. The time allotted to electricity and electronics in the curricula of meteorologists is still inadequate, partly because of the lack of written material on electrotechnical disciplines which is aimed at the specific needs of meteorology. This section ''Basic electronics for the meteorologist", has been written to help achieve the training objectives stemming from the increasing application of electricity and electronics in meteorology. About thirty years ago, the electronic tube was the fundamental circuit component in electronics. At present a number of semiconductor devices are of primary importance but the basic laws governing the behaviour of electronic circuits are valid for circuits with electron tubes as well as with transistors. The differences between tubes and semiconductors are accounted for through consideration of their respective performance parameters. "Basic electronics for the meteorologist" places emphasis on semiconductor devices whilst giving due attention in the discussions to electron tubes as circuit elements, limiting their treatment to fundamental facts of practical value only. This seems to be justified for two reasons: (a) vacuum tubes are still in use in meteorological equipment; and (b) a discussion of vacuum tubes helps to understand the basic principles of electronics in bridging the historical gap, which might otherwise present problems. In Cbapter 18, emphasis is lent to the fundamental laws of electricity and, wherever possible, certain meteorological applications of the topic under discussion are included. Electronic circuits are discussed in Chapter 19. More attention is paid to circuits with a specific meteorological application which, although inevitably introducing a certain imbalance in the contents of the lecture notes, seems necessary in an attempt to impart to students knowledge of practical value for application in the general field of meteorology. Figures in brackets after section headings indicate the appropriate reference-work number, Routine technical information for use in the practical application of the principles of electricity and electronics in meteorology is given in the annex.
C H A P T E R 18
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
18.1
General {5), (lO)L (31)
Engineering is an applied science dealing with physical quantities. The science of electrical engineerihg is basE':'d on a few experimentally established fundamental laws. The electric circuit theory is based on the results achieved by Coulomb (1785), Ohm (1827), Faraday (1831) and Kirchoff (1857). The whole subject matter of electromagnetic devices and electromagnetic energy conversion can be treated by the application of Ampere's law (1825) and Faraday's law of induction (1831). These fundamental laws are mathematical expressions of the experimentally established relationships between physical quantities. The physical quantities are measured in an established and universally accepted system of units. These units are permanent, reproducible and amenable to precise comparison. The system is based on a few selected units which are fundamental to it. All others are expressed in terms of the fundamental ones. The fundamental laws can be expressed in different unit systems but the most suitable for engineering work and scientific research is the universally accepted SI system (Systeme international). The SI system is based on the following units:
•
Metre (m) - length, dimension (L);
·•
Kilogram (kg) - mass (M);
•
Second ("5)
•
Ampere (A) - electrical current \I) i
• Kelvin
•
-
time (T);
(K) - degree of temperature (9)
Candela (Cd)
-
intensity of light (J).
As regards the selection of the fundamental units two considerations are viewed as important: (a)
The basic relationships between the various quantities should involve a minimum number of constants;
(b)
The measuring units should be of a practical size.
The various units - or rather physical quantities - can be expressed in terms of the fundamental ones, using dimensional relationships which show the fundamental units as factors and exponents. The mechanical quantities of force, linear velocity and linear acceleration are expressed, using the dimensional relationships, as follows:
CHAPTER 18
Quantity
. Mathematical expression
Dimensional relationship
s. I.
newton (N)
units
Force
f
= m.a
f (M T-2 L)
Linear velocity
V
= s/t
V
(L r-1)
(m s-1)
Linear acceleration
a
= (v2-vl)/t
a (L T-2)
(m s-2)
The subdivisions and multiples of the physical quantities are as follows: deci-(d)
-. 10-1
deka-(dk}
centi- (c)
= 10-2
hecto-(h)
= 101 2 = 10
milli-(m)
kilo-(k)
= 103
micro-(p,)
= 10-3 = 10-6
mega-(M)
nano-(n)
= 10-9
giga-(G)
pico-(p)
10-12 =
tera-(T)
= 10 9 = 10 = 1012
6
All electrical quantities can be expressed in the SI system of units {as in the present lecture notes) but in practical work, electrical quantities measured in other unit systems may be encountered. A conversion factor for the conversion of the other system units into the SI system can be used if necessary. Such conversion factors are given throughout the lecture notes whenever necessary.
18.2
The nature of electricity .,. units of charge, current, voltage ...: resistance-reiated laws (5), (10), (31)
18.2.1 The concept of electricity is based on the existence of elementary electrical charges. The electrical charge of the negative carrier of electricity, the electron, has been found to be 1.602 x lo-19 C. The unit coulomb (C) is an SI unit of electrical charge and it may appear at present .so~ewhat arbitrdry. It was named after Charles Augustin Coulomb (1736-1806), a noted scientist. In order to clarify the situation we may resort to the converse. stqtement·concerning the unit of charge: about 6.4 x 1018 electrons are required to form the quantity of electricity equal to one coulomb. The presence of electrical charges indicates the possible manifestation of a force in the region SUfrounding them. It is a proven fact that like charges (in sign) will repel and unlike charges will attract each other. This is a manifestation of a field of force called an electrical field. Coulomb's law gives an expression for the force of between point charges: 01.02 2 f
= -~-'-::-2 4 n.E.r
where:
(N) (L M
r-
)
attraction or repulsion
(l)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
369
0 .0 =electrical charges 1n coulombs, C; 1 2
=a
constant related to the surrounding medium, known as permitivity. The constant is measured in farads per metre (F m-1) and of dimen-
E
sion L- 3 M-l T4 I 2 . The value of the permitivity of "free space", . 7 2 calculated from E = 10 /4 7T. c
f.'.
°
= 8.854
x 10-
12
1 (F m- ) is
0
where:
-1
c
= v~locity of light (m s
4n
= proportionality constant; = distance between the charges
r
);
in metres (m);
(known as relative permitivity).
At this stage, we are more interested in the charges in motion, because they bring about energy transfer. We are particularly interested in those situations where motion is confined to a definite path formed by materials (electrical conductors).
In terms of their electrical charge conductivity, materials can be divided .into three major groups: (a)
(b)
Conductors, having good electrical conduction properties: {i)
First class (all metals and carbon);
(ii)
Second class (electrolytes);
Semiconductors, having poor conduction properties; I
(c)
Insulators (dielectrics), having extremely poor conduction properties.
The paths along which electrical charges move in matter are generally called electrical circuits. The rate of motion of the electrical charges is called electrical current. The flow of a charge at the rate of one coulomb per second is known as an ampere (A), the SI unit of electrical current:
I
= 0/t
(A)
where: Q
=
electrical charge in coulombs, (c);
t
=
time in seconds (s).
(2)
CHAPTER 18
370
18 An electrical current of one ampere would mean the passage of 6,4 x 10 electrons per second through a given conductor cross-section. An electrical current of one ampere passed through a solution of silver nitrate will deposit metallic silver at the negative pole at a rate of 1.118 mg s-1, The positive direction of flow of an electrical current is taken, according to convention, as the opposite direction to that of the movement of negative charges (electrons), i.e. from the positive terminal of the electrical source to the negative terminal. The electrical current is measured by a measuring instrument known as an AMMETER, which is connected in the path of the electrical current, in a series with the electrical circuit, The principle of action of the ammeter will be discussed later. At this stage, it suffices to know its graphical presentation, as used in drawings of electrical circuits (Figure 231).
I
Figure 231 - Graphical presentation of an ammeter From equation (2), by a simple re-arrangement one can obtain: Q
= I.t
of the quantities involved,
(C)
(3)
With I in amperes and t in seconds, a measure for the quantity of electricity that flows through the circuit is obtained in AMPERESECONDS (As). For most practical purposes the following multiples are used: 1 amperehour (Ah)
=3
600 ampereseconds (As)
The ratio of the current to the cross-section area at a specific place in the electrical circuit is known as the CURRENT DENSITY at that place: j
= I/S
(A m-2) ,
( I L-2)
Electrical circuits usually make use of copper conductors whose crosssectional area in most practical cases is measured in square millimetres:
Accordingly, equation (4) can be re-written: j
= I/S
( A m-2) -= 10 -4 ( A mm -2)
Current density in copper conductors used for electrotechnical purposes is limited to 5 A mm-2
(4)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
371
EXERCISES 1.
Question
Find the force acting between two point charges of one coulomb each placed in free space one metre apart. Answer From equation (1) and the va.lue for f
1
= 4
X
n
X
X
1
8.854
X
10
_12
E
X
given, we obtain:
0
1
2
= 1010· N
=
1 megaton
It has been demonstrated that the coulomb is an exorbitant quantity, not practical for ordinary engineering calculations. 2.
Question
What electrical charge will be accumulated in the storage battery of an ordinary portable radio if the electric source to which it is connected charges it at a current of Ool A for 16 hours? Answer Following equation (3): Q 3.
= I.t = 0.1
X
16
= 1.6
(Ah)
Question
A platinum wire of 0.1 mm diameter of a platinum resistance thermometer takes a current of 0.5 x l0-3 A with a maximum allowable overheating of O.l°C. What is the maximum allowable current density in the wire? Answer Following equation (4): j
= I/S = 0.5
3 2 x 10- /nx 0.05
= 0.06
2 (A/mm )
=6
x 10-
6
2 (A/m )
The following table can be used to obtain the necessary information for electroplating:
4.
Metal
Character Valency of electro~ plating
,Copper Acid Cyanide Copper Nickel Acid Chromium Acid Cyanide Silver Gold Cyanide
2 1 2 6 1 1
Ions
Atomic weight
Equivalent weight
Cu++ Cu+ Ni++ Cr++++++ A+ g+ Au
63.57 63.57 58.69 52.02 107.88 197.20
31.78 63.57 29.35 8.67 107.88 197.20
Electra chemical
Thickness of deposit, mm for j=l .(A/dm) and l(Ah) 0.013 0.017 0.010 0.0006 0.037 0.320
1.19 2.372 1.095 0.323 4.025 7.358
'
CHAPTER 18
372
N.B. 1 Ah deposits on the cathode (1 g equivalent/26.8 g) of metal, known as the electrochemical equivalent of the deposited metal, a quantity to be found in the seventh column of the table. 18.2.2 The motion of electrical charges is accompanied by an energy change or energy transfer. If we consider a positive charge Q placed at a point between two plates b and c made of a conducting material and connected to the terminals of an electric source G, a force will manifest itself in moving the charge in the direction of the negatively charged plate, b (see Figure 232). + c
+ +
Q
9a I
I
__.,,
+ +
Figure 232 - Positive charge in an electric field
+
I
As already mentioned, part of the space in which a mechanical force is acting on an electric charge is called an electric field. A quantity is defined for the electric field, E, called electric field strength, such that: E
= F/Q
(volts/metre)
(5)
where: F (newton)
= mechanical
Q (coulomb)
= electric
force acting on the charge;
charge.
The electric field strength is numerically equal to the force acting on a unit charge placed in the field. If, under the action of the field force, the positive charge Q has moved a distance 1, the work which has been done can be expressed as follows: A
= F.l = Q.E.l
(joule~)
(6)
and the quantity U obtained from equation (6): U = A/Q (volts)
(7)
is called the voltage between the end points of the charge's motion. If the work done on the charge by its motion from a to d has increased the potential energy of the charge, a voltage rise exists in the direction of d. Conversely, if a decrease of the charge's potential energy occurs by its change of position, a voltage drop exists in that direction.
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
373
With the arrangement shown in Figure 232 we have a voltage drop in the direction from plate c to plate b and the direction of the electric field will be in the direction of movement of the positive charge 0. When several alternati"ve paths exist between the end points a and d, the above statement will be true, regardless of which path is chosen. The dimensions of equation (7) can be obtained, based on the following reasoning: voltage
(8)
The units of measurement of the field strength and (8): E
E can be obtained from equations (7)
= F/0 = A/0.1 = U.0/0.1 = U/1
(V/m)
(8')
Equation (7) can be re-written in the following way: A/T
U = O/t
(9)
(V)
= W/I
where:
W(W) is the electrical power;
and I (A) is the current associated with it.
Equation (9) is an important relationship. Its terms, re-arranged, demonstrate that the electrical power in a circuit between two points, having a voltage drop V and a steady current flow I, is the product of U and I: W = U. I (W)
(10)
The ratio of the energy given from the electric source to an electron, to the quantity of the electron charge is known as the electromotive force (e.m.f), again measured in volts (V). If, between the terminals of the electric source shown in Figure 232 a receiver of electric energy is connected (i.e. a sink), a closed circuit is obtained, with an electric current circulating in it. In such a case, the entire e.m.f. of the source is spent along the circuit: part of it as an internal voltage drop (inside the source), U and partly as an external voltage drop, U. Therefore: 0
e.m. f.
= U0
+ U
(11)
The movement of electric charges gives rise to another type of field of force in. the vicinity of the circuit. This field, called the magnetic field, exists simultaneously with the electric field causing forces to act on another current carrying elements or on pieces of iron. (The magnetic field will be discussed in a later chapter.) 18.2.3
Units of electrical resistance - resistors
One type of circuit element requires a voltage directly proportional to the current through it, The constant of proportionality is called resistance. This circuit constant is closely related to energy dissipation as heat in the circuit, resembling the friction element of the pipe in the hydraulic analogue of the electrical phenomenon.
CHAPTER 18
374 Expressed quantitatively:
U = R.I
(12)
or R
= U/I
(13)
(ohm,f2) • (U in V and I in A)
A physical device whose main electric characteristic is electrical resistance is called a resistor. Its graphical presentation is shown in Figure 233.
I+ ~
C:JJ---1
_: u:._ '
.
Figure 233 - Symbol of a resistor
The voltage/current relationship in equation (12) is known as Ohm's law" The electrical resistance R of an electrical conductor is directly proportional to its length,£, inversely proportional to its cross-sectional area and a function of its material: R
= J!.4. s
(14)
(Q)
where:
p = resistivity of the material of the conductor (Q.m);
£=
length of the conductor (m);
s=
cross-sectional area of the conductor (m ).
2
In engineering practice, another expression for the resistivity pis deeply rooted, one having the following dimensions: flmm2 P (--) = 10-6 (Q. m) m
A quantity reciprocal to the resistivity
Y = £/p
(siemens/metre
9
P is
the conductivity
Y
5/m)
(15)
The engineering dimensions of conductivity are:
2 (m/ Q. mm )
= 106 (5/m)
The resistance of the conducting material is also dependent upon the temperature of the material: (16)
where:
Rl
= resistance = resistance
Cl:'l
= temperature
R2
0
of the conductor at -a Celsius temperature t2; 0
of the conductor at a Celsius temperature tl; 0 coefficient of resistance of the material at tl.
375
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
For a standard annealed copper conductor, a convenient empirical equation for the relationship between R and R is the following: 2 1 0 234.5 + t2 R2
= Rl
234.5 +
t~
(17)
The values of resistivity, conductivity and the temperature coefficient of resistance are shown in the table below:
-----------------------------------------------------------------------------------. a Material
=
L1T
0° - 100°C (l/°C)
Aluminium
0.0278
36
0.00423
Tungsten
0.0612
16.34
0.00464
Steel
0.13 to 0.20
7.7 to 5.0
0.00625
Cons tan tan
0.49
2.04
0.00004
Brass
0.04
25.0
0.0020
Copper
0.0175
57.0
0.004
Manganin
0.42
2.28
0.000006
Nichrome
0.98
1.01
0.0003
German silver
0.47
2.1
0.0004
Lead
0.221
4.52
0.00411
Carbon
7.25
0.138
0.0003
Resistors, as presently used, are a large family of electronic components. The major types of resistor are shown in Figure 234 by "family tree": RESISTORS Constant
I
Carbon
I
Layer
~
Solid
valu~
\
Wire~wound
/
Potentiometers
t
Trimmers
~
Temperature-sensitive
~
Light-sensitive
~ .. Vo1 tage-sens1.t1.ve Figure 234 - The major types of resistor The constant-value resistors are manufactured as small cylinders of resistor material having two copper conductors attached at each end. The copper conductors, which may be tin- or silver-plated, are used to solder the resistor in place in the electrical circuit. The constant-v.alue resistors differ in size, according to their power rating. The standard power ratings of constant-value resistors are as follows: 0.125 W, 0.25 W, 0.5 W, 1 W, 2 W, 5 W, 10 W and 25 W.
376
CHAPTER 18
The resistahce value of a resistor is indicated on its cylindrical surface in two main ways: (a)
Digitally, e.g. 1500 (150 ohm), lOk.O (10 kilohm),
2MQ (2megohm);
(b)
Through a colour code, using the following coding colours per digit: black for 0; brown for 1; red for 2; orange for 3; yellow for 4; green for 5; blue for 6; violet for 7; grey for 8 and white for 9.
The coding colours are printed on the cylindrical surface of the resistor as colour bands, or dots, starting from one end of the body. The first coding colour indicates the first digit of the value of th~ resistance, the second the second digit, the third indicates the number of zeros and the fourth (either silver or gold) indicates the deviation of the resistance value from the nominal one: silver 10 per cent, gold five per cent. Carbon~layer resistors, have their resistance material in a thin layer on top of the surface of a ceramic cylinder. A heat-resistance paint protects the resistance layer of the resistor.
Solid carbon resistors have their resistance material in the form of hardpressed cylinders - usually plastic protected material. Wirewound resistors use low thermal resistance coefficient wire (manganin, constantan) wound on ceramic cylinders which may be covered with paint or enamel for protection. Potentiometers may be wirewound or carbon layer. The value of their resistance can be changed through a handle-actuated slide contact. The trimmer-potentiometers are variable resistors, which can be used for adjusting the resistance value of a circuit. They have a screwdriver-actuated sliding contact. As the name implies, the temperature-) light- or voltage-sensitive resistors alter their values according to the values of the sensed parameters: temperature, light or voltage. According to the scatter of the resistance value of the resistors around nominal value indicated on them, they fall into one of the following groups: the (a)
Resistors of a 5 per cent scatter of the indicated value;
(b)
Resistors of a 10 per cent scatter of the indicated value;
(c)
Resistors of a 20 per cent scatter of the indicated value;
(d)
The resistors have a so-called nominal power-dissipation rating which is determined according to the resistor's ability to preserve its nominal value under a specific allowable overheating (40°) above the ambient temperature.
The temperature of overheating of a resistor can be found based on the semiempirical formula: p
a.s
(18)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM where:
377
a= coefficient accounting for the combined convective and radiation heat
exchange (W cm-2 oc-1). Its value depends on the resistor's surface, the surrounding components, pressure, etc. and has an average value of 1.5 to 2.0 X lo-3 (W cm-2 oc-1); 2
S
= heat
P
= power
exchange surface area of the resistor (cm ); dissipated by the resistor (W).
It has been experimentally established,_2hat an overheating of 40°-50°C is obtained with a ratio of P/S = 0.1 to 0.15 (W cm ). Non-metallic resistance material resistors can work safely up to 120°C. wirewound resistors have a higher safe working-temperature threshold. 10- 4Q °C-
The
fhe temperature-resistance coefficient of car~on resfstors is from 2 to 20 x and for the Wirewound resistors 0 - 2 X 10- 0°C- •
Resistors are sources of thermal noise, which may affect the operation of electronic equipment. The thermal noise voltage of a carbon resistor can be assessed using the relationship: ( pV)
,_,
for R(kQ )
and
llf(kHz).
EXERCISES 1.
Question
Find the internal resistance of an electric source and the resistance R of the load if the source's e.m.f. is 4.5 V and the voltage at its terminals for a current drain from the load of 0.5 A is 4 V. Answer Applying equation (11): e.m. f. 4.5V U
0
= U0 +
= U0
= 0.5
U
+ 4 V
V, the internal voltage drop.
Applying Ohm's law to the circuit: e.m.f. U
0
= R.I, 0
= 0.5
for the internal part, R
0
Similarly, for the external part: R 2. R
= I(R
0
V/0.5A
+ R),
= 1!2.
= U/I = 4V/0.5A = B!l.
Question
= lOOQ
Find the power dissipated by a platinum resistance thermometer of resistance and a sensor current of 0.5 mA.
Answer The _voltage drop across the lOO[]
U = 50 mV.
resistance is U = 0.0005 A x 100:
CHAPTER 18
378
-6 The power dissipated as heat is: P = U x I = 0.05 x 0.0005 = 25 x 10 W. With this amount of power dissipated from the sensor's surface the overheating remains 0 less than 0.01 c.. 3.
Question
Find the resistance R o and R_ o of the platinum resistance thermometer 50 50 and the from the previous example knowing that its resistance at 0°C is R = lOOQ 0 thermal coefficient of resistance is a = 0.0038. Answer Applying equation (16): Substituting the values of R and a:
:: R (1 +a x 50). 0
0
R o = lOO (1 + 0.0038 x 50) = 1190; similarly: 50 R_
50
o = 100(1
0,0038
X
50)
= 81 0.
The percentage change of resistance per degree Celsius is 0.38 per cent per degree.
4.
Question
Find the resistance increment for a 10° change of temperature of a two-lead copper cord connecting the temperature sensor to the instrument if the initial resistance of the cord is one ohm, assuming the temperature interval is T =20° and 0 2· . T =10 • 1 Answer Applying the empirical equation (17); 234.5 + 20 234.5 + 10
5.
= 1.0409; .1R
= R2
- R 1
= 0.04090.
Question
Find the change of resistance of a carbon resistor having R o = 10 k 0 if its temperature is i~creased to 50°C, assuming a negative temperature resistance coefficient of 20 X 10- Q °C-l, · Answer
= R0
(1 - 0.002 x t
= 10.000 18.2.4
0 )
(1 - 0.002 x 50)
=9
000 .Q - a drop of 10 per cent!
lh! !a~s_o£ Qh~ ~n£ ~i£cho£f_-_the_l~w_g£V!r~i~g_the_c£n~e£sio~ £f_d~c~ £O~e£ in!o_h~a! £O~n!xio~ £f_r!sis!a~c!s_a~d_d~c~ !l!c!ric~l_S£U£C!s_i~
= =£i£c~i! !l!m!n!s
~e£i~s_a~d_p~r~l!el
(5),
(10),
(31)
When an electricity source's terminals are connected by a conductor, an electric field will be established, causing electrical charges to move along the conductor in a.direction depending on the charge's sign and the orientation of the field. If the conductor's length is 1 and the voltage between the terminals of the voltage source is U, the field intensity will be:
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
379
u
(8')
E =-
£
The electrical current I through the conductor will be proportional to the field intensity E, the conductivity V of the conductor and its cross-section S, according to the relationship:
= VE
I
(19)
S
Substituting E with its equal from equation (8') into equation 19, taking into account that 1/V = p (the conductor's resistivity) gives: I=
u u vT.s = ..L
(20)
PS Equation (20) can be presented differently:
I
= U/R
(21)
where R is defined as the conductor's resistance, R = £/S. p. sents Ohm's law written differently.
Equation (21) repre-
Another electricity law is illustrated diagrammatically in Figure 235. The current-carrying conductors are connected at one junction, a. It seems almost obvious that as many electricity carriers are coming into the junction, as are going out:
(22)
Figure 235 - A current node There can be no accumulation of charges along the conductors. presentation of equation (22) is as follows:
A different
n
LI.=O 1
(23)
i-1
Equation (23) states that the algebraic sum of all the currents directed towards and going out from a junction point is zero. This statement is Kirchoff's first law or Kirchoff's current law • . Kirchoff's second or voltage law is illustrated in Figure 236: here the closed electrical circuit in Figure 236 consists of one voltage source and several load resistances. Following the energy conservation principle we can write: (24)
I
R3
Figure 236 - Voltage source and load resistance
380
CHAPTER 18 Noting that the quantities I.R. (i = 1, 2, 3 •. ) are the voltage drops (or ~
voltage rises) across the circuit element R., we can rewrite equation (24) as follows: ~
(25) Equation (25) is equivalent to the statement that the algebraic sum of all voltage drops taken in a specific direction around a closed circuit is zero, This ~s Kirchoff's second or voltage law, which can easily be applied to more complicated electrical circuits. In a previous paragraph we have already discussed the electrical power relationship W = U. I (W) (U(V), I(A)) (equation (10)), Bearing in mind that the power is the rate of working, the following expression can be written for the energy (work):
A= U.I.t 1
j
(L 2MT- 2 )
(26)
= 1 ws -1
In engineering practice the unit for work, the joule, is more easily used in the form (W s-1). The multiples of the latter are:
o (kWh) kilowatt-hour
= 100 W x 3 = 1 000 W x
• (MWh) megawatt-hour
= 106
• (hWh) hectowatt-hour
600 s; 3 600 s;
W x 3 600 s.
A different expression for electrical power can be derive-d from equation ( 10) : W
= IJ.I
but
U
= I.R
2 hence _ W = I .R
or
(27)
Taking into account the relationship between electrical and heat energy units: 1 W.s
= 0.24
col
1 kW.s = 0.24 kcal we arrive at Joule-Lentz's law:
Q = 0.24 x I x U x t
= 0.24
2 2 x I R x t = 0.24(U /R) x t (col) (t in s) (28)
Q = 0.864 x U x I x t (kcal) (with t in h)
The following relationships are applicable for the total resistance of a number of resistors (in this case three) connected in series (Figure 237) or in parallel (Figure 238): I
R3 Figure 237 - Connexion of resistors in series
Figure 238 - Connexion of resistors in parallel
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM Rser tot
= R1
+
R R 2 + 3
1 Rpar tot
381 (29)
or
(30)
The following relationships are applicable for a series (Figure 239) and for a parallel (Figure 240) connexion respectively of electrical d.c. sources: (31)
Provided the discharge capacities of the three sources are equal, the electrical discharge capacity in Ah of the series combination of these remains the same as that of the individual source. The parallel connexion of d.c. sources is applicable by equality of the voltages of the sources.
u
+- - - ser Figure 240 - Connexion of d.c. sources in parallel
Figure 239 - Connexion of d.c. sources in series
The voltage of the parallel combination remains the same as that of the individual source:
The discharge capacity in Ah of the parallel combination of d.c. soUrces, in the case of eoual individual discharge capacities, is increased as many times as the number of sources in the parallel connexion. Thus far, one circuit element has been discussed: the resistive circuit element, which requires a voltage directly proportional to the current through it. The proportionality constant is called resistance, R: u
= R.i
The resistance R is related to the energy dissipation as heat in the circuit. The second type of circuit element reauires a voltage directly proportional to the time derivative (or rate of change) of the current. The proportionality constant is called induction, usually denoted by L. This circuit element is closely related to the magnetic field of the circuit. u
= L. di/dt
382
CHAPTER 18
The third type of circuit element reauires a current proportional to the time derivative of the voltage. The constant of proportionality is called capacitance, C: i
= C.du/dt
This circuit element is closely associated with the electric field of the circuit. Induction and capacitance will be discussed later in the lecture notes.
EXERCISES
1.
Question
In the circuit of figure 241 the voltage E is common to all elements and the current (total) is 10 A. Find the individual currents in the branches, as well as the voltage E. Find the total load resistance.
Figure 241 - Resistive circuit Answer The voltage drops in the three branches are all the same: (a) (b)
The simultaneous solution of (a) and (b) gives the values of the individual currents:
I1
= 2(A)
I
= 5(A) = 3(A)
I
2 3
The total resistance of the three resistors connected in parallel is:
1/R
=
10
X
4 + 10
X
6.67 + 4
10
X
4
X
X
6.67
266.8 R = 133.38 :20 The voltage E is:
E
= I.R = 10
X
2
= 20 v.
6.67
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM 2.
383
Question
Find the power taken from ~he source and dissipated in the form of heat by the resistive load of the circuit of the previous example. Answer 2 The total power is P = U.I = I R = lOO x 2 = 200 W, which is all dissipated as heat by the resistive load. Expressed in units of heat it will be: Q
3.
= 0.24
P
' = 0.24
x 200
= 48
-1
(cal s
)
Question Determine the unknown voltages and currents in the circuit of figure 242.
+
E
6 -
+
56
Volt
Figure 242 - More complicated resistive circuit Answer The Kirchoff voltage-law equations are written for loops I, II, III: E + E + E 3 1 2 --E
2
= 56
(loop I)
=0
(loop II)
- E + E 4 5
E + E - E 6 1 4
=0
(loop III)
The Kirchoff current-law equations are written for the nodes A, B, C: (l/2)E (l/2)E
+ (l/10)E
1 1
(l/10)E
6
- (l/2)E
6
- E 4
=0
=0 (l/4)E = 2 5
+ E - (l/5)E
4
3
2
(node A) (node B) (node C)
The simultaneous solution of these six equations will give: E 1
= 16
V; E 2
E4 = -4 V; E5
= 20 = 16
= 20
V;
V; E6 = 20
~
V; E3
The corresponding currents can be found from Figure 242 by the application of Ohm's law. A different approach is based on the calculation of the current values rather than the voltages:
384
CHAPTER 18 2I 1 + 5I -5I 10I The
corresp~ding
2 6
2
- I
+ 2I
4
= 56
(loop I)
0 5 =
(loop II)
3
+ 4I
+ I4 - 2I 1
=0
(loop III)
nodal equations: Il + I6 - I3 = 0
(node A)
Il + I4 - I2 = 0
(node B)
I6 - I4 - I5 = 2
(node C)
The simultaneous solution of the six enuations gives: I
= 8 A;. I2.= 4 A;
!3 = 10 A;
I4
= 4 A;
I6 = 2 A.
I5
= 4 A;
The values are found through the application of Ohm's law to the elements of the circuit in F.igure 242. 18.2.5
D.c. electric sources: carbon type dry cell lead-acid storage battery.~~~=§~~=~§!~!~~§~~~~rs;;-rro;-----------------------------------------
The electrolytes, solutions of acids, salts and bases, are conductors m electricity known as second-class conductors. The molecules of the electrolyte split into positive and negative ions. The hydrogen and the metals give the positive ions, the non-metallic components negative ions. If an electrical field is applied to an electrolyte through two electrodes, anode (positive) and cathode (negative), the ions in the electrolyte are set into motion. The positive ions are attracted by the cathode and the negaiive ions by the anode. Both electrolytic charges (ions) contribute to the electrical current in the elect~olyte. In the process a certain amount of matter is transferred to the electrodes according to the sign of the respective ions. The amount of matter transported can be obtained from the equation: G
where: c
= c • I. t
( mg)
= coefficient
of proportionality electrochemical equivalent of an ion (the mass of the ion liberated or deposited onto the electrode by 1 C of electricity) !.l Values o~ c for Ag = ljl8 mg c-l; for Cu = 0.329 mg C ; N~ 0.304 mg C ;
=
I =electrical current in amperes (A); t =time of action of the electrical curre.nt in seconds (s). A potential difference appears between two electrodes immersed in an electrolyte. This phenomenon is made use of in the galvanic cells - primary electric batteries. One primary battery in widespread-use is the dry cell invented by Leclanche (Figure 243). The electrodes of the dry cell are carbon (positive) and zinc (negative). The electrolyte is a viscous solution of ammonium chloride (NH Cl) and inert material. 4
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
385
-metal cap
~carbon electrode (+) (-)
~ ~: ~::
--:
;..
..--
-- ·.
-
-~
:-.··.. •,•' ,._.•!
electrolyte depolarizer (Mn0 ) 2
--- ·-. I
•:
Figure 243 - Dry-cell battery In order to prevent polarization of the cell (gas deposition on the anode), a depolarization compound is used (manganese dioxide) pressed around the anode. Long use of the dry cell causes the zinc cartridge to erode and the electrolyte which is corrosive, eventually to leak. Modern dry cells are leak-proof. The e.m.f. of the dry cell is 1.5 V. depends on the size of the battery.
The electrical discharge capacity
Storage batteries are secondary electrical sources and must be charged electrically before use. The battery charger is usually a mains-fed rectifier of appropriate voltage and current ratings. One type of storage battery widely used in automobiles is the lead-acid battery. Its positive electrode is made of a lead-dioxide (Pb0 ) active mass pressed 2 into a lead frame. The negative electrode is made of spongy lead in order to increase its active surface. Wooden or plastic separators are inserted between the electrodes to prevent direct contact between them. The electrolyte is a 24 - 26 per cent solution of distilled water and sulphuric acid (H so ). 2 4 The e.m.f. of a single battery cell is 2 V. Most widely used are batteries in a series connexion of three cells (6 V) or six cells (12 V). The amperehour capacity of the battery is related to the size of its electrodes. Through discharge of the battery, both the Pb0 and Pb electrodes are 2 gradually converted into PbS0 4 , the electrolyte thinning in the process because of the build-up of water, The discharge should be stopped before the voltage of a single cell drops below 1.8 V. Further discharge below 1.8 V is likely to make the process irreversible because of a heavy build-up of PbS0 , when the battery will be 4 damaged permanently. During the charging of the battery, through its connexion to the rectifier (mind the proper polarity!), the electrical current passing through the battery restores the PbO? at the positive electrode and the Pb at the negative one. The proper density of the electrolyte is restored as well. When the battery is completely charged, the electrolyte's density is totally restored by a rise of the voltage at the terminals of the single cell to 2.6 - 2.7 V, Excessive bubbling of the electrolyte, arising from a build-up of gas, accompanies the final phase of battery charging. The charging voltage is slightly higher than the battery voltage. The charging current is usually selected to be not greater than one-tenth of the numerical value of the nominal discharging capacity of the battery. Thus, a 50 Ah batteDY is charged by a five ampere current for at least 12 - 15 hours.
CHAPTER 18
386
Newly-produced lead-acid batteries, if intended for prolonged storage, are released from the factory without the electrolyte. The necessary electrolyte is prepared in the knowledge that 310 cm3 of concentrated H2so 4 dissolved in 1 000 cm 3 of distilled water gives the proper electrolyte concentration of 24° Bom~ (at 15°C temperature of the electrolyte). The total liquid volume of the battery is approximately assessed and the electrolyte prepared in a lead or ebony container in the necessary quantity, observing the above proportions. The electrolyte is poured into the battery after it has cooled down to at least 25°C. The electrical charging of the battery is commenced no sooner than 12 hours after the electrolyte has been poured into it. The first charging is followed by d gradual discharge (discharge current less than one-tenth of the battery capacity) and a second charging. The average brand of lead-acid batteries is usually operational for at least 600 charge/discharge cycles before showing a marked reduction of electrical capacity. The coefficient of efficiency of a good battery is about 0.8. Maintenance tips for lead-acid batteries:
• Keep the electrolyte of the battery always about 10 mm above the electrode plates;
•
After a drop in the level of the electrolyte because of evaporation, add distilled water to restore the level;
• Check the density of the electrolyte in each battery cell frequently, using a special hydrometer;
• Keep the battery clean externally, but be careful not to let the washing water leak into tbe inside of the battery and spoil . the electrolyter
•
If not used for prolonged periods (more than a month), the battery should be discharged to 1.8 V and then charged again regularly (once a month) to keep it fit and prevent irreversible changes in the electrodes;
• Careful operation is required for a battery in ambient temperatures
below 0°C. It should be borne in mind that at -10°C the battery's capacity is reduced to about 60 per cent of its normal value, which sets a limit on the discharge current rating;
• A discharged battery should be charged again as soon as possible • The positive electrode of the nickel-cadmium storage battery is Ni(OH) 3 contained in a· pocketed bron frame, the negative electrode being spongy cadmium. The electrode plates are separated by insulating inserts.. The electrolyte is 21 per cent water solution of KOH. The e.m.f. per cell is 1.4 V. During discharge, the positive electrode is gradually reduced to Ni(OH) 2 and the negative electrode to Cd(OH) • The electrolyte density remains constant 2 during the discharge. The threshold discharge voltage is 1.15 V per cell. During the charging process, both electrodes are restored to their initial form, the e.m.f. rising to 1.8 per cell. The charging current is again one-tenth of the battery's capacity roting.
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
387
The internal resistance Of NiCd storage batteries is higher than that of lead-acid ones. They are less liable to short-circu~t and can be left uncharged for longer periods without detriment but they have a lower coefficient of efficiency of about 0.6. The NiCd batteries require less care in operation. Their electrolyte is sensitive to an exposure to C0 and is easily spoiled by it. The electrolyte of 2 this type of battery, which is not gas-tight, is changed periodically. Solution of simple
18.2.6
ele~trical
circuits - principle of superposition
!~~~~~~~=~!=!~!~~~!~=~~~=~~~!~~~r5J;-rioJ;-r~IJ
____________________
'
Network simplification is a method of analysis which is very helpful in reducing the complexity of a circuit. An electrical source can be presented in a simplified form, assuming a volt/ampere characteristic such as the one presented in Figure 244. The terminal voltage of the source E has an open circuit value of E (the circuit current I = 0), a linear decrease with the increasing circ~it I oc and a zero value at thgcmaximum circuit current I , that of a short circuit. se E.
I+
E
Source
--
Load
I
(b)
(a)
Figure 244 - Circuit simplification The graph in Figure 244(b) can be mathematically expressed by the following equation:
E = Eoc -R 0 .I
where: R
= the
(1)
ratio of the open-circuit voltage to short-circuit current.
An equivalent circuit of the source shown in Figure 244 can be described, following equation (1) (Figure 245).
Figure 245 - Voltage source equivalent circuit
The equivalent circuit in Figure 245 is known as a "voltage source equivalent circuit". An alternative presentation can be found if equation (1) is re-written in a form in which the current is dependent on the voltage. Dividing equation (1) by R and substituting E/R yields:
0
0
by G .E, where G =1/R , 0
0
0
CHAPTER 18
388
(2)
Following equation (2), the alternative "current source equivalent circuit" is as shown in Figur~ 246.
~
+ G
0
+
E
+_
Figure 246 - Current source equivalent circuit Since the circuits of Figures 245 and 246 both have the same volt/ampere characteristics shown in Figure 244(b), either one can be used to represent a practical circuit source. The terminal characteristics of the circuits of Figures 245 and 246 are identical and one can be used to represent the other. The circuit to be used in the solution of a practical problem depends on the method of solution planned, according to convenience considerations. One method of circuit simplification is provided by the so-called circuit theorems. The principle of superposition is a very useful method of circuit simplification because it extends the applicability of Ohm's law to circuits having more than one source. This theorem states: In a network with two or resistance components, the current as an algebraic sum of the effects position is possible in any system such that
more sources of current or voltage and constant or voltage produced in the circuit can be considered produced by each source acting separately. Superin which the cause-effect relationship y = f(x) is
The summarized requirements for the application of the principle of superposition are: all the components must be linear and bilateral. By linear is meant that the current should be proportional to the applied voltage. By bilateral is meant that the current is the same amount for opposite polarities of the applied voltage. Networks with resistors, capacitors and air-core inductors are generally. linear and bilateral. Active components, such as transistors, diodes and tubes are never bilateral and often non-linear. In order to use one source at a time in the principle of superposition, all other sources of the circuit must be ''disabled" temporarily. A voltage source is disabled by assuming a short circuit across its terminals. As an example of the application of the superposition principle, a voltage divider having two voltage sources is considered. The problem is to find the voltage at point P to the chassis ground for the circuit shown in Figure 247(a). The two steps in the application of the method are illustrated by the circuits ·shown in Figure 247(b) and (c).
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
(a)
(b)
(c)
Actual circuit
V -short circuited 2
V -short circuited 1
389
I
Figure 247 - The principle of superposition applied to a voltage divider Considering the circuit in Figure 247(b), the voltage at P is obtained from the equation: 60k
30k + 60k
• 240V
= 160V
Considering the circuit in Figure 247(c): V
p2 =
R2 Rl + R2
. v2
=
30k 30k + 60k
(-90)V
= -30V
Finally, the actual voltage is obtained as a result of the superposition of the two voltages acting separately:
= l30V The same procedure can be followed with more than two voltage sources. Named after M. L. Thevenin, a French engineer, Thevenin's theorem enables the representation of a complicated electrical circuit, consisting of many components and sources, by an equivalent series circuit with respect to any desired pair of terminals in the network (Figure 248)0
a
a
Network b
b
Figure 248 - Actual "network" and Thevenin's equivalent In the equivalent circuit shown in Figure 248, V h is the open circuit voltage _across the terminals a and b. The polarity of Vt~ is the same as in the original network presented as a box. The resistance Rtb is the open-circuit resistance across the terminals a and b, with all sources of the network "disabled". This is the actual resistance between the terminals a and b to be measured if the internal sources are shortcircuited.
390
CHAPTER 18 R1
R1
a
a
J 6
L
V b=24V
-t>
Ja
b
b (b)
(a)
~
a
fa
R2
--£>
R b=2
1
va.b
~
b (d)
(c)
b
R
:6vt~· t
·Vth 12V
---o
(e)
Figure 249 - Steps in the "thevenizing" of a circuit To "thevenize" the circuit in Figure 249(a), in which we want to find the voltage VL across the load resistance RL =2 Q and the current through it IL, we must first mentally open the circuit disconnecting RL (Figure 249(b)). In the second step we find the Thevenin equivalent of the remainder of the circuit that is still connected to a and b. The Thevenin equivalent circuit always consists of a single voltage source Vth in series with a single resistance Rth" As a result of opening the circuit a voltage divider is obtained, consisting of the resistances R and R • The Thevenin voltage is the voltage across R : 1 2 2 VR 2
= (6/9).36V = 24V = Vth
To find the Thevenin resistance R±h' the RL remains disconnected but the source V is short-circuited (Figure 249(c)). Now R comes in parallel with R2 , therefore: 1 Rth = R1 // R2 = 18/(3+6) = 2[2 (Figure 249(d)). In order to fina the voltage drop across fhe load VL and the current I , we finally connect R between the terminals of 1 the "thevenized" circuit in Figure 2~9(d), thus obtaining Fl.gure 249(e). Applying the voltage divider formula we then obtain: VL= (2/4).24 = 12V
REVIEW OF THE FU\JDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
391
Note that 6 A flow through Rth as well. The same result could be obtained by solving the series/parallel circuit applying Ohm's law. Thevenin's solution, however, has the advantage of enabling the calculation of the effect of RL for a number of different values. Using Ohm's law, a complete new solution is required, each time the RL value is changed.
(d)
(c)
Figure 250 - Thevenizing a two-source circuit The circuit in Figure 250(a) having two sources is thevenized in the following way (Vth and Rth are sought between the terminals (a and b)): (l)
Disconnect R • 3
(2)
Short-circuit v , with v acting: 2 1 R2
(the contribution of vl to vab). (3)
. v
= -16.BV
The polarity
Short-circuit V , with V acting: 2 1 Rl
v
Rl + R2 (4)
1
2
lS
negative at terminal a.
= -16.BV.
Find the resultant Vab'
As both V and V produce the same polarity volta9e 2 1 across ab (-16.8V), they are added together: Vab = Vth = V1 + V2 = -33.6V (a minus sign means the terminal is negative).
(5)
Calculate Rth:
short-circuit V and v 1
parallel with R :
RR
2
1 2
Rth =
R' + R 1 2
=
2
as 1n Figure 250(c).
36 12 + 3
= 2.4!2v
Then R comes 1n 1
392
CHAPTER 18
(6)
Draw the circuit diagram of Thevenin's equivalent, using Rth = 2.4!2 and Vth = 33.6 V (Figure 250(d)).
(7)
Find the current through R = RL re-connecting it between the terminals of 3 Thevenin's equivalent circuit, Figure 250(d):
As a further example, let us
thevenize
the bridge circuit in Figure 251.
(b) l.)
.l,\-~.:b
---t>
Rl
V, short-circuited /
\
\c)
Figure 251 - Thevenizing a bridge circuit The following steps are taken: (1)
Disconnect RL.
(2)
Find the voltages across the two voltage dividers R /R and R /R : 3 4 1 2
(The bottom line is voltage reference-ground.) (3)
Find the voltage across the terminals a and b:
(Terminal a is more negative than terminal b.) (4)
Find the equivalent Thevenin resistance Rth between the terminals a and b, by short-circuiting V (Figure 25l(c)),
The resistances R and R appear in paral4 3 lel as do the resistances R and R and the two pairs are in series: 1 2
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
(Figure 25l(d))
RTa + RTb = 2 + 2.4 = 4.4fl = Rth (5)
393
Now connect RL across a and b of the Thevenin equivalent circuit of the bridge (Figure 25l(d)) and find the current through the load resistance IL using Ohm's law: 8/(4.4 + 2)
= 1.25
A
The bridge can be solved, considering the current loops on hand in the circuit, thus obtaining the necessary number of equations which are solved simultaneously. This is a more complicated procedure (see the paragraph on "thermistor bridge" based on the Wheatstone bridge.) Norton's theorem is named after E. l. Norton, a Bell Laboratoriei scientist. It is used as a circuit simplification method, this time in terms of the currents instead of voltages as in the previous theorem. A network can be reduced through Norton's theorem to a simple parallel circuit with a current source (Figure 252(b)): I I
a1
1
(a) b
Figure 252 - (a) Thevenin and (b) Norton equivalent circuits The value of the current IN in the Norton equivalent circuit is the shortcircuit current through the shorted terminals a and b and the value of RN is the resistance of the circuit looking back from the opened terminals a and b, with the source short-circuited. In fact, RN is the same as Rth but is placed in parallel to the current source, while the latter is placed in a series with the voltage source.
fol2A
3hort /circuit jwnper
(c)
(b)
(a)
R1
~:--:)
a
a -{>
c[E} . !1t=2 12A ·--{)
b (d)
(e)
(:f)
Figure 253 - Steps in "nortonizing" a circuit
CHAPTER 18
394
The "nortonizing" of the circuit in Figure 253 is carried out in the following steps: (1)
Select the terminals a and b for application of the Norton theorem.
(2)
Short-circuit the terminals a ~nd b and find the short-circuit current IN.
(3)
With terminals a and b opened, short-circuit the source in order to find the Norton equivalent resistance RN.
(4)
With the current IN and the equivalent re~istance RN known, draw the Norton equivalent circuit diagram.
(5)
Reconnect RL across the terminals a and b an~ find the load current IL.
It should be kept in mind that the short-circuit current I~ flowing through the short-circuit wire jumper (imagined). flows through all the circuit components connected in serie~ wlth the terminals a and b. The Thevenin theorem states that any network can be represented as eonsisting of a voltage source and a series resistance, The Norton theorem states that any network can be of a current source and.a shunt resistance.
re~resented
as consisting
A conversion of a Norton into a Thevenin form of network.is possible, either following the reasoning 6utlined in the beginning of the present p~ragraph, or simply making use of the following formulae: Norton into Thevenin;
Rth
= RN
The Thevenin-into-Norton conversion is made using the formulae: RN
=
IN
= Vth/Rth
Rth
Any voltage source with its series resistance can be converted into an equivalent current source with the same resistance in parallel.
EXERCISES 1\
Wheatstone bridge configuration, balanced version
The balanced Wheatstone bridge, shown in Figure 254, is used to measure the value of an unknown resistance, R , through the comparison between it and the known value of a variable adjustable ~esistance. The comparison is made through the bridge, using a sensitive "zero current" indicator, usually a galvanometer,
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
395
2
u + b
1
Figure 254 - The balanced Wheatstone bridge circuit The two legs of the balanced bridge are in fact two voltage dividers consisting of the resistances: Rx' R and Rv' R • 1 2 If the voltage drop across the R adjustable resistor is made equal to that of R , there will be no voltage differencev between the points (3) and (4) of the · brid~e and, therefore, no current through the indicator connected across (3) and (4). With these conditions fulfilled, the following relationship is valid:
The values of R and R are fixed and for the value of the unknown resistance Rx we 2 1 obtain: The ratio of R and R being constant enables the dial of the adjustable resistor RV 1 2 to be graduated directly in resistance units, giving the value of the measured resistance. With this simple arrangement, if R is a temperature-sensitive resistance X
and the dial of the adjustable resistance is graduated in degrees of temperature following a calibration, the Wheatstone bridge can be used for temperature measurements. The temperature-sensing resistance may be a platinum wire sensor having a 0°C resistance of lOOfl and a temperature/resistance algorithm of the kind:
As seen from the above relationship, a change of almost 0,4 per cent in the sensor's resistance is obtained for a change of temperature of 1°. The sensor R may be a semiconductor thermoresistor - a thermistor, X
An
expression relating the resistance of a thermistor and its absolute temperature, frequently used for meteorological purposes, is the following one:
where: a, A = constants depending on the semiconductor material.
R =
. . . (1 dR - .9._) Th e negat~ve temperature coe ff.~c~ent t herm~stors dT T have a 2 temperature sensitivity of about four per cent per degree Celsius,
396
CHAPTER 18
Care should be taken with bridge temperature measurements not to go beyond the temperature sensor's measurement current ratings. A failure to do so might contribute to significant measurement errors due to overheating (or even destruction) of the sensor,
2.
The Wheatstone unbalanced bridge
The unbalanced version of the bridge has a current meter connected in its diagonal between the points (3) and (4), gen~rally having an input resistance R. The current r through R is a measure of the value of the temperature-sensitive 2 resistance rt, hence a measure of the temperature itself, provided the instrument is calibrated against a standard thermometer. The unbalanced version of the Wheatstone bridge is shown in Figure 255. The "current loops" method and Kirchoff's law are employed in analysing the operation of the bridge (a more efficient way of treating the circuit shown has been discussed in the previous paragraph). 2
I(~~t u - r/ ~4 r1
1 Figure 255 - Unbalanced Wheatstone bridge
Following the circuit diagram in Figure 255 and considering the current loops: Loop (1) consisting of U, r , r • 1 2 Loop (2) consisting of r , r , R. 1 3 Loop (3) consisting of r , r , r , rt. 2 1 3 By the use of Kirchoff's law,l:I.R= l:U the following equations can be written: U
= r1
0 - r 0 = r
1 2
(I (I (I
1 2 3
(1)
- I 2 - I 3 ) + r 2 (11 - I 2 ) +
r3 - r1)
+ r
3
- I ) + r (I + 1 2 1
(I
2
+
r3 )
r 3 - I 1)
+ R.I + r
3
(2)
2 (I
2
+
r 3)
+ rti
3
(3)
After re-arranging the variables of equations (1), (2) and (3), we obtain:
u = Il
(rl + r2) + I2 (-rl) + I3 (-r2-rl)
0 = Il (-r ) + I (r + r + R) + I3 (rl + r3) 2 3 1 1 0
= Il
(-r2 -rl) + I2 (rl + r3) + I3 (r2 + rl + r3 + rt)
(4) (5) (6)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AMD MAGNETISM
397
Solving equations (4), (5) and (6) together for I
yields the relationship 2 between the meter current and the value of the temperature-sensitive resistance (sensor). The expression for I
2
using determinants is given:
rl + r2
u
-r2-rl
-rl
0
-rl + r3
-r2-rl
0
r2 + rl + r3 + rt ( 6' )
or, in a more explicit form: (7)
It is evident from (7) that the relationship I 2 = f(rt) is a non-linear one but, through proper selection of r , r , r , the shape of the curve can be improved 3 2 1 considerably. As already pointed out, the unbalanced bridge can be solved using the Thevenin-theorem simplification. This approach to the circuit solution is briefly outlined below: (See Figure 251 and compare it to Figure 255.) (8) where: (9)
Equation (9) presents the Thevenin equivalent voltage. rtr3
r2rl Rth = RTa + RTb = r2 + rl where:
+
r2rl RTa = ri/rl = r2 + rl rtr3
RTb = r//r 3
= rt
+ r3
rt + r3
(10)
398
CHAPTER 18 By substituting the values of Vth and
Rthas obtained from equations (9)
and (10) into equation (8) and after the necessary re-arrangement of the terms, equation (7) is again obtained.
3.
Linearization of the response of a thermistor sensor
Thermistors are used frequently in temperature measurements using the unbalanced Wheatstone bridge configuration circuit. This arrangement has one serious shortcoming: its non-linear response to temperature. One remedy is the sub-range linearization of the thermistor response, dividing the total temperature-measuring range of the thermistor thermometer into l0°C sub-ranges. The linearization circuit 1s obtained, according to the following reasoning: Considering the series combination of the thermistor's resistance rt and a temperature stable resistance R inserted between points (3) and (2) of the bridge n
leg (Figure 255), the following relationship for the measurement current will be valid (see Figure 256): . I
u = _ __;;,o-::::--
(l)
rt + Rn
X
u.
/\2
\~~0 ~ ' x~· Rn . 3v
rt
Figure 256 - Thermistor and serieslinearizing resistor where:
u0 = the I
X
= the
voltage between points (2) and (3) of the circuit in Figure 255; current through rt and R . n'
rt = A.ea/T is the thermistor's temperature response algorithm. Substituting the expression for rt into equation (l) gives:
u
I
= --~0:---X
(2)
A.ea/T + R
n
The graph of equation ~) has an inflexion at a point determined by the value of the thermistor's resistance, r t. In the vicinity of the inflexion point, 0
the graph can be considered almost linear. The presence of an inflexion 1s indicated by the second derivative of equation (2) in terms of T being equal to zero:
ii dt 2 x
= 2U0
(rt + Rn)-
3
2 2 2 (drt/dTf- U0 (Rn + rt)- .(d rt/dT )
=0
(3)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
= -rt.
Bearing in mind that drt/dT
399
a 2 2 2 2 T2 and d rt/dT =rt(a/T ) ,(1+2T/a),
the following result 1s obtained from (3): (4)
2(r t + R )-lr t = 1 + 2T/a o
n
o
where: r t = the resistance of the thermistor at the point of inflexion in the 0 vicinity of which the current through the thermistor may be considered as a linear function of the temperature. The value of the series-linearizing resistance R is obtained from (4): n
1 - E
Rn = rot 1
(5)
-1- E
where: £
= 2T/a
The constant a (together with A) are obtained from the thermistor's 0 response equation rt=A.ea/T for two known values of the temperature T = 273 + t • In order to keep the self-heating of the thermistor from the measurement_ 2 current within safe limits (0.1°C), the product r .I 2 must be kept below 1-3 mW cm of the thermistor's surface. t x
4. One way to obtain a thermistor wide-range linearization is to use it as a base resistor in a transistor-multivibrator circuit. The pulse frequency of the generator would depend on the thermistor's resistance and the collector-to-collector d.c. voltage will be a linear function of ~he temperature of the sensor. This approach to the linearization of the thermistor response is discussed in the chapter dealing with pulse generators. A different way of attaining wide-range linearization is to use a combination of tw~ thermistors T and T and two passive resistive elements R and R, 1 2 1 as shown in Figure 257.
E
0
Figure 257 - A 0.2% deviation from linearity thermistor combination The principle of operation of the combination of active and passive elements for the linearization of the temperature response is based on the following: ~ith properly selected components, the output of the combination R and T approximates 2 the straight line of the response curve in the lower temperature range wnile becoming virtually ineffective at high temperatures. The thermistor T approximates the de1 sired curve at high temperatures, becoming ineffectual at low temperatures. In the
CHAPTER 18
400
middle temperature range the thermistor T and the combination T and R contribute 1 2 nearly equally. The appropriate value for the elements in the circuit shown in Figure 257 can be selected according to the solution of the following set of equations: E
0
-nL1E o =E.:tn
RTl(RT2 + R) + Rl(RTl + RT2 + R) RT
= RTo
exp [ 8(1/T - l/T 0
)
n
= 0,
+ C(l/T - l/T 0
1, 2
2 )
]
E
0
I ,1 E0
(1) (2)
Equation (2) gives the thermistor's response at temperature T (kelvin); 8 and C are thermistor constants. For a temperature range of 0°-l00°C, the maximum deviation from linearity of the response of the thermistor combination is about + 0.2 per cent, with the following selection of component values:
Rn = 300
at 25°C·
R = 6 2500
=6
at 25°C·I
Rl
RT2
kD
'
=3
200!2
A three-element combination (Figure 258) having a 0.04 per cent deviation from linearity within the 0°-l00°C interval has the following values of the elements:
E.ln
Figure 258 - Three-element linear temperature response thermistor
Rn = 45 RT2 = 15 RT3
=
kD
at 25°C·I
R
=2
160!2
kD
at 25°C·I
Rl
=4
031!2
3 kD
at 25°C;
R2 = 9 025 D
Although its advantages are attractive, this linearization method is not very easy to realize materially. The main difficulties are connected ~ith the solution of the set of equations and the selection of the desired values of resistance of the thermistors. The multi-element linear thermistor should have its elements placed in a common cartridge which is filled with a high thermal-conductivity material, thus securing the same temperature for all the active elements of the combination.
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
401
The thermistor temperature sensor has a number of positive features:
• •
High sensitivity;
•
Possibilities for miniaturization;
Easily attainable matching of the resistance of the thermistor to the signal converter (wide range of thermistor resistance at 20° values are available industrially);
Low
cost~
The semiconductor sensor exhibits a number of negative features as well, however:
•
Ageing (gradual change of the resistance at 20° with time); Considerable scatter of the resistance values of thermistors of a given type;
• •
Exponential response of the sensor; Sensitivity to damage by thermal runaway in the case of circuit failures •
Thermistors are frequently used in electronic telemetering circuits, which usually require sensors having resistance values above several kD
18.3
The nature of magnetism - magnetic quantities and units of measurement (5), (10), (31)
The use of the compass in direction-finding is well known and is based, as usually stated, on the interaction between the "magnetized needle" of the compass and the "Earth's magnetic field". A magnet is a body that attracts iron and certain other metals by virtue of a surrounding field of force. A natural form of iron ore consisting principally of an oxide of iron (Fe o ) and called magnetite has been known for its magnetic 3 4 properties for many years. Steel can be magnetized and is used for the compass needle which, in the absence of disturbing factors such as nearby iron objects, will set itself so that its t~o ends point to the respective magnetic poles of the Earth. The end pointing towards the North Pole is called the north-seeking or simply north pole; the other end is called the south pole. If the ends of two magnetic needles (or bar magnets) are brought close to one another, attraction or repulsion will occur, depending on whether the adjacent poles are of opposite polarity or the same polarity. The region surrounding a magnet within which magnetic forces are acting is called a magnetic field. The magnetic field is presented by its lines of force of flux indicating the direction in which the magnetic forces of flux cannot be seen and by our standards cannot be considered as material, they can be visualized by placing a sheet of thin card placed over a magnet and sprinkling it with iron filings. By convention it is assumed that the direction of the lines of force is from the riorth to the south pole in the region surrounding the magnet and from the south to the north
CHAPTER 18
402
pole within the body of the magnet (Figure 259).
Figure 259 - Magnet and lines of force of flux The field distribution around various magnets is such that at no point do the flux lines of the individual magnets cross. The compass needle always points in a northerly-southerly direction, so that its own field lines are parallel to that of the Earth. The magnetic field and the electric current are linked together! a conductorcarrying current is always accompanied by a magnetic field; a magnetic field varying with time and space, its lines of force intersecting a conductor, gives rise to an electric current.
A force of attraction will be experienced by a piece of iron anywhere in the vicinity of a magnet, provided the distance is sufficiently small, but its magnitude will vary with position. It will be strongest around the poles of the magnet and relatively weak near the centre. In addition, it will decrease with distance from the magnet in all directions. A straight conductor of length 1, carrying current I and placed in a magnetic field, will experience a mechanical force on itself F, which can be expressed in terms of the magnetic field characteristic B, known as magnetic flux density, according to the following equation:
= B.I.l
F
where:
I in amperes; 1 T
=1
V s
-1
1 :1.n metres;
-2
m
=1
-2
Wb m
(N)
(1)
B in teslas (abbreviated (T)); •
In the olde: CGS sy~~em of ~2its,,the magnetic flux density is measured in 4 gauss (Gs): 1 Gs = 10 Wb m = 10 T. A magnetic field exists around a current-carrying conductor as long as the current is maintained. The magnetic lines of force circle a straight conductor and can be visualized by iron filings sprinkled on a card through which the conductor is threaded. The lines of force thus visuali~ed will look like concentric circleso ~tic
If looked at in the direction of the current flow, the direction of the field around the current-carrying conductor will be clock-wise (Figure 260).
The magnetic flux density B is linked with another magnetic quantity H, known as a magnetizing force: (2)
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM where:
p
= absolute 1
t"''
~~
permeability of the medium measured in Q
s -1 m-1
=1
s
-1
403 -1 m
HenryI metre, abbreviated H m-1 . I
Figure 260 - Lines of force around a current-carrying conductor As seen from equation (2), the magnetic effect at a point 1n space depends upon the matter or medium at that point. The permeability of free space:
Po
=4 n
x 10 -7{ H m-1)
= (3)
where:
= relative
permeability of the medium, equals 1 for free space, less than 1 for diamagnetics, greater than 1 for paramagnetics and much greater than 1 for ferromagnetics.
The magnetic effect produced at a point in space near a current-carrying conductor depends on the value of the current I and the distance from the centre of the conductor, r: H
=2
I
n.r
(4)
where: r
= the
distance, i.e. the radius, of the line of force along which the magnetic force is considered. Equation (4) could be presented in the following form:
H=
I L
(5)
where: L = the length of the line of force. Equations (4) and (5) are valid for the case of a single, straight and infinitely long conductor. If a number N of conductors are carrying current in the same direction (e.g. the turns of a coil):
N. I H= L
(6)
404
CHAPTER 18
The product I.N. is known as magnetomotive force (m.m.f), measured in ampere-turns (At). When a straight current-carrying conductor of length 1 moves under the action of a force F in a plane perpendicular to a homogeneous magnetic field, along a distance b (Figure 261), mechanical work will be done according to the relationship: A= F.b = B.I.l.b (J) Bearing in mind that S B.S
(7)
l.b is the area covered by the moving conductor and
= /Sl = 0.00432/0.0036 = 1.2
(Wb m-2) i
B2
= ajs2 = 0.00432/0.0048 = 0.9
(Wb m- 2 );
83
-= Cl>jS3
(Wb m-2).
= 0.00432/000036
= 1.2
REVIEW OF THE FUNDAMENTAL LAWS OF ELECTRICITY AND MAGNETISM
411
From the graph in Figure 267, available commercially, and the values of B 1 obtained from the above relationships, the values of H are readily derived:
1
-2 Wb m 2.0
cast ·steel
1 .o
Am
0.5 1
1. 5
2
2. 5
X
-1
3 10 H
Figure 267- 8/H relationship, cast-steel
H1
= 900
(A m-1);
H2
= 700
(A m-1);
H3 = 83/~0
= 1,2/(1.256 X 10- 6 ) = 0,955 X 106 = 955 000 (A m-1)
Substituting these values in equation (20) yields: I.N. = 900 X 0.545 + 700 = 5 384.
X
0.17 + 955 000
X
0.005
= 490
+ 119 + 4 775
From equation (21) and substituting the values for the m.m.f. and the current through the winding I gives: m.mof./I= N = 5 384/5 = 1077 turns. Thus, a winding of 1 077 turns carrying a current of 5 A will produce an of 5 384 A which, in turn with the magnetic circuit parameters assumed, will give rise to a magnetic flux amounting to 0.00432 Wb.
m.m.~
Consider a ferrite core, used for magnetic core storage 1n digital computers, having the following characteristics: Shape: closed ring (toroidal); Outside diameter: Oo002 m; Inside diameter: 0.00127 m; Difference in diameters: Oo00381 m; Width of toroid: 0,00635 m; Cross-sectional area of toroid: 0.00381 x 0.00635 = 2.42 x 10 -7 m2 The desired magnetic flux density 1n the core 1s B = 0.15
Wb m-2
412
CHAPTER 18 Find the flux$ and the necessary ampere-turns
NI~
Answer Following the relationship:
Cl>
= B.S
and substituting in it the values for B and S yields:
ID
= 0.15
X
2.42
X
lQ-7 = 3,63
X
The mean length of the magnetic flux path L
10-8 Wb
= 5.18
x lo-3 m.
Following the expressions: m.m.f. = -----1NI-----