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Chan, Shu-Park “Section I – Circuits”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
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© 2000 by CRC Press LLC
© 2000 by CRC Press LLC
I
Circuits
1 Passive Components
M. Pecht, P. Lall, G. Ballou, C. Sankaran, N. Angelopoulos
Resistors • Capacitors and Inductors • Transformers • Electrical Fuses
2 Voltage and Current Sources
R.C. Dorf, Z. Wan, C.R. Paul, J.R. Cogdell
Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals • Ideal and Practical
Sources • Controlled Sources
3 Linear Circuit Analysis
M.D. Ciletti, J.D. Irwin, A.D. Kraus, N. Balabanian,
T.A. Bickart, S.P. Chan, N.S. Nise
Voltage and Current Laws • Node and Mesh Analysis • Network Theorems • Power and
Energy • Three-Phase Circuits • Graph Theory • Two Port Parameters and Transformations
4 Passive Signal Processing
W.J. Kerwin
Low-Pass Filter Functions • Low-Pass Filters • Filter Design
5 Nonlinear Circuits
J.L. Hudgins, T.F. Bogart, Jr., K. Mayaram, M.P. Kennedy,
G. Kolumbán
Diodes and Rectifiers • Limiters • Distortion • Communicating with Chaos
6 Laplace Transform
R.C. Dorf, Z. Wan, D.E. Johnson
Definitions and Properties • Applications
7 State Variables: Concept and Formulation
W.K. Chen
State Equations in Normal Form • The Concept of State and State Variables and Normal
Tree • Systematic Procedure in Writing State Equations • State Equations for Networks Described
by Scalar Differential Equations • Extension to Time-Varying and Nonlinear Networks
8 The z-Transform
R.C. Dorf, Z. Wan
Properties of the
z
-Transform • Unilateral
z
-Transform •
z
-Transform Inversion • Sampled Data
9T-
P
Equivalent Networks
Z. Wan, R.C. Dorf
Three-Phase Connections • Wye
⇔
Delta Transformations
10 Transfer Functions of Filters
R.C. Dorf, Z. Wan
Ideal Filters • The Ideal Linear-Phase Low-Pass Filter • Ideal Linear-Phase Bandpass
Filters • Causal Filters • Butterworth Filters • Chebyshev Filters
11 Frequency Response
P. Neudorfer
Linear Frequency Response Plotting • Bode Diagrams • A Comparison of Methods
12 Stability Analysis
F. Szidarovszky, A.T. Bahill
Using the State of the System to Determine Stability • Lyapunov Stability Theory • Stability of
Time-Invariant Linear Systems • BIBO Stability • Physical Examples
13 Computer Software for Circuit Analysis and Design
J.G. Rollins, P. Bendix
Analog Circuit Simulation • Parameter Extraction for Analog Circuit Simulation
© 2000 by CRC Press LLC
Shu-Park Chan
International Technological University
HIS SECTION PROVIDES A BRIEF REVIEW of the definitions and fundamental concepts used in the
study of linear circuits and systems. We can describe a
circuit
or
system
, in a broad sense, as a collection
of objects called
elements
(
components, parts,
or
subsystems
) which form an entity governed by certain
laws or constraints. Thus, a physical system is an entity made up of physical objects as its elements or
components. A subsystem of a given system can also be considered as a system itself.
A mathematical model describes the behavior of a physical system or device in terms of a set of equations,
together with a schematic diagram of the device containing the symbols of its elements, their connections, and
numerical values. As an example, a physical electrical system can be represented graphically by a network which
includes resistors, inductors, and capacitors, etc. as its components. Such an illustration, together with a set of
linear differential equations, is referred to as a model system.
Electrical circuits may be classified into various categories. Four of the more familiar classifications are
(a) linear and nonlinear circuits, (b) time-invariant and time-varying circuits, (c) passive and active circuits,
and (d) lumped and distributed circuits. A
linear
circuit can be described by a set of linear (differential)
equations; otherwise it is a nonlinear circuit. A
time-invariant
circuit or system implies that none of the
components of the circuit have parameters that vary with time; otherwise it is a
time-variant
system. If the
total
energy delivered to a given circuit is nonnegative at any instant of time, the circuit is said to be
passive
;
otherwise it is
active
. Finally, if the dimensions of the components of the circuit are small compared to the
wavelength of the highest of the signal frequencies applied to the circuit, it is called a
lumped
circuit; otherwise
it is referred to as a
distributed
circuit.
There are, of course, other ways of classifying circuits. For example, one might wish to classify circuits
according to the number of accessible terminals or terminal pairs (ports). Thus, terms such as
n-terminal circuit
and
n-port
are commonly used in circuit theory. Another method of classification is based on circuit configu-
rations (topology),
1
which gives rise to such terms as
ladders, lattices, bridged-T circuits,
etc.
As indicated earlier, although the words
circuit
and
system
are synonymous and will be used interchangeably
throughout the text, the terms
circuit theory
and
system theory
sometimes denote different points of view in
the study of circuits or systems. Roughly speaking,
circuit theory
is mainly concerned with interconnections of
components (circuit topology) within a given system, whereas
system theory
attempts to attain generality by
means of abstraction through a generalized (input-output state) model.
One of the goals of this section is to present a unified treatment on the study of linear circuits and systems.
That is, while the study of linear circuits with regard to their topological properties is treated as an important
phase of the entire development of the theory, a generality can be attained from such a study.
The subject of circuit theory can be divided into two main parts, namely, analysis and synthesis. In a broad
sense,
analysis
may be defined as “the separating of any material or abstract entity [system] into its constituent
elements;” on the other hand,
synthesis
is “the combining of the constituent elements of separate materials or
abstract entities into a single or unified entity [system].”
2
It is worth noting that in an analysis problem, the solution is always
unique
no matter how difficult it may
be, whereas in a synthesis problem there might exist an infinite number of solutions or, sometimes,
none at all!
It should also be noted that in some network theory texts the words
synthesis
and
design
might be used
interchangeably throughout the entire discussion of the subject. However, the term
synthesis
is generally used
to describe
analytical
procedures that can usually be carried out step by step, whereas the term
design
includes
practical (design) procedures (such as trial-and-error techniques which are based, to a great extent, on the
experience of the designer) as well as analytical methods.
In analyzing the behavior of a given physical system, the first step is to establish a mathematical model. This
model is usually in the form of a set of either differential or difference equations (or a combination of them),
1
Circuit topology or graph theory deals with the way in which the circuit elements are interconnected. A detailed discussion
on elementary applied graph theory is given in Chapter 3.6.
2
The definitions of analysis and synthesis are quoted directly from
The Random House Dictionary of the English Language,
2nd ed., Unabridged, New York: Random House, 1987.
T
© 2000 by CRC Press LLC
the solution of which accurately describes the motion of the physical systems. There is, of course, no exception
to this in the field of electrical engineering. A physical electrical system such as an amplifier circuit, for example,
is first represented by a circuit drawn on paper. The circuit is composed of resistors, capacitors, inductors, and
voltage and/or current sources,
1
and each of these circuit elements is given a symbol together with a mathe-
matical expression (i.e., the voltage-current or simply
v-i
relation) relating its terminal voltage and current at
every instant of time. Once the network and the
v-i
relation for each element is specified, Kirchhoff’s voltage
and current laws can be applied, possibly together with the physical principles to be introduced in Chapter 3.1,
to establish the mathematical model in the form of differential equations.
In Section I, focus is on analysis only (leaving coverage of synthesis and design to Section III, “Electronics”).
Specifically, the passive circuit elements—resistors, capacitors, inductors, transformers, and fuses—as well as
voltage and current sources (active elements) are discussed. This is followed by a brief discussion on the elements
of linear circuit analysis. Next, some popularly used passive filters and nonlinear circuits are introduced. Then,
Laplace transform, state variables,
z
-transform, and T and
p
configurations are covered. Finally, transfer
functions, frequency response, and stability analysis are discussed.
Nomenclature
1
Here, of course, active elements such as transistors are represented by their equivalent circuits as combinations of resistors
and dependent sources.
Symbol Quantity Unit
A
area m
2
B
magnetic flux density Tesla
C
capacitance F
e
induced voltage V
e
dielectric constant F/m
e
ripple factor
f
frequency Hz
F
force Newton
f
magnetic flux weber
I
current A
J
Jacobian
k
Boltzmann constant 1.38
´
10
–23
J/K
k
dielectric coefficient
K
coupling coefficient
L
inductance H
l
eigenvalue
M
mutual inductance H
n
turns ratio
n
filter order
Symbol Quantity Unit
w
angular frequency rad/s
P
power W
PF
power factor
q
charge C
Q
selectivity
R
resistance
W
R
(
T
) temperature coefficient
W
/
°
C
of resistance
r
resistivity
W
m
s
Laplace operator
t
damping factor
q
phase angle degree
v
velocity m/s
V
voltage V
W
energy J
X
reactance
W
Y admittance S
Z impedance W
Pecht, M., Lall, P., Ballou, G., Sankaran, C., Angelopoulos, N. “Passive Components”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
© 2000 by CRC Press LLC
1
Passive Components
1.1 Resistors
Resistor Characteristics•Resistor Types
1.2 Capacitors and Inductors
Capacitors•Types of Capacitors•Inductors
1.3 Transformers
Types of Transformers•Principle of
Transformation•Electromagnetic Equation•Transformer
Core•Transformer Losses•Transformer
Connections•Transformer Impedance
1.4 Electrical Fuses
Ratings•Fuse Performance•Selective
Coordination•Standards•Products•Standard—Class H•
HRC•Trends
1.1 Resistors
Michael Pecht and Pradeep Lall
The resistor is an electrical device whose primary function is to introduce resistance to the flow of electric
current. The magnitude of opposition to the flow of current is called the resistance of the resistor. A larger
resistance value indicates a greater opposition to current flow.
The resistance is measured in ohms. An ohm is the resistance that arises when a current of one ampere is
passed through a resistor subjected to one volt across its terminals.
The various uses of resistors include setting biases, controlling gain, fixing time constants, matching and
loading circuits, voltage division, and heat generation. The following sections discuss resistor characteristics
and various resistor types.
Resistor Characteristics
Voltage and Current Characteristics of Resistors
The resistance of a resistor is directly proportional to the
resistivity
of the material and the length of the resistor
and inversely proportional to the cross-sectional area perpendicular to the direction of current flow. The
resistance
R
of a resistor is given by
(1.1)
where
r
is the resistivity of the resistor material (
W
· cm),
l
is the length of the resistor along direction of current
flow (cm), and
A
is the cross-sectional area perpendicular to current flow (cm
2
) (Fig. 1.1). Resistivity is an
inherent property of materials. Good resistor materials typically have resistivities between 2
´
10
–6
and 200
´
10
–6
W
· cm.
R
l
A
=
r
Michael Pecht
University of Maryland
Pradeep Lall
Motorola
Glen Ballou
Ballou Associates
C. Sankaran
Electro-Test
Nick Angelopoulos
Gould Shawmut Company
© 2000 by CRC Press LLC
The resistance can also be defined in terms of sheet resistivity. If
the sheet resistivity is used, a standard sheet thickness is assumed
and factored into resistivity. Typically, resistors are rectangular in
shape; therefore the length
l
divided by the width
w
gives the number
of squares within the resistor (Fig. 1.2). The number of squares
multiplied by the resistivity is the resistance.
(1.2)
where
r
sheet
is the sheet resistivity (
W
/square),
l
is the length of resistor (cm),
w
is the width of the resistor (cm),
and
R
sheet
is the sheet resistance (
W
).
The resistance of a resistor can be defined in terms of the
voltage drop
across the resistor and current through
the resistor related by Ohm’s law,
(1.3)
where
R
is the resistance (
W
),
V
is the voltage across the resistor (V), and
I
is the current through the resistor
(A). Whenever a current is passed through a resistor, a voltage is dropped across the ends of the resistor.
Figure 1.3 depicts the symbol of the resistor with the Ohm’s law relation.
All resistors dissipate power when a voltage is applied. The power dissipated by the resistor is represented by
(1.4)
where
P
is the power dissipated (W),
V
is the voltage across the resistor (V), and
R
is the resistance (
W
). An
ideal resistor dissipates electric energy without storing electric or magnetic energy.
Resistor Networks
Resistors may be joined to form networks. If resistors are joined in series, the effective resistance (
R
T
) is the
sum of the individual resistances (Fig. 1.4).
(1.5)
FIGURE 1.1Resistance of a rectangular
cross-section resistor with cross-sectional
area A and length L.
R
l
w
sheet sheet
=r
R
V
I
=
P
V
R
=
2
RR
Ti
i
n
=
=
å
1
FIGURE 1.2Number of squares in a rectangular resistor.
FIGURE 1.3A resistor with
resistance R having a current I
flowing through it will have a
voltage drop of IR across it.
© 2000 by CRC Press LLC
If resistors are joined in parallel, the effective resistance (
R
T
) is the reciprocal of the sum of the reciprocals
of individual resistances (Fig. 1.5).
(1.6)
Temperature Coefficient of Electrical Resistance
The resistance for most resistors changes with temperature. The tem-
perature coefficient of electrical resistance is the change in electrical
resistance of a resistor per unit change in temperature. The
tempera-
ture coefficient of resistance
is measured in
W
/
°
C. The temperature
coefficient of resistors may be either positive or negative. A positive
temperature coefficient denotes a rise in resistance with a rise in tem-
perature; a negative temperature coefficient of resistance denotes a
decrease in resistance with a rise in temperature. Pure metals typically
have a positive temperature coefficient of resistance, while some metal
alloys such as constantin and manganin have a zero temperature coef-
ficient of resistance. Carbon and graphite mixed with binders usually
exhibit negative temperature coefficients, although certain choices of
binders and process variations may yield positive temperature coeffi-
cients. The temperature coefficient of resistance is given by
R
(
T
2
) =
R
(
T
1
)[1 +
a
T
1
(
T
2
–
T
1
)] (1.7)
where
a
T
1
is the temperature coefficient of electrical resistance at reference temperature
T
1
,
R
(
T
2
) is the resistance
at temperature
T
2
(
W
), and
R
(
T
1
) is the resistance at temperature
T
1
(
W
). The reference temperature is usually
taken to be 20
°
C. Because the variation in resistance between any two temperatures is usually not linear as
predicted by Eq. (1.7), common practice is to apply the equation between temperature increments and then
to plot the resistance change versus temperature for a number of incremental temperatures.
High-Frequency Effects
Resistors show a change in their resistance value when subjected
to ac voltages. The change in resistance with voltage frequency is
known as the
Boella effect.
The effect occurs because all resistors
have some inductance and capacitance along with the resistive
component and thus can be approximated by an equivalent circuit
shown in Fig. 1.6. Even though the definition of useful frequency
range is application dependent, typically, the useful range of the
resistor is the highest frequency at which the impedance differs
from the resistance by more than the tolerance of the resistor.
The frequency effect on resistance varies with the resistor construction. Wire-wound resistors typically exhibit
an increase in their impedance with frequency. In composition resistors the capacitances are formed by the
many conducting particles which are held in contact by a dielectric binder. The ac impedance for film resistors
remains constant until 100 MHz (1 MHz = 10
6
Hz) and then decreases at higher frequencies (Fig. 1.7). For
film resistors, the decrease in dc resistance at higher frequencies decreases with increase in resistance. Film
resistors have the most stable high-frequency performance.
FIGURE 1.4
Resistors connected in series.
11
1
RR
Ti
i
n
=
=
å
FIGURE 1.5Resistors connected in
parallel.
FIGURE 1.6Equivalent circuit for a resistor.
© 2000 by CRC Press LLC
The smaller the diameter of the resistor the better is its frequency response. Most high-frequency resistors
have a length to diameter ratio between 4:1 to 10:1. Dielectric losses are kept to a minimum by proper choice
of base material.
Voltage Coefficient of Resistance
Resistance is not always independent of the applied voltage. The
voltage coefficient of resistance
is the change
in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. The
voltage coefficient is given by the relationship
(1.8)
where
R
1
is the resistance at the rated voltage
V
1
and
R
2
is the resistance at 10% of rated voltage
V
2
.
Noise
Resistors exhibit electrical noise in the form of small ac voltage fluctuations when dc voltage is applied. Noise
in a resistor is a function of the applied voltage, physical dimensions, and materials. The total noise is a sum
of Johnson noise, current flow noise, noise due to cracked bodies, and loose end caps and leads. For variable
resistors the noise can also be caused by the jumping of a moving contact over turns and by an imperfect
electrical path between the contact and resistance element.
The Johnson noise is temperature-dependent thermal noise (Fig. 1.8). Thermal noise is also called “white
noise” because the noise level is the same at all frequencies. The magnitude of thermal noise,
E
RMS
(V), is
dependent on the resistance value and the temperature of the resistance due to thermal agitation.
(1.9)
where
E
RMS
is the root-mean-square value of the noise voltage (V),
R
is the resistance (
W
),
K
is the Boltzmann
constant (1.38
´
10
–23
J/K),
T
is the temperature (K), and Df is the bandwidth (Hz) over which the noise energy
is measured.
Figure 1.8 shows the variation in current noise versus voltage frequency. Current noise varies inversely with
frequency and is a function of the current flowing through the resistor and the value of the resistor. The
magnitude of current noise is directly proportional to the square root of current. The current noise magnitude
is usually expressed by a noise index given as the ratio of the root-mean-square current noise voltage (E
RMS
)
FIGURE 1.7Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors.
Voltage coefficient=
100(
(
1
21
RR
RVV
–)
–)
2
2
E kRTf
RMS
= 4 D
[...]... different colors The rst band is the most signicant gure, the second band is the second signicant gure, the third band is the multiplier or the number of zeros that have to be added after the rst two signicant gures, and the fourth band is the tolerance on the resistance value If the fourth band is not present, the resistor tolerance is the standard 20% above and below the rated value When the color code... minimize the ratio of the impedance looking from the voltmeter to the impedance of the test circuit The LC circuit is then tuned to resonate and the resultant voltage measured The value of Q may then be equated Q = resonant voltage across C voltage across R (1.50) The Q of any coil may be approximated by the equation 2pf L R XL = R Q = (1.51) where f = the frequency, Hz; L = the inductance, H; R = the dc... current Grasp the conductor in the right hand with the thumb extending along the conductor pointing in the direction of the current With the ngers partly closed, the nger tips will point in the direction of the magnetic eld Maxwells rule states, If the direction of travel of a right-handed corkscrew represents the direction of the current in a straight conductor, the direction of rotation of the corkscrew... surface The metal on which the oxide lm is formed serves as the anode or positive terminal of the capacitor; the oxide lm is the dielectric, and the cathode or negative terminal is either a conducting liquid or a gel The equivalent circuit of an electrolytic capacitor is shown in Fig 1.13, where A and B are the capacitor terminals, C is the effective capacitance, and L is the self-inductance of the capacitor... inductance, H; and R = resistance, W The Q of the coil can be measured using the circuit of Fig 1.20 for frequencies up to 1 MHz The voltage across the inductance (L) at resonance equals Q(V) (where V is the voltage developed by the oscillator); therefore, it is only necessary to measure the output voltage from the oscillator and the voltage across the inductance The oscillator voltage is driven across... resistance: The change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage Voltage drop: The difference in potential between the two ends of the resistor measured in the direction of ow of current The voltage drop is V = IR, where V is the voltage across the resistor, I is the current through the resistor, and R is the resistance Voltage rating: The. .. two points, an electric eld exists that is the result of the separation of unlike charges The strength of the eld will depend on the amount the charges have been separated Capacitance is the concept of energy storage in an electric eld and is restricted to the area, shape, and spacing of the capacitor plates and the property of the material separating them When electrical current ows into a capacitor,... After a capacitor is charged, the rectier stops conducting and the capacitor discharges into the load, as shown in Fig 1.17, until the next cycle Then the capacitor recharges again to the peak voltage The De is equal to the total peak-to-peak ripple voltage and is a complex wave containing many harmonics of the fundamental ripple frequency, causing the noticeable heating of the capacitor Tantalum Capacitors... distance between the plates, and the dielectric constant of the insulating material between the plates [see Eq (1.21)] In tantalum electrolytics, the distance between the plates is the thickness of the tantalum pentoxide lm, and since the dielectric constant of the tantalum pentoxide is high, the capacitance of a tantalum capacitor is high â 2000 by CRC Press LLC Tantalum capacitors contain either liquid... in the conductor A conductor that moves at right angles to the lines of force cuts the maximum number of lines per inch per second, therefore creating a maximum voltage The right-hand rule determines direction of the induced electromotive force (emf) The emf is in the direction in which the axis of a right-hand screw, when turned with the velocity vector, moves through the smallest angle toward the .
;
otherwise it is
active
. Finally, if the dimensions of the components of the circuit are small compared to the
wavelength of the highest of the. LLC
the solution of which accurately describes the motion of the physical systems. There is, of course, no exception
to this in the field of electrical engineering.
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