Chan, Shu-Park “Section I – Circuits”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
The Intel Pentium® processor, introduced at speeds of up to 300 MHz, combines the architectural advances
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The Pentium II processor is the newest member of the P6 processor family, but certainly not the last in
the line of high performance processors. (Courtesy of Intel Corporation.)
© 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
9 T-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
T
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.
2The 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.
© 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
Symbol Quantity Unit Symbol Quantity Unit
A area m2 w angular frequency rad/s
B magnetic flux density Tesla P power W
C capacitance F PF power factor
e induced voltage V q charge C
e dielectric constant F/m Q selectivity
e ripple factor R resistance W
f frequency Hz R(T) temperature coefficient W/°C
F force Newton of resistance
f magnetic flux weber r resistivity Wm
I current A s Laplace operator
J Jacobian t damping factor
k Boltzmann constant 1.38 ´ 10–23 J/K q phase angle degree
k dielectric coefficient v velocity m/s
K coupling coefficient V voltage V
L inductance H W energy J
l eigenvalue X reactance W
M mutual inductance H Y admittance S
n turns ratio Z impedance W
n filter order
1Here, of course, active elements such as transistors are represented by their equivalent circuits as combinations of resistors
and dependent sources.
© 2000 by CRC Press LLC
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
1
Passive Components
Michael Pecht 1.1 Resistors
University of Maryland Resistor Characteristics • Resistor Types
1.2 Capacitors and Inductors
Pradeep Lall Capacitors • Types of Capacitors • Inductors
Motorola 1.3 Transformers
Types of Transformers • Principle of
Glen Ballou
Transformation • Electromagnetic Equation • Transformer
Ballou Associates
Core • Transformer Losses • Transformer
C. Sankaran Connections • Transformer Impedance
Electro-Test 1.4 Electrical Fuses
Ratings • Fuse Performance • Selective
Nick Angelopoulos Coordination • Standards • Products • Standard— Class H •
Gould Shawmut Company 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
rl
R = (1.1)
A
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 (cm2) (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.
© 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.
FIGURE 1.1 Resistance of a rectangular
l cross-section resistor with cross-sectional
Rsheet = rsheet (1.2)
w area A and length L.
where rsheet is the sheet resistivity (W/square), l is the length of resistor (cm), w is the width of the resistor (cm),
and Rsheet 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,
V
R = (1.3)
I
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
V2
P = (1.4)
R
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 (RT) is the
sum of the individual resistances (Fig. 1.4).
n
RT = åR i (1.5)
i =1
FIGURE 1.3 A resistor with
resistance R having a current I
flowing through it will have a
FIGURE 1.2 Number of squares in a rectangular resistor. voltage drop of IR across it.
© 2000 by CRC Press LLC
FIGURE 1.4 Resistors connected in series.
If resistors are joined in parallel, the effective resistance (RT) is the reciprocal of the sum of the reciprocals
of individual resistances (Fig. 1.5).
n
1 1
RT
= åR (1.6)
i =1 i
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 FIGURE 1.5 Resistors connected in
exhibit negative temperature coefficients, although certain choices of parallel.
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 + aT1(T 2 – T 1)] (1.7)
where aT1 is the temperature coefficient of electrical resistance at reference temperature T1, R(T2) is the resistance
at temperature T2 (W), and R(T1) is the resistance at temperature T1 (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
FIGURE 1.6 Equivalent circuit for a resistor.
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 = 106 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.
© 2000 by CRC Press LLC
FIGURE 1.7 Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors.
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
100(R1 – R2 )
Voltage coefficient = (1.8)
R2 (V1 – V2 )
where R1 is the resistance at the rated voltage V1 and R2 is the resistance at 10% of rated voltage V2.
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, ERMS (V), is
dependent on the resistance value and the temperature of the resistance due to thermal agitation.
ERMS = 4kRTDf (1.9)
where ERMS 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 (ERMS)
© 2000 by CRC Press LLC
FIGURE 1.8 The total resistor noise is the sum of current noise and thermal noise. The current noise approaches the
thermal noise at higher frequencies. (Source: Phillips Components, Discrete Products Division, 1990–91 Resistor/Capacitor
Data Book, 1991. With permission.)
over one decade bandwidth to the average voltage caused by a specified constant current passed through the
resistor at a specified hot-spot temperature [Phillips, 1991].
æ Noise voltage ö
N.I. = 20 log10 ç ÷ (1.10)
è dc voltage ø
æf ö
E RMS = Vdc ´ 10N.I. / 20 log ç 2 ÷ (1.11)
è f1 ø
where N.I. is the noise index, Vdc is the dc voltage drop across the resistor, and f1 and f2 represent the frequency
range over which the noise is being computed. Units of noise index are mV/V. At higher frequencies, the current
noise becomes less dominant compared to Johnson noise.
Precision film resistors have extremely low noise. Composition resistors show some degree of noise due to
internal electrical contacts between the conducting particles held together with the binder. Wire-wound resistors
are essentially free of electrical noise unless resistor terminations are faulty.
Power Rating and Derating Curves
Resistors must be operated within specified temperature limits to avoid permanent damage to the materials.
The temperature limit is defined in terms of the maximum power, called the power rating, and derating curve.
The power rating of a resistor is the maximum power in watts which the resistor can dissipate. The maximum
power rating is a function of resistor material, maximum voltage rating, resistor dimensions, and maximum
allowable hot-spot temperature. The maximum hot-spot temperature is the temperature of the hottest part on
the resistor when dissipating full-rated power at rated ambient temperature.
The maximum allowable power rating as a function of the ambient temperature is given by the derating
curve. Figure 1.9 shows a typical power rating curve for a resistor. The derating curve is usually linearly drawn
from the full-rated load temperature to the maximum allowable no-load temperature. A resistor may be
operated at ambient temperatures above the maximum full-load ambient temperature if operating at lower
than full-rated power capacity. The maximum allowable no-load temperature is also the maximum storage
temperature for the resistor.
© 2000 by CRC Press LLC
FIGURE 1.9 Typical derating curve for resistors.
Voltage Rating of Resistors
The maximum voltage that may be applied to the resistor is called the voltage rating and is related to the power
rating by
V = PR (1.12)
where V is the voltage rating (V), P is the power rating (W), and R is the resistance (W). For a given value of
voltage and power rating, a critical value of resistance can be calculated. For values of resistance below the
critical value, the maximum voltage is never reached; for values of resistance above the critical value, the power
dissipated is lower than the rated power (Fig. 1.10).
Color Coding of Resistors
Resistors are generally identified by color coding or direct digital marking. The color code is given in Table 1.1.
The color code is commonly used in composition resistors and film resistors. The color code essentially consists
of four bands of different colors. The first band is the most significant figure, the second band is the second
significant figure, the third band is the multiplier or the number of zeros that have to be added after the first
two significant figures, 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 is
used on fixed wire-wound resistors, the first band is applied in double width.
FIGURE 1.10 Relationship of applied voltage and power above and below the critical value of resistance.
© 2000 by CRC Press LLC
TABLE 1.1 Color Code Table for Resistors
Fourth Band
Color First Band Second Band Third Band Tolerance, %
Black 0 0 1
Brown 1 1 10
Red 2 2 100
Orange 3 3 1,000
Yellow 4 4 10,000
Green 5 5 100,000
Blue 6 6 1,000,000
Violet 7 7 10,000,000
Gray 8 8 100,000,000
White 9 9 1,000,000,000
Gold 0.1 5%
Silver 0.01 10%
No band 20%
Blanks in the table represent situations which do not exist in the color code.
Resistor Types
Resistors can be broadly categorized as fixed, variable, and special-purpose. Each of these resistor types is
discussed in detail with typical ranges of their characteristics.
Fixed Resistors
The fixed resistors are those whose value cannot be varied after manufacture. Fixed resistors are classified into
composition resistors, wire-wound resistors, and metal-film resistors. Table 1.2 outlines the characteristics of
some typical fixed resistors.
Wire-Wound Resistors. Wire-wound resistors are made by winding wire of nickel-chromium alloy on a
ceramic tube covering with a vitreous coating. The spiral winding has inductive and capacitive characteristics
that make it unsuitable for operation above 50 kHz. The frequency limit can be raised by noninductive winding
so that the magnetic fields produced by the two parts of the winding cancel.
Composition Resistors. Composition resistors are composed of carbon particles mixed with a binder. This
mixture is molded into a cylindrical shape and hardened by baking. Leads are attached axially to each end, and
the assembly is encapsulated in a protective encapsulation coating. Color bands on the outer surface indicate
the resistance value and tolerance. Composition resistors are economical and exhibit low noise levels for
resistances above 1 MW. Composition resistors are usually rated for temperatures in the neighborhood of 70°C
for power ranging from 1/8 to 2 W. Composition resistors have end-to-end shunted capacitance that may be
noticed at frequencies in the neighborhood of 100 kHz, especially for resistance values above 0.3 MW.
Metal-Film Resistors. Metal-film resistors are commonly made of nichrome, tin-oxide, or tantalum nitride,
either hermetically sealed or using molded-phenolic cases. Metal-film resistors are not as stable as the
TABLE 1.2 Characteristics of Typical Fixed Resistors
Operating
Resistor Types Resistance Range Watt Range Temp. Range a, ppm/°C
Wire-wound resistor
Precision 0.1 to 1.2 MW 1/8 to 1/4 –55 to 145 10
Power 0.1 to 180 kW 1 to 210 –55 to 275 260
Metal-film resistor
Precision 1 to 250 MW 1/20 to 1 –55 to 125 50–100
Power 5 to 100 kW 1 to 5 –55 to 155 20–100
Composition resistor
General purpose 2.7 to 100 MW 1/8 to 2 –55 to 130 1500
© 2000 by CRC Press LLC
wire-wound resistors. Depending on the application, fixed resistors are manufactured as precision resistors,
semiprecision resistors, standard general-purpose resistors, or power resistors. Precision resistors have low
voltage and power coefficients, excellent temperature and time stabilities, low noise, and very low reactance.
These resistors are available in metal-film or wire constructions and are typically designed for circuits having
very close resistance tolerances on values. Semiprecision resistors are smaller than precision resistors and are
primarily used for current-limiting or voltage-dropping functions in circuit applications. Semiprecision resistors
have long-term temperature stability. General-purpose resistors are used in circuits that do not require tight
resistance tolerances or long-term stability. For general-purpose resistors, initial resistance variation may be in
the neighborhood of 5% and the variation in resistance under full-rated power may approach 20%. Typically,
general-purpose resistors have a high coefficient of resistance and high noise levels. Power resistors are used
for power supplies, control circuits, and voltage dividers where operational stability of 5% is acceptable. Power
resistors are available in wire-wound and film constructions. Film-type power resistors have the advantage of
stability at high frequencies and have higher resistance values than wire-wound resistors for a given size.
Variable Resistors
Potentiometers. The potentiometer is a special form of variable resistor with three terminals. Two terminals
are connected to the opposite sides of the resistive element, and the third connects to a sliding contact that can
be adjusted as a voltage divider.
Potentiometers are usually circular in form with the movable contact attached to a shaft that rotates.
Potentiometers are manufactured as carbon composition, metallic film, and wire-wound resistors available in
single-turn or multiturn units. The movable contact does not go all the way toward the end of the resistive
element, and a small resistance called the hop-off resistance is present to prevent accidental burning of the
resistive element.
Rheostat. The rheostat is a current-setting device in which one terminal is connected to the resistive element
and the second terminal is connected to a movable contact to place a selected section of the resistive element
into the circuit. Typically, rheostats are wire-wound resistors used as speed controls for motors, ovens, and
heater controls and in applications where adjustments on the voltage and current levels are required, such as
voltage dividers and bleeder circuits.
Special-Purpose Resistors
Integrated Circuit Resistors. Integrated circuit resistors are classified into two general categories: semicon-
ductor resistors and deposited film resistors. Semiconductor resistors use the bulk resistivity of doped semi-
conductor regions to obtain the desired resistance value. Deposited film resistors are formed by depositing
resistance films on an insulating substrate which are etched and patterned to form the desired resistive network.
Depending on the thickness and dimensions of the deposited films, the resistors are classified into thick-film
and thin-film resistors.
Semiconductor resistors can be divided into four types: diffused, bulk, pinched, and ion-implanted. Table 1.3
shows some typical resistor properties for semiconductor resistors. Diffused semiconductor resistors use resis-
tivity of the diffused region in the semiconductor substrate to introduce a resistance in the circuit. Both n-type
and p-type diffusions are used to form the diffused resistor.
A bulk resistor uses the bulk resistivity of the semiconductor to introduce a resistance into the circuit.
Mathematically the sheet resistance of a bulk resistor is given by
re
Rsheet = (1.13)
d
where Rsheet is the sheet resistance in (W/square), re is the sheet resistivity (W/square), and d is the depth of the
n-type epitaxial layer.
Pinched resistors are formed by reducing the effective cross-sectional area of diffused resistors. The reduced
cross section of the diffused length results in extremely high sheet resistivities from ordinary diffused resistors.
© 2000 by CRC Press LLC
TABLE 1.3 Typical Characteristics of Integrated Circuit Resistors
Temperature
Sheet Resistivity Coefficient
Resistor Type (per square) (ppm/°C)
Semiconductor
Diffused 0.8 to 260 W 1100 to 2000
Bulk 0.003 to 10 kW 2900 to 5000
Pinched 0.001 to 10 kW 3000 to 6000
Ion-implanted 0.5 to 20 kW 100 to 1300
Deposited resistors
Thin-film
Tantalum 0.01 to 1 kW m100
SnO2 0.08 to 4 kW –1500 to 0
Ni-Cr 40 to 450 W m100
Cermet (Cr-SiO) 0.03 to 2.5 kW m150
Thick-film
Ruthenium-silver 10 W to 10 MW m200
Palladium-silver 0.01 to 100 kW –500 to 150
Ion-implanted resistors are formed by implanting ions on the semiconductor surface by bombarding the
silicon lattice with high-energy ions. The implanted ions lie in a very shallow layer along the surface (0.1 to
0.8 mm). For similar thicknesses ion-implanted resistors yield sheet resistivities 20 times greater than diffused
resistors. Table 1.3 shows typical properties of diffused, bulk, pinched, and ion-implanted resistors. Typical
sheet resistance values range from 80 to 250 W/square.
Varistors. Varistors are voltage-dependent resistors that show a high degree of nonlinearity between their
resistance value and applied voltage. They are composed of a nonhomogeneous material that provides a
rectifying action. Varistors are used for protection of electronic circuits, semiconductor components, collectors
of motors, and relay contacts against overvoltage.
The relationship between the voltage and current of a varistor is given by
V = kI b (1.14)
where V is the voltage (V), I is the current (A), and k and b are constants that depend on the materials and
manufacturing process. The electrical characteristics of a varistor are specified by its b and k values.
Varistors in Series. The resultant k value of n varistors connected in series is nk. This can be derived by
considering n varistors connected in series and a voltage nV applied across the ends. The current through each
varistor remains the same as for V volts over one varistor. Mathematically, the voltage and current are expressed
as
nV = k 1 I b (1.15)
Equating the expressions (1.14) and (1.15), the equivalent constant k1 for the series combination of varistors
is given as
k 1 = nk (1.16)
Varistors in Parallel. The equivalent k value for a parallel combination of varistors can be obtained by
connecting n varistors in parallel and applying a voltage V across the terminals. The current through the varistors
will still be n times the current through a single varistor with a voltage V across it. Mathematically the current
and voltage are related as
V = k 2(nI) b (1.17)
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From Eqs. (1.14) and (1.17) the equivalent constant k2 for the series combination of varistors is given as
k
k2 = (1.18)
nb
Thermistors. Thermistors are resistors that change their resistance exponentially with changes in temperature.
If the resistance decreases with increase in temperature, the resistor is called a negative temperature coefficient
(NTC) resistor. If the resistance increases with temperature, the resistor is called a positive temperature coef-
ficient (PTC) resistor.
NTC thermistors are ceramic semiconductors made by sintering mixtures of heavy metal oxides such as
manganese, nickel, cobalt, copper, and iron. The resistance temperature relationship for NTC thermistors is
RT = Ae B/T (1.19)
where T is temperature (K), RT is the resistance (W), and A, B are constants whose values are determined by
conducting experiments at two temperatures and solving the equations simultaneously.
PTC thermistors are prepared from BaTiO3 or solid solutions of PbTiO3 or SrTiO3. The resistance temperature
relationship for PTC thermistors is
RT = A + Ce BT (1.20)
where T is temperature (K), RT is the resistance (W), and A, B are constants determined by conducting
experiments at two temperatures and solving the equations simultaneously. Positive thermistors have a PTC
only between certain temperature ranges. Outside this range the temperature is either zero or negative. Typically,
the absolute value of the temperature coefficient of resistance for PTC resistors is much higher than for NTC
resistors.
Defining Terms
Doping: The intrinsic carrier concentration of semiconductors (e.g., Si) is too low to allow controlled charge
transport. For this reason some impurities called dopants are purposely added to the semiconductor.
The process of adding dopants is called doping. Dopants may belong to group IIIA (e.g., boron) or group
VA (e.g., phosphorus) in the periodic table. If the elements belong to the group IIIA, the resulting
semiconductor is called a p-type semiconductor. On the other hand, if the elements belong to the group
VA, the resulting semiconductor is called an n-type semiconductor.
Epitaxial layer: Epitaxy refers to processes used to grow a thin crystalline layer on a crystalline substrate. In
the epitaxial process the wafer acts as a seed crystal. The layer grown by this process is called an epitaxial
layer.
Resistivity: The resistance of a conductor with unit length and unit cross-sectional area.
Temperature coefficient of resistance: The change in electrical resistance of a resistor per unit change in
temperature.
Time stability: The degree to which the initial value of resistance is maintained to a stated degree of certainty
under stated conditions of use over a stated period of time. Time stability is usually expressed as a percent
or parts per million change in resistance per 1000 hours of continuous use.
Voltage coefficient of 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
flow 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 maximum voltage that may be applied to the resistor.
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Related Topics
22.1 Physical Properties • 25.1 Integrated Circuit Technology • 51.1 Introduction
References
Phillips Components, Discrete Products Division, 1990–91 Resistor/Capacitor Data Book, 1991.
C.C. Wellard, Resistance and Resistors, New York: McGraw-Hill, 1960.
Further Information
IEEE Transactions on Electron Devices and IEEE Electron Device Letters: Published monthly by the Institute of
Electrical and Electronics Engineers.
IEEE Components, Hybrids and Manufacturing Technology: Published quarterly by the Institute of Electrical and
Electronics Engineers.
G.W.A. Dummer, Materials for Conductive and Resistive Functions, New York: Hayden Book Co., 1970.
H.F. Littlejohn and C.E. Burckel, Handbook of Power Resistors, Mount Vernon, N.Y.: Ward Leonard Electric
Company, 1951.
I.R. Sinclair, Passive Components: A User’s Guide, Oxford: Heinmann Newnes, 1990.
1.2 Capacitors and Inductors
Glen Ballou
Capacitors
If a potential difference is found between two points, an electric field exists that is the result of the separation
of unlike charges. The strength of the field will depend on the amount the charges have been separated.
Capacitance is the concept of energy storage in an electric field and is restricted to the area, shape, and
spacing of the capacitor plates and the property of the material separating them.
When electrical current flows into a capacitor, a force is established between two parallel plates separated by
a dielectric. This energy is stored and remains even after the input is removed. By connecting a conductor (a
resistor, hard wire, or even air) across the capacitor, the charged capacitor can regain electron balance, that is,
discharge its stored energy.
The value of a parallel-plate capacitor can be found with the equation
x [(N – 1)A]
C = ´ 10 –13 (1.21)
d
where C = capacitance, F; = dielectric constant of insulation; d = spacing between plates; N = number of plates;
A = area of plates; and x = 0.0885 when A and d are in centimeters, and x = 0.225 when A and d are in inches.
The work necessary to transport a unit charge from one plate to the other is
e = kg (1.22)
where e = volts expressing energy per unit charge, g = coulombs of charge already transported, and k =
proportionality factor between work necessary to carry a unit charge between the two plates and charge already
transported. It is equal to 1/C, where C is the capacitance, F.
The value of a capacitor can now be calculated from the equation
q
C = e (1.23)
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where q = charge (C) and e is found with Eq. (1.22).
The energy stored in a capacitor is
CV 2
W = (1.24)
2
where W = energy, J; C = capacitance, F; and V = applied voltage, V. TABLE 1.4 Comparison of Capacitor
The dielectric constant of a material determines the electrostatic Dielectric Constants
energy which may be stored in that material per unit volume for a given K
voltage. The value of the dielectric constant expresses the ratio of a Dielectric (Dielectric Constant)
capacitor in a vacuum to one using a given dielectric. The dielectric of Air or vacuum 1.0
Paper 2.0–6.0
air is 1, the reference unit employed for expressing the dielectric constant. Plastic 2.1–6.0
As the dielectric constant is increased or decreased, the capacitance will Mineral oil 2.2–2.3
Silicone oil 2.7–2.8
increase or decrease, respectively. Table 1.4 lists the dielectric constants Quartz 3.8–4.4
of various materials. Glass 4.8–8.0
Porcelain 5.1–5.9
The dielectric constant of most materials is affected by both temper- Mica 5.4–8.7
ature and frequency, except for quartz, Styrofoam, and Teflon, whose Aluminum oxide 8.4
Tantalum pentoxide 26
dielectric constants remain essentially constant. Ceramic 12–400,000
The equation for calculating the force of attraction between two plates Source: G. Ballou, Handbook for Sound
is Engineers, The New Audio Cyclopedia, Car-
mel, Ind.: Macmillan Computer Publish-
AV 2 ing Company, 1991. With permission.
F = (1.25)
k (1504S )2
where F = attraction force, dyn; A = area of one plate, cm2; V = potential energy difference, V; k = dielectric
coefficient; and S = separation between plates, cm.
The Q for a capacitor when the resistance and capacitance is in series is
1
Q = (1.26)
2p f RC
where Q = ratio expressing the factor of merit; f = frequency, Hz; R = resistance, W; and C = capacitance, F.
When capacitors are connected in series, the total capacitance is
1
CT = (1.27)
1/C1 + 1/C 2 + × × × + 1/Cn
and is always less than the value of the smallest capacitor.
When capacitors are connected in parallel, the total capacitance is
CT = C1 + C2 + · · · + Cn (1.28)
and is always larger than the largest capacitor.
When a voltage is applied across a group of capacitors connected in series, the voltage drop across the
combination is equal to the applied voltage. The drop across each individual capacitor is inversely proportional
to its capacitance.
V AC X
VC = (1.29)
CT
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where VC = voltage across the individual capacitor in the series (C1, C 2, ...,Cn ), V; VA = applied voltage, V; CT =
total capacitance of the series combination, F; and CX = capacitance of individual capacitor under consideration, F.
In an ac circuit, the capacitive reactance, or the impedance, of the capacitor is
1
XC = (1.30)
2pfC
where XC = capacitive reactance, W ; f = frequency, Hz; and C = capacitance, F. The current will lead the voltage
by 90° in a circuit with a pure capacitor.
When a dc voltage is connected across a capacitor, a time t is required to charge the capacitor to the applied
voltage. This is called a time constant and is calculated with the equation
t = RC (1.31)
where t = time, s; R = resistance, W; and C = capacitance, F.
In a circuit consisting of pure resistance and capacitance, the time constant t is defined as the time required
to charge the capacitor to 63.2% of the applied voltage.
During the next time constant, the capacitor charges to 63.2% of the remaining difference of full value, or
to 86.5% of the full value. The charge on a capacitor can never actually reach 100% but is considered to be
100% after five time constants. When the voltage is removed, the capacitor discharges to 63.2% of the full value.
Capacitance is expressed in microfarads (mF, or 10–6 F) or picofarads (pF, or 10–12 F) with a stated accuracy
or tolerance. Tolerance may also be stated as GMV (guaranteed minimum value), sometimes referred to as
MRV (minimum rated value).
All capacitors have a maximum working voltage that must not be exceeded and is a combination of the dc
value plus the peak ac value which may be applied during operation.
Quality Factor (Q)
Quality factor is the ratio of the capacitor’s reactance to its resistance at a specified frequency and is found by
the equation
1
Q =
2 pfCR (1.32)
1
=
PF
where Q = quality factor; f = frequency, Hz; C = value of capacitance, F; R = internal resistance, W; and PF =
power factor
Power Factor (PF)
Power factor is the preferred measurement in describing capacitive losses in ac circuits. It is the fraction of
input volt-amperes (or power) dissipated in the capacitor dielectric and is virtually independent of the capac-
itance, applied voltage, and frequency.
Equivalent Series Resistance (ESR)
Equivalent series resistance is expressed in ohms or milliohms (W , mW) and is derived from lead resistance,
termination losses, and dissipation in the dielectric material.
Equivalent Series Inductance (ESL)
The equivalent series inductance can be useful or detrimental. It reduces high-frequency performance; however,
it can be used in conjunction with the internal capacitance to form a resonant circuit.
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Dissipation Factor (DF)
The dissipation factor in percentage is the ratio of the effective series resistance of a capacitor to its reactance
at a specified frequency. It is the reciprocal of quality factor (Q) and an indication of power loss within the
capacitor. It should be as low as possible.
Insulation Resistance
Insulation resistance is the resistance of the dielectric material and determines the time a capacitor, once charged,
will hold its charge. A discharged capacitor has a low insulation resistance; however once charged to its rated
value, it increases to megohms. The leakage in electrolytic capacitors should not exceed
I L = 0.04C + 0.30 (1.33)
where IL = leakage current, mA, and C = capacitance, mF.
Dielectric Absorption (DA)
The dielectric absorption is a reluctance of the dielectric to give up stored electrons when the capacitor is
discharged. This is often called “memory” because if a capacitor is discharged through a resistance and the
resistance is removed, the electrons that remained in the dielectric will reconvene on the electrode, causing a
voltage to appear across the capacitor. DA is tested by charging the capacitor for 5 min, discharging it for 5 s,
then having an open circuit for 1 min after which the recovery voltage is read. The percentage of DA is defined
as the ratio of recovery to charging voltage times 100.
Types of Capacitors
Capacitors are used to filter, couple, tune, block dc, pass ac, bypass, shift phase, compensate, feed through,
isolate, store energy, suppress noise, and start motors. They must also be small, lightweight, reliable, and
withstand adverse conditions.
Capacitors are grouped according to their dielectric material and mechanical configuration.
Ceramic Capacitors
Ceramic capacitors are used most often for bypass and coupling applications (Fig. 1.11). Ceramic capacitors
can be produced with a variety of K values (dielectric constant). A high K value translates to small size and
less stability. High-K capacitors with a dielectric constant >3000 are physically small and have values between
0.001 to several microfarads.
FIGURE 1.11 Monolythic® multilayer ceramic capacitors. (Courtesy of Sprague Electric Company.)
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