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
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
=
=
?
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
T1
(T
2
– T
1
)] (1.7)
where a
T1
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
Tii
n
=
=
?
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
? 2000 by CRC Press LLC
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].
(1.10)
(1.11)
where N.I. is the noise index, V
dc
is the dc voltage drop across the resistor, and f
1
and f
2
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.
FIGURE 1.8The 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.)
N.I.
Noise voltage
dc voltage
=
?
è
?
?
?
÷
20
10
log
EV
f
f
RMS dc
N.I.
=′
?
è
?
?
?
÷
10
20
2
1
/
log
? 2000 by CRC Press LLC
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
(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 1.9Typical derating curve for resistors.
FIGURE 1.10Relationship of applied voltage and power above and below the critical value of resistance.
VPR=
? 2000 by CRC Press LLC
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 1.1Color 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.
TABLE 1.2Characteristics 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
(1.13)
where R
sheet
is the sheet resistance in (W/square), r
e
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.
R
d
e
sheet
=
r
? 2000 by CRC Press LLC
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 k
1
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)
TABLE 1.3Typical 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
SnO
2
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
? 2000 by CRC Press LLC
From Eqs. (1.14) and (1.17) the equivalent constant k
2
for the series combination of varistors is given as
(1.18)
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
R
T
= Ae
B/T
(1.19)
where T is temperature (K), R
T
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 BaTiO
3
or solid solutions of PbTiO
3
or SrTiO
3
. The resistance temperature
relationship for PTC thermistors is
R
T
= A + Ce
BT
(1.20)
where T is temperature (K), R
T
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.
k
k
n
2
=
b
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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
(1.21)
where C = capacitance, F; e = 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
(1.23)
C
xNA
d
=′
e[( –)]
–
1
10
13
C
q
e
=
? 2000 by CRC Press LLC
where q = charge (C) and e is found with Eq. (1.22).
The energy stored in a capacitor is
(1.24)
where W = energy, J; C = capacitance, F; and V = applied voltage, V.
The dielectric constant of a material determines the electrostatic
energy which may be stored in that material per unit volume for a given
voltage. The value of the dielectric constant expresses the ratio of a
capacitor in a vacuum to one using a given dielectric. The dielectric of
air is 1, the reference unit employed for expressing the dielectric constant.
As the dielectric constant is increased or decreased, the capacitance will
increase or decrease, respectively. Table 1.4 lists the dielectric constants
of various materials.
The dielectric constant of most materials is affected by both temper-
ature and frequency, except for quartz, Styrofoam, and Teflon, whose
dielectric constants remain essentially constant.
The equation for calculating the force of attraction between two plates
is
(1.25)
where F = attraction force, dyn; A = area of one plate, cm
2
; 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
(1.26)
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
(1.27)
and is always less than the value of the smallest capacitor.
When capacitors are connected in parallel, the total capacitance is
C
T
= C
1
+ C
2
+ · · · + C
n
(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.
(1.29)
W
CV
=
2
2
TABLE 1.4Comparison of Capacitor
Dielectric Constants
K
Dielectric (Dielectric Constant)
Air or vacuum 1.0
Paper 2.0–6.0
Plastic 2.1–6.0
Mineral oil 2.2–2.3
Silicone oil 2.7–2.8
Quartz 3.8–4.4
Glass 4.8–8.0
Porcelain 5.1–5.9
Mica 5.4–8.7
Aluminum oxide 8.4
Tantalum pentoxide 26
Ceramic 12–400,000
Source: G. Ballou, Handbook for Sound
Engineers, The New Audio Cyclopedia, Car-
mel, Ind.: Macmillan Computer Publish-
ing Company, 1991. With permission.
F
AV
kS
=
2
2
1504()
Q
fRC
=
1
2p
C
CC C
T
n
=
+ +×××+
1
11 1
12
// /
V
VC
C
C
AX
T
=
? 2000 by CRC Press LLC
where V
C
= voltage across the individual capacitor in the series (C
1
, C
2
, ...,C
n
), V; V
A
= applied voltage, V; C
T
=
total capacitance of the series combination, F; and C
X
= capacitance of individual capacitor under consideration, F.
In an ac circuit, the capacitive reactance, or the impedance, of the capacitor is
(1.30)
where X
C
= 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
(1.32)
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.
X
fC
C
=
1
2p
Q
fCR
PF
=
=
1
2
1
p
? 2000 by CRC Press LLC
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 I
L
= 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 1.11Monolythic? multilayer ceramic capacitors. (Courtesy of Sprague Electric Company.)
? 2000 by CRC Press LLC
Good temperature stability requires capacitors to have a K value between 10 and 200. If high Q is also
required, the capacitor will be physically larger. Ceramic capacitors with a zero temperature change are called
negative-positive-zero (NPO) and come in a capacitance range of 1.0 pF to 0.033 mF.
An N750 temperature-compensated capacitor is used when accurate capacitance is required over a large
temperature range. The 750 indicates a 750-ppm decrease in capacitance with a 1°C increase in temperature
(750 ppm/°C). This equates to a 1.5% decrease in capacitance for a 20°C temperature increase. N750 capacitors
come in values between 4.0 and 680 pF.
Film Capacitors
Film capacitors consist of alternate layers of metal foil and one or more layers of a flexible plastic insulating
material (dielectric) in ribbon form rolled and encapsulated (see Fig. 1.12).
Mica Capacitors
Mica capacitors have small capacitance values and are usually used in high-frequency circuits. They are con-
structed as alternate layers of metal foil and mica insulation, which are stacked and encapsulated, or are silvered
mica, where a silver electrode is screened on the mica insulators.
Paper-Foil-Filled Capacitors
Paper-foil-filled capacitors are often used as motor capacitors and are rated at 60 Hz. They are made of alternate
layers of aluminum and paper saturated with oil that are rolled together. The assembly is mounted in an oil-
filled, hermetically sealed metal case.
Electrolytic Capacitors
Electrolytic capacitors provide high capacitance in a tolerable size; however, they do have drawbacks. Low
temperatures reduce performance, while high temperatures dry them out. The electrolytes themselves can leak
and corrode the equipment. Repeated surges above the rated working voltage, excessive ripple currents, and
high operating temperature reduce performance and shorten capacitor life.
Electrolytic capacitors are manufactured by an electrochemical formation of an oxide film on a metal surface.
The metal on which the oxide film is formed serves as the anode or positive terminal of the capacitor; the oxide
film 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 caused by terminals,
electrodes, and geometry.
FIGURE 1.12Film-wrapped film capacitors. (Courtesy of Sprague Electric Company.)
? 2000 by CRC Press LLC
The shunt resistance (insulation resistance) R
s
accounts for the
dc leakage current. Heat is generated in the ESR from ripple
current and in the shunt resistance by voltage. The ESR is due to
the spacer-electrolyte-oxide system and varies only slightly except
at low temperature, where it increases greatly.
The impedance of a capacitor (Fig. 1.14) is frequency-depen-
dent. The initial downward slope is caused by the capacitive reac-
tance X
C
. The trough (lowest impedance) is almost totally resistive, and the upward slope is due to the capacitor’s
self-inductance X
L
. An ESR plot would show an ESR decrease to about 5–10 kHz, remaining relatively constant
thereafter.
Leakage current is the direct current that passes through a capacitor when a correctly polarized dc voltage
is applied to its terminals. It is proportional to temperature, becoming increasingly important at elevated
ambient temperatures. Leakage current decreases slowly after voltage is applied, reaching steady-state condi-
tions in about 10 min.
If a capacitor is connected with reverse polarity, the oxide film is forward-biased, offering very little resistance
to current flow. This causes overheating and self-destruction of the capacitor.
The total heat generated within a capacitor is the sum of the heat created by the I
leakage
′ V
applied
and the I
2
R
losses in the ESR.
The ac ripple current rating is very important in filter applications because excessive current produces
temperature rise, shortening capacitor life. The maximum permissible rms ripple current is limited by the
internal temperature and the rate of heat dissipation from the capacitor. Lower ESR and longer enclosures
increase the ripple current rating.
Capacitor life expectancy is doubled for each 10°C decrease in operating temperature, so a capacitor operating
at room temperature will have a life expectancy 64 times that of the same capacitor operating at 85°C (185°F).
The surge voltage specification of a capacitor determines its ability to withstand high transient voltages that
generally occur during the starting up period of equipment. Standard tests generally specify a short on and
long off period for an interval of 24 h or more, and the allowable surge voltage levels are generally 10% above
the rated voltage of the capacitor.
Figure 1.15 shows how temperature, frequency, time, and applied voltage affect electrolytic capacitors.
Aluminum Electrolytic Capacitors. Aluminum electrolytic capacitors use aluminum as the base material
(Fig. 1.16). The surface is often etched to increase the surface area as much as 100 times that of unetched foil,
resulting in higher capacitance in the same volume.
Aluminum electrolytic capacitors can withstand up to 1.5 V of reverse voltage without detriment. Higher
reverse voltages, when applied over extended periods, lead to loss of capacitance. Excess reverse voltages applied
for short periods cause some change in capacitance but not to capacitor failure.
Large-value capacitors are often used to filter dc power supplies. After a capacitor is charged, the rectifier
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. Tantalum electrolytics are
the preferred type where high reliability and long
service life are paramount considerations.
Tantalum capacitors have as much as three times
better capacitance per volume efficiency than alumi-
num electrolytic capacitors, because tantalum pen-
toxide has a dielectric constant three times greater
than that of aluminum oxide (see Table 1.4).
The capacitance of any capacitor is determined by
the surface area of the two conducting plates, the
FIGURE 1.13Simplified equivalent circuit of
an electrolytic capacitor.
Figure 1.14Impedance characteristics of a capacitor.
? 2000 by CRC Press LLC
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 film,
and since the dielectric constant of the tantalum pentoxide is high, the capacitance of a tantalum capacitor is
high.
FIGURE 1.15 Variations in aluminum electrolytic characteristics caused by temperature, frequency, time, and applied
voltage. (Courtesy of Sprague Electric Company.)
FIGURE 1.16 Verti-lytic? miniature single-ended aluminum electrolytic capacitor. (Courtesy of Sprague Electric
Company.)
? 2000 by CRC Press LLC
Tantalum capacitors contain either liquid or solid electrolytes. The liquid
electrolyte in wet-slug and foil capacitors, generally sulfuric acid, forms the
cathode (negative) plate. In solid-electrolyte capacitors, a dry material, man-
ganese dioxide, forms the cathode plate.
Foil Tantalum Capacitors. Foil tantalum capacitors can be designed to
voltage values up to 300 V dc. Of the three types of tantalum electrolytic
capacitors, the foil design has the lowest capacitance per unit volume and is
best suited for the higher voltages primarily found in older designs of equip-
ment. It is expensive and used only where neither a solid-electrolyte (Fig. 1.18)
nor a wet-slug (Fig. 1.19) tantalum capacitor can be employed.
FIGURE 1.18Tantalex? solid electrolyte tantalum capacitor. (Courtesy of Sprague Electric Company.)
FIGURE 1.19Hermetically sealed sintered-anode tantalum capacitor. (Courtesy of Sprague Electric Company.)
FIGURE 1.17 Full-wave capaci-
tor charge and discharge.
? 2000 by CRC Press LLC
Foil tantalum capacitors are generally designed for operation over the temperature range of –55 to +125°C
(–67 to +257°F) and are found primarily in industrial and military electronics equipment.
Solid-electrolyte sintered-anode tantalum capacitors differ from the wet versions in their electrolyte, which is
manganese dioxide.
Another variation of the solid-electrolyte tantalum capacitor encases the element in plastic resins, such as
epoxy materials offering excellent reliability and high stability for consumer and commercial electronics with
the added feature of low cost.
Still other designs of “solid tantalum” capacitors use plastic film or sleeving as the encasing material, and
others use metal shells that are backfilled with an epoxy resin. Finally, there are small tubular and rectangular
molded plastic encasements.
Wet-electrolyte sintered-anode tantalum capacitors, often called “wet-slug” tantalum capacitors, use a pellet
of sintered tantalum powder to which a lead has been attached, as shown in Fig. 1.19. This anode has an
enormous surface area for its size.
Wet-slug tantalum capacitors are manufactured in a voltage range to 125 V dc.
Use Considerations. Foil tantalum capacitors are used only where high-voltage constructions are required or
where there is substantial reverse voltage applied to a capacitor during circuit operation.
Wet sintered-anode capacitors, or “wet-slug” tantalum capacitors, are used where low dc leakage is required.
The conventional “silver can” design will not tolerate reverse voltage. In military or aerospace applications
where utmost reliability is desired, tantalum cases are used instead of silver cases. The tantalum-cased wet-slug
units withstand up to 3 V reverse voltage and operate under higher ripple currents and at temperatures up to
200°C (392°F).
Solid-electrolyte designs are the least expensive for a given rating and are used where their very small size is
important. They will typically withstand a reverse voltage up to 15% of the rated dc working voltage. They also
have good low-temperature performance characteristics and freedom from corrosive electrolytes.
Inductors
Inductance is used for the storage of magnetic energy. Magnetic energy is stored as long as current keeps
flowing through the inductor. In a perfect inductor, the current of a sine wave lags the voltage by 90°.
Impedance
Inductive reactance X
L
, the impedance of an inductor to an ac signal, is found by the equation
X
L
= 2pfL (1.34)
where X
L
= inductive reactance, W; f = frequency, Hz; and L = inductance, H.
The type of wire used for its construction does not affect the inductance of a coil. Q of the coil will be
governed by the resistance of the wire. Therefore coils wound with silver or gold wire have the highest Q for a
given design.
To increase inductance, inductors are connected in series. The total inductance will always be greater than
the largest inductor.
L
T
= L
1
+ L
2
+ · · · + L
n
(1.35)
To reduce inductance, inductors are connected in parallel.
(1.36)
The total inductance will always be less than the value of the lowest inductor.
L
LL L
T
n
=
+ +×××+
1
11 1
12
// /
? 2000 by CRC Press LLC
Mutual Inductance
Mutual inductance is the property that exists between two conductors carrying current when their magnetic
lines of force link together.
The mutual inductance of two coils with fields interacting can be determined by the equation
(1.37)
where M = mutual inductance of L
A
and L
B
, H; L
A
= total inductance, H, of coils L
1
and L
2
with fields aiding;
and L
B
= total inductance, H, of coils L
1
and L
2
with fields opposing.
The coupled inductance can be determined by the following equations. In parallel with fields aiding,
(1.38)
In parallel with fields opposing,
(1.39)
In series with fields aiding,
L
T
= L
1
+ L
2
+ 2M (1.40)
In series with fields opposing,
L
T
= L
1
+ L
2
– 2M (1.41)
where L
T
= total inductance, H; L
1
and L
2
= inductances of the individual coils, H; and M = mutual inductance,
H.
When two coils are inductively coupled to give transformer action, the coupling coefficient is determined by
(1.42)
where K = coupling coefficient; M = mutual inductance, H; and L
1
and L
2
= inductances of the two coils, H.
An inductor in a circuit has a reactance equal to j2pfL W. Mutual inductance in a circuit has a reactance
equal to j2pfL W. The operator j denotes that the reactance dissipates no energy; however, it does oppose
current flow.
The energy stored in an inductor can be determined by the equation
(1.43)
where W = energy, J (W · s); L = inductance, H; and I = current, A.
M
LL
AB
=
–
4
L
LMLM
T
=
+
+
+
1
11
12
L
LMLM
T
=
1
11
12
–
–
–
K
M
LL
=
12
W
LI
=
2
2
? 2000 by CRC Press LLC
Coil Inductance
Inductance is related to the turns in a coil as follows:
1.The inductance is proportional to the square of the turns.
2.The inductance increases as the length of the winding is increased.
3.A shorted turn decreases the inductance, affects the frequency response, and increases the insertion loss.
4.The inductance increases as the permeability of the core material increases.
5.The inductance increases with an increase in the cross-sectional area of the core material.
6.Inductance is increased by inserting an iron core into the coil.
7.Introducing an air gap into a choke reduces the inductance.
A conductor moving at any angle to the lines of force cuts a number of lines of force proportional to the
sine of the angles. Thus,
V = bLv sin q ′ 10
–8
(1.44)
where b = flux density; L = length of the conductor, cm; and v = velocity, cm/s, of conductor moving at an
angle q.
The maximum voltage induced in a conductor moving in a magnetic field is proportional to the number of
magnetic lines of force cut by that conductor. When a conductor moves parallel to the lines of force, it cuts no
lines of force; therefore, no current is generated 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 flux density vector.
The magnetomotive force (mmf) in ampere-turns produced by a coil is found by multiplying the number
of turns of wire in the coil by the current flowing through it.
(1.45)
where T = number of turns; V = voltage, V; and R = resistance, W.
The inductance of a single layer, a spiral, and multilayer coils can be calculated by using either Wheeler’s or
Nagaoka’s equations. The accuracy of the calculation will vary between 1 and 5%. The inductance of a single-
layer coil can be calculated using Wheeler’s equation:
(1.46)
For the multilayer coil,
(1.47)
For the spiral coil,
(1.48)
Ampere-turns=T
V
R
TI
?
è
?
?
?
÷
=
L
BN
BA
=
+
22
910
mH
L
BN
BAC
=
++
08
6910
22
.
mH
L
BN
BC
=
+
22
81
mH
? 2000 by CRC Press LLC
where B = radius of the winding, N = number of turns in the coil, A = length of the winding, and C = thickness
of the winding.
Q
Q is the ratio of the inductive reactance to the internal resistance of the coil and is affected by frequency,
inductance, dc resistance, inductive reactance, the type of winding, the core losses, the distributed capacity,
and the permeability of the core material.
The Q for a coil where R and L are in series is
(1.49)
where f = frequency, Hz; L = 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 oscil-
lator); 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 a low value of resistance, R,
about 1/100 of the anticipated rf resistance of the LC combination, to
assure that the measurement will not be in error by more than 1%. For
most measurements, R will be about 0.10 W and should have a voltage
of 0.1 V. Most oscillators cannot be operated into this low impedance,
so a step-down matching transformer must be employed. Make C as large as convenient to 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
(1.50)
The Q of any coil may be approximated by the equation
(1.51)
where f = the frequency, Hz; L = the inductance, H; R = the dc resistance, W (as measured by an ohmmeter);
and X
L
= the inductive reactance of the coil.
Time Constant
When a dc voltage is applied to an RL circuit, a certain amount of time is required to change the circuit [see
text with Eq. (1.31)]. The time constant can be determined with the equation
(1.52)
where R = resistance, W; L = inductance, H; and T = time, s.
Q
fL
R
=
2p
FIGURE 1.20Circuit for measuring
the Q of a coil.
Q
C
R
=
resonant voltage across
voltage across
Q
fL
R
X
R
L
=
=
2p
T
L
R
=
? 2000 by CRC Press LLC
The right-hand rule is used to determine the direction of a magnetic field around a conductor carrying a
direct current. Grasp the conductor in the right hand with the thumb extending along the conductor pointing
in the direction of the current. With the fingers partly closed, the finger tips will point in the direction of the
magnetic field.
Maxwell’s 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 will represent the direction of the
magnetic lines of force.”
Impedance
The total impedance created by resistors, capacitors, and inductors in circuits can be determined with the
following equations.
For resistance and capacitance in series,
(1.53)
(1.54)
For resistance and inductance in series,
(1.55)
(1.56)
For inductance and capacitance in series,
(1.58)
(1.57)
For resistance, inductance, and capacitance in series,
(1.59)
(1.60)
For capacitance and resistance in parallel,
(1.61)
For resistance and inductance in parallel,
(1.62)
ZRX
C
=+
22
q=arctan
X
R
C
ZRX
L
=+
22
q=arctan
X
R
L
Z
XX X X
XX X X
LC L C
CL C L
=
>
>
ì
í
?
?
?
–
–
when
when
ZRXX
LC
=+-
22
()
q=arctan
XX
R
LC
–
Z
RX
RX
C
C
=
+
22
Z
RX
RX
L
L
=
+
22
? 2000 by CRC Press LLC
For capacitance and inductance in parallel,
(1.63)
(1.64)
For inductance, capacitance, and resistance in parallel,
(1.65)
(1.66)
For inductance and series resistance in parallel with resistance,
(1.67)
(1.68)
For inductance and series resistance in parallel with capacitance,
(1.69)
(1.70)
For capacitance and series resistance in parallel with inductance and series resistance,
(1.71)
(1.72)
Z
XX
XX
XX
XX
XX
XX
LC
LC
LC
CL
CL
CL
=
>
>
ì
í
?
?
?
?
?
–
–
when
when
Z
RX X
XX R X X
LC
LC L C
=
+
22 2 2
(–)
q=arctan
RX X
XX
LC
LC
(–)
ZR
RX
RR X
L
L
=
+
++
2
1
22
12
22
()
q=
++
arctan
XR
RXRR
L
L
2
1
22
12
ZX
RX
RXX
C
L
LC
=
+
+-
22
()
q=
-
arctan
XX X R
RX
LC L
C
(–)
2
Z
RXRX
RR X X
LC
LC
=
++
++
()()
()(–)
1
22
2
22
12
22
q=
+- +
++ +
arctan
XR X XR X
RR X R R X
LCCL
CL
()()
()()
2
22
1
22
12
22
21
22
? 2000 by CRC Press LLC
where Z = impedance, W; R = resistance, W; L = inductance, H; X
L
= inductive reactance, W; X
C
= capacitive
reactance, W; and q = phase angle, degrees, by which current leads voltage in a capacitive circuit or lags voltage
in an inductive circuit (0° indicates an in-phase condition).
Resonant Frequency
When an inductor and capacitor are connected in series or parallel, they form a resonant circuit. The resonant
frequency can be determined from the equation
(1.73)
where f = frequency, Hz; L = inductance, H; C = capacitance, F; and X
L
, X
C
= impedance, W.
The resonant frequency can also be determined through the use of a reactance chart developed by the Bell
Telephone Laboratories (Fig. 1.21). This chart can be used for solving problems of inductance, capacitance,
frequency, and impedance. If two of the values are known, the third and fourth values may be found with its use.
Defining Terms
Air capacitor: A fixed or variable capacitor in which air is the dielectric material between the capacitor’s plates.
Ambient temperature: The temperature of the air or liquid surrounding any electrical part or device. Usually
refers to the effect of such temperature in aiding or retarding removal of heat by radiation and convection
from the part or device in question.
Ampere-turns:The magnetomotive force produced by a coil, derived by multiplying the number of turns of
wire in a coil by the current (A) flowing through it.
Anode: The positive electrode of a capacitor.
Capacitive reactance: The opposition offered to the flow of an alternating or pulsating current by capacitance
measured in ohms.
Capacitor: An electrical device capable of storing electrical energy and releasing it at some predetermined
rate at some predetermined time. It consists essentially of two conducting surfaces (electrodes) separated
by an insulating material or dielectric. A capacitor stores electrical energy, blocks the flow of direct
current, and permits the flow of alternating current to a degree dependent essentially upon capacitance
and frequency. The amount of energy stored, E = 0.5 CV
2
.
Cathode: The capacitor’s negative electrode.
Coil:Anumber of turns of wire in the form of a spiral. The spiral may be wrapped around an iron core or
an insulating form, or it may be self-supporting. A coil offers considerable opposition to ac current but
very little to dc current.
Conductor:Abare or insulated wire or combination of wires not insulated from one another, suitable for
carrying an electric current.
Dielectric: The insulating (nonconducting) medium between the two electrodes (plates) of a capacitor.
Dielectric constant:The ratio of the capacitance of a capacitor with a given dielectric to that of the same
capacitor having a vacuum dielectric.
Disk capacitor: A small single-layer ceramic capacitor with a dielectric insulator consisting of conductively
silvered opposing surfaces.
Dissipation factor (DF):The ratio of the effective series resistance of a capacitor to its reactance at a specified
frequency measured in percent.
Electrolyte:Current-conducting solution between two electrodes or plates of a capacitor, at least one of which
is covered by a dielectric.
f
LC
CX
X
L
C
L
=
=
=
1
2
1
2
2
p
p
p
? 2000 by CRC Press LLC
Electrolytic capacitor: A capacitor solution between two electrodes or plates of a capacitor, at least one of
which is covered by a dielectric.
Equivalent series resistance (ESR): All internal series resistance of a capacitor concentrated or “lumped” at
one point and treated as one resistance of a capacitor regardless of source, i.e., lead resistance, termination
losses, or dissipation in the dielectric material.
Farad: The basic unit of measure in capacitors. Acapacitor charged to 1 volt with a charge of 1 coulomb
(1 ampere flowing for 1 second) has a capacitance of 1 farad.
Field: Ageneral term referring to the region under the influence of a physical agency such as electricity,
magnetism, or a combination produced by an electrical charged object.
Impedance (Z): Total opposition offered to the flow of an alternating or pulsating current measured in ohms.
(Impedance is the vector sum of the resistance and the capacitive and inductive reactance, i.e., the ratio
of voltage to current.)
Inductance: The property which opposes any change in the existing current. Inductance is present only when
the current is changing.
Inductive reactance (X
L
): The opposition to the flow of alternating or pulsating current by the inductance
of a circuit.
FIGURE 1.21 Reactance chart. (Courtesy AT&T Bell Laboratories.)
? 2000 by CRC Press LLC
Inductor: Aconductor used to introduce inductance into a circuit.
Leakage current: Stray direct current of relatively small value which flows through a capacitor when voltage
is impressed across it.
Magnetomotive force: The force by which the magnetic field is produced, either by a current flowing through
a coil of wire or by the proximity of a magnetized body. The amount of magnetism produced in the first
method is proportional to the current through the coil and the number of turns in it.
Mutual inductance: The property that exists between two current-carrying conductors when the magnetic
lines of force from one link with those from another.
Negative-positive-zero (NPO): An ultrastable temperature coefficient (±30 ppm/°C from –55 to 125°C)
temperature-compensating capacitor.
Phase: The angular relationship between current and voltage in an ac circuit. The fraction of the period which
has elapsed in a periodic function or wave measured from some fixed origin. If the time for one period
is represented as 360° along a time axis, the phase position is called phase angle.
Polarized capacitor: An electrolytic capacitor in which the dielectric film is formed on only one metal
electrode. The impedance to the flow of current is then greater in one direction than in the other. Reversed
polarity can damage the part if excessive current flow occurs.
Power factor (PF): The ratio of effective series resistance to impedance of a capacitor, expressed as a percentage.
Quality factor (Q): The ratio of the reactance to its equivalent series resistance.
Reactance (X): Opposition to the flow of alternating current. Capacitive reactance (X
c
) is the opposition
offered by capacitors at a specified frequency and is measured in ohms.
Resonant frequency: The frequency at which a given system or object will respond with maximum amplitude
when driven by an external sinusoidal force of constant amplitude.
Reverse leakage current: Anondestructive current flowing through a capacitor subjected to a voltage of
polarity opposite to that normally specified.
Ripple current: The total amount of alternating and direct current that may be applied to an electrolytic
capacitor under stated conditions.
Temperature coefficient (TC): A capacitor’s change in capacitance per degree change in temperature. May
be positive, negative, or zero and is usually expressed in parts per million per degree Celsius (ppm/°C)
if the characteristics are linear. For nonlinear types, TC is expressed as a percentage of room temperature
(25°C)capacitance.
Time constant: In a capacitor-resistor circuit, the number of seconds required for the capacitor to reach
63.2% of its full charge after a voltage is applied. The time constant of a capacitor with a capacitance (C)
in farads in series with a resistance (R) in ohms is equal to R ′ C seconds.
Winding: Aconductive path, usually wire, inductively coupled to a magnetic core or cell.
Related Topic
55.5 Dielectric Materials
References
Exploring the capacitor, Hewlett-Packard Bench Briefs, September/October 1979. Sections reprinted with per-
mission from Bench Briefs, a Hewlett-Packard service publication.
Capacitors, 1979 Electronic Buyer’s Handbook, vol. 1, November 1978. Copyright 1978 by CMP Publications,
Inc. Reprinted with permission.
W. G. Jung and R. March, “Picking capacitors,” Audio, March 1980.
“Electrolytic capacitors: Past, present and future,” and “What is an electrolytic capacitor,” Electron. Des., May
28, 1981.
R.F. Graf, “Introduction To Aluminum Capacitors,” Sprague Electric Company. Parts reprinted with permission.
“Introduction To Aluminum Capacitors,” Sprague Electric Company. Parts reprinted with permission.
Handbook of Electronics Tables and Formulas, 6th ed., Indianapolis: Sams, 1986.
? 2000 by CRC Press LLC
1.3 Transformers
C. Sankaran
The electrical transformer was invented by an American electrical engineer, William Stanley, in 1885 and was
used in the first ac lighting installation at Great Barrington, Massachusetts. The first transformer was used to
step up the power from 500 to 3000 V and transmitted for a distance of 1219 m (4000 ft). At the receiving end
the voltage was stepped down to 500 V to power street and office lighting. By comparison, present transformers
are designed to transmit hundreds of megawatts of power at voltages of 700 kV and beyond for distances of
several hundred miles.
Transformation of power from one voltage level to another is a vital operation in any transmission, distri-
bution, and utilization network. Normally, power is generated at a voltage that takes into consideration the
cost of generators in relation to their operating voltage. Generated power is transmitted by overhead lines many
miles and undergoes several voltage transformations before it is made available to the actual user. Figure 1.22
shows a typical power flow line diagram.
Types of Transformers
Transformers are broadly grouped into two main categories: dry-type and liquid-filled transformers. Dry-type
transformers are cooled by natural or forced circulation of air or inert gas through or around the transformer
enclosure. Dry-type transformers are further subdivided into ventilated, sealed, or encapsulated types depending
upon the construction of the transformer. Dry transformers are extensively used in industrial power distribution
for rating up to 5000 kVA and 34.5 kV.
Liquid-filled transformers are cooled by natural or forced circulation of a liquid coolant through the windings
of the transformer. This liquid also serves as a dielectric to provide superior voltage-withstand characteristics.
The most commonly used liquid in a transformer is a mineral oil known as transformer oil that has a continuous
operating temperature rating of 105°C, a flash point of 150°C, and a fire point of 180°C. A good grade
transformer oil has a breakdown strength of 86.6 kV/cm (220 kV/in.) that is far higher than the breakdown
strength of air, which is 9.84 kV/cm (25 kV/in.) at atmospheric pressure.
Silicone fluid is used as an alternative to mineral oil. The breakdown strength of silicone liquid is over
118 kV/cm (300 kV/in.) and it has a flash point of 300°C and a fire point of 360°C. Silicone-fluid-filled
transformers are classified as less flammable. The high dielectric strengths and superior thermal conductivities
of liquid coolants make them ideally suited for large high-voltage power transformers that are used in modern
power generation and distribution.
FIGURE 1.22Power flow line diagram.
? 2000 by CRC Press LLC
Principle of Transformation
The actual process of transfer of electrical power from a voltage of V
1
to a voltage of V
2
is explained with the
aid of the simplified transformer representation shown in Fig. 1.23. Application of voltage across the primary
winding of the transformer results in a magnetic field of f
1
Wb in the magnetic core, which in turn induces
a voltage of V
2
at the secondary terminals. V
1
and V
2
are related by the expression V
1
/V
2
= N
1
/N
2
, where N
1
and N
2
are the number of turns in the primary and secondary windings, respectively. If a load current of I
2
A
is drawn from the secondary terminals, the load current establishes a magnetic field of f
2
Wb in the core and
in the direction shown. Since the effect of load current is to reduce the amount of primary magnetic field, the
reduction in f
1
results in an increase in the primary current I
1
so that the net magnetic field is almost restored
to the initial value and the slight reduction in the field is due to leakage magnetic flux. The currents in the
two windings are related by the expression I
1
/I
2
= N
2
/N
1
. Since V
1
/V
2
= N
1
/N
2
= I
2
/I
1
, we have the expression
V
1
· I
1
= V
2
· I
2
. Therefore, the voltamperes in the two windings are equal in theory. In reality, there is a slight
loss of power during transformation that is due to the energy necessary to set up the magnetic field and to
overcome the losses in the transformer core and windings. Transformers are static power conversion devices
and are therefore highly efficient. Transformer efficiencies are about 95% for small units (15 kVA and less),
and the efficiency can be higher than 99% for units rated above 5 MVA.
Electromagnetic Equation
Figure 1.24 shows a magnetic core with the area of cross section A = W · D m
2
. The transformer primary
winding that consists of N turns is excited by a sinusoidal voltage v = V sin(wt), where w is the angular frequency
given by the expression w = 2pf and f is the frequency of the applied voltage waveform. f is magnetic field in
the core due to the excitation current i:
FIGURE 1.23Electrical power transfer.
FIGURE 1.24Electromagnetic relation.
? 2000 by CRC Press LLC
Induced voltage in the winding
Maximum value of the induced voltage
E = NwF
The root-mean-square value
where flux F (webers) is replaced by the product of the flux density B (teslas) and the area of cross section of
the core.
This fundamental design equation determines the size of the transformer for any given voltage and frequency.
Power transformers are normally operated at flux density levels of 1.5 T.
Transformer Core
The transformer core is the medium that enables the transfer of power from the primary to the secondary to
occur in a transformer. In order that the transformation of power may occur with the least amount of loss, the
magnetic core is made up of laminations which have the highest permeability, permeability being a measure
of the ease with which the magnetic field is set up in the core.
The magnetic field reverses direction every one half cycle of the applied voltage and energy is expended in
the core to accomplish the cyclic reversals of the field. This loss component is known as the hysteresis loss P
h
:
P
h
= 150.7V
e
fB
1.6
W
where V
e
is the volume of the core in cubic meters, f is the frequency, and B is the maximum flux density in teslas.
As the magnetic field reverses direction and cuts across the core structure, it induces a voltage in the
laminations known as eddy voltages. This phenomenon causes eddy currents to circulate in the laminations.
The loss due to eddy currents is called the eddy current loss P
e
:
P
e
= 1.65V
e
B
2
f
2
t
2
/r
where V
e
is the volume of the core in cubic meters, f is the frequency, B is the maximum flux density in teslas,
t is thickness of the laminations in meters, and r is the resistivity of the core material in ohm-meters.
Hysteresis losses are reduced by operating the core at low flux densities and using core material of high
permeability. Eddy current losses are minimized by low flux levels, reduction in thickness of the laminations,
and high resistivity core material.
Cold-rolled, grain-oriented silicon steel laminations are exclusively used in large power transformers to
reduce core losses. A typical silicon steel used in transformers contains 95% iron, 3% silicon, 1% manganese,
0.2% phosphor, 0.06% carbon, 0.025% sulphur, and traces of other impurities.
fw
p
w=-
?
è
?
?
?
÷
=-FFsin cos( )tt
2
eN
d
dt
N
dt
dt
Nt=- = =-
fw
ww
[ cos( )]
sin( )
F
F
E
EfN
f
rms
NBA== =
2
2
2
444
pF
.
? 2000 by CRC Press LLC
Transformer Losses
The heat developed in a transformer is a function of the losses that occur during transformation. Therefore,
the transformer losses must be minimized and the heat due to the losses must be efficiently conducted away
from the core, the windings, and the cooling medium. The losses in a transformer are grouped into two
categories: (1) no-load losses and (2) load losses. The no-load losses are the losses in the core due to excitation
and are mostly composed of hysteresis and eddy current losses. The load losses are grouped into three categories:
(1) winding I
2
R losses, (2) winding eddy current losses, and (3) other stray losses. The winding I
2
R losses are
the result of the flow of load current through the resistance of the primary and secondary windings. The winding
eddy current losses are caused by the magnetic field set up by the winding current, due to formation of eddy
voltages in the conductors. The winding eddy losses are proportional to the square of the rms value of the
current and to the square of the frequency of the current. When transformers are required to supply loads that
are rich in harmonic frequency components, the eddy loss factor must be given extra consideration. The other
stray loss component is the result of induced currents in the buswork, core clamps, and tank walls by the
magnetic field set up by the load current.
Transformer Connections
A single-phase transformer has one input (primary) winding and one output (secondary) winding. A conven-
tional three-phase transformer has three input and three output windings. The three windings can be connected
in one of several different configurations to obtain three-phase connections that are distinct. Each form of
connection has its own merits and demerits.
Y Connection (Fig. 1.25)
In the Y connection, one end of each of the three windings is connected together
to form a Y, or a neutral point. This point is normally grounded, which limits
the maximum potential to ground in the transformer to the line to neutral voltage
of the power system. The grounded neutral also limits transient overvoltages in
the transformer when subjected to lightning or switching surges. Availability of
the neutral point allows the transformer to supply line to neutral single-phase
loads in addition to normal three-phase loads. Each phase of the Y-connected
winding must be designed to carry the full line current, whereas the phase voltages
are only 57.7% of the line voltages.
Delta Connection (Fig. 1.26)
In the delta connection, the finish point of each winding is connected
to the start point of the adjacent winding to form a closed triangle, or
delta. A delta winding in the transformer tends to balance out unbal-
anced loads that are present on the system. Each phase of the delta
winding only carries 57.7% of the line current, whereas the phase
voltages are equal to the line voltages.
Large power transformers are designed so that the high-voltage side
is connected in Y and the low-voltage side is connected in delta. Dis-
tribution transformers that are required to supply single-phase loads
are designed in the opposite configuration so that the neutral point is
available at the low-voltage end.
Open-Delta Connection (Fig. 1.27)
An open-delta connection is used to deliver three-phase power if
one phase of a three-phase bank of transformers fails in service.
When the failed unit is removed from service, the remaining units
can still supply three-phase power but at a reduced rating. An open-
delta connection is also used as an economical means to deliver
three-phase power using only two single-phase transformers. If P
FIGURE 1.25Y connection.
FIGURE 1.26Delta connection.
FIGURE 1.27Open-delta connection.
? 2000 by CRC Press LLC
is the total three-phase kVA, then each transformer of the open-delta bank must have a rating of P/ kVA.
The disadvantage of the open-delta connection is the unequal regulation of the three phases of the transformer.
T Connection (Fig. 1.28)
The T connection is used for three-phase power trans-
formation when two separate single-phase transformers
with special configurations are available. If a voltage
transformation from V
1
to V
2
volts is required, one of
the units (main transformer) must have a voltage ratio
of V
1
/V
2
with the midpoint of each winding brought
out. The other unit must have a ratio of 0.866V
1
/0.866V
2
with the neutral point brought out, if needed.
The Scott connection is a special type of T connection
used to transform three-phase power to two-phase
power for operation of electric furnaces and two-phase
motors. It is shown in Fig. 1.29.
Zigzag Connection (Fig. 1.30)
This connection is also called the interconnected star connection where the winding of each phase is divided
into two halves and interconnected to form a zigzag configuration. The zigzag connection is mostly used to
derive a neutral point for grounding purposes in three-phase, three-wire systems. The neutral point can be
used to (1) supply single-phase loads, (2) provide a safety ground, and (3) sense and limit ground fault currents.
Transformer Impedance
Impedance is an inherent property in a transformer that results in a voltage drop as power is transferred from
the primary to the secondary side of the power system. The impedance of a transformer consists of two parts:
resistance (R) and reactance (X). The resistance component is due to the resistance of the material of the winding
and the percentage value of the voltage drop due to resistance becomes less as the rating of the transformer
increases. The reactive component, which is also known as leakage reactance, is the result of incomplete linkage
of the magnetic field set up by the secondary winding with the turns of the primary winding, and vice versa.
The net impedance of the transformer is given by Z = . The impedance value marked on the trans-
former is the percentage voltage drop due to this impedance under full-load operating conditions:
3
FIGURE 1.28T connection.
R
2
X
2
+
% impedance zIZ
V
=
?
è
?
?
?
÷
100
FIGURE 1.29Three-phase–two-phase transformation. FIGURE 1.30Zigzag connection.
? 2000 by CRC Press LLC
where I is the full-load current of the transformer, Z is the impedance in ohms of the transformer, and V is
the voltage rating of the transformer winding. It should be noted that the values of I and Z must be referred
to the same side of the transformer as the voltage V.
Transformers are also major contributors of impedance to limit the fault currents in electrical power systems.
Defining Terms
Breakdown strength:Voltage gradient at which the molecules of medium break down to allow passage of
damaging levels of electric current.
Dielectric: Solid, liquid, or gaseous substance that acts as an insulation to the flow of electric current.
Harmonic frequency:Integral multiples of fundamental frequency. For example, for a 60-Hz supply the
harmonic frequencies are 120, 180, 240, 300, . . .
Magnetic field: Magnetic force field where lines of magnetism exist.
Magnetic flux:Term for lines of magnetism.
Regulation:The change in voltage from no-load to full-load expressed as a percentage of full-load voltage.
Related Topics
9.3 Wye ? Delta Transformations?36.1 Magnetism?61.6 Protection?64.1 Transformer Construction
References and Further Information
Bean, Chackan, Moore and Wentz, Transformers for the Electric Power Industry, New York: McGraw-Hill, 1966.
General Electric, Transformer Connections, 1960.
A. Gray, Electrical Machine Design, New York: McGraw-Hill.
IEEE, C57 Standards on Transformers, New York: IEEE Press, 1992.
IEEE Transactions on Industry Applications.
R. R. Lawrence, Principles of Alternating Current Machinery, New York: McGraw-Hill, 1920.
Power Engineering Review.
C. Sankaran, Introduction to Transformers, New York: IEEE Press, 1992.
S. A. Stigant and A.C. Franklin, The J & P Transformer Book, London: Newnes-Butterworths, 1973.
1.4 Electrical Fuses
Nick Angelopoulos
The fuse is a simple and reliable safety device. It is second to none in its ease of application and its ability to
protect people and equipment.
The fuse is a current-sensitive device. It has a conductor with a reduced cross section (element) normally
surrounded by an arc-quenching and heat-conducting material (filler). The entire unit is enclosed in a body
fitted with end contacts. A basic fuse element design is illustrated in Fig. 1.32.
Ratings
Most fuses have three electrical ratings: ampere rating, voltage rating, and interrupting rating. The ampere
rating indicates the current the fuse can carry without melting or exceeding specific temperature rise limits.
The voltage rating, ac or dc, usually indicates the maximum system voltage that can be applied to the fuse. The
interrupting rating (I.R.) defines the maximum short-circuit current that a fuse can safely interrupt. If a fault
current higher than the interrupting rating causes the fuse to operate, the high internal pressure may cause the
fuse to rupture. It is imperative, therefore, to install a fuse, or any other type of protective device, that has an
interrupting rating not less than the available short-circuit current. A violent explosion may occur if the
interrupting rating of any protective device is inadequate.
? 2000 by CRC Press LLC
A fuse must perform two functions. The first, the “passive” function, is one that tends to be taken for granted.
In fact, if the fuse performs the passive function well, we tend to forget that the fuse exists at all. The passive
function simply entails that the fuse can carry up to its normal load current without aging or overheating.
Once the current level exceeds predetermined limits, the “active” function comes into play and the fuse operates.
It is when the fuse is performing its active function that we become aware of its existence.
In most cases, the fuse will perform its active function in response to two types of circuit conditions. The
first is an overload condition, for instance, when a hair dryer, teakettle, toaster, and radio are plugged into the
same circuit. This overload condition will eventually cause the element to melt. The second condition is the
overcurrent condition, commonly called the short circuit or the fault condition. This can produce a drastic,
almost instantaneous, rise in current, causing the element to melt usually in less than a quarter of a cycle.
Factors that can lead to a fault condition include rodents in the electrical system, loose connections, dirt and
moisture, breakdown of insulation, foreign contaminants, and personal mistakes. Preventive maintenance and
care can reduce these causes. Unfortunately, none of us are perfect and faults can occur in virtually every
electrical system—we must protect against them.
Fuse Performance
Fuse performance characteristics under overload conditions are published in the form of average melting
time–current characteristic curves, or simply time-current curves. Fuses are tested with a variety of currents, and
the melting times are recorded. The result is a graph of time versus current coordinates that are plotted on log-
log scale, as illustrated in Fig. 1.33.
Under short-circuit conditions the fuse operates and fully opens the circuit in less than 0.01 s. At 50 or
60 Hz, this represents operation within the first half cycle. The current waveform let-through by the fuse is the
shaded, almost triangular, portion shown in Fig. 1.34(a). This depicts a fraction of the current that would have
been let through into the circuit had a fuse not been installed.
FIGURE 1.31 A variety of plug, cartridge, and blade type fuses.
FIGURE 1.32 Basic fuse element.
? 2000 by CRC Press LLC
Fuse short-circuit performance characteristics are published in the form of peak let-through (I
p
) graphs and
I
2
t graphs. I
p
(peak current) is simply the peak of the shaded triangular waveform, which increases as the fault
current increases, as shown in Fig. 1.34(b). The electromagnetic forces, which can cause mechanical damage
to equipment, are proportional to I
p
2
.
I
2
t represents heat energy measured in units of A
2
s (ampere squared seconds) and is documented on I
2
t
graphs. These I
2
t graphs, as illustrated in Fig. 1.34(c), provide three values of I
2
t: minimum melting I
2
t, arcing
I
2
t, and total clearing I
2
t. I
2
t and I
p
short-circuit performance characteristics can be used to coordinate fuses
and other equipment. In particular, I
2
t values are often used to selectively coordinate fuses in a distribution
system.
FIGURE 1.33Time-current characteristic curves.
? 2000 by CRC Press LLC
Selective Coordination
In any power distribution system, selective coordination exists when the fuse immediately upstream from a
fault operates, leaving all other fuses further upstream unaffected. This increases system reliability by isolating
the faulted branch while maintaining power to all other branches. Selective coordination is easily assessed by
FIGURE 1.34 (a) Fuse short-circuit operation. (b) Variation of fuse peak let-through current I
p
. (c) I
2
t graph.
? 2000 by CRC Press LLC
comparing the I
2
t characteristics for feeder and branch circuit fuses. The branch fuse should have a total clearing
I
2
t value that is less than the melting I
2
t value of the feeder or upstream fuse. This ensures that the branch
fuse will melt, arc, and clear the fault before the feeder fuse begins to melt.
Standards
Overload and short-circuit characteristics are well documented by fuse manufacturers. These characteristics
are standardized by product standards written in most cases by safety organizations such as CSA (Canadian
Standards Association) and UL (Underwriters Laboratories). CSA standards and UL specify product designa-
tions, dimensions, performance characteristics, and temperature rise limits. These standards are used in con-
junction with national code regulations such as CEC (Canadian Electrical Code) and NEC (National Electrical
Code) that specify how the product is applied.
IEC (International Electrotechnical Commission—Geneva, Switzerland) was founded to harmonize electrical
standards to increase international trade in electrical products. Any country can become a member and
participate in the standards-writing activities of IEC. Unlike CSA and UL, IEC is not a certifying body that
certifies or approves products. IEC publishes consensus standards for national standards authorities such as
CSA (Canada), UL (USA), BSI (UK) and DIN (Germany) to adopt as their own national standards.
Products
North American low-voltage distribution fuses can be classified under two types: Standard or Class H, as
referred to in the United States, and HRC (high rupturing capacity) or current-limiting fuses, as referred to
in Canada. It is the interrupting rating that essentially differentiates one type from the other.
Most Standard or Class H fuses have an interrupting rating of 10,000 A. They are not classified as HRC or
current-limiting fuses, which usually have an interrupting rating of 200,000 A. Selection is often based on the
calculated available short-circuit current.
In general, short-circuit currents in excess of 10,000 A do not exist in residential applications. In commercial
and industrial installations, short-circuit currents in excess of 10,000 A are very common. Use of HRC fuses
usually means that a fault current assessment is not required.
Standard—Class H
In North America, Standard or Class H fuses are available in 250- and 600-V ratings with ampere ratings up
to 600 A. There are primarily three types: one-time, time-delay, and renewable. Rating for rating, they are all
constructed to the same dimensions and are physically interchangeable in standard-type fusible switches and
fuse blocks.
One-time fuses are not reusable once blown. They are used for general-purpose resistive loads such as lighting,
feeders, and cables.
Time-delay fuses have a specified delay in their overload characteristics and are designed for motor circuits.
When started, motors typically draw six times their full load current for approximately 3 to 4 seconds. This
surge then decreases to a level within the motor full-load current rating. Time-delay fuse overload characteristics
are designed to allow for motor starting conditions.
Renewable fuses are constructed with replaceable links or elements. This feature minimizes the cost of
replacing fuses. However, the concept of replacing fuse elements in the field is not acceptable to most users
today because of the potential risk of improper replacement.
HRC
HRC or current-limiting fuses have an interrupting rating of 200 kA and are recognized by a letter designation
system common to North American fuses. In the United States they are known as Class J, Class L, Class R, etc.,
and in Canada they are known as HRCI-J, HRC-L, HRCI-R, and so forth. HRC fuses are available in ratings
up to 600 V and 6000 A. The main differences among the various types are their dimensions and their short-
circuit performance (I
p
and I
2
t) characteristics.
? 2000 by CRC Press LLC
One type of HRC fuse found in Canada, but not in the United States, is the HRCII-C or Class C fuse. This
fuse was developed originally in England and is constructed with bolt-on-type blade contacts. It is available in
a voltage rating of 600 V with ampere ratings from 2 to 600 A. Some higher ampere ratings are also available
but are not as common. HRCII-C fuses are primarily regarded as providing short-circuit protection only.
Therefore, they should be used in conjunction with an overload device.
HRCI-R or Class R fuses were developed in the United States. Originally constructed to Standard or Class H
fuse dimensions, they were classified as Class K and are available in the United States with two levels of short-
circuit performance characteristics: Class K1 and Class K5. However, they are not recognized in Canadian
Standards. Under fault conditions, Class K1 fuses limit the I
p
and I
2
t to lower levels than do Class K5 fuses.
Since both Class K1 and K5 are constructed to Standard or Class H fuse dimensions, problems with inter-
changeability occur. As a result, a second generation of these K fuses was therefore introduced with a rejection
feature incorporated in the end caps and blade contacts. This rejection feature, when used in conjunction with
rejection-style fuse clips, prevents replacement of these fuses with Standard or Class H 10-kA I.R. fuses. These
rejection style fuses are known as Class RK1 and Class RK5. They are available with time-delay or non-time-
delay characteristics and with voltage ratings of 250 or 600 V and ampere ratings up to 600 A. In Canada, CSA
has only one classification for these fuses, HRCI-R, which have the same maximum I
p
and I
2
t current-limiting
levels as specified by UL for Class RK5 fuses.
HRCI-J or Class J fuses are a more recent development. In Canada, they have become the most popular HRC
fuse specified for new installations. Both time-delay and non-time-delay characteristics are available in ratings
of 600 V with ampere ratings up to 600 A. They are constructed with dimensions much smaller than HRCI-R
or Class R fuses and have end caps or blade contacts which fit into 600-V Standard or Class H-type fuse clips.
However, the fuse clips must be mounted closer together to accommodate the shorter fuse length. Its shorter
length, therefore, becomes an inherent rejection feature that does not allow insertion of Standard or HRCI-R
fuses. The blade contacts are also drilled to allow bolt-on mounting if required. CSA and UL specify these fuses
to have maximum short-circuit current-limiting I
p
and I
2
t limits lower than those specified for HRCI-R and
HRCII-C fuses. HRCI-J fuses may be used for a wide variety of applications. The time-delay type is commonly
used in motor circuits sized at approximately 125 to 150% of motor full-load current.
HRC-L or Class L fuses are unique in dimension but may be considered as an extension of the HRCI-J fuses
for ampere ratings above 600 A. They are rated at 600 V with ampere ratings from 601 to 6000 A. They are
physically larger and are constructed with bolt-on-type blade contacts. These fuses are generally used in low-
voltage distribution systems where supply transformers are capable of delivering more than 600 A.
In addition to Standard and HRC fuses, there are many other types designed for specific applications. For
example, there are medium- or high-voltage fuses to protect power distribution transformers and medium-
voltage motors. There are fuses used to protect sensitive semiconductor devices such as diodes, SCRs, and triacs.
These fuses are designed to be extremely fast under short-circuit conditions. There is also a wide variety of
dedicated fuses designed for protection of specific equipment requirements such as electric welders, capacitors,
and circuit breakers, to name a few.
Trends
Ultimately, it is the electrical equipment being protected that dictates the type of fuse needed for proper
protection. This equipment is forever changing and tends to get smaller as new technology becomes available.
Present trends indicate that fuses also must become smaller and faster under fault conditions, particularly as
available short-circuit fault currents are tending to increase.
With free trade and the globalization of industry, a greater need for harmonizing product standards exists.
The North American fuse industry is taking big steps toward harmonizing CSA and UL fuse standards, and at
the same time is participating in the IEC standards process. Standardization will help the electrical industry to
identify and select the best fuse for the job—anywhere in the world.
? 2000 by CRC Press LLC
Defining Terms
HRC (high rupturing capacity): A term used to denote fuses having a high interrupting rating. Most low-
voltage HRC-type fuses have an interrupting rating of 200 kA rms symmetrical.
I
2
t (ampere squared seconds): A convenient way of indicating the heating effect or thermal energy which is
produced during a fault condition before the circuit protective device has opened the circuit. As a
protective device, the HRC or current-limiting fuse lets through far less damaging I
2
t than other protective
devices.
Interrupting rating (I.R.): The maximum value of short-circuit current that a fuse can safely interrupt.
Related Topic
1.1 Resistors
References
R.K. Clidero and K.H. Sharpe, Application of Electrical Construction, Ontario, Canada: General Publishing Co.
Ltd., 1982.
Gould Inc., Shawmut Advisor, Circuit Protection Division, Newburyport, Mass.
C. A. Gross, Power Systems Analysis, 2nd ed., New York: Wiley, 1986.
E. Jacks, High Rupturing Capacity Fuses, New York: Wiley, 1975.
A. Wright and P.G. Newbery, Electric Fuses, London: Peter Peregrinus Ltd., 1984.
Further Information
For greater detail the “Shawmut Advisor” (Gould, Inc., 374 Merrimac Street, Newburyport MA 01950) or the
“Fuse Technology Course Notes” (Gould Shawmut Company, 88 Horner Avenue, Toronto, Canada M8Z-5Y3)
may be referred to for fuse performance and application.
? 2000 by CRC Press LLC
