THE ENGINEERING THE HANDBOOK - SECOND EDITION II (end) ppt

111

Filters (Passive)

111.1 Fundamentals

111.2 Applications.

Simple RL and RC Filters • Simple RLC Filters • Compound
Filters • Constant-

k

Filters •

m

-Derived Filters

A

filter

is a frequency-sensitive two-port circuit that transmits with or without amplification signals in
a band of frequencies and rejects (or attenuates) signals in other bands. The electric filter was invented
during the First World War by two engineers working independently of each other — the American
engineer G. A. Campbell and the German engineer K. W. Wagner. O. Zobel followed in the 1920s. These
devices were developed to serve the growing areas of telephone and radio communication. Today, filters
are found in all types of electrical and electronic applications from power to communications. Filters
can be both active and passive. In this section we will confine our discussion to those filters that employ

no active devices for their operation. The main advantage of passive filters over active ones is that they
require no power (other than the signal) to operate. The disadvantage is that they often employ inductors
that are bulky and expensive.

111.1 Fundamentals

The basis for filter analysis involves the determination of a filter circuit’s sinusoidal steady state response
from its transfer function

T

(

j

w

). Some references use

H

(

j

w

) for the transfer function. The filter’s transfer
function

T

(

j

w

) is a complex function and can be represented through its gain

Ω

T

(

j

w

)

Ω

and phase

–

T

(

j

w

) characteristics. The gain and phase responses show how the filter alters the amplitude and phase
of the input signal to produce the output response. These two characteristics describe the

frequency
response

of the circuit since they depend on the frequency of the input sinusoid. The signal-processing
performance of devices, circuits, and systems is often specified in terms of their frequency response. The
gain and phase functions can be expressed mathematically or graphically as

frequency-response

plots.
Figure 111.1 shows examples of gain and phase responses versus frequency,

w

.
The terminology used to describe the frequency response of circuits and systems is often based on the
form of the gain plot. For example, at high frequencies the gain in Figure 111.1 falls off so that output
signals in this frequency range are reduced in amplitude. The range of frequencies over which the output
is significantly attenuated is called the

stopband.

At low frequencies the gain is essentially constant and
there is relatively little attenuation. The frequency range over which there is little attenuation is called a

passband.

The frequency associated with the boundary between a passband and an adjacent stopband is
called the

cutoff frequency

(

w

C



=



2

p

f

C

)

. In general, the transition from the passband to the stopband,
called the

transition band

, is relatively gradual, so the precise location of the cutoff frequency is a matter
of definition. The most widely used approach defines the cutoff frequency as the frequency at which the
gain has decreased by a factor of from its maximum value in the passband.
120707/.=

Albert J. Rosa

University of Denver

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111

-2

The Engineering Handbook, Second Edition

This particular definition is based on the fact that the power delivered to a resistor by a sinusoidal
current or voltage waveform is proportional to the square of its amplitude. At a cutoff frequency the gain
is reduced by a factor of and the square of the output amplitude, and thusly also its power, is
reduced by a factor of one half. For this reason the cutoff frequency is also called the

half-power frequency.

There are four prototypical filters. These are

low pass

(LP),

high pass

(HP),

band pass

(BP), and

bandstop

(BS). Figure 111.2 shows how the amplitude of an input signal consisting of three separate equal-
amplitude frequencies is altered by each of the four-prototypical filter responses. The low-pass filter
passes frequencies below its cutoff frequency

w

C

, called its

passband,

and attenuates the frequencies above
the cutoff, called its

stopband.

The high-pass filter passes frequencies above the cutoff frequency

w

C

and
attenuates those below. The band-pass filter passes those frequencies that lie between two cutoff frequen-
cies,

w

C

1

and

w

C

2

, its passband, and attenuates those frequencies that lie outside the passband. Finally,
the bandstop filter attenuates those frequencies that lie in its reject or stopband, between

w

C

1

and

w

C

2

,
and passes all others.
The

bandwidth

of a gain characteristic is defined as the frequency range spanned by its passband. For
the band-pass case in Figure 111.2, the bandwidth is the difference in the two cutoff frequencies.
BW

=

w

C

2



-



w

C

1

(111.1)
This equation applies to the low-pass response with the lower cutoff frequency

w

C

1

set to zero. In other
words, the bandwidth of a low-pass circuit is equal to its cutoff frequency (BW

=



w

C

). The bandwidth
of a high-pass characteristic is infinite since the upper cutoff frequency

w

C

1

is infinity. For the bandstop
case, Equation (111.1) defines the bandwidth of the stopband rather than the passbands.
Frequency-response plots are usually made using logarithmic scales for the frequency variable because
the frequency ranges of interest often span several orders of magnitude. A logarithmic frequency scale
compresses the data range and highlights important features in the gain and phase responses. The use
of a logarithmic frequency scale involves some special terminology. A frequency range whose end points
have a 2:1 ratio is called an

octave

and one with 10:1 ratio is called a

decade.

Straight-line approximations

FIGURE 111.1

Low-pass filter characteristics showing passband, stopband, and the cutoff frequency,

w

C

.
Gain
Phase
−45°
0°
−90°
A
A
2
0
Passband
Stopband
ω
c
ω
ω
ω
c

√
12/

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© 2005 by CRC Press LLC

Filters (Passive)

111

-3

of these plots, called Bode (Bow-dee) plots, are often used to describe the general behavior of the devices,
circuits, or systems.
In frequency-response plots the gain

Ω

T

(

j

w

)

Ω

is often expressed in

decibels

(dB), defined as

Ω

T

(

j

w

)

Ω

dB



=

20 log

10

Ω

T

(

j

w

)

Ω

(111.2)
Gains expressed in decibels can be either positive, negative, or zero. A gain of zero dB means that

Ω

T

(

j

w

)

Ω

=

1 — that is, the input and output amplitudes are equal. A positive dB gain means the output amplitude
exceeds the input since

Ω

T

(

j

w

)

Ω



>

1, whereas a negative dB gain means the output amplitude is smaller
than the input since

Ω

T

(

j

w

)

Ω



<

1. A cutoff frequency usually occurs when the gain is reduced from its
maximum passband value by a factor or 3 dB.
Figure 111.3 shows the asymptotic gain characteristics of ideal and real low-pass filters. The gain of
the

ideal filter

is unity (0 dB) throughout the passband and zero (

-•

dB) in the stopband. It also has an
infinitely narrow transition band. The asymptotic gain responses of real low-pass filters show that we can
only approximate the ideal response. As the order of the filter or number of poles

n

increases, the
approximation improves since the asymptotic slope or “rolloff” in the stopband is

-

20

¥



n

dB/decade.
On the other hand, adding poles requires additional stages in a cascade realization, so there is a trade-off
between (1) filter complexity and cost and (2) how closely the filter gain approximates the ideal response.
Figure 111.4 shows how low-pass filter requirements are often specified. To meet the specification, the
gain response must lie within the unshaded region in the figure, as illustrated by the two responses shown
in Figure 111.4. The parameter

T

max

is the

passband gain.

In the passband the gain must be within 3 dB

of

T

max

and must equal at the cutoff frequency

w

C

. In the stopband the gain must decrease
and remain below a gain of

T

min

for all

w



≥



w

min

. A low-pass filter design requirement is usually defined
by specifying values for these four parameters. The parameters

T

max



and



w

C

define the passband response,
whereas

T

min

and

w

min

specify how rapidly the stopband response must decrease.

FIGURE 111.2

Four prototype filters and their effects on an input signal consisting of three frequencies.
Passband
Stopband
LOW PASS
GAIN
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
Input
Transfer function
Output
GAIN
GAIN
GAIN
ω
ω
1
ω
1
ω
c

ω
2
ω
2
ω
3
ω
3
ω
1
ω
2
ω
3
ω
1
ω
2
ω
3
ω
1
ω
2
ω
3
ω
1
ω
2

ω
3
ω
1
ω
2
ω
3
ω
1
ω
2
ω
3
ω
1
ω
2
ω
3
ω
ω
ω
ω
ω
ω
ω
ω
Amplitude
Amplitude

Amplitude
Passband
Stopband
Passband
Passband
Stopband
ω
1
ω
c
ω
2
ω
3
ω
Stopband Stopband
ω
1
ω
c1
ω
2
ω
c2
ω
3
ω
Passband
HIGH PASS
BAND PASS

BANDSTOP
12/
T
max
/2

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© 2005 by CRC Press LLC

111

-4

The Engineering Handbook, Second Edition

111.2 Applications

Simple RL and RC Filters

A first-order LP filter has the following transfer function:
(111.3)

FIGURE 111.3

The effect of increasing the order

n

of a filter relative to an ideal filter.

FIGURE 111.4

Parameters for specifying low-pass filter requirements.
|T(jω)|
dB
Passband
Stopband
−40
0.1 1.0 10
0
−20
−60
Ideal
n = 1
n = 2
n = 3
Real
ω
C
ω
|T(jω)|
dB
T
MAX
T
MIN
3dB
Passband
Stopban
d

ω
MIN
ω
c
ω
Ts
K
s
()=
+a

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© 2005 by CRC Press LLC

Filters (Passive) 111-5
where for a passive filter K £ a and a = w
C
. This transfer function can be realized in several ways including
using either of the two circuits shown in Figure 111.5.
For sinusoidal response the respective transfer functions are
(111.4)
For these filters the passband gain is equal to one and the cutoff frequency is determined by R/L for the
RL filter and 1/RC for the RC filter. The gain ΩT( jw)Ω and phase –T( jw) plots of these circuits are
shown back in Figure 111.1.
A first-order HP filter is given by the following transfer function:
(111.5)
where, for a passive filter, K £ 1 and a is the cutoff frequency. This transfer function can also be realized
in several ways including using either of the two circuits shown in Figure 111.6. For sinusoidal response
the respective transfer functions are
(111.6)

For the LP filters the passband gain is one and the cutoff frequency is determined by R/L for the RL filter
and 1/RC for the RC filter. The gain ΩT( jw)Ω and phase –T( jw) plots of these circuits are shown in
Figure 111.7.
FIGURE 111.5 Single-pole LP filter realizations: (a) RL, (b) RC.
FIGURE 111.6 Single-pole HP filter realizations: (a) RL, (b) RC.
(a) (b)
R
C
L
R
R
C
L
(a)
(b)
R
Tj
RL
jRL
Tj
/RC
j/RC
RL RC
()
()
; ( )
()
w
w
w

w
=
+
=
+
/
/
1
1
Ts
Ks
s
()=
+a
Tj
j
jRL
Tj
j
jRC
RL RC
()
()
; ( )
()
w
w
w
w
w

w
=
+
=
+//1
1586_book.fm Page 5 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
111-6 The Engineering Handbook, Second Edition
Simple RLC Filters
Simple second-order LP, HP, or BP filters can be made using series or parallel RLC circuits. Series or
parallel RLC circuits can be connected to produce the following transfer functions:
(111.7)
where for a series RLC circuit and . w
0
is called the undamped natural frequency
and is related to the cutoff frequency in the HP and LP case and is the center frequency in the BP case.
z is called the damping ratio and determines the nature of the roots of the equation that translates to
how quickly a transition is made from the passband to the stopband. z in the BP case helps define the
bandwidth of the circuit, that is, . Figure 111.8 shows how a series RLC circuit can be
connected to achieve the transfer functions given in Equation 111.7. The gain ΩT( jw)Ω and phase –T( jw)
plots of these circuits are shown in Figure 111.9 through Figure 111.11.
FIGURE 111.7 High-pass filter characteristics showing passband, stopband, and the cutoff frequency, w
c
.
Gain
Phase
45°
0°
90°
A

A
2
0
Passband
Stopband
ω
c
ω
ω
ω
c
√
Tj
K
j
LP
()w
wzwww
=
-+ +
2
00
2
2
Tj
K
j
HP
()w
w

wzwww
=
-
-+ +
2
2
00
2
2
Tj
Kj
j
BP
()w
w
wzwww
=
-+ +
2
00
2
2
w
0
= LC z=
RC
L2
Bw = 2
0
zw

1586_book.fm Page 6 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
Filters (Passive) 111-7
FIGURE 111.8 RLC circuit connections to achieve LP, HP, or BP responses.
FIGURE 111.9 Second-order low-pass gain responses.
FIGURE 111.10 Second-order band-pass gain responses.
R CL
HP BP LP
+ V
OHP
–
+
V
OBP
–+
V
OLP
–
+
V
IN
–
0.01 0.1 1.0 10 100 10000.001
|T(0)|
|T(0)|
|T(0)|
|T(0)|
dB
+ 20 dB
|T(0)|

dB
+ 0 dB
|T(0)|
dB
− 20 dB
|T(0)|
dB
− 40 dB
10
100
Gain
Gain (dB)
10|T(0)|
ζ = 0.5
ω
ω
0
ζ = 5
ζ = 0.05
Asymptote
s
0.01 0.1 1.0 10 100 10000.001
Gain
Gain (dB)
|K|
10 ω
0
K
dB
+ 20dB

ω
0
K
dB
− 20dB
ω
0
K
dB
− 40dB
ω
0
K
dB
+ 0dB
ω
0
ω
0
ω
|K|
10
ω
0
|K|
ω
0
|K|
100 ω
0

B
ζ=5
ζ = 0.5
ζ = 5
B
ζ=0.05
ζ = 0.05
B
ζ=0.5
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© 2005 by CRC Press LLC
111-8 The Engineering Handbook, Second Edition
Compound Filters
Compound filters are higher-order filters obtained by cascading lower-order designs. Ladder circuits are
an important class of compound filters. Two of the more common passive ladder circuits are the constant-
k and the m-derived filters (either of which can be configured using a T-section, p-section, or L-section,
or combinations thereof), the bridge-T network and parallel-T network, and the Butterworth and
Chebyshev realizations. Only the first two known as image-parameter filters will be discussed in this
section. Figure 111.12(a) shows a standard ladder network consisting of two impedances, Z
1
and Z
2
,
organized as an L-section filter. Figure 111.12(b) and Figure 111.12(c) show how the circuit can be
redrawn to represent a T-section or ’-section filter, respectively.
T- and ’-section filters (also referred to as “full sections”) are usually designed to be symmetrical so
that either can have its input and output reversed without changing its behavior. The “L-section” (also
known as a “half section”) is unsymmetrical, and orientation is important. Since cascaded sections “load”
each other, the choice of termination impedance is important. The image impedance, Z
i

, of a symmetrical
filter is the impedance with which the filter must be terminated in order to “see” the same impedance
at its input terminals. In general the image impedance is the desired load or source impedance to which
the filter matches. The image impedance of a filter can be found from
(111.8)
where Z
1O
is the input impedance of the filter with the output terminals open circuited, and Z
1S
is its
input impedance with the output terminals short-circuited. For symmetrical filters the output and input
can be reversed without any change in its image impedance — that is,
(111.9)
The concept of matching filter sections and terminations to a common image impedance permits the
development of symmetrical filter designs.
The image impedances of T- and ’-section filters are given as
FIGURE 111.11 Second-order high-pass gain responses.
0.01 0.1 1.0 10 100 10000.001
|T(∞)|
|T(∞)|
dB
+ 20 dB
|T(∞)|
dB
+ 0 dB
|T(∞)|
dB
− 20 dB
|T(∞)|
dB

− 40 dB
|T(∞)|
10
|T(∞)|
100
Gain Gain (dB)
10|T(∞)|
ζ = 5
ω
0
ω
Asymptotes
ζ = 0.05
ζ = 0.5
ZZZ
iOS
=
11
ZZZ ZZZ
ZZZ
iOS i OS
iii
111 2 22
12
==
==


and
ZZZ ZZZ

iT O S
==+
11 1
2
12
1
4
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© 2005 by CRC Press LLC
Filters (Passive) 111-9
and
(111.10)
These expressions also describe the input (or output) impedance when the filter’s output (or input) is
terminated with appropriate image impedance, i.e., Z
¸
or Z
T
.
The image impedance of an L-section filter, being unsymmetrical, depends on the terminal pair being
calculated. For the L-section shown in Figure 111.12(a), image impedances are
(111.11)
These equations show that the image impedance of an L-section at its input is equal to the image
impedance of a T-section, whereas the image impedance of an L-section at its output is equal to the
image impedance of a ’-section. This relationship is important in achieving an optimum termination
when cascading L-sections with T- and/or ’-sections to form a composite filter.
Since Z
1
and Z
2
vary significantly with frequency, the image impedances of T- and ’-sections will also

change. This condition does not present any particular problem in combining any number of equivalent
filter sections together, since their impedances va4ry equally at all frequencies. But this does make it
difficult to terminate these filters exactly, causing a limitation of these types of filters. However, there is
a frequency within the filter’s passband where the image impedance becomes purely resistive. It is useful
FIGURE 111.12 Ladder networks: (a) standard L-section, (b) T-section, (c) ’-section.
Z
1
2Z
2
2Z
2
Z
1
2Z
2
2Z
2
Z
1
2Z
2
2Z
2
2Z
2
2
2
1
1
Z

1
Z
2
Z
1
Z
2
Z
1
Z
2
2
2
1
1
Z
1
/2
Z
2
2
2
1
1
Z
1
/2
Z
1
/2

Z
2
Z
1
/2
Z
1
/2
Z
2
Z
1
/2
(a)
(b)
(c)
ZZZ
ZZ
ZZZ
iOS’
==
+
11
12
1
2
12
14(/)
ZZZZZ Z
ZZ

ZZZ
Z
iL iT iL i11
2
12 2
12
1
2
12
1
4
14
=+= =
+
=
’

(/)
and
1586_book.fm Page 9 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
111-10 The Engineering Handbook, Second Edition
to terminate the filter with this value of resistance since it provides good matching over much of the
filter’s passband.
To develop the theory of constant-k and m-derived filters, consider the circuit of Figure 111.13. The
current transfer function in the sinusoidal steady state is given by T( jw) = ΩT( jw)Ω–T( jw) = I
2
/I
1
:

(111.12)
where a is the attenuation in dB, b is the phase shift in radians, and g is the image transfer function.
For the circuit shown in Figure 111.13, the following relationship can be derived:
(111.13)
This relationship and those in Equation (111.12) will be used to develop the constant-k and m-derived
filters.
Constant-k Filters
During the 1920s O. Zobel developed an important class of symmetrical filters called constant-k filters
with the conditions that Z
1
and Z
2
are purely reactive — that is, ±X( jw) and
Z
1
Z
2
= k
2
= R
2
(111.14)
In modern references an R replaces the k. Note that the units of k are ohms. The advantage of this type
of filter is that the image impedance in the passband is a pure resistance, whereas in the stopband it is
purely reactive. Hence if the termination is a pure resistance and equal to the image impedance, all the
power will be transferred to the load since the filter itself is purely reactive. Unfortunately, the value of
the image impedance varies significantly with frequency, and any termination used will result in a
mismatch except at one frequency.
In LC constant-k filters, Z
1

and Z
2
have opposite signs, so that . The
image impedances become
(111.15)
Therefore, in the stopband and passband, we have the following relations for standard T- or ’-sections,
where n represents the number of identical sections:
FIGURE 111.13 Circuit for determining the transfer function of a T-section filter.
2
21
1
Z
i
Z
1
/2
Z
2
Z
1
/2
Z
i
V
I
1
I
2
+
−

Tj Tj Tj
Ij
Ij
ee e
j
() () ()

()
()
www
w
w
ab g
=–
===
-
2
1
tanh /g= ZZ
SO11
ZZ jX jX X X R
12 1 2 12
2
=± =+ =m
ZR ZZ Z
R
Z
iT i
= =

’
14
1
12
1
(/)and
/4Z
2

()
1586_book.fm Page 10 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
Filters (Passive) 111-11
(111.16)
The ultimate roll-off of constant-k filters is 20 dB per decade per reactive element or 60 dB per decade
for the T- or ’-section, 40 dB per decade per L-section. Figure 111.14 shows normalized plots of a and
b versus . These curves are generalized and apply to low-pass, high-pass, band-pass, or band-
reject filters. Figure 111.15 shows examples of a typical LP ’-section, an HP T-section, and a BP T-section.
m-Derived Filters
The need to develop a filter section that could provide high attenuation in the stopband near the cutoff
frequency prompted the development of the m-derived filter. O. Zobel developed a class of filters that
had the same image impedance as the constant-k but had a higher attenuation near the cutoff frequency.
The impedances in the m-derived filter were related to those in the constant-k as
FIGURE 111.14 Normalized plots of attenuation and phase angle for various numbers of sections n.
FIGURE 111.15 Typical sections: (a) LP ’-section, (b) HP T-section, (c) BP T-section.
0246810
0
50
100
150

200
250
300
350
400
450
500
550
600
Normalized Frequency (rad/sec)
Normalized Frequency (rad/sec)
Attenuation (dB)
Phase Shift (radians)
0123 5
4
768910
0
5
10
15
20
25
30
35
40
n = 10
n = 10
n = 7
n = 7
n = 4

n = 4
n = 1
n = 1
L
1
/2C
1
/2C
L
2C 2C
L
C
2C
(a)
1
/2L2C
1
/2L
(b)
(c)

cosh /
sin /
Stopband Passband

, , . . . =
aa
bpp b
=- =
=± ± -

-
-
24 0
324
1
12
1
12
nZZ
nn n ZZ
-ZZ
12
4/
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© 2005 by CRC Press LLC
111-12 The Engineering Handbook, Second Edition
(111.17)
where m is a positive constant £ 1. If m = 1 then the impedances reduce to those of the constant-k. Figure
111.16 shows generalized m-derived T- and ’-sections.
The advantage of the m-derived filter is that it gives rise to infinite attenuation at a selectable frequency,
w
˚
, just beyond cutoff, w
C
. This singularity gives rise to a more rapid attenuation in the stopband than
can be obtained using constant-k filters. Equation 111.18 relates the cutoff frequency to the infinite
attenuation frequency.
(111.18)
Figure 111.17 shows the attenuation curve for a single m-derived LP stage for two values of m. The
smaller m becomes, the steeper the attenuation near the cutoff, but also the lesser the attenuation at

higher frequencies.
Constant-k filters have image impedance in the passband that is always real but that varies with
frequency, making the choice of an optimum termination difficult. The impedance of an m-derived filter
also varies, but how it varies depends on m. Figure 111.18 shows how the impedance varies with frequency
(both normalized) and m. In most applications, m is chosen to be 0.6, keeping the image impedance
nearly constant over about 80% of the passband.
FIGURE 111.16 m-derived filters: (a) T-section, (b) ’-section.
1
/2mZ
1
(b)(a)
1
/2mZ
1
(1 − m)Z
1
4m
Z
2
m
(1 − m)Z
1
4m
mZ
1
2Z
2
m
2Z
2

m
ZmZ Z
m
m
Z
m
Z
mk m k k11 2
2
12
1
4
1
==
-
Ê
Ë
Á
ˆ
¯
˜
+ and
m
C
=-
•
1
2
2
w

w
1586_book.fm Page 12 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
Filters (Passive) 111-13
Defining Terms
Bridge-T network — A two-port network that consists of a basic T-section and another element con-
nected so as to “bridge across” the two arms. Such networks find applications as band rejection
filters, calibration bridges, and feedback networks.
Butterworth filters — Ladder networks that enjoy a unique passband frequency response characteristic
that remains very constant until near the cutoff, hence the designation “maximally flat.” This
FIGURE 111.17 Attenuation curves for a single-stage filter with m = 0.6 and m = 0.9.
FIGURE 111.18 Z
iT
/R and Z
i’
/R versus normalized frequency for various values of m.
Attenuation (dB)
100
75
50
25.57
12.03
25
0
0 0.5 1 1.25 1.5 2 2.29 2.5
ω/ω
0
3 3.5 4 4.5
5
m = 0.9

m = 0.6
Z
in
/R
Z
in
/R
2
1.5
1
0.5
0
2
1.5
1
0.5
0
0 0.2 0.4 0.6
ω/ω
0
ω/ω
0
0.8 1
1.2510.750.50.250
m = 1
m = 0
m = 0.4
m = 0.6
m = 0.8
m = 1

m = 0.8
m = 0.6
m = 0.4
m = 0
1586_book.fm Page 13 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
111-14 The Engineering Handbook, Second Edition
filter has its critical frequency remain fixed regardless of the number of stages employed. It
obtains this characteristic by realizing a transfer function built around a Butterworth polyno-
mial.
Chebyshev filters — A variant of the Butterworth design that achieves a significantly steeper transition
band about its critical frequency for the same number of poles. Although the Chebyshev filter
also maintains the integrity of its critical frequency regarding the number of poles, it trades the
steeper roll-off for a fixed ripple — usually specified as 1 dB or 3 dB — in the passband.
Chebyshev filters are also called equal-ripple or stagger-tuned filters. They are designed by
realizing a transfer function using a Chebyshev polynomial.
Parallel-T networks — A two-port network that consists of two separate T-sections in parallel with only
the ends of the arms and the stem connected. Parallel-T networks have applications similar to
those of the bridge-T but can produce narrower attenuation bandwidths.
References
Herrero, J. L. and Willoner, G. 1966. Synthesis of Filters, Prentice Hall, Englewood Cliffs, NJ.
Thomas, R. E. and Rosa, A. J. 2004. The Analysis and Design of Linear Circuits, John Wiley & Sons,
Hoboken, NJ.
Van Valkenburg, M. E. 1955. Two-terminal-pair reactive networks (filters). In Network Analysis, Prentice
Hall, Englewood Cliffs, NJ.
Weinberg, L. 1962. Network Analysis and Synthesis, W. L. Everitt (ed.) McGraw-Hill, New York.
Williams, A. B. 1981. Electronic Filter Design Handbook, McGraw-Hill, New York.
Zobel, O. J. 1923. Theory and Design of Uniform and Composite Electric Wave Filters. Bell Telephone
Syst. Tech. J. 2:1.
Further Information

Huelsman, L. P. 1993. Active and Passive Analog Filter Design — An Introduction, McGraw-Hill, New York.
Good current introductory text covering all aspects of active and passive filter design.
Sedra, A. S. and Brackett, P. O. 1978. Filter Theory and Design: Active and Passive, Matrix, Beaverton, OR.
Modern approach to filter theory and design.
1586_book.fm Page 14 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC

112

Power Distribution

112.1 Equipment

112.2 System Divisions and Types

112.3 Electrical Analysis, Planning, and Design

Phase Balancing • Fault Analysis • Protection and
Coordination • Reliability Analysis

112.4 System Control

112.5 Operations

The function of power distribution is to deliver to consumers economic, reliable, and safe electrical
energy in a manner that conforms to regulatory standards. Power distribution systems receive electric
energy from high-voltage transmission systems and deliver energy to consumer service-entrance equip-
ment. Systems typically supply alternating current at voltage levels ranging from 120 V to 46 kV.
Figure 112.1 illustrates aspects of a distribution system. Energy is delivered to the distribution substa-
tion (shown inside the dashed line) by three-phase transmission lines. A transformer in the substation

steps the voltage down to the distribution primary system voltage — in this case, 12.47 kV. Primary
distribution lines leave the substation carrying energy to consumers. The substation contains a breaker
that may be opened to disconnect the substation from the primary distribution lines. If the breaker is
opened, outside the substation there is normally an open supervisory switch that may be closed in order
to provide an alternate source of power for the customers normally served by the substation. The
substation also contains a capacitor bank used for either voltage or power factor control.
Four types of customers, along with representative distribution equipment, are shown in Figure 112.1.
A set of loads requiring high reliability of service is shown being fed from an underground three-phase
secondary network cable grid. A single fault does not result in an interruption to this set of loads. A
residential customer is shown being supplied from a two-wire, one-phase overhead lateral. Commercial
and industrial customers are shown being supplied from the three-phase, four-wire, overhead primary
feeder. At the industrial site, a capacitor bank is used to control the power factor. Except for the industrial
customer, all customers shown have 240/120 V service. The industrial customer has 480Y/277 V service.
For typical electric utilities in the U.S., investment in distribution ranges from 35 to 60% of total
capital investment.

112.1 Equipment

Figure 112.1 illustrates a typical arrangement of some of the most common equipment. Equipment may
be placed into the general categories of transmission, protection, and control.
Arresters protect distribution system equipment from transient over-voltages due to lightning or
switching operations. In over-voltage situations the arrester provides a low-resistance path to ground for
currents to follow.

Robert Broadwater

Virginia Polytechnic Institute and
State University

Albert Sargent

Entergy Corporation

Murat Dilek

Electrical Distribution Design, Inc.

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© 2005 by CRC Press LLC

112

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The Engineering Handbook, Second Edition

Capacitor banks are energy storage devices primarily used to control voltage and power factor. System
losses are reduced by the application of capacitors.
Conductors are used to transmit energy and may be either bare or insulated. Bare conductors have
better thermal properties and are generally used in overhead construction where contact is unlikely.
Insulated cables are used in underground/conduit construction and in overhead applications where
minimum right-of-way is available. Concentric neutral and tape-shielded cables provide both a phase
conductor and a return path conductor in one unit.

FIGURE 112.1

Distribution system schematic.
Fault
Limiters
Underground

Cable
High
Reliability
Load
High
Reliability
Load
Distribution Substation
3-Phase, 4-Wire Overhead
Combination
Switch
3-Phase, 4-Wire Lateral
3-Phase
Pad-Mounted
Distribution
Transformer
Commercial Customer
Network
Transformer
Network
Protector
Secondary Network
Network
Transformer
Network
Protector
Feed from alternate source
Supervisory
Switch
(Normally Open)

Voltage Regulator
Power
Fuse
Capacitor
Bank
Three Phase
Pad Mounted
Distribution
Transformer
Switch Gear
Industrial Customer
Fused
Cutout
1-Phase, 2-Wire Lateral
Single Phase
Pole Mounted
Distribution
Transformer
Residential Customer
Transmission
Line
Transmission
Line
Power
Transformer
Capacitor
Bank
Disconnect
Disconnect
Disconnect

Disconnect
12.47 KV Bus
Breaker
Recloser

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Power Distribution

112

-3

Distribution lines are made up of conductors and are classified according to primary voltage, the
number of phases, number of conductors, and return path. The three-phase, four-wire, multi-grounded
system is the most common primary system, where one conductor is installed for each of the three phases
and the fourth conductor is a neutral that provides a return current path. Multi-grounded means that
the neutral is grounded at many points, so that the earth provides a parallel path to the neutral for return
current. Three-phase, three-wire primary systems, or delta-connected systems, are rarely used because
faults therein are more difficult to detect. A lateral is a branch of the system that is shorter in length,
more lightly loaded, or has a smaller conductor size than the primary feeder.
Distribution transformers step the voltage down from the primary circuit value to the customer
utilization level, thus controlling voltage magnitude. Sizes range from 1.5 to 2500 kVA. Distribution
transformers are installed on poles, ground-level pads, or in underground vaults. A specification of 7200/
12,470Y V for the high-voltage winding of a single-phase transformer means the transformer may be
connected in a line-to-neutral “wye” connection for a system with a line-to-line voltage of 12,470 V or
in a line-to-line “delta” connection for a system with a line-to-line voltage of 7200 V. A specification of
240/120 V for the low-voltage winding means the transformer provides a three-wire connection with
120 V mid-tap voltage and 240 V full-winding voltage. A specification of 480Y/277 V for the low voltage

winding means the winding is permanently wye-connected with a fully insulated neutral avilable for a
three-phase, four-wire service to deliver 480 V line-to-line and 277 V line-to-neutral.
Distribution substations consist of one or more step-down power transformers configured with switch
gear, protective devices, and voltage regulation equipment for the purpose of supplying, controlling,
switching, and protecting the primary feeder circuits. The voltage is stepped down for safety and flexibility
of handling in congested consumer areas. Over-current protective devices open and interrupt current
flow in order to protect people and equipment from fault current. Switches are used for control to
interrupt or redirect power flow. Switches may be operated manually, automatically with PLC control,
or remotely with supervisory control. Switches are usually rated to interrupt load current and may be
either pad or pole mounted.
Power transformers are used to control and change voltage level. Power transformers equipped with
tap-changing mechanisms can control secondary voltage over a typical range of plus or minus 10%.
Voltage regulators are autotransformers with tap-changing mechanisms that may be used throughout
the system for voltage control. If the voltage at a remote point is to be controlled, then the regulator can
be equipped with a line drop compensator that may be set to regulate the voltage at the remote point
based upon local voltage and current measurements. Modern microprocessor-based controls enable
regulators and line capacitors to work together to provide optimal voltage regulation.

112.2 System Divisions and Types

Distribution transformers separate the primary system from the secondary. Primary circuits transmit
energy from the distribution substation to customer distribution transformers. Three-phase distribution
lines that originate at the substation are referred to as primary feeders or primary circuits. Primary feeders
are illustrated in Figure 112.1. Secondary circuits transmit energy from the distribution transformer to
the customer’s service entrance. Line-to-line voltages range from 208 to 600 V.
Radial distribution systems provide a single path of power flow from the substation to each individual
customer. This is the least costly system to build and operate and thus the most widely used.
Primary networks contain at least one loop that generally may receive power from two distinct sources.
This design results in better continuity of service. A primary network is more expensive than the radial
system design because more protective devices, switches, and conductors are required.

Secondary networks are normally underground cable grids providing multiple paths of power flow to
each customer. A secondary network generally covers a number of blocks in a downtown area. Power is
supplied to the network at a number of points via network units, consisting of a network transformer
in series with a network protector. A network protector is a circuit breaker connected between the
secondary winding of the network transformer and the secondary network itself. When the network is

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© 2005 by CRC Press LLC

112

-4

The Engineering Handbook, Second Edition

operating properly, energy flows into the network. The network protector opens when reverse energy
flow is detected, such as may be caused by a fault in the primary system.

112.3 Electrical Analysis, Planning, and Design

The distribution system is planned, designed, constructed, and operated based on the results of electrical
analysis. Generally, computer-aided analysis is used.
Line impedances are needed by most analysis applications. Distribution lines are electrically unbal-
anced due to loads, unequal distances between phases, dissimilar phase conductors, and single-phase or
two-phase laterals. Currents flow in return paths due to the imbalance in the system. Three-phase, four-
wire, multigrounded lines have two return paths — the neutral conductor and earth. Three-phase,
multigrounded concentric neutral cable systems have four return paths. The most accurate modeling of
distribution system impedance is based upon Carson’s equations. With this approach a 5

¥

5 impedance
matrix is derived for a system with two return paths, and a 7

¥

7 impedance matrix is derived for a
system with four return paths. For analysis, these matrices are reduced to 3

¥

3 matrices that relate phase
voltage drops (i.e., , , ) to phase currents (i.e., , , ) as indicated by
Load analysis forms the foundation of system analysis. Load characteristics are time varying and depend
on many parameters, including connected consumer types and weather conditions. The load demand
for a given customer or group of customers is the load averaged over an interval of time, say 15 min.
The peak demand is the largest of all demands. The peak demand is of particular interest since it represents
the load that the system must be designed to serve. Diversity relates to multiple loads having different
time patterns of energy use. Due to diversity, the peak demand of a group of loads is less than the sum
of the peak demands of the individual loads. For a group of loads,
Loads may be modeled as either lumped parameter or distributed. Lumped parameter load models
include constant power, constant impedance, constant current, voltage-dependent, and combinations
thereof. Generally, equivalent lumped parameter load models are used to model distributed loads.
Consider the line section of length L shown in Figure 112.2(a), with a uniformly distributed load current
that varies along the length of the line as given by
The total load current drawn by the line section is thus
I

L

=

I

2



-

I

1


An equivalent lumped parameter model for the uniformly distributed current load is shown in Figure
112.2(b).
Metered load values are used for analysis when available. Otherwise, estimated loads are calculated
using kWHr-to-Peak-KW conversion factors, daily load shapes, and diversity factor curves based on
DV
A
DV
B
DVC I
A
I
B
I

C
D
D
D
V
V
V
ZZZ
ZZZ
ZZZ
I
I
I
A
B
C
AA AB AC
BA BB BC
CA CB CC
A
B
C
È
Î
Í
Í
Í
Í
˘
˚

˙
˙
˙
˙
=
È
Î
Í
Í
Í
Í
˘
˚
˙
˙
˙
˙
È
Î
Í
Í
Í
Í
˘
˚
˙
˙
˙
˙

Diversity factor =
Sum of individual load peaks
Group peak
ix
II
L
xI()=
-
+
21
1

1586_book.fm Page 4 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC

Power Distribution

112

-5

customer class. These estimation parameters are derived from load research data gathered from a set of
randomly selected test customers. A representative diversified load curve for a residential customer type
is shown in Figure 112.3, and a representative diversity factor curve for a residential customer type is
shown in Figure 112.4.
Load forecasting is concerned with determining load magnitudes during future years from customer
growth projections. Short-range forecasts generally have time horizons of approximately five years,
whereas long-range forecasts project to around 20 years.

Power flow analysis determines system voltages, currents, and power flows. Power flow results are
checked to ensure that voltages fall within allowable limits, that equipment overloads do not exist, and
that phase imbalances are within acceptable limits. For primary and secondary networks, power flow

FIGURE 112.2

(a)



Line section model with distributed load current; (b) lumped parameter equivalent model.

FIGURE 112.3

Representative diversified load curve for a residential customer type.
L
0.75 L
I
1
I
1
I
2
I
2
(a) Line Section Model With Distributed Load Current
(b) Lumped Parameter Equivalent Model
Distributed Load Current
2I
L

3
I
L
3
0
0
24 681012 14 16 18 20 22 24
300
600
900
1500
1200
Diversified kW
Hours
Peak Day of the Month Hourly Load Curves

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112

-6

The Engineering Handbook, Second Edition

methods used in transmission system analysis are applied. For radially operated systems, the ladder
method is used. The actual implementation of the ladder method may vary with the type of load models
used. All ladder load flow methods assume the substation bus voltage is known. An algorithm for the
ladder method consists of the following five steps:
1.

Step 1.

Assume a value for all node voltages throughout the circuit. Generally, assumed voltages
are set equal to the substation voltage.
2.

Step 2.

At each load in the circuit, calculate the current from the known load value and assumed
voltage.
3.

Step 3.

Starting at the ending nodes, sum load currents to obtain line section current estimates,
performing summation until the substation is reached.
4.

Step 4.

Having estimates of all line section currents, start at the substation and calculate line
section voltage drops and new estimates of node voltages.
5.

Step 5.

Compare new node voltages with estimates of previous iteration values. The algorithm
has converged if the change in voltage is sufficiently small. If the algorithm has not converged,
return to Step 2.

Dynamic load analysis includes such studies as motor-starting studies. Rapid changes in large loads
can result in large currents, with a resultant drop in system voltage. If the dip in voltage is too large or
too frequent, then other loads are adversely affected, such as in an annoying flicker of lights. This study
generally employs a power flow calculation that is run at a number of points along the dynamic charac-
teristic of the load.
Planning involves using load forecasting and other analysis calculations to evaluate voltage level,
substation locations, feeder routes, transformer/conductor sizes, voltage/power factor control, and res-
toration operations. Decisions are based upon considerations of efficiency, reliability, peak demand, and
life cycle cost.

Phase Balancing

Phase balancing is used to balance the current or power flows on the different phases of a line section.
This results in improved efficiency and primary voltage level balance. The average current in the three
phases is defined as
The maximum deviation from I

avg

is given by

FIGURE 112.4

Representative diversity factor curve for a residential customer type.
6111621263136411
1
1.20
1.40
1.60
kW Load Scaling

Customers
I
III
ABC
avg
=
++
3

1586_book.fm Page 6 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC

Power Distribution

112

-7

Phase imbalance is defined as

Fault Analysis

Fault analysis provides the basis for protection system design. Thus, in the model used to calculate fault
currents, load currents are neglected. Sources of fault current are the substation bus, distributed resource
generators, and large synchronous motors on the feeder or neighboring feeders. A variety of fault
conditions are considered at each line section, including three-phase-to-ground, single-phase-to-ground,
and separate phases contacting one another. In performing the calculations, both bolted (i.e., zero-
impedance) faults and faults with impedance in the fault path are considered. Of interest are the maxi-
mum and minimum phase and return path fault currents, as well as the fault types that result in these
currents.

At a multi-phase grounded node in a linear distribution system, post-fault voltages are related to pre-
fault voltages and fault currents as given by

V

f

=

V

0

–

Z

th

I

(112.1)
where

V

f

denotes phase voltages (voltages between phase A and ground, B and ground, and C and ground)
of the node during the fault,

V

0

is the array of phase voltages before the fault occurs,

I

is the array of
fault currents that will flow out of the phases of the node during the fault, and

Z

th

(a 3

¥

3 matrix)
represents the phase Thevenin matrix looking into the node.
Once

Z

th

and pre-fault voltages at the node are available,

V

f

can be written in terms of

I

depending
upon



conditions imposed by the fault. Then Equation (112.1) can be solved for

I

. The pre-fault system
model represents the system behavior before the fault occurs. On the pre-fault model, a power flow
calculation may be used to obtain the voltages

V

0

.
The post-fault model represents the system behavior during the fault. For fault calculations, the circuit
model used is modified in several ways from the pre-fault model. Usually, load currents are neglected in

the post-fault model, and instead superposition is used to add load currents obtained from the pre-fault
power flow analysis to fault currents. For the fault calculations, the circuit model is assumed to be linear.
Other changes for the post-fault circuit analysis include neglecting slow-acting control devices (such as
substation transformer tap changers) that do not have time to react during the time of the fault; and
inserting appropriate Thevenin equivalent source impedances, representing the Thevenin impedance seen
by the distribution substation looking back into the transmission or subtransmission system. Using the
post-fault circuit model assumptions, a power flow calculation using constant current injections at the
fault point may be used to perform the fault calculations. This is the approach described here.
The calculation of

Z

th

may be performed by inserting a small test load sequentially at every individual
phase of the faulted node. Prior to any test load insertion, the phase voltages of the node are obtained
from a power flow solution. Let these voltages be

V

i

. After inserting a test load, the power flow is used
again to obtain the current flowing into the test load and the voltages at all phases of the node. Ratios
of changes in

V

i

to the current drawn by the test load constitute the columns of

Z

th

. For instance, if the
test load is inserted between phase A and ground at grounded node

N,

the results calculated are the first
column of

Z

th

.
To elaborate, refer to Figure 112.5 where a grounded node

N

is considered. Here, a general node at
which any phase may exist is assumed. The power flow is run on the post-fault system model, and phase-
to-neutral voltages at node

N

are obtained as

V

an

,

V

bn

, and

V

cn

for the phases A, B, and C, respectively
DIIIIIII
ABCdev avg avg avg
maximum of =
{}
,,
Phase imbalance
dev
avg
=
DI
I

1586_book.fm Page 7 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC

112

-8

The Engineering Handbook, Second Edition

[Figure 112.5(a)]. The neutral

n

is regarded to be the same as ground. A test load will be inserted between
A and

n

, B and

n

, and C and

n

sequentially [Figure 112.5(b) through Figure 112.5(d)]. During each load
insertion, line currents and phase-to-neutral voltages may be obtained from a power flow calculation.
The elements

z

ij

of

Z

th

may be determined in the following manner:
, ,
, ,
, ,

Z

th

represents the relationship between the voltage changes and the current changes at

N.

Suppose phase-
to-neutral voltages at

N

before the fault are . Assume that a fault occurs at

N

and causes
currents to flow out of phases A, B, and C, respectively, resulting in phase-to-neutral voltages
.
Then voltage changes at

N

are related to the currents drawn, via

Z

th

as
(112.2)
where
Equation (112.2) denotes a general case. Suppose node

N

is a double-phase location having phases A
and B but no phase C. Then, all the elements in the third row and third column of

Z

th

are zero.

FIGURE 112.5

Constructing

Z

th

at a grounded node N. (a) Voltages before inserting any test load. (b) A test load
being inserted between phase A and ground. (c) A test load being inserted between phase B and ground. (d) A test
load being inserted between phase C and ground.
Post-fault
System
Model
Post-fault
System
Model
Phases present
at node N
a
+
b
+
c
+
n
V
an
V

bn
V
cn
V
an
V
an
V
bn
V
bn
V
cn
V
cn
Post-fault
System
Model
Post-fault
System
Model
n
I
a

= 0
I
a

= 0

I
b

= 0
I
b

= 0
I
a

=
I
a
I
c

= 0
I
c

= 0
I
b

=
I
b
(2)
(2)

(2)
(2)
V
an
(1)
(1)
(1)
(1)
V
bn
V
cn
(3)
(3)
(3) (3)
(a)
(b)
(c)
(d)
−
−
n
−
n
−
I
c

=
I

c
z
VV
I
an an
a
11
1
1
=
-
()
()
z
VV
I
bn bn
a
21
1
1
=
-
()
()
z
VV
I
cn cn
a

31
1
1
=
-
()
()
z
VV
I
an an
b
12
2
2
=
-
()
()
z
VV
I
bn bn
b
22
2
2
=
-
()

()
z
VV
I
cn cn
b
32
2
2
=
-
()
()
z
VV
I
an an
c
13
3
3
=
-
()
()
z
VV
I
bn bn
c

23
3
3
=
-
()
()
z
VV
I
cn cn
c
33
3
3
=
-
()
()
VV V
an
i
bn
i
cn
i
,, and
II I
ab c
, and

VV V
an
f
bn
f
cn
f
,, and
D
D
D
V
V
V
zzz
zzz
zzz
I
I
I
an
bn
cn
a
b
c
È
Î
Í
Í

Í
˘
˚
˙
˙
˙
=
È
Î
Í
Í
Í
˘
˚
˙
˙
˙
È
Î
Í
Í
Í
˘
˚
˙
˙
˙
11 12 13
21 22 23
31 32 33

DVVV
kn kn
i
kn
f
=- = for k a,b,c.
1586_book.fm Page 8 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
Power Distribution 112-9
Var ious fault cases at N are shown in Figure 112.6. A general case of a three-phase-to-ground fault at
N is represented in Figure 112.6(a). Here, each phase has its own fault impedance (Z
a
, Z
b
, and Z
c
for
phases A, B, and C, respectively) to the common point p. Z
f
is the impedance between p and n. Consider
solving for a three-phase-to-ground fault. Let V
i
kn
and V
f
kn
denote pre-fault and post-fault phase-to-
ground voltages of phase k, respectively. Then the boundary conditions are:
, (112.3)

, (112.4)
, (112.5)
. (112.6)
First using Equation (112.3) in Equation (112.4), Equation (112.5), and Equation (112.6), then sub-
stituting Equation (112.4), Equation (112.5), and Equation (112.6) into Equation (112.2) gives the fault
currents as
Any fault event imposes a set of boundary conditions. Initial voltages (pre-fault voltages) can be readily
calculated from the power flow. The final voltages are expressed under the boundary conditions in terms
of the fault currents and fault impedances. Then, Equation (112.2) is solved for fault currents. Using this
approach, fault currents for cases b through d shown in Figure 112.6 may be evaluated. For an ungrounded
node, the phase-to-phase voltages instead of phase-to-neutral voltages are employed.
FIGURE 112.6 Var ious faults at a grounded node N. (a) Three-phase fault. (b) Phase-to-phase fault. (c) Double-
phase-to-ground fault. (d) Single-phase-to-ground fault.
n −
n − n −
n −
(a)
(b)
(c) (d)

Z
a
Z
b
Z
c
Z
b
Z
c

I
a
I
a
I
a
I
b
I
b
I
c
I
c
= 0
I
c
I
b
I
a
= 0
I
c
= 0
I
b
= 0

I

f
I
f
I
f
I
f

Z
f
Z
f
Z
f

Z
f
+
+
+
p
p

Post-fault
System
Model
Post-fault
System
Model
Post-fault

System
Model
Post-fault
System
Model
V
an
f
V
an
f
V
bn
f
V
cn
f
V
an
f
V
bn
f
V
cn
f
V
an
f
V

bn
f
V
cn
f
V
bn
f
V
cn
f
I III
fabc
=++
DVVVV IZIZ
an an
i
an
f
an
i
aa f f
=-=- +
()
DVVVV IZIZ
bn bn
i
bn
f
bn

i
bb f f
=-=- +
()
DVVVV IZIZ
cn cn
i
cn
f
cn
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cc f f
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zZZ zZ zZ
zZ zZZ zZ
zZ zZ zZZ
V
V
V
a
b
c
af f f
fbff
ffcf
an

i
bn
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cn
i
È
Î
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Í
Í
˘
˚
˙
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=
++ + +
++++
++++
È
Î
Í
Í
Í
˘
˚
˙
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È

Î
Í
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˘
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11 12 13
21 22 23
31 32 33
-1

1586_book.fm Page 9 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC
112-10 The Engineering Handbook, Second Edition
Protection and Coordination
Over-current protection is the most common protection applied to the distribution system. With over-
current protection, the protective device trips when a large current is detected. The time to trip is a
function of the magnitude of the fault current. The larger the fault current is, the quicker the operation.
Var ious types of equipment are used. A circuit breaker is a switch designed to interrupt fault current,
the operation of which is controlled by relays. An over-current relay, upon detecting fault current, sends
a signal to the breaker to open. A recloser is a switch that opens and then recloses a number of times
before finally locking open. A fuse is a device with a fusible member, referred to as a fuse link, which in
the presence of an over current melts, thus opening up the circuit.
Breakers may be connected to reclosing relays, which may be programmed for a number of opening
and reclosing cycles. With a recloser or a reclosing breaker, if the fault is momentary, then the power
interruption is also momentary. If the fault is permanent, then after a specified number of attempts at
reclosing the device locks open. Breakers are generally more expensive than comparable reclosers. Breakers

are used to provide more sophisticated protection, which is available via choice of relays. Fuses are
generally used in the protection of laterals.
Protective equipment sizing and other characteristics are determined from the results of fault analysis.
Moving away from the substation in a radial circuit, both load current and available fault current decrease.
Protective devices are selected based on this current grading. Protective devices are also selected to have
different trip-delay times for the same fault current. With this time grading, protective devices are
coordinated to work together such that the device closest to a permanent fault clears the fault. Thus
reclosers can be coordinated to protect load-side fuses from damage due to momentary faults.
Reliability Analysis
Reliability analysis involves determining indices that relate to continuity of service to the customer.
Reliability is a function of tree conditions, lightning incidence, equipment failure rates, equipment repair
times, and circuit design. The reliability of a circuit generally varies from point to point due to protection
system design, placement of switches, and availability of alternative feeds. Many indices are used in
evaluating system reliability. Common ones include system average interruption frequency index (SAIFI),
system average interruption duration index (SAIDI), customer average interruption frequency index
(CAIFI), and customer average interruption duration index (CAIDI) as defined by
SAIFI =
Total number of customer interruptions
Total number of customers served
SAIDI =
Sum of customer interruption durations
Total number of customers
CAIFI =
Total number of customer interruptions
Total number of customers affected
CAIDI =
Sum of customer interruption durations
Total number of customers affected
1586_book.fm Page 10 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC

Power Distribution 112-11
112.4 System Control
Voltage control is required for proper operation of customer equipment. For instance, in the U.S., “voltage
range A” for single-phase residential users specifies that the voltage may vary at the service entrance from
114/228 V to 126/252 V. Regulators, tap-changing under load transformers, and switched capacitor banks
are used in voltage control.
Power factor control is used to improve system efficiency. Due to the typical load being inductive,
power factor control is generally achieved with fixed and/or switched capacitor banks.
Power flow control is achieved with switching operations. Such switching operations are referred to
as system reconfiguration. Reconfiguration may be used to balance a load among interconnected distri-
bution substations. Such switching operations reduce losses while maintaining proper system voltage.
Load control may be achieved with voltage control and also by remotely operated switches that
disconnect load from the system. Generally, load characteristics are such that if the voltage magnitude
is reduced, then the power drawn by the load will decrease for some period of time. Load control with
remotely operated switches is also referred to as load management.
Effective system control is essential to provide adequate power quality. Power quality may be defined
as the absence of service interruptions, voltage dips and sags, and voltage spikes and surges. Proper
control of system voltage is more critical now than ever before because many microprocessor-based
controls and adjustable speed drives have voltage tolerances less than 10%.
112.5 Operations
The operations function includes system maintenance, construction, and service restoration. Mainte-
nance, such as trimming trees to prevent contact with overhead lines, is important to ensure a safe and
reliable system. Interruptions may be classified as momentary or permanent. A momentary interruption
is one that disappears very quickly — for instance, a recloser operation due to a fault from a tree limb
briefly touching an overhead conductor. Power restoration operations are required to repair damage
caused by permanent interruptions.
While damaged equipment is being repaired, power restoration operations often involve reconfigura-
tion in order to restore power to interrupted areas. With reconfiguration, power flow calculations may
be required to ensure that equipment overloads are not created from the switching operations.
Defining Terms

Current return path — The path that current follows from the load back to the distribution substation.
This path may consist of either a conductor (referred to as the neutral) or earth, or the parallel
combination of a neutral conductor and the earth.
Fault — A conductor or equipment failure or unintended contact between conductors or between con-
ductors and grounded objects. If not interrupted quickly, fault current can severely damage
conductors and equipment.
Phase — Relates to the relative angular displacement of the three sinusoidally varying voltages produced
by the three windings of a generator. For instance, if phase A voltage is 120– 0˚ V, phase B
voltage 120– -120˚ V, and phase C voltage 120– 120˚ V, the phase rotation is referred to as
ABC. Sections of the system corresponding to the phase rotation of the voltage carried are
commonly referred to as phase A, B, or C.
Tap-changing mechanism — A control device that varies the voltage transformation ratio between the
primary and secondary sides of a transformer. The taps may only be changed by discrete
amounts, say 0.625%.
1586_book.fm Page 11 Monday, May 10, 2004 3:53 PM
© 2005 by CRC Press LLC