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V
RY
+ 
V
YB
+ 
V
BR
=
0 implies that line voltage 
phasors in a three-phase system will have to form a closed triangle.]
30. The line voltages of an ideal three-phase unbalanced source are measured to be 420 V rms, 390 V 
rms and 380 V rms between R–Y, Y–B and B–R, respectively. This source drives a balanced RLC 
load through a 3-wire connection with wire-impedance of 1
+ 
j3 
W
in each wire. If the magnitude 
of positive sequence component of line voltage across load is found to be 380 V rms find the 
magnitude of negative sequence component of line voltage across the load.

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Introduction 
9.1
D y n a m i c C i r c u i t s
w i t h P e r i o d i c I n p u t s –
A n a l y s i s b y F o u r i e r 
S e r i e s
ChAPter ObjeCtIveS
• (i) To explain how a periodic waveform can be expanded in terms of sinusoids and why such 
an expansion is necessary, (ii) to show how such an expansion may be obtained for a given 
periodic waveform; and (iii) to show how the expansion can be used to solve for the forced 
response of a circuit. 
• To discuss the difference between steady-state response and forced response of a circuit to 
set the background for application of sinusoidal expansion of periodic waveforms in Circuit 
Analysis.
• To bring out the important properties of Fourier Series expansion through solved examples.
IntroductIon
This chapter goes into the determination of steady-state response component in the output of 
dynamic circuits when they are excited by a periodic input waveform. The problem of expressing a 
periodic waveform as a sum of pure sinusoidal components is addressed first. Subsequently, the use 
of frequency response data to solve for forced response when the forcing function is such a sum of 
sinusoidal components will be taken up.
Chapter 
9

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9.2
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
A waveform 
v 
(
t
) is periodic on 
t
with a period of 
T
if and only if the following condition is 
satisfied by it for all 
t 
and 
n
where 
n 
is a positive integer.
v t
v t
nT
t
n n
( )
(
),
;
,
,
,
.
=
±
=
for any 
and for any 
1 2 3
…
This implies that the values of v(t) at similarly positioned time points, at an interval of T seconds 
between adjacent points, will be same.
Obviously, if v(t) is periodic it must extend from 
-∞
to 
+∞
in the time-axis, since the value of integer n 
in the definition of periodicity is not limited to any finite number. Therefore, it is equally obvious that, there 
is no strictly periodic waveform in nature. All waveforms in electrical circuits start at some time instant 
and stop at some other time instant. Hence, all practical circuit waveforms are necessarily aperiodic. 
Fourier series expansion deals with periodic waveforms. It resolves a periodic waveform into pure 
sinusoidal components. Equivalently, it expands the periodic waveform as a linear combination of 
infinitely many harmonically related sinusoidal waveforms. Two sinusoids are harmonically related if 
their frequencies are integer multiples of some common frequency value. Fourier series expresses the 
periodic waveform as a sum of infinitely many sinusoids with frequencies which are integer multiples 
of the frequency of that waveform. This frequency is called the fundamental frequency. The sinusoidal 
component that is at the same frequency as that of the periodic waveform is termed as the fundamental 
component and all the other sinusoidal components with frequencies which are integral multiples of 
this frequency are called the harmonic components. 
Granted that a periodic waveform can be expanded in this manner and that frequency response 
information for a circuit can then be employed to find the steady-state response of the circuit for such 
a periodic input, two questions come up at this point.
• Why the obsession with sinusoids? Is it because a periodic waveform can be expanded only in 
terms of sinusoids?
• If a periodic waveform is only a mathematical entity with no corresponding physical counterpart, 
why bother to study the steady-state response of circuits to such a hypothetical input?