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# Which has fourier series such that ntthe fourier transform

300

Chapter 9

t
e α
for t 0 , ( α 0

lim
α→0

e

0

for t 0 , ( α 0
1 δ ( ) 1 Sign
2 2 (9.205)
1 δ ( )

1

2 j

2 π f

V e n j n
(9.206)
T

V f ( )

∫��

v t e j 2 π ft

n���

V e n j 2 π n t e j 2 π ft
T

V n ∫ ��� e j 2 π n t e j 2 π ft
T
V n

∫��

e j 2 π ( f n ) t dt
T (9.207)
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Frequency Domain Circuit Analysis

301

1 e j 2 πft dt
(9.208)
j 2 π ( f n ) t n

j f T ) δ ⎜⎜ f
∫ � ⎜⎜⎝ T

⎟⎟⎠

V f ( ) �
V δ⎛⎜ f n
(9.210)
n��� n ⎜⎜⎝ T

Example 9.14
Determine the spectrum V ( f ) of the periodic voltage waveform, v ( t ) of Figure 9.35 with τ�T /3.

Solution
From equations 9.117 and 9.210 we can write V ( f ) as follows:

V f ( ) �
V A τ sinc ⎛⎜ n τ ⎞⎟⎟⎟⎟⎠ δ ⎛⎜ f n
n��� T ⎜⎜⎝ T ⎜⎜⎝ T
(9.211)

V A sinc ⎛⎜ n ⎞⎟⎟⎟⎟⎠ δ ⎛⎜⎜⎜⎝ f n
n��� 3 ⎜⎜⎝ 3 T

9.3.6.6 Rayleigh’s Energy Theorem

This theorem states that the energy, Ex , of a signal x ( t ) can be calculated from its spectrum X( f ) according to the following equation:

E x
x t ( ) 2 dt
| X f ( ) | 2
(9.212)
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