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9.2
the exPonentIal fourIer serIes
Let v(t) be a periodic function of time and let it satisfy the conditions required to be satisfied for its
expansion in terms of sinusoids to exist. Let the period of v(t) be T s and let its angular frequency
be
w
o
rad/s
(
).
w
p
o
=
2
T Then Fourier theorem, in effect, states that v(t) may be represented by the
infinite series
v t
v e
v e
v e
v
v e
v
j
t
j
t
j
t
j
t
o
o
o
o
( )
= +
+
+
+ +
+
−
−
−
−
−
−
3
3
2
2
1
0
1
2
w
w
w
w
ee
v e
j
t
j
t
o
o
2
3
3
w
w
+
+
Using summation notation we write this series as
v t
v e
n
n
jn t
n
n
o
( )
;
=
=−∞
=∞
∑
w
integer
(9.2-1)
This equation states that the periodic function v(t) can be constructed or synthesised from
infinitely many complex exponential functions of time drawn from j
w
axis in the signal plane.
v
n
in Eqn. 9.2-1 are called the coefficients of exponential Fourier series.
w
o
is the fundamental
frequency.
v(t) is usually a real function of time since it represents some voltage or current waveform in a
circuit. Therefore we expect that
v
-
n
will turn out to be
v
n
*
for any non-zero value of n.
Let
Then
n
n
n
n
n
n
v
a
j
b
v
a
j
b
=
−
=
+
2
2
2
2
.
.
*
9.8
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
Then the contribution of n
th
harmonic to v(t) can be expressed as
=
+
=
+
=
+
−
−
−
v e
v e
v e
v e
a
j
b
n
jn t
n
jn t
n
jn t
n
jn t
n
n
o
o
o
o
w
w
w
w
*
2
2
+
−
=
+
−
e
a
j
b
e
a
n t
b
n t
jn t
n
n
jn t
n
n
o
o
w
w
w
w
2
2
2
2
cos
sin
o
o
−−
−
+
+
j
a
n t
b
n t
a
n t
b
n t
n
n
n
n
2
2
2
2
sin
cos
cos
sin
w
w
w
w
o
o
o
o
++
−
=
+
j
a
n t
b
n t
a
n t b
n t
n
n
n
n
2
2
sin
cos
cos
sin
w
w
w
w
o
o
o
o
Hence
v
v
n
n
−
=
*
will result in the two complex exponential contributions adding up to yield a real
function of time.
Equation 9.2-1 tells us how to construct the periodic waveform v(t) from its harmonic components.
But how do we get the exponential Fourier series coefficients
v
n
given the function v(t) ?
We proceed as follows.
First we introduce a new index variable k in the place of n in Eqn. 9.2-1 and restate that equation
as follows:
v t
v e
k
jk
t
k
k
( )
=
=−∞
=∞
∑
w
o
Then we multiply both sides by
e
jn t
-
w
o
where n is a particular value of k.
v t e
e
v e
v
v e
jn t
jn t
k
jk
t
k
k
n
k
j k n
t
k
k
o
( )
(
)
−
−
=−∞
=∞
−
=−∞
=
=
∑
w
w
w
w
o
o
o
≠≠
=∞
=−∞
≠
=∞
∑
∑
=
−
+
−
n
k
n
k
k
k n
k
v
v
k n
t
j
k n
t
[cos(
)
sin(
)
]
w
w
o
o
We wish to extract
v
n
.
We remember an interesting property of sinusoids – the area under a
sinusoidal curve over one period is zero, since the area accumulated under the positive half-cycle
is cancelled exactly by the area accumulated under the negative half-cycle. More generally, the area
under a sinusoid over any time interval equal to its period or integer multiples of its period will be
zero.
k
-
n is an integer. Thus, a sinusoid with angular frequency of (k
-
n)
w
o
will have integral number
of cycles in T seconds since T
=
2
p
/
w
o
. Therefore,
cos(
)
sin(
)
.
k n
t dt
k n
t dt
k
n
t
t T
t
t T
−
=
−
=
≠
+
+
∫
∫
w
w
o
o
and
for
0
0
trigonometric Fourier Series
9.9
We make use of this fact to extract
v
n
as,
v t e
dt
v dt
v
k n
t
j
k n
t dt
v T
jn t
n
k
n
( )
[cos(
)
sin(
)
]
−
=
+
−
+
−
=
+
w
w
w
o
o
o
0
tt
t T
k
k n
k
t
t T
t
t T
n
jn t dt
t
t T
v
T
v t e
+
=−∞
≠
=∞
+
+
−
+
∫
∑
∫
∫
∫
∴ =
1
( )
w
o
The required integration can be carried out over any interval of width T. However, this interval is
usually chosen to be [
-
T/2,
+
T/2] in order to exploit certain symmetries that the waveform v(t) may
possess. Therefore,
v
T
v t e
dt
n
jn t
T
T
=
−
−
−
∫
1
2
2
( )
w
o
analysis equation
(9.2-2)
Equations 9.2-1 and 9.2-2 are called the synthesis equation and the analysis equation, respectively
and the two together form the Fourier series pair.
We expect that
v
n
-
will turn out to be
v
n
*
for any non-zero value of n. We show that it is indeed so.
v
T
v t e
dt
T
v t e
dt
T
v t e
n
j
n
t
T
T
jn t
T
T
o
o
−
− −
−
−
=
=
=
∫
∫
1
1
1
2
2
2
2
( )
( )
( )
(
)
w
w
−−
−
−
−
(
)
=
∫
∫
jn t
T
T
jn t
T
T
o
o
dt
T
v t e
dt
v
w
w
*
*
( )
(
2
2
2
2
1
(since
tt
t
v
n
)
*
is a real function )
=
The value n
=
0 is a special one. The harmonic coefficient at n
=
0 appears alone without a conjugate
companion. We examine this coefficient further.
v
T
v t e
dt
T
v t dt
j
t
T
T
T
T
o
o
=
=
− ⋅ ⋅
−
−
∫
∫
1
1
0
2
2
2
2
( )
( )
w
Thus,
v
o
is a real value representing the cycle average value of v(t). The area of v(t) in one cycle is
divided by the period to arrive at
v
o
.
It represents the DC content in the waveform. If this DC content
is removed from v(t), it becomes a pure AC signal that has zero area under one cycle.
9.3
trIgonometrIc fourIer serIes
The trigonometric form of Fourier series affords better insight into how sinusoids combine to produce
the periodic waveform v(t). This form is derived from the exponential form below as follows.
Let
Then
v
a
j
b
v
a
j
b
n
n
n
n
n
n
=
−
=
+
2
2
2
2
.
.
*
Then,
9.10
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
v t
v
v e
v e
v
v e
v
n
jn t
n
jn t
n
n
jn t
n
o
o
o
( )
[
]
[
*
=
+
+
=
+
+
−
−
=
∞
−
∑
0
1
0
w
w
w
ee
v
a
j
b
e
a
j
b
e
jn t
n
n
n
jn t
n
n
jn
o
o
w
w
w
]
=
∞
−
∑
=
+
+
+
−
1
0
2
2
2
2
oo
t
n
n
o
n
o
n
n
v
a
n t
b
n t
=
+
+
=
∞
=
∞
=
∞
∑
∑
∑
1
0
1
1
cos
sin
w
w
∴
=
+
+
=
=
=
∞
=
∞
∑
∑
v t
a
a
n t
b
n t
a
v
T
v t d
n
o
n
o
n
n
o
( )
cos
sin
( )
o
o
where
w
w
1
1
1
tt
a
v
v
v
v
v
T
v t
n t dt
T
T
n
n
n
n
n
n
o
,
Re( )
( ) cos
*
−
−
∫
=
+
=
+
=
=
2
2
2
2
w
for
nn
b
v
v
v
v
v
T
v t
n
T
T
n
n
n
n
n
n
=
= − +
= − +
= −
=
−
−
∫
1 2 3
2
2
2
2
, ,
Im( )
( )sin
*
…
w
w
o
T
T
t dt
n
for
=
−
∫
1 2 3
2
2
, ,
…
(9.3-1)
This can be written in the following form by combining the cosine and sine contributions for a
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