1
EE102
Systems and Signals
Fall Quarter 2010
Jin Hyung Lee
Homework #4 Solutions
Due: Wednesday, October 27, 2010 at 5 PM.
1. Suppose that
f
(
t
)
is a periodic signal with period
T
0
, and that
f
(
t
)
has a Fourier series. If
τ
is a real number, show that
f
(
t

τ
)
can be expressed as a Fourier series identical to that
for
f
(
t
)
except for the multiplication by a complex constant, which you must find.
Solution:
f
(
t

τ
) =
∞
X
n
=
∞
D
n,τ
e
janω
0
t
where
D
n,τ
=
Z
t
0
+
T
0
t
0
f
(
t

τ
)
e

jω
0
nt
dt
Let
t
0
=
t

τ
,
D
n,τ
=
Z
t
0
+
τ
+
T
0
t
0
+
τ
f
(
t
0
)
e

jω
0
n
(
t
0
+
τ
)
dt
0
=
e

jnω
0
τ
Z
t
0
+
τ
+
T
0
t
0
+
τ
f
(
t
0
)
e

jω
0
nt
0
dt
0
=
e

jnω
0
τ
D
n
where we have used the fact that
f
(
t
)
is periodic in the last step. We get the same Fourier
series no matter where we choose the one period to integrate over. The Fourier series is
then
f
(
t

τ
) =
∞
X
n
=
∞
D
n
e

jnω
0
τ
e
janω
0
t
2. Switching amplifiers are a very efficient way to generate a timevarying output voltage
from a fixed supply voltage. They are particularly useful in highpower applications.
The basic idea is that an output voltage
a
is generated by rapidly switching between zero
and the supply voltage
A
. The output is then lowpass filtered to remove the harmonics
generated by the switching operation. For our purposes we can consider the lowpass filter
as an integrator over many switching cycles, so the output voltage is the average value
of the switching waveform.. Varying the switching rate varies the output voltage. In this
problem we will only consider the case where the desired output voltage is constant.
We can analyze this system with the Fourier series. If the output pulses are spaced by
T
,
the waveform the amplifier generates immediately before the lowpass filter is
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2
T
2
T

2
T

T
0
t
α
T
A
The duty cycle of the switching amplifier is
α
, and the width of the pulses is
αT
. When
α
= 1
, the amplifier is constantly on and produces its maximum output
A
.
(a) Reducing the duty cycle reduces the output voltage. After the lowpass filter, only the
zerofrequency spectral component
D
0
remains, and this will be the output voltage.
Find the value of
D
0
as a function of the duty cycle
α
. If the desired output voltage is
a
and the supply voltage is
A
, what should
α
be?
Solution:
The zero frequency spectral component is
D
0
=
1
T
Z
T/
2

T/
2
f
(
t
)
dt
=
1
T
Z
αT/
2

αT/
2
Adt
=
αTA
T
=
αA.
If we want an output of
a
volts, we want
αA
=
a
or
α
=
a/A.
(b) The lowpass filter must suppress (average out) the harmonics generated by the switch
ing waveform. The first harmonic is the most difficult to suppress since it tends to be
large, and is closest in frequency. Find an expression for the amplitude of the first
harmonic
D
1
as a function of the duty cycle
α
.
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 Fall '08
 Levan
 Fourier Series, duty cycle, Fourier series coefficients

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