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EE 350
CONTINUOUSTIME LINEAR SYSTEMS
FALL 2007
Laboratory #2: Sinusoidal SteadyState Analysis
The objectives of the second laboratory are to
•
Experimentally measure the frequency response function
H
(
jω
)o
ftwonetworks
.
•
Use MATLAB to plot the magnitude and angle of
H
(
jω
)in
decibels
as a function of frequency
ω
.
1
Background
If a boundedinput boundedoutput (BIBO) stable linear timeinvariant (LTI) system is driven by a sinusoid of
frequency
ω
, then the steadystate response of the circuit is also a sinusoid of frequency
ω
. Figure 1 shows a BIBO
stable LTI system with input
f
(
t
) and response
y
(
t
).
Figure 1: BIBO stable LTI system with input
f
(
t
) and output
y
(
t
).
Suppose a sinusoidal input
f
(
t
)=
F
m
cos(
ωt
+
θ
f
)
,
where
F
m
and
θ
f
are constants, is applied to the system. The resulting zerostate response can be expressed as the
sum of a natural response
y
n
(
t
) and forced response
y
φ
(
t
)
y
(
t
y
n
(
t
)+
y
φ
(
t
)
.
Because the system is BIBO stable, the natural response
y
n
(
t
) will exponentially relax towards zero so that the
steadystate response of the circuit is the forced response
y
ss
(
t
y
φ
(
t
)
.
As the forcing function is a sinusoid of frequency
ω
, the steadystate response is also a sinusoid of frequency
ω
y
ss
(
t
Y
m
(
ω
)cos(
+
θ
y
(
ω
))
.
It is important to note that the amplitude
Y
m
and phase
θ
y
of the response depends on the frequency
ω
of the input.
The sinusoidal input
f
(
t
) and the sinusoidal steadystate response
y
ss
(
t
) can be represented by phasors
˜
F
(
ω
F
m
e
jθ
f
˜
Y
(
ω
Y
m
(
ω
)
e
jθ
y
(
ω
)
.
Given this representation, the
frequency response function
H
(
jω
) is de±ned as
H
(
jω
˜
Y
(
ω
)
˜
F
(
ω
)
.
Once we know
H
(
jω
), we can ±nd the sinusoidal steadystate response
y
ss
(
t
) for any sinusoidal input
f
(
t
)us
ingthe
relationships
˜
Y
(
ω
H
(
jω
)
˜
F
(
ω
)
and
y
ss
(
t
)=Re
{
˜
Y
(
ω
)
e
jωt
}
.
In order to understand the e²ect of the circuit on a sinusoidal input, it is useful to plot the magnitude of
H
(
jω
)
as a function of
ω
. The magnitude of the frequency response function can vary over many orders of magnitude
as frequency is varied, and so it is useful to plot the logarithm of

H
(
jω
)

. Electrical engineers typically express
the magnitude of the frequency response function in
decibels
(abbreviated dB) that are de±ned as 20 log
10

H
(
jω
)

.
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB
 Frequency

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