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EE 350
CONTINUOUSTIME LINEAR SYSTEMS ANALYSIS
FALL 2007
Laboratory #1: Time Response Characteristics
The objectives of the Frst laboratory are to:
•
Review the representation of passive circuits using ODE models.
•
Utilize MATLAB to Fnd the roots of the characteristic equation, numerically solve ODEs, and determine
timeresponse characteristics.
•
Estimate component values using time response characteristics acquired with MATLAB and the dSPACE data
acquisition system.
Results from Laboratory #1 are required to complete Problem Set 4.
1
ODE Representation of Passive Circuits
The passive circuits in ±igure 1 produce an output voltage
y
(
t
) in response to a driving voltage
f
(
t
), and may be
represented by an ODE of the form
Q
(
D
)
y
(
t
)=
P
(
D
)
f
(
t
)
(
D
n
+
a
n

1
D
n

1
+
···
+
a
1
D
+
a
0
)
y
(
t
(
b
m
D
m
+
b
m

1
D
m

1
+
+
b
1
D
+
b
0
)
f
(
t
)
where
D
k
≡
d
k
dt
k
is the di²erential operator, and
a
i
and
b
i
are constant coeﬃcients. The parameter
n
is the highest derivative of
y
(
t
)
and represents the order of the system. ±or a circuit, the number of independent energy storage elements determines
the system order. The adjective
independent
speciFes that capacitors (inductors) connected in series or parallel count
as a single energy storage element.The laboratory instructor will review determination of the polynomials
Q
(
D
)and
P
(
D
) using circuit analysis techniques from EE 210 and will help you complete Table 1 for the circuits shown in
±igure 1.
Circuit
Q
(
D
)
P
(
D
)
Frstorder RC
Frstorder RL
secondorder RLC
Table 1: Polynomials
P
and
Q
representing the passive circuits in ±igure 1.
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View Full Document Figure 1: Passive circuits considered in Laboratory #1: (A) RC circuit, (B) RL circuit, and (C) RLC circuit.
2
Simulation and Analysis using a MATLAB Mfle
This section introduces several MATLAB commands that are useful for studying ODEs and their solutions. The
MATLAB command
roots
determines the roots of the characteristic equation
Q
(
λ
)=
λ
n
+
a
n

1
λ
n

1
+
···
+
a
1
λ
+
a
o
=0
.
For example, if
Q
(
D
D
3
+4
D
2
+
D
−
3
,
enter
>> Q
=[1
,
4
,
1
,
−
3];
roots(
Q
)
and MATLAB will return the roots of
Q
(
λ
) = 0. Note that the polynomial
Q
(
D
) is entered as a row vector in
MATLAB, where the ±rst element is the coeﬃcient associated with
D
3
.
The command
step
determines the unitstep response of a system. For example, suppose we want to determine
the unitstep response of a secondorder system described by the ODE
d
2
y
dt
2
dy
dt
+ 100
y
(
t
) = 200
f
(
t
)
.
First represent the ODE as
Q
(
D
)
y
(
t
P
(
D
)
f
(
t
)
,
where
Q
(
D
D
2
D
+ 100
P
(
D
)
=
200
.
Represent these polynomials in MATLAB as row vectors
>> Q
=[
1
,
4
,
100];
>> P
=
[200];
The MATLAB command
>>
step(
P, Q
)
generates a plot of the unitstep response over a time interval determined by MATLAB. To generate the unitstep
response at 200 points over a speci±ed time interval, say 0
≤
t
≤
5, use the commands
>>
Q
,
4
,
100];
>>
P
= [200];
>>
t
= linspace(0
,
5
,
200);
>>
step(
P, Q, t
)
You can also save the response to a vector
y
using
>>
Q
,
4
,
100];
>>
P
= [200];
>>
t
= linspace(0
,
5
,
200);
>>
y
=step(
P, Q, t
);
MATLAB provides a command,
lsim
, for determining the zerostate response of a LTI system to an arbitrary
input. As an example, the zerostate response of the system
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This note was uploaded on 07/23/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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