ECE 382 – NOTES ON ANALOGOUS SYSTEMS
Systems that are governed by the same type of differential equations are called
analogous
systems
. If the response of one physical system to a given excitation is found, then the response of
all other systems that are described by the same set of equations are known for the same excitation
function.
We study analogous systems for the following reasons:
•
Compact size of electric components (
R
,
L
,
C
);
•
Circuit theory is well understood;
•
Ease of setting up experimental verification.
We shall approach this study of analogous systems using the
impedance approach
.
1
Series Connection of Electrical Components
Consider a
series
connection of three electrical components (an inductor, a resistor, and a capacitor)
as shown below (see Figure 1).
~
L
R
C
i
+
_
e
Figure 1:
Series connection of RLC
electrical components.
Using the impedance approach, the equation describ
ing this system is
E
(
s
) = (
sL
+
R
+
1
sC
)
I
(
s
)
4
=
Z
eg
I
(
s
)
.
(1)
Or it can be written as
E
(
s
) = (
s
2
L
+
sR
+
1
C
)
I
(
s
)
s
.
(2)
Thus, for seriallyconnected electrical elements, the
total (or equivalent) impedance is the
sum
of the individ
ual impedance of each element.
2
Parallel Connection of Electrical Components
Next consider the three basic electrical components connected in
parallel
as shown below (see
Figure 2).
i
i
L
i
L
i
R
R
C
i
C
e
Figure 2: Parallel connection of RLC electri
cal components.
Using the impedance approach again,
the
equation describing the system in the Laplace
transform domain is
I
(
s
) = (
1
R
+
1
sL
+
1
1
sC
)
E
(
s
) = (
1
R
+
1
sL
+
sC
)
E
(
s
)
.
(3)
Or it can be rewritten as
I
(
s
) = (
s
2
C
+
s
R
+
1
L
)
E
(
s
)
s
.
(4)
Thus, for parallellyconnected electrical elements, the total impedance is equal to one divided
by the sum of the reciprocal of the individual impedance of each element.
1
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3
Series Connection of Mechanical Components
Similar to the electrical components
(
R
,
L
,
C
)
that can be connected either in series or parallel,
mechanical elements (mass,
M
; dashpot,
f
; and coil spring,
K
) can also be connected in either
series or parallel arrangement.
x
F
M
f
K
1
K
2
(a)
x
F
f
K
1
K
2
M
(b)
Figure 3:
Series connection of mechanical
components.
We
shall
use
the
concept
of
equivalent
“groundedchair” representation.
Figure 3(a) is
represented by its groundedchair representation
in Fig.
3(b).
Figure 3(b) shows that the mass
is in
series
with the other elements.
The inertia
force
(
M
¨
x
(
t
))
is always taken with respect to the
ground.
For seriallyconnected mechanical ele
ments, the force
F
is equal to the summation of the
forces acting on each individual component and
each component undergoes the
same displacement
F
(
t
) =
M
¨
x
(
t
)+
f
˙
x
(
t
)+(
K
1
+
K
2
)
x
(
t
)
.
(5)
Its Laplace transform equivalence is
F
(
s
) = (
Ms
2
+
fs
+
K
1
+
K
2
)
X
(
s
)
4
=
Z
eg
X
(
s
)
.
(6)
The equivalent impedance is
Z
eq
=
Ms
2
+
fs
+
K
1
+
K
2
.
Thus, for seriallyconnected mechanical elements, the total (or equivalent) impedance is the
sum
of the individual impedance of each element. (Note that the mechanical elements look like
are connected in “parallel” when viewed as electrical connection, but they are actually connected
in series)
4
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 Spring '08
 Staff
 Inductor, Mechanical Components, series connection, Analogous Systems

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