76 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems
EXAMPLE 3–7 Consider again the system shown in Figure 3–8. Obtain the transfer function Eo(s)/Ei(s)by use
of the complex impedance approach. (Capacitors C 1 andC 2 are not charged initially.)
The circuit shown in Figure 3–8 can be redrawn as that shown in Figure 3–10(a), which can be
further modified to Figure 3–10(b).
In the system shown in Figure 3–10(b) the current Iis divided into two currents I 1 andI 2.
Noting that
we obtain
Noting that
we obtain
SubstitutingZ 1 =R 1 , Z 2 =1/AC 1 sB,Z 3 =R 2 ,andZ 4 =1/AC 2 sBinto this last equation, we get
which is the same as that given by Equation (3–33).
=
1
R 1 C 1 R 2 C 2 s^2 +AR 1 C 1 +R 2 C 2 +R 1 C 2 Bs+ 1
Eo(s)
Ei(s)
=
1
C 1 s
1
C 2 s
R 1 a
1
C 1 s
+R 2 +
1
C 2 s
b+
1
C 1 s
aR 2 +
1
C 2 s
b
Eo(s)
Ei(s)
=
Z 2 Z 4
Z 1 AZ 2 +Z 3 +Z 4 B+Z 2 AZ 3 +Z 4 B
Eo(s)=Z 4 I 2 =
Z 2 Z 4
Z 2 +Z 3 +Z 4
I
Ei(s)=Z 1 I+Z 2 I 1 = cZ 1 +
Z 2 AZ 3 +Z 4 B
Z 2 +Z 3 +Z 4
dI
I 1 =
Z 3 +Z 4
Z 2 +Z 3 +Z 4
I, I 2 =
Z 2
Z 2 +Z 3 +Z 4
I
Z 2 I 1 =AZ 3 +Z 4 BI 2 , I 1 +I 2 =I
Z 1 Z 3
Z 2 Z 4
Z (^1) I
2
I 1
Z 2
Z 3
Z 4
I
Ei(s)
Eo(s)
Ei(s) Eo(s)
(a) (b)
Figure 3–10
(a) The circuit of
Figure 3–8 shown in
terms of impedances;
(b) equivalent circuit
diagram.
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