Modern Control Engineering

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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