Modern Control Engineering

Example Problems and Solutions 93

or

(3–39)

Since

(3–40)

by substituting Equation (3–40) into Equation (3–39), we obtain

from which we get the transfer function Eo(s)/Ei(s)to be

(3–41)

To find the transfer function Eo(s)/Ei(s)of the circuit shown in Figure 3–26, we substitute

into Equation (3–41). The result is

which is, as a matter of course, the same as that obtained in Problem A–3–6.

Eo(s)

Ei(s)

=-

R 1 R 2 - R 1

1

Cs

R 1 a

1

Cs

+R 2 b

=-

R 2 Cs- 1

R 2 Cs+ 1

Z 1 =

1

Cs

, Z 2 =R 2 , Z 3 =R 1 , Z 4 =R 1

Eo(s)

Ei(s)

=-

Z 4 Z 2 - Z 3 Z 1

Z 3 AZ 1 +Z 2 B

c

Z 4 Z 1 +Z 4 Z 2 - Z 4 Z 1 - Z 3 Z 1

Z 4 AZ 1 +Z 2 B

dEi(s)=-

Z 3

Z 4

Eo(s)

EA(s)=EB(s)=

Z 1

Z 1 +Z 2

Ei(s)

Ei(s)- a 1 +

Z 3

Z 4

bEA(s)=-

Z 3

Z 4

Eo(s)

A

B

ei eo

Z 3

Z 1

Z 2

Z 4

• 
  

+

Figure 3–27
Operational-
amplifier circuit.