Modern Control Engineering

Section 3–3 / Mathematical Modeling of Electrical Systems 81

Impedance Approach to Obtaining Transfer Functions. Consider the op-amp

circuit shown in Figure 3–17. Similar to the case of electrical circuits we discussed ear-

lier, the impedance approach can be applied to op-amp circuits to obtain their transfer

functions. For the circuit shown in Figure 3–17, we have

Since we have

(3–34)

Eo(s)

Ei(s)

=-

Z 2 (s)

Z 1 (s)

E¿(s)0,

Ei(s)-E¿(s)

Z 1

=

E¿(s)-Eo(s)

Z 2

+

Eo(s)

I(s)

Ei(s)

E 9 (s)

Z 1 (s)

Z 2 (s)

Figure 3–17
Operational-
amplifier circuit.

EXAMPLE 3–9 Referring to the op-amp circuit shown in Figure 3–16, obtain the transfer function Eo(s)/Ei(s)by

use of the impedance approach. The complex impedances Z 1 (s)andZ 2 (s)for this circuit are

and

The transfer function Eo(s)/Ei(s)is, therefore, obtained as

which is, of course, the same as that obtained in Example 3-8.

Eo(s) Ei(s)

=-

Z 2 (s) Z 1 (s)

=-

R 2

R 1

1

R 2 Cs+ 1

Z 2 (s)=

1

Cs+

1

R 2

=

R 2