Modern Control Engineering

82 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems

Lead or Lag Networks Using Operational Amplifiers. Figure 3–18(a) shows an

electronic circuit using an operational amplifier. The transfer function for this circuit

can be obtained as follows: Define the input impedance and feedback impedance as Z 1

andZ 2 ,respectively. Then

Hence, referring to Equation (3–34), we have

(3–35)

Notice that the transfer function in Equation (3–35) contains a minus sign. Thus, this circuit

is sign inverting. If such a sign inversion is not convenient in the actual application, a sign

inverter may be connected to either the input or the output of the circuit of Figure 3–18(a).

An example is shown in Figure 3–18(b). The sign inverter has the transfer function of

The sign inverter has the gain of Hence the network shown in Figure 3–18(b)

has the following transfer function:

=Kc a (3–36)

Ts+ 1

aTs+ 1

=Kc

s+

1

T

s+

1

aT

Eo(s)

Ei(s)

=

R 2 R 4

R 1 R 3

R 1 C 1 s+ 1

R 2 C 2 s+ 1

=

R 4 C 1

R 3 C 2

s+

1

R 1 C 1

s+

1

R 2 C 2

- R 4 R 3.

Eo(s)

E(s)

=-

R 4

R 3

E(s)

Ei(s)

=-

Z 2

Z 1

=-

R 2

R 1

R 1 C 1 s+ 1

R 2 C 2 s+ 1

=-

C 1

C 2

s+

1

R 1 C 1

s+

1

R 2 C 2

Z 1 =

R 1

R 1 C 1 s+ 1

, Z 2 =

R 2

R 2 C 2 s+ 1

+

• 
  

+

• 
  

+

• 
  

(a) (b)

Z 1

C 1

Z 2

C 2

R 2

i 1 i 2

R 1

Ei(s)

E 9 (s)

E(s)

C 1

C 2

Ei(s)

E(s) Eo(s)

R 1

R 2

R 3

R 4

Lead or lag network Sign inverter

Figure 3–18

(a) Operational-amplifier circuit; (b) operational-amplifier circuit used as a lead or lag compensator.

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