Modern Control Engineering

where

Notice that

This network has a dc gain of

Note that this network, whose transfer function is given by Equation (3–36), is a lead

network if or a<1.It is a lag network if

PID Controller Using Operational Amplifiers. Figure 3–19 shows an electronic

proportional-plus-integral-plus-derivative controller (a PID controller) using opera-

tional amplifiers. The transfer function is given by

where

Thus

Noting that

Eo(s)

E(s)

=-

R 4

R 3

E(s)

Ei(s)

=-a

R 2 C 2 s+ 1

C 2 s

ba

R 1 C 1 s+ 1

R 1

b

Z 1 =

R 1

R 1 C 1 s+ 1

, Z 2 =

R 2 C 2 s+ 1

C 2 s

E(s)

Ei(s)

=-

Z 2

Z 1

E(s)Ei(s)

R 1 C 17 R 2 C 2 , R 1 C 16 R 2 C 2.

Kc a=R 2 R 4 AR 1 R 3 B.

Kc a=

R 4 C 1

R 3 C 2

R 2 C 2

R 1 C 1

=

R 2 R 4

R 1 R 3

, a=

R 2 C 2

R 1 C 1

T=R 1 C 1 , aT=R 2 C 2 , Kc=

R 4 C 1

R 3 C 2

Section 3–3 / Mathematical Modeling of Electrical Systems 83

+

+

• 
  
  
  
  • 
    
  
  
  
  

Z 1

C 1

Z 2

R 2 C^2

R 1

Ei(s)

E(s) Eo(s)

R 3

R 4

Figure 3–19
Electronic PID
controller.