Modern Control Engineering

Section 3–3 / Mathematical Modeling of Electrical Systems 85

1 2 3 4 5 6 7

P

I

PD

PI

PID

Lead or lag

Lag–lead

Control

Action G(s)= Operational-Amplifier Circuits

Eo(s)

Ei(s)

R 4

R 3

R 2

R 1

1

R 1 C 2 s

R 4

R 3

R 4

R 3

R 2

R 1 (R^1 C^1 s+ 1)

R 4

R 3

R 2

R 1

R 2 C 2 s+ 1

R 2 C 2 s

R 4

R 3

R 2

R 1

(R 1 C 1 s+ 1) (R 2 C 2 s+ 1)

R 2 C 2 s

R 4

R 3

R 2

R 1

R 1 C 1 s+ 1

R 2 C 2 s+ 1

R 6

R 5

R 4

R 3

[(R 1 +R 3 )C 1 s+ 1] (R 2 C 2 s+ 1)

(R 1 C 1 s+ 1) [(R 2 +R 4 )C 2 s+ 1]

eo

eo

ei

ei

+

• 
  
  
  
  • 
    
  
  
  
  
  
  
  • 
    
  
  
  
  

+

• 
  
  
  
  • 
    
  
  
  
  
  
  
  • 
    
  
  
  
  

R 1

R 2

R 2

R 3

R 4

R (^1) R
3
C 2 R 4
eo
ei

• 
  

• 
  
  
  
  • 
    
    
    
    • R 3
    
    
    
    
  
  
  
  

C 1 R 4

R 2

R 1

R 1

ei eo

+

• 
  
  
  
  • 
    
  
  
  
  
  
  
  • R 3
  
  
  
  

C 2 R 4

R 2

R 1

ei eo

+

• 
  
  
  
  • 
    
  
  
  
  
  
  
  • R 3
  
  
  
  

C 1 C 2 R 4

R 2

R 1

eo

ei +

• 
  
  
  
  • 
    
    
    
    • R 3
    
    
    
    
  
  
  
  

R 4

C 2

C 1

R 4

R 2

R 1

R 3

eo

ei +

• 
  
  
  
  • 
    
  
  
  
  
  
  
  • R 5
  
  
  
  

R 6

C 2

C 1

Table 3–1 Operational-Amplifier Circuits That May Be Used as Compensators