Modern Control Engineering

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

A–3–8. Obtain the transfer function of the operational-amplifier circuit shown in Figure 3–28.

Solution.We will first obtain currents i 1 ,i 2 ,i 3 ,i 4 , and i 5. Then we will use node equations at nodes

AandB.

At node A, we have i 1 =i 2 +i 3 +i 4 ,or

(3–42)

At node B, we get i 4 =i 5 ,or

(3–43)

By rewriting Equation (3–42), we have

(3–44)

From Equation (3–43), we get

(3–45)

By substituting Equation (3–45) into Equation (3–44), we obtain

Taking the Laplace transform of this last equation, assuming zero initial conditions, we obtain

from which we get the transfer function as follows:

Eo(s)

Ei(s)

=-

1

R 1 C 1 R 2 C 2 s^2 +CR 2 C 2 +R 1 C 2 +AR 1 R 3 BR 2 C 2 Ds+AR 1 R 3 B

Eo(s)Ei(s)

• C 1 C 2 R 2 s^2 Eo(s)+a

1

R 1

+

1

R 2

+

1

R 3

bA-R 2 C 2 BsEo(s)-

1

R 3

Eo(s)=

Ei(s)

R 1

C 1 a-R 2 C 2

d^2 eo

dt^2

b+ a

1

R 1

+

1

R 2

+

1

R 3

bA-R 2 C 2 B

deo

dt

=

ei

R 1

+

eo

R 3

eA=-R 2 C 2

deo

dt

C 1

deA

dt

+ a

1

R 1

+

1

R 2

+

1

R 3

beA=

ei

R 1

+

eo

R 3

eA

R 2

=C 2

• deo
  dt

ei-eA

R 1

=

eA-eo

R 3

+C 1

deA

dt

+

eA

R 2

i 4 =

eA

R 2

, i 5 =C 2

• deo
  dt

i 1 =

ei-eA

R 1

; i 2 =

eA-eo

R 3

, i 3 =C 1

deA

dt

Eo(s)Ei(s)

i 1 R 1

i 2

i 4

i 3

A

ei C 1 eo

R 3

i 5 C 2

R B

2

• 
  

• 
  

Figure 3–28

Operational-

amplifier circuit.

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