Modern Control Engineering

Example Problems and Solutions 91

Hence

Then the voltages Ei(s)andEo(s)can be obtained as

Hence, the transfer function Eo(s)/Ei(s)of the network shown in Figure 3–25(a) is obtained as

(3–38)

For the bridged T network shown in Figure 3–24(a), substitute

into Equation (3–38). Then we obtain the transfer function Eo(s)/Ei(s)to be

Similarly, for the bridged T network shown in Figure 3–24(b), we substitute

into Equation (3–38). Then the transfer function Eo(s)/Ei(s)can be obtained as follows:

=

R 1 CR 2 Cs^2 + 2 R 1 Cs+ 1

R 1 CR 2 Cs^2 +A 2 R 1 C+R 2 CBs+ 1

Eo(s)

Ei(s)

=

1

Cs

1

Cs

+R 1 a

1

Cs

+

1

Cs

+R 2 b

R 1 a

1

Cs

+

1

Cs

+R 2 b+

1

Cs

1

Cs

+R 2

1

Cs

Z 1 =

1

Cs

, Z 2 =R 1 , Z 3 =

1

Cs

, Z 4 =R 2

=

RC 1 RC 2 s^2 + 2 RC 2 s+ 1

RC 1 RC 2 s^2 +A 2 RC 2 +RC 1 Bs+ 1

Eo(s)

Ei(s)

=

R^2 +

1

C 1 s

aR+R+

1

C 2 s

b

1

C 1 s

aR+R+

1

C 2 s

b+R^2 +R

1

C 2 s

Z 1 =R, Z 2 =

1

C 1 s

, Z 3 =R, Z 4 =

1

C 2 s

Eo(s)

Ei(s)

=

Z 3 Z 1 +Z 2 AZ 1 +Z 3 +Z 4 B

Z 2 AZ 1 +Z 3 +Z 4 B+Z 1 Z 3 +Z 1 Z 4

=

Z 3 Z 1 +Z 2 AZ 1 +Z 3 +Z 4 B

Z 1 +Z 3 +Z 4

I 1

=

Z 3 Z 1

Z 1 +Z 3 +Z 4

I 1 +Z 2 I 1

Eo(s)=Z 3 I 3 +Z 2 I 1

=

Z 2 AZ 1 +Z 3 +Z 4 B+Z 1 AZ 3 +Z 4 B

Z 1 +Z 3 +Z 4

I 1

= cZ 2 +

Z 1 AZ 3 +Z 4 B

Z 1 +Z 3 +Z 4

dI 1

Ei(s)=Z 1 I 2 +Z 2 I 1

I 2 =

Z 3 +Z 4

Z 1 +Z 3 +Z 4

I 1 , I 3 =

Z 1

Z 1 +Z 3 +Z 4

I 1