People.clarkson.edu

Example: Consider this circuit and corresponding asymptotic magnitude Bode plot. The input to this circuit

is the voltage of the voltage source, v1 (t ) . The output is the capacitor voltage vo (t ) . Design this

circuit to have this Bode plot.

Solution: Represent the circuit in the frequency domain:

Using voltage division twice gives:

jL R2

( ) ( ) V2 ()

V1 ( )

=

R2 + jL

R1

+

jL R2 R2 + jL

=

R1 R 2

+

jL R2 jL R1

+ R2

=L

j

R1 1+

L j

R1 + R2

R1 R 2

and

R4

A R4

Vo V 2

( ) ( )

=

1+ j C R4

R3

+

1+

R4 j C

R4

A=

A R4

R3 + R4 + j C R3R4

= R3 + R4 1+ j C R3R4 R3 + R4

Combining these equations gives

1

( ) ( ) H( ) =

Vo ( ) Vi ( )

=

R1

ALR4 R3 + R4

L

1+ j

j

R1 + R2 R1 R 2

1+

j

CR3R4 R3 + R4

The Bode plot corresponds to the network function:

H( )

=

1 +

j

k j

p1

1

+

j

p2

=

1 +

j

k j

200

1 +

j

20000

H

(

)

k j = k j 11

k j j 1

=

k

p1

p1

k j

p1

j j

p2

=

k

p1 p2 j

p1 p1 p2

p2

This equation indicates that |H()|=k p1 when p1 p2. The Bode plot indicates that |H()|=20 dB = 10 when p1 p2. Consequently

Finally,

k = 10 = 10 = 0.05 p1 200

H (

)

=

1 +

j

0.05 j

200

1

+

j

20000

Comparing the equation for H() obtained from the circuit to the equation for H() obtained from the Bode plot gives:

( ) ( ) 0.05 = ALR4 , 200 = R1R2 and 20000 = R3 + R4

R1 R3 + R4

L R1 + R2

C R3R4

Pick L = 1 H, and R1 = R2 , then R1 = R2 = 400 . Let C = 0.1 F and R3 = R4 , then R3 = R4 = 1000 . Finally, A=40.

2

Example: Consider this circuit and corresponding asymptotic magnitude Bode plot. The input to this circuit

is the voltage of the voltage source, vin (t ) . The output is the voltage vo (t ) . Design this circuit to

have this Bode plot.

Solution: Represent the circuit in the frequency domain:

Mesh equations:

Solving yields:

Vin () = I( ) [R1 + ( j L1 - j M ) + (- j M + j L2 ) + R2 ] Vo () = I() [(- j M + j L2 )+R2 ]

H() = Vo () =

R2 + j(L2 -M )

Vin () R1 + R2 + j(L1 + L2 -2M )

Comparing to the given Bode plot yields:

K1 =

lim |H()|

= L2 -M L1 + L2 -2M

= 0.75

and

K2 =

lim | H() |

0

=

R2 R1 + R2

= 0.2

z = R2 = 333 rad s and p = R1 +R2 = 1250 rad s

L2 -M

L1 + L2 -2M

3

Example: Consider this circuit and corresponding asymptotic magnitude Bode plot. The input to this circuit

is the voltage of the voltage source, v1 (t ) . The output is the capacitor voltage vo (t ) . Design this

circuit to have this Bode plot.

Solution: From Table 13.3-2:

( ) R2 = k = 32 dB = 40

R1

R2 = 40 10?103 = 400 k

( ) 1 = p = 400 rad/s

C2 R2

C2 = (400)

1 400 ?103

= 6.25 nF

( ) 1 = z = 4000 rad/s

C1 R1

C1

=

( 4000 )

1 10

? 103

= 25 nF

4

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download