0195136047.pdf

190 TIME-DEPENDENT CIRCUIT ANALYSIS

vS(t)

+

−

+

−

RL = 100 Ω vL(t)

RS = 100 Ω

C

L

Figure P3.4.5

vS(t)

+

−

+

−

C RL vL(t)

RS L

Figure P3.4.6

vS(t)

+

−

+

−

L RL vL(t)

RS C

Figure P3.4.7

3.4.10Reconsider Problem 3.4.9 and the corresponding

asymptotic Bode plots.

(a) FindH ̄ 1 ,H ̄ 2 ,andH ̄ 3 atω=5 rad/s.

(b) At what angular frequencyωis the magni-

tude ofH ̄ 4 (j ω)one-half of the magnitude of

H ̄ 4 (j 5 )?

(c) Determine the angular frequency at which

H 6 (ω) is 0 dB and the angular frequency at

whichθ 6 (ω)=−180°.

(d) LetH 5 (s)=V 2 /V 1. Forv 1 (t)= 0 .1 cos 20t,

find the steady-state value ofv 2 (t).

3.4.11Sketch the idealized (asymptotic) Bode plot for

the transfer function

H ̄(jω)=^10 (^1 +j^2 ω)

( 1 +j 10 ω)( 1 +j 0. 25 ω)

Find the angular frequency at whichH(ω)is0dB

and the angular frequency at whichθ(ω)=−60°.

*3.4.12The loop gain of an elementary feedback control
system (see Figure 3.4.12) is given byG(s)·H(s),
which is 10/( 1 +s/ 2 )( 1 +s/ 6 )( 1 +s/ 50 ). Sketch
the asymptotic Bode plot of the loop-gain function.
Gain margin(GM) is defined by [−20 log|G(ω ̄ π)
H(ω ̄ π)|], which is the negative of the magnitude
of the loop gain atω=ωπ,ωπrepresents the
angular frequency at which the loop gain reaches
a phase of−π.Ifωurepresents the value ofω

where the loop gain has a magnitude of unity, then

phase margin(PM) is defined by the phase of the

loop gain atω=ωuplusπ. Evaluate GM and PM

for this case from the asymptotic Bode plot.

3.4.13Sketch the asymptotic Bode plots for the follow-

ing loop-gain functions, and find the approximate

values of gain and phase margins in each case. (For

definitions of GM and PM, see Problem 3.4.12.)

(a)G 1 (s)H 1 (s)=

12 ( 0. 7 +s)

(^0.^003 +s)(^0.^04 +s)(^7 +s)

(b)G 2 (s)H 2 (s)=

100 ( 1 +s/ 3. 9607 )

s( 1 + 2 s)( 1 +s/ 39. 607 )

3.4.14(a) For a seriesRLCresonant circuit, find an ex-

pression for the voltage across the resistance

VRand obtain the ratioVR/VS, whereVSis the

applied voltage. Identify the expressions for

the series resonant frequency and bandwidth.

(b) Determine the resonant frequency and band-

width, given the voltage transfer function to

be 10^3 /(s^2 + 103 s+ 1010 ).

3.4.15A simple parallel resonant circuit withL= 50 μH

is used to perform the frequency selection. The cir-

cuit is to be tuned to the first station at a frequency

of 1000 kHz. In order to minimize the interaction