0195136047.pdf

106 TIME-DEPENDENT CIRCUIT ANALYSIS

EXAMPLE 3.1.2

Consider aGLCparallel circuit excited byi(t)=Iestin the time domain. Assume no initial

inductive current or capacitive voltage att=0. Draw the transformed network in the frequency

s-domain and solve for the frequency-domain forced response of the resultant voltage. Then find

thet-domain forced responsev(t).

Solution

The forced response is produced by the particular excitation applied. The KCL equation for the

circuit of Figure E3.1.2(a) is

v(t)

−

+

V

−

+

i(t) G L C

(a)

I G Cs

(b)

1

Ls

Figure E3.1.2GLCparallel circuit withi(t)=Iest.(a)Time domain.(b)Transformed network in

s-domain.

i(t)=Gv(t)+

1

L

∫t

−∞

v(τ)dτ+C

dv(t)

dt

The corresponding transformed network in thes-domain is shown in Figure E3.1.2(b), for which

the following KCL relation holds. (Note that the initial inductive current att=0 is assumed to

be zero.)

I=GV+

1

Ls

V+CsV

Solving forV, one gets

V=

I

G+( 1 /Ls)+Cs

=

I

Y(s)

whereY(s) can be seen to be the addition of each parallel admittance of the elements. The time

function corresponding to the frequency-domain response is given by

v(t)=Vest=

I

G+( 1 /Ls)+Cs

est

which is also an exponential with the same exponent contained ini(t).

Note that impedances in series are combined like resistances in series, whereas admittances in

parallel are combined like conductances in parallel. Series–parallel impedance/admittance com-

binations can be handled in the same way as series–parallel resistor/conductance combinations.

Notice that in the dc case whens=0, the impedance of the capacitor 1/Cstends to be infinite,

signifying an open circuit, whereas the impedance of the inductorLsbecomes zero, signifying a

short circuit.

Forced Response to Sinusoidal Excitation

Consider an excitation of the form

v(t)=Vmcos(ωt+φ) (3.1.17)