PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 355

R.4.100 The voltage across a capacitor C, denoted by vC(t) is expressed in the time domain

and its equivalent in the frequency domain as follows by (using FT properties):

vt

C

id Vs

Is

sC

V

Ccc s

t

()

^10 ( ) ↔ () () C( )

∞

∫



Recall that VC(0) denotes the initial voltage vC(t) at t = 0.

R.4.101 For example, let us analyze the case of a capacitor C of 2 F, charged with an initial

voltage of +5 V. Then its frequency domain representation, using the LT is given by

Vs

Is

C ss

()

()



2

5

R.4.102 Note that if VC(s) =

I(s)

___

sC

, then the impedance of the capacitor C in the frequency

domain is given by

Xs

C sC

()

1

R.4.103 The equivalent circuit models of a capacitor in the time and frequency domain are

shown in Figure 4.10 using either a voltage source in series with the impedance

XC(s) = 1/(sC), or by source transformation, a current source in parallel with XC(s),

assuming that its initial voltage is vC(0) = V 0 V.

R.4.104 The voltage across an inductor L, denoted by vL(t), is expressed as follows in the

time and frequency domain by

vt L

di t

dt

LL() Vs sLIs LI

()

←→ () () 0

Recall that i( 0 ) = I 0 denotes the initial current through L at t = 0.

R.4.105 Note that if VL(s) = sLI(s), then the impedance of the inductor L in the frequency

domain is given by

XL(s) = sL Ω

R.4.106 The equivalent circuit model of an inductor L in the time and frequency domains

are shown in Figure 4.11, using either a voltage source in series with the impedance

Time domain s-domain

C

+

−

V 0 V

1/ (sC)

1/ (sC)

V 0 /s

C V 0

FIGURE 4.10

Time–frequency domain relation for C.