PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 357

Time domain s-domain

i(t) = I 0

I(s) = I 0 /s

i(t) = I 0 cos(w 0 t)

i(t) =

(amp) (amp)

(amp) (amp)

s^2 + w 02

I 0 s

FIGURE 4.13

Time–frequency domain relation for DC and AC current sources.

v

R

Vc(0) = V 0

sw closes at t = 0

i(t)

C

FIGURE 4.14

Network of R.4.109.

FIGURE 4.15

Frequency domain representation of Figure 4.14.

V/s

Vr(s) = I(s)R

Vc(s) =

sw closes at t = 0

I(s)

C

sC

I(s) V 0

s

+

Note that these type of problems referred as transients were presented, discussed,

and solved in Chapter 2, using DEs (time domain).

ANALYTICAL Solution

The fi rst step in the solution process is to redraw the circuit diagram shown in

Figure 4.14 in the frequency domain (using the LTs) illustrated in Figure 4.15.

Observe that the voltage source V in the time domain is transformed to V/s in the

frequency domain, due to the step characteristics of the switch (sw). Its transform is

given by Vu(t) → V/s. Also observe that the initial voltage of the capacitor also acts as

a step function in time, therefore, its transform is given by

Vut 0 ()→Vs^0