Introduction to Electric Circuits

194 Transient analysis

These transform pairs, together with a few other commonly encountered ones,
are given in Table 8.1.

Table 8.1

f(t) Description L[f(t)] = F(s)

1 exp (-at) exponential function 1/,(s + a)

2 1 unit step function 1/s

3 A step function of amplitude A/s

A

4 d{f(t)}/dt differential of f(t) sF(s) - f(O)

t

5 ff(t)dt

0 integral off(t) (1/s)F(s) + f(O)/s

6 t ramp function 1/s 2

7 sin oJt sinusoidal function w/'(s 2 + w 2)

8 cos ox cosinusoidal function s/(s 2 + w 2)

9 exp (-a) sin wt exponentially decaying ~o/[(s + a) 2 + w 2]

sinusoidal function

10 exp (-at) cos tot exponentially decaying (s Jr- O~)/[(S -[- 0~) 2 "Jr- (.0 2]

cosinusoidal function

Application to electrical circuit transient analysis

For transient circuit analysis we convert the original circuit into a transform

circuit, and to do this we need to be able to transform the circuit elements R, L

and C.

Resistance

If a resistor R has a current flowing through it given by i = f(t) then the voltage

across it will also be a function of time, given by v(t) = Ri(t). Taking Laplace

transforms we have

L[v(t)] = RL[i(t)]

v(s) = RI(s)

Thus resistance is the same in the s-domain as it is in the time domain.

Inductance

If an inductor L has a current flowing through it given by i(t) then the voltage

across it will be given by v(t)- Ldi/dt. Taking Laplace transforms we have,

from number 4 in Table 8.1.

v(s) : Lsi(s)- Li(O)

If the current is initially zero the second term on the right-hand side disappears

(Li(O) : O)