Introduction to Electric Circuits

8.5 Self-assessment test 201

Completing the square of the denominator we have

i(s) = (500/L)/[{s + (R/2L)} 2 + {(1/CL) - (R/2L)2}]

Putting L = 250 txH, n/2L = a, and {(lICE) - (R/2L) 2} = ~o 2, we have

i(s) = 2/[(s + a) 2 + of] IXS

Multiplying numerator and denominator by o~ we have

i(s) = (2/w){w/[(s + a) 2 + 2]}

From transform pair number 7 in Table 8.1 we see that w/[(s + a) 2 -~- 0) 2] is the
transform of exp (-at)sin o)t so that, as a function of time we have for the
current,

i(t) = (2/~o) exp (-at) sin o)t

This is of the form shown in Fig. 8.40 which is an exponentially decaying sine
wave. It is said to be underdamped.

o

Figure 8.40

This means that the current undergoes a period of oscillation before reaching
its new required value (zero in this case).
This result was obtained assuming that of is positive (i.e. (1//CL) > (R/2L)2).
There are other possibilities (~o 2 could be negative or zero) leading to other
results associated with overdamping and critical damping, respectively, in which
the current reaches zero more or less rapidly and without oscillation.

8.5 SELF-ASSESSMENT TEST

1 Give two conditions which could lead to the transient operation of an

electric circuit.

2 Explain why transient conditions do not exist in purely resistive circuits.

3 State the reason why there could be periods of transient operation in

inductive circuits.

4 Explain why there is a period of transient operation immediately after

switching on a circuit containing capacitance.