Question

The switch in the circuit has been closed for a long time and is opened at t = 0. 

 a. Calculate the initial value of I

 b. Calculate the initial energy stored in the inductor.

 c. What is the time constant of the circuit for t ≥ 0?

 d. What is the numerical expression for i() for t20?

 e. What percentage of the initial energy stored has been dissipated in the 4 Ω resistor 5ms after the switch has been opened?

Answer 1

a) Since the switch is closed for a long time before t =0 , the circuit can be considered to be reached at steady state at t <0. The inductor at steady state behaves as a short circuit .Hence 4ohm resistor across inductor gets shorted.Then we can calculate the total current I as

I = 100/(1 + 20||5) = 20A

Hence the current through inductor by current division rule is i = 20 × [20 /(20+5)] = 16 A.

At t =0 , when the switch gets opened, the inductor will resist the change in current as current cannot change instantaneously in an inductor.So

Initial current, is 16A flowing into the inductor

Hence from figure i = -16A ( direction of i is moving out from inductor)

b) initial energy stored in the inductor is given by

Putting values of L and i, we get

energy in inductor = 1.28J

c)For t>0 , as the switch is opened ,the part of circuit with inductor and 4ohm resistor will be cutt off from the rest circuit.The inductor here has energy stored and the time constant can be calculated as the ratio of inductance of inductor to the equivalent resistance across that inductor,

Hence time constant for t>0 = L /R = 10mH / 4ohm = 2.5msec

d) Expression for i(t) = i (at infinity) + [ i(0+) - i(infinity) ]

Where, . i(t) = current through inductor at any time t

i(infinity) = current through inductor at steady state when t>0

i(0+) = current through inductor at initial state(>0)

T = time constant

When t >0 ,it becomes source free circuit. Whole energy of the inductor will get dissipated through the resistor.Hence at new steady state, current through inductor will be zero.

i(t)= 0 + [ 16 - 0 ]

i(t) = 16 A