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Question 19.SP.3: In the circuit shown (Fig. 19-8), find (a) the total inducta......

In the circuit shown (Fig. 19-8), find (a) the total inductance and (b) the time constant. Assume that a current of 10 A is flowing when the switch S is opened. Find (c) the current 2 s later.

19.8
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(a) Reduce the series-parallel combination .of inductances to its equivalent.

Parallel:   L_a=\frac{L_1 L_2}{L_1+L_2}=\frac{5(1)}{5+1}=\frac{5}{6}=0.83\ H

Parallel:   L_b=\frac{L_3 L_4}{L_3+L_4}=\frac{4(4)}{4+4}=\frac{16}{8}=2.0\ H

Then series:   L_T=L_a+L_b=0.83+2.0=2.83\ H

(b) The time constant of the circuit is the ratio of total inductance to the total resistance of the circuit.

T=\frac{L_T}{R}=\frac{2.83}{5}=0.57\ s

(c) Write the formula for decaying current. The V/R value is 10 A.

i=\frac{V}{R} e^{-R t / L}    (19-7)

\begin{aligned}-\frac{R t}{L_T} & =-\frac{5(2)}{2.83}=-3.5 \\ e^{-3.5} & =0.030\end{aligned}

Then i = 10(0.030) = 0.30 A

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