GATE 2017 EC – Question 45

Networks, Signals and Systems · Time and Frequency Domain Analysis of Linear Circuits · 2 marks · Numerical answer

In the circuit shown, the voltage $V_{IN}(t)$ is described by:

$$V_{IN}(t) = \begin{cases} 0, & \text{for } t < 0 \\ 15 \text{ Volts}, & \text{for } t \geq 0 \end{cases}$$

where $t$ is in seconds. The time (in seconds) at which the current $I$ in the circuit will reach the value 2 Amperes is ________.

A voltage source $V_{IN}(t)$ in series with a $1\ \Omega$ resistor feeds two inductors in parallel, a $1$ H and a $2$ H inductor. The current $I$ is marked on the wire leading to the $2$ H inductor.

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Correct answer: 0.33 to 0.35

Explanation

The two inductors share the same voltage, so the current divides in the ratio of the inverse inductances: the 1 H inductor carries twice as much as the 2 H inductor, and the 2 H inductor carries one third of the total. The two in parallel are equivalent to $\frac{2}{3}$ H, so the total current is $15\left(1 - e^{-t/\tau}\right)$ with $\tau = \frac{L}{R} = \frac{2}{3}$ s. The current $I$ is a third of this, $5(1 - e^{-1.5t})$. Setting this equal to 2 gives $e^{-1.5t} = 0.6$, so $t = \frac{\ln(1/0.6)}{1.5} = 0.34$ s.