The switch in the circuit of Fig. 8.55 has been closed a...

The switch in the circuit of Fig. 8.55 has been closed a ridiculously long time before suddenly being thrown open at $t=0$. (a) Obtain expressions for $i_L$ and $v$ in the circuit of Fig. 8.55 which are valid for all $t \geq 0$. (b) Calculate $i_L(t)$ and $v(t)$ at the instant just prior to the switch opening, at the instant just after the switch opening, and at $t=470 \mu \mathrm{s}$.

Key Concepts

Time Constants and Exponential Decay

In first-order transient circuits, such as those involving an inductor and resistor, the time constant (a product of resistance and inductance, in RL circuits) characterizes the rate at which the circuit variables exponentially approach their new steady-state values. The overall response is typically expressed as a simple exponential decay (or growth) function with this time constant.

Inductor Behavior and Initial Conditions

Inductors resist changes in current due to their stored magnetic energy, ensuring the current through them is continuous even if the circuit is suddenly altered. This principle is central to correctly determining initial conditions after switching events, as the inductor current immediately after an abrupt change remains equal to the pre-switch value, while the voltage across it can change abruptly.

Steady-State Analysis

This concept deals with the behavior of circuits after they have been operating for a long period, such that all transient responses have died out and the circuit variables become constant. In this context, determining the inductor current just before the switch is opened relies on the steady?state condition reached during long-term operation, where the inductor acts as a short circuit in DC circuits.

Transient Response in Circuits

The study of how circuit variables (like current and voltage) change immediately after a sudden change, such as switching, is known as transient analysis. It involves solving differential equations that characterize the natural response of the circuit away from the steady state, emphasizing the system’s behavior immediately after the switch is thrown.

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