The circuit of Figure P5.49 is in steady state at t=0^-. Find v for t>0 if L is (a) 2.4 H , (b)

step 1 First up, we can calculate the inductive time constant of the circuit, Tor. Tor is simply given

The circuit of Figure P5.49 is in steady state at t=0^-....

Key Concepts

Inductor Voltage-Current Relationship

The fundamental relationship for an inductor is given by v = L(di/dt), indicating that the voltage across an inductor is proportional to the rate of change of current. This principle is key to deriving and solving the differential equations that describe the transient behavior of circuits containing inductors.

Time Constant

The time constant, usually denoted by tau (?), is a critical parameter in transient analysis. In an RL circuit, it is defined as ? = L/R, where L is the inductance and R is the resistance in the circuit. This constant determines the speed of the exponential response: a larger time constant results in a slower rate of change in current and voltage.

Transient Response of RL Circuits

When a sudden change, such as a switch closing or opening, occurs, an RL circuit does not immediately settle to a new steady state. Instead, it exhibits a transient response where the current and voltage change over time according to a differential equation. This response is typically exponential in nature, reflecting the gradual adjustment from the initial to the final state.

Steady-State Analysis

In circuit analysis, steady-state refers to the condition where all circuit variables (currents and voltages) have reached constant values. For circuits containing inductors, this means that after a long time with constant excitation, the inductor behaves as a short circuit due to its inability to sustain a voltage drop when the current is unchanging.

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