Questions: Transient Response in RL Circuits

Question 1

A switch closes at t = 0, connecting a 12V source to a series RL circuit with R = 4Ω and L = 2H. The inductor was initially unenergized. What is the current immediately after the switch closes?

• A. 3 A — the steady-state value V/R
• B. 6 A — the initial rate of change V/L
• C. 0 A — the inductor opposes instantaneous current change
• D. 1.5 A — the average of initial and final values

Answer: (C)

Inductor current cannot change instantaneously because an instantaneous jump would require infinite voltage (v = L di/dt). Since the initial current is 0 A before the switch closes, it must remain 0 A immediately after. The current then grows exponentially toward the steady-state value of 3 A (V/R = 12/4) with time constant τ = L/R = 0.5 s. The misconception in option A is treating the circuit as having instantly reached DC steady state.

Question 2

An inductor carrying 500 mA is suddenly disconnected by opening a switch, with no alternative current path provided. What happens?

• A. The current drops instantly to zero as the magnetic field collapses
• B. The current decays exponentially to zero through the inductor's winding resistance
• C. A large voltage spike appears across the switch terminals as the inductor attempts to sustain current with no available path
• D. The inductor stores the energy indefinitely until a path is provided

Answer: (C)

An inductor cannot accept instantaneous current interruption — it will generate whatever voltage is necessary to maintain current flow. With no path available, the voltage at the switch terminals climbs rapidly (theoretically to infinity) until an arc forms or the energy is dissipated through parasitic breakdown. This is the inductive voltage spike problem. In practice it destroys unprotected transistors in switching circuits. The solution is a freewheeling diode that provides a controlled current path for the decaying inductor current.

Question 3

After exactly one time constant (τ = L/R), the current in an RL circuit has reached 100% of its final steady-state value.

• T. True
• F. False

Answer: False

After one time constant, the current has reached approximately 63.2% of its final value — not 100%. The exponential formula i(t) = I_f(1 − e^(−t/τ)) approaches I_f asymptotically; it never reaches I_f exactly at a finite time. In practice, the transient is considered 'over' after about 5τ, when the current is within 1% of I_f. This is a common misreading of time constant definitions.

Question 4

The current through an inductor cannot change instantaneously because any instantaneous change would require infinite voltage.

• T. True
• F. False

Answer: True

This follows directly from v = L di/dt. An instantaneous current change means dt → 0 while di is finite, making v → ∞. Since infinite voltage is physically unrealizable, current must change continuously and smoothly. This is the single physical constraint that generates all RL transient behavior: the circuit is forced to negotiate a smooth exponential transition from initial to final current, governed by the time constant τ = L/R.

Question 5

Explain the 'three-element recipe' for solving any first-order RL transient. What are the three quantities, and why is each one determined independently?

Model answer: The three elements are: (1) I_i, the initial current — determined by the inductor's current just before the switching event, because inductor current cannot jump; (2) I_f, the final (steady-state) current — determined by DC analysis of the new circuit after switching, treating the inductor as a short circuit since di/dt → 0 at steady state; (3) τ = L/R — the time constant determined by the Thévenin resistance seen by the inductor in the new circuit after switching. With these three values, the complete transient is i(t) = I_f + (I_i − I_f) · e^(−t/τ).

The three quantities are independent because each answers a different question about the circuit: where we start (initial condition via continuity), where we end up (DC steady state), and how fast we get there (the ratio of energy storage to dissipation). Any first-order RL circuit — regardless of source configuration or initial conditions — is fully characterized by these three numbers.