RL Circuit: Time Constant τ = L/R, Transient Response

Every switch-mode power supply, relay driver, motor coil, and transformer winding presents an inductor to its driving circuit, and every time that circuit switches, the current cannot jump instantaneously — it slews at a rate set by L/R. Ignoring this is how designers get flyback spikes that blow MOSFETs, and respecting it is how snubbers, freewheeling diodes, and gate drivers get sized correctly. The RL circuit's transient response is the textbook model behind all of it.

The Governing Equation

Consider a series RL circuit driven by a step input — a DC source V suddenly connected through a switch, resistor R, and inductor L:

   R          L
V ─/\/\/─┬───UUUU───┐
         │           │
        (switch)     │
         │           │
         └───────────┘  (loop, current i(t))

Apply KVL around the loop for t ≥ 0 (switch closes at t = 0):

V = i(t)·R + L·di(t)/dt

This is a first-order linear ODE. The inductor's defining relation, V_L = L·di/dt, is what makes current — not voltage — the state variable that cannot change instantaneously. Physically this follows from energy storage: energy in an inductor is E = ½·L·i², and a real inductor cannot absorb or release energy in zero time without infinite instantaneous power, so i(t) must be continuous even though v(t) across it can jump.

Solving for the Current: Charging Case

Assume zero initial current, i(0) = 0 (inductor fully "discharged"). Solving the ODE:

i(t) = (V/R)·(1 − e^(−t/τ)),   where τ = L/R

τ, the time constant, has units of seconds: [H]/[Ω] = [V·s/A]/[V/A] = s. It sets the pace of the exponential — the current rises toward its final value I_final = V/R (the steady-state condition where the inductor behaves as a short circuit, since di/dt → 0 means V_L → 0).

The voltage across the inductor is found from V_L = L·di/dt:

V_L(t) = V·e^(−t/τ)

It starts at the full source voltage V (inductor initially behaves as an open circuit, opposing the sudden current change) and decays to 0.

Solving for the Discharge Case

Now consider the inductor discharging into R alone (source removed, loop closed through R), with initial current I₀:

i(t) = I₀·e^(−t/τ)
V_R(t) = I₀·R·e^(−t/τ)
V_L(t) = −I₀·R·e^(−t/τ)

Note the polarity flip on V_L — the inductor now acts as a source, driving current in the same direction it was flowing, with the voltage reversing sign relative to the charging case to maintain current continuity.

The Universal Exponential Form

Both cases collapse into one general formula, useful for any RL transient regardless of initial and final conditions:

x(t) = x(∞) + [x(0) − x(∞)]·e^(−t/τ)

where x(t) can be current or voltage, x(0) is the initial value, x(∞) is the final (steady-state) value, and τ = L/R. This is the same form used for RC transients (with τ = RC instead), which reflects the duality between the two energy-storage elements.

τ = L/R: Physical Meaning and Percentages

τ is the time for the current to cover 63.2% of the remaining distance to its final value. Multiples of τ give standard checkpoints:

5τ is the standard engineering rule of thumb for "fully settled" — beyond this the residual error is under 1%, usually smaller than component tolerances anyway.

A larger R makes τ smaller (faster transient) — more resistance dissipates energy faster, forcing the current to settle quicker. A larger L makes τ larger (slower transient) — more stored energy per unit current takes longer to build up or bleed off. This is the opposite scaling from RC circuits' capacitor-charging resistance dependence, worth keeping straight: in RL, R speeds things up; in RC, R slows things down.

Worked Example

A relay coil has L = 100 mH and winding resistance R = 50 Ω, driven from a V = 24 V supply through a switch.

Step 1 — time constant:

τ = L/R = 0.1 H / 50 Ω = 2 ms

Step 2 — final current:

I_final = V/R = 24 V / 50 Ω = 0.48 A

Step 3 — current at t = 1 ms (half of τ):

i(1 ms) = 0.48·(1 − e^(−1/2)) = 0.48·(1 − 0.6065) = 0.48·0.3935 = 0.189 A

Step 4 — voltage across the inductor at t = 1 ms:

V_L(1 ms) = V·e^(−t/τ) = 24·e^(−0.5) = 24·0.6065 = 14.56 V

Check via KVL at t = 1 ms:

V_R = i·R = 0.189 A × 50 Ω = 9.45 V
V_R + V_L = 9.45 + 14.56 = 24.01 V ≈ V  ✓ (rounding)

KVL closes to within rounding error, confirming the solution is consistent.

Step 5 — settling time: 5τ = 10 ms. After the switch opens, if the coil discharges through a flyback diode (assume ~0.7 V forward drop, small compared to the 24 V driving the buildup — approximated here as ideal short for R-only decay), current decays with the same τ = 2 ms, reaching near-zero by 10 ms.

Practical Implications

• Inductive kickback: opening the switch abruptly forces di/dt to be enormous if no discharge path exists, producing V_L = L·di/dt → ∞ in the ideal limit — real circuits see this as a large negative voltage spike that can exceed a switch's or MOSFET's breakdown rating. This is why relay drivers and inductive loads always need a freewheeling diode, snubber (RC or RCD), or TVS clamp across the coil.
• Switching speed vs. dissipation trade-off: faster current buildup (smaller τ, i.e., higher R) means faster response but more resistive loss; motor and relay drive circuits balance this against power budget.
• Gate driver design: MOSFET gate drive loops have parasitic inductance in series with gate resistance; the L/R time constant of this parasitic loop affects turn-on/turn-off speed and ringing, especially at high switching frequencies.
• Transformers and motor windings: the L/R time constant of a winding determines how quickly flux (and hence torque, in a motor) can build after a voltage step is applied — critical in servo and motor-control loop bandwidth calculations.
• EMI and snubber sizing: τ estimates how long a decaying inductive current will continue to source energy into a snubber network after a switch opens, informing snubber capacitor/resistor sizing.

Key Takeaways

• The RL series circuit's step response is governed by τ = L/R, the time constant, with units of seconds.
• Current through an inductor cannot change instantaneously (energy continuity); voltage across it can, and does, jump at switching instants.
• Charging: i(t) = (V/R)(1 − e^(−t/τ)); discharging: i(t) = I₀·e^(−t/τ) — both are special cases of the universal form x(t) = x(∞) + [x(0) − x(∞)]e^(−t/τ).
• 5τ is the standard "settled" threshold (>99% of final value); 1τ marks 63.2% progress.
• Unlike RC circuits, increasing R in an RL circuit shortens τ — more resistance forces faster current decay/rise, the opposite scaling from capacitive charging.
• Abrupt interruption of inductor current without a discharge path causes large voltage spikes — the root cause of inductive kickback, and the reason flyback diodes/snubbers are mandatory across relay coils and switched inductive loads.