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RL Time Constant Calculator

Calculate the time constant and settling time of an RL circuit from inductance and resistance

Frequently Asked Questions

What is the RL time constant?

It is the inductance divided by the resistance and sets how quickly current rises. After one time constant the current reaches about sixty-three percent of its final value.

Why is settling time five time constants?

Each time constant closes about sixty-three percent of the remaining gap, so after five constants less than one percent of the gap remains.

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How This Calculator Works

An inductor resists sudden changes in current, so when an RL circuit is switched on the current rises gradually toward its final value. The time constant equals the inductance divided by the resistance and sets the pace of that rise. After one time constant the current reaches about sixty-three percent of its steady value.

How To Use It

Enter the inductance in millihenries and the series resistance in ohms. The calculator converts the inductance to henries internally, divides by the resistance, and returns the time constant in milliseconds. It also reports the time to reach roughly ninety-nine percent of the steady-state current, which is five time constants.

Why Five Time Constants

Each time constant closes about sixty-three percent of the remaining gap to the final current. After five time constants less than one percent of the gap remains, so engineers treat the circuit as fully settled at that point. The same five-constant rule applies to the decay when the source is removed.

Tips

Larger inductance slows the current rise.
Higher resistance speeds the rise toward steady state.
Five time constants is the practical settling time.

RL vs. RC Time Constants - Key Differences

Both RL and RC circuits produce exponential transitions, but the quantity that exponentiates is different. In an RC circuit, the voltage across the capacitor builds exponentially with time constant τ = RC: v(t) = V × (1 - e^(-t/τ)). The capacitor resists changes in voltage. In an RL circuit, the current through the inductor builds exponentially with time constant τ = L/R: i(t) = (V/R) × (1 - e^(-t/τ)). The inductor resists changes in current. Both reach 63.2% of their final value after exactly one time constant and are considered fully settled after five time constants (99.3%). The symmetry is deliberate: capacitors and inductors are electrical duals, and their circuits mirror each other mathematically.

Energy Storage

A capacitor stores energy electrostatically in the electric field between its plates: E = ½CV². An inductor stores energy magnetically in the field threading its coil: E = ½LI². When a switch opens in an RL circuit, the magnetic energy stored in the inductor must go somewhere - it cannot disappear instantly. The inductor forces current to continue flowing, which generates a voltage spike across the switch contacts. This is why relay coils, motor windings, and solenoids require flyback protection diodes (also called freewheeling or snubber diodes) to provide a safe path for the inductor current to decay.

Charging and Discharging Waveforms

When a voltage V is applied to a series RL circuit at t = 0, the current ramps up as i(t) = (V/R)(1 - e^(-t/τ)). At t = 0 the current is zero; as t approaches infinity the current approaches V/R. The voltage across the inductor is the remainder: v_L(t) = V × e^(-t/τ). At t = 0 the inductor takes the full supply voltage; at t = infinity the inductor voltage is zero and the resistor takes the full supply. When the source is removed and the circuit is shorted (or when the switch opens and a freewheeling path exists), the current decays as i(t) = I_0 × e^(-t/τ).

RL Time Constant in Power Electronics

In a flyback converter, the primary inductor current ramps up during the switch-on period at a rate di/dt = V_in / L. The energy stored (E = ½LI²) is transferred to the secondary during the switch-off period. The peak current at switch-off is I_pk = V_in × t_on / L. The RL time constant of the primary winding (where R includes winding resistance and reflected load) determines how quickly the inductor resets between cycles; incomplete reset in continuous conduction mode means the next cycle starts with nonzero current.

Inductive Kickback

When a switch in series with an inductive load opens, the inductor tries to maintain the current that was flowing. With no low-impedance path available, the voltage across the inductor rises rapidly: V = L × di/dt. A fast current collapse (large di/dt) can generate hundreds or thousands of volts in microseconds, destroying transistors, damaging insulation, and generating EMI. The standard protection is a freewheeling diode (also called a flyback, snubber, or clamp diode) placed in parallel with the inductive load, oriented to conduct when the switch opens. This provides a low-impedance path for inductor current to continue flowing, clamping the voltage at one diode drop above the supply rail. For faster clamping, a transient voltage suppressor (TVS) or RC snubber can be used. Relay coil demagnetization circuits must discharge the stored magnetic energy safely before the relay can be re-energized.

Motor Starting

Electric motors have a combination of resistance and inductance in their stator windings; the RL time constant governs how quickly current builds when voltage is applied. At the instant of start-up, only winding resistance limits current (back-EMF is zero at standstill), so inrush current can be 5 to 10 times the rated running current. Soft starters ramp the applied voltage gradually to limit this current. Variable frequency drives (VFDs) start the motor by ramping up both frequency and voltage from zero, which keeps slip and current within controlled limits throughout acceleration.

RL Filters and Impedance

An inductor's impedance is Z_L = jωL, which increases linearly with frequency. A series RL circuit acts as a low-pass filter when the output is taken across the resistor: at low frequencies the inductor has low impedance and passes most of the voltage to the resistor; at high frequencies the inductor has high impedance and blocks the signal. The -3 dB cutoff frequency is ω_c = R/L (in radians per second) or f_c = R/(2πL) in hertz. Alternatively, taking the output across the inductor produces a high-pass filter. RL filters are used in power line filtering, motor drive output filters, and audio crossovers where the current-handling capability of an inductor is preferred over a capacitor.

Ferrite Bead as RL Filter

A ferrite bead looks like a small inductor at low frequency but its behavior changes above its self-resonant frequency. Rather than acting as a pure reactive inductor (which would reflect energy back to the source), ferrite beads are specifically designed to be resistive at high frequencies - they absorb and dissipate EMI energy as heat in the ferrite core. This makes them highly effective at damping resonances and absorbing high-frequency noise on power rails and signal lines without creating the resonance problems that a purely reactive inductor could cause.

Inductor vs. Capacitor Filtering

Inductors and capacitors filter by complementary mechanisms. A series inductor blocks high-frequency current changes - it is effective against current-source type interference and is placed in series with the supply current path. A shunt capacitor absorbs high-frequency voltage changes - it is effective against voltage-source type interference and is placed from the supply rail to ground. Together in an LC pi filter (capacitor - inductor - capacitor), they provide steep attenuation of both types of interference, which is why pi filters appear at the input of virtually every switched-mode power supply and sensitive analog circuit.

Frequently Asked Questions

Why does the inductor voltage spike when a switch opens? The fundamental relationship is V = L × di/dt. If the current changes rapidly (large di/dt), the voltage generated by the inductor is large. When a switch opens with no freewheeling path, current attempts to drop to zero in the microseconds it takes the switch contacts to separate. Even a modest inductance of 1 mH dropping 1 A in 1 µs produces V = 0.001 × 1/0.000001 = 1000 V. This explains why unsuppressed relay coils and solenoids routinely destroy the transistors driving them.

How do I choose between RL and RC filtering? Consider the source impedance of the interference. If the noise source is high impedance (tends to impress a voltage), a shunt RC filter (capacitor to ground) is most effective. If the noise source is low impedance (tends to drive a current), a series RL filter (inductor in the current path) is most effective. For power supply filtering where both types of interference coexist, an LC filter that uses both elements provides the best rejection. Also consider that inductors can handle more current than capacitors without self-heating and do not require polarization, while capacitors are generally smaller and less expensive for a given filtering requirement.