RL Circuit Calculator

How this calculator analyzes an RL circuit

A series RL circuit behaves differently depending on the type of source. With a DC source and a switch, the inductor resists sudden current changes, producing an exponential rise (or decay) in current described by the RL time constant. With a steady AC source, the circuit settles into a steady state where inductive reactance combines with resistance to form impedance, and current lags voltage by a phase angle determined by their ratio.

Select DC transient mode to see the time constant and the current/voltage at a specific elapsed time after switching. Select AC steady-state mode to see reactance, impedance, phase angle, power factor, and power at a given frequency and RMS supply voltage.

Formula steps

Safety guidance: back-EMF, heating, and AC resistance

• Back-EMF on switching: interrupting current through an inductor abruptly produces a voltage spike proportional to the current being interrupted and how quickly it's cut off. Relay coils, solenoids, motor windings, and other inductive loads commonly need flyback diodes or snubber circuits across the switching device to prevent arcing or component damage.
• Continuous vs. transient current rating: the resistor and inductor in an RL circuit should each be rated for the actual steady-state (or worst-case transient) current calculated here, not just the initial or average current — resistors dissipate I²R continuously once current reaches steady state.
• AC resistance and skin effect: at higher AC frequencies, a conductor's effective resistance can be higher than its DC resistance due to skin effect (current crowding toward the conductor's surface). This calculator uses a single resistance value and does not model this increase, which can matter for high-frequency inductor windings.

Code cautions

This calculator performs component-level circuit analysis and does not verify code compliance. Switching devices, relays, and contactors used with inductive loads are typically given separate ratings for resistive versus inductive loads by their manufacturer and relevant standards (such as UL and IEC ratings), and should be selected accordingly rather than assuming a resistive-load rating applies. For motor circuits, transformer primaries, and other significant inductive loads in fixed installations, conductor and overcurrent protection sizing should follow the applicable electrical code (for example, NEC Article 430 for motors) rather than this calculator's simplified analysis.

Design limitations of this calculator

• Ideal, linear components assumed: core saturation, hysteresis and eddy current losses in the inductor, and resistor tolerance/temperature coefficient are not modeled.
• Single time constant / single frequency: DC mode assumes a single switching event from a fixed source; AC mode assumes a pure sinusoidal steady-state at one frequency, not harmonics or transient startup behavior.
• No mutual inductance or coupling: circuits with transformers or coupled inductors are not modeled.
• No wiring or contact resistance: connections are assumed ideal; real-world contact resistance is not included.
• Series topology only: this calculator models a single series RL branch, not parallel or mixed RL networks.

Frequently asked questions

• What is an RL circuit?
  An RL circuit contains a resistor (R) and an inductor (L) connected together, most commonly in series. Its behavior differs depending on the source: with a DC source and a switch, it exhibits a transient current rise or decay described by an exponential curve; with an AC source, it reaches a steady state where current and voltage are related by impedance and a phase angle rather than a simple resistance.
• How do you calculate the time constant of an RL circuit?
  The RL time constant is τ = L ÷ R, measured in seconds. It represents how quickly current rises toward (or decays from) its final value after a DC voltage is applied or removed — after about 5 time constants, the current is considered to have essentially reached steady state.
• How do you calculate impedance in an AC RL circuit?
  Total impedance combines resistance and inductive reactance using the Pythagorean relationship, since they are 90° out of phase: Z = √(R² + Xl²), where Xl = 2π × f × L is the inductive reactance at the operating frequency. Impedance is measured in ohms, just like resistance and reactance individually.
• What is the phase angle in an RL circuit?
  In an AC RL circuit, current lags voltage by a phase angle θ = arctan(Xl ÷ R). At very low frequency (Xl near zero), the circuit behaves almost purely resistively and the phase angle approaches 0°. At very high frequency (Xl much larger than R), the phase angle approaches 90°, meaning current lags voltage by nearly a quarter cycle.
• What is power factor in an RL circuit?
  Power factor is the cosine of the phase angle, PF = cos(θ), and represents the fraction of apparent power (V × I) that does real, useful work. A purely resistive circuit has a power factor of 1 (unity); an RL circuit always has a lagging power factor below 1, since some of the apparent power is reactive power stored and returned by the inductor rather than consumed.
• Why does interrupting current in an RL circuit create a voltage spike?
  An inductor opposes any sudden change in current. If a switch interrupts current abruptly, the rate of change of current (dI/dt) becomes very large for an instant, and since inductor voltage is V = L × (dI/dt), this produces a brief but potentially very high voltage spike — often called inductive kickback or back-EMF — which is why flyback diodes or snubbers are standard practice for switched inductive loads.
• What happens to current in an RL circuit right after power is applied?
  At the instant a DC voltage is applied, an inductor initially opposes the change and current starts at zero, then rises exponentially toward its final steady-state value (V ÷ R) following I(t) = Ifinal × (1 − e^(−t/τ)). This is the opposite of a capacitor, which initially behaves like a short circuit before charging up.
• How is real power different from apparent power in an RL circuit?
  Real power (P, in watts) is the power actually dissipated as heat in the resistance: P = I² × R. Apparent power (S, in volt-amps) is simply voltage times current: S = V × I. Because current lags voltage in an RL circuit, S is always greater than P, and the difference is accounted for by reactive power (Q, in VAR) stored and released by the inductor: Q = I² × Xl.