RL Circuit Calculator

Calculate time constant, resistance, inductance, current at any time, or time to reach target current in RL circuits.

RL Circuit Formulas: Time Constant: τ = L / R Where: • τ (tau) = time constant (seconds) • L = inductance (henries, H) • R = resistance (ohms, Ω) Current Rise (starting from 0A): I(t) = If × (1 - e^(-t/τ)) where If = Vs/R (final current) Current Decay (starting from I0): I(t) = I0 × e^(-t/τ) Time to reach current: • Rising: t = -τ × ln(1 - I/If) • Decaying: t = -τ × ln(I/I0) Inductor Voltage: VL(t) = Vs × e^(-t/τ) (during rise) VL = L × (di/dt) (general form) Resistor Voltage: VR(t) = I(t) × R Energy Stored in Inductor: E = ½ × L × I² Key Time Points (Rising): • 1τ: 63.2% of final current • 2τ: 86.5% of final current • 3τ: 95.0% of final current • 4τ: 98.2% of final current • 5τ: 99.3% of final current
Example 1 (Time Constant - Relay Coil): L = 500 mH, R = 100 Ω, Vs = 12V τ = 0.5 / 100 = 0.005 s = 5 ms If = 12 / 100 = 0.12 A = 120 mA Time to 99% = 5τ = 25 ms Energy stored = 0.5 × 0.5 × 0.12² = 3.6 mJ Example 2 (Current at Time - Motor Start): L = 50 mH, R = 10 Ω, Vs = 24V, t = 10 ms τ = 0.05 / 10 = 0.005 s = 5 ms If = 24 / 10 = 2.4 A I(10ms) = 2.4 × (1 - e^(-10/5)) I(10ms) = 2.4 × (1 - e^(-2)) I(10ms) = 2.4 × 0.865 = 2.08 A Example 3 (Time to Current - Solenoid): L = 200 mH, R = 50 Ω, Vs = 12V, Target = 0.15 A τ = 0.2 / 50 = 0.004 s = 4 ms If = 12 / 50 = 0.24 A t = -4 × ln(1 - 0.15/0.24) t = -4 × ln(0.375) t = -4 × (-0.981) = 3.92 ms Example 4 (Inductor Voltage at Time): L = 100 mH, R = 20 Ω, Vs = 5V, t = 3 ms τ = 0.1 / 20 = 0.005 s = 5 ms VL = 5 × e^(-3/5) VL = 5 × e^(-0.6) VL = 5 × 0.549 = 2.74V VR = 5 - 2.74 = 2.26V I = 2.26 / 20 = 0.113 A Example 5 (Current Decay - Relay Release): L = 1 H, R = 100 Ω, I0 = 0.12 A, t = 20 ms τ = 1 / 100 = 0.01 s = 10 ms I(20ms) = 0.12 × e^(-20/10) I(20ms) = 0.12 × e^(-2) I(20ms) = 0.12 × 0.135 = 0.016 A = 16 mA Example 6 (Buck Converter Inductor): L = 22 µH, R = 0.05 Ω (ESR), Vs = 12V τ = 0.000022 / 0.05 = 0.00044 s = 0.44 ms If = 12 / 0.05 = 240 A (very high!) Note: Real converters use switching, not DC analysis

What is the RL time constant?

The RL time constant (τ or tau) is the time required for current in an inductor to reach 63.2% of its final value or decay to 36.8% during current decrease. It's calculated as τ = L/R, where L is inductance in henries and R is resistance in ohms.

Why does an inductor oppose current changes?

Inductors generate a back EMF (electromotive force) proportional to the rate of current change: V = L(di/dt). This opposes sudden current changes, making current rise and fall exponentially rather than instantly, following the time constant τ = L/R.

What is the current rise formula in RL circuits?

I(t) = If × (1 - e^(-t/τ)), where I(t) is current at time t, If is final steady-state current (Vs/R), τ is time constant (L/R), and e is Euler's number. Current starts at zero and exponentially approaches If.

What is the current decay formula in RL circuits?

I(t) = I0 × e^(-t/τ), where I(t) is current at time t, I0 is initial current, and τ is time constant (L/R). When the voltage source is removed, current decays exponentially from I0 to zero through the resistor.

How long does it take for current to reach steady state?

Theoretically infinite, but practically after 5 time constants (5τ), current reaches 99.3% of final value, considered steady state. For τ = 10 ms, steady state is reached in about 50 ms.

What is the voltage across an inductor during current change?

VL(t) = Vs × e^(-t/τ) during rise, where Vs is source voltage. Initially VL equals Vs (opposing all voltage), then exponentially decays to zero as current stabilizes. The inductor voltage plus resistor voltage always equals source voltage.

What are practical applications of RL circuits?

RL circuits are used in inductive loads (motors, solenoids, relays), power supplies (buck/boost converters), electromagnetic braking, spark suppression (snubbers), low-pass filters, and energy storage in switching regulators. Inductors smooth current in power electronics.

Why do relays and solenoids need flyback diodes?

When current through an inductor is suddenly interrupted, the inductor generates a large voltage spike (V = L × di/dt) that can damage transistors/switches. A flyback diode provides a current path, allowing safe energy dissipation and preventing voltage spikes.

How does inductance affect the time constant?

Larger inductance increases time constant, making current changes slower. For R = 100 Ω: L = 10 mH gives τ = 0.1 ms, but L = 1 H gives τ = 10 ms. Large inductors resist current changes more strongly, storing more magnetic energy.

How does resistance affect current rise time?

Higher resistance decreases time constant (faster response). For L = 100 mH: R = 10 Ω gives τ = 10 ms, but R = 100 Ω gives τ = 1 ms. Higher resistance limits final current but allows faster approach to steady state.

What is inductive reactance and how does it relate to RL circuits?

Inductive reactance XL = 2πfL opposes AC current flow. In AC circuits, total impedance Z = √(R² + XL²). At DC (f=0), XL = 0, so only resistance matters. The RL time constant applies to transient DC response, not AC steady-state.

Can I measure inductance with an RL time constant?

Yes! Apply known voltage through known resistance, measure time for current to reach 63.2% of final value (that's τ), then calculate L = τ × R. Or measure current rise curve and fit to exponential equation. Accurate for millihenries to henries range.

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