RLC Circuit Simulator

Three components, one equation

Go around the loop and add the voltages. The supply pushes, the resistor drops IR, the inductor drops L·dI/dt, and the capacitor holds q/C. Kirchhoff says they balance, which gives L·d²q/dt² + R·dq/dt + q/C = V.

That is the damped driven oscillator with different letters. Inductance behaves as mass, resistance as the damping coefficient, and one over capacitance as the spring constant. The mass spring simulator on this site solves the identical equation, and putting the two side by side is the quickest way to believe that one piece of mathematics describes two things that look nothing alike.

Why the capacitor overshoots the supply

Charge the capacitor through a resistor alone and its voltage climbs to the supply and stops. Add an inductor and it sails past. The reason is that an inductor resists any change in current, so when the capacitor reaches the supply voltage the current does not obligingly stop. It keeps flowing, keeps delivering charge, and drives the voltage above the supply until the capacitor’s excess over the supply, which reverses the voltage across the inductor, finally turns it around.

The size of the overshoot is exp(−πζ/√(1−ζ²)), which depends on the damping ratio and nothing else. At ζ = 0.1 it is 73 percent, at 0.5 it is 16 percent, and at 1 it is zero. That is a real hazard rather than a curiosity: a 5 V logic supply that overshoots to 8.6 V on power-up can destroy whatever it is feeding.

The current starts at zero, not at V over R

This surprises people who have only met RC circuits. In an RC circuit the current jumps instantly to V/R at the moment of switching. Here it starts at exactly zero, because a step change in current through an inductor would need an infinite voltage across it.

Watch the labels at the instant the switch closes: the entire supply voltage sits across the inductor, and none across the resistor, because no current is flowing yet. That is what makes the inductor the component setting the initial behaviour.

What critical damping is for

The critical resistance is 2√(L/C), and it separates two qualitatively different behaviours.

• Below it. The circuit rings at f₀√(1 − ζ²) and overshoots, and well below it takes a while to settle down.
• At it. The fastest possible approach to the final voltage with no overshoot at all, which is what an instrument or a control loop needs wherever any overshoot is a false reading. It is not the fastest to settle: at ζ around 0.8 the circuit overshoots by about 1.5 percent but settles within 2 percent of the supply sooner.
• Above it. No overshoot, but slower, and getting slower the more resistance you add. Past critical, more damping is worse, not safer.

Set the resistance to the critical value shown in the readouts and watch the last of the ringing vanish. Then push it to three times critical and watch the settling time grow.

Where the energy goes

The bar under the schematic shows how the stored energy splits between the capacitor’s electric field and the inductor’s magnetic field. Set the resistance to zero and the split sloshes back and forth indefinitely, exactly as kinetic and potential energy do in an undamped spring. The resistor is the only component that removes energy from the circuit; it turns it into heat, which is why the ringing dies away.

What this model leaves out

Ideal components. Real inductors have winding resistance, often tens of ohms, which adds to whatever R you set and damps the circuit more than you asked for; that alone explains many circuits that ring less than predicted. Real capacitors have series resistance and leak. Neither is included here.

The supply is also ideal, with no internal resistance and an instantaneous switch. And everything is linear: no component saturates, and the inductor’s inductance never falls as the current rises, which a real ferrite core does.

Common mistakes

• Expecting the capacitor to stop at the supply voltage. Only if the resistance is at or above critical.
• Using the resonant frequency as the ringing frequency. The circuit rings at f₀√(1 − ζ²), which is lower, and does not ring at all past critical.
• Thinking more resistance always settles things faster. Broadly true well below critical, but the quickest settling comes a little under it, at ζ around 0.8, and past critical more resistance only slows it.
• Forgetting the inductor’s own resistance. It adds to R, and for a small inductor it can dominate whatever resistor you fitted.
• Mixing prefixes. Millihenries with microfarads, or henries with nanofarads, throws √(LC) out by a factor of a thousand or more. The sliders here are in mH and nF and the conversion is done for you.
• Assuming no current means no stored energy. At the peak of the overshoot the current is zero, so the inductor stores nothing, yet the capacitor holds more energy than it will once the circuit has settled.