Physics
RLC circuit calculator: impedance, resonance and transients
Reactances, impedance, resonance, power factor and the RC and RL time constants. Clear the unknown and the formula rearranges — the frequency too, and the component itself.
Opposite reactances, and the frequency where they cancel
At direct current a capacitor is an open circuit and a coil is just a wire. At alternating current both oppose the flow, but in opposite senses and depending on the frequency: the inductive reactance grows with f, the capacitive one falls. It is not true resistance, because it dissipates no energy: it stores energy for a quarter of a cycle and hands it back in the next.
Because the two reactances shift the current in opposite directions, they enter the impedance with a minus sign: Z = √(R² + (Xʟ − Xᴄ)²). There is therefore a frequency at which they cancel exactly, and the impedance falls to its lowest possible value, the resistance alone. That is resonance, f₀ = 1/(2π√(LC)), and it is how a radio picks one station out of all those reaching the aerial.
The power factor measures how much of the flowing current is doing any good. An industrial motor with cos φ = 0.7 draws 43% more current from the grid than the same useful power would need, and the losses along the lines grow with the square of the current. That is why large consumers are charged a penalty on their power factor and install capacitor banks to correct it.
Common mistakes
- Adding the reactances to the impedance as if they were resistances: they must first be subtracted from each other, then combined with R by Pythagoras.
- Confusing apparent power with real power: the first, in volt-amperes, is what the supply has to deliver; the second, in watts, is what the load actually converts.
- Assuming that more resistance makes an RL transient slower: the constant is L/R, so a larger resistance makes it faster. For RC the opposite is true.
Frequently asked questions
How do you work out the impedance of a series RLC circuit?
Z = √(R² + (Xʟ − Xᴄ)²), with Xʟ = 2πfL and Xᴄ = 1/(2πfC). The reactances subtract because they shift the current in opposite directions, and only the result is combined with R.
What is the formula for the resonant frequency?
f₀ = 1/(2π√(LC)). At that frequency the two reactances are equal and cancel, and the impedance falls to the resistance alone: the current is at its maximum.
What is the power factor?
It is cos φ, the cosine of the phase angle between voltage and current. It says what fraction of the apparent power becomes real power: at cos φ = 1 all of it, at cos φ = 0 none.
How long does an RC transient last?
The time constant is τ = R·C. After τ the capacitor is at 63%, after 3τ at 95% and after 5τ at 99%: by convention it is treated as finished after five time constants.
How this calculation works
Inductive reactance: Xʟ = 2π·f·L = ω·L. Capacitive reactance: Xᴄ = 1/(2π·f·C) = 1/(ω·C). Impedance of a series RLC: Z = √(R² + (Xʟ − Xᴄ)²). Phase angle: tan φ = (Xʟ − Xᴄ)/R. Resonant frequency: f₀ = 1/(2π·√(L·C)), where Xʟ = Xᴄ and Z = R. Real power: P = V·I·cos φ, in watts. Apparent power: S = V·I, in volt-amperes. Reactive power: Q = V·I·sin φ, in var. RC transient: τ = R·C, with charge q(t) = Q(1 − e^(−t/τ)). RL transient: τ = L/R, with current i(t) = I(1 − e^(−t/τ)). After τ it reaches 63%, after 3τ 95%, after 5τ 99%.
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