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Reminder sheet

RLC circuits

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Coils and capacitors resist alternating current without heating: they are reactances. Combined with R they form the impedance, which Fresnel draws so well.

In AC a coil and a capacitor oppose the current without dissipating energy: their reactance is X_L = L·ω (grows with f) and X_C = 1/(C·ω) (decreases with f). The current lags the voltage by 90° in a coil, leads by 90° in a capacitor.

The impedance Z (Ω) generalises resistance: U = Z·I. In series, Z = √(R² + (X_L − X_C)²) and tan φ = (X_L − X_C)/R. When X_L = X_C the circuit is at resonance: Z = R, maximum current.

Formulas

ReactancesXL = L · ω
XC = 1C · ω
QuantitySymbolUnit
ReactanceXΩ
InductanceLH
CapacitanceCF
Angular frequencyωrad/s
Series impedanceZ = √(R² + (XL − XC)²)
QuantitySymbolUnit
ImpedanceZΩ
ResistanceRΩ
Ohm's law in ACU = Z · I
QuantitySymbolUnit
RMS voltageUV
RMS currentIA
Phase shifttan φ = XL − XCR
cos φ = RZ
QuantitySymbolUnit
Phase shiftφrad

Worked example

A series circuit R = 100 Ω, L = 0.2 H, C = 20 µF is supplied at 230 V, 50 Hz. Find the reactances, the impedance, the current and the phase shift.

ω = 2π × 50 = 314 rad/s.

X_L = L · ω = 0.2 × 314 = 62.8 Ω; X_C = 1 / (C · ω) = 1 / (20 × 10⁻⁶ × 314) = 159 Ω.

Z = √(R² + (X_L − X_C)²) = √(100² + (−96.2)²) = 139 Ω.

I = U / Z = 230 / 139 = 1.66 A.

tan φ = (X_L − X_C) / R = −0.962 → φ = −44°: X_C dominates, the circuit is capacitive, the current leads the voltage.

Common mistakes

  • Adding R, X_L and X_C like series resistors: they are not in phase, they combine as vectors (Pythagoras).
  • Using f instead of ω in the reactances: X_L = L·ω with ω = 2π·f.
  • Forgetting the sign of the phase shift: X_L > X_C, inductive circuit, current lags; X_C > X_L, capacitive circuit, current leads.

IEC symbols

Resistor
Coil
Capacitor
AC source

Illustration

RX_L − X_CZφ

Impedance triangle: Z² = R² + (X_L − X_C)².

Frequently asked questions

What is resonance?
The frequency at which X_L = X_C: the two reactances cancel, Z = R, the current is maximum and in phase with the voltage. It is f₀ = 1 / (2π√(LC)); this is how a radio receiver selects a station.
Do a coil or a capacitor consume power?
No active power: they store energy during a quarter period and give it back during the next. Only the resistance turns energy into heat. They do, however, exchange reactive power with the mains.

21 exercises in this topic

This chapter is covered by the paths: General electricity, ISL-oriented exercises

All sheets · All formulas · Glossary of electrical quantities · Standard electrical symbols (IEC 60617)

Practise this topic
Simulated time00:00:00
Q = I·t-
W = P·t-

Workbench

Circuit diagram CEI 60617

How does it work?

The perfboard: each big hole is a circuit node. You push a single leg of a component (resistor, lamp...) into it. The 4 small holes around it are connected to it: that is where chips and leads plug in.

  1. Drag a component onto the board: it snaps onto the grid. R rotates it.
  2. To join neighbouring big holes, click Jumper chip in the palette (or C) then drag on the board: a line of chips is laid at once. Bridged holes become a single node.
  3. For long links and for the power supply (placed next to the board), drag a lead from one big hole to another (or to a terminal).
  4. As soon as a loop is closed, the circuit is solved in real time (Ohm + Kirchhoff) and the diagram follows.
  5. Zoom with the wheel (or + / −, two-finger pinch); drag the background to pan, 0 fits the board. The + buttons around the board add a row or a column.

Del delete · R rotate · C chip brush · Esc cancel · wheel zoom · drag the background to pan

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