Series LCR Circuit and Resonance: The Complete Guide

A series LCR circuit connects a resistor, an inductor and a capacitor in series with an AC source. Because the same current flows through all three, their voltage drops add as phasors: the resistor drop is in phase with the current, the inductor drop leads by 90 degrees and the capacitor drop lags by 90 degrees. At one special frequency the inductor and capacitor effects cancel exactly, and this condition is called resonance.

Resonance matters far beyond the textbook. It is how a radio receiver selects one station from hundreds, how induction heaters and wireless chargers transfer energy efficiently, and why certain frequencies can cause dangerously large voltages in power systems. The series LCR circuit is the single most important AC network to understand well.

Key Concepts

• Impedance (Z): the total opposition of the series combination to AC, measured in ohms; it combines resistance and the net reactance.
• Resonant frequency (f₀): the frequency at which inductive and capacitive reactances are equal and cancel.
• Resonance condition: XL = XC, impedance is minimum and purely resistive, current is maximum, and voltage and current are in phase.
• Voltage magnification: at resonance the voltages across L and C individually can be many times the supply voltage, though they cancel each other.
• Sharpness of resonance: how selectively the circuit responds near f₀, described by the quality factor.

Formulas

The impedance of the series circuit is Z = √(R² + (XL − XC)²), where R is in ohms, XL = 2πfL and XC = 1/(2πfC). The RMS current is Irms = Vrms / Z and the phase angle satisfies tan φ = (XL − XC) / R. Resonance occurs when XL = XC, which gives the resonant frequency f₀ = 1 / (2π√(LC)) hertz, with L in henrys and C in farads. At resonance Z = R (minimum) and Irms is maximum.

Worked Example

A series circuit has R = 10 Ω, L = 0.1 H and C = 100 μF. Find the resonant frequency and the RMS current at resonance when a 230 V supply is connected, and also the voltage across the inductor at resonance.

1. f₀ = 1 / (2π√(LC)) = 1 / (2π√(0.1 × 100 × 10⁻⁶)) = 1 / (2π√(10⁻⁵)) = 1 / (2π × 3.16 × 10⁻³) ≈ 50.3 Hz.
2. At resonance Z = R = 10 Ω, so Irms = 230 / 10 = 23 A.
3. XL at resonance = 2πf₀L = 2π × 50.3 × 0.1 ≈ 31.6 Ω.
4. Voltage across inductor VL = Irms × XL = 23 × 31.6 ≈ 727 V.

Although the supply is only 230 V, the inductor (and equally the capacitor) has about 727 V across it at resonance, roughly three times the supply. The two large voltages are 180 degrees out of phase with each other, so they cancel and the supply still sees only the 230 V dropped across the resistor.

Common Misconceptions

• At resonance the impedance becomes zero. Wrong. At resonance the reactances cancel, but the resistance remains, so the minimum impedance is Z = R and the current, though maximum, is finite.
• The large voltages across L and C violate energy conservation. Wrong. They are equal and opposite in phase, so their sum is zero at every instant. Energy simply oscillates between the inductor magnetic field and the capacitor electric field.
• Resonance happens at the supply frequency automatically. Wrong. Resonance happens only when the supply frequency equals 1/(2π√(LC)). Changing L or C shifts the resonant frequency, which is exactly how tuning works.

Real-World Applications

Radio and television tuners vary a capacitor so the circuit resonates at the frequency of the desired station, rejecting all others. Induction cookers and metal heaters operate near resonance to deliver large currents efficiently. Wireless phone chargers use resonant coupling between coils. In power systems, engineers avoid accidental resonance because the magnified voltages can puncture insulation. Band-pass filters in audio and communication equipment are practical LCR resonant circuits.

In a series LCR circuit the impedance is minimum and the current maximum at the resonant frequency f₀ = 1/(2π√(LC)), where the inductor and capacitor voltages cancel. This one condition explains radio tuning, efficient power transfer and a few electrical hazards, making it the centrepiece of AC theory.