Resonance in Series RLC Circuit

A series RLC circuit connects a resistor, inductor and capacitor in one path across an AC voltage source. The ideal series RLC circuit reaches resonance when its inductive and capacitive reactances have equal magnitudes.
The inductor and capacitor store energy in different fields.

1. An inductor carrying current stores energy in its magnetic field.
2. A charged capacitor stores energy in its electric field.

In a source-free ideal LC circuit, energy alternates between the inductor’s magnetic field and the capacitor’s electric field. Resistance dissipates energy and damps that natural oscillation. In a sinusoidally driven series RLC circuit, resonance occurs when inductive reactance equals capacitive reactance in magnitude, so their opposite imaginary impedance components cancel. The source then sees the minimum series impedance permitted by the circuit losses.

Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of Inductive Reactance Vs Frequency

For an ideal inductor, inductive reactance follows X_L = 2πfL, so X_L increases linearly with frequency. The value is zero in sinusoidal steady state at f = 0 and grows without bound as frequency approaches infinity. A real inductor also has winding resistance, parasitic capacitance, core loss and a finite self-resonant frequency, so it does not remain an ideal short or open over every frequency.

Variation of Capacitive Reactance Vs Frequency

For an ideal capacitor, capacitive reactance follows X_C = 1/(2πfC), so its magnitude decreases as frequency rises. The value is unbounded at f = 0 and approaches zero as frequency approaches infinity. A real capacitor has leakage, equivalent series resistance, equivalent series inductance and voltage limits, so the ideal open- and short-circuit descriptions apply only within an appropriate model and frequency range.

Inductive Reactance and Capacitive Reactance Vs Frequency

At low frequency, X_L is small and X_C is large. As frequency rises, the first increases and the second decreases. Their curves cross at the ideal resonant frequency f_r.
At resonance, X_L = X_C:


Setting f = f_r and solving gives the ideal series-resonance frequency:

Variation of Impedance Vs Frequency

At resonance in a series RLC circuit, the ideal inductive and capacitive reactances cancel. The total impedance becomes purely real and equals the total series resistance, Z = R, its minimum value for fixed R, L and C in the ideal model. Below f_r, the net reactance is capacitive; above it, the net reactance is inductive. With a fixed-amplitude voltage source, current magnitude is maximum at resonance.

Resonant Current

The source voltage in a series RLC circuit is the phasor sum of the resistor, inductor and capacitor voltages. At resonance in series RLC circuit, the inductor and capacitor voltages are equal in magnitude and 180° apart, so their phasor sum is zero. The resistor voltage therefore equals the source voltage, V = V_r.
Because I = V/Z, the ideal resonant current equals I = V/R for a fixed-amplitude source. This current reaches its maximum only when R and the source magnitude are treated as constant across the frequency sweep. The individual inductor and capacitor voltages can each exceed the source voltage by the circuit quality factor.
Below resonance, the magnitude of capacitive reactance Z_c falls as frequency rises, so current approaches its peak. Above resonance, inductive impedance Z_L dominates; as Z_L rises, current falls. Real component loss and parasitics set the actual peak, bandwidth and high-frequency behavior.

Power Factor at Resonance

At ideal series resonance, the opposite reactive voltage components cancel in the source-voltage phasor sum. The input impedance is purely resistive, so V_r, source voltage V and circuit current I are in phase. The input phase angle is zero and the displacement power factor is unity. Non-sinusoidal sources or frequency-dependent losses require a fuller power analysis.

Application of Series RLC Resonant Circuit

Because resonance in series RLC circuit produces a frequency-selective current peak, designers use it in tuning, band-pass networks, impedance matching and sensing. The function depends on where the output is taken and how the source and load connect. High-Q circuits can create large internal voltages, so component voltage and current ratings still apply.
Summary
For the ideal series RLC model driven by a fixed-amplitude sinusoidal source, resonance has these properties:

1. Inductive reactance X_L is equal to capacitive reactance X_C.
2. Total impedance is at its minimum and equals the total series resistance: Z = R.
3. Circuit current reaches its maximum value, I = V/R, for the fixed source magnitude.
4. The inductor and capacitor voltage phasors cancel, so resistor voltage V_r equals supply voltage V.
5. Net reactance is zero, so source voltage and current are in phase and their phase angle is zero.
6. Displacement power factor is unity for a sinusoidal source.
7. The ideal resonant frequency is given by the following expression.