Series Resonance

Resonance in electrical circuits is the frequency condition at which the reactive effects of inductance and capacitance exactly cancel each other. In a series RLC circuit, this cancellation causes the circuit impedance to collapse to its minimum value, which is purely resistive. This results in maximum current flow for a given applied voltage. Series resonance is fundamental to the design of filters, oscillators, tuned amplifiers, and communication receivers, and it is one of the highest-weightage topics in GATE Network Analysis.

Core Concept Explanation

In a series RLC circuit connected to an AC voltage source, the total impedance is Z = R + jXL - jXC = R + j(ωL - 1/ωC). The inductive reactance XL increases linearly with frequency, while the capacitive reactance XC decreases with frequency. At a particular frequency, these two reactive components become equal in magnitude, XL = XC, and they cancel each other completely. This frequency is the resonant frequency ω0 (or f0), at which Z = R (purely resistive, minimum possible).

Since the impedance is minimum at resonance, the current I = V/Z = V/R is maximum at resonance for a fixed applied voltage V. Below resonance, XC dominates (XC > XL), so the circuit is capacitive and the current leads the voltage. Above resonance, XL dominates, so the circuit is inductive and the current lags the voltage. At resonance, the current is exactly in phase with the applied voltage.

The voltages across L and C individually at resonance can be much larger than the applied source voltage. The voltage across the inductor at resonance is VL = IXL = (V/R)(ω0L). Similarly, VC = IXC = (V/R)(1/ω0C). Since ω0L = 1/ω0C at resonance, VL = VC, and they are 180 degrees out of phase, so they cancel. Their common magnitude is Q times the applied voltage V, where Q is the quality factor.

Mathematical Expression

Setting XL = XC gives ω0L = 1/ω0C, so ω0² = 1/LC. Therefore the resonant frequency is ω0 = 1/sqrt(LC) rad/s, or in hertz: f0 = 1/(2π sqrt(LC)). These are the most important formulas for series resonance.

The quality factor Q measures the sharpness of the resonance peak. It is defined as Q = ω0L/R = 1/(ω0CR) = (1/R) sqrt(L/C). A high Q circuit has a sharp, tall resonance peak and low bandwidth. The bandwidth BW is defined as the frequency range between the two half-power frequencies f1 and f2 (at which the current falls to 1/sqrt(2) of its maximum, or equivalently, the power falls to half the maximum). BW = f2 - f1 = f0/Q = R/(2πL).

At resonance, the stored energy oscillates between L and C without any net energy exchange with the source. The source only supplies the energy dissipated in R. The ratio of energy stored to energy dissipated per radian is exactly Q, which is the physical interpretation of the quality factor.

Practical Understanding

Series resonant circuits are used as bandpass filters in radio receivers to select a specific carrier frequency out of many. The circuit is tuned to the station frequency by adjusting C (variable capacitor), and the bandwidth must be narrow enough to reject adjacent stations. A high-Q circuit provides better selectivity (narrower bandwidth) but requires lower resistance, which means less dissipation.

In power systems, accidental series resonance between line inductance and power factor correction capacitors can occur at harmonic frequencies. This creates dangerously large currents and overvoltages, damaging equipment. Understanding series resonance conditions helps engineers design detuning reactors that shift the resonant frequency away from harmonic frequencies.

Given:
Series RLC circuit: R = 10 Ω, L = 0.1 H, C = 10 µF
Applied voltage V = 50 V (rms)

Why this formula applies:
At resonance, XL = XC, Z = R minimum, current is maximum

Formula:
f0 = 1 / (2π√LC)
I_max = V / R
Q = ω0L / R
BW = f0 / Q

Substitution:
ω0 = 1/√(0.1 × 10×10⁻⁶) = 1/√(10⁻⁶) = 1000 rad/s
f0 = 1000 / (2π) = 159.2 Hz
I_max = 50 / 10 = 5 A (rms)
Q = 1000 × 0.1 / 10 = 100/10 = 10

Calculation:
BW = f0 / Q = 159.2 / 10 = 15.92 Hz
VL at resonance = VL = I × ω0L = 5 × 1000 × 0.1 = 500 V
VC at resonance = VC = I × (1/ω0C) = 5 × 1/(1000×10⁻⁵) = 500 V
(Note: VL = VC = Q × V = 10 × 50 = 500 V — voltage magnification)

Final Answer: f0 = 159.2 Hz, Imax = 5 A, Q = 10, BW = 15.92 Hz, VL = VC = 500 V

Exam Tip: In GATE, voltage across L or C at resonance equals Q times the source voltage. This voltage magnification can make VL or VC far exceed the supply voltage — do not assume they are less than the applied voltage. Also, BW = R/(2πL) in Hz or BW = R/L in rad/s are direct formulas worth memorizing.

Resonance Mechanism in Detail

• Resonant frequency: ω0 = 1/√LC (rad/s) or f0 = 1/(2π√LC) (Hz).
• At f0: XL = XC, net reactance = 0, Z = R (minimum), I = V/R (maximum).
• Below f0: circuit is capacitive (XC > XL), current leads voltage.
• Above f0: circuit is inductive (XL > XC), current lags voltage.
• Quality factor Q = ω0L/R = 1/(ω0CR): measures sharpness of resonance.
• Bandwidth BW = f0/Q = R/(2πL): frequency range between half-power points.
• Voltage magnification: VL = VC = Q × V at resonance (can greatly exceed supply).

Quick Revision

• Resonant frequency: f0 = 1/(2π√LC); ω0 = 1/√LC.
• At resonance: Z = R (minimum), I = V/R (maximum), circuit is purely resistive.
• Quality factor: Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C).
• Bandwidth: BW = f0/Q = R/(2πL) Hz; or BW = R/L rad/s.
• Voltage across L and C at resonance: VL = VC = Q × Vsource (voltage magnification).
• Higher Q means narrower bandwidth and sharper selectivity.
• GATE trap: Series resonance gives maximum current, not minimum — minimum current is for parallel resonance.