Resonance in AC Circuits — Core Principles

Core Principles

Resonance in AC circuits occurs when the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$). This specific frequency is called the resonant frequency ($f_0 = 1/(2\pi\sqrt{LC})$). In a series RLC circuit, resonance leads to minimum impedance (Z=R), maximum current, and unity power factor.

The voltages across L and C can be much larger than the source voltage (voltage magnification). This circuit acts as an 'acceptor' for current at $f_0$. In contrast, a parallel RLC circuit at resonance exhibits maximum impedance, minimum line current, and unity power factor.

The currents circulating between L and C can be much larger than the source current (current magnification). This circuit acts as a 'rejector' for current at $f_0$. The Quality Factor (Q-factor) describes the sharpness of resonance, with higher Q indicating a narrower bandwidth and greater selectivity.

Resonance is fundamental to tuning circuits, filters, and oscillators, allowing specific frequencies to be selected or rejected.

Important Differences

vs Parallel RLC Resonance

Aspect	This Topic	Parallel RLC Resonance
Impedance at Resonance	Minimum (Z = R)	Maximum (ideally infinite, practically very high)
Current from Source at Resonance	Maximum ($I = V/R$)	Minimum ($I = V/Z_{max}$)
Phase Angle ($\phi$)	Zero (voltage and current in phase)	Zero (voltage and current in phase)
Power Factor (cos$\phi$)	Unity (1)	Unity (1)
Voltage/Current Magnification	Voltage magnification ($V_L, V_C > V_{source}$)	Current magnification ($I_L, I_C > I_{source}$)
Circuit Behavior	Acceptor circuit (accepts current at $f_0$)	Rejector circuit (rejects current at $f_0$)
Q-factor Formula	$Q = (\omega_0 L)/R = 1/(\omega_0 CR)$	$Q = R/(\omega_0 L) = \omega_0 CR$

While both series and parallel RLC circuits exhibit resonance at the same frequency $f_0 = 1/(2\pi\sqrt{LC})$ and achieve unity power factor, their fundamental behaviors regarding impedance and current are diametrically opposite. Series resonance leads to minimum impedance and maximum current, acting as an acceptor circuit with voltage magnification. Parallel resonance results in maximum impedance and minimum line current, acting as a rejector circuit with current magnification. Understanding these differences is crucial for selecting the appropriate circuit for specific applications like filters or oscillators.