8.3 Resonance in AC circuits

Resonance in AC circuits describes the condition where inductive and capacitive reactances cancel each other out, allowing maximum energy transfer at a specific frequency. This concept is central to how radios select stations, how filters separate signals, and how power systems operate efficiently.

Fundamentals of AC Circuits

AC circuits use time-varying voltages and currents, which creates behavior you won't see in DC circuits. Before diving into resonance, you need a solid grasp of the basic components and how they interact with alternating current.

Components of AC Circuits

• Resistors dissipate electrical energy as heat and follow Ohm's law in both AC and DC circuits.
• Capacitors store energy in electric fields. Their opposition to current (capacitive reactance) decreases as frequency increases: $X_C = \frac{1}{\omega C}$.
• Inductors store energy in magnetic fields. Their opposition to current (inductive reactance) increases with frequency: $X_L = \omega L$.
• AC voltage sources generate sinusoidal waveforms defined by amplitude, frequency, and phase.

The frequency dependence of capacitors and inductors is what makes resonance possible. At some frequency, their opposing effects balance perfectly.

Alternating Current vs. Direct Current

AC periodically reverses direction, while DC maintains constant polarity. AC's big advantage is that transformers can step voltage up or down, enabling efficient long-distance power transmission. DC is common in low-voltage electronics and battery-powered devices.

When describing AC quantities, you'll use RMS (Root Mean Square) values rather than peak values. RMS gives you the equivalent DC value that would deliver the same power to a resistor.

Phasor Diagrams

Phasors are rotating vectors in the complex plane that represent AC quantities. The magnitude of a phasor corresponds to the amplitude of the sinusoidal waveform, and the angle represents the phase difference between voltage and current.

Phasor addition simplifies circuit analysis significantly. Instead of adding sinusoidal functions directly, you add vectors, which is much easier to work with when multiple components or sources are involved.

Resonance Phenomenon

Resonance occurs when a system's natural frequency matches the frequency of an applied signal. In AC circuits, this means the energy stored in the inductor's magnetic field and the capacitor's electric field swaps back and forth with maximum efficiency.

Definition of Resonance

At resonance, the inductive reactance and capacitive reactance cancel each other out. The result is that impedance drops to its minimum value (in a series circuit), current reaches its maximum, and the circuit behaves as though only the resistor is present. On a frequency response curve, resonance appears as a sharp peak.

Conditions for Resonance

For resonance to occur, you need both inductance and capacitance in the circuit, and:

$X_L = X_C$

Expanding the reactance formulas:

$\omega L = \frac{1}{\omega C}$

When this condition is met, the imaginary part of the impedance vanishes, and the circuit impedance becomes purely resistive.

Resonant Frequency

Solving $\omega L = \frac{1}{\omega C}$ for $\omega$:

1. Multiply both sides by $\omega$: $\omega^2 L = \frac{1}{C}$
2. Solve for $\omega$: $\omega_r = \frac{1}{\sqrt{LC}}$
3. Convert to ordinary frequency: $f_r = \frac{1}{2\pi\sqrt{LC}}$

The resonant frequency depends only on $L$ and $C$. Changing either value shifts where resonance occurs, which is exactly how a radio tuner works (you adjust a variable capacitor to change $f_r$).

Series RLC Circuits

A series RLC circuit has a resistor, inductor, and capacitor all in a single loop with an AC source. This configuration is the foundation for many filters and tuning circuits.

Impedance in Series RLC

The total impedance combines resistance and the net reactance:

$Z = R + j(X_L - X_C)$

The magnitude is:

$|Z| = \sqrt{R^2 + (X_L - X_C)^2}$

And the phase angle between voltage and current is:

$\theta = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$

At resonance, $X_L = X_C$, so the reactive terms cancel. Impedance reduces to just $R$, and the phase angle drops to zero.

Resonance in Series RLC

At the resonant frequency:

• Current reaches its maximum value: $I_{max} = \frac{V_{source}}{R}$
• Voltage and current are perfectly in phase (the circuit looks purely resistive)
• The voltage across the inductor and capacitor can individually exceed the source voltage. This is called voltage magnification, and it's why resonance can be both useful and dangerous.

The voltages across $L$ and $C$ are equal in magnitude but opposite in phase, so they cancel in the loop equation while each can be quite large on its own.

Quality Factor

The quality factor ($Q$) measures how sharp the resonance peak is:

$Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r CR}$

A high $Q$ means a narrow, tall resonance peak, which translates to better selectivity (the circuit responds strongly to a narrow band of frequencies). A low $Q$ gives a broader, flatter peak. The voltage magnification at resonance equals $Q$, so the voltage across $L$ or $C$ is $Q$ times the source voltage.

Parallel RLC Circuits

In a parallel RLC circuit, the resistor, inductor, and capacitor are each connected across the same AC source. The resonance behavior here is essentially the opposite of the series case.

Admittance in Parallel RLC

For parallel circuits, it's easier to work with admittance ($Y$), the reciprocal of impedance:

$Y = \frac{1}{R} + j\left(\omega C - \frac{1}{\omega L}\right)$

The magnitude is:

$|Y| = \sqrt{\left(\frac{1}{R}\right)^2 + \left(\omega C - \frac{1}{\omega L}\right)^2}$

The real part is called conductance ($G = 1/R$), and the imaginary part is susceptance ($B$).

Resonance in Parallel RLC

At resonance in a parallel circuit:

• The inductive and capacitive susceptances cancel, so admittance is minimized
• Impedance is maximized (opposite of the series case)
• Total current from the source reaches its minimum
• Large circulating currents can flow between $L$ and $C$ internally, even though the source current is small

This makes parallel RLC circuits useful for impedance matching and power factor correction.

Bandwidth

Bandwidth is the frequency range over which the circuit response stays within 3 dB of its peak value:

$BW = \frac{f_r}{Q}$

This formula works for both series and parallel RLC circuits. A higher $Q$ gives a narrower bandwidth, meaning greater selectivity. There's always a trade-off: a very selective circuit passes a very narrow range of frequencies, which can be too restrictive for some applications.

Power in Resonant Circuits

Power Factor

The power factor is the ratio of real power (watts actually consumed) to apparent power (the product of RMS voltage and current):

$PF = \cos\theta$

At resonance in a series RLC circuit, $\theta = 0$, so the power factor equals 1 (unity). This means all the power delivered by the source is consumed by the resistor, with no reactive power wasted. Parallel RLC circuits are often used in industry for power factor correction, bringing the power factor closer to unity.

Maximum Power Transfer

Maximum power transfer to a load occurs when the load impedance is the complex conjugate of the source impedance. In a series RLC circuit, this condition is naturally satisfied at resonance: the reactive components cancel, and maximum power is delivered to the resistor. Impedance matching networks in real systems frequently use resonant circuits to achieve this.

Efficiency Considerations

Resonant circuits can be highly efficient because energy oscillates back and forth between the inductor and capacitor rather than being dissipated. Losses occur primarily in the resistance. Higher $Q$ means a larger ratio of stored energy to energy lost per cycle, so high-$Q$ circuits are generally more efficient but have limited bandwidth.

Applications of Resonance

Radio and Television Tuning

LC tuning circuits select a specific station's frequency from the entire broadcast spectrum. A variable capacitor adjusts the resonant frequency to match the desired station. Superheterodyne receivers use multiple resonant stages for improved selectivity, and antenna matching networks use resonant circuits to maximize signal reception.

Filters and Signal Processing

• Bandpass filters use series or parallel RLC circuits to pass a specific frequency range while rejecting others
• Notch filters (band-stop) reject a narrow band of unwanted frequencies using anti-resonance
• Active filters combine resonant circuits with amplifiers for sharper roll-off and better performance

Wireless Power Transfer

Resonant inductive coupling enables efficient wireless energy transmission. When the transmitter and receiver are tuned to the same resonant frequency, power transfer is maximized. Applications include wireless phone chargers, electric vehicle charging pads, and powering implantable medical devices. Resonant systems achieve significantly higher efficiency than non-resonant inductive approaches.

Damping in Resonant Circuits

Damping determines how quickly oscillations die out and directly shapes the frequency response. The amount of resistance in the circuit controls the damping level.

Underdamped vs. Overdamped Systems

• Underdamped systems oscillate with gradually decreasing amplitude before settling to steady state. These circuits have sharper resonance peaks and higher $Q$.
• Overdamped systems return to steady state without oscillating, but they do so more slowly. These circuits are more stable but less selective.

Critical Damping

Critical damping is the boundary between underdamped and overdamped behavior. It produces the fastest return to steady state without any oscillation. The damping ratio $\zeta = 1$ at critical damping. This is often the target in control systems and measurement instruments where you want a quick, stable response.

Damping Ratio

The damping ratio ($\zeta$) quantifies the damping level for a series RLC circuit:

$\zeta = \frac{R}{2\sqrt{L/C}}$

• $\zeta < 1$: underdamped (oscillatory)
• $\zeta = 1$: critically damped
• $\zeta > 1$: overdamped (sluggish, no oscillation)

The damping ratio directly influences both the shape of the frequency response curve and the transient behavior when the circuit is first energized or disturbed.

Frequency Response

Frequency response describes how a circuit's output changes as you sweep the input frequency. It tells you everything about the circuit's selectivity, gain, and phase behavior.

Amplitude Response

The amplitude response plots the output magnitude relative to the input across a range of frequencies, typically on a logarithmic scale in decibels (dB). Resonant circuits show a clear peak (for bandpass) or dip (for notch) in this plot. You can read the bandwidth and $Q$ factor directly from the amplitude response curve by finding the 3 dB points.

Phase Response

The phase response shows the phase shift between input and output as a function of frequency. Near resonance, the phase changes rapidly. In a series RLC circuit, the phase shifts from $+90°$ (capacitive, below resonance) through $0°$ (at resonance) to $-90°$ (inductive, above resonance). This rapid phase change near $f_r$ is characteristic of resonant systems.

Bode Plots

Bode plots combine amplitude and phase information in two stacked graphs:

• Magnitude plot: amplitude in dB vs. log frequency
• Phase plot: phase in degrees vs. log frequency

Asymptotic straight-line approximations make it possible to sketch Bode plots quickly, which simplifies the analysis of complex transfer functions.

Resonance in Coupled Circuits

When two or more resonant circuits interact through magnetic or electric coupling, new phenomena emerge that are important in transformer design, wireless communication, and advanced filter design.

Mutual Inductance

Mutual inductance ($M$) measures the magnetic coupling between two inductors:

$M = k\sqrt{L_1 L_2}$

where $k$ is the coupling coefficient (ranging from 0 for no coupling to 1 for perfect coupling). Mutual inductance allows energy transfer between primary and secondary circuits and directly affects the resonant behavior of the coupled system.

Transformer Resonance

Transformers have parasitic winding capacitance that can form unintended resonant circuits with the leakage inductance. If not managed, this can cause voltage spikes and reduced efficiency. However, resonant transformers intentionally exploit this effect to generate high voltages, as in Tesla coils and certain power supply topologies.

Coupled Resonators

Systems of two or more interacting resonant circuits exhibit mode splitting: a single resonant frequency splits into two (or more) closely spaced frequencies. The coupling strength determines the separation between these split frequencies and affects both the bandwidth and efficiency of energy transfer. Coupled resonators are used in wireless power transfer systems and in multi-pole bandpass filters for communications.

Practical Considerations

Component Tolerances

Real components deviate from their nominal values due to manufacturing variations, often by 5% to 20% depending on the component grade. Temperature changes cause further drift. For resonant circuits where precise tuning matters, designers use trimming (adjustable components) or self-tuning feedback loops to compensate.

Parasitic Effects

Every real component carries unwanted parasitic elements: inductors have parasitic capacitance, capacitors have parasitic inductance, and both have some series resistance. These parasitics become significant at high frequencies and can shift the actual resonant frequency away from the calculated value. Careful PCB layout and component selection help minimize these effects.

Stability and Sensitivity

High-$Q$ resonant circuits are inherently sensitive to small changes in component values, temperature, and operating conditions. Over time, component aging can drift the resonant frequency. For applications requiring stable operation, feedback mechanisms or temperature-compensated components are used. There's always a trade-off between high selectivity (narrow bandwidth, high $Q$) and robust, stable performance.