Resonance in AC Circuits
Imagine pushing a child on a swing. If you push at exactly the right moment — matching the swing's natural rhythm — you add energy efficiently and the amplitude grows dramatically. Push at the wrong frequency and your effort is wasted. This is resonance, and it appears in electrical circuits with striking mathematical precision. In an AC circuit containing a resistor (R), an inductor (L), and a capacitor (C) connected in series, there exists one special frequency at which the circuit responds with the maximum possible current for a given voltage. At most frequencies, inductors and capacitors oppose current flow in opposite ways. An inductor's inductive reactance (X_L = ωL) increases with frequency, while a capacitor's capacitive reactance (X_C = 1/ωC) decreases with frequency. These two reactances partially cancel each other. At the resonant frequency, they cancel completely, leaving only the resistance R to limit current. The circuit behaves as if the inductor and capacitor have vanished — only the resistor remains. This is the condition for resonance. Resonance in AC circuits is not just a theoretical curiosity. It underlies the operation of radio and television tuners (which select one station from thousands), the filters in audio equalizers, the transmitters in mobile phones, and the energy transfer systems in wireless charging pads. Understanding resonance means understanding how engineers extract a single signal from a cacophony of electromagnetic noise — one of the most practically powerful ideas in all of electronics.
- Impedance in a Series RLC Circuit
- The Resonance Condition
- Resonant Frequency in Hertz
- Phase Angle at Resonance
- The Quality Factor Q
Impedance in a Series RLC Circuit
In a series RLC circuit driven by an AC source of angular frequency ω (rad/s), the total impedance is Z = √(R² + (X_L − X_C)²), where X_L = ωL is the inductive reactance and X_C = 1/(ωC) is the capacitive reactance. Current amplitude is I₀ = V₀/Z. Because Z ≥ R always, current is maximum when the term (X_L − X_C)² is minimised — ideally zero.
The Resonance Condition
Resonance occurs when X_L = X_C, i.e., ωL = 1/(ωC). Solving for ω gives the resonant angular frequency ω₀ = 1/√(LC). At this frequency, Z = R (its minimum possible value), so the current I₀ = V₀/R reaches its maximum. The voltage across L and across C can individually be much larger than the source voltage — they are equal in magnitude but opposite in phase, so they cancel in the loop.
Resonant Frequency in Hertz
Since ω₀ = 2πf₀, the resonant frequency in hertz is f₀ = 1/(2π√(LC)). This depends only on the inductance L and capacitance C, not on the resistance R. Resistance affects how sharp or broad the resonance peak is (captured by the quality factor Q = ω₀L/R = (1/R)√(L/C)), but not where the peak sits. A high-Q circuit is very selective — it responds strongly to a narrow band of frequencies. A low-Q circuit responds to a wider band.
Phase Angle at Resonance
Away from resonance, the circuit has a non-zero phase angle φ = arctan((X_L − X_C)/R) between voltage and current. At resonance, X_L − X_C = 0, so φ = 0. Voltage and current are perfectly in phase, meaning the source delivers power as efficiently as if it were driving a purely resistive load. All the power dissipated goes into R; the inductor and capacitor continuously exchange energy with each other but return it to the circuit, consuming no net power.
The Quality Factor Q
The sharpness of the resonance peak is described by the quality factor Q = ω₀L/R. A high Q means the resonance peak is narrow and tall — the circuit is selective. The bandwidth BW = f₀/Q gives the range of frequencies around f₀ at which current stays above 1/√2 of its maximum (the half-power points). For example, a radio tuner needs a high Q to distinguish stations separated by only 10 kHz. An audio crossover filter might deliberately use a low Q for a broad, smooth response.
The Key Equation
- f_0 — Resonant frequency (Hz (hertz)): The frequency at which inductive and capacitive reactances are equal and impedance is minimum.
- L — Inductance (H (henry)): The inductor's opposition to changing current; stores energy in a magnetic field.
- C — Capacitance (F (farad)): The capacitor's ability to store charge; stores energy in an electric field.
Worked Example
Problem. A series RLC circuit has R = 20 Ω, L = 50 mH, and C = 200 μF. (a) Find the resonant frequency f₀. (b) Find the impedance and peak current at resonance if the source has a peak voltage V₀ = 100 V. (c) Find the quality factor Q.
Answer. Resonant frequency ≈ 50.3 Hz; impedance at resonance = 20 Ω; peak current = 5.0 A; Q ≈ 0.79.
Origins · 1 minHeinrich Hertz and how this was discovered
Hertz was the first to conclusively demonstrate the existence of electromagnetic waves predicted by Maxwell, using resonant LC circuits as both transmitter and receiver. His transmitter sparked across a gap connected to an inductive loop; his receiver was a small resonant loop that would spark only when tuned to match the transmitter's frequency — a direct experimental observation of electromagnetic resonance. His work proved that light and radio waves are the same phenomenon and laid the experimental foundation for all wireless communication. The SI unit of frequency, the hertz (Hz), is named in his honour. When asked about the practical applications of his discovery of radio waves, Hertz famously replied 'Nothing, I guess' — he died at 36, just before Marconi used his work to invent wireless telegraphy and transform world communication.
Related Concepts
- Impedance in AC Circuits — Resonance is defined as the frequency at which total impedance is minimised to Z = R; understanding impedance is essential to understanding resonance.
- Inductive and Capacitive Reactance — The resonance condition X_L = X_C directly involves both reactances; you must understand how they each depend on frequency.
- Power in AC Circuits — At resonance, the power factor cos φ = 1 and maximum real power is delivered to the resistor — connecting resonance to AC power analysis.
- Phasor Diagrams — Phasor diagrams show visually why X_L and X_C cancel at resonance, making the net reactive component zero and the impedance purely real.