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What is Resonant circuit?

A circuit that behaves in a special way at the frequency at which the reactances of a coil and a capacitor cancel (the resonant frequency f0 = 1/(2π√(LC))). At series resonance the impedance drops to its minimum R and the current peaks; at parallel resonance the impedance peaks and the source current is at its minimum. The sharpness of resonance is expressed by the Q factor, and the bandwidth is f0/Q. Resonant circuits are used in filters, oscillators and radio tuning circuits.

Read the beginner explainer: AC vs DC Explained: Impedance, Phase and Power Factor

Japanese name: 共振回路

Resonant circuit appears in 9 practice questions across 4 topics, including Circuit and electromagnetism basics, Circuit Analysis: AC, Linear Circuits and Transients, Analog and electronic circuits.

Field: Circuit Analysis: AC, Linear Circuits and Transients

Series and parallel resonance

The impedance of a series RLC circuit is Z = R + j(ωL − 1/(ωC)); its imaginary part becomes 0 at ω0 = 1/√(LC), where the current V/R peaks. In a parallel RLC circuit, the imaginary part of the admittance Y = 1/R + j(ωC − 1/(ωL)) becomes 0 and the impedance peaks. With L = 1 mH and C = 10 nF, f0 ≈ 50.3 kHz in both cases.

Q factor and magnification

For series resonance Q = (1/R)√(L/C), and for parallel resonance Q = R√(C/L). At series resonance the voltages across L and C are Q times the source voltage, and at parallel resonance the currents in L and C are Q times the source current. A small source can produce large voltages or currents, but care is needed not to exceed the voltage or current ratings of the parts.

Where they are used

A radio tuning circuit uses LC resonance to pick out only the signal at the desired frequency. LC oscillators and crystal oscillators use the sharpness of resonance to produce a stable frequency. On the other hand, when the parasitic inductance and capacitance of power wiring and components on a printed circuit board form an unintended resonance, noise and voltage fluctuations grow at particular frequencies.

Try it: questions about Resonant circuit

Which is closest to the resonant frequency of a series resonant circuit with L = 1 mH and C = 10 nF?

  1. AAbout 50 kHz
  2. BAbout 316 kHz
  3. CAbout 5 kHz
Show answer and explanation

Answer: A. About 50 kHz

The resonant frequency is f0 = 1/(2π√(LC)). Since LC = 1×10⁻³ × 10×10⁻⁹ = 1×10⁻¹¹, √(LC) ≈ 3.16×10⁻⁶ s and f0 ≈ 1/(2π × 3.16×10⁻⁶) ≈ 50.3 kHz. At this frequency the inductor and capacitor reactances are equal, both √(L/C) ≈ 316 Ω. The resonant frequency is inversely proportional to the square root of LC, so making C 100 times larger lowers it to 1/10.

Which is closest to the bandwidth (the difference between the two frequencies at which the current falls to 1/√2 of its resonant value) of a series RLC resonant circuit with R = 10 Ω, L = 1 mH and C = 10 nF?

  1. AAbout 1.6 kHz
  2. BAbout 10 kHz
  3. CAbout 50 kHz
Show answer and explanation

Answer: A. About 1.6 kHz

The resonant frequency is f0 = 1/(2π√(LC)) ≈ 50.3 kHz and the Q factor is Q = (1/R)√(L/C) = (1/10) × √(1×10⁻³/(10×10⁻⁹)) ≈ 316/10 ≈ 31.6. The bandwidth is BW = f0/Q ≈ 50.3 kHz/31.6 ≈ 1.59 kHz. For a series RLC circuit it can also be found directly as BW = R/(2πL) = 10/(2π × 1×10⁻³) ≈ 1.59 kHz. A smaller resistance narrows the bandwidth and sharpens frequency selectivity.

FAQ

Why does the current not become infinite at the resonant frequency?

Because the resistance R remains even when the reactances cancel at series resonance. The current is set by V/R, and the smaller R is, the higher the Q and the sharper the current peak. The winding resistance of real coils and the losses of capacitors also act as R.

Does no current flow in L and C at parallel resonance?

It does. The currents in L and C are equal and opposite, so they cancel as seen from the source, but a current Q times the source current circulates between L and C. Energy is exchanged back and forth between the magnetic field of the coil and the electric field of the capacitor.

Resonant circuit in the explanations

Questions that mention Resonant circuit (9)

  1. Which is closest to the resonant frequency of a series resonant circuit with L = 1 mH and C = 10 nF?
  2. Which is closest to the bandwidth (the difference between the two frequencies at which the current falls to 1/√2 of its resonant value) of a series RLC resonant circuit with R = 10 Ω, L = 1 mH and C = 10 nF?
  3. What does a high Q (quality factor) of a resonant circuit mean?
  4. Which correctly describes a series RLC circuit driven by a sinusoidal voltage at its resonant frequency?
  5. In the parallel RLC circuit in the figure, the source frequency is set to the resonant frequency f0 = 1/(2π√(LC)). Which of the following correctly describes the circuit?
  6. In the series RLC circuit in the figure, R = 10 Ω, L = 10 mH and C = 1 µF, and the frequency of a 1 V RMS sinusoidal voltage is set to the resonant frequency. Which of the following is the RMS voltage VC across the capacitor?
  7. In the parallel RLC circuit in the figure, R = 20 kΩ, L = 1 mH and C = 10 nF. Near the resonant frequency, which of the following is closest to the difference between the two frequencies at which the magnitude of the circuit impedance falls to 1/√2 of its value at resonance (the bandwidth)?
  8. Which expression gives the resonant frequency of a series LC circuit?
  9. Which element is essential for a superconducting qubit (transmon) to work as a qubit?

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Related terms

Q factor (quality factor) Impedance Reactance Admittance Phasor Qubit Transmon Josephson junction Mesh analysis Nodal analysis

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