Series RLC resonance | Engimations

By the end of this topic you will be able to

Calculate inductive and capacitive reactance and the impedance of a series RLC circuit.
Find the resonant frequency, quality factor and bandwidth.
Explain why the current peaks at resonance and what the phase angle shows.

Reactance

In an AC circuit, inductors and capacitors oppose current by an amount that depends on frequency. This opposition is called reactance.

X_L = 2\pi f L, \qquad X_C = \frac{1}{2\pi f C}

An inductor's reactance grows with frequency (it resists rapid changes in current); a capacitor's falls with frequency (it passes rapid changes more easily). Both are in ohms.

Series RLC impedance

The total opposition of a resistor, inductor and capacitor in series is the impedance:

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

$X_L$ and $X_C$ work against each other: the inductor's voltage and the capacitor's voltage are in opposite phase, so they partly cancel.

Resonance

At the particular frequency where $X_L = X_C$ , the two reactances cancel completely and the impedance is just $R$ , its minimum value. The current is then at its maximum, $V/R$ , and is in phase with the supply voltage. This is resonance:

f_0 = \frac{1}{2\pi\sqrt{LC}}

Quality factor and bandwidth

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

Q = \frac{1}{R}\sqrt{\frac{L}{C}}, \qquad \text{bandwidth} = \frac{f_0}{Q}

A small resistance gives a high $Q$ , a tall narrow peak, which is good for a radio tuner that must pick one station among many. The bandwidth is the width of the band over which the current is at least $1/\sqrt2$ (0.707) of its peak value (the half-power points).

Phase

Below $f_0$ the circuit is capacitive (current leads voltage, $\varphi < 0$ ); above $f_0$ it is inductive (current lags, $\varphi > 0$ ). At resonance $\varphi = 0$ .

Interactive modelSlide the frequency across resonance, then change R to see the peak widen or sharpen.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Worked example

A series circuit has $R = 10\ \Omega$ , $L = 10$ mH and $C = 10\ \mu\text{F}$ and is driven by a 10 V source. Find $f_0$ , $Q$ , the bandwidth and the current at resonance.

Resonant frequency. $f_0 = \dfrac{1}{2\pi\sqrt{0.01\times10\times10^{-6}}} = \dfrac{1}{2\pi\times3.162\times10^{-4}} = 503$ Hz.

Quality factor. $Q = \dfrac{1}{10}\sqrt{\dfrac{0.01}{10\times10^{-6}}} = \dfrac{31.6}{10} = 3.16$ .

Bandwidth. $\text{BW} = 503/3.16 = 159$ Hz.

Current at resonance. $I = V/R = 10/10 = 1.0$ A, and at the reactances $X_L = X_C = 2\pi\times503\times0.01 = 31.6\ \Omega$ . Each reactive voltage is $31.6$ V, over three times the 10 V supply!

Practice problems

Find the resonant frequency for $L = 1$ mH and $C = 1\ \mu\text{F}$ .

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$f_0 = 1/(2\pi\sqrt{10^{-9}}) = 5033$ Hz.
A circuit has $R = 5\ \Omega$ , $L = 10$ mH and $C = 10\ \mu\text{F}$ . Find $Q$ .

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$Q = (1/5)\sqrt{0.01/(10\times10^{-6})} = 31.6/5 = 6.3$ .
In the model, if you double R, what happens to the peak current and to the bandwidth?

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The peak current halves ( $V/R$ ), and the bandwidth doubles (BW $= f_0/Q$ and $Q \propto 1/R$ ).
At 0.5 $f_0$ is the circuit inductive or capacitive?

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Capacitive, as $X_C > X_L$ below resonance (the current leads the voltage).

Key formulae

Inductive reactance: $X_L = 2\pi f L$
Capacitive reactance: $X_C = \frac{1}{2\pi f C}$
Impedance: $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$
Resonant frequency: $f_0 = \frac{1}{2\pi\sqrt{LC}}$
Quality factor and bandwidth: $Q = \frac{1}{R}\sqrt{\frac{L}{C}}, \quad \text{BW} = \frac{f_0}{Q}$
Phase angle: $\varphi = \tan^{-1}\frac{X_L - X_C}{R}$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (45 min). 5 min: tuning a radio; 15 min: reactance, impedance, cancelling; 10 min: resonance and Q; 15 min: model (frequency sweep, resistance sweep).

Activity. Challenge: choose L and C to resonate at 1 kHz with a Q of 5; check in the model.

Discussion. Where are resonant circuits used? (Tuners, filters, wireless charging, induction heating.)

Assessment idea. Calculate bandwidth and half-power frequencies for given R, L, C.