Resonance & Q-Factor

PHYAC Circuits, Power & Resonance

A series LCR circuit resonates when $X_L = X_C$ ; the net reactance vanishes, the impedance is minimum ( $Z = R$ ) and the current is maximum. The resonant frequency depends only on $L$ and $C$ , and the sharpness of the peak is measured by the quality factor $Q$ .

Students compute $f_0$ , $\omega_0$ and $Q$ — short, formula-driven marks.

Resonant angular frequency

\omega_0 = \dfrac{1}{\sqrt{LC}}

\omega_0

in rad/s,

L

in H,

C

in F; the condition

X_L = X_C

gives this directly.

Resonant frequency

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

f_0

in Hz; equal to

\omega_0/2\pi

and independent of

R

Quality factor

Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}

Q

= sharpness of resonance (dimensionless); a larger

Q

means a narrower, sharper peak, so a small

R

gives high

Q

At resonance $X_L = X_C$ , so $Z = R$ (minimum) and the current is maximum; the circuit is purely resistive with power factor 1.
$f_0$ depends only on $L$ and $C$ , NOT on $R$ — changing $R$ alters the sharpness, not the resonant frequency.
$Q$ measures sharpness: a high $Q$ (small $R$ ) gives a narrow bandwidth and selective tuning (as in radios).
Bandwidth $\Delta\omega = R/L = \omega_0/Q$ ; a higher $Q$ means a smaller bandwidth.
Convert $\mu$ F to F and mH to H before substituting into $\sqrt{LC}$ .
$Q$ can be written equivalently as $\omega_0 L/R$ , $1/(\omega_0 RC)$ , or $\tfrac1R\sqrt{L/C}$ — all equal at resonance.
At resonance the voltages across $L$ and $C$ are equal and opposite (each can be $Q$ times the source voltage) and so cancel.

Where the marks go

Confusing $\omega_0$ (rad/s) with $f_0$ (Hz) — they differ by the $2\pi$ factor.
Putting $R$ into the resonant-frequency formula — $f_0$ is independent of $R$ .
Leaving $C$ in $\mu$ F or $L$ in mH inside $\sqrt{LC}$ .
Writing $Q = R/\omega_0 L$ (inverted) instead of $\omega_0 L/R$ .

How the board asks it

Numericalresonant frequency and quality factor formulas
A series $LCR$ circuit has $L = 2.0\,\text{H}$ , $C = 32\,\mu\text{F}$ and $R = 10\,\Omega$ . Calculate the resonant frequency $f_0$ and the quality factor $Q$ of the circuit.
Define / statequality factor as a measure of sharpness
Define the quality factor $Q$ of a series resonant circuit and state how it is related to the resonant angular frequency $\omega_0$ and the bandwidth $\Delta\omega$ .
Derive / prove $X_L = X_C$ at resonance
Obtain an expression for the resonant frequency $f_0$ of a series $LCR$ circuit, showing that at resonance the impedance is minimum and equal to $R$ .
Give reasons $f_0$ is independent of $R$
In a series $LCR$ circuit the resistance $R$ is increased while $L$ and $C$ are kept unchanged. Give reasons why the resonant frequency $f_0$ remains the same but the circuit becomes less selective.
Diagram / graphsharpness of the current peak
Sketch the variation of current with frequency for a series $LCR$ circuit for two different values of resistance, and state which curve corresponds to the higher quality factor $Q$ .
Applicationhigh $Q$ gives narrow bandwidth and selective tuning
Explain how resonance in a series $LCR$ circuit is used to tune a radio receiver to a desired station, and why a high $Q$ factor is desirable.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.