E5A

Resonance and Q

Find a circuit's resonant frequency, say what series and parallel circuits do there, and use Q to get bandwidth.

13 pool questions1 on the exam

Syllabus: Resonance and Q: characteristics of resonant circuits; series and parallel resonance; definitions and effects of Q; half-power bandwidth

Resonance is the frequency where XL equals XC

XL grows with frequency and XC shrinks, so there is exactly one frequency where they are equal. Solve XL = XC for f and you get the resonant frequency. R does not change it.

Resonant frequencyf₀ = 1 ÷ (2π × √(L × C))

Where the two lines cross, XL = XC. That frequency is the resonant frequency.

Inductance (L)50 µHCapacitance (C)40 pF

f₀ = 1 ÷ (2π√(L×C))3.56 MHz

L	C	√(L×C)	f₀
50 µH	40 pF	4.472 × 10⁻⁸	1 ÷ (2π × 4.472 × 10⁻⁸) = 3.56 MHz
50 µH	10 pF	2.236 × 10⁻⁸	1 ÷ (2π × 2.236 × 10⁻⁸) = 7.12 MHz

Remember4× the capacitance (or inductance) halves f₀. Use henries and farads, then µ = 10⁻⁶ and p = 10⁻¹².

At resonance, series Z dips and parallel Z peaks

At f₀ the inductor and capacitor effects cancel, leaving only R. Voltage and current are in phase, because there is no net reactance to shift them.

Same two parts, opposite behaviour at resonance.

Series resistance (R)10 Ω

Impedance at f₀10 Ω

At resonance	Series RLC	Parallel RLC
Impedance	minimum, about R	maximum, about R
Current from the source	maximum	minimum
Current circulating between L and C	n/a	maximum

Memory hookSeries is the easy path: low Z, big current. Parallel is a roadblock: high Z, little line current, yet a big current swings round inside the loop.

Higher Q means a narrower, taller response

Q measures how little a circuit wastes per cycle. Half-power bandwidth is the width between the points where power has fallen to half.

BandwidthBW = f₀ ÷ Q

Bandwidth is measured between the half-power (−3 dB) points: 70.7% of the peak voltage.

Quality factor (Q)50

BW = f₀ ÷ Q = 7.1 MHz ÷ 50142 kHz

Series circuit: voltage across L or C50 × the source

Series: Q = X ÷ R. Parallel: Q = R ÷ X.
Higher Q: narrower bandwidth, including in an impedance-matching circuit.
Series circuit: voltage across L or C is Q times the source voltage, so it can exceed it. Higher Q raises these internal voltages.

RememberBW = f₀ ÷ Q: 7.1 MHz ÷ 150 = 47.3 kHz. Series Q is X over R, parallel Q is R over X.

Loading lesson…

Go deeper

Compendium articles on the ideas in this lesson, beyond what the exam asks.

Check what stuck

Practise the 13 pool questions for this group. Answers are shuffled, and every one comes with a one-line "why".

Practise E5A →

√(L×C)

f₀

50 µH

40 pF

4.472 × 10⁻⁸

1 ÷ (2π × 4.472 × 10⁻⁸) = 3.56 MHz

50 µH

10 pF

2.236 × 10⁻⁸

1 ÷ (2π × 2.236 × 10⁻⁸) = 7.12 MHz

At resonance

Series RLC

Parallel RLC

Impedance

minimum, about R

maximum, about R

Current from the source

maximum

minimum

Current circulating between L and C

n/a

maximum