1.10 Resonance, Q Factor and Tuned Circuits

ACMA Amateur Radio Standard Recognition Certificate - Theory Exam

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1 Chapter 1: Electrical Principles and Components

1.1 Mathematics, Units and Metric Prefixes 1.2 DC Circuits, Voltage, Current, and Ohm's Law 1.3 Power, Energy, and Resistance Calculations 1.4 Resistor Colour Code and Component Symbol Identification 1.5 Capacitors: Construction, Ratings and Safety 1.6 Inductors: Construction, Factors Affecting Inductance and Combinations 1.7 Capacitor and Inductor Combination Calculations 1.8 AC Circuits, Frequency, Wavelength, and RMS Values 1.9 Reactance and Impedance in AC Circuits 1.10 Resonance, Q Factor and Tuned Circuits 1.11 Transformers: Turns Ratio, Impedance Matching and Eddy Currents 1.12 Rectifiers, PIV and Power Supply Smoothing 1.13 Active Components: Semiconductors, Diodes, and Transistors 1.14 Transistor and FET Symbols, Configurations and External Behaviour

2 Chapter 2: Transmitters and Receivers

2.1 Superheterodyne Receiver Architecture and Block Diagrams 2.2 Frequency Converters, IF Amplifiers and Transceiver Sharing 2.3 Receiver Performance: Sensitivity, Selectivity, and AGC 2.4 Signal-to-Noise Ratio and Receiver Noise Performance 2.5 Transmitter Architecture, Oscillators and Frequency Synthesisers 2.6 Transmission Quality: Frequency Drift, Parasitic Oscillation and Over-Modulation 2.7 Modulation Techniques: AM, FM, and SSB 2.8 Amplifier Classes and Power Amplifiers (Class A, B, C, and AB) 2.9 Modulation, Duty Cycle and Rated Transmitter Power 2.10 Transceiver Control: ALC, Squelch, and Digital Modes 2.11 Digital Transmission Modes and Bandwidth

3 Chapter 3: Transmission Lines and Antennas

3.1 Transmission Lines: Coaxial, Twin-Lead, and Characteristic Impedance 3.2 Standing Wave Ratio (SWR) and Reflection Loss 3.3 Impedance Matching, Baluns, and Antenna Tuning Units (ATUs) 3.4 Resonant Antennas: Dipoles, Verticals, and End-Fed Antennas 3.5 Angle of Radiation, Antenna Height and Current Distribution 3.6 Directional Antennas: Yagis, Gain and Radiation Patterns 3.7 Effective Radiated Power: System Gains and Losses

4 Chapter 4: Radio Wave Propagation

4.1 Electromagnetic Waves: Velocity, Wavelength and Polarisation 4.2 Ground Wave, Direct Wave and Line-of-Sight Propagation 4.3 Ionospheric Layers (D, E, F1, F2) and Skywave Propagation 4.4 Critical Frequency, Maximum Usable Frequency (MUF), and Fading 4.5 Solar Activity, Sunspot Cycles, and VHF Anomalous Propagation 4.6 Seasonal and Diurnal Effects on Band Selection

5 Chapter 5: Electrical and RF Measurements

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6 Chapter 6: Interference and Electromagnetic Compatibility (EMC)

6.1 Harmonic Emissions, Spurious Signals, and Transmitter Filtering 6.2 How Interference Enters Television and Broadcast Receivers 6.3 Masthead Amplifier Overload and Audio-Stage Demodulation 6.4 Electromagnetic Interference (EMI), Breakthrough, Ferrites, and EMC 6.5 Filter Types, Response Curves and EMC Filter Selection 6.6 EMC Good Practice: Minimum Power, Balanced Antennas and Feedline Routing 6.7 Vehicle EMC and Resolving Community Interference Complaints

7 Chapter 7: Safety and Environmental Hazards

7.1 Mains Voltage Safety, Earthing, and High-Voltage Capacitors 7.2 Equipment Approval, Mains Wiring and Fuse Practice 7.3 Station Layout, Cabling and Emergency Shut-Off 7.4 Lightning Protection and Station Security 7.5 Electric Shock: Prevention, Rescue and Emergency Response 7.6 Battery, Chemical and Hearing Safety in the Amateur Station 7.7 RF Exposure: EMR Hazards and Safe Distance 7.8 Antenna Erection, Siting and Power Line Safety

1.10 Resonance, Q Factor and Tuned Circuits

Key Takeaways Explain This Topic AI

At resonance the inductive and capacitive reactance are equal and cancel, leaving a circuit that appears purely resistive to the source.

The resonant frequency is 1 divided by 2 pi times the square root of LC, so multiplying either L or C by four halves the resonant frequency.

A series tuned circuit presents minimum impedance and maximum current at resonance, acting as an acceptor or trap to earth.

A parallel tuned circuit, or tank, presents maximum impedance and draws minimum supply current at resonance, acting as a rejector or amplifier load.

Q equals reactance divided by loss resistance and is an indicator of circuit losses; bandwidth is the resonant frequency divided by Q.

Last updated: July 2026

1.10 Resonance, Q Factor and Tuned Circuits

ACMA Exam Focus: Syllabus items 4.21, 4.22 and 4.23 — understand that resonance occurs when inductive and capacitive reactance are equal and the circuit becomes resistive, understand Q as an indicator of the losses in a tuned circuit, and know that a series tuned circuit presents a low impedance at resonance while a parallel tuned circuit presents a high impedance.

The condition for resonance (item 4.21)

Inductive reactance rises with frequency and capacitive reactance falls with it. Put an inductor and a capacitor into the same circuit and there is exactly one frequency at which the two are numerically equal: X L = X C X_L = X_C X L ​ = X C ​

That frequency is the resonant frequency , f 0 f_0 f 0 ​ . At resonance the two reactances are equal in magnitude but opposite in phase, so they cancel one another completely. The net reactance is zero and the circuit looks purely resistive to the source — voltage and current are back in phase, and the only opposition left is the real resistance of the components and of the load.

Setting 2 π f 0 L = 1 / ( 2 π f 0 C ) 2\pi f_0 L = 1/(2\pi f_0 C) 2 π f 0 ​ L = 1/ ( 2 π f 0 ​ C ) and solving for f 0 f_0 f 0 ​ gives the formula worth committing to memory: f 0 = 1 2 π L C f_0 = \frac{1}{2\pi\sqrt{LC}} f 0 ​ = 2 π L C ​ 1 ​

with f 0 f_0 f 0 ​ in hertz, L L L in henries and C C C in farads.

Worked example 1. A VFO tank circuit uses L = 2.2 L = 2.2 L = 2.2 µH with C = 100 C = 100 C = 100 pF.

L C = ( 2.2 × 10 − 6 ) × ( 100 × 10 − 12 ) = 2.2 × 10 − 16 LC = (2.2 \times 10^{-6}) \times (100 \times 10^{-12}) = 2.2 \times 10^{-16} L C = ( 2.2 × 1 0 − 6 ) × ( 100 × 1 0 − 12 ) = 2.2 × 1 0 − 16

L C = 1.483 × 10 − 8 \sqrt{LC} = 1.483 \times 10^{-8} L C ​ = 1.483 × 1 0 − 8

f 0 = 1 / ( 2 π × 1.483 × 10 − 8 ) = 1 / ( 9.32 × 10 − 8 ) ≈ 1.07 × 10 7 f_0 = 1 / (2\pi \times 1.483 \times 10^{-8}) = 1 / (9.32 \times 10^{-8}) \approx 1.07 \times 10^{7} f 0 ​ = 1/ ( 2 π × 1.483 × 1 0 − 8 ) = 1/ ( 9.32 × 1 0 − 8 ) ≈ 1.07 × 1 0 7 Hz, that is 10.7 MHz .

Because both L L L and C C C sit under a square root, multiplying either one by four halves the resonant frequency , and multiplying either by 100 divides f 0 f_0 f 0 ​ by ten. That proportional relationship is examined far more often than the arithmetic itself.

Series tuned circuits: low impedance at resonance (item 4.23)

Wire L L L and C C C in series and feed the pair from a signal source. Away from resonance one reactance or the other dominates and the pair opposes current strongly. At f 0 f_0 f 0 ​ the reactances cancel and all that remains is the small series resistance of the coil wire, so:

• impedance is at a minimum , equal to the circuit resistance R R R ;
• current drawn from the source is at a maximum ;

the circuit behaves like a near short circuit at f 0 f_0 f 0 ​ .

That makes a series tuned circuit the natural acceptor , or a trap to earth : a series LC across a receiver input shunts one unwanted frequency to chassis and leaves everything else alone.

Parallel tuned circuits: high impedance at resonance (item 4.23)

Wire L L L and C C C in parallel — the classic tank circuit — and the picture inverts. At f 0 f_0 f 0 ​ the two branch currents are equal and opposite, so they circulate around the loop between the inductor and the capacitor instead of being drawn from the source:

• impedance is at a maximum , and it is resistive;
• current drawn from the source is at a minimum ;

the circuit behaves like a near open circuit at f 0 f_0 f 0 ​ .

A parallel tank is therefore used as a rejector and as the load of a tuned amplifier stage. A transmitter power-amplifier tank presents a high impedance at the operating frequency and a low impedance at the harmonics, which is one reason a properly tuned PA stage is comparatively clean.

• Series LC Parallel LC (tank)
• Impedance at f 0 f_0 f 0 ​ minimum , equal to R R R maximum
• Current from the supply at f 0 f_0 f 0 ​ maximum minimum
• Behaves like a short circuit an open circuit
• Typical station use acceptor, trap to earth tuned amplifier load, rejector, tank

A memory hook that survives exam nerves: the components are in series, so the current has one path and it flows freely — series means low impedance . In the parallel circuit the current goes round and round inside the loop rather than through the supply — parallel means high impedance .

Q factor as an indicator of losses (item 4.22)

Neither the capacitor nor the inductor is perfect. The coil has wire resistance, core loss and, at HF, skin effect; the capacitor has a small equivalent series resistance. All of that appears as a series resistance R R R that dissipates energy every cycle. The quality factor, Q , compares the energy stored in the reactance with the energy lost in that resistance: Q = X L R Q = \frac{X_L}{R} Q = R X L ​ ​

Q is a pure ratio and has no units.

High Q means low losses: a sharp, narrow resonance curve and good selectivity. Air-cored coils on ceramic formers, silvered mica capacitors and thick low-resistance conductors all raise Q.

Low Q means high losses: a broad, flat resonance curve and poor selectivity. Lossy cores, thin wire, damp or dust-covered formers, and any resistance placed across the circuit all pull Q down.

Q is also degraded by whatever the tuned circuit is connected to. A high-Q tank feeding a low-impedance amplifier input has its loaded Q dragged down, and sometimes that is deliberate: a receiver intermediate-frequency filter needs enough bandwidth to pass a whole SSB signal, so damping is designed in on purpose.

Bandwidth follows directly from Q (item 4.22)

The bandwidth of a tuned circuit is the frequency span between the two points at which the response has fallen to half power, that is 3 dB down. It follows from the resonant frequency and Q: BW = f 0 Q \text{BW} = \frac{f_0}{Q} BW = Q f 0 ​ ​

Worked example 2. A 40 m tank circuit resonates at 7.10 MHz with a Q of 50. BW = 7100 kHz 50 = 142 kHz \text{BW} = \frac{7100\ \text{kHz}}{50} = 142\ \text{kHz} BW = 50 7100 kHz ​ = 142 kHz

Worked example 3. An 80 m tuned circuit uses a coil whose reactance is 106 Ω at 3.60 MHz, with a total loss resistance of 2.5 Ω.

Q = 106 / 2.5 = 42.4 Q = 106 / 2.5 = 42.4 Q = 106/2.5 = 42.4

BW = 3600 kHz / 42.4 ≈ 85 kHz \text{BW} = 3600\ \text{kHz} / 42.4 \approx 85\ \text{kHz} BW = 3600 kHz /42.4 ≈ 85 kHz

Note the direction of that relationship. Raising Q narrows the bandwidth, which is exactly what a receiver front end needs when it has to reject a nearby broadcast station — but a transmitter tank with too high a Q becomes fussy to tune and circulates very large currents through the tank components.