1.10 Resonance, Q Factor and Tuned Circuits

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.

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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:

$X_L = X_C$

That frequency is the resonant frequency, $f_0$. 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 $2\pi f_0 L = 1/(2\pi f_0 C)$ and solving for $f_0$ gives the formula worth committing to memory:

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

with $f_0$ in hertz, $L$ in henries and $C$ in farads.

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

1. $LC = (2.2 \times 10^{-6}) \times (100 \times 10^{-12}) = 2.2 \times 10^{-16}$
2. $\sqrt{LC} = 1.483 \times 10^{-8}$
3. $f_0 = 1 / (2\pi \times 1.483 \times 10^{-8}) = 1 / (9.32 \times 10^{-8}) \approx 1.07 \times 10^{7}$ Hz, that is 10.7 MHz.

Because both $L$ and $C$ sit under a square root, multiplying either one by four halves the resonant frequency, and multiplying either by 100 divides $f_0$ 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 $L$ and $C$ 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 $f_0$ 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 $R$;
• current drawn from the source is at a maximum;
• the circuit behaves like a near short circuit at $f_0$.

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 $L$ and $C$ in parallel — the classic tank circuit — and the picture inverts. At $f_0$ 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 $f_0$.

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.

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 $R$ that dissipates energy every cycle. The quality factor, Q, compares the energy stored in the reactance with the energy lost in that resistance:

$Q = \frac{X_L}{R}$

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:

$\text{BW} = \frac{f_0}{Q}$

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

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.

1. $Q = 106 / 2.5 = 42.4$
2. $\text{BW} = 3600\ \text{kHz} / 42.4 \approx 85\ \text{kHz}$

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.