Q Factor and Bandwidth

Not every resonant circuit responds the same way. Some ring sharply at almost exactly one frequency and ignore everything else; others respond lazily over a broad range. The Q factor is the single number that captures this difference — and it's the reason your radio can find one station without picking up its neighbour.

The energy (voltage) response curve

The previous lesson looked at a parallel LC circuit oscillating freely, at its own natural resonant frequency $f_0$. In practice, a resonant circuit is usually connected to a source that drives it continuously at some chosen frequency $f$, and what matters then is how strongly the circuit responds — how large a voltage or how much energy builds up in it — as that driving frequency is varied.

Plotting the circuit's response (its energy, or equivalently its voltage amplitude) against the driving frequency $f$ produces a curve that peaks sharply at $f = f_0$ and falls away on either side. The shape of that peak is what distinguishes one resonant circuit from another:

• A narrow, sharply-peaked response curve means the circuit only responds strongly to frequencies very close to $f_0$ — it is highly selective.
• A broad, gently-peaked response curve means the circuit still responds reasonably well over a much wider spread of frequencies around $f_0$ — it is much less selective.

Both curves can have exactly the same peak position $f_0$, set by the same $L$ and $C$ — what differs between them is purely how quickly the response falls away as you move off resonance.

Defining bandwidth and the Q factor

To turn "how sharply peaked" into a number you can calculate, the specification defines the bandwidth, $f_B$, as the width of the response curve measured between the two points where the energy has fallen to 50% of its peak value. These are sometimes called the half-power points. The bandwidth is the frequency range over which the circuit is considered to be responding strongly — outside this range, the response has dropped below half its peak energy and is treated as having fallen away significantly.

The Q factor (quality factor) then combines the resonant frequency and the bandwidth into a single dimensionless number describing how sharp the resonance is:

Because $Q$ is a ratio of the peak position to the width of the peak, it's a measure of shape rather than of absolute frequency — two circuits resonating at very different frequencies can still have the same Q factor if their bandwidths scale in proportion. A high Q means a narrow bandwidth relative to $f_0$: a sharply selective circuit. A low Q means a broad bandwidth relative to $f_0$: a circuit that responds over a wide range and discriminates poorly between nearby frequencies.

Selectivity in tuning circuits

The practical importance of all this shows up directly in radio reception. A radio receiver's tuning circuit is, at its core, a parallel LC resonant circuit connected to the aerial. Turning the tuning dial changes $C$ (or sometimes $L$), which changes the circuit's resonant frequency $f_0$ so that it matches the carrier frequency broadcast by the station you want to listen to.

But real broadcast bands are crowded: many stations transmit on frequencies only a small amount apart. A tuning circuit that responded broadly around $f_0$ — a low-Q circuit — would pick up several nearby stations at once, overlapping and interfering with each other. A tuning circuit with a high Q factor, by contrast, responds strongly only extremely close to the target station's frequency and rejects signals even a relatively small distance away, allowing the receiver to isolate the desired station cleanly.

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