Q Factor: What is it? (And How Do You Measure It?)

Contents

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Key learnings:

Q Factor Definition: The Q factor is defined as a dimensionless parameter describing the efficiency and energy loss in an oscillator or resonator.
Measurement with Q Meter: A Q meter measures the Q factor by assessing energy dissipation in radio frequency circuits, crucial for understanding circuit performance.
Series Resonance Principle: The Q meter operates on series resonance, where equal inductance and capacitance cause energy oscillation between fields.
Q Factor Formula: The Q factor formula is the ratio of resonance frequency to bandwidth, adapting to different circuits for precise measurements.
Damping and Q Factor: The Q factor determines damping conditions (overdamped, underdamped, critically damped), impacting system stability and response.

What is Q Factor?

Q factor (also known as Quality Factor or Q-factor) is a dimensionless measure of how lightly damped an oscillator or resonator is. For a coil, a capacitor or an inductor it reports losses relative to stored energy and, for a resonator, the ratio of centre frequency to bandwidth.

K. S. Johnson of Western Electric introduced the symbol Q while judging coil loss. He picked Q because other letters were already taken. The name quality factor came later, from colleagues who used his symbol.

Q reports energy loss in a resonant device. That device can be a pendulum, another mechanical resonator or an electrical tuned circuit.

A higher Q means less energy lost per cycle, lower damping and longer ringing. The oscillation frequency is set by the resonator, not by Q. What lasts longer is the amplitude envelope.

In electronic resonators the loss is resistance in the loop. Coil series resistance is often the largest term. The same Q also sets the 3 dB bandwidth around the centre frequency: Q = f0 / bandwidth.

How to Measure Q Factor?

A Q-meter is the laboratory instrument used to read Q of RF coils and capacitors. It reports the resonance Q, which shows how much energy is lost each cycle, not the radio frequency itself.

Working Principle of Q meter

A classic Q-meter uses series resonance. Resonance here means the inductive and capacitive reactances are equal in magnitude. Energy then swaps between the capacitor electric field and the inductor magnetic field. The meter reading comes from that series R, L and C combination.

At series resonance,

$\begin{align*} X_{L} = X_{C}\end{align*}$

$\begin{align*}E_{L} = IX_{L}\; \; \; \; \; E_{C} = IX_{C}\; \; \; \; E=IR\end{align*}$

where E is the applied voltage, $E_{C}$ is the capacitor voltage, $E_{L}$ is the inductor voltage, $E_{L}$ also equals I times the inductive reactance XL, $X_{C}$ is the capacitive reactance, R is the coil resistance and I is the circuit current.

Hence the series Q is

$\begin{align*} Q =\frac{X_{L}}{R} = \frac{X_{C}}{R} = \frac{E_{C}}{E}\end{align*}$

If E is held constant, a voltmeter across the capacitor can be scaled to read Q directly.

Q Factor Formula

For a driven resonator, Q is the resonance frequency divided by the 3 dB bandwidth.

$\begin{align*}Q = \frac{Resonance Frequency}{Bandwidth}\end{align*}$

Use the same unit for both frequency and bandwidth.

The same idea is written in component values for the circuits below.

Q Factor of Inductor

The Q of a reactive part depends on the test frequency. That frequency is usually the resonance of the circuit that contains it.

The Q factor of an inductor with series loss resistance is the Q of a resonant circuit made from that inductor and an ideal capacitor:

$\begin{align*}Q_{L}=\frac{X_{L}}{R_{L}}=\frac{\omega_{0}L}{R_{L}}\end{align*}$

where $\omega_{0}$ is the angular resonance frequency in radians per second, L is the inductance, $X_{L}$ is the inductive reactance and $R_{L}$ is the series resistance of the inductor.

Q Factor of Capacitor

The Q factor of a capacitor with series loss resistance is the same as the Q of a resonant circuit using that capacitor and an ideal inductor:

$\begin{align*}Q_{C}=\frac{-X_{C}}{R_{C}} = \frac{1}{\omega_{0}CR_{C}}\end{align*}$

where $\omega_{0}$ is the angular resonance frequency in radians per second, C is the capacitance, $X_{C}$ is the capacitive reactance and $R_{C}$ is the series resistance of the capacitor.

For a series inductor and capacitor, the resonator Q follows from the two component Q values, whether the loss is series resistance or another equivalent series loss.

$\begin{align*} Q = \frac{1}{\frac{1}{Q_{L}}+\frac{1}{Q_{C}}}\end{align*}$

Q Factor of LC Circuits

In a parallel LC tank the inductor’s own resistance R is still in series with L. For that common model the tank Q has the same algebraic form as the series case:

$\begin{align*}Q=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{\omega_{0}L}{R} = \frac{1}{\omega_{0}RC}\end{align*}$

where R, L and C are the resistance, inductance and capacitance of that model.

Q Factor of RLC Circuits

A tuned radio-frequency (TRF) receiver uses one or more tuned RF amplifier stages, then a demodulator and usually an audio amplifier.

For an ideal series RLC circuit, and for a TRF tuned stage, Q is

$\begin{align*}Q=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{\omega_{0}L}{R} = \frac{1}{\omega_{0}RC}\end{align*}$

where R, L and C are the series resistance, inductance and capacitance. A larger series R gives a smaller Q.

For a parallel RLC circuit, with R across L and C, the Q factor is the reciprocal form of the series case.

$\begin{align*}Q = R\sqrt{\frac{C}{L}}\frac{R}{\omega_{0}L}= \omega_{0}RC\end{align*}$

If R, L and C are all in parallel, a smaller R damps the tank more and lowers Q. Designers use that form when they set filter bandwidth from a parallel resistance.

Q Factor Transfer Function

A filter response can be written as an s-domain transfer function. The variable s comes from the Laplace transformation and is complex frequency. Second-order filters also have a Q, often written through $\alpha$ .

$\begin{align*}\alpha = \frac{1}{Q}\end{align*}$

The stored line calls that quantity the damping ratio: $\xi = 2\alpha$ . Standard texts more often write the damping ratio as 1/(2Q).

Q Factor Low Pass Filter

$\begin{align*} H(s)=\frac{K}{1+\frac{s}{\omega_{0}}}\end{align*}$

The first formula is the first-order low-pass filter. A second-order low-pass uses Q and $\omega_{0}$ as below.

$\begin{align*}H(s)= \frac{H}{s^2+\frac{\omega_0 S}{Q}+\omega_0^2}\end{align*}$

where $H_0$ is the pass-band gain and $\omega_0$ is the natural or cutoff frequency used in that prototype.

Q Factor High Pass Filter

A second-order high pass filter is obtained by changing the low-pass numerator to $H_0s^2$ . The magnitude shape is the low-pass shape reversed on a log-frequency axis.

$\begin{align*}H(s)= \frac{H_0s^2}{s^2+\frac{\omega_0 S}{Q}+\omega_0^2}\end{align*}$

Q Factor Band Pass Filter

The stored bandpass filter numerator is written $H_0\omega_0s^2$ . Many textbooks use a first-power s term such as H0 (ω0/Q) s. The formula block below is left as stored.

$\begin{align*}H(s)= \frac{H_0\omega_0s^2}{s^2+\frac{\omega_0 S} {Q}+\omega_0^2}\end{align*}$

where $\omega_0$ is the centre frequency of the passband. $H_0$ is a gain constant of the circuit.

$\begin{align*} H_0 = \frac{H}{Q}\end{align*}$

In band-pass design, Q is the selectivity.

$\begin{align*}Q=\frac{F_0}{F_H - F_L}\end{align*}$

where $F_H$ and $F_L$ are the frequencies where the response is 3 dB below the peak

Q Factor Notch (Bandstop) Filter

Replacing the bandpass numerator with $s^2 + \omega_z^2$ gives a band-stop, also called a notch filter or band-reject filter. A narrow stop is usually called a notch. A wide stop is usually called a band-reject.

A stored band-reject transfer function is

$\begin{align*}H(s)= \frac{H_0(\omega_z^2+s^2)}{s^2+\frac{\omega_0 S} {Q}+\omega_0^2}\end{align*}$

Q factor and Damping

Q also classifies simple damped oscillators. Damping is any process that removes stored oscillation energy.

Overdamped Condition: The system is overdamped when Q is low ( $Q < \frac{1}{2}$ ). It does not oscillate. After a step it returns by exponential decay and only approaches the final value.

Its impulse response is the sum of two decaying exponentials at different rates. A second-order low-pass with very low Q has an almost first-order phase shape. After a step, the output rises slowly toward the final value.

Underdamped Condition: The system is underdamped when Q is above one half ( $Q >\frac{1}{2}$ ). It rings at its natural frequency while the amplitude decays.

A lightly underdamped system may ring only once or twice. As Q rises, damping falls. A high-Q second-order low-pass overshoots a step, rings, then settles to the final value.

Critically Damped Condition: The system is critically damped at ( $Q =\frac{1}{2}$ ). Like the overdamped case it does not ring. It also does not overshoot the final value.

The step response is faster than a heavily overdamped case and still reaches the final value without overshoot.

Quality factors of common systems

A unit-gain Sallen-Key low-pass with equal resistors and equal capacitors is a common critically damped example. Its Q is $Q =\frac{1}{2}$

A second-order Butterworth filter is underdamped, with $Q =\frac{1}{\sqrt{2}}$

Effects of Q Factor

In RF tuned circuits Q sets selectivity, loss and ringing. A high Q is useful when a narrow band is wanted. Other designs need a lower Q for a wider passband.

Bandwidth:

A higher Q narrows the 3 dB bandwidth. Lower loss keeps more energy in the tank, so the resonance peak is sharper and selectivity rises.

The chosen Q is a trade: wide enough for the wanted band, narrow enough to reject nearby signals.

Oscillator phase noise:

An oscillator also produces phase noise: random phase fluctuations that spread as sidebands around the carrier. Designers try to keep that noise down. A higher resonator Q is one of the usual ways to reduce close-in phase noise.

Ringing:

Higher Q also means less loss, so a free oscillation dies more slowly and the circuit rings longer. That helps an oscillator start and stay in oscillation because less energy has to be replaced each cycle.