Quality Factor and Cut-off Frequency Defintion

Suppose we have

$$H(s) = \dfrac{ k \frac{\omega_o}{Q}s}{s^2 + \frac{\omega_o}{Q}s + \omega_n^2}$$

What is the definition of the quality factor? My understanding is that it is the gain at the cut-off frequency. I know how to find this from a bode plot by extending a line from the band pass gain and extending a line when there is a drop off in gain and finding their intersection. This gives us the cut-off frequency and hence we can find the gain at the cut-off frequency, which is the quality factor.

Also, if I had a transfer function as above, how would I find the cut-off frequency mathematically without a bode plot? If I set $|{H(\omega_o j)}| = \dfrac{1}{\sqrt{2}}$, I can find the cut off frequency, but doesn't that assume that the quality factor is $\dfrac{1}{\sqrt{2}}$? Also, to do this, doesn't the gain have to be 1?

Thanks.

• No - your understanding is not correct (see my detailed answer). Example: In your eqation (bandpass) set w=wn. And the result is H(w=wn)= k.
– LvW
Commented Jun 6, 2018 at 7:33
• 14tim4 there is a small error in your transfer function: Replace wn² by wo². The frequency wo is called "pole frequency".
– LvW
Commented Jun 6, 2018 at 8:21

For second-order circuits (lowpass, bandpass, highpass) the quality factor Q appears in the transfer function as shown in the given bandpass function.

What is the meaning of the Q-factor?

Answer: It is a measure for the magnitude of the function at the pole frequency wo. (Simply introduce w=wo in the transfer function to see the effect, wn² must read: wo²). More than that. it is one of two figures which characterize the position of the pole in the left half of the complex s-plane.

The pole frequency wo is nothing else than the length of the pointer from the origin to the actual pole position and the Q-factor (also called "pole-Q") is a measure of the distance to the Im-axis (which is important for the stability margin of the system).

The factor Q is defined as Q=wo/2*R(p) (R(p)=real part of the pole). From this definition follows that we also have Q=1/2d (with d=damping factor).

(1) Bandpass: It can be shown that this definition for Q gives a value which is equal to the classical bandpass Q=fo/BW (BW: 3dB-bandwidth).

(2) Lowpass and Highpass: There are different Q-factors for the various forms resp. alternatives of the filter (approxinations).

Examples: Q=0.5773 (Thomson-Bessel), Q=0.7071 (Butterworth), Q=0.9565 (Chebyshev, ripple 1 dB), Q=1.3065 (Chebyshev, ripple 3 dB) .

Q-factor is a historical term. In early radios, which were developed enough to have LC resonant circuit tuning, the frequency selectivity was highly dependent on how low were losses in the LC resonant circuit. Anything possible was tried to get high enough quality to the tuning. Practical tests and math analysis proved that especially losses in the coils and also output load that was connected to LC circuits limited the frequency selectivity.

Mathematically Q factor is Fres/B where Fres is the passband mid frequency and B is the bandwidth of a bandpass filter.

In 2nd order bandpass filter and LC resonant circuit Q factor also tells how slowly oscillation energy vanishes when there's no energy input, only some old input still oscillates in the circuit as sinusoidal AC signal. Q = Emax/Els where Emax is the energy stored in the circuit at moment t=0 and Els = is the oscillation energy reduction due losses during one radian ie. during time interval t = 0...1/(2*Pi*Fres).

For reference read this: https://en.wikipedia.org/wiki/RLC_circuit