Q-Factor

March 29, 2018

The Q-factor of a resonance is the ratio of the resonance’s center frequency to its half-power bandwidth. It tells us how sharp, or steep, the resonance is.

(1) $\begin{equation*} \text{Transmissibility}=\frac{\text{Ch}1}{\text{Control}}=\frac{yG}{xG} \end{equation*}$

(2) $\begin{equation*} \text{Q-factor}=\frac{\text{Resonant frequency}}{3dB \text{ bandwidth}}=\frac{f_{0}\text{Hz}}{f_{2}-f_{1}\text{Hz}} \end{equation*}$

Example

For example, if the center frequency of a resonance value was 97.91Hz, the equation to solve the Q-factor would be:

(3) $\begin{equation*} Q=\frac{\text{resonance center frequency }(f_c)}{-3dB\text{ bandwidth }(BW)}=\frac{97.91Hz}{100.3Hz-96.89Hz}=\frac{97.91Hz}{3.41Hz}\approx 28.7 \end{equation*}$

calculation for the Q-factor of a resonant frequency

Figure 4.2. Q-factor of a resonance (with some approximation).

Resonance Sharpness

In Figure 4.3, the left graph displays resonances with the same amplitude and center frequency but varying bandwidths. Notice the difference in sharpness between the three curves.

graphical representation of the Q-factor for different resonant frequencies

Figure 4.3. Q-factor for several resonances. On the left are identical frequencies with varying bandwidths. On the right are identical bandwidths with varying center frequencies (logarithmic axes).

The right graph displays resonances with the same amplitude and bandwidth but varying center frequencies. On a linear axis, the difference in sharpness is not evident. However, it is on a logarithmic axis.

Example

Let’s refer back to Figure 4.1. The Q-factor of the 67.99Hz resonance was 25.4, 12.4 for the 142.1Hz resonance, and 68.5 for the 538.1Hz resonance. The 538.1Hz resonance was the sharpest.

The sharper the resonance, the quicker the resonance will resonate during a sine sweep. The Q-factor measures the difference between a resonance that slowly ramps up during a sine sweep (a wide resonance with a low Q-factor) and a resonance that quickly ramps up (a sharp resonance with a high Q-factor).

The sharper the resonance, the harder it is to control. For an in-depth discussion on this topic, see the following webinar on SRTD testing.