# Q-factor

A measure of how good a resonant system is can be found in terms of the Quality (or Q) factor. This is defined as

The smaller the losses the larger the Q factor and the sharper the resonance. For small damping the Q factor can be identified with

Where the Bandwidth (BW) is defined as the distance between those frequencies around the resonance for which the amplitude drops to 1/sqrt(2) times the value at resonance. This Q-factor is equal, in this case of small losses, to w 0 /(2b).

Q-factor depends only on the characteristics of the system (w 0 and b) and it can be considered as an amplification factor between the average energy lost per cycle and the average energy stored in the system, that are always proportional. When a system starts vibrating because an external force is acting on it, the stored energy increases with time and so do the energy lost; when time passes and this energy lost is equal to the energy being fed into the system by the external action, the steady state is reached and the amplitude of the vibrations remains constant.

The effect of the value of Q on the response of the oscillatory system is shown in the following picture.

### Measuring the Q-factor

The measurement of the Q-factor is straightforward when its value is in the order of 10 or higher. In this case

and the Q measurement is reduced to the determination of the resonant frequency (natural frequency of the undamped system) and the Bandwidth.

The measurement procedure, once the resonant frequency is known, is reduced to the measurement of the bandwidth. This is made by changing the frequency around resonance until the Amplitude, A, is equal to the amplitude at resonance, A 0 , divided by the square root of 2. That is to say, we have to detect one frequency w where the corresponding amplitude is A=A 0 /sqrt(2). The bandwidth is then equal to 2*abs(w-w 0 ), where abs means absolute value. Remember that the resonance curve is symmetrical around w 0 , and there will be 2 such frequencies.

We shall work with the following system:

The relevant parameters for this experiment are:
• VLoss factor: 0.002
• TimeStep:1/128
• Resonant Period: To= 0.62812. Resonant frequency w 0 =10.
• Note: Remember that to reach the steady state, you have to allow the system to oscillate for times in the order of 10/(2*b) (around 40 time units in our case).
The steady state amplitude measured at resonance is A 0 =257.

Changing the period of the Electric field to T=0.6365 the amplitude (steady state) of the oscillations is A=181, equal to A 0 /sqrt(2).

After some simple calculations we get a Q factor of Q=38.

The theoretical value is (in the high-Q approximation): Q=w 0 /(2b) = 39.1, in very good agreement with the measured value.

Of course, to select the frequency corresponding to A=A 0 /sqrt(2), we used the theory here reported (because we could compute the Bandwidth in terms of the known characteristics of the system). In experimental work the measuring procedure is as we have told: change the frequency, measure A, and so on until the desired amplitude is located.