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.

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.