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Quality Factor
The energy loss rate of a weakly damped (i.e.,
) harmonic oscillator is conveniently
characterized in terms of a parameter,
, which is known as the
quality factor. This quantity is defined to be
times the
energy stored in the oscillator, divided by the energy lost in a single
oscillation period. If the oscillator is weakly damped then
the energy lost per period is relatively small, and
is therefore
much larger than unity. Roughly speaking,
is the number of oscillations
that the oscillator typically completes, after being set in motion, before its
amplitude decays to a negligible value. For instance, the quality factor for the
damped oscillation shown in Figure 7 is
.
Let us find an expression for
.
As we have seen, the motion of a weakly damped
harmonic oscillator is specified by [see Equation (72)]

(81) 
It follows that

(82) 
Thus, making use of Equation (78), the energy lost during a single oscillation period is
In the weakly damped limit,
, the exponential factor is approximately
unity in the interval
to
, so that

(84) 
where
.
Thus,

(85) 
because, as is readily demonstrated,
The energy stored in the oscillator
(at
) is [cf., Equation (16)]

(88) 
Hence, we obtain

(89) 
Next: LCR Circuits
Up: Damped and Driven Harmonic
Previous: Damped Harmonic Oscillation
Richard Fitzpatrick
20130408