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During my study of oscillators I have encountered two different ways do define the Quality factor $Q$, which gives information about the amount of damping on a system. Suppose the oscillator equation is $$ \frac{d^2 x}{dt^2}+2\beta \frac{dx}{dt}+\omega_0^2 x = 0 $$ The resonant angular frequency can be found to be $$ \omega_r = \sqrt{\omega_0^2-\beta^2}$$ by setting the derivative of the amplitude of the particular solution with respect to angular frequency to be equal to zero and hence find its maximum. The quality factor is always defined so that it grows larger as the damping factor (in this case $\beta$) decreases.

The two different definitions I have seen are $$Q = \frac{\omega_r}{2\beta}$$ and $$ Q = \frac{\omega_0}{2\beta}.$$ Since $\omega_r \neq \omega_0$ unless there is no damping, these two definitions are not equivalent. Which one is the most useful or the one that is used more frequently, and why are there discrepancies in this definition? Also, how do these discrepancies translate into the determination of the type of oscillator based on $Q$?

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  • $\begingroup$ In order to be able to talk about resonant frequency the oscillator needs to be driven by an external force. Are you sure that the equation of motion is set equal to zero? $\endgroup$ – Urb Jun 22 at 18:05
  • $\begingroup$ All this is just Wikipedia indulging in its common habit of counting angels on pinheads instead than being a practical source of information. And nobody (except a few physicists) puts a factor of 2 in the damping term of the equation of motion either! As the answer says, there is no practical difference, and "resonant frequency" never means "damped resonant frequency" in practice - it is always "undamped resonant frequency". $\endgroup$ – alephzero Jun 22 at 18:56
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You ask about which definition is most useful. This is the right way to ask the question.

The answer is as follows. For $\beta \ll \omega_0$ you can see that $\omega_r \approx \omega_0$. Under these conditions we have that both definitions of $Q$ are approximately equal, that is, $Q_r \approx Q_0$. We also have, under this condition, that $Q_r, Q_0 \gg 1$.

So for low damping and high quality factor we have that the two definitions basically agree. In this circumstance it doesn't matter which definition you use, they are equally useful. I would tend to use $Q_0$ so that I don't need to bother to calculate $\omega_r$ (for the applications I work with the difference between $\omega_r$ and $\omega_0$ is typically not something I care about).

The two definition can disagree when $\beta \approx \omega_0$ for $\beta \gg \omega_0$. However, in this case, the oscillator is overdamped and doesn't undergo any oscillation so it doesn't really make sense to talk about an oscillation frequency even.

Here I've plotted both $Q_r$ and $Q_0$ as a function of $\frac{\beta}{\omega_0}$. You see very close agreement for $\beta \ll \omega_0$ but as $\beta$ approaches $\omega_0$ the two definitions begin to diverge.

enter image description here

The TLDR version of this answer would be: The definition of quality factor only really matters for low damping/high quality factor. When this condition is met the two definitions basically agree so it doesn't matter which you use.

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As you will have found, there is more than one way of defining what is meant by the resonance frequency. That for which the amplitude is a maximum, and that for which the peak power dissipation is a maximum are two favourite ones. For amplitude resonance, $\omega_{res}^2 =\omega_0^2-2\beta^2$, whereas for 'power resonance', $\omega_{res} =\omega_0$. Power resonance occurs at the frequency when the body's peak velocity is greatest.

In series LCR circuits power resonance occurs for $\omega_{res} = \omega_0=\frac{1}{\sqrt{LC}}$, when the peak current is greatest – clearly a frequency of interest. [In an electrical circuit the analogue of amplitude resonance is when the peak charge on the capacitor plates is greatest.] Electrical engineers define $Q$ for LCR circuits by $$Q=\frac{\omega_0 L}{R}.$$ This is exactly equivalent to your $$Q=\frac{\omega_0}{2 \beta},$$ as you can see by comparing your differential equation for $x$ with no driving force with that for the charge, q, on the capacitor in the LCR series circuit with no applied voltage: $$\frac {d^2 q}{dt^2} +\frac RL \frac {dq}{dt} +\frac{1}{LC} q=0$$

For power resonance (that is velocity resonance or current resonance) your two definitions of Q are equivalent. And Q has a delightful simple significance for the resonance curve. If $\omega_2$ and $\omega_1$ are the upper and lower angular frequencies at which the power has fallen to 1/2 its value at resonance, then $$Q=\frac{\omega_0}{\omega_2-\omega_1}$$ Q is therefore a beautiful measure of the sharpness of the resonance curve.

So what about amplitude resonance? Does one define Q differently in this case? I've never seen this done. And even a somewhat altered definition of Q wouldn't, as far as I know, allow us to retrieve an exact relationship as simple as $Q=\frac{\omega_0}{\omega_2-\omega_1}$ for amplitude resonance. So there's no motivation to define Q in any other way.

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