Resonances, waves and fields: 3.11 Q

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3.11 Q - a review and more

We have discussed the quality factor or Q a number of places in previous postings ([1], Eqn.(35), [2], Eqn.(63), [3] are a few). Reference [1] is an introductory overview of some of the basic concepts. Figure 22 below summarizes the more important equations discussed previously:

Quality factor, Q - a review

Impulse driven resonator

Sinusoidally driven resonator with an increasing drive frequency

Fig. 22a. After an impulse to start it, a resonator will oscillate in a free decay mode. Mouse over the animation to activate it and click on it to restart it.

Fig. 22b. Above we show a resonator excited by a sinusoidal driving force which increases in frequency with time. The amplitude of the resonator peaks at its resonant frequency (2.5Hz in this case). Mouse over the image to start the animation, mouse off to suspend it, or click on it to restart it.

Equations for Fig. 22a - impact excited resonator.

1. The general equation for the decay of a shock excited resonator is given by:

$x(t)=A e^{-t/\tau} \cos \omega_{decay}t$ , (109)

where x is the variable that happens to be oscillating in the particular resonator at hand, A is the initial amplitude at t = 0 , ω_decay is the angular frequency of the decaying oscillations, and t is time.

2. If the resonator has no damping, its resonant frequency is given by [link]:

, (110)

where k is the spring constant and m is the mass of the bob.

3. When there is damping, the resonant frequency becomes:

, (111)

where τ is the decay time of the amplitude (time for the amplitude to fall to 1/e or 37% of its original value) and is given by:

, (112)

where R is the damping factor of a previous posting and Q is the quality factor.

4. The decay time constant for the energy [2^ndlink] in the resonator (sum of kinetic and potential energies) is half that of the above time constant, because the energy is proportional to the square of the amplitude:

. (113)

There is a little ripple [ref] in the energy decay because the damping is often not uniform throughout the resonator's oscillation cycle but instead peaks when the bob's velocity is maximum.

The two decay times are shown in the above animation.

5. Using (113) we show next that the quality factor Q equals π times the number of oscillations the resonator undergoes within one time constant:

$Q= \frac{\tau\omega_0}{2}=\frac{\tau(2 \pi f_0)}{2}=\pi \times(f_0\tau)=\pi \times \text{ (number of oscillations during }\tau)$

= π × (number of oscillations during τ ) . (114a)

From (113) we can also see that Q equals the number of radians required for the energy in the resonator to decay to 1/e of its initial value:

Q = ω₀τ_energy = number of radians in time τ . (114b)

You can understand (114a) and (114b) from their units:

f₀τ =	cycles	× seconds = cycles
	second

and (114c)

ω₀τ_energy =	radians	× seconds = radians
	second

6. The quality factor also relates the power lost during free decay to the energy inside the resonator [ref].

. (115)

Equations for Fig. 22b - sinusoidally excited resonator.

Looking at the above animation, we see that the resonance peak (or curve) is the main feature. It has the following properties:

1. Its horizontal position, i.e. the frequency of the peak. It is approximately equal to ω₀ as given by (110). This approximation is very good at high Q values. A somewhat better approximation is given by w_decay, i.e. (111). The exact peak actually occurs at [ref]:

, (116)

at least for the setup shown above.

2. Another feature of the resonance curve is its height. In terms of the variables of the above resonator the height is given by:

. (117)

Thus the height is proportional to the quality factor Q.

3. Peak width is also called the bandwidth. The bandwidth is defined to be the width, in frequency units, of the curve at a height of half its peak value in the energy graph (the top graph in Fig. 22b above.) Since the amplitude is proportional to the square root of the energy in the resonator, the two half power (i.e. energy) points on the energy graph will correspond to the amplitude being 71% (which is 1/√2) of its peak value. The bandwidths are shown in the above graphs. In the high Q case the bandwidth is given by:

. (118)

4. So we see that higher Q's mean more peaked resonance curves: higher and more narrow.

At the same time, we need to realize that the resonance curve only represents the equilibrium response of the resonator. If we do a frequency sweep, as in the above animation, the frequency must be swept slowly enough that equilibrium is established at each frequency. The non-equilibrium response of a resonator is much more complicated, especially for high Q resonators driven off resonance [ref].

Various sources of loss and various Q's

In most real resonators there are several sources of damping (i.e. of energy loss or dissipation). Examples of losses in the mass spring resonator in Fig. 22a might be the friction of moving the bob through the air and it might simultaneously be the monkey not acting as a rigid solid, but instead flexing with the motions of the bob. In the resonator of Fig. 22b losses might be due to air friction, a somewhat flexible support hook, and the drive motor.

In an earlier posting (Equation (22)) we found the basic differential equation for a resonator with losses was:

, (117)

which has damped sinusoid solutions in the form of (109) above. Additional complexity in the motion can add all sorts of complexity to the defining equation, but in terms of adding damping the important addition come in the form of additional terms containing a first derivative of x. That is, the total damping force becomes:

, (118)

and (117) becomes:

. (119)

A little manipulation of (112) and (110) above shows us that the Q for a single damping source is given by:

. (120)

With the additional damping sources of (118) and (119), equation (120) becomes:

. (121)

multiple Q's:

It is common to define a Q for each source of damping, i.e.:

. . . . (122)

Q's add as inverses:

The Q's add as inverses (similar to parallel resistors):

. (123)

The utility of breaking down the Q into these partial Q's is that a person can get a sense of how each damping means affects the total Q. Because of the property of addition by inverses, the partial Q having the lowest value will affect the total Q the most. Thus a resonator having two damping sources with partial Q's of Q₁ = 100 and Q₂ = 500 will by (123) have a total Q of 83, mostly determined by Q₁. If we were to reduce the second source of damping and raise Q₂ by 100 to 600, the total Q would only rise to 86. On the other hand if we concentrated on the first damping source and raised Q₁ by 100 to 200, then the total Q would rise to 143.

power loss of each partial Q:

In many cases, the power loss from the resonator associated with each damping means is clearly distinct and separated. For example, the power of a mass spring resonator might lose power to friction with the air, to acoustic radiation, and power carried off by a flexible support frame. Each of these mechanisms is clearly separateable and could be measured independently.

Using (115) and (123) above we can write:

, (124)

where U is the total energy in the resonator and:

. (125)

We see that (124) is consistent with conservation of power:

P_total = P₁ + P₂ + P₃ + etc. . (126)

Equation (125) indicates that each power loss is proportional to the total energy in the resonator, as should be the case for linear systems.

We should stress that our beginning equation, (117), had no source term and therefore appropriate for free decay of a resonator. If we add a constant sinusoidal driving force, then it turns out that we can still separate out the various losses and Q's as in (121), (122)and (123). At the same time, the power loss equations are, in general, not separable as in (124) and (125). Therefore in calculating the Q's from power loss, it is best to consider the case in which the source is turned off and the resonator is in free decay.

Coupling Q

Of particular importance to experts that study resonators is the "coupling Q". This is the Q associated with damping of the resonator due to the drive mechanism. Coupling Q is not an issue for an impact driven resonator if the object doing the impacting only makes momentary contact with the resonator as in [link]. However in the impact case of Fig. 22a where the contact is more than monentary or in most sinusoidally driven cases, such as that shown in Fig. 22b, the coupling Q (and the associated coupling losses) are very important.

In the case of a driven resonator, the coupling power loss is especially confusing because the drive mechanism is supplying energy to drive the resonance. So where is the loss?

One way to explain this is using the series-of-impacts model discussed earlier in postings [link]. There we considered the driving sinusoid to be equivalent to a series of impulses, where each impulse involves the driver being momentarily operated and then turned off, but not disconnected from the resonator. The total response of the resonator is the sum of the "ringing" from all these impulses. If the frequency of the driving impulses was "off resonance" the ringings partially destructively interfere and the response is less than when it was "on resonance". Any damping caused by the non-energized driver will shorten the ringing times (time constant τ) and therefore reduce the total response and broaden the resonance curve.

Fig. 23. Elemental electronic LRC resonator with an AC driver having output resistance.

A mathematical explanation is that if there is a dissipative term involving the driver in the defining differential equation, then there will be "coupling losses" due to this term and the peak height will be reduced and peak width broadened as discussed above. Fig. 23 shows the equivalent circuit of an LRC circuit with a series driving source. Note the "output resistance" of the source (most real, non-ideal, electronic drivers have a finite output resistance) that is in series with the resistance of the resonator part of the circuit. This output resistance will add to the resonator resistance in determining the total dissipation and decay time constant of the circuit.

In most simple drive circuits (or drive mechanisms in the mass/spring or other mechanical resonators) the output resistance of the driver is the resistance to current (or motion) exerted by the driver when it (the driver) is turned off. We might consider the situation where the driver is sinusoidally driving the resonator for an interval of time [as in an earlier posting] then is turned off. We could measure the decay time and from that calculate the Q resulting from combined damping of the driver and the resonator using (112) solved for Q:

. (127)

To be able to measure the losses in the resonator alone, an experimenter must minimize the dissipation due to the resonator driver. Using an impact source that is truly disconnected from the resonator after the impact is a good approach. Another approach is to minimize the coupling between the driver and resonator. A third approach is to use a driver that causes minimal damping when it is switched off. For example, for the driver in Fig. 22b, a motor and linkages that are very rigid and cannot be pushed around by the resonator's motion, would cause very little damping of the resonance, even when the motor is switched off. Its coupling Q would be very high and have little impact on the total Q.

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