\( Q \) (quality factor)
\( Q \) (quality factor) is a property that relates the total stored energy to the energy dissipated (loss) —
\begin{align} \label{eq:13601a} Q = 2\pi \left[ \frac{\textsf{Energy stored}}{\textsf{Energy dissipated per cycle}} \right] \end{align}
The "energy dissipated per cycle" is the external energy (work) that would be needed in order to maintain a fixed amount of oscillation.
If the numerator and denominator are multiplied by the frequency \( f \) (cycles/sec), the denominator becomes "Energy dissipated/sec" (i.e., Power), so the above equation can also be written as —
\begin{align} \label{eq:13602a} Q = 2\pi \, f \left[ \frac{\textsf{Energy stored}}{\textsf{Power}} \right] \end{align}
\( Q \) has no units.
\( Q \) can be determined for any system (electrical, mechanical, etc.) or any single component or portion of a system. Note that for a "perfect" system (i.e., no energy dissipation), \( Q \) is infinite.
Interpretation
Resonant system
For a freely oscillating resonant system, \( Q \) indicates the rate at which oscillation decays due to energy loss. A high \( Q \) indicates relatively slow decay (example of figure 1) whereas a low \( Q \) indicates relatively rapid decay (example of figure 2  see log decrement ).






In an ultrasonic system the energy loss may come from internal friction within a vibrating material (which will show up as a temperature rise), energy that is transferred to the support structure or air, or energy that is transferred to the load. In addition, a transducer will have electrical (resistive) losses due to current flow.
Nonresonant system
Although Q is often applied to resonant systems, this is not a requirement. In fact, equations \eqref{eq:13601a} and \eqref{eq:13602a} can equally be applied to nonresonant systems — in particular, to capacitors and inductors.
A nonideal capacitor (i.e., a capacitor with loss) can be represented as an ideal capacitor \( C \) in series with a resistor \( R_C \).
For a capacitor —
\begin{align} \label{eq:13603a} Q &= \frac{R_C}{(1/2\pi \, f \, C)} \\[0.7em]%eqn_interline_spacing &= R \, (2\pi \, f \, C) \nonumber \end{align}
For an inductor —
\begin{align} \label{eq:13604a} Q &= \frac{R_L}{(2\pi \, f \, L)} \end{align}
where —
\( R_C \)  = equivalent resistance of capacitor 
\( R_L \)  = equivalent resistance of inductor 
\( C \)  = capacitance 
\( L \)  = inductance 
\( f \)  = frequency 
Q for resonator materials
For materials used in ultrasonic resonators, the following are relevant.
 \( Q \) is a material property (in a similar manner to the modulus of elasticity). However, the test conditions must be specified because \( Q \) may depend on:
 The material's heat treatment. For example, D2 tool steel has significantly lower loss (power), and hence higher \( Q \), when it is hardened.
 The strain to which the part is subjected. Often \( Q \) will decrease as the strain increases but this depends on the particular material.
 \( Q \) is generally independent of frequency.
 For piezoelectric cermics \( Q \) depends on the operating conditions (static compression, electrical field, temperature, etc.).
If \( Q \) of a material is known then the theoretical power dissipated by a resonator can be calculated if the stored energy (either potential energy (strain energy) or kinetic energy) can be determined from equation \eqref{eq:13602a} —
\begin{align} \label{eq:13605a} \textsf{Power} = 2\pi \, f \left[ \frac{\textsf{Energy stored}}{Q} \right] \end{align}
Data for materials
Neppiras (table 4, p. 146) cites the following \( Q \) values.
Material  \( Q \) 

Naval brass  3000 
Aluminum bronze  17000 
Kmonel  5300 
Duralumin (DTD 363)  50000 
Hiduminium (?)  100000 
Tool steel (KE 672)  1400 
Titanium (318A == Ti6Al4V)  24000 
Wuchinich (1) (figure 10) measured the following values by striking a flexing "chime". However, the strains were low so the cited values may not be relevant at typical ultrasonic strains.
Material  Condition  \( Q \) 

Ti6Al4V  As received  2000  6000 
Ti6Al4V  Annealed  18000  22000 
174PH stainless steel  Annealed  7000 
PH157Mo stainless  Annealed  17000 
PH157Mo stainless  Hardened  17000 
PH138Mo stainless  Annealed  10000 
Custom 455 stainless  H900  10000 
MACOR machineable ceramic  As received  5000 
\( Q \) measurement
For a resonant system \( Q \) can be determined by —
Also see
Attenuation
Damping ratio
Loss tangent
Structural damping