Editing Q factor (section)

== {{mvar|Q}}-factor and damping ==
{{main|Damping|Linear time-invariant system|l2=linear time invariant (LTI) system}}

The {{mvar|Q}}-factor determines the [[qualitative data|qualitative]] behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see [[harmonic oscillator]] and [[LTI system|linear time invariant (LTI) system]].)

Starting from the stored energy definition for, it can be shown that <math> Q = \frac{1}{2\zeta}</math>, where <math>\zeta</math> is the [[Damping#Damping ratio|damping ratio]]. There are three key distinct cases:

* A system with '''low quality factor''' ({{math|''Q'' < {{small|{{sfrac|1|2}}}}}}) is said to be '''overdamped'''. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by [[exponential decay]], approaching the steady state value [[asymptotic]]ally. It has an [[impulse response]] that is the sum of two [[exponential decay|decaying exponential functions]] with different rates of decay. As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system. A second-order [[low-pass filter]] with a very low quality factor has a nearly first-order step response; the system's output responds to a [[Heaviside step function|step input]] by slowly rising toward an asymptote.
* A system with '''high quality factor''' ({{math|''Q'' > {{small|{{sfrac|1|2}}}}}}) is said to be '''underdamped'''. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal. Underdamped systems with a low quality factor (a little above {{math|1=''Q'' = {{small|{{sfrac|1|2}}}}}}) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. More generally, the output of a second-order [[low-pass filter]] with a very high quality factor responds to a step input by quickly rising above, oscillating around, and eventually converging to a steady-state value.
* A system with an '''intermediate quality factor''' ({{math|1=''Q'' = {{small|{{sfrac|1|2}}}}}}) is said to be '''critically damped'''. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response (approach to the final value) possible without overshoot. Real system specifications usually allow some overshoot for a faster initial response or require a slower initial response to provide a [[factor of safety|safety margin]] against overshoot.

In [[negative feedback]] systems, the dominant closed-loop response is often well-modeled by a second-order system. The [[phase margin]] of the open-loop system sets the quality factor {{mvar|Q}} of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor).

=== Some examples ===
{{Anchor|Quality factors of common systems|Some examples}}

{{bulleted list
| A unity-gain [[Sallen–Key topology#Application: low-pass filter|Sallen–Key lowpass filter topology]] with equal capacitors and equal resistors is critically damped (i.e., {{math|''Q'' {{=}} {{small|{{sfrac|1|2}}}}}}).
| A second-order [[Bessel filter]] (i.e., continuous-time filter with flattest [[group delay]]) has an underdamped {{math|''Q'' {{=}} {{small|{{sfrac|{{sqrt|3}}}}}}}}.
| A second-order [[Butterworth filter]] (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped {{math|''Q'' {{=}} {{small|{{sfrac|{{sqrt|2}}}}}}}}.<ref>{{cite book |last1=Sabah |first1=Nassir H. |title=Circuit Analysis with PSpice: A Simplified Approach |date=2017 |publisher=CRC Press |isbn=9781315402215 |page=446 |url=https://books.google.com/books?id=B7W8DgAAQBAJ&pg=PA446}}</ref>
| A pendulum's {{mvar|Q}}-factor is: {{math|''Q'' {{=}} ''Mω''/''Γ''}}, where {{mvar|M}} is the mass of the bob, {{math|''ω'' {{=}} 2''π''/''T''}} is the pendulum's radian frequency of oscillation, and {{mvar|Γ}} is the frictional damping force on the pendulum per unit velocity.
| The design of a high-energy (near [[terahertz (unit)|terahertz]]) [[gyrotron]] considers both diffractive Q-factor, <math display="inline">Q_D \approx 30 \left(\frac{L}{\lambda}\right)^2</math> as a function of resonator length {{mvar|L}}, wavelength {{mvar|λ}}, and ohmic {{mvar|Q}}-factor ({{math|TE{{sub|''m,p''}}}}–modes)
<math display="block">Q_\Omega = \frac{R_\mathrm{w}}{\delta} \frac{1 - m^2}{v^2_{m,p}},</math>
where {{math|''R''{{sub|w}}}} is the cavity wall radius, {{mvar|δ}} is the [[skin depth]] of the cavity wall, {{mvar|v{{sub|m,p}}}} is the [[eigenvalue]] scalar ({{mvar|m}} is the azimuth index, {{mvar|p}} is the radial index; in this application, skin depth is {{nowrap|<math display="inline">\delta = {1}/{\sqrt{ \pi f \sigma u_o}}</math>)}}<ref>{{cite web |title=Near THz Gyrotron: Theory, Design, and Applications |url=https://ireap.umd.edu/sites/ireap.umd.edu/files/documents/ruifengpu-ms.pdf |website=The Institute for Research in Electronics and Applied Physics |publisher=University of Maryland |access-date=5 January 2021}}</ref>
| In [[medical ultrasonography]], a transducer with a high {{mvar|Q}}-factor is suitable for [[doppler ultrasonography]] because of its long ring-down time, where it can measure the velocities of blood flow. Meanwhile, a transducer with a low {{mvar|Q}}-factor has a short ring-down time and is suitable for organ imaging because it can receive a broad range of reflected echoes from bodily organs.<ref>{{cite book |last1=Curry |first1=TS |last2=Dowdey |first2=JE |last3=Murry |first3=RC |title=Christensen's Physics of Diagnostic Radiology |date=1990 |publisher=Lippincott Williams & Wilkins |isbn=9780812113105 |page=331 |url=https://books.google.com/books?id=W2PrMwHqXl0C |access-date=22 January 2023}}</ref>
}}