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| {{For|other uses of the terms '''Q''', '''Q factor''', and '''Quality factor'''|Q value (disambiguation)}}
| | This is my favorite workout, all you require is a good pair of boots along with a jump rope. This is a fast paced, full body workout. Skipping needs a bit of practice and after you become skilled in skipping, you are burning more calories plus building your muscles inside a right technique. You can burn an incredible 200 calories in a mere 12 minute exercise session of skipping. This exercise is even better than running and is the best cardio exercise to get rid of fat.<br><br>This change by itself may assist we cut a sodium intake, clean a colon, shed fat, and lose weight fast. All animal goods, even lean meats, contain excess calories, sodium plus saturated fat. As an example, rather of 8 ounces of grilled poultry breast with 440 calories plus 8 grams of saturated fat, you have the same amount of firm tofu with 328 calories plus 0 grams of saturated fat. Why starve oneself when you are able to have the cake plus eat it too?<br><br>In regards to the quickest means to lose weight, sleep is an often overlooked component. People simply talk about eating and working out to get rid of weight. Not everybody ever states sleeping is important for weight loss. Yet somehow, this really is regarded as the most important components inside the weight-loss equation.<br><br>First of all, numerous folks mistakenly think that HCG is simply another fancy designer drug. In truth, HCG is a hormone that happens naturally in every of our bodies. Whenever women are expecting, they have a very large supply of the hormone. Because of the, whenever a woman takes a pregnancy test, if a certain focus of HCG is found inside the circulation, then the woman is considered positive for the test.<br><br>Have two slices of wheat bread with some soy cheese spread on them. Prepare a fruit salad that contains pineapple chunks, kiwi fruit, papaya, passion fruit with several [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight] topping of muesli. Grill this mixture for five minutes. This breakfast should contribute about 288 calories.<br><br>Exercising initial thing each morning eliminates these feelings. You have finished before the day begins. We is happier because we will learn which the rest of the day is free because we will have already done what you required to. There is no guilt and no dreading exercise considering it may absolutely be completed.<br><br>All inside all, this is a quite quick weight loss so you should be prepared to accept the possibility that 2 weeks from now we can not be 10 lbs lighter. However, should you choose the right fat loss program plus stick to it, we will see quickly plus continuous results. |
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| [[Image:Bandwidth.svg|thumb|350px|The [[Bandwidth (signal processing)|bandwidth]], <math>\Delta f</math>, or ''f''<sub>1</sub> to ''f''<sub>2</sub>, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is <math>f_c/\Delta f</math>. The higher the Q, the narrower and 'sharper' the peak is.]]
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| In [[physics]] and [[engineering]] the '''quality factor''' or '''Q factor''' is a [[Dimensionless quantity|dimensionless]] parameter that describes how [[damping|under-damped]] an [[harmonic oscillator|oscillator]] or [[resonator]] is,<ref>{{cite book
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| | title = Electric power transformer engineering
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| | author = James H. Harlow
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| | publisher = CRC Press
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| | year = 2004
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| | isbn = 978-0-8493-1704-0
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| | pages = 2–216
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| | url = http://books.google.com/books?id=DANXjaoaucYC&pg=PT241&dq=q-factor+damping#v=onepage&q=q-factor%20damping&f=false
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| }}</ref> or equivalently, characterizes a resonator's [[bandwidth (signal processing)|bandwidth]] relative to its center frequency.<ref>{{cite book
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| | title = Electronic circuits: fundamentals and applications
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| | author = Michael H. Tooley
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| | publisher = Newnes
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| | year = 2006
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| | isbn = 978-0-7506-6923-8
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| | pages = 77–78
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| | url = http://books.google.com/books?id=8fuppV9O7xwC&pg=PA77&dq=q-factor+bandwidth#v=onepage&q=q-factor%20bandwidth&f=false
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| }}</ref>
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| Higher ''Q'' indicates a lower rate of energy loss relative to the stored energy of the resonator; the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high ''Q'', while a pendulum immersed in oil has a low one. Resonators with high quality factors have low [[damping]] so that they ring longer.
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| ==Explanation==
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| [[sine wave|Sinusoidally]] driven [[resonator]]s having higher Q factors [[resonance|resonate]] with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high-Q [[RLC circuit|tuned circuit]] in a radio receiver would be more difficult to tune, but would have more [[selectivity (electronic)|selectivity]]; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High-Q oscillators oscillate with a smaller range of frequencies and are more stable. (See [[oscillator phase noise]].)
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| The quality factor of oscillators varies substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q near ½. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability have high quality factors. Tuning forks have quality factors around 1000. The quality factor of [[atomic clock]]s, [[Superconducting Radio Frequency|superconducting RF]] cavities used in accelerators, and some high-Q [[optical cavity|lasers]] can reach as high as 10<sup>11</sup><ref>
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| [http://www.rp-photonics.com/q_factor.html Encyclopedia of Laser Physics and Technology:Q factor]
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| </ref> and higher.<ref>
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| [http://tf.nist.gov/general/enc-q.htm Time and Frequency from A to Z: Q to Ra]
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| </ref>
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| There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the [[damping ratio]], [[bandwidth (signal processing)|relative bandwidth]], [[oscillator linewidth|linewidth]] and bandwidth measured in [[Octave (electronics)|octave]]s.
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| The concept of "Q" originated with K.S. Johnson of Western Electric Company's Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol Q was only because all other letters of the alphabet were taken. The term was not intended as an abbreviation for "quality" or "quality factor", although these terms have grown to be associated with it.<ref>http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf</ref><ref>B. Jeffreys, Q.Jl R. astr. Soc. (1985) 26, 51-52</ref><ref name="Paschotta">{{cite book
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| | last = Paschotta
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| | first = Rüdiger
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| | authorlink =
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| | coauthors =
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| | title = Encyclopedia of Laser Physics and Technology, Vol. 1: A-M
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| | publisher = Wiley-VCH
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| | year = 2008
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| | location =
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| | pages = 580
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| | url = http://books.google.com/books?id=hdkJ5ASTFjcC&pg=PA580&dq=%22Q+factor%22+definition+history
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| | doi =
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| | isbn = 3527408282}}</ref>
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| == Definition of the quality factor ==
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| In the context of resonators, ''Q'' is defined in terms of the ratio of the energy stored in the resonator to the energy supplied by a generator, per cycle, to keep signal [[amplitude]] constant, at a frequency (the [[resonant frequency]]), ''f<sub>r</sub>'', where the stored energy is constant with time:
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| :<math>
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| Q = 2 \pi \times \frac{\mbox{Energy Stored}}{\mbox{Energy dissipated per cycle}} = 2 \pi f_r \times \frac{\mbox{Energy Stored}}{\mbox{Power Loss}}. \,
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| </math>
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| The factor 2''π'' makes ''Q'' expressible in simpler terms, involving only the coefficients of the second-order differential equation describing most resonant systems, electrical or mechanical. In electrical systems, the stored energy is the sum of energies stored in lossless [[inductors]] and [[capacitors]]; the lost energy is the sum of the energies dissipated in [[resistors]] per cycle. In mechanical systems, the stored energy is the maximum possible stored energy, or the total energy, i.e. the sum of the [[potential energy|potential]] and [[kinetic energy|kinetic]] energies at some point in time; the lost energy is the work done by an external [[conservative force]], per cycle, to maintain amplitude.
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| For high values of ''Q'', the following definition is also mathematically accurate:
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| :<math>Q = \frac{f_r}{\Delta f} = \frac{\omega_r}{\Delta \omega}, \,</math>
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| where ''f<sub>r</sub>'' is the resonant frequency, Δ''f'' is the half-power bandwidth i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ''ω<sub>r</sub>'' = 2''πf<sub>r</sub>'' is the [[angular frequency|angular]] resonant frequency, and Δ''ω'' is the angular half-power bandwidth.
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| More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of ''Q'' is used:<ref>{{cite book|title=Electric Circuits|isbn=0-201-17288-7|author=James W. Nilsson|year=1989}}</ref>
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| :<math>
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| Q(\omega) = \omega \times \frac{\mbox{Maximum Energy Stored}}{\mbox{Power Loss}}, \,
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| </math>
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| where ''ω'' is the [[angular frequency]] at which the stored energy and power loss are measured. This definition is consistent with its usage in describing circuits with a single reactive element (capacitor or inductor), where it can be shown to be equal to the ratio of [[reactive power]] to [[real power]]. (''See'' [[#Individual reactive components|Individual reactive components]].)
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| == Q factor and damping ==
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| {{main|damping|LTI system theory|l2=linear time invariant (LTI) system}}
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| The ''Q'' factor determines the [[qualitative data|qualitative]] behavior of simple [[damping|damped]] oscillators. (For mathematical details about these systems and their behavior see [[harmonic oscillator]] and [[LTI system|linear time invariant (LTI) system]].)
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| * A system with '''low quality factor''' (''Q'' < ½) is said to be '''[[overdamping|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]].
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| * A system with '''high quality factor''' (''Q'' > ½) is said to be '''[[underdamping|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 ''Q'' = ½) 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.
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| * A system with an '''intermediate quality factor''' (''Q'' = ½) 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.
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| 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 ''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).
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| === Quality factors of common systems ===
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| * A unity gain [[Sallen–Key topology|Sallen–Key filter topology]] with equivalent capacitors and equivalent resistors is critically damped (i.e., <math>Q = 1/2</math>).{{Citation needed|date=October 2008}}
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| * A second order [[Butterworth filter]] (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped <math>Q = 1/\sqrt{2}</math>.<ref>http://opencourseware.kfupm.edu.sa/colleges/ces/ee/ee303/files%5C5-Projects_Sample_Project3.pdf</ref>
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| * A [[Bessel filter]] (i.e., continuous-time filter with flattest [[group delay]]) has an underdamped <math>Q = 1/\sqrt{3}</math>.{{Citation needed|date=December 2007}}
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| == Physical interpretation of Q ==
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| Physically speaking, ''Q'' is <math>2\pi</math> times the ratio of the total energy stored divided by the energy lost in a single cycle or equivalently the ratio of the stored energy to the energy dissipated over one radian of the oscillation.<ref>
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| {{cite book | last = Jackson | first = R. | title = Novel Sensors and Sensing | publisher = Institute of Physics Pub | location = Bristol | year = 2004 | isbn = 0-7503-0989-X | pages = 28 }}
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| </ref>
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| It is a dimensionless parameter that compares the [[Exponential decay#Mean_lifetime|exponential time constant]] τ for decay of an [[oscillating]] physical system's [[amplitude]] to its oscillation [[Frequency|period]]. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy.
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| Equivalently (for large values of ''Q''), the ''Q'' factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to <math>e^{-2\pi}</math>, or about 1/535 or 0.2%, of its original energy.<ref>{{cite web | title = Vibrations and Waves | work = Light and Matter online text series | author = Benjamin Crowell |year=2006 | url = http://www.lightandmatter.com/html_books/3vw/ch02/ch02.html }}, Ch.2</ref>
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| The width (bandwidth) of the resonance is given by
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| :<math>
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| \Delta f = \frac{f_0}{Q} \,
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| </math>,
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| where <math>f_0</math> is the [[resonant frequency]], and <math>\Delta f</math>, the [[Bandwidth (signal processing)|bandwidth]], is the width of the range of frequencies for which the energy is at least half its peak value.
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| The resonant frequency is often expressed in natural units (radians per second), rather than using the <math>f_0</math> in [[hertz]], as
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| :<math>\omega_0 = 2 \pi f_0</math>.
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| The factors ''Q'', [[damping ratio]] ζ, [[Exponential decay|attenuation rate]] α, and [[Exponential decay#Mean_lifetime|exponential time constant]] τ are related such that:<ref name=Siebert>{{cite book | title = Circuits, Signals, and Systems | author = William McC. Siebert | publisher = MIT Press }}</ref>
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| :<math>
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| Q = \frac{1}{2 \zeta} = { \omega_0 \over 2 \alpha } = { \tau \omega_0 \over 2 },
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| </math>
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| and the [[damping ratio]] can be expressed as:
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| :<math>
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| \zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 } = { 1 \over \tau \omega_0 }.
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| </math>
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| The envelope of oscillation decays proportional to <math>e^{-\alpha t}</math> or <math>e^{-t / \tau}</math>, where α and τ can be expressed as:
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| :<math>\alpha = { \omega_0 \over 2 Q } = \zeta \omega_0 = {1 \over \tau}</math>
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| and
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| :<math>\tau = { 2 Q \over \omega_0 } = {1 \over \zeta \omega_0} = {1 \over \alpha} </math>.
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| The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as <math>e^{-2\alpha t}</math> or <math>e^{-2t / \tau}</math>.
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| For a two-pole lowpass filter, the [[transfer function]] of the filter is<ref name=Siebert/>
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| :<math>
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| H(s) = \frac{ \omega_0^2 }{ s^2 + \underbrace{ \frac{ \omega_0 }{Q} }_{2 \zeta \omega_0 = 2 \alpha }s + \omega_0^2 } \,
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| </math>
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| For this system, when <math>Q > 0.5</math> (i.e., when the system is underdamped), it has two complex [[complex conjugate|conjugate]] poles that each have a [[real part]] of <math>-\alpha</math>. That is, the attenuation parameter <math>\alpha</math> represents the rate of [[exponential decay]] of the oscillations (that is, of the output after an [[impulse response|impulse]]) into the system. A higher quality factor implies a lower attenuation rate, and so high-Q systems oscillate for many cycles. For example, high-quality bells have an approximately [[pure tone|pure sinusoidal tone]] for a long time after being struck by a hammer.
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| <center>
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| {|
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| |-----
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| {| class="wikitable" style="text-align: center;"
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| |-
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| !Filter type (2nd order)!!Transfer function<ref>[http://www.analog.com/library/analogdialogue/archives/43-09/edch%208%20filter.pdf Chapter 8 – Analog Filters – Analog Devices]</ref>
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| |-
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| !Lowpass
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| |<math>H(s) = \frac{ \omega_0^2 }{ s^2 + \frac{ \omega_0 }{Q}s + \omega_0^2 }</math>
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| |-
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| !Bandpass
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| |<math>H(s) = \frac{ \frac{\omega_0}{Q}s}{ s^2 + \frac{ \omega_0 }{Q}s + \omega_0^2 }</math>
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| |-
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| !Notch
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| |<math>H(s) = \frac{ s^2 + \omega_z^2}{ s^2 + \frac{ \omega_0 }{Q}s + \omega_0^2 }</math>
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| |-
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| !Highpass
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| |<math>H(s) = \frac{ s^2 }{ s^2 + \frac{ \omega_0 }{Q}s + \omega_0^2 }</math>
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| |}
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| |}</center>
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| == Electrical systems ==
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| [[Image:bandwidth.svg|right|350px|thumb|A graph of a filter's gain magnitude, illustrating the concept of -3 dB at a voltage gain of 0.707 or half-power bandwidth. The frequency axis of this symbolic diagram can be linear or [[logarithm]]ically scaled.]]
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| For an electrically resonant system, the ''Q'' factor represents the effect of [[electrical resistance]] and, for electromechanical resonators such as [[Crystal oscillator|quartz crystals]], mechanical [[friction]].
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| === RLC circuits ===
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| In an ideal series [[RLC circuit]], and in a [[tuned radio frequency receiver]] (TRF) the ''Q'' factor is:
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| :<math>
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| Q = \frac{1}{R} \sqrt{\frac{L}{C}} = \frac{\omega_0 L}{R}
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| </math>, | |
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| where <math>R</math>, <math>L</math> and <math>C</math> are the [[electrical resistance|resistance]], [[inductance]] and [[capacitance]] of the tuned circuit, respectively. The larger the series resistance, the lower the circuit Q.
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| For a parallel RLC circuit, the Q factor is the inverse of the series case:<ref>[http://fourier.eng.hmc.edu/e84/lectures/ch3/node8.html Series and Parallel Resonance]</ref>
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| :<math>
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| Q = R \sqrt{\frac{C}{L}} = \frac{R}{\omega_0 L}
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| </math>,
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| Consider a circuit where R, L and C are all in parallel. The lower the parallel resistance, the more effect it will have in damping the circuit and thus the lower the Q. This is useful in filter design to determine the bandwidth.
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| In a parallel LC circuit where the main loss is the resistance of the inductor, R, in series with the inductance, L, ''Q'' is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve Q and narrow the bandwidth is the desired result.
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| === Individual reactive components ===
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| The Q of an individual reactive component depends on the frequency at which it is evaluated, which is typically the resonant frequency of the circuit that it is used in. The Q of an inductor with a series loss resistance is the same as the Q of a resonant circuit using that inductor with a perfect capacitor:<ref name=dipaolo>
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| {{cite book
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| | title = Networks and Devices Using Planar Transmission Lines
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| | author = Franco Di Paolo
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| | publisher = CRC Press
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| | year = 2000
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| | isbn = 9780849318351
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| | pages = 490–491
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| | url = http://books.google.com/books?id=z9CsA1ZvwW0C&pg=PA489
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| }}</ref>
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| :<math>Q_L = \frac{X_L}{R_L}=\frac{\omega_0 L}{R_L}</math>
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| Where:
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| * <math>\omega_0</math> is the resonance frequency in radians per second,
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| * <math>L</math> is the inductance,
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| * <math>X_L</math> is the [[Electrical reactance#Inductive reactance|inductive reactance]], and
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| * <math>R_L</math> is the series resistance of the inductor.
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| The Q of a capacitor with a series loss resistance is the same as the Q of a resonant circuit using that capacitor with a perfect inductor:<ref name=dipaolo/>
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| :<math>Q_C = \frac{X_C}{R_C}=\frac{1}{\omega_0 C R_C}</math>
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| Where:
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| * <math>\omega_0</math> is the resonance frequency in radians per second,
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| * <math>C</math> is the capacitance,
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| * <math>X_C</math> is the [[Electrical reactance#Capacitive reactance|capacitive reactance]], and
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| * <math>R_C</math> is the series resistance of the capacitor.
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| In general, the Q of a resonator involving a series combination of a capacitor and an inductor can be determined from the Q values of the components, whether their losses come from series resistance or otherwise:<ref name=dipaolo/>
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| : <math> Q = \frac{1}{(1/Q_L + 1/Q_C)} </math>
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| == Mechanical systems ==
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| For a single damped mass-spring system, the ''Q'' factor represents the effect of simplified [[viscosity|viscous]] damping or [[Drag (physics)|drag]], where the damping force or drag force is proportional to velocity. The formula for the Q factor is:
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|
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| :<math>
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| Q = \frac{\sqrt{M k}}{D}, \,
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| </math>
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| <!-- To derive the above equation, go to the reference link and substitute eqn (2) into eqn (3), then simplify -->
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| where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation <math>F_{\text{damping}}=-Dv</math>, where <math>v</math> is the velocity.<ref>[http://units.physics.uwa.edu.au/__data/page/115450/lecture5_(amplifier_noise_etc).pdf Methods of Experimental Physics – Lecture 5: Fourier Transforms and Differential Equations]</ref>
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| == Optical systems ==
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| In [[optics]], the ''Q'' factor of a [[resonant cavity]] is given by
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| :<math>
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| Q = \frac{2\pi f_o\,\mathcal{E}}{P}, \,
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| </math>
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| where <math>f_o</math> is the resonant frequency, <math>\mathcal{E}</math> is the stored energy in the cavity, and <math>P=-\frac{dE}{dt}</math> is the power dissipated. The optical ''Q'' is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonant [[photon]] in the cavity is proportional to the cavity's ''Q''. If the ''Q'' factor of a [[laser|laser's]] cavity is abruptly changed from a low value to a high one, the laser will emit a [[Pulse (physics)|pulse]] of light that is much more intense than the laser's normal continuous output. This technique is known as [[Q-switching]].
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| == See also ==
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| * [[Damping ratio]]
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| * [[Attenuation]]
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| * [[Phase margin]]
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| * [[Bandwidth (signal processing)|Bandwidth]]
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| * [[Q meter]]
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| * [[Dissipation factor]]
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| {{refbegin}}
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| * {{Cite book|last1=Agarwal|first1=Anant|authorlink1=Anant Agarwal|last2=Lang|first2=Jeffrey|title=Foundations of Analog and Digital Electronic Circuits|year=2005|publisher=Morgan Kaufmann|isbn=1-55860-735-8|url = http://books.google.com/books?id=83onAAAACAAJ&dq=intitle:%22Foundations+of+Analog+and+Digital+Electronic+Circuits%22 }}
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| {{refend}}
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| == External links ==
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| {{Commons category|Quality factor}}
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| * [http://www.sengpielaudio.com/calculator-cutoffFrequencies.htm Calculating the cut-off frequencies when center frequency and Q factor is given]
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| * [http://www.techlib.com/reference/q.htm Explanation of Q factor in radio tuning circuits]
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| [[Category:Electrical parameters]]
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| [[Category:Linear filters]]
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| [[Category:Mechanics]]
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| [[Category:Optics]]
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