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edit a unity gain sallen key lowpass filter topology with equal capacitors and equal resistors is critically damped i e q 1 2 a second order bessel filter i e continuous time filter with flattest group delay has an underdamped q 1 3 a second order butterworth filter i e continuous time filter with the flattest passband frequency response is underdamped q 1 2 11 a pendulum s q factor is q mω γ where m is the mass of the bob ω 2 π t is the pendulum s radian frequency of oscillation and γ is the frictional damping force on the pendulum per unit velocity the design of a high energy near terahertz gyrotron considers both diffractive q factor q d 30 l λ 2 textstyle q_ d approx 30 left frac l lambda right 2 as a function of resonator length l wavelength λ and ohmic q factor te m p modes q ω r w δ 1 m 2 v m p 2 displaystyle q_ omega frac r_ mathrm w delta frac 1 m 2 v_ m p 2 where r w is the cavity wall radius δ is the skin depth of the cavity wall v m p is the eigenvalue scalar m is the azimuth index p is the radial index in this application skin depth is δ 1 π f σ u o textstyle delta 1 sqrt pi f sigma u_ o 12 in medical ultrasonography a transducer with a high 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 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 13 physical interpretation edit physically speaking q is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation or nearly equivalently at high enough q values 2 π times the ratio of the total energy stored and the energy lost in a single cycle 14 it is a dimensionless parameter that compares the exponential time constant τ for decay of an oscillating physical system s amplitude to its oscillation period equivalently it compares the frequency at which a system oscillates to the rate at which it dissipates its energy more precisely the frequency and period used should be based on the system s natural frequency which at low q values is somewhat higher than the oscillation frequency as measured by zero crossings 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 e 2 π or about 1 535 or 0 2 of its original energy 15 this means the amplitude falls off to approximately e π or 4 of its original amplitude 16 the width bandwidth of the resonance is given by approximately δ f f n q displaystyle delta f frac f_ mathrm n q where f n is the natural frequency and δ f the bandwidth is the width of the range of frequencies for which the energy is at least half its peak value the resonant frequency is often expressed in natural units radians per second rather than using the f n in hertz as ω n 2 π f n displaystyle omega _ mathrm n 2 pi f_ mathrm n the factors q damping ratio ζ natural frequency ω n attenuation rate α and exponential time constant τ are related such that 17 page needed q 1 2 ζ ω n 2 α τ ω n 2 displaystyle q frac 1 2 zeta frac omega _ mathrm n 2 alpha frac tau omega _ mathrm n 2 and the damping ratio can be expressed as ζ 1 2 q α ω n 1 τ ω n displaystyle zeta frac 1 2q alpha over omega _ mathrm n 1 over tau omega _ mathrm n the envelope of oscillation decays proportional to e αt or e t τ where α and τ can be expressed as α ω n 2 q ζ ω n 1 τ displaystyle alpha omega _ mathrm n over 2q zeta omega _ mathrm n 1 over tau and τ 2 q ω n 1 ζ ω n 1 α displaystyle tau 2q over omega _ mathrm n 1 over zeta omega _ mathrm n frac 1 alpha the energy of oscillation or the power dissipation decays twice as fast that is as the square of the amplitude as e 2 αt or e 2 t τ for a two pole lowpass filter the transfer function of the filter is 17 h s ω n 2 s 2 ω n q 2 ζ ω n 2 α s ω n 2 displaystyle h s frac omega _ mathrm n 2 s 2 underbrace frac omega _ mathrm n q _ 2 zeta omega _ mathrm n 2 alpha s omega _ mathrm n 2 for this system when q 1 2 i e when the system is underdamped it has two complex conjugate poles that each have a real part of α that is the attenuation parameter α represents the rate of exponential decay of the oscillations that is of the output after an 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 sinusoidal tone for a long time after being struck by a hammer transfer functions for 2nd order filters filter type 2nd order transfer function h s 18 lowpass ω n 2 s 2 ω n q s ω n 2 displaystyle frac omega _ mathrm n 2 s 2 frac omega _ mathrm n q s omega _ mathrm n 2 bandpass ω n q s s 2 ω n q s ω n 2 displaystyle frac frac omega _ mathrm n q s s 2 frac omega _ mathrm n q s omega _ mathrm n 2 notch bandstop s 2 ω n 2 s 2 ω n q s ω n 2 displaystyle frac s 2 omega _ mathrm n 2 s 2 frac omega _ mathrm n q s omega _ mathrm n 2 highpass s 2 s 2 ω n q s ω n 2 displaystyle frac s 2 s 2 frac omega _ mathrm n q s omega _ mathrm n 2 electrical systems edit 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 logarithmically scaled for an electrically resonant system the q factor represents the effect of electrical resistance and for electromechanical resonators such as quartz crystals mechanical friction relationship between q and bandwidth edit the 2 sided bandwidth relative to a resonant frequency of f 0 hz is f 0 q displaystyle frac f_ 0 q for example an antenna tuned to have a q value of 10 and a centre frequency of 100 khz would have a 3 db bandwidth of 10 khz in audio bandwidth is often expressed in terms of octaves then the relationship between q and bandwidth is q 2 b w 2 2 b w 1 1 2 sinh 1 2 ln 2 b w displaystyle q frac 2 frac bw 2 2 bw 1 frac 1 2 sinh left frac 1 2 ln 2 bw right where bw is the bandwidth in octaves 19 rlc circuits edit in an ideal series rlc circuit and in a tuned radio frequency receiver trf the q factor is 20 q 1 r l c ω 0 l r 1 ω 0 r c displaystyle q frac 1 r sqrt frac l c frac omega _ 0 l r frac 1 omega _ 0 rc where r l and c are the resistance inductance and capacitance of the tuned circuit respectively larger series resistances correspond to lower circuit q values for a parallel rlc circuit the q factor is the inverse of the series case 21 20 q r c l r ω 0 l ω 0 r c displaystyle q r sqrt frac c l frac r omega _ 0 l omega _ 0 rc 22 consider a circuit where r l and c are all in parallel the lower the parallel resistance is the more effect it will have in damping the circuit and thus result in lower q this is useful in filter design to determine the bandwidth 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 individual reactive components edit 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 q of a resonant circuit using that inductor including its series loss and a perfect capacitor 23 q l x l r l ω 0 l r l displaystyle q_ l frac x_ l r_ l frac omega _ 0 l r_ l where ω 0 is the resonance frequency in radians per second l is the inductance x l is the inductive reactance and r l is the series resistance of the inductor 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 23 q c x c r c 1 ω 0 c r c displaystyle q_ c frac x_ c r_ c frac 1 omega _ 0 cr_ c where ω 0 is the resonance frequency in radians per second c is the capacitance x c is the capacitive reactance and r c is the series resistance of the capacitor 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 23 q 1 1 q l 1 q c displaystyle q frac 1 frac 1 q_ l frac 1 q_ c mechanical systems edit for a single damped mass spring system the q factor represents the effect of simplified viscous damping or drag where the damping force or drag force is proportional to velocity the formula for the q factor is q m k d displaystyle q frac sqrt mk d where m is the mass k is the spring constant and d is the damping coefficient defined by the equation f damping dv where v is the velocity 24 acoustical systems edit the q of a musical instrument is critical an excessively high q in a resonator will not evenly amplify the multiple frequencies an instrument produces for this reason string instruments often have bodies with complex shapes so that they produce a wide range of frequencies fairly evenly the q of a brass instrument or wind instrument needs to be high enough to pick one frequency out of the broader spectrum buzzing of the lips or reed by contrast a vuvuzela is made of flexible plastic and therefore has a very low q for a brass instrument giving it a muddy breathy tone instruments made of stiffer plastic brass or wood have higher q values an excessively high q can make it harder to hit a note q in an instrument may vary across frequencies but this may not be desirable helmholtz resonators have a very high q as they are designed for picking out a very narrow range of frequencies optical systems edit in optics the q factor of a resonant cavity is given by q 2 π f o e p displaystyle q frac 2 pi f_ o e p where f o is the resonant frequency e is the stored energy in the cavity and p de dt 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 s cavity is abruptly changed from a low value to a high one the laser will emit a pulse of light that is much more intense than the laser s normal continuous output this technique is known as q switching q factor is of particular importance in plasmonics where loss is linked to the damping of the surface plasmon resonance 25 while loss is normally considered a hindrance in the development of plasmonic devices it is possible to leverage this property to present new enhanced functionalities 26 see also edit acoustic resonance attenuation chu harrington limit list of piezoelectric materials phase margin q meter q multiplier dissipation factor references edit tooley michael h 2006 electronic circuits fundamentals and applications newnes pp 77 78 isbn 978 0 7506 6923 8 archived from the original on 2016 12 01 hickman ian 2013 analog electronics analog circuitry explained newnes p 42 isbn 9781483162287 encyclopedia of laser physics and technology q factor archived 2009 02 24 at the wayback machine time and frequency from a to z q to ra archived 2008 05 04 at the wayback machine 1 2 3 4 green estill i october 1955 the story of q pdf american scientist 43 584 594 archived pdf from the original on 2012 12 03 retrieved 2012 11 21 b jeffreys q jl r astr soc 1985 26 51 52 paschotta rüdiger 2008 encyclopedia of laser physics and technology vol 1 a m wiley vch p 580 isbn 978 3527408283 archived from the original on 2018 05 11 1 2 slyusar v i 60 years of electrically small antennas theory proceedings of the 6 th international conference on antenna theory and techniques 17 21 september 2007 sevastopol ukraine pp 116 118 antenna theory and techniques pdf archived pdf from the original on 2017 08 28 retrieved 2017 09 02 1 2 u a bakshi a v bakshi 2006 network analysis technical publications p 228 isbn 9788189411237 james w nilsson 1989 electric circuits addison wesley publishing company isbn 0 201 17288 7 sabah nassir h 2017 circuit analysis with pspice a simplified approach crc press p 446 isbn 9781315402215 near thz gyrotron theory design and applications pdf the institute for research in electronics and applied physics university of maryland retrieved 5 january 2021 curry ts dowdey je murry rc 1990 christensen s physics of diagnostic radiology lippincott williams wilkins p 331 isbn 9780812113105 retrieved 22 january 2023 jackson r 2004 novel sensors and sensing bristol institute of physics pub p 28 isbn 0 7503 0989 x benjamin crowell 2006 light and matter archived from the original on 2011 05 19 ch 18 anant agarwal 2005 foundations of analog digital electronic circuits lang jeffrey jeffrey h amsterdam elsevier p 647 isbn 9781558607354 oclc 60245509 1 2 siebert william mcc circuits signals and systems mit press analog dialogue technical journal analog devices pdf www analog com archived pdf from the original on 2016 08 04 dennis bohn rane january 2008 bandwidth in octaves versus q in bandpass filters www rane com archived from the original on 2019 10 21 retrieved 2019 11 20 1 2 u a bakshi a v bakshi 2008 electric circuits technical publications pp 2 79 isbn 9788184314526 permanent dead link complete response i constant input fourier eng hmc edu archived from the original on 2012 01 10 frequency response resonance bandwidth q factor archived 2014 12 06 at the wayback machine pdf 1 2 3 di paolo franco 2000 networks and devices using planar transmission lines crc press pp 490 491 isbn 9780849318351 archived from the original on 2018 05 11 methods of experimental physics lecture 5 fourier transforms and differential equations archived 2012 03 19 at the wayback machine pdf tavakoli mehdi jalili yousef seyed elahi seyed mohammad 2019 04 28 rayleigh wood anomaly approximation with fdtd simulation of plasmonic gold nanohole array for determination of optimum extraordinary optical transmission characteristics superlattices and microstructures 130 454 471 bibcode 2019sumi 130 454t doi 10 1016 j spmi 2019 04 035 s2cid 150365680 chen gang mahan gerald meroueh laureen huang yi tsurimaki yoichiro tong jonathan k ni george zeng lingping cooper thomas alan 2017 12 31 losses in plasmonics from mitigating energy dissipation to embracing loss enabled functionalities advances in optics and photonics 9 4 775 827 arxiv 1802 01469 bibcode 2017adop 9 775b doi 10 1364 aop 9 000775 issn 1943 8206 further reading edit agarwal anant lang jeffrey 2005 foundations of analog and digital electronic circuits morgan kaufmann isbn 1 55860 735 8 external links edit wikimedia commons has media related to quality factor calculating the cut off frequencies when center frequency and q factor is given explanation of q factor in radio tuning circuits authority control databases gnd retrieved from https en wikipedia org w index 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