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tails about these systems and their behavior see harmonic oscillator and linear time invariant lti system starting from the stored energy definition it can be shown that q 1 2 ζ displaystyle q frac 1 2 zeta where ζ displaystyle zeta is the damping ratio there are three key distinct cases a system with low quality factor q 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 asymptotically it has an impulse response that is the sum of two 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 step input by slowly rising toward an asymptote a system with high quality factor q 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 q 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 q 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 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 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 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 ...
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