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oscillator resonates is called the bandwidth thus a high q tuned circuit in a radio receiver would be more difficult to tune but would have more 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 the quality factor of devices varies substantially from system to system depending on their function and design systems for which damping is important such as dampers keeping a door from slamming shut have q near 1 2 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 clocks superconducting rf cavities used in accelerators and some high q lasers can reach 10 11 3 and higher 4 there are many alternative quantities used by physicists and engineers to describe how damped an oscillator is important examples include the damping ratio relative bandwidth linewidth and bandwidth measured in octaves 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 at the time 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 5 6 7 definition edit the definition of q since its first use in 1914 has been generalized to apply to coils and condensers resonant circuits resonant devices resonant transmission lines cavity resonators 5 it has expanded beyond the electronics field to apply to dynamical systems in general mechanical and acoustic resonators material q and quantum systems such as spectral lines and particle resonances and optical resonators such as laser cavities there are two common definitions for q which apply generally to all these systems but are not exactly equivalent they become approximately equivalent as q becomes larger meaning the resonator becomes less damped bandwidth definition edit one of these definitions which is more applicable to high q devices is the frequency to bandwidth ratio of the resonator 5 q def f r δ f ω r δ ω displaystyle q mathrel stackrel text def frac f_ mathrm r delta f frac omega _ mathrm r delta omega where f r is the resonant frequency δ f is the resonance width or full width at half maximum fwhm i e the bandwidth over which the power of vibration is greater than half the power at the resonant frequency ω r 2 πf r is the angular resonant frequency and δ ω is the angular half power bandwidth under this definition q is the reciprocal of fractional bandwidth stored energy definition edit the other common nearly equivalent definition for q is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes 8 9 5 q def 2 π energy stored energy dissipated per cycle 2 π f r energy stored power loss displaystyle q mathrel stackrel text def 2 pi times frac text energy stored text energy dissipated per cycle 2 pi f_ mathrm r times frac text energy stored text power loss 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 sum of the potential and kinetic energies at some point in time the lost energy is the work done by an external force per cycle to maintain amplitude more generally and in the context of reactive component specification especially inductors the frequency dependent definition of q is used 8 10 failed verification see discussion 9 q ω ω maximum energy stored power loss displaystyle q omega omega times frac text maximum energy stored text power loss 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 q factor and damping edit main articles damping and linear time invariant lti system the q factor is a measure of the qualitative behavior of simple damped oscillators for mathematical details 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...
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