Saturday, February 20, 2010

Q factor



The bandwidth, Δf, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is f0 / Δf. The higher the Q, the narrower and 'sharper' the peak is.
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-dampedan oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency.Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; 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. Oscillators with high quality factors have low damping so that they ring longer.
Sinusoidally driven resonators having higher Q factors 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 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. (See oscillator phase noise.)
The quality factor of oscillators vary substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q = ½. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors aroundQ = 1000. The quality factor of atomic clocks and some high-Q lasers can reach as high as 1011and higher.
There are many alternate quantities used by physicists and engineers to describe how damped an oscillator is and that are closely related to the quality factor. Important examples include: the damping ratio, relative bandwidth, linewidth and bandwidth measured in octaves.
The concept of Q factor originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator.


Definition of the quality factor

There are two separate definitions of the quality factor that are equivalent for high Q resonators but are different for strongly damped oscillators.
Generally Q is defined in terms of the ratio of the energy stored in the resonator to that of the energy being lost in one cycle:
Q = 2 \pi \times \frac{\mbox{Energy Stored}}{\mbox{Energy dissipated per cycle}}. \,
The factor of 2π is used to keep this definition of Q consistent (for high values of Q) with the second definition:
Q = \frac{f_r}{\Delta f} = \frac{\omega_r}{\Delta \omega}, \,
where fr is the resonant frequency, Δf is the bandwidth, ωr is the angular resonant frequency, and Δω is the angular bandwidth.
The definition of Q in terms of the ratio of the energy stored to that of the energy dissipated per cycle can be rewritten as:
Q = \omega \times \frac{\mbox{Energy Stored}}{\mbox{Power Loss}} \,
where ω is defined to be the angular frequency of the circuit (system), and the energy stored and power loss are properties of a system under consideration.


Q factor and damping


The Q factor determines 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.)
  • A system with low quality factor (Q < ½) 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-orderlow-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 > ½) 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 = 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.
  • 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 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).


Quality factors of common systems


Physical interpretation of Q

Physically speaking, Q is 2π 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 per one radian of the oscillation.
It is a dimensionless parameter that compares the time constant 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.
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 1 / e2π, or about 1/535, of its original energy.
The width (bandwidth) of the resonance is given by
\Delta f = \frac{f_0}{Q} \,,
where f0 is the resonant frequency, and Δf, the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.
The factors Q, damping ratio ζ, and attenuation α are related such that
\zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 }.
So the quality factor can be expressed as
Q = \frac{1}{2 \zeta} = { \omega_0 \over 2 \alpha },
and the exponential attenuation rate can be expressed as
\alpha = \zeta \omega_0 = { \omega_0 \over 2 Q }.
For any 2nd order low-pass filter, the response function of the filter is
H(s) = \frac{ \omega_c^2 }{ s^2 + \underbrace{ \frac{ \omega_c }{Q} }_{2 \zeta \omega_c = 2 \alpha }s + \omega_c^2 } \,
For this system, when Q > 0.5 (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 (e.g., after an impulse) of the system. A higher quality factor implies a lower attenuation, and so high Q systems oscillate for long times. For example, high quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer.


Electrical systems

A graph of a filter's gain magnitude, illustrating the concept of -3 dB at a 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.


RLC circuits

In an ideal series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is:
Q = \frac{1}{R} \sqrt{\frac{L}{C}} \,,
where R, L and C are the resistance, inductance and capacitance of the tuned circuit, respectively.
In a parallel RLC circuit, Q is the same.


Complex impedances

\tilde{Z} = R + j\Chi \,
the Q factor is the ratio of the reactance to the resistance (or equivalently, the absolute value of the ratio of reactive powerto real power), that is:
Q = \left | \frac{\Chi}{R} \right | \,
Thus, one can also calculate the Q factor for a complex impedance by knowing just the power factor of the circuit
Q = \frac{\left | \sin \phi \right |}{\left | \cos \phi \right |} = \frac{\sqrt{1-PF^2}}{PF} = \sqrt{\frac{1}{PF^2}-1} \,
or just the tangent of the phase angle
Q = \left | \tan \phi \right |\,
where φ is the phase angle and PF is the power factor of the circuit.


Mechanical systems

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 = \frac{\sqrt{M k}}{D} \,,
where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation Fdamping = − Dv, where v is the velocity.


Optical systems

In optics, the Q factor of a resonant cavity is given by
Q = \frac{2\pi f_o\,\mathcal{E}}{P} \,,
where fo is the resonant frequency, \mathcal{E} is the stored energy in the cavity, and P=-\frac{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.
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Resonance


Increase of amplitude as damping decreases and frequency approaches resonance frequency of a dampedsimple harmonic oscillator.[1][2]
In physics, resonance is the tendency of a system (usually a linear system) to oscillate at larger amplitude at some frequenciesthan at others. These are known as the system's resonant frequencies (or resonance frequencies). At these frequencies, even small periodic driving forces can produce large amplitude oscillations.
Resonances occur when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a pendulum). However, there are some losses from cycle to cycle, calleddamping. When damping is small, the resonant frequency is approximately equal to a natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.
Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, acoustic resonance,electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g. musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies.
Resonance was discovered by Galileo Galilei with his investigations of pendulums and musical strings beginning in 1602.


Examples

Pushing a person in a swing is a common example of resonance. The loaded swing, a pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate.
One familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonance frequency) will make the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo will result in smaller arcs. This is because the energy the swing absorbs is maximized when the pushes are 'in phase' with the swing's oscillations, while some of the swing's energy is actually extracted by the opposing force of the pushes when they are not.
Resonance occurs widely in nature, and is exploited in many man-made devices. It is the mechanism by which virtually all sinusoidalwaves and vibrations are generated. Many sounds we hear, such as when hard objects of metal, glass, or wood are struck, are caused by brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples are:
Mechanical and acoustic resonance
Atomic, particle, and molecular resonance


Theory

"Universal Resonance Curve", a symmetric approximation to the normalized response of a resonant circuit; abscissa values are deviation from center frequency, in units of center frequency divided by 2Q; ordinate is relative amplitude, and phase in cycles; dashed curves compare the range of responses of real two-pole circuits for a Q value of 5; for higher Q values, there is less deviation from the universal curve. Crosses mark the edges of the 3-dB bandwidth (gain 0.707, phase shift 45 degrees or 0.125 cycle).
The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system, and is usually not exactly symmetric about the resonant frequency, as illustrated for the simple harmonic oscillator above. For a lightly damped linear oscillator with a resonant frequency Ω, the intensity of oscillations Iwhen the system is driven with a driving frequency ω is typically approximated by a formula that is symmetric about the resonant frequency:I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^2 + \left( \frac{\Gamma}{2} \right)^2 }.
The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, and this response is found in many physical situations involving resonant systems. Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is inversely proportional to theQ factor, which is a measure of the sharpness of the resonance.
In electrical engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick E. Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.


Resonators

A physical system can have as many resonance frequencies as it has degrees of freedom; each degree of freedom can vibrate as a harmonic oscillator. Systems with one degree of freedom, such as a mass on a spring, pendulums, balance wheels, and LC tuned circuits have one resonance frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers can have two resonance frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to the next becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.
Extended objects that experience resonance due to vibrations inside them are called resonators, such as organ pipes, vibrating strings, quartz crystals, microwave cavities, and laser rods. Since these can be viewed as being made of millions of coupled moving parts (such as atoms), they can have millions of resonance frequencies. The vibrations inside them travel as waves, at an approximately constant velocity, bouncing back and forth between the sides of the resonator. If the distance between the sides is d\,, the length of a round trip is 2d\,. In order to cause resonance, the phase of a sinusoidal wave after a round trip has to be equal to the initial phase, so the waves will reinforce. So the condition for resonance in a resonator is that the round trip distance, 2d\,, be equal to an integer number of wavelengths \lambda\, of the wave:
2d = N\lambda,\qquad\qquad N \in \{1,2,3...\}
If the velocity of a wave is v\,, the frequency is f = v / \lambda\, so the resonance frequencies are:
f = \frac{Nv}{2d}\qquad\qquad N \in \{1,2,3...\}
So the resonance frequencies of resonators, called normal modes, are equally spaced multiples of a lowest frequency called the fundamental frequency. The multiples are often calledovertones. There may be several such series of resonance frequencies, corresponding to different modes of vibration.


Q factor


The quality factor or Q factor is a dimensionless parameter that describes how damped an oscillator or resonator is,[8] or equivalently, characterizes a resonator's bandwidth relative to its center frequency.[9] Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; 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. Oscillators with high quality factors have low damping so that they ring longer.
Sinusoidally driven resonators having higher Q factors 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 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. (See oscillator phase noise.)
The quality factor of oscillators vary substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q = ½. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors around Q = 1000. The quality factor of atomic clocks and some high-Q lasers can reach as high as 1011 and higher.
There are many alternate quantities used by physicists and engineers to describe how damped an oscillator is and that are closely related to the quality factor. Important examples include: the damping ratio, relative bandwidth, linewidth and bandwidth measured in octaves.


Examples of resonance


Mechanical and acoustic resonance


Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, and airplanes. Engineerswhen designing objects must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other oscillating parts a phenomenon known as resonance disaster.
Avoiding resonance disasters is a major concern in every building, tower and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 730-ton pendulum — a tuned mass damper — to cancel resonance. Furthermore, the structure is designed to resonate at a frequency which does not typically occur. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, Engineers designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.
Many clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal
Acoustic resonance is a branch of mechanical resonance that is concerned the mechanical vibrations in the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 12 Hz and 20,000 Hz (20 kHz),
Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in aflute, and the shape of a drum membrane. Acoustic resonance is also important for hearing. For example, resonance of a stiff structural element, called the basilar membrane within thecochlea of the inner ear allows hairs on the membrane to detect sound. (For mammals the membrane by having different resonance on either end so that high frequencies are concentrated on one end and low frequencies on the other.)
Like mechanical resonance, acoustic resonance can result in catastrophic failure of the vibrator. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass; although this is difficult in practice.


Electrical resonance


Electrical resonance occurs in an electric circuit at a particular resonance frequency when the impedance between the input and output of the circuit is at a minimum (or when the transfer function is at a maximum). Often this happens when the impedance between the input and output of the circuit is almost zero and when the transfer function is close to one.


Optical resonance


An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflects multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency whiletransverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam. Ring resonators and whispering galleries are example of optical resonators which do not form standing waves.
Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them. (Flat mirrors are not often used because of the difficulty of aligning them to the needed precision.) The geometry (resonator type) must be chosen so that the beam remains stable (that the size of the beam does not continually grow with multiple reflections. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.
Optical cavities are designed to have a very large Q factor; a beam will reflect a very large number of times with little attenuation. Therefore the frequency line width of the beam is very small indeed compared to the frequency of the laser.
Additional optical resonances are Guided-mode resonances and surface plasmon resonance, which result in anomalus reflection and high evanescent fields at resonance. In this case the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating.


Orbital resonance


In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances withSaturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.


Atomic, particle, and molecular resonance


900MHz, 21.2 T NMR Magnet at HWB-NMR, Birmingham, UK
Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon involving the observation of specific quantum mechanical magnetic properties of an atomic nucleus in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals and non-crystalline materials through NMR spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).
All nuclei that contain odd numbers of nucleons have an intrinsic magnetic moment and angular momentum. A key feature of NMR is that the resonance frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonance frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely from its resonance frequency.
Electron paramagnetic resonance, otherwise known as Electron Spin Resonance (ESR) is a spectroscopic technique similar to NMR used with unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be paramagnetic.
The Mössbauer effect (German: Mößbauer [Meß-Bauer]) is a physical phenomenon discovered by Rudolf Mößbauer in 1957; it refers to the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in a solid form.
Resonance (particle physics): In quantum mechanics and quantum field theory resonances may appear in similar circumstances to classical physics. However, they can also be thought of as unstable particles, with the formula above still valid if the Γ is the decay rate andΩ replaced by the particle's mass M. In that case, the formula just comes from the particle's propagator, with its mass replaced by thecomplex number M + iΓ. The formula is further related to the particle's decay rate by the optical theorem.


Failure of the original Tacoma Narrows Bridge


The dramatically visible, rhythmic twisting that resulted in the 1940 collapse of "Galloping Gertie," the original Tacoma Narrows Bridge, has sometimes been characterized in physics textbooks as a classical example of resonance; however, this description is misleading. The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to a more complicated oscillation between the bridge and the winds passing through it — a phenomenon known as aeroelastic flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding.


Resonance causing a vibration on the International Space Station

The rocket engines for the International Space Station are controlled by autopilot. Ordinarily the uploaded parameters for controlling the engine control system for the Zvezda module will cause the rocket engines to boost the International Space Station to a higher orbit. The rocket engines are hinge-mounted, and ordinarily the operation is not noticed by the crew. But on January 14, 2009, the uploaded parameters caused the autopilot to swing the rocket engines in larger and larger oscillations, at a frequency of 0.5Hz. These oscillations were captured on video, and lasted for 142 seconds.
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