Q factor

Last updated
The bandwidth
D
f
=
f
2
-
f
1
{\displaystyle \Delta f=f_{2}-f_{1}}
of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is
f
c
D
f
{\displaystyle {\frac {f_{c}}{\Delta f}}}
. The higher the Q, the narrower and 'sharper' the peak is. Bandwidth.svg
The bandwidth of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is . 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 underdamped an oscillator or resonator is, [1] and characterizes a resonator's bandwidth relative to its centre frequency. [2] 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 or vibrate longer.

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Engineering applied science

Engineering is the use of scientific principles to design and build machines, structures, and other things, including bridges, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering.

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of one unit that is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Contents

Explanation

Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). 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.)

Resonance phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies

In mechanical systems, resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

Sine wave Mathematical curve that describes a smooth repetitive oscillation; continuous wave

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

The quality factor of oscillators varies substantially from system to system, depending on their construction. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q near 12. 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 as high as 1011 [3] and higher. [4]

Atomic clock extremely accurate reference clock used as a standard for timekeeping

An atomic clock is a clock device that uses a hyperfine transition frequency in the microwave, or electron transition frequency in the optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element. Atomic clocks are the most accurate time and frequency standards known, and are used as primary standards for international time distribution services, to control the wave frequency of television broadcasts, and in global navigation satellite systems such as GPS.

Optical cavity arrangement of mirrors that forms a standing wave cavity resonator for light waves

An optical cavity, resonating 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 while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam.

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.

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes.

Bandwidth (signal processing) difference between the upper and lower frequencies in a continuous set of frequencies

Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator has energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero bandwidth. The more phase noise the oscillator exhibits, the wider the bandwidth of each harmonic.

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

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, material Q and spectral lines. [5]

Resonant devices

In the context of resonators, there are two common definitions for Q, which aren't exactly equivalent. They become approximately equivalent as Q becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator: [5]

where fr is the resonant frequency, Δf 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, ωr = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth.

Coils and condensers

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]

The factor 2π 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 and kinetic energies at some point in time; the lost energy is the work done by an external conservative 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]

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

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.)

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

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 over one radian of the oscillation. [12]

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.

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−2π, or about 1535 or 0.2%, of its original energy. [13] This means the amplitude falls off to approximately eπ or 4% of its original amplitude. [14]

The width (bandwidth) of the resonance is given by

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 resonant frequency is often expressed in natural units (radians per second), rather than using the f0 in hertz, as

The factors Q, damping ratio ζ, attenuation rate α, and exponential time constant τ are related such that: [15]

and the damping ratio can be expressed as:

The envelope of oscillation decays proportional to eαt or et/τ, where α and τ can be expressed as:

and

The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as e−2αt or e−2t|τ.

For a two-pole lowpass filter, the transfer function of the filter is [15]

For this system, when Q > 12 (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.

Filter type (2nd order)Transfer function [16]
Lowpass
Bandpass
Notch (Bandstop)
Highpass

Electrical systems

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. Bandwidth.svg
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

The 2-sided bandwidth relative to a resonant frequency of F0 Hz is F0/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

where BW is the bandwidth in octaves. [17]

RLC circuits

In an ideal series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is: [18]

where R, L and C are the resistance, inductance and capacitance of the tuned circuit, respectively. The larger the series resistance, the lower the circuit Q.

For a parallel RLC circuit, the Q factor is the inverse of the series case: [19] [18]

[20]

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.

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

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. [21]

where:

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: [21]

where:

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: [21]

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:

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. [22]

Acoustical systems

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. 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

In optics, the Q factor of a resonant cavity is given by

where fo 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 [23] . 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 [24] .

See also

Related Research Articles

Resonator Device or system that exhibits resonance

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones. Another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency.

LC circuit

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.

Q meter

A Q meter is a piece of equipment used in the testing of radio frequency circuits. It has been largely replaced in professional laboratories by other types of impedance measuring device, though it is still in use among radio amateurs. It was developed at Boonton Radio Corporation in Boonton, New Jersey in 1934 by William D. Loughlin.

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the imaginary parts of impedances or admittances of circuit elements cancel each other

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedances or admittances of circuit elements cancel each other. In some circuits, this happens when the impedance between the input and output of the circuit is almost zero and the transfer function is close to one.

Parametric oscillator

A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency of the oscillator. A simple example of a parametric oscillator is a child pumping a playground swing by periodically standing and squatting to increase the size of the swing's oscillations. The child's motions vary the moment of inertia of the swing as a pendulum. The "pump" motions of the child must be at twice the frequency of the swing's oscillations. Examples of parameters that may be varied are the oscillator's resonance frequency and damping .

Vackář oscillator

A Vackář oscillator is a wide range variable frequency oscillator (VFO) that strives for a near constant output amplitude over its frequency range. It is similar to a Colpitts oscillator or a Clapp oscillator, but those designs do not have a constant output amplitude when tuned.

Prototype filter

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

Vibration mechanical phenomenon whereby oscillations occur about an equilibrium point; precisely used to describe mechanical oscillation

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

Microwave cavity

A microwave cavity or radio frequency (RF) cavity is a special type of resonator, consisting of a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. The structure is either hollow or filled with dielectric material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's resonant frequencies they reinforce to form standing waves in the cavity. Therefore, the cavity functions similarly to an organ pipe or sound box in a musical instrument, oscillating preferentially at a series of frequencies, its resonant frequencies. Thus it can act as a bandpass filter, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

RLC circuit Resistor Inductor Capacitor Circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

Circuit quantum electrodynamics provides a means of studying the fundamental interaction between light and matter. As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation.

Staggered tuning is a technique used in the design of multi-stage tuned amplifiers whereby each stage is tuned to a slightly different frequency. In comparison to synchronous tuning it produces a wider bandwidth at the expense of reduced gain. It also produces a sharper transition from the passband to the stopband. Both staggered tuning and synchronous tuning circuits are easier to tune and manufacture than many other filter types.

A double-tuned amplifier is a tuned amplifier with transformer coupling between the amplifier stages in which the inductances of both the primary and secondary windings are tuned separately with a capacitor across each. The scheme results in a wider bandwidth and steeper skirts than a single tuned circuit would achieve.

Cavity optomechanics

Cavity optomechanics is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications.

Loop-gap resonator

A loop-gap resonator (LGR) is an electromagnetic resonator that operates in the radio and microwave frequency ranges. The simplest LGRs are made from a conducting tube with a narrow slit cut along its length. The LGR dimensions are typically much smaller than the free-space wavelength of the electromagnetic fields at the resonant frequency. Therefore, relatively compact LGRs can be designed to operate at frequencies that are too low to be accessed using, for example, cavity resonators. These structures can have very sharp resonances making them useful for electron spin resonance (ESR) experiments and precision measurements of electromagnetic material properties.

References

  1. Harlow, James H. (2004). Electric power transformer engineering. CRC Press. pp. 2–216. ISBN   978-0-8493-1704-0. Archived from the original on 2016-12-02.
  2. 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.
  3. Encyclopedia of Laser Physics and Technology: Q factor Archived 2009-02-24 at the Wayback Machine
  4. Time and Frequency from A to Z: Q to Ra Archived 2008-05-04 at the Wayback Machine
  5. 1 2 3 4 Green, Estill I. (October 1955). "The Story of Q" (PDF). American Scientist. 43: 584–594. Archived from the original (PDF) on 2012-12-03.
  6. B. Jeffreys, Q.Jl R. astr. Soc. (1985) 26, 51–52
  7. Paschotta, Rüdiger (2008). Encyclopedia of Laser Physics and Technology, Vol. 1: A-M. Wiley-VCH. p. 580. ISBN   3527408282. Archived from the original on 2018-05-11.
  8. 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. "Archived copy" (PDF). Archived (PDF) from the original on 2017-08-28. Retrieved 2017-09-02.CS1 maint: Archived copy as title (link)
  9. 1 2 U.A.Bakshi, A. V. Bakshi (2006). Network Analysis. Technical Publications. p. 228. ISBN   9788189411237.
  10. James W. Nilsson (1989). Electric Circuits. ISBN   0-201-17288-7.
  11. Sabah, Nassir H. (2017). Circuit Analysis with PSpice: A Simplified Approach. CRC Press. p. 446. ISBN   9781315402215.
  12. Jackson, R. (2004). Novel Sensors and Sensing. Bristol: Institute of Physics Pub. p. 28. ISBN   0-7503-0989-X.
  13. Benjamin Crowell (2006). "Light and Matter". Archived from the original on 2011-05-19., Ch. 18
  14. Anant., Agarwal (2005). Foundations of analog & digital electronic circuits. Lang, Jeffrey (Jeffrey H.). Amsterdam: Elsevier. p. 647. ISBN   9781558607354. OCLC   60245509.
  15. 1 2 Siebert, William McC. Circuits, Signals, and Systems. MIT Press.
  16. "Analog Dialogue Technical Journal - Analog Devices" (PDF). www.analog.com. Archived (PDF) from the original on 2016-08-04.
  17. http://www.rane.com/note170.html
  18. 1 2 U.A.Bakshi; A.V.Bakshi (2008). Electric Circuits. Technical Publications. pp. 2–79. ISBN   9788184314526.
  19. "Complete Response I - Constant Input". fourier.eng.hmc.edu. Archived from the original on 2012-01-10.
  20. Frequency Response: Resonance, Bandwidth, Q Factor Archived 2014-12-06 at the Wayback Machine
  21. 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.
  22. Methods of Experimental Physics – Lecture 5: Fourier Transforms and Differential Equations Archived 2012-03-19 at the Wayback Machine
  23. 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. doi:10.1016/j.spmi.2019.04.035.
  24. 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. doi:10.1364/AOP.9.000775. ISSN   1943-8206.

Further reading