Polynomial-time algorithm for approximating the volume of convex bodies

2013-12-05T13:11:15Z

117.206.127.232:

{{ref improve|date=March 2013}}
A '''microwave cavity''' or ''[[radio frequency]] (RF) cavity'' is a special type of [[resonator]], consisting of a closed (or largely closed) metal structure that confines [[electromagnetic fields]] in the [[microwave]] region of the spectrum. The structure is either hollow or filled with [[dielectric]] material.

A microwave cavity acts similarly to a [[resonant circuit]] with extremely low loss at its [[frequency]] of operation, resulting in [[quality factor]]s up to the order of 10<sup>6</sup>, compared to 10<sup>2</sup> for circuits made with separate [[inductors]] and [[capacitors]] at the same frequency. They are used in [[electronic oscillator|oscillator]]s and [[transmitter]]s to create microwave signals, and as [[electronic filter|filter]]s to separate a signal at a given frequency from other signals, in equipment such as [[radar]] equipment, [[microwave relay]] stations, satellite communications, and [[microwave oven]]s.

In addition to their use in electrical networks, RF cavities can manipulate [[charged particle]]s passing through them by application of [[acceleration voltage]] and are thus used in [[particle accelerator]]s.

== Theory of operation ==

Most resonant cavities are made from closed (or short-circuited) sections of [[Waveguide (electromagnetism)|waveguide]] or high-[[permittivity]] [[dielectric]] material (see [[dielectric resonator]]). Electric and magnetic energy is stored in the cavity and the only losses are due to finite [[Electrical resistivity and conductivity|conductivity]] of cavity walls and [[Loss tangent|dielectric losses]] of material filling the cavity. Every cavity has numerous resonant frequencies that correspond to electromagnetic field modes satisfying necessary boundary conditions on the walls of the cavity. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half-wavelength at resonance.<ref name="pozar">David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.</ref> Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength [[transmission line]] resonator.<ref name="pozar" /> [[Q factor]] of a resonant cavity can be calculated using [[Cavity Perturbation Theory|cavity perturbation theory]] and expressions for stored electric and magnetic energy.

The electromagnetic fields in the cavity are excited via external coupling. An external power source is usually coupled to the cavity by a small [[aperture]], a small wire probe or a loop.<ref name="collin">R. E. Collin, Foundations for Microwave Engineering, 2nd edition, IEEE Press, New York, NY, 2001.</ref> External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.<ref name="montgomery">Montgomery, C. G. & Dicke, Robert H. & Edward M. Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987.</ref>

=== Quality factor ===

The [[quality factor]] <math>\scriptstyle Q</math> of a cavity can be decomposed into three parts, representing different power loss mechanisms.

*<math>\scriptstyle Q_c</math>, resulting from the power loss in the walls which have finite conductivity{{clarify|What is l?|date=October 2012}}
:{{NumBlk|:|<math>Q_c = \frac{(kad)^3b\eta}{2\pi^2R_s} \cdot \frac{1}{l^2a^3\left(2b + d\right) + \left(2b + a\right)d^3}\,</math>|{{EquationRef|3}}}}
*<math>\scriptstyle Q_d</math>, resulting from the power loss in the lossy [[dielectric]] material filling the cavity.
:{{NumBlk|:|<math>Q_d = \frac{1}{\tan \delta}\,</math>|{{EquationRef|4}}}}
*<math>\scriptstyle Q_{ext}</math>, resulting from power loss through unclosed surfaces (holes) of the cavity geometry.

Total Q factor of the cavity can be found as<ref name="pozar" />
{{NumBlk|:|<math>Q = \left( \frac{1}{Q_c}+\frac{1}{Q_d}\right) ^{-1}\,</math>|{{EquationRef|2}}}}

where k is the [[wavenumber]], <math>\scriptstyle \eta</math> is the [[Wave impedance|intrinsic impedance]] of the dielectric, <math>\scriptstyle R_s</math> is the [[Electrical resistivity and conductivity|surface resistivity]] of the cavity walls, <math>\scriptstyle \mu_r</math> and <math>\scriptstyle \epsilon_r</math> are relative [[Permeability (electromagnetism)|permeability]] and [[permittivity]] respectively and <math>\scriptstyle \tan \delta</math> is the [[loss tangent]] of the dielectric.

=== Cavity geometry ===

==== Rectangular cavity ====

[[File:Rectangular cavity.JPG|thumb|Rectangular cavity]]

Resonance frequencies of a rectangular microwave cavity for any [[Transverse mode|<math>\scriptstyle TE_{mnl}</math>]] or [[Transverse mode|<math>\scriptstyle TM_{mnl}</math>]] resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by<ref name="pozar" />

{{NumBlk|:|<math>\begin{align}
f_{mnl} &= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\cdot k_{mnl}\\
&= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 + \left(\frac{l\pi}{d}\right)^2}\\
&= \frac{c}{2\sqrt{\mu_r\epsilon_r}}\sqrt{\left( \frac{m}{a}\right) ^2+\left(\frac{n}{b}\right) ^2 + \left(\frac{l}{d}\right) ^2}
\end{align}</math>|{{EquationRef|1}}}}

where <math>\scriptstyle k_{mnl}</math> is the [[wavenumber]], with <math>\scriptstyle m</math>, <math>\scriptstyle n</math>, <math>\scriptstyle l</math> being the mode numbers and <math>\scriptstyle a</math>, <math>\scriptstyle b</math>, <math>\scriptstyle d</math> being the corresponding dimensions; c is the speed of light in vacuum; and <math>\scriptstyle \mu_r</math> and <math>\scriptstyle \epsilon_r</math> are relative [[Permeability (electromagnetism)|permeability]] and [[permittivity]] respectively.

==== Cylindrical cavity ====

[[File:Cylindrical cavity.svg|thumb|Cylindrical cavity]]

The field solutions of a cylindrical cavity of length <math>\scriptstyle L</math> and radius <math>\scriptstyle R</math> follow from the solutions of a cylindrical [[waveguide]] with additional electric boundary conditions at the position of the enclosing plates. The resonance frequencies are different for TE and TM modes.

;TM modes:<ref name="wangler">T. Wangler, ''RF linear accelerators'', Wiley (2008)</ref> <math>f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2}</math>

;TE modes:<ref name="wangler" /> <math>f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X'_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2}</math>

Here, <math>\scriptstyle X_{mn}</math> denotes the <math>\scriptstyle n</math>-th zero of the <math>\scriptstyle m</math>-th [[Bessel function]], and <math>\scriptstyle X'_{mn}</math> denotes the <math>\scriptstyle n</math>-th zero of the ''derivative'' of the <math>\scriptstyle m</math>-th Bessel function.

== Comparison to LC circuits ==

[[File:LC cavity.JPG|thumb|LC cavity|LC circuit equivalent for microwave resonant cavity]]

Microwave resonant cavities can be represented and thought of as simple [[LC circuit]]s.<ref name="montgomery" /> For a microwave cavity, the stored electric energy is equal to the stored magnetic energy at resonance as is the case for a resonant [[LC circuit]]. In terms of inductance and capacitance, the resonant frequency for a given <math>\scriptstyle mnl</math> mode can be written as<ref name="montgomery" />

{{NumBlk|:|<math>L_{mnl} = \mu k_{mnl}^2V\,</math>|{{EquationRef|6}}}}

{{NumBlk|:|<math>C_{mnl} = \frac{\epsilon}{k_{mnl}^4V}\,</math>|{{EquationRef|7}}}}

{{NumBlk|:|<math>\begin{align}
f_{mnl} &= \frac{1}{2\pi\sqrt{L_{mnl}C_{mnl}}}\\
&= \frac{1}{2\pi\sqrt{\frac{1}{k_{mnl}^2} \mu\epsilon}}
\end{align}</math>|{{EquationRef|5}}}}

where V is the cavity volume, <math>\scriptstyle k_{mnl}</math> is the mode wavenumber and <math>\scriptstyle \epsilon</math> and <math>\scriptstyle \mu</math> are permittivity and permeability respectively.

To better understand the utility of resonant cavities at microwave frequencies, it is useful to note that the losses of conventional inductors and capacitors start to increase with frequency in the [[VHF]] range. Similarly, for frequencies above one [[gigahertz]], Q factor values for transmission-line resonators start to decrease with frequency.<ref name="collin" /> Because of their low losses and high Q factors, cavity resonators are preferred over conventional LC and transmission-line resonators at high frequencies.

=== Losses in LC resonant circuits ===

Conventional inductors are usually wound from wire in the shape of a [[helix]] with no core. [[Skin effect]] causes the high frequency resistance of inductors to be many times their [[direct current]] resistance. In addition, capacitance between turns causes [[dielectric]] losses in the [[Insulator (electrical)|insulation]] which coats the wires. These effects make the high frequency resistance greater and decrease the Q factor.

Conventional capacitors use [[air]], [[mica]], [[ceramic]] or perhaps [[teflon]] for a dielectric. Even with a low loss dielectric, capacitors are also subject to skin effect losses in their [[lead (electronics)|leads]] and [[plate (electronics)|plates]]. Both effects increase their [[equivalent series resistance]] and reduce their Q.

Even if the Q factor of VHF inductors and capacitors is high enough to be useful, their [[Parasitic element (electrical networks)|parasitic]] properties can significantly affect their performance in this frequency range. The shunt capacitance of an inductor may be more significant than its desirable series inductance. The series inductance of a capacitor may be more significant than its desirable shunt capacitance. As a result, in the VHF or microwave regions, a capacitor may appear to be an inductor and an inductor may appear to be a capacitor. These phenomena are better known as [[Parasitic element (electrical networks)|parasitic inductance]] and [[parasitic capacitance]].

=== Losses in cavity resonators ===

Dielectric loss of air is extremely low for high frequency electric or magnetic fields. Air-filled microwave cavities confine electric and magnetic fields to the air spaces between their walls. Electric losses in such cavities are almost exclusively due to currents flowing in cavity walls. While losses from wall currents are small, cavities are frequently [[plated]] with [[silver]] to increase their [[electrical conductivity]] and reduce these losses even further. [[Copper]] cavities frequently [[oxidize]], which increases their loss. Silver or [[gold]] plating prevents oxidation and reduces electrical losses in cavity walls. Even though gold is not quite as good a conductor as copper, it still prevents oxidation and the resulting deterioration of Q factor over time. However, because of its high cost, it is used only in the most demanding applications.

Some satellite resonators are silver plated and covered with a gold flash layer. The current then mostly flows in the high-conductivity silver layer, while the gold flash layer protects the silver layer from oxidizing.

== References ==
{{Reflist}}

[[Category:Microwave technology]]
[[Category:Accelerator physics]]

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Polynomial-time algorithm for approximating the volume of convex bodies