# Microwave cavity

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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. 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.

Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around twenty thousand times per second to around three hundred billion times per second. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies; these are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves. Different sources specify different upper and lower bounds for the frequency range.

A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. 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.

Microwaves are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter; with frequencies between 300 MHz (1 m) and 300 GHz (1 mm). Different sources define different frequency ranges as microwaves; the above broad definition includes both UHF and EHF bands. A more common definition in radio engineering is the range between 1 and 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are often referred to by their IEEE radar band designations: S, C, X, Ku, K, or Ka band, or by similar NATO or EU designations.

## Contents

A microwave cavity acts similarly to a resonant circuit with extremely low loss at its frequency of operation, resulting in quality factors (Q factors) up to the order of 106, compared to 102 for circuits made with separate inductors and capacitors at the same frequency. They are used in place of resonant circuits at microwave frequencies, since at these frequencies discrete resonant circuits cannot be built because the values of inductance and capacitance needed are too low. They are used in oscillators and transmitters to create microwave signals, and as filters to separate a signal at a given frequency from other signals, in equipment such as radar equipment, microwave relay stations, satellite communications, and microwave ovens.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave. Oscillators convert direct current (DC) from a power supply to an alternating current (AC) signal. They are widely used in many electronic devices. Common examples of signals generated by oscillators include signals broadcast by radio and television transmitters, clock signals that regulate computers and quartz clocks, and the sounds produced by electronic beepers and video games.

In electronics and telecommunications, a transmitter or radio transmitter is an electronic device which produces radio waves with an antenna. The transmitter itself generates a radio frequency alternating current, which is applied to the antenna. When excited by this alternating current, the antenna radiates radio waves.

RF cavities can also manipulate charged particles passing through them by application of acceleration voltage and are thus used in particle accelerators and microwave vacuum tubes such as klystrons and magnetrons.

In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, which are all believed to have the same charge. Another charged particle may be an atomic nucleus devoid of electrons, such as an alpha particle.

In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. If not specified further, the term is likely to refer to the longitudinal effective acceleration voltage .

A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams.

## Theory of operation

Most resonant cavities are made from closed (or short-circuited) sections of 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 conductivity of cavity walls and 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. [1] Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength transmission line resonator. [1] Q factor of a resonant cavity can be calculated using cavity perturbation theory and expressions for stored electric and magnetic energy.

In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves. This type of waveguide is used as a transmission line mostly at microwave frequencies, for such purposes as connecting microwave transmitters and receivers to their antennas, in equipment such as microwave ovens, radar sets, satellite communications, and microwave radio links.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

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. [2] External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis. [3]

In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane.

### Resonant frequencies

The resonant frequencies of a cavity can be calculated from its dimensions.

#### Rectangular cavity

Resonance frequencies of a rectangular microwave cavity for any ${\displaystyle \scriptstyle TE_{mnl}}$ or ${\displaystyle \scriptstyle TM_{mnl}}$ resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by [1]

A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.

{\displaystyle {\begin{aligned}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{aligned}}}

(1)

where ${\displaystyle \scriptstyle k_{mnl}}$ is the wavenumber, with ${\displaystyle \scriptstyle m}$, ${\displaystyle \scriptstyle n}$, ${\displaystyle \scriptstyle l}$ being the mode numbers and ${\displaystyle \scriptstyle a}$, ${\displaystyle \scriptstyle b}$, ${\displaystyle \scriptstyle d}$ being the corresponding dimensions; c is the speed of light in vacuum; and ${\displaystyle \scriptstyle \mu _{r}}$ and ${\displaystyle \scriptstyle \epsilon _{r}}$ are relative permeability and permittivity of the cavity filling respectively.

#### Cylindrical cavity

The field solutions of a cylindrical cavity of length ${\displaystyle \scriptstyle L}$ and radius ${\displaystyle \scriptstyle R}$ 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
[4] ${\displaystyle f_{mnp}={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {X_{mn}}{R}}\right)^{2}+\left({\frac {p\pi }{L}}\right)^{2}}}}$
TE modes
[4] ${\displaystyle f_{mnp}={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {X'_{mn}}{R}}\right)^{2}+\left({\frac {p\pi }{L}}\right)^{2}}}}$

Here, ${\displaystyle \scriptstyle X_{mn}}$ denotes the ${\displaystyle \scriptstyle n}$-th zero of the ${\displaystyle \scriptstyle m}$-th Bessel function, and ${\displaystyle \scriptstyle X'_{mn}}$ denotes the ${\displaystyle \scriptstyle n}$-th zero of the derivative of the ${\displaystyle \scriptstyle m}$-th Bessel function.

### Quality factor

The quality factor ${\displaystyle \scriptstyle Q}$ of a cavity can be decomposed into three parts, representing different power loss mechanisms.

• ${\displaystyle \scriptstyle Q_{c}}$, resulting from the power loss in the walls which have finite conductivity[ clarification needed ]

${\displaystyle Q_{c}={\frac {(kad)^{3}b\eta }{2\pi ^{2}R_{s}}}\cdot {\frac {1}{l^{2}a^{3}\left(2b+d\right)+\left(2b+a\right)d^{3}}}\,}$

(3)

• ${\displaystyle \scriptstyle Q_{d}}$, resulting from the power loss in the lossy dielectric material filling the cavity.

${\displaystyle Q_{d}={\frac {1}{\tan \delta }}\,}$

(4)

• ${\displaystyle \scriptstyle Q_{ext}}$, resulting from power loss through unclosed surfaces (holes) of the cavity geometry.

Total Q factor of the cavity can be found as [1]

${\displaystyle Q=\left({\frac {1}{Q_{c}}}+{\frac {1}{Q_{d}}}\right)^{-1}\,}$

(2)

where k is the wavenumber, ${\displaystyle \scriptstyle \eta }$ is the intrinsic impedance of the dielectric, ${\displaystyle \scriptstyle R_{s}}$ is the surface resistivity of the cavity walls, ${\displaystyle \scriptstyle \mu _{r}}$ and ${\displaystyle \scriptstyle \epsilon _{r}}$ are relative permeability and permittivity respectively and ${\displaystyle \scriptstyle \tan \delta }$ is the loss tangent of the dielectric.

## Comparison to LC circuits

Microwave resonant cavities can be represented and thought of as simple LC circuits. [3] 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 ${\displaystyle \scriptstyle mnl}$ mode can be written as [3]

${\displaystyle L_{mnl}=\mu k_{mnl}^{2}V\,}$

(6)

${\displaystyle C_{mnl}={\frac {\epsilon }{k_{mnl}^{4}V}}\,}$

(7)

{\displaystyle {\begin{aligned}f_{mnl}&={\frac {1}{2\pi {\sqrt {L_{mnl}C_{mnl}}}}}\\&={\frac {1}{2\pi {\sqrt {{\frac {1}{k_{mnl}^{2}}}\mu \epsilon }}}}\end{aligned}}}

(5)

where V is the cavity volume, ${\displaystyle \scriptstyle k_{mnl}}$ is the mode wavenumber and ${\displaystyle \scriptstyle \epsilon }$ and ${\displaystyle \scriptstyle \mu }$ 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. [2] 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 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 leads and 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 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 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.

## Related Research Articles

In telecommunications and electrical engineering, electrical length refers to the length of an electrical conductor in terms of the phase shift introduced by transmission over that conductor at some frequency.

In mechanical systems, resonance is a phenomenon that 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.

The relative permittivity of a material is its (absolute) permittivity expressed as a ratio relative to the vacuum permittivity.

In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is, and characterizes a resonator's bandwidth relative to its centre frequency. 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.

In physics, the dissipation factor (DF) is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of quality factor, which represents the "quality" or durability of oscillation.

In microwave and radio-frequency engineering, a stub or resonant stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

Capacitors are manufactured in many forms, styles, lengths, girths, and from many materials. They all contain at least two electrical conductors separated by an insulating layer. Capacitors are widely used as parts of electrical circuits in many common electrical devices.

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.

A dielectric resonator is a piece of dielectric (nonconductive) material, usually ceramic, that is designed to function as a resonator for radio waves, generally in the microwave and millimeter wave bands. The microwaves are confined inside the resonator material by the abrupt change in permittivity at the surface, and bounce back and forth between the sides. At certain frequencies, the resonant frequencies, the microwaves form standing waves in the resonator, oscillating with large amplitudes. Dielectric resonators generally consist of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. The resonant frequency is determined by the overall physical dimensions of the resonator and the dielectric constant of the material.

Foster's reactance theorem is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always strictly monotonically increases with frequency. It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed. The proof of the theorem was presented by Ronald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues at American Telephone & Telegraph.

A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator. The original name is still widely used in many languages, but not commonly in English.

A bias tee is a three-port network used for setting the DC bias point of some electronic components without disturbing other components. The bias tee is a diplexer. The low-frequency port is used to set the bias; the high-frequency port passes the radio-frequency signals but blocks the biasing levels; the combined port connects to the device, which sees both the bias and RF. It is called a tee because the 3 ports are often arranged in the shape of a T.

Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angleδ or the corresponding loss tangent tan δ. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

A ceramic capacitor is a fixed-value capacitor where the ceramic material acts as the dielectric. It is constructed of two or more alternating layers of ceramic and a metal layer acting as the electrodes. The composition of the ceramic material defines the electrical behavior and therefore applications. Ceramic capacitors are divided into two application classes:

A dielectric resonator antenna (DRA) is a radio antenna mostly used at microwave frequencies and higher, that consists of a block of ceramic material of various shapes, the dielectric resonator, mounted on a metal surface, a ground plane. Radio waves are introduced into the inside of the resonator material from the transmitter circuit and bounce back and forth between the resonator walls, forming standing waves. The walls of the resonator are partially transparent to radio waves, allowing the radio power to radiate into space.

Cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator. These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity or a small deformation of its boundary.

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.

## References

1. David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.
2. R. E. Collin, Foundations for Microwave Engineering, 2nd edition, IEEE Press, New York, NY, 2001.
3. 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.
4. T. Wangler, RF linear accelerators, Wiley (2008)