Microwave cavity

A microwave cavity has a fundamental mode, which exhibits the lowest resonant frequency of all possible resonant modes. For example, the fundamental mode of a cylindrical cavity is the TM₀₁₀ mode. For certain applications, there is motivation to reduce the dimensions of the cavity. This can be done by using a loaded cavity, where a capacitive or an inductive load is integrated in the cavity's structure.

The precise resonant frequency of a loaded cavity must be calculated using finite element methods for Maxwell's equations with boundary conditions.

Loaded cavities are particularly suited for accelerating low velocity charged particles. This application for many types of loaded cavities, Some common types are listed below.

• The reentrant cavity[3][4]
• The helical resonator
• The spiral resonator[5]
• The split-ring resonator[6]
• The quarter wave resonator[7]
• The half wave resonator[8]

The Q factor of a resonant cavity can be calculated using 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.[9] External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.[10]

The resonant frequencies of a cavity are a function of its geometry.

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[2]

${\displaystyle {\begin{aligned}f_{mnl}&={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}\cdot k_{mnl}\\&={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {m\pi }{a}}\right)^{2}+\left({\frac {n\pi }{b}}\right)^{2}+\left({\frac {l\pi }{d}}\right)^{2}}}\\&={\frac {c}{2{\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {m}{a}}\right)^{2}+\left({\frac {n}{b}}\right)^{2}+\left({\frac {l}{d}}\right)^{2}}}\end{aligned}}}$

(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

[11] ${\displaystyle f_{mnp}={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {X_{mn}}{R}}\right)^{2}+\left({\frac {p\pi }{L}}\right)^{2}}}}$

TE modes

[11] ${\displaystyle f_{mnp}={\frac {c}{2\pi {\sqrt {\mu _{r}\epsilon _{r}}}}}{\sqrt {\left({\frac {X'_{mn}}{R}}\right)^{2}+\left({\frac {p\pi }{L}}\right)^{2}}}}$

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.

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

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

(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[2]

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

(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.

An absorption wavemeter. It consists of an adjustable cavity calibrated in frequency. When the resonant frequency of the cavity reaches the frequency of the applied microwaves it absorbs energy, causing a dip in the output power. Then the frequency can be read off the scale.

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.

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.