A microwave cavity or ''
radio frequency
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 to around . This is roughly between the up ...
(RF) cavity'' is a special type of
resonator
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 ...
, consisting of a closed (or largely closed) metal structure that confines
electromagnetic fields
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
in the
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
region of the spectrum. The structure is either hollow or filled with
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's
resonant frequencies
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
they reinforce to form
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s 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
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
. Thus it can act as a
bandpass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-p ...
, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.
A microwave cavity acts similarly to a
resonant circuit with extremely low loss at its
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of operation, resulting in
quality factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
s (Q factors) up to the order of 10
6, compared to 10
2 for circuits made with separate
inductors and
capacitors
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of a ...
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
oscillator
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s and
transmitter
In electronics and telecommunications, a radio transmitter or just 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 ...
s to create microwave signals, and as
filters to separate a signal at a given frequency from other signals, in equipment such as
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
equipment,
microwave relay stations, satellite communications, and
microwave oven
A microwave oven (commonly referred to as a microwave) is an electric oven that heats and cooks food by exposing it to electromagnetic radiation in the microwave frequency range. This induces polar molecules in the food to rotate and produce ...
s.
RF cavities can also manipulate
charged particle
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 pa ...
s passing through them by application of
acceleration voltage and are thus used in
particle accelerator
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.
Large accelerators are used for fundamental research in particle ...
s and microwave
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
s such as
klystrons and
magnetrons.
Theory of operation
Most resonant cavities are made from closed (or short-circuited) sections of
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
or high-
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
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.
[ David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.] Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
resonator.
Q factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
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
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.
An ...
, a small wire probe or a loop.
[R. E. Collin, Foundations for Microwave Engineering, 2nd edition, IEEE Press, New York, NY, 2001.] External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.
[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.]
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
or
resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by
where
is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
, with
,
,
being the mode numbers and
,
,
being the corresponding dimensions; c is the speed of light in vacuum; and
and
are relative
permeability and
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the cavity filling respectively.
Cylindrical cavity
The field solutions of a cylindrical cavity of length
and radius
follow from the solutions of a cylindrical
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
with additional electric boundary conditions at the position of the enclosing plates. The resonance frequencies are different for TE and TM modes.
;TM modes:
[T. Wangler, ''RF linear accelerators'', Wiley (2008)]
;TE modes:
Here,
denotes the
-th zero of the
-th
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
, and
denotes the
-th zero of the ''derivative'' of the
-th Bessel function.
Quality factor
The
quality factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
of a cavity can be decomposed into three parts, representing different power loss mechanisms.
*
, resulting from the power loss in the walls which have finite conductivity
*
, resulting from the power loss in the lossy
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
material filling the cavity.
*
, resulting from power loss through unclosed surfaces (holes) of the cavity geometry.
Total Q factor of the cavity can be found as
where k is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
,
is the
intrinsic impedance The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
of the dielectric,
is the
surface resistivity of the cavity walls,
and
are relative
permeability and
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
respectively and
is the
loss tangent of the dielectric.
Comparison to LC circuits
Microwave resonant cavities can be represented and thought of as simple
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 a ...
s.
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
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 a ...
. In terms of inductance and capacitance, the resonant frequency for a given
mode can be written as
where V is the cavity volume,
is the mode wavenumber and
and
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
Very high frequency (VHF) is the ITU designation for the range of radio frequency electromagnetic waves (radio waves) from 30 to 300 megahertz (MHz), with corresponding wavelengths of ten meters to one meter.
Frequencies immediately below VHF ...
range. Similarly, for frequencies above one
gigahertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
, Q factor values for transmission-line resonators start to decrease with frequency.
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
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
with no core.
Skin effect
Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the co ...
causes the high frequency resistance of inductors to be many times their
direct current
Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or ev ...
resistance. In addition, capacitance between turns causes
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
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
Micas ( ) are a group of silicate minerals whose outstanding physical characteristic is that individual mica crystals can easily be split into extremely thin elastic plates. This characteristic is described as perfect basal cleavage. Mica is ...
,
ceramic
A ceramic is any of the various hard, brittle, heat-resistant and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porcelain, ...
or perhaps
teflon
Polytetrafluoroethylene (PTFE) is a synthetic fluoropolymer of tetrafluoroethylene that has numerous applications. It is one of the best-known and widely applied PFAS. The commonly known brand name of PTFE-based composition is Teflon by Chemo ...
for a dielectric. Even with a low loss dielectric, capacitors are also subject to skin effect losses in their
leads
Lead is a chemical element with symbol Pb and atomic number 82.
Lead or The Lead may also refer to:
Animal handling
* Leash, or lead
* Lead (leg), the leg that advances most in a quadruped's cantering or galloping stride
* Lead (tack), a lin ...
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
Parasitism is a close relationship between species, where one organism, the parasite, lives on or inside another organism, the host, causing it some harm, and is adapted structurally to this way of life. The entomologist E. O. Wilson ha ...
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
Parasitic capacitance is an unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other. When two electrical conductors at different voltages ...
.
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
Plating is a surface covering in which a metal is deposited on a conductive surface. Plating has been done for hundreds of years; it is also critical for modern technology. Plating is used to decorate objects, for corrosion inhibition, to improv ...
with
silver
Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical ...
to increase their
electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
and reduce these losses even further.
Copper
Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pink ...
cavities frequently
oxidize
Redox (reduction–oxidation, , ) is a type of chemical reaction in which the oxidation states of substrate change. Oxidation is the loss of electrons or an increase in the oxidation state, while reduction is the gain of electrons or ...
, which increases their loss. Silver or
gold
Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile ...
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
Microwave technology
Accelerator physics