Microwave cavity
Encyclopedia
A microwave cavity is a closed metal structure that confines electromagnetic fields in the microwave
region of the spectrum. Such cavities act as resonant circuits with extremely low loss at their frequency
of operation. Their Q factor
may reach several million compared to a few hundred for resonant circuits made with inductors and capacitors at the same frequency.
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. Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength transmission line
resonator. 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. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.
or
resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by:
where
is the wavenumber
, c is the speed of light in vacuum, and
and 
are relative permeability
and permittivity
respectively.
Q factor of rectangular microwave cavity can be decomposed in two parts. First part is
, the Q factor of the cavity with lossy walls filled with lossless dielectric
, and the second part is
, the Q factor of the cavity with perfectly conducting wall filled with lossy dielectric
. Total Q factor of the cavity can be found as:
and 
are calculated as follows:
where k is the wavenumber
,
is the intrinsic impedance
of the dielectric,
is the surface resistivity
of the cavity walls,
and 
are relative permeability
and permittivity
respectively and
is the loss tangent
of the dielectric.
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
. 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 range. Similarly, for frequencies above one gigahertz, 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.
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 capacititor 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
.
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.
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
region of the spectrum. Such cavities act as resonant circuits with extremely low loss at their frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of operation. Their Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
may reach several million compared to a few hundred for resonant circuits made with inductors and capacitors at the same frequency.
Theory of Operation
Resonant cavities are usually made from closed (or short-circuited) sections of waveguideWaveguide (electromagnetism)
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...
or high-permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
dielectric
Dielectric
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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
material (see dielectric resonator
Dielectric resonator
A dielectric resonator is an electronic component that exhibits resonance for a narrow range of frequencies, generally in the microwave band. The resonance is similar to that of a circular hollow metallic waveguide, except that the boundary is defined by large change in permittivity rather than by...
). Electric and magnetic energy is stored in the cavity and the only losses are due to finite conductivity
Conductivity
Conductivity may refer to:*Electrical conductivity, a measure of a material's ability to conduct an electric current*Conductivity , also the specific conductance, is a measurement of the electrical conductance per unit distance in an electrolytic or aqueous solution*Ionic conductivity, a measure of...
of cavity walls and dielectric losses
Loss tangent
The loss tangent is a parameter of a dielectric material that quantifies its inherent dissipation of electromagnetic energy. The term refers to the tangent of the angle in a complex plane between the resistive component of an electromagnetic field and its reactive component.-Electromagnetic...
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. Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
resonator. Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
of a resonant cavity can be calculated using cavity perturbation theory
Cavity Perturbation Theory
Cavity perturbation theorydescribes 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.-General...
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
Aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...
, a small wire probe or a loop. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.
Rectangular Microwave Cavity
Resonant frequency of a rectangular microwave cavity for anyTransverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
or
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by:
where
is the wavenumberWavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
, c is the speed of light in vacuum, and
and are relative permeabilityPermeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
and permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
respectively.
Q factor of rectangular microwave cavity can be decomposed in two parts. First part is
, the Q factor of the cavity with lossy walls filled with lossless dielectricDielectric
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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
, and the second part is
, the Q factor of the cavity with perfectly conducting wall filled with lossy dielectricDielectric
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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
. Total Q factor of the cavity can be found as:
and are calculated as follows:where k is the wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
,
is the intrinsic impedanceWave impedance
The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields . For a transverse-electric-magnetic plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium...
of the dielectric,
is the surface resistivityElectrical resistivity and conductivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
of the cavity walls,
and are relative permeabilityPermeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
and permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
respectively and
is the loss tangentLoss tangent
The loss tangent is a parameter of a dielectric material that quantifies its inherent dissipation of electromagnetic energy. The term refers to the tangent of the angle in a complex plane between the resistive component of an electromagnetic field and its reactive component.-Electromagnetic...
of the dielectric.
Comparison to LC Circuits
Microwave resonant cavities can be represented and thought of as simple LC circuitLC circuit
An LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
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
LC circuit
An LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
. 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 range. Similarly, for frequencies above one gigahertz, 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 helixHelix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
with no core. Skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...
causes the high frequency resistance of inductors to be many times their direct current
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
resistance. In addition, capacitance between turns causes dielectric
Dielectric
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 in a conductor, but only slightly shift from their average equilibrium positions causing 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
Mica
The mica group of sheet silicate minerals includes several closely related materials having highly perfect basal cleavage. All are monoclinic, with a tendency towards pseudohexagonal crystals, and are similar in chemical composition...
, ceramic
Ceramic
A ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...
or perhaps teflon for a dielectric. Even with a low loss dielectric, capacitors are also subject to skin effect losses in their leads
Lead (electronics)
In electronics, a lead is an electrical connection consisting of a length of wire or soldering pad that comes from a device. Leads are used for physical support, to transfer power, to probe circuits , and to transmit information...
and plates. Both effects increase their equivalent series resistance
Equivalent series resistance
Practical capacitors and inductors as used in electric circuits are not ideal components with only capacitance or inductance. However, they can be treated to a very good approximation as ideal capacitors and inductors in series with a resistance; this resistance is defined to be the equivalent...
and reduce their Q.
Even if the Q factor of VHF inductors and capacitors is high enough to be useful, their parasitic
Parasitic element (electrical networks)
In electrical networks, a parasitic element is a circuit element that is possessed by an electrical component but which it is not desirable for it to have for its intended purpose. For instance, a resistor is designed to possess resistance, but will also possess unwanted 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 capacititor may appear to be an inductor and an inductor may appear to be a capacitor. These phenomena are better known as parasitic inductance
Parasitic element (electrical networks)
In electrical networks, a parasitic element is a circuit element that is possessed by an electrical component but which it is not desirable for it to have for its intended purpose. For instance, a resistor is designed to possess resistance, but will also possess unwanted parasitic...
and parasitic capacitance
Parasitic capacitance
In electrical circuits, parasitic capacitance, stray capacitance or, when relevant, self-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...
.
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 silverSilver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...
to increase their electrical conductivity and reduce these losses even further. Copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...
cavities frequently oxidize, which increases their loss. Silver or gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...
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.