# Dielectric resonator

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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.

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.

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

Dielectric resonators function similarly to cavity resonators, hollow metal boxes that are also widely used as resonators at microwave frequencies, except that the radio waves are reflected by the large change in permittivity rather than by the conductivity of metal. At millimeter wave frequencies, metal surfaces become lossy reflectors, so dielectric resonators are used at these frequencies. Dielectric resonators' main use is in millimeter-wave electronic oscillators (dielectric resonator oscillator, DRO) to control the frequency of the radio waves generated. They are also used as bandpass filters as well as antennas.

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.

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 radio engineering, an antenna is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

## Historical overview

In the late 19th century, Lord Rayleigh demonstrated that an infinitely long cylindrical rod made up of dielectric material could serve as a waveguide. Additional theoretical and experimental work done in Germany in early 20th century, offered further insight into the behavior of electromagnetic waves in dielectic rod waveguides. Since a dielectric resonator can be thought of as a truncated dielectric rod waveguide, this research was essential for scientific understanding of electromagnetic phenomena in dielectric resonators. In 1939 Robert D. Richtmyer published a study in which he showed that dielectric structures can act just as metallic cavity resonators. He appropriately named these structures dielectric resonators. Richtmyer also demonstrated that, if exposed to free space, dielectic resonators must radiate because of the boundary conditions at the dielectric-to-air interface. These results were later used in development of DRA (Dielectric Resonator Antenna). Due to World War II, lack of advanced materials and adequate manufacturing techniques, dielectric resonators fell in relative obscurity for another two decades after Richtmyer's study was published. However, in the 1960s, as high-frequency electronics and modern communications industry started to take off, dielectric resonators gained in significance. They offered a size-reducing design alternative to bulky waveguide filters and lower-cost alternatives for electronic oscillator, frequency selective limiter and slow-wave circuits. In addition to cost and size, other advantages that dielectric resonators have over conventional metal cavity resonators are lower weight, material availability, and ease of manufacturing. There is a vast availability of different dielectric resonators on the market today with unloaded Q factor on the order of 10000s.

Robert Davis Richtmyer was an American physicist, mathematician, educator, author, and musician.

World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945. The vast majority of the world's countries—including all the great powers—eventually formed two opposing military alliances: the Allies and the Axis. A state of total war emerged, directly involving more than 100 million people from over 30 countries. The major participants threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. World War II was the deadliest conflict in human history, marked by 50 to 85 million fatalities, most of whom were civilians in the Soviet Union and China. It included massacres, executions, the genocide of the Holocaust, strategic bombing, premeditated death from starvation and disease, and the only use of nuclear weapons in war.

A waveguide filter is an electronic filter that is constructed with waveguide technology. Waveguides are hollow metal tubes inside which an electromagnetic wave may be transmitted. Filters are devices used to allow signals at some frequencies to pass, while others are rejected. Filters are a basic component of electronic engineering designs and have numerous applications. These include selection of signals and limitation of noise. Waveguide filters are most useful in the microwave band of frequencies, where they are a convenient size and have low loss. Examples of microwave filter use are found in satellite communications, telephone networks, and television broadcasting.

## Theory of operation

Although dielectric resonators display many similarities to resonant metal cavities, there is one important difference between the two: while the electric and magnetic fields are zero outside the walls of the metal cavity (i.e. open circuit boundary conditions are fully satisfied), these fields are not zero outside the dielectric walls of the resonator (i.e. open circuit boundary conditions are approximately satisfied). Even so, electric and magnetic fields decay from their maximum values considerably when they are away from the resonator walls. Most of the energy is stored in the resonator at a given resonant frequency for a sufficiently high dielectric constant ${\displaystyle \varepsilon _{r}}$. Dielectric resonators can exhibit extremely high Q factor that is comparable to a metal walled cavity.

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.

There are three types of resonant modes that can be excited in dielectric resonators: transverse electric (TE), transverse magnetic (TM) or hybrid electromagnetic (HEM) modes. Theoretically, there is an infinite number of modes in each of the three groups, and desired mode is usually selected based on the application requirements. Generally, ${\displaystyle TE_{01n}}$ mode is used in most non-radiating applications, but other modes can have certain advantages for specific applications.

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.

Approximate resonant frequency of ${\displaystyle TE_{01n}}$ mode for an isolated cylindrical dielectric resonator can be calculated as:

${\displaystyle f_{GHz}={\frac {34}{a{\sqrt {\varepsilon _{r}}}}}\left({\frac {a}{L}}+3.45\right)}$

Where ${\displaystyle a}$ is the radius of the cylindrical resonator and ${\displaystyle L}$ is its length. Both ${\displaystyle a}$ and ${\displaystyle L}$ are in millimeters. Resonant frequency ${\displaystyle f_{GHz}}$ is in gigahertz. This formula is accurate to about 2% in the range:

${\displaystyle 0.5<{\frac {a}{L}}<2}$

${\displaystyle 30<\varepsilon _{r}<50}$

However, since a dielectric resonator is usually enclosed in a conducting cavity for most applications, the real resonant frequencies are different from the one calculated above. As conducting walls of the enclosing cavity approach the resonator, change in boundary conditions and field containment start to affect resonant frequencies. The size and type of the material encapsulating the cavity can drastically impact the performance of the resonant circuit. This phenomenon can be explained using cavity perturbation theory. If a resonator is enclosed in a metallic cavity, resonant frequencies change in following fashion:

• if the stored energy of the displaced field is mostly electric, its resonant frequency will decrease;
• if the stored energy of the displaced field is mostly magnetic, its resonant frequency will increase. This happens to be the case for ${\displaystyle TE_{01n}}$ mode.

The most common problem exhibited by dielectric resonator circuits is their sensitivity to temperature variation and mechanical vibrations. Even though recent improvements in materials science and manufacturing mitigated some of these issues, compensating techniques still may be required to stabilize the circuit performance over temperature and frequency.

## Common applications

The most common applications, of dielectric resonators are:

## Notes

1. Lord Rayleigh, “On the Passage of Waves Through Tubes, or the Vibration of Dielectric Cylinders”, Philosophical Magazine, Vol. 43, pp. 125-132, February 1897.
2. D. Hondros, “Ueber elektromagnetische Drahtwelle,” Annalen der Physik, Vol. 30, pp. 905-949, 1909.
3. H. Zahn, “Ueber den Nachweis elektromagnetischer Wellen an dielektrischen Draehten,”, Annalen der Physik, vol. 37, pp. 907-933, 1916.
4. R.D. Richtmyer, “Dielectric Resonators”, J.Appl. Phys., Vol. 10, pp. 391-398, June 1939.
5. Darko Kajfez and Piere Guillon, Dielectric Resonators, Artech House, Dedham, MA, 1986.
6. Marian W. Pospieszalski, “Cylindrical Dielectric Resonators and Their Applications in TEM Line Microwave Circuits”, IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 233-238, March 1979.
7. A. Okaya and L.F. Barash, “The Dielectric Microwave Resonator”, Proc. IRE, Vol. 50, pp. 2081-2092, October 1962.
8. M.J. Loboda, T.E. Parker and G.K. Montress, "Temperature sensitivity of dielectric resonators and dielectric resonator oscillators," Proc. of the 42nd Annual Freq. Cont. Symp., pp.263-271, Jun 1988.
9. J.K. Plourde and C. Ren, “Application of Dielectric Resonators in Microwave Components”, IEEE Trans. Microwave Theory Tech., Vol. MTT-29, pp. 754-769, August 1981.

## Related Research Articles

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.

A klystron is a specialized linear-beam vacuum tube, invented in 1937 by American electrical engineers Russell and Sigurd Varian, which is used as an amplifier for high radio frequencies, from UHF up into the microwave range. Low-power klystrons are used as oscillators in terrestrial microwave relay communications links, while high-power klystrons are used as output tubes in UHF television transmitters, satellite communication, radar transmitters, and to generate the drive power for modern particle accelerators.

A Gunn diode, also known as a transferred electron device (TED), is a form of diode, a two-terminal passive semiconductor electronic component, with negative resistance, used in high-frequency electronics. It is based on the "Gunn effect" discovered in 1962 by physicist J. B. Gunn. Its largest use is in electronic oscillators to generate microwaves, in applications such as radar speed guns, microwave relay data link transmitters, and automatic door openers.

A Goubau line or Sommerfeld-Goubau line, or G-line for short, is a single wire transmission line used to conduct radio waves at UHF and microwave frequencies. The dielectric coated transmission line was invented by F. Harms in 1907 and George J. E. Goubau in 1950, based on work on surface waves on wires from 1899 by Arnold Sommerfeld. It is used as a feedline at UHF to link high frequency transmitters and receivers to their antennas, and in scientific research.

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.

A backward wave oscillator (BWO), also called carcinotron or backward wave tube, is a vacuum tube that is used to generate microwaves up to the terahertz range. Belonging to the traveling-wave tube family, it is an oscillator with a wide electronic tuning range.

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.

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.

An opto-electronic oscillator (OEO) is an optoelectronic circuit that produces repetitive electronic sine wave and/or modulated optical continuous wave signals.

A microwave cavity or radio frequency (RF) cavity is a special type of resonator, consisting of a 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.

A distributed element filter is an electronic filter in which capacitance, inductance and resistance are not localised in discrete capacitors, inductors and resistors as they are in conventional filters. Its purpose is to allow a range of signal frequencies to pass, but to block others. Conventional filters are constructed from inductors and capacitors, and the circuits so built are described by the lumped element model, which considers each element to be "lumped together" at one place. That model is conceptually simple, but it becomes increasingly unreliable as the frequency of the signal increases, or equivalently as the wavelength decreases. The distributed element model applies at all frequencies, and is used in transmission line theory; many distributed element components are made of short lengths of transmission line. In the distributed view of circuits, the elements are distributed along the length of conductors and are inextricably mixed together. The filter design is usually concerned only with inductance and capacitance, but because of this mixing of elements they cannot be treated as separate "lumped" capacitors and inductors. There is no precise frequency above which distributed element filters must be used but they are especially associated with the microwave band.

Metamaterial antennas are a class of antennas which use metamaterials to increase performance of miniaturized antenna systems. Their purpose, as with any electromagnetic antenna, is to launch energy into free space. However, this class of antenna incorporates metamaterials, which are materials engineered with novel, often microscopic, structures to produce unusual physical properties. Antenna designs incorporating metamaterials can step-up the antenna's radiated power.

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.

Planar transmission lines are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Transmission lines are more than simply interconnections. With simple interconnections, the propagation of the electromagnetic wave along the wire is fast enough to be considered instantaneous, and the voltages at each end of the wire can be considered identical. If the wire is longer than a large fraction of a wavelength, these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled so that its electrical behaviour is highly predictable. At lower frequencies, these considerations are only necessary for the cables connecting different pieces of equipment, but at microwave frequencies the distance at which transmission line theory becomes necessary is measured in millimetres. Hence, transmission lines are needed within circuits.

Coplanar waveguide is a type of electrical planar transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. On a smaller scale, coplanar waveguide transmission lines are also built into monolithic microwave integrated circuits. Conventional coplanar waveguide (CPW) consists of a single conducting track printed onto a dielectric substrate, together with a pair of return conductors, one to either side of the track. All three conductors are on the same side of the substrate, and hence are coplanar. The return conductors are separated from the central track by a small gap, which has an unvarying width along the length of the line. Away from the cental conductor, the return conductors usually extend to an indefinite but large distance, so that each is notionally a semi-infinite plane.

## References

• Lord Rayleigh, “On the Passage of Waves Through Tubes, or the Vibration of Dielectric Cylinders”, Philosophical Magazine, Vol. 43, pp. 125–132, February 1897.
• D. Hondros, “Ueber elektromagnetische Drahtwelle,” Annalen der Physik, Vol. 30, pp. 905–949, 1909.
• H. Zahn, “Ueber den Nachweis elektromagnetischer Wellen an dielektrischen Draehten,”, Annalen der Physik, vol. 37, pp. 907–933, 1916.
• R.D. Richtmyer, “Dielectric Resonators”, J.Appl. Phys., Vol. 10, pp. 391–398, June 1939.
• Darko Kajfez and Piere Guillon, Dielectric Resonators, Artech House, Dedham, MA, 1986.
• Marian W. Pospieszalski, “Cylindrical Dielectric Resonators and Their Applications in TEM Line Microwave Circuits”, IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 233–238, March 1979.
• A. Okaya and L.F. Barash, “The Dielectric Microwave Resonator”, Proc. IRE, Vol. 50, pp. 2081–2092, October 1962.
• M.J. Loboda, T.E. Parker and G.K. Montress, "Temperature sensitivity of dielectric resonators and dielectric resonator oscillators," Proc. of the 42nd Annual Freq. Cont. Symp., pp. 263–271, Jun 1988.
• J.K. Plourde and C. Ren, “Application of Dielectric Resonators in Microwave Components”, IEEE Trans. Microwave Theory Tech., Vol. MTT-29, pp. 754–769, August 1981.