Resonator

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A resonator is a device or system that exhibits

to produce oscillations of very precise frequency.

A cavity resonator is one in which waves exist in a hollow space inside the device. In electronics and radio,

.

Explanation

A physical system can have as many

crystal lattice

composed of N atoms bound together can have N resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to the next becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.

The term resonator is most often used for a homogeneous object in which vibrations travel as waves, at an approximately constant velocity, bouncing back and forth between the sides of the resonator. The material of the resonator, through which the waves flow, can be viewed as being made of millions of coupled moving parts (such as atoms). Therefore, they can have millions of resonant frequencies, although only a few may be used in practical resonators. The oppositely moving waves

in the resonator. If the distance between the sides is ${\displaystyle d\,}$, the length of a round trip is ${\displaystyle 2d\,}$. To cause resonance, the

sinusoidal

wave after a round trip must be equal to the initial phase so the waves self-reinforce. The condition for resonance in a resonator is that the round trip distance, ${\displaystyle 2d\,}$, is equal to an integer number of wavelengths ${\displaystyle \lambda \,}$ of the wave:

${\displaystyle 2d=N\lambda ,\qquad \qquad N\in \{1,2,3,\dots \}}$

If the velocity of a wave is ${\displaystyle c\,}$, the frequency is ${\displaystyle f=c/\lambda \,}$ so the resonant frequencies are:

${\displaystyle f={\frac {Nc}{2d}}\qquad \qquad N\in \{1,2,3,\dots \}}$

So the resonant frequencies of resonators, called

. There may be several such series of resonant frequencies in a single resonator, corresponding to different modes of vibration.

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Resonant circuits

An electrical circuit composed of discrete components can act as a resonator when both an

after the circuit symbols for the components.

A distributed-parameter resonator has capacitance, inductance, and resistance that cannot be isolated into separate lumped capacitors, inductors, or resistors. An example of this, much used in filtering, is the helical resonator.

An inductor consisting of a coil of wire, is self-resonant at a certain frequency due to the parasitic capacitance between its turns. This is often an unwanted effect that can cause parasitic oscillations in RF circuits. The self-resonance of inductors is used in a few circuits, such as the Tesla coil.

Cavity resonators

A cavity resonator is a hollow closed conductor such as a metal box or a cavity within a metal block, containing

, and the cavity stores electromagnetic energy.

Since the cavity's lowest resonant frequency, the fundamental frequency, is that at which the width of the cavity is equal to a half-wavelength (λ/2), cavity resonators are only used at microwave frequencies and above, where wavelengths are short enough that the cavity is conveniently small in size.

Due to the low resistance of their conductive walls, cavity resonators have very high

. Their resonant frequency can be tuned by moving one of the walls of the cavity in or out, changing its size.

Cavity magnetron

The cavity magnetron is a vacuum tube with a filament in the center of an evacuated, lobed, circular cavity resonator. A perpendicular magnetic field is imposed by a permanent magnet. The magnetic field causes the electrons, attracted to the (relatively) positive outer part of the chamber, to spiral outward in a circular path rather than moving directly to this anode. Spaced about the rim of the chamber are cylindrical cavities. The cavities are open along their length and so they connect with the common cavity space. As electrons sweep past these openings they induce a resonant high frequency radio field in the cavity, which in turn causes the electrons to bunch into groups. A portion of this field is extracted with a short antenna that is connected to a waveguide (a metal tube usually of rectangular cross section). The waveguide directs the extracted RF energy to the load, which may be a cooking chamber in a microwave oven or a high gain antenna in the case of radar.

Klystron

The klystron, tube waveguide, is a beam tube including at least two apertured cavity resonators. The beam of charged particles passes through the apertures of the resonators, often tunable wave reflection grids, in succession. A collector electrode is provided to intercept the beam after passing through the resonators. The first resonator causes bunching of the particles passing through it. The bunched particles travel in a field-free region where further bunching occurs, then the bunched particles enter the second resonator giving up their energy to excite it into oscillations. It is a particle accelerator that works in conjunction with a specifically tuned cavity by the configuration of the structures.

The

is a klystron utilizing only a single apertured cavity resonator through which the beam of charged particles passes, first in one direction. A repeller electrode is provided to repel (or redirect) the beam after passage through the resonator back through the resonator in the other direction and in proper phase to reinforce the oscillations set up in the resonator.

Application in particle accelerators

On the beamline of an accelerator system, there are specific sections that are cavity resonators for radio frequency (RF) radiation. The (charged) particles that are to be accelerated pass through these cavities in such a way that the microwave electric field transfers energy to the particles, thus increasing their kinetic energy and thus accelerating them. Several large accelerator facilities employ superconducting niobium cavities for improved performance compared to metallic (copper) cavities.

Loop-gap resonator

The loop-gap resonator (LGR) is made by cutting a narrow slit along the length of a conducting tube. The slit has an effective capacitance and the bore of the resonator has an effective inductance. Therefore, the LGR can be modeled as an RLC circuit and has a resonant frequency that is typically between 200 MHz and 2 GHz. In the absence of radiation losses, the effective resistance of the LGR is determined by the resistivity and electromagnetic skin depth of the conductor used to make the resonator.

One key advantage of the LGR is that, at its resonant frequency, its dimensions are small compared to the free-space wavelength of the electromagnetic fields. Therefore, it is possible to use LGRs to construct a compact and high-Q resonator that operates at relatively low frequencies where cavity resonators would be impractically large.

Dielectric resonators

If a piece of material with large dielectric constant is surrounded by a material with much lower dielectric constant, then this abrupt change in dielectric constant can cause confinement of an electromagnetic wave, which leads to a resonator that acts similarly to a cavity resonator.^[1]

Transmission-line resonators

or open circuit, connected in series or parallel with a main transmission line.

Planar transmission-line resonators are commonly employed for coplanar, stripline, and microstrip transmission lines. Such planar transmission-line resonators can be very compact in size and are widely used elements in microwave circuitry. In cryogenic solid-state research, superconducting transmission-line resonators contribute to solid-state spectroscopy ^[2] and quantum information science.^[3][4]

Optical cavities

In a

). This allows standing wave modes to exist with little loss.

Mechanical

Mechanical resonators are used in

.

Mechanical resonators can also be used to induce a standing wave in other media. For example, a multiple degree of freedom system can be created by imposing a base excitation on a cantilever beam. In this case the

Acoustic

The most familiar examples of acoustic resonators are in

, and the sound boxes of stringed instruments are examples of acoustic cavity resonators.

Automobiles

The exhaust pipes in automobile

Percussion instruments

In many

is achieved via a mechanism that opens and shuts the resonators.

Stringed instruments

String instruments such as the bluegrass banjo may also have resonators. Many five-string banjos have removable resonators, so players can use the instrument with a resonator in bluegrass style, or without it in folk music style. The term resonator, used by itself, may also refer to the resonator guitar.

The modern

.

See also

• Coupling coefficient of resonators
• Crab cavity
• Nuclear magnetic resonance
• Optical ring resonators
• Superconducting RF

References and notes

External links

• Media related to Resonators at Wikimedia Commons