n signals, parking brake, headlights, transmission position).

Cautions may be displayed

for special problems (fuel low, check engine, tire pressure low, door ajar, seat belt
unfastened). Problems are recorded so they can be reported to diagnostic equipment.
Navigation systems can provide voice commands to reach a destination. Automotive
instrumentation must be cheap and reliable over long periods in harsh environments.
There may be independent airbag systems that contain sensors, logic and actuators.
Anti-skid braking systems use sensors to control the brakes, while cruise control affects

throttle position. A wide v Cathode

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Diagram of a copper cathode in a galvanic cell (e.g., a battery). Positively charged cations move
towards the cathode allowing a positive current i to flow out of the cathode.

A cathode is the electrode from which a conventional current leaves a polarized

electrical device. This definition can be recalled by using the mnemonic CCD for
Cathode Current Departs. A conventional current describes the direction in which
positive charges move. Electrons have a negative electrical charge, so the movement
of electrons is opposite to that of the conventional current flow. Consequently, the
mnemonic cathode current departs also means that electrons flow into the device's
cathode from the external circuit. For example, the end of a household battery marked
with a + (plus) is the cathode.
The electrode through which conventional current flows the other way, into the device,
is termed an anode.

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Formula
Justification
Occurrences

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Gravitation
Electrostatics
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Example

Sound in a gas

Field theory interpretation

Non-Euclidean implications
History
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References
External links

Inverse-square law
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 S represents the light source, while r represents the measured points. The lines represent the
flux emanating from the sources and fluxes. The total number of flux lines depends on the
strength of the light source and is constant with increasing distance, where a greater density of
flux lines (lines per unit area) means a stronger energy field. The density of flux lines is inversely
proportional to the square of the distance from the source because the surface area of a sphere
increases with the square of the radius. Thus the field intensity is inversely proportional to the
square of the distance from the source.

In science, an inverse-square law is any scientific law stating that the observed
"intensity" of a specified physical quantity is inversely proportional to the square of the
distance from the source of that physical quantity. The fundamental cause for this can
be understood as geometric dilution corresponding to point-source radiation into three-
dimensional space.

Radar energy expands during both the signal transmission and the reflected return, so
the inverse square for both paths means that the radar will receive energy according to
the inverse fourth power of the range.

To prevent dilution of energy while propagating a signal, certain methods can be used
such as a waveguide, which acts like a canal does for water, or how a gun barrel
restricts hot gas expansion to one dimension in order to prevent loss of energy transfer
to a bullet.

Formula[edit]
In mathematical notation the inverse square law can be expressed as an intensity (I)
varying as a function of distance (d) from some centre. The intensity is proportional (see
∝) to the reciprocal of the square of the distance thus:

intensity ∝ 1distance2

It can also be mathematically expressed as :

intensity1intensity2=distance22distance12

or as the formulation of a constant quantity:

intensity1×distance12=intensity2×distance22
The divergence of a vector field which is the resultant of radial inverse-square law fields
with respect to one or more sources is proportional to the strength of the local sources,
and hence zero outside sources. Newton's law of universal gravitation follows an
inverse-square law, as do the effects of electric, light, sound, and radiation phenomena.

Justification[edit]
The inverse-square law generally applies when some force, energy, or other conserved
quantity is evenly radiated outward from a point source in three-dimensional space.
2
Since the surface area of a sphere (which is 4πr ) is proportional to the square of the
radius, as the emitted radiation gets farther from the source, it is spread out over an
area that is increasing in proportion to the square of the distance from the source.
Hence, the intensity of radiation passing through any unit area (directly facing the point
source) is inversely proportional to the square of the distance from the point source.
Gauss's law for gravity is similarly applicable, and can be used with any physical
quantity that acts in accordance with the inverse-square relationship.

Occurrences[edit]
Gravitation[edit]

Gravitation is the attraction between objects that have mass. Newton's law states:

The gravitational attraction force between two point masses is directly proportional to
the product of their masses and inversely proportional to the square of their separation
[citation
distance. The force is always attractive and acts along the line joining them.
needed]

If the distribution of matter in each body is spherically symmetric, then the objects can
be treated as point masses without approximation, as shown in the shell theorem.
Otherwise, if we want to calculate the attraction between massive bodies, we need to
add all the point-point attraction forces vectorially and the net attraction might not be
exact inverse square. However, if the separation between the massive bodies is much
larger compared to their sizes, then to a good approximation, it is reasonable to treat
the masses as a point mass located at the object's center of mass while calculating the
gravitational force.

As the law of gravitation, this law was suggested in 1645 by Ismaël Bullialdus. But
Bullialdus did not accept Kepler's second and third laws, nor did he appreciate
Christiaan Huygens's solution for circular motion (motion in a straight line pulled aside
by the central force). Indeed, Bullialdus maintained the sun's force was attractive at
aphelion and repulsive at perihelion. Robert Hooke and Giovanni Alfonso Borelli both
[1]
expounded gravitation in 1666 as an attractive force. Hooke's lecture "On gravity" was
[2]
at the Royal Society, in London, on 21 March. Borelli's "Theory of the Planets" was
[3]
published later in 1666. Hooke's 1670 Gresham lecture explained that gravitation
applied to "all celestiall bodys" and added the principles that the gravitating power
decreases with distance and that in the absence of any such power bodies move in
straight lines. By 1679, Hooke thought gravitation had inverse square dependence and
[4]
communicated this in a letter to Isaac Newton: my supposition is that the attraction
[5]
always is in duplicate proportion to the distance from the center reciprocall.

Hooke remained bitter about Newton claiming the invention of this principle, even
though Newton's 1686 Principia acknowledged that Hooke, along with Wren and Halley,
[6]
had separately appreciated the inverse square law in the solar system, as well as
[7]
giving some credit to Bullialdus.

Electrostatics[edit]

Main article: Electrostatics

The force of attraction or repulsion between two electrically charged particles, in
addition to being directly proportional to the product of the electric charges, is inversely
proportional to the square of the distance between them; this is known as Coulomb's
15 [8]
law. The deviation of the exponent from 2 is less than one part in 10 .

F=keq1q2r2

Light and other electromagnetic radiation[edit]

The intensity (or illuminance or irradiance) of light or other linear waves radiating from a
point source (energy per unit of area perpendicular to the source) is inversely
proportional to the square of the distance from the source, so an object (of the same
size) twice as far away receives only one-quarter the energy (in the same time period).

More generally, the irradiance, i.e., the intensity (or power per unit area in the direction
of propagation), of a spherical wavefront varies inversely with the square of the distance
from the source (assuming there are no losses caused by absorption or scattering).

For example, the intensity of radiation from the Sun is 9126 watts per square meter at
the distance of Mercury (0.387 AU); but only 1367 watts per square meter at the
distance of Earth (1 AU)—an approximate threefold increase in distance results in an
approximate ninefold decrease in intensity of radiation.

For non-isotropic radiators such as parabolic antennas, headlights, and lasers, the
effective origin is located far behind the beam aperture. If you are close to the origin,
you don't have to go far to double the radius, so the signal drops quickly. When you are
far from the origin and still have a strong signal, like with a laser, you have to travel very
far to double the radius and reduce the signal. This means you have a stronger signal
or have antenna gain in the direction of the narrow beam relative to a wide beam in all
directions of an isotropic antenna.

In photography and stage lighting, the inverse-square law is used to determine the “fall
off” or the difference in illumination on a subject as it moves closer to or further from the
light source. For quick approximations, it is enough to remember that doubling the
[9]
distance reduces illumination to one quarter; or similarly, to halve the illumination
increase the distance by a factor of 1.4 (the square root of 2), and to double
illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a
point source, the inverse square rule is often still a useful approximation; when the size
of the light source is less than one-fifth of the distance to the subject, the calculation
[10]
error is less than 1%.

The fractional reduction in electromagnetic fluence (Φ) for indirectly ionizing radiation
with increasing distance from a point source can be calculated using the inverse-square
law. Since emissions from a point source have radial directions, they intercept at a
2
perpendicular incidence. The area of such a shell is 4πr where r is the radial distance
from the center. The law is particularly important in diagnostic radiography and
radiotherapy treatment planning, though this proportionality does not hold in practical
situations unless source dimensions are much smaller than the distance. As stated in
Fourier theory of heat “as the point source is magnification by distances, its radiation is
dilute proportional to the sin of the angle, of the increasing circumference arc from the
point of origin”.

Example[edit]

Let P be the total power radiated from a point source (for example, an omnidirectional
isotropic radiator). At large distances from the source (compared to the size of the
source), this power is distributed over larger and larger spherical surfaces as the
distance from the source increases. Since the surface area of a sphere of radius r is A =
2
4πr , the intensity I (power per unit area) of radiation at distance r is
 I=PA=P4πr2.

The energy or intensity decreases (divided by 4) as the distance r is doubled; if

measured in dB would decrease by 6.02 dB per doubling of distance. When referring to
measurements of power quantities, a ratio can be expressed as a level in decibels by
evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the
reference value.

Sound in a gas[edit]

In acoustics, the sound pressure of a spherical wavefront radiating from a point source
decreases by 50% as the distance r is doubled; measured in dB, the decrease is still
6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to
power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):

p ∝ 1r

The same is true for the component of particle velocity

that is in-phase with the instantaneous sound pressure

p
 :

v ∝1r

In the near field is a quadrature component of the particle velocity that is 90° out of
phase with the sound pressure and does not contribute to the time-averaged energy or
the intensity of the sound. The sound intensity is the product of the RMS sound
pressure and the in-phase component of the RMS particle velocity, both of which are
inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:

I = pv ∝ 1r2.

Field theory interpretation[edit]

For an irrotational vector field in three-dimensional space, the inverse-square law
corresponds to the property that the divergence is zero outside the source. This can be
generalized to higher dimensions. Generally, for an irrotational vector field in n-
dimensional Euclidean space, the intensity "I" of the vector field falls off with the distance "r"
th
following the inverse (n − 1) power law

I∝1rn−1,
 [citation needed]
given that the space outside the source is divergence free.

Non-Euclidean implications[edit]
The inverse-square law, fundamental in Euclidean spaces, also applies to non-
Euclidean geometries, including hyperbolic space. The inherent curvature in these
spaces impacts physical laws, underpinning various fields such as cosmology, general
[11]
relativity, and string theory.

John D. Barrow, in his 2020 paper "Non-Euclidean Newtonian Cosmology," elaborates on the
behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He illustrates that F and
Φ obey the formulas F ∝ 1 / R^2 sinh^2(r/R) and Φ ∝ coth(r/R), where R and r represent the
[11]
curvature radius and the distance from the focal point, respectively.

The concept of the dimensionality of space, first proposed by Immanuel Kant, is an

[12]
ongoing topic of debate in relation to the inverse-square law. Dimitria Electra Gatzia
and Rex D. Ramsier, in their 2021 paper, argue that the inverse-square law pertains
[12]
more to the symmetry in force distribution than to the dimensionality of space.

Within the realm of non-Euclidean geometries and general relativity, deviations from the
inverse-square law might not stem from the law itself but rather from the assumption
that the force between bodies depends instantaneously on distance, contradicting
special relativity. General relativity instead interprets gravity as a distortion of
spacetime, causing freely falling particles to traverse geodesics in this curved
[13]
spacetime.

History[edit]
John Dumbleton of the 14th-century Oxford Calculators, was one of the first to express
functional relationships in graphical form. He gave a proof of the mean speed theorem
stating that "the latitude of a uniformly difform movement corresponds to the degree of
the midpoint" and used this method to study the quantitative decrease in intensity of
illumination in his Summa logicæ et philosophiæ naturalis (ca. 1349), stating that it was
not linearly proportional to the distance, but was unable to expose the Inverse-square
[14]
law.

German astronomer Johannes Kepler discussed the inverse-square law and how it affects the
intensity of light.

In proposition 9 of Book 1 in his book Ad Vitellionem paralipomena, quibus astronomiae

pars optica traditur (1604), the astronomer Johannes Kepler argued that the spreading
[15][16]
of light from a point source obeys an inverse square law:

Sicut se Just as [the ratio

habent of] spherical
spharicae surfaces, for
superificies, which the
quibus origo source of light is
 lucis pro centro the center, [is]
est, amplior ad from the wider to
angustiorem: the narrower, so
ita se habet the density or
fortitudo seu fortitude of the
densitas lucis rays of light in
radiorum in the narrower
angustiori, ad [space], towards
illamin in laxiori the more
sphaerica, hoc spacious
est, conversim. spherical
Nam per 6. 7. surfaces, that is,
tantundem inversely. For
lucis est in according to
angustiori [propositions] 6
sphaerica & 7, there is as
superficie, much light in the
quantum in narrower
fusiore, tanto spherical
ergo illie surface, as in
stipatior & the wider, thus it
densior quam is as much more
hic. compressed and
dense here than
there.

In 1645, in his book Astronomia Philolaica ..., the French astronomer Ismaël Bullialdus
[17]
(1605–1694) refuted Johannes Kepler's suggestion that "gravity" weakens as the
inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse
[18][19]
square of the distance:

Virtus autem As for the

illa, qua Sol power by which
prehendit seu the Sun seizes
harpagat or holds the
planetas, planets, and
corporalis quae which, being
ipsi pro manibus corporeal,
est, lineis rectis functions in the
in omnem manner of
mundi hands, it is
amplitudinem emitted in
emissa quasi straight lines
species solis throughout the
cum illius whole extent of
corpore rotatur: the world, and
cum ergo sit like the species
corporalis of the Sun, it
imminuitur, & turns with the
extenuatur in body of the
maiori spatio & Sun; now,
intervallo, ratio seeing that it is
autem huius corporeal, it
imminutionis becomes
eadem est, ac weaker and
luminus, in attenuated at a
ratione nempe greater
dupla distance or
intervallorum, interval, and
 sed eversa. the ratio of its
decrease in
strength is the
same as in the
case of light,
namely, the
duplicate
proportion, but
inversely, of
the distances
[that is, 1/d²].

In England, the Anglican bishop Seth Ward (1617–1689) publicized the ideas of
Bullialdus in his critique In Ismaelis Bullialdi astronomiae philolaicae fundamenta
inquisitio brevis (1653) and publicized the planetary astronomy of Kepler in his book
Astronomia geometrica (1656).

In 1663–1664, the English scientist Robert Hooke was writing his book Micrographia
(1666) in which he discussed, among other things, the relation between the height of
the atmosphere and the barometric pressure at the surface. Since the atmosphere
surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any
unit area of the Earth's surface is a truncated cone (which extends from the Earth's
center to the vacuum of space; obviously only the section of the cone from the Earth's

surface Waveguide
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 An example of a waveguide: A section of flexible waveguide used for RADAR that has a flange.

Electric field Ex component of the TE31 mode inside an x-band hollow metal waveguide.

A waveguide is a structure that guides waves by restricting the transmission of energy

to one direction. Common types of waveguides include acoustic waveguides which
direct sound, optical waveguides which direct light, and radio-frequency waveguides
which direct electromagnetic waves other than light like radio waves.

Without the physical constraint of a waveguide, waves would expand into three-
dimensional space and their intensities would decrease according to the inverse square
law.

There are different types of waveguides for different types of waves. The original and
most common meaning is a hollow conductive metal pipe used to carry high frequency
[1]
radio waves, particularly microwaves. Dielectric waveguides are used at higher radio
frequencies, and transparent dielectric waveguides and optical fibers serve as
waveguides for light. In acoustics, air ducts and horns are used as waveguides for
sound in musical instruments and loudspeakers, and specially-shaped metal rods
conduct ultrasonic waves in ultrasonic machining.

The geometry of a waveguide reflects its function; in addition to more common types
that channel the wave in one dimension, there are two-dimensional slab waveguides
which confine waves to two dimensions. The frequency of the transmitted wave also
dictates the size of a waveguide: each waveguide has a cutoff wavelength determined
by its size and will not conduct waves of greater wavelength; an optical fiber that guides
light will not transmit microwaves which have a much larger wavelength. Some naturally
occurring structures can also act as waveguides. The SOFAR channel layer in the
[2]
ocean can guide the sound of whale song across enormous distances. Any shape of
cross section of waveguide can support EM waves. Irregular shapes are difficult to
analyse. Commonly used waveguides are rectangular and circular in shape.

Uses[edit]

Waveguide supplying power for the Argonne National Laboratory Advanced Photon Source.

The uses of waveguides for transmitting signals were known even before the term was
coined. The phenomenon of sound waves guided through a taut wire have been known
for a long time, as well as sound through a hollow pipe such as a cave or medical
stethoscope. Other uses of waveguides are in transmitting power between the
components of a system such as radio, radar or optical devices. Waveguides are the
fundamental principle of guided wave testing (GWT), one of the many methods of non-
[3]
destructive evaluation.

Specific examples:

● Optical fibers transmit light and signals for long distances with low attenuation
and a wide usable range of wavelengths.
● In a microwave oven a waveguide transfers power from the magnetron,
where waves are formed, to the cooking chamber.
● In a radar, a waveguide transfers radio frequency energy to and from the
antenna, where the impedance needs to be matched for efficient power
transmission (see below).
● Rectangular and circular waveguides are commonly used to connect feeds of
parabolic dishes to their electronics, either low-noise receivers or power
amplifier/transmitters.
● Waveguides are used in scientific instruments to measure optical, acoustic
and elastic properties of materials and objects. The waveguide can be put in
contact with the specimen (as in a medical ultrasonography), in which case
the waveguide ensures that the power of the testing wave is conserved, or
the specimen may be put inside the waveguide (as in a dielectric constant
measurement, so that smaller objects can be tested and the accuracy is
[4]
better.
[5]
● A transmission line is a commonly used specific type of waveguide.

History[edit]

This section
duplicates the
scope of other
articles,
 specifically
Waveguide
(electromagnetis
m)#History.
Please discuss
this issue and
help introduce a
summary style
to the section by
replacing the
section with a
link and a
summary or by
splitting the
content into a
new article.
(November
2020)

The first structure for guiding waves was proposed by J. J. Thomson in 1893, and was
first experimentally tested by Oliver Lodge in 1894. The first mathematical analysis of
[6]: 8
electromagnetic waves in a metal cylinder was performed by Lord Rayleigh in 1897.
For sound waves, Lord Rayleigh published a full mathematical analysis of propagation
[7]
modes in his seminal work, "The Theory of Sound". Jagadish Chandra Bose
researched millimeter wavelengths using waveguides, and in 1897 described to the
[8][9]
Royal Institution in London his research carried out in Kolkata.

The study of dielectric waveguides (such as optical fibers, see below) began as early as
the 1920s, by several people, most famous of which are Rayleigh, Sommerfeld and
 [10]
Debye. Optical fiber began to receive special attention in the 1960s due to its
importance to the communications industry.

The development of radio communication initially occurred at the lower frequencies

because these could be more easily propagated over large distances. The long
wavelengths made these frequencies unsuitable for use in hollow metal waveguides
because of the impractically large diameter tubes required. Consequently, research into
hollow metal waveguides stalled and the work of Lord Rayleigh was forgotten for a time
and had to be rediscovered by others. Practical investigations resumed in the 1930s by
George C. Southworth at Bell Labs and Wilmer L. Barrow at MIT. Southworth at first
took the theory from papers on waves in dielectric rods because the work of Lord
Rayleigh was unknown to him. This misled him somewhat; some of his experiments
failed because he was not aware of the phenomenon of waveguide cutoff frequency
already found in Lord Rayleigh's work. Serious theoretical work was taken up by John

R. Carson and Sallie P. Mead. This work led to the discovery that for the TE01 mode in
circular waveguide losses go down with frequency and at one time this was a serious
[11]: 544–548
contender for the format for long-distance telecommunications.

The importance of radar in World War II gave a great impetus to waveguide research,
at least on the Allied side. The magnetron, developed in 1940 by John Randall and
Harry Boot at the University of Birmingham in the United Kingdom, provided a good
power source and made microwave radar feasible. The most important centre of US
research was at the Radiation Laboratory (Rad Lab) at MIT but many others took part in
the US, and in the UK such as the Telecommunications Research Establishment. The
head of the Fundamental Development Group at Rad Lab was Edward Mills Purcell. His
researchers included Julian Schwinger, Nathan Marcuvitz, Carol Gray Montgomery,
and Robert H. Dicke. Much of the Rad Lab work concentrated on finding lumped
element models of waveguide structures so that components in waveguide could be
analysed with standard circuit theory. Hans Bethe was also briefly at Rad Lab, but while
there he produced his small aperture theory which proved important for waveguide
cavity filters, first developed at Rad Lab. The German side, on the other hand, largely
ignored the potential of waveguides in radar until very late in the war. So much so that
when radar parts from a downed British plane were sent to Siemens & Halske for
analysis, even though they were recognised as microwave components, their purpose
could not be identified.

At that time, microwave techniques were badly neglected in Germany. It was generally
believed that it was of no use for electronic warfare, and those who wanted to do
research work in this field were not allowed to do so.

— H. Mayer, wartime vice-president of Siemens & Halske

German academics were even allowed to continue publicly publishing their research in
[12]: 548–554 [13]: 1055, 1057
this field because it was not felt to be important.

Immediately after World War II waveguide was the technology of choice in the
microwave field. However, it has some problems; it is bulky, expensive to produce, and
the cutoff frequency effect makes it difficult to produce wideband devices. Ridged
waveguide can increase bandwidth beyond an octave, but a better solution is to use a
technology working in TEM mode (that is, non-waveguide) such as coaxial conductors
since TEM does not have a cutoff frequency. A shielded rectangular conductor can also
be used and this has certain manufacturing advantages over coax and can be seen as
the forerunner of the planar technologies (stripline and microstrip). However, planar
technologies really started to take off when printed circuits were introduced. These
methods are significantly cheaper than waveguide and have largely taken its place in
most bands. However, waveguide is still favoured in the higher microwave bands from
[12]: 556–557 [14]: 21–27, 21–50
around Ku band upwards.

Properties[edit]
Propagation modes and cutoff frequencies[edit]
A propagation mode in a waveguide is one solution of the wave equations, or, in other
[10]
words, the form of the wave. Due to the constraints of the boundary conditions, there
are only limited frequencies and forms for the wave function which can propagate in the
waveguide. The lowest frequency in which a certain mode can propagate is the cutoff
frequency of that mode. The mode with the lowest cutoff frequency is the fundamental
[15]:
mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.
38

Propagation modes are computed by solving the Helmholtz equation alongside a set of
boundary conditions depending on the geometrical shape and materials bounding the
region. The usual assumption for infinitely long uniform waveguides allows us to
assume a propagating form for the wave, i.e. stating that every field component has a
known dependency on the propagation direction (i.e.

). More specifically, the common approach is to first replace all unknown time-varying
fields

u(x,y,z,t)

(assuming for simplicity to describe the fields in cartesian components)

with their complex phasors representation

U(x,y,z)

, sufficient to fully describe any infinitely long single-tone signal at frequency

, (angular frequency
ω=2πf

), and rewrite the Helmholtz equation and boundary conditions accordingly.

Then, every unknown field is forced to have a form like

U(x,y,z)=U^(x,y)e−γz

, where the

term represents the propagation constant (still unknown) along the direction along
which the waveguide extends to infinity. The Helmholtz equation can be rewritten to
accommodate such form and the resulting equality needs to be solved for

and

U^(x,y)

, yielding in the end an eigenvalue equation for

and a corresponding eigenfunction

U^(x,y)γ

[16]
for each solution of the former.
The propagation constant

of the guided wave is complex, in general. For a lossless case, the propagation
constant might be found to take on either real or imaginary values, depending on the
chosen solution of the eigenvalue equation and on the angular frequency

. When

is purely real, the mode is said to be "below cutoff", since the amplitude of the field
phasors tends to exponentially decrease with propagation; an imaginary

, instead, represents modes said to be "in propagation" or "above cutoff", as the

complex amplitude of the phasors does not change with

[17]
.

Impedance matching[edit]

In circuit theory, the impedance is a generalization of electrical resistance in the case of

alternating current, and is measured in ohms (
Ω

[10]
). A waveguide in circuit theory is described by a transmission line having a length
[18]: 2–3, 6–12 [19]: 14 [20]
and characteristic impedance. In other words, the impedance
indicates the ratio of voltage to current of the circuit component (in this case a
waveguide) during propagation of the wave. This description of the waveguide was
originally intended for alternating current, but is also suitable for electromagnetic and
sound waves, once the wave and material properties (such as pressure, density,
dielectric constant) are properly converted into electrical terms (current and impedance
[21]: 14
for example).

Impedance matching is important when components of an electric circuit are connected

(waveguide to antenna for example): The impedance ratio determines how much of the
wave is transmitted forward and how much is reflected. In connecting a waveguide to
an antenna a complete transmission is usually required, so an effort is made to match
[20]
their impedances.

The reflection coefficient can be calculated using:

Γ=Z2−Z1Z2+Z1

, where

(Gamma) is the reflection coefficient (0 denotes full transmission, 1 full reflection, and
0.5 is a reflection of half the incoming voltage),

Z1
 and

Z2

are the impedance of the first component (from which the wave enters) and the
[22]
second component, respectively.

An impedance mismatch creates a reflected wave, which added to the incoming waves
creates a standing wave. An impedance mismatch can be also quantified with the
standing wave ratio (SWR or VSWR for voltage), which is connected to the impedance
ratio and reflection coefficient by:

VSWR=|V|max|V|min=1+|Γ|1−|Γ|

, where

|V|min/max

are the minimum and maximum values of the voltage absolute value, and
the VSWR is the voltage standing wave ratio, which value of 1 denotes full
transmission, without reflection and thus no standing wave, while very large values
[20]
mean high reflection and standing wave pattern.

Electromagnetic waveguides[edit]
Radio-frequency waveguides[edit]

Main articles: Waveguide (radio frequency) and Transmission line

Waveguides can be constructed to carry waves over a wide portion of the
electromagnetic spectrum, but are especially useful in the microwave and optical
frequency ranges. Depending on the frequency, they can be constructed from either
conductive or dielectric materials. Waveguides are used for transferring both power and
[15]: 1–3 [23]: xiii–xiv
communication signals.

In this military radar, microwave radiation is transmitted between the source and the reflector by
a waveguide. The figure suggests that microwaves leave the box in a circularly symmetric mode
(allowing the antenna to rotate), then they are converted to a linear mode, and pass through a
flexible stage. Their polarisation is then rotated in a twisted stage and finally they irradiate the
parabolic antenna.

Optical waveguides[edit]

Main article: Waveguide (optics)

Waveguides used at optical frequencies are typically dielectric waveguides, structures

in which a dielectric material with high permittivity, and thus high index of refraction, is
surrounded by a material with lower permittivity. The structure guides optical waves by
[24]
total internal reflection. An example of an optical waveguide is optical fiber.

Other types of optical waveguide are also used, including photonic-crystal fiber, which
guides waves by any of several distinct mechanisms. Guides in the form of a hollow
tube with a highly reflective inner surface have also been used as light pipes for
illumination applications. The inner surfaces may be polished metal, or may be covered
with a multilayer film

nts that are used in design and configuration of automated systems in areas such as
electrical and pneumatic domains, and the control of quantities being measured. They
typically work for industries

ngineering Standing wave ratio

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 SWR of a vertical HB9XBG Antenna for the 40m-band as a function of frequency

In radio engineering and telecommunications, standing wave ratio (SWR) is a

measure of impedance matching of loads to the characteristic impedance of a
transmission line or waveguide. Impedance mismatches result in standing waves along
the transmission line, and SWR is defined as the ratio of the partial standing wave's
amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the
line.

[1][2]
Voltage standing wave ratio (VSWR) (pronounced "vizwar" ) is the ratio of
maximum to minimum voltage on a transmission line . For example, a VSWR of 1.2
means a peak voltage 1.2 times the minimum voltage along that line, if the line is at
least one half wavelength long.

A SWR can be also defined as the ratio of the maximum amplitude to minimum
amplitude of the transmission line's currents, electric field strength, or the magnetic field
strength. Neglecting transmission line loss, these ratios are identical.

[3]
The power standing wave ratio (PSWR) is defined as the square of the VSWR,
however, this deprecated term has no direct physical relation to power actually involved
in transmission.

SWR is usually measured using a dedicated instrument called an SWR meter. Since
SWR is a measure of the load impedance relative to the characteristic impedance of the
transmission line in use (which together determine the reflection coefficient as
described below), a given SWR meter can interpret the impedance it sees in terms of
SWR only if it has been designed for the same particular characteristic impedance as
the line. In practice most transmission lines used in these applications are coaxial
cables with an impedance of either 50 or 75 ohms, so most SWR meters correspond to
one of these.

Checking the SWR is a standard procedure in a radio station. Although the same
information could be obtained by measuring the load's impedance with an impedance
analyzer (or "impedance bridge"), the SWR meter is simpler and more robust for this
purpose. By measuring the magnitude of the impedance mismatch at the transmitter
output it reveals problems due to either the antenna or the transmission line.

Impedance matching[edit]
Main article: Impedance matching

SWR is used as a measure of impedance matching of a load to the characteristic

impedance of a transmission line carrying radio frequency (RF) signals. This especially
applies to transmission lines connecting radio transmitters and receivers with their
antennas, as well as similar uses of RF cables such as cable television connections to
TV receivers and distribution amplifiers. Impedance matching is achieved when the
source impedance is the complex conjugate of the load impedance. The easiest way of
achieving this, and the way that minimizes losses along the transmission line, is for the
imaginary part of the complex impedance of both the source and load to be zero, that
is, pure resistances, equal to the characteristic impedance of the transmission line.
When there is a mismatch between the load impedance and the transmission line, part
of the forward wave sent toward the load is reflected back along the transmission line
towards the source. The source then sees a different impedance than it expects which
can lead to lesser (or in some cases, more) power being supplied by it, the result being
very sensitive to the electrical length of the transmission line.

Such a mismatch is usually undesired and results in standing waves along the
transmission line which magnifies transmission line losses (significant at higher
frequencies and for longer cables). The SWR is a measure of the depth of those
standing waves and is, therefore, a measure of the matching of the load to the
transmission line. A matched load would result in an SWR of 1:1 implying no reflected
wave. An infinite SWR represents complete reflection by a load unable to absorb
electrical power, with all the incident power reflected back towards the source.

It should be understood that the match of a load to the transmission line is different from
the match of a source to the transmission line or the match of a source to the load seen
through the transmission line. For instance, if there is a perfect match between the load
*
impedance Zload and the source impedance Zsource = Z load, that perfect match will
remain if the source and load are connected through a transmission line with an
electrical length of one half wavelength (or a multiple of one half wavelengths) using a
transmission line of any characteristic impedance Z0. However the SWR will generally
not be 1:1, depending only on Zload and Z0. With a different length of transmission line,
the source will see a different impedance than Zload which may or may not be a good
match to the source. Sometimes this is deliberate, as when a quarter-wave matching
section is used to improve the match between an otherwise mismatched source and
load.
However typical RF sources such as transmitters and signal generators are designed to
look into a purely resistive load impedance such as 50Ω or 75Ω, corresponding to
common transmission lines' characteristic impedances. In those cases, matching the
load to the transmission line, Zload = Z0, always ensures that the source will see the
same load impedance as if the transmission line weren't there. This is identical to a 1:1
SWR. This condition (Zload = Z0) also means that the load seen by the source is
independent of the transmission line's electrical length. Since the electrical length of a
physical segment of transmission line depends on the signal frequency, violation of this
condition means that the impedance seen by the source through the transmission line
becomes a function of frequency (especially if the line is long), even if Zload is
frequency-independent. So in practice, a good SWR (near 1:1) implies a transmitter's
output seeing the exact impedance it expects for optimum and safe operation.

Relationship to the reflection

coefficient[edit]

Incident wave (blue) is fully reflected (red wave) out of phase at short-circuited end of transmission line,
creating a net voltage (black) standing wave. Γ = −1, SWR = ∞.
 Standing waves on transmission line, net voltage shown in different colors during one period of
oscillation. Incoming wave from left (amplitude = 1) is partially reflected with (top to bottom) Γ = 0.6,
−0.333, and 0.8 ∠60°. Resulting SWR = 4, 2, 9.

The voltage component of a standing wave in a uniform transmission line consists of

the forward wave (with complex amplitude

Vf

) superimposed on the reflected wave (with complex amplitude

Vr

).

A wave is partly reflected when a transmission line is terminated with an impedance

unequal to its characteristic impedance. The reflection coefficient

can be defined as:

Γ=VrVf.
or

Γ=ZL−ZoZL+Zo

is a complex number that describes both the magnitude and the phase shift of the
reflection. The simplest cases with

measured at the load are:

● Γ=−1
● : complete negative reflection, when the line is short-circuited,
● Γ=0
● : no reflection, when the line is perfectly matched,
● Γ=+1
● : complete positive reflection, when the line is open-circuited.

The SWR directly corresponds to the magnitude of

At some points along the line the forward and reflected waves interfere constructively,
exactly in phase, with the resulting amplitude

Vmax

given by the sum of those waves' amplitudes:

|Vmax|=|Vf|+|Vr|=|Vf|+|ΓVf|=(1+|Γ|)|Vf|.
At other points, the waves interfere 180° out of phase with the amplitudes partially
cancelling:

|Vmin|=|Vf|−|Vr|=|Vf|−|ΓVf|=(1−|Γ|)|Vf|.

The voltage standing wave ratio is then

VSWR=|Vmax||Vmin|=1+|Γ|1−|Γ|.

Since the magnitude of

always falls in the range [0,1], the SWR is always greater than or equal to unity. Note
that the phase of Vf and Vr vary along the transmission line in opposite directions to
each other. Therefore, the complex-valued reflection coefficient

varies as well, but only in phase. With the SWR dependent only on the complex
magnitude of

, it can be seen that the SWR measured at any point along the transmission line
(neglecting transmission line losses) obtains an identical reading.
Since the power of the forward and reflected waves are proportional to the square of
the voltage components due to each wave, SWR can be expressed in terms of forward
and reflected power:

SWR=1+Pr/Pf1−Pr/Pf.

By sampling the complex voltage and current at the point of insertion, an SWR meter is
able to compute the effective forward and reflected voltages on the transmission line for
the characteristic impedance for which the SWR meter has been designed. Since the
forward and reflected power is related to the square of the forward and reflected
voltages, some SWR meters also display the forward and reflected power.

In the special case of a load RL, which is purely resistive but unequal to the
characteristic impedance of the transmission line Z0, the SWR is given simply by their
ratio:

SWR=max{RLZ0,Z0RL}

with the ratio or its reciprocal is chosen to obtain a value greater than unity.

The standing wave pattern[edit]

Using complex notation for the voltage amplitudes, for a signal at frequency f, the actual
(real) voltages Vactual as a function of time t are understood to relate to the complex
voltages according to:

Vactual=Re(ei2πftV) .

Thus taking the real part of the complex quantity inside the parenthesis, the actual
voltage consists of a sine wave at frequency f with a peak amplitude equal to the
complex magnitude of V, and with a phase given by the phase of the complex V. Then
with the position along a transmission line given by x, with the line ending in a load
located at xo, the complex amplitudes of the forward and reverse waves would be
written as:

Vfwd(x)=e−ik(x−xo)AVrev(x)=Γeik(x−xo)A

for some complex amplitude A (corresponding to the forward wave at xo that some
treatments use phasors where the time dependence is according to

e−i2πft

and spatial dependence (for a wave in the +x direction) of

e+ik(x−xo) .

Either convention obtains the same result for Vactual.

According to the superposition principle the net voltage present at any point x on the
transmission line is equal to the sum of the voltages due to the forward and reflected
waves:

Vnet(x)=Vfwd(x)+Vrev(x)=e−ik(x−xo)(1+Γei2k(x−xo))A

Since we are interested in the variations of the magnitude of Vnet along the line (as a
function of x), we shall solve instead for the squared magnitude of that quantity, which
simplifies the mathematics. To obtain the squared magnitude we multiply the above
quantity by its complex conjugate:

|Vnet(x)|2=Vnet(x)Vnet∗(x)=e−ik(x−xo)(1+Γei2k(x−xo))Ae+ik(x−xo)
(1+Γ∗e−i2k(x−xo))A∗=[ 1+|Γ|2+2 Re⁡(Γei2k(x−xo)) ]|A|2
Depending on the phase of the third term, the maximum and minimum values of Vnet
(the square root of the quantity in the equations) are

(1+|Γ|)|A|

and

(1−|Γ|)|A| ,

respectively, for a standing wave ratio of:

SWR=|Vmax||Vmin|=1+|Γ|1−|Γ|

|Γ|= SWR−1 SWR+1

as earlier asserted. Along the line, the above expression for

|Vnet(x)|2

is seen to oscillate sinusoidally between

|Vmin|2

and

|Vmax|2
 with a period of

2π

2k

. This is half of the guided wavelength λ =

2π

for the frequency f . That

[7][8]
following NSSL's research. In Canada, Environment Canada constructed the King
[9]
City station, with a 5 cm research Doppler radar, by 1985; McGill University
dopplerized its radar (J. S. Marshall Radar Observatory) in 1993. This led to a complete
[10]
Canadian Doppler network between 1998 and 2004. France and other European
countries had switched to Doppler networks by the early 2000s. Meanwhile, rapid
advances in computer technology led to algorithms to detect signs of severe weather,
and many applications for media outlets and researchers.

After 2000, research on dual polarization technology moved into operational use,
increasing the amount of information available on precipitation type (e.g. rain vs. snow).
"Dual polarization" means that microwave radiation which is polarized both horizontally
and vertically (with respect to the ground) is emitted. Wide-scale deployment was done
by the end of the decade or the beginning of the next in some countries such as the
[11]
United States, France, and Canada. In April 2013, all United States National Weather
[12]
Service NEXRADs were completely dual-polarized.

Since 2003, the U.S. National Oceanic and Atmospheric Administration has been
experimenting with phased-array radar as a replacement for conventional parabolic
antenna to provide more time resolution in atmospheric sounding. This could be
significant with severe thunderstorms, as their evolution can be better evaluated with
more timely data.
Also in 2003, the National Science Foundation established the Engineering Research
Center for Collaborative Adaptive Sensing of the Atmosphere (CASA), a
multidisciplinary, multi-university collaboration of engineers, computer scientists,
meteorologists, and sociologists to conduct fundamental research, develop enabling
technology, and deploy prototype engineering systems designed to augment existing
radar systems by sampling the generally undersampled lower troposphere with
inexpensive, fast scanning, dual polarization, mechanically scanned and phased array
radars.

In 2023, the private American company Tomorrow.io launched a Ka-band space-based

[13][14]
radar for weather observation and forecasting.

Principle[edit]
Sending radar pulses[edit]

A radar beam spreads out as it moves away from the radar station, covering an increasingly
large volume.

Weather radars send directional pulses of microwave radiation, on the order of one
microsecond long, using a cavity magnetron or klystron tube connected by a waveguide
to a parabolic antenna. The wavelengths of 1 – 10 cm are approximately ten times the
diameter of the droplets or ice particles of interest, because Rayleigh scattering occurs
at these frequencies. This means that part of the energy of each pulse will bounce off
these small particles, back towards the Weather
radar
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Weather radar in Norman, Oklahoma with rainshaft

 Weather (WF44) radar dish

University of Oklahoma OU-PRIME C-band, polarimetric, weather radar during construction

Weather radar, also called weather surveillance radar (WSR) and Doppler weather
radar, is a type of radar used to locate precipitation, calculate its motion, and estimate
its type (rain, snow, hail etc.). Modern weather radars are mostly pulse-Doppler radars,
capable of detecting the motion of rain droplets in addition to the intensity of the
precipitation. Both types of data can be analyzed to determine the structure of storms
and their potential to cause severe weather.

During World War II, radar operators discovered that weather was causing echoes on
their screens, masking potential enemy targets. Techniques were developed to filter
them, but scientists began to study the phenomenon. Soon after the war, surplus radars
were used to detect precipitation. Since then, weather radar has evolved and is used by
national weather services, research departments in universities, and in television
stations' weather departments. Raw images are routinely processed by specialized
software to make short term forecasts of future positions and intensities of rain, snow,
hail, and other weather phenomena. Radar output is even incorporated into numerical
weather prediction models to improve analyses and forecasts.

History[edit]

Typhoon Cobra as seen on a ship's radar screen in December 1944.

During World War II, military radar operators noticed noise in returned echoes due to
rain, snow, and sleet. After the war, military scientists returned to civilian life or
continued in the Armed Forces and pursued their work in developing a use for those
[1]
echoes. In the United States, David Atlas at first working for the Air Force and later for
MIT, developed the first operational weather radars. In Canada, J.S. Marshall and R.H.
[2][3]
Douglas formed the "Stormy Weather Group" in Montreal. Marshall and his doctoral
student Walter Palmer are well known for their work on the drop size distribution in mid-
latitude rain that led to understanding of the Z-R relation, which correlates a given radar
reflectivity with the rate at which rainwater is falling. In the United Kingdom, research
continued to study the radar echo patterns and weather elements such as stratiform rain
and convective clouds, and experiments were done to evaluate the potential of different
wavelengths from 1 to 10 centimeters. By 1950 the UK company EKCO was
[4]
demonstrating its airborne 'cloud and collision warning search radar equipment'.
1960s radar technology detected tornado producing supercells over the Minneapolis-Saint Paul
metropolitan area.

Between 1950 and 1980, reflectivity radars, which measure the position and intensity of
precipitation, were incorporated by weather services around the world. The early
meteorologists had to watch a cathode ray tube. In 1953 Donald Staggs, an electrical
engineer working for the Illinois State Water Survey, made the first recorded radar
[5]
observation of a "hook echo" associated with a tornadic thunderstorm.

The first use of weather radar on television in the United States was in September 1961.
As Hurricane Carla was approaching the state of Texas, local reporter Dan Rather,
suspecting the hurricane was very large, took a trip to the U.S. Weather Bureau WSR-
57 radar site in Galveston in order to get an idea of the size of the storm. He convinced
the bureau staff to let him broadcast live from their office and asked a meteorologist to
draw him a rough outline of the Gulf of Mexico on a transparent sheet of plastic. During
the broadcast, he held that transparent overlay over the computer's black-and-white
radar display to give his audience a sense both of Carla's size and of the location of the
storm's eye. This made Rather a national name and his report helped in the alerted
population accepting the evacuation of an estimated 350,000 people by the authorities,
which was the largest evacuation in US history at that time. Just 46 people were killed
thanks to the warning and it was estimated that the evacuation saved several thousand
lives, as the smaller 1900 Galveston hurricane had killed an estimated 6000-12000
[6]
people.

During the 1970s, radars began to be standardized and organized into networks. The
first devices to capture radar images were developed. The number of scanned angles
was increased to get a three-dimensional view of the precipitation, so that horizontal
cross-sections (CAPPI) and vertical cross-sections could be performed. Studies of the
organization of thunderstorms were then possible for the Alberta Hail Project in Canada
and National Severe Storms Laboratory (NSSL) in the US in particular.

The NSSL, created in 1964, began experimentation on dual polarization signals and on
Doppler effect uses. In May 1973, a tornado devastated Union City, Oklahoma, just
west of Oklahoma City. For the first time, a Dopplerized 10 cm wavelength radar from
 [7]
NSSL documented the entire life cycle of the tornado. The researchers discovered a
mesoscale rotation in the cloud aloft before the tornado touched the ground – the
tornadic vortex signature. NSSL's research helped convince the National Weather
[7]
Service that Doppler radar was a crucial forecasting tool. The Super Outbreak of
tornadoes on 3–4 April 1974 and their devastating destruction might have helped to get
[citation needed]
funding for further developments.

NEXRAD in South Dakota with a supercell in the background.

Between 1980 and 2000, weather radar networks became the norm in North America,
Europe, Japan and other developed countries. Conventional radars were replaced by
Doppler radars, which in addition to position and intensity could track the relative
velocity of the particles in the air. In the United States, the construction of a network
consisting of 10 cm radars, called NEXRAD or WSR-88D (Weather Surveillance Radar
[7][8]
1988 Doppler), was started in 1988 following NSSL's research. In Canada,
[9]
Environment Canada constructed the King City station, with a 5 cm research Doppler
radar, by 1985; McGill University dopplerized its radar (J. S. Marshall Radar
[10]
Observatory) in 1993. This led to a complete Canadian Doppler network between
1998 and 2004. France and other European countries had switched to Doppler
networks by the early 2000s. Meanwhile, rapid advances in computer technology led to
algorithms to detect signs of severe weather, and many applications for media outlets
and researchers.

After 2000, research on dual polarization technology moved into operational use,
increasing the amount of information available on precipitation type (e.g. rain vs. snow).
"Dual polarization" means that microwave radiation which is polarized both horizontally
and vertically (with respect to the ground) is emitted. Wide-scale deployment was done
by the end of the decade or the beginning of the next in some countries such as the
[11]
United States, France, and Canada. In April 2013, all United States National Weather
[12]
Service NEXRADs were completely dual-polarized.
Since 2003, the U.S. National Oceanic and Atmospheric Administration has been
experimenting with phased-array radar as a replacement for conventional parabolic
antenna to provide more time resolution in atmospheric sounding. This could be
significant with severe thunderstorms, as their evolution can be better evaluated with
more timely data.

Also in 2003, the National Science Foundation established the Engineering Research
Center for Collaborative Adaptive Sensing of the Atmosphere (CASA), a
multidisciplinary, multi-university collaboration of engineers, computer scientists,
meteorologists, and sociologists to conduct fundamental research, develop enabling
technology, and deploy prototype engineering systems designed to augment existing
radar systems by sampling the generally undersampled lower troposphere with
inexpensive, fast scanning, dual polarization, mechanically scanned and phased array
radars.

In 2023, the private American company Tomorrow.io launched a Ka-band space-based

[13][14]
radar for weather observation and forecasting.

Principle[edit]
Sending radar pulses[edit]

A radar beam spreads out as it moves away from the radar station, covering an increasingly
large volume.

Weather radars send directional pulses of microwave radiation, on the order of one
microsecond long, using a cavity magnetron or klystron tube connected by a waveguide
to a parabolic antenna. The wavelengths of 1 – 10 cm are approximately ten times the
diameter of the droplets or ice particles of interest, because Rayleigh scattering occurs
at these frequencies. This means that part of the energy of each pulse will bounce off
[15]
these small particles, back towards the radar station.

Shorter wavelengths are useful for smaller particles, but the signal is more quickly
attenuated. Thus 10 cm (S-band) radar is preferred but is more expensive than a 5 cm
C-band system. 3 cm X-band radar is used only for short-range units, and 1 cm Ka-
band weather radar is used only for research on small-particle phenomena such as
[15]
drizzle and fog. W band (3 mm) weather radar systems have seen limited university
use, but due to quicker attenuation, most data are not operational.
Radar pulses diverge as they move away from the radar station. Thus the volume of air
that a radar pulse is traversing is larger for areas farther away from the station, and
smaller for nearby areas, decreasing resolution at farther distances. At the end of a 150
– 200 km sounding range, the volume of air sca

[15]
adar station.

Shorter wavelengths are useful for smaller particles, but the signal is more quickly
attenuated. Thus 10 cm (S-band) radar is preferred but is more expensive than a 5 cm
C-band system. 3 cm X-band radar is used only for short-range units, and 1 cm Ka-
band weather radar is used only for research on small-particle phenomena such as
[15]
drizzle and fog. W band (3 mm) weather radar systems have seen limited university
use, but due to quicker attenuation, most data are not operational.

Radar pulses diverge as they move away from the radar station. Thus the volume of air
that a radar pulse is traversing is larger for areas farther away from the station, and
smaller for nearby areas, decreasing resolution at farther distances. At the end of a 150
– 200 km sounding range, the volume of aiContents hide

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Construction and operation
Toggle Construction and operation subsection
Conventional tube design
Hull or single-anode magnetron
Split-anode magnetron
Cavity magnetron
Common features
Applications
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Radar
Heating
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History
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See also
References
External links

Cavity magnetron
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"Magnetron" redirects here. Not to be confused with Megatron, Metatron, or Magneton

(disambiguation).

Magnetron with section removed to exhibit the cavities. The cathode in the center is not visible.
The antenna emitting microwaves is at the left. The magnets producing a field parallel to the long
axis of the device are not shown.
 A similar magnetron with a different section removed. Central cathode is visible; antenna
conducting microwaves at the top; magnets are not shown.

Obsolete 9 GHz magnetron tube and magnets from a Soviet aircraft radar. The tube is embraced
between the poles of two horseshoe-shaped alnico magnets (top, bottom), which create a
magnetic field along the axis of the tube. The microwaves are emitted from the waveguide
aperture (top) which in use is attached to a waveguide conducting the microwaves to the radar
antenna. Modern tubes use rare-earth magnets, electromagnets or ferrite magnets which are
much less bulky.

The cavity magnetron is a high-power vacuum tube used in early radar systems and
subsequently in microwave ovens and in linear particle accelerators. A cavity
magnetron generates microwaves using the interaction of a stream of electrons with a
magnetic field, while moving past a series of cavity resonators, which are small, open
cavities in a metal block. Electrons pass by the cavities and cause microwaves to
oscillate within, similar to the functioning of a whistle producing a tone when excited by
an air stream blown past its opening. The resonant frequency of the arrangement is
determined by the cavities' physical dimensions. Unlike other vacuum tubes, such as a
klystron or a traveling-wave tube (TWT), the magnetron cannot function as an amplifier
for increasing the intensity of an applied microwave signal; the magnetron serves solely
as an electronic oscillator generating a microwave signal from direct current electricity
supplied to the vacuum tube.

The use of magnetic fields as a means to control the flow of an electric current was
spurred by the invention of the Audion by Lee de Forest in 1906. Albert Hull of General
Electric Research Laboratory, USA, began development of magnetrons to avoid de
[1]
Forest's patents, but these were never completely successful. Other experimenters
picked up on Hull's work and a key advance, the use of two cathodes, was introduced
by Habann in Germany in 1924. Further research was limited until Okabe's 1929
Japanese paper noting the production of centimeter-wavelength signals, which led to
worldwide interest. The development of magnetrons with multiple cathodes was
proposed by A. L. Samuel of Bell Telephone Laboratories in 1934, leading to designs by
Postumus in 1934 and Hans Hollmann in 1935. Production was taken up by Philips,
General Electric Company (GEC), Telefunken and others, limited to perhaps 10 W
output. By this time the klystron was producing more power and the magnetron was not
widely used, although a 300W device was built by Aleksereff and Malearoff in the USSR
[1]
in 1936 (published in 1940).

The cavity magnetron was a radical improvement introduced by John Randall and Harry
[2]: 24–26 [3]
Boot at the University of Birmingham, England in 1940. Their first working
example produced hundreds of watts at 10 cm wavelength, an unprecedented
[4][5]
achievement. Within weeks, engineers at GEC had improved this to well over a
kilowatt, and within months 25 kilowatts, over 100 kW by 1941 and pushing towards a
megawatt by 1943. The high power pulses were generated from a device the size of a
small book and transmitted from an antenna only centimeters long, reducing the size of
[6]
practical radar systems by orders of magnitude. New radars appeared for night-
[6]
fighters, anti-submarine aircraft and even the smallest escort ships, and from that
point on the Allies of World War II held a lead in radar that their counterparts in
Germany and Japan were never able to close. By the end of the war, practically every
Allied radar was based on the magnetron.

The magnetron continued to be used in radar in the post-war period but fell from favour
in the 1960s as high-power klystrons and traveling-wave tubes emerged. A key
characteristic of the magnetron is that its output signal changes from pulse to pulse,
both in frequency and phase. This renders it less suitable for pulse-to-pulse
comparisons for performing moving target indication and removing "clutter" from the
[7]
radar display. The magnetron remains in use in some radar systems, but has become
much more common as a low-cost source for microwave ovens. In this form, over one
[7][8]
billion magnetrons are in use today.

Construction and operation[edit]

Conventional tube design[edit]

In a conventional electron tube (vacuum tube), electrons are emitted from a negatively
charged, heated component called the cathode and are attracted to a positively charged
component called the anode. The components are normally arranged concentrically,
placed within a tubular-shaped container from which all air has been evacuated, so that
the electrons can move freely (hence the name "vacuum" tubes, called "valves" in
British English).

If a third electrode (called a control grid) is inserted between the cathode and the anode,
the flow of electrons between the cathode and anode can be regulated by varying the
voltage on this third electrode. This allows the resulting electron tube (called a "triode"
because it now has three electrodes) to function as an amplifier because small
variations in the electric charge applied to the control grid will result in identical
variations in the much larger current of electrons flowing between the cathode and
[9]
anode.

Hull or single-anode magnetron[edit]

The idea of using a grid for control was invented by Philipp Lenard, who received the
Nobel Prize for Physics in 1905. In the USA it was later patented by Lee de Forest,
resulting in considerable research into alternate tube designs that would avoid his
patents. One concept used a magnetic field instead of an electrical charge to control
current flow, leading to the development of the magnetron tube. In this design, the tube
was made with two electrodes, typically with the cathode in the form of a metal rod in
the center, and the anode as a cylinder around it. The tube was placed between the
[10][better source needed]
poles of a horseshoe magnet arranged such that the magnetic field
was aligned parallel to the axis of the electrodes.

With no magnetic field present, the tube operates as a diode, with electrons flowing
directly from the cathode to the anode. In the presence of the magnetic field, the
electrons will experience a force at right angles to their direction of motion (the Lorentz
force). In this case, the electrons follow a curved path between the cathode and anode.
The curvature of the path can be controlled by varying either the magnetic field using an
electromagnet, or by changing the electrical potential between the electrodes.

At very high magnetic field settings the electrons are forced back onto the cathode,
preventing current flow. At the opposite extreme, with no field, the electrons are free to
flow straight from the cathode to the anode. There is a point between the two extremes,
the critical value or Hull cut-off magnetic field (and cut-off voltage), where the electrons
just reach the anode. At fields around this point, the device operates similar to a triode.
However, magnetic control, due to hysteresis and other effects, results in a slower and
less faithful response to control current than electrostatic control using a control grid in a
conventional triode (not to mention greater weight and complexity), so magnetrons saw
limited use in conventional electronic designs.

It was noticed that when the magnetron was operating at the critical value, it would emit
energy in the radio frequency spectrum. This occurs because a few of the electrons,
instead of reaching the anode, continue to circle in the space between the cathode and
the anode. Due to an effect now known as cyclotron radiation, these electrons radiate
radio frequency energy. The effect is not very efficient. Eventually the electrons hit one
of the electrodes, so the number in the circulating state at any given time is a small
percentage of the overall current. It was also noticed that the frequency of the radiation
depends on the size of the tube, and even early examples were built that produced
signals in the microwave regime.

Early conventional tube systems were limited to the high frequency bands, and although
very high frequency systems became widely available in the late 1930s, the ultra high
frequency and microwave bands were well beyond the ability of conventional circuits.
The magnetron was one of the few devices able to generate signals in the microwave
band and it was the only one that was able to produce high power at centimeter
wavelengths.

Split-anode magnetron[edit]

Split-anode magnetron (c. 1935). (left) The bare tube, about 11 cm high. (right) Installed for use
between the poles of a strong permanent magnet

The original magnetron was very difficult to keep operating at the critical value, and
even then the number of electrons in the circling state at any time was fairly low. This
meant that it produced very low-power signals. Nevertheless, as one of the few devices
known to create microwaves, interest in the device and potential improvements was
widespread.

The first major improvement was the split-anode magnetron, also known as a
negative-resistance magnetron. As the name implies, this design used an anode that
was split in two—one at each end of the tube—creating two half-cylinders. When both
were charged to the same voltage the system worked like the original model. But by
slightly altering the voltage of the two plates, the electrons' trajectory could be modified
so that they would naturally travel towards the lower voltage side. The plates were
connected to an oscillator that reversed the relative voltage of the two plates at a given
[10]
frequency.

At any given instant, the electron will naturally be pushed towards the lower-voltage side
of the tube. The electron will then oscillate back and forth as the voltage changes. At the
same time, a strong magnetic field is applied, stronger than the critical value in the
original design. This would normally cause the electron to circle back to the cathode, but
due to the oscillating electrical field, the electron instead follows a looping path that
[10]
continues toward the anodes.

Since all of the electrons in the flow experienced this looping motion, the amount of RF
energy being radiated was greatly improved. And as the motion occurred at any field
level beyond the critical value, it was no longer necessary to carefully tune the fields
and voltages, and the overall stability of the device was greatly improved. Unfortunately,
the higher field also meant that electrons often circled back to the cathode, depositing
their energy on it and causing it to heat up. As this normally causes more electrons to
[10]
be released, it could sometimes lead to a runaway effect, damaging the device.

Cavity magnetron[edit]

The great advance in magnetron design was the resonant cavity magnetron or
electron-resonance magnetron, which works on entirely different principles. In this
design the oscillation is created by the physical shape of the anode, rather than external
circuits or fields.
 A cross-sectional diagram of a resonant cavity magnetron. Magnetic lines of force are parallel to
the geometric axis of this structure.

Mechanically, the cavity magnetron consists of a large, solid cylinder of metal with a
hole drilled through the centre of the circular face. A wire acting as the cathode is run
down the center of this hole, and the metal block itself forms the anode. Around this
hole, known as the "interaction space", are a number of similar holes ("resonators")
drilled parallel to the interaction space, connected to the interaction space by a short
channel. The resulting block looks something like the cylinder on a revolver, with a
[11]
somewhat larger central hole. Early models were cut using Colt pistol jigs.
Remembering that in an AC circuit the electrons travel along the surface, not the core,
of the conductor, the parallel sides of the slot acts as a capacitor while the round holes
form an inductor: an LC circuit made of solid copper, with the resonant frequency
defined entirely by its dimensions.

The magnetic field is set to a value well below the critical, so the electrons follow arcing
paths towards the anode. When they strike the anode, they cause it to become
negatively charged in that region. As this process is random, some areas will become
more or less charged than the areas around them. The anode is constructed of a highly
conductive material, almost always copper, so these differences in voltage cause
currents to appear to even them out. Since the current has to flow around the outside of
the cavity, this process takes time. During that time additional electrons will avoid the
hot spots and be deposited further along the anode, as the additional current flowing
around it arrives too. This causes an oscillating current to form as the current tries to
[12]
equalize one spot, then another.

The oscillating currents flowing around the cavities, and their effect on the electron flow
within the tube, cause large amounts of microwave radiofrequency energy to be
generated in the cavities. The cavities are open on one end, so the entire mechanism
forms a single, larger, microwave oscillator. A "tap", normally a wire formed into a loop,
extracts microwave energy from one of the cavities. In some systems the tap wire is
replaced by an open hole, which allows the microwaves to flow into a waveguide.

As the oscillation takes some time to set up, and is inherently random at the start,
subsequent startups will have different output parameters. Phase is almost never
preserved, which makes the magnetron difficult to use in phased array systems.
Frequency also drifts from pulse to pulse, a more difficult problem for a wider array of
radar systems. Neither of these present a problem for continuous-wave radars, nor for
microwave ovens.

Common features[edit]
 Cutaway drawing of a cavity magnetron of 1984. Part of the righthand magnet and copper anode
block is cut away to show the cathode and cavities. This older magnetron uses two horseshoe
shaped alnico magnets, modern tubes use rare-earth magnets.

All cavity magnetrons consist of a heated cylindrical cathode at a high (continuous or

pulsed) negative potential created by a high-voltage, direct-current power supply. The
cathode is placed in the center of an evacuated, lobed, circular metal chamber. The
walls of the chamber are the anode of the tube. A magnetic field parallel to the axis of
the cavity is imposed by a permanent magnet. The electrons initially move radially
outward from the cathode attracted by the electric field of the anode walls. The
magnetic field causes the electrons to spiral outward in a circular path, a consequence
of the Lorentz force. Spaced around the rim of the chamber are cylindrical cavities.
Slots are cut along the length of the cavities that open into the central, common cavity
space. As electrons sweep past these slots, they induce a high-frequency radio field in
each resonant cavity, which in turn causes the electrons to bunch into groups. A portion
of the radio frequency energy is extracted by a short coupling loop 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.

The sizes of the cavities determine the resonant frequency, and thereby the frequency
of the emitted microwaves. However, the frequency is not precisely controllable. The
operating frequenc