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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Electrostatics
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Example

Sound in a gas

Field theory interpretation

Non-Euclidean implications
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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 andContents hide

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From Wikipedia, the free encyclopedia

(Redirected from Metal-oxide-semiconductor FET)

Two power MOSFETs in D2PAK surface-mount packages. Operating as switches, each of these
components can sustain a blocking voltage of 120 V in the off state, and can conduct a conti-
nuous current of 30 A in the on state, dissipating up to about 100 W and controlling a load of
over 2000 W. A matchstick is pictured for scale.
In electronics, the metal–oxide–semiconductor field-effect transistor (MOSFET,
MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly
fabricated by the controlled oxidation of silicon. It has an insulated gate, the voltage of
which determines the conductivity of the device. This ability to change conductivity with
the amount of applied voltage can be used for amplifying or switching electronic signals.
The term metal–insulator–semiconductor field-effect transistor (MISFET) is almost
synonymous with MOSFET. Another near-synonym is insulated-gate field-effect
transistor (IGFET).

The basic principle of the field-effect transistor was first patented by Julius Edgar
[1]
Lilienfeld in 1925.

The main advantage of a MOSFET is that it requires almost no input current to control
the load current, when compared to bipolar junction transistors (BJTs). In an
enhancement mode MOSFET, voltage applied to the gate terminal increases the
conductivity of the device. In depletion mode transistors, voltage applied at the gate
[2]
reduces the conductivity.

The "metal" in the name MOSFET is sometimes a misnomer, because the gate material
can be a layer of polysilicon (polycrystalline silicon). Similarly, "oxide" in the name can
also be a misnomer, as different dielectric materials are used with the aim of obtaining
strong channels with smaller applied voltages.

The MOSFET is by far the most common transistor in digital circuits, as billions may be
included in a memory chip or microprocessor. Since MOSFETs can be made with either
p-type or n-type semiconductors, complementary pairs of MOS transistors can be used
to make switching circuits with very low power consumption, in the form of CMOS logic.
 A cross-section through an nMOSFET when the gate voltage VGS is below the threshold for
making a conductive channel; there is little or no conduction between the terminals drain and
source; the switch is off. When the gate is more positive, it attracts electrons, inducing an n-type
conductive channel in the substrate below the oxide (yellow), which allows electrons to flow
between the n-doped terminals; the switch is on.

Simulation of formation of inversion channel (electron density) and attainment of threshold vol-
tage (IV) in a nanowire MOSFET. Note: Threshold voltage for this device lies around 0.45 V.

History[edit]
The basic principle of this kind of transistor was first patented by Julius Edgar Lilienfeld
[1]
in 1925.

The structure resembling the MOS transistor was proposed by Bell scientists William
Shockley, John Bardeen and Walter Houser Brattain, during their investigation that led
to discovery of the transistor effect. The structure failed to show the anticipated effects,
due to the problem of surface state: traps on the semiconductor surface that hold
electrons immobile. In 1955 Carl Frosch and L. Derick accidentally grew a layer of
silicon dioxide over the silicon wafer. Further research showed that silicon dioxide could
prevent dopants from diffusing into the silicon wafer. Building on this work Mohamed M.
Atalla showed that silicon dioxide is very effective in solving the problem of one
[3]
important class of surface states.

Following this research, Mohamed Atalla and Dawon Kahng demonstrated in the 1960s
[4]
a device that had the structure of a modern MOS transistor. The principles behind the
device were the same as the ones that were tried by Bardeen, Shockley and Brattain in
their unsuccessful attempt to build a surface field-effect device.

The device was about 100 times slower than contemporary bipolar transistors and was
initially seen as inferior. Nevertheless, Kahng pointed out several advantages of the
[5]
device, notably ease of fabrication and its application in integrated circuits.

Composition[edit]

Photomicrograph of two metal-gate MOSFETs in a test pattern. Probe pads for two gates and
three source/drain nodes are labeled.

Usually the semiconductor of choice is silicon. Some chip manufacturers, most notably
IBM and Intel, use an alloy of silicon and germanium (SiGe) in MOSFET channels.
[citation needed]
Many semiconductors with better electrical properties than silicon, such as
gallium arsenide, do not form good semiconductor-to-insulator interfaces, and thus are
not suitable for MOSFETs. Research continues on creating insulators with acceptable
electrical characteristics on other semiconductor materials.

To overcome the increase in power consumption due to gate current leakage, a high-κ
dielectric is used instead of silicon dioxide for the gate insulator, while polysilicon is
[6]
replaced by metal gates (e.g. Intel, 2009).

The gate is separated from the channel by a thin insulating layer, traditionally of silicon
dioxide and later of silicon oxynitride. Some companies use a high-κ dielectric and
metal gate combination in the 45 nanometer node.

When a voltage is applied between the gate and the source, the electric field generated
penetrates through the oxide and creates an inversion layer or channel at the
semiconductor-insulator interface. The inversion layer provides a channel through
which current can pass between source and drain terminals. Varying the voltage
between the gate and body modulates the conductivity of this layer and thereby controls
the current flow between drain and source. This is known as enhancement mode.

Operation[edit]

Metal–oxide–semiconductor structure on p-type silicon

Metal–oxide–semiconductor structure[edit]
The traditional metal–oxide–semiconductor (MOS) structure is obtained by growing a
layer of silicon dioxide (SiO
2) on top of a silicon substrate, commonly by thermal oxidation and depositing a layer of
metal or polycrystalline silicon (the latter is commonly used). As silicon dioxide is a
dielectric material, its structure is equivalent to a planar capacitor, with one of the
electrodes replaced by a semiconductor.

When a voltage is applied across a MOS structure, it modifies the distribution of

charges in the semiconductor. If we consider a p-type semiconductor (with NA the

density of acceptors, p the density of holes; p = NA in neutral bulk), a positive voltage,

VG, from gate to body (see figure) creates a depletion layer by forcing the positively
charged holes away from the gate-insulator/semiconductor interface, leaving exposed a

carrier-free region of immobile, negatively charged acceptor ions (see doping). If VG is

high enough, a high concentration of negative charge carriers forms in an inversion
layer located in a thin layer next to the interface between the semiconductor and the
insulator.

Conventionally, the gate voltage at which the volume density of electrons in the
inversion layer is the same as the volume density of holes in the body is called the

threshold voltage. When the voltage between transistor gate and source (VG) exceeds

the threshold voltage (Vth), the difference is known as overdrive voltage.

This structure with p-type body is the basis of the n-type MOSFET, which requires the
addition of n-type source and drain regions.

MOS capacitors and band diagrams[edit]

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The MOS capacitor structure is the heart of the MOSFET. Consider a MOS capacitor
where the silicon base is of p-type. If a positive voltage is applied at the gate, holes
which are at the surface of the p-type substrate will be repelled by the electric field
generated by the voltage applied. At first, the holes will simply be repelled and what will
remain on the surface will be immobile (negative) atoms of the acceptor type, which
creates a depletion region on the surface. A hole is created by an acceptor atom, e.g.,
boron, which has one less electron than a silicon atom. Holes are not actually repelled,
being non-entities; electrons are attracted by the positive field, and fill these holes. This
creates a depletion region where no charge carriers exist because the electron is now
fixed onto the atom and immobile.

As the voltage at the gate increases, there will be a point at which the surface above
the depletion region will be converted from p-type into n-type, as electrons from the bulk
area will start to get attracted by the larger electric field. This is known as inversion. The
threshold voltage at which this conversion happens is one of the most important
parameters in a MOSFET.

In the case of a p-type MOSFET, bulk inversion happens when the intrinsic energy level
at the surface becomes smaller than the Fermi level at the surface. This can be seen on
a band diagram. The Fermi level defines the type of semiconductor in discussion. If the
Fermi level is equal to the Intrinsic level, the semiconductor is of intrinsic, or pure type.
If the Fermi level lies closer to the conduction band (valence band) then the
semiconductor type will be of n-type (p-type).

[clarify]
When the gate voltage is increased in a positive sense (for the given example),
this will shift the intrinsic energy level band so that it will curve downwards towards the
valence band. If the Fermi level lies closer to the valence band (for p-type), there will be
a point when the Intrinsic level will start to cross the Fermi level and when the voltage
reaches the threshold voltage, the intrinsic level does cross the Fermi level, and that is
what is known as inversion. At that point, the surface of the semiconductor is inverted
from p-type into n-type.

If the Fermi level lies above the intrinsic level, the semiconductor is of n-type, therefore
at inversion, when the intrinsic level reaches and crosses the Fermi level (which lies
closer to the valence band), the semiconductor type changes at the surface as dictated
by the relative positions of the Fermi and Intrinsic energy levels.

Structure and channel formation[edit]

See also: Field effect (semiconductor)

Channel formation in nMOS MOSFET shown as band diagram: Top panels: An applied gate
voltage bends bands, depleting holes from surface (left). The charge inducing the bending is
balanced by a layer of negative acceptor-ion charge (right). Bottom panel: A larger applied
voltage further depletes holes but conduction band lowers enough in energy to populate a
conducting channel.

C–V profile for a bulk MOSFET with different oxide thickness. The leftmost part of the curve
corresponds to accumulation. The valley in the middle corresponds to depletion. The curve on
the right corresponds to inversion.
A MOSFET is based on the modulation of charge concentration by a MOS capacitance
between a body electrode and a gate electrode located above the body and insulated
from all other device regions by a gate dielectric layer. If dielectrics other than an oxide
are employed, the device may be referred to as a metal-insulator-semiconductor FET
(MISFET). Compared to the MOS capacitor, the MOSFET includes two additional
terminals (source and drain), each connected to individual highly doped regions that are
separated by the body region. These regions can be either p or n type, but they must
both be of the same type, and of opposite type to the body region. The source and drain
(unlike the body) are highly doped as signified by a "+" sign after the type of doping.

If the MOSFET is an n-channel or nMOS FET, then the source and drain are n+ regions
and the body is a p region. If the MOSFET is a p-channel or pMOS FET, then the
source and drain are p+ regions and the body is a n region. The source is so named
because it is the source of the charge carriers (electrons for n-channel, holes for p-
channel) that flow through the channel; similarly, the drain is where the charge carriers
leave the channel.

The occupancy of the energy bands in a semiconductor is set by the position of the
Fermi level relative to the semiconductor energy-band edges.

See also: Depletion region

With sufficient gate voltage, the valence band edge is driven far from the Fermi level,
and holes from the body are driven away from the gate.

At larger gate bias still, near the semiconductor surface the conduction band edge is
brought close to the Fermi level, populating the surface with electrons in an inversion
layer or n-channel at the interface between the p region and the oxide. This conducting
channel extends between the source and the drain, and current is conducted through it
when a voltage is applied between the two electrodes. Increasing the voltage on the
gate leads to a higher electron density in the inversion layer and therefore increases the
current flow between the source and drain. For gate voltages below the threshold value,
the channel is lightly populated, and only a very small subthreshold leakage current can
flow between the source and the drain.
When a negative gate-source voltage (positive source-gate) is applied, it creates a p-
channel at the surface of the n region, analogous to the n-channel case, but with
opposite polarities of charges and voltages. When a voltage less negative than the
threshold value (a negative voltage for the p-channel) is applied between gate and
source, the channel disappears and only a very small subthreshold current can flow
between the source and the drain. The device may comprise a silicon on insulator
device in which a buried oxide is formed below a thin semiconductor layer. If the
channel region between the gate dielectric and the buried oxide region is very thin, the
channel is referred to as an ultrathin channel region with the source and drain regions
formed on either side in or above the thin semiconductor layer. Other semiconductor
materials may be employed. When the source and drain regions are formed above the
channel in whole or in part, they are referred to as raised source/drain regions.

Parameter nMOSFET pMOSFET

Source/drain n-type p-type

type

Channel type n-type p-type

(MOS
capacitor)

G Polysilico n+ p+
n
 t
Metal φm ~ Si φm ~ Si
conduction valence band
band

Well type p-type n-type

Threshold Positive Negative

voltage, Vth (enhanc (enhanc

ement) ement)
Negative Positive
(depleti (depleti
on) on)

Band-bending Downwards Upwards

Inversion layer Electrons Holes

carriers

Substrate type p-type n-type

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 Sensi Schottky

diode
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From Wikipedia, the free encyclopedia

"FET" redirects here. For other uses, see FET (disambiguation).

Cross-sectional view of a field-effect transistor, showing source, gate and drain terminals

The field-effect transistor (FET) is a type of transistor that uses an electric field to
control the flow of current in a semiconductor. It comes in two types: junction FET
(JFET) and metal-oxide-semiconductor FET (MOSFET). FETs have three terminals:
source, gate, and drain. FETs control the flow of current by the application of a voltage
to the gate, which in turn alters the conductivity between the drain and source.

FETs are also known as unipolar transistors since they involve single-carrier-type
operation. That is, FETs use either electrons (n-channel) or holes (p-channel) as charge
carriers in their operation, but not both. Many different types of field effect transistors
exist. Field effect transistors generally display very high input impedance at low
frequencies. The most widely used field-effect transistor is the MOSFET (metal–oxide–
semiconductor field-effect transistor).
History[edit]
Further information: History of the transistor

Julius Edgar Lilienfeld, who proposed the concept of a field-effect transistor in 1925.

The concept of a field-effect transistor (FET) was first patented by the Austro-Hungarian
[1]
born physicist Julius Edgar Lilienfeld in 1925 and by Oskar Heil in 1934, but they were
unable to build a working practical semiconducting device based on the concept. The
transistor effect was later observed and explained by John Bardeen and Walter Houser
Brattain while working under William Shockley at Bell Labs in 1947, shortly after the 17-
year patent expired. Shockley initially attempted to build a working FET by trying to
modulate the conductivity of a semiconductor, but was unsuccessful, mainly due to
problems with the surface states, the dangling bond, and the germanium and copper
compound materials. In the course of trying to understand the mysterious reasons
behind their failure to build a working FET, it led to Bardeen and Brattain instead
inventing the point-contact transistor in 1947, which was followed by Shockley's bipolar
[2][3]
junction transistor in 1948.

The first FET device to be successfully built was the junction field-effect transistor
[2] [4]
(JFET). A JFET was first patented by Heinrich Welker in 1945. The static induction
transistor (SIT), a type of JFET with a short channel, was invented by Japanese
engineers Jun-ichi Nishizawa and Y. Watanabe in 1950. Following Shockley's
theoretical treatment on the JFET in 1952, a working practical JFET was built by
[5]
George C. Dacey and Ian M. Ross in 1953. However, the JFET still had issues
[6]
affecting junction transistors in general. Junction transistors were relatively bulky
devices that were difficult to manufacture on a mass-production basis, which limited
them to a number of specialised applications. The insulated-gate field-effect transistor
(IGFET) was theorized as a potential alternative to junction transistors, but researchers
were unable to build working IGFETs, largely due to the troublesome surface state
[6]
barrier that prevented the external electric field from penetrating into the material. By
the mid-1950s, researchers had largely given up on the FET concept, and instead
[7]
focused on bipolar junction transistor (BJT) technology.

The foundations of MOSFET technology were laid down by the work of William
Shockley, John Bardeen and Walter Brattain. Shockley independently envisioned the
FET concept in 1945, but he was unable to build a working device. The next year
Bardeen explained his failure in terms of surface states. Bardeen applied the theory of
surface states on semiconductors (previous work on surface states was done by
Shockley in 1939 and Igor Tamm in 1932) and realized that the external field was
blocked at the surface because of extra electrons which are drawn to the semiconductor
surface. Electrons become trapped in those localized states forming an inversion layer.
Bardeen's hypothesis marked the birth of surface physics. Bardeen then decided to
make use of an inversion layer instead of the very thin layer of semiconductor which
Shockley had envisioned in his FET designs. Based on his theory, in 1948 Bardeen
patented the progenitor of MOSFET, an insulated-gate FET (IGFET) with an inversion
layer. The inversion layer confines the flow of minority carriers, increasing modulation
and conductivity, although its electron transport depends on the gate's insulator or
quality of oxide if used as an insulator, deposited above the inversion layer. Bardeen's
patent as well as the concept of an inversion layer forms the basis of CMOS technology
today. In 1976 Shockley described Bardeen's surface state hypothesis "as one of the
[8]
most significant research ideas in the semiconductor program".

After Bardeen's surface state theory the trio tried to overcome the effect of surface
states. In late 1947, Robert Gibney and Brattain suggested the use of electrolyte placed
between metal and semiconductor to overcome the effects of surface states. Their FET
device worked, but amplification was poor. Bardeen went further and suggested to
rather focus on the conductivity of the inversion layer. Further experiments led them to
replace electrolyte with a solid oxide layer in the hope of getting better results. Their
goal was to penetrate the oxide layer and get to the inversion layer. However, Bardeen
suggested they switch from silicon to germanium and in the process their oxide got
inadvertently washed off. They stumbled upon a completely different transistor, the
point-contact transistor. Lillian Hoddeson argues that "had Brattain and Bardeen been
working with silicon instead of germanium they would have stumbled across a
[8][9][10][11][12]
successful field effect transistor".

By the end of the first half of the 1950s, following theoretical and experimental work of
Bardeen, Brattain, Kingston, Morrison and others, it became more clear that there were
two types of surface states. Fast surface states were found to be associated with the
bulk and a semiconductor/oxide interface. Slow surface states were found to be
associated with the oxide layer because of adsorption of atoms, molecules and ions by
the oxide from the ambient. The latter were found to be much more numerous and to
have much longer relaxation times. At the time Philo Farnsworth and others came up
with various methods of producing atomically clean semiconductor surfaces.
In 1955, Carl Frosch and Lincoln Derrick accidentally covered the surface of silicon
wafer with a layer of silicon dioxide. They showed that oxide layer prevented certain
dopants into the silicon wafer, while allowing for others, thus discovering the passivating
effect of oxidation on the semiconductor surface. Their further work demonstrated how
to etch small openings in the oxide layer to diffuse dopants into selected areas of the
silicon wafer. In 1957, they published a research paper and patented their technique
summarizing their work. The technique they developed is known as oxide diffusion
masking, which would later be used in the fabrication of MOSFET devices. At Bell Labs,
the importance of Frosch's technique was immediately realized. Results of their work
circulated around Bell Labs in the form of BTL memos before being published in 1957.
At Shockley Semiconductor, Shockley had circulated the preprint of their article in
[6][13][14]
December 1956 to all his senior staff, including Jean Hoerni.

In 1955, Ian Munro Ross filed a patent for a FeFET or MFSFET. Its structure was like
that of a modern inversion channel MOSFET, but ferroelectric material was used as a
dielectric/insulator instead of oxide. He envisioned it as a form of memory, years before
the floating gate MOSFET. In February 1957, John Wallmark filed a patent for FET in
which germanium monoxide was used as a gate dielectric, but he didn't pursue the idea.
In his other patent filed the same year he described a double gate FET. In March 1957,
in his laboratory notebook, Ernesto Labate, a research scientist at Bell Labs, conceived
of a device similar to the later proposed MOSFET, although Labate's device didn't
[15][16][17][18]
explicitly use silicon dioxide as an insulator.

Metal-oxide-semiconductor FET (MOSFET)[edit]

Main article: MOSFET

Mohamed Atalla (left) and Dawon Kahng (right) invented the MOSFET (MOS field-effect
transistor) in 1959.

A breakthrough in FET research came with the work of Egyptian engineer Mohamed
[3]
Atalla in the late 1950s. In 1958 he presented experimental work which showed that
growing thin silicon oxide on clean silicon surface leads to neutralization of surface
states. This is known as surface passivation, a method that became critical to the
semiconductor industry as it made mass-production of silicon integrated circuits
[19][20]
possible.

The metal–oxide–semiconductor field-effect transistor (MOSFET) was then invented by

[21][22]
Mohamed Atalla and Dawon Kahng in 1959. The MOSFET largely superseded
[2]
both the bipolar transistor and the JFET, and had a profound effect on digital
[23][22] [24]
electronic development. With its high scalability, and much lower power
[25]
consumption and higher density than bipolar junction transistors, the MOSFET made
[26]
it possible to build high-density integrated circuits. The MOSFET is also capable of
[27]
handling higher power than the JFET. The MOSFET was the first truly compact
[6]
transistor that could be miniaturised and mass-produced for a wide range of uses. The
[20]
MOSFET thus became the most common type of transistor in computers, electronics,
[28]
and communications technology (such as smartphones). The US Patent and
Trademark Office calls it a "groundbreaking invention that transformed life and culture
[28]
around the world".

CMOS (complementary MOS), a semiconductor device fabrication process for

MOSFETs, was developed by Chih-Tang Sah and Frank Wanlass at Fairchild
[29][30]
Semiconductor in 1963. The first report of a floating-gate MOSFET was made by
[31]
Dawon Kahng and Simon Sze in 1967. A double-gate MOSFET was first
demonstrated in 1984 by Electrotechnical Laboratory researchers Toshihiro Sekigawa
[32][33]
and Yutaka Hayashi. FinFET (fin field-effect transistor), a type of 3D non-planar
multi-gate MOSFET, originated from the research of Digh Hisamoto and his team at
[34][35]
Hitachi Central Research Laboratory in 1989.

Basic information