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==Coordinate system==


A '''semiconductor diode''' is a two-terminal device that conducts current in only one direction, made by joining a ''p''-type semiconducting layer to an ''n''-type semiconducting layer.
The coordinates of a point '''r''' in an ''n''-dimensional real numerical space ℝ<sup>n</sup> or a complex ''n''-space ℂ<sup>n</sup>  are simply an ordered set of ''n'' real or complex numbers:<ref name=Korn>


==Electrical behavior==
{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1
{{Image|Nonideal diode current-voltage behavior.PNG|right|250px|Nonideal ''pn''-diode current-voltage characteristics.}}
|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}}
The ideal diode has zero resistance for the ''forward bias polarity'', and infinite resistance (conducts zero current) for the ''reverse voltage polarity''. The ''pn-diode'' is not ideal. As shown in the figure, the diode does not conduct appreciably until a nonzero ''knee voltage'' (also called the ''turn-on voltage'') is reached. Above this voltage the slope of the current-voltage curve is not infinite (on-resistance is not zero). In the reverse direction the diode conducts a nonzero leakage current (exaggerated by a smaller scale in the figure) and at a sufficiently large reverse voltage below the ''breakdown voltage'' the current increases very rapidly with more negative reverse voltages.


==Types==
</ref><ref name=Morita>[http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#v=onepage&q&f=false Morita]
Semiconductor diodes come in a large variety of types:
</ref><ref name=Fritzche>
*''pn''-diode: The ''pn'' junction diode consists of an ''n''-type semiconductor joined to a ''p''-type semiconductor.
*Zener diode: The [[Zener diode]] is a special type of ''pn''-diode made to operate in the reverse breakdown region, and used often as a voltage regulator. The breakdown voltage in these didoes is sometimes called the ''Zener voltage''. Depending upon the voltage range designed for, the diode may break down by either [[Zener breakdown]], an [[electron tunneling]] behavior, or by [[avalanche breakdown]].
*Schottky diode: The [[Schottky diode]] is made using a metal such as aluminum or platinum, on a lightly doped semiconductor substrate.
*Metal-oxide varistor: The [[varistor]] is intended to provide a voltage controlled resistance. Its resistance under small voltage variations is set by the choice of a bias voltage.
*Tunnel diode: Like the Zener diode, the [[tunnel diode]] (or Esaki diode) is made up of heavily doped ''n-'' and ''p''-type layers with a very abrupt transition between the two types. Conduction takes place by electron tunneling.
*Light-emitting diode: The [[Light Emitting Diode|light-emitting diode]] is designed to convert electrical current into light.
*''pin''-diode: The ''pin''-diode is made of three layers: an intrinsic (undoped) layer between the ''p''- and ''n''-type layers. Because of its rapid switching characteristics it is used in microwave and radio-frequency applications.
*Gunn diode: The [[Gunn diode]] is a ''transferred electron device''  based upon the [[Gunn effect]] in III-V semiconductors, and is used to generate microwave oscillations.


==Operation==
[http://books.google.com/books?id=jSeRz36zXIMC&pg=PA155&dq=complex+%22coordinate+system%22&hl=en&ei=LA2JTYD1MYfWtQP2j92NDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q=complex%20%22coordinate%20system%22&f=false Fritzche]</ref>
Here, the operation of the simple ''pn'' junction diode is considered. The objective is to explain the various bias regimes in the figure. Operation is described using band-bending diagrams that show how the lowest conduction band energy and the highest valence band energy vary with position inside the diode under various bias conditions.
:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ , x^n] \ .</math>
===Zero bias===
Coordinate surfaces, coordinate lines, and [[Basis (linear algebra)|basis vectors]] are components of a '''coordinate system'''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |page=84  |isbn=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig  }}</ref>
{{Image|Pn-junction zero bias.PNG|right|250px|Band-bending diagram for ''pn''-junction diode at zero applied voltage.}}
As shown in the band-banding diagram at the right, when a ''p''-type and an ''n''-type region of the same semiconductor are brought into contact, the Fermi occupancy level is situated at a constant level. This level insures that in the field-free bulk on both sides of the junction the hole and electron occupancies are correct. However, a flat Fermi level requires the bands on the ''p''-type side to move higher than the corresponding bands on the ''n''-type side. Consequently, a region near the junction becomes depleted of both holes and electrons, forming an insulating region with no mobile charges. However, the absence of mobile charge means that these charges are not present to compensate for the immobile charges contributed as a negative charge on the ''p''-type side dues to acceptor dopant and as a positive charge on the ''n''-type side due to donor dopant. The dimensions of this depletion region adjust so the negative acceptor charge on the ''p''-side exactly balances the positive donor charge on the ''n''-side, so there is no electric field outside the depletion region on either side. In this band configuration no voltage is applied and no current flows through the diode. To force current through the diode a ''forward bias'' must be applied, as described next.


===Forward bias===
==Manifolds==
{{Image|Pn-junction forward bias.PNG|right|120px|Band-bending diagram for pn-diode in forward bias}}
A coordinate system in mathematics is a facet of [[geometry]] or of [[algebra]], in particular, a property of [[Manifold (geometry)|manifold]]s (for example, in physics, [[configuration space]]s or [[phase space]]s).<ref name=Hawking>
In forward bias the occupancy level for holes tends to stay at the level of the bulk ''p''-type semiconductor wile the occupancy level for electrons follows that for the bulk ''n-''-type. Naturally, the ''p''-type bulk is raised in energy by the applied voltage compared to the ''n''-type, so the two bulk occupancy levels are separated by an energy determined by the applied voltage. (The band bending diagram is made in units of volts, so no electron charge appears to convert ''V'' to energy.) As shown in the diagram, this behavior means the depletion region narrows as holes are pushed into it from the ''p''-side and electrons from the ''n''-side.


===Reverse bias===
According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." {{cite book |title=The Large Scale Structure of Space-Time |author=Stephen W. Hawking & George Francis Rayner Ellis |isbn=0521099064 |year=1973 |publisher=Cambridge University Press |pages=p. 11 |url=http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&lr=&as_brr=0&sig=ACfU3U1q-iaRTBDo6J8HMEsyPeFi8cJNWg#PPA11,M1
{{Image|Pn-junction reverse bias.PNG|right|150px|Band-bending for ''pn''-diode in forward bias}}
}} A mathematical definition is: ''A connected [[Hausdorff space]] ''M'' is called an ''n''-dimensional manifold if each point of ''M'' is contained in an open set that is homeomorphic to an open set in Euclidean ''n''-dimensional space.''
 
</ref><ref name=Morita2>
{{cite book |title=Geometry of Differential Forms |author=Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu |pages=p. 12 |url=http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1
|isbn=0821810456 |year=2001 |publisher=American Mathematical Society Bookstore  }}
 
</ref> The coordinates of a point '''r''' in an ''n''-dimensional space are simply an ordered set of ''n'' numbers:<ref name=Korn>
 
{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1
|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}}
 
</ref>
:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ ,  x^n] \ .</math>
 
In a general [[Banach space]], these numbers could be (for example) coefficients in a functional expansion like a [[Fourier series]]. In a physical problem, they could be [[spacetime]] coordinates or [[normal mode]] amplitudes. In a [[Robotics|robot design]], they could be angles of relative rotations, linear displacements, or deformations of [[linkage (mechanical)|joints]].<ref name=Yamane>
 
{{cite book |author=Katsu Yamane |title=Simulating and Generating Motions of Human Figures |isbn=3540203176 |year=2004 |publisher=Springer  |pages=12–13 |url=http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22&lr=&as_brr=0&sig=ACfU3U3LRGJJTAHs21CHdOvuu08vw0cAuw#PPA13,M1  }}
 
</ref> Here we will suppose these coordinates can be related to a [[Cartesian coordinate]] system by a set of functions:
:<math>x^j = x^j (x,\  y,\  z,\  \dots)\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  </math>
 
where ''x'', ''y'', ''z'', ''etc.'' are the ''n'' Cartesian coordinates of the point. Given these functions,  '''coordinate surfaces''' are defined by the relations:
 
:<math> x^j (x, y, z, \dots) = \mathrm{constant}\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  .</math>
 
The intersection of these surfaces define '''coordinate lines'''. At any selected point, tangents to the intersecting coordinate lines at that point define a set of '''basis vectors''' {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, …, '''e'''<sub>n</sub>} at that point. That is:
 
:<math>\mathbf{e}_i(\mathbf{r}) =\lim_{\epsilon \rightarrow 0} \frac{\mathbf{r}\left(x^1,\  \dots,\  x^i+\epsilon,\  \dots ,\  x^n \right) - \mathbf{r}\left(x^1,\  \dots,\  x^i,\  \dots ,\  x^n \right)}{\epsilon }\ ,</math>
 
which can be normalized to be of unit length. For more detail see [[curvilinear coordinates]].
 
Coordinate surfaces, coordinate lines, and [[Basis (linear algebra)|basis vectors]] are components of a '''coordinate system'''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |page=84  |isbn=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig  }}</ref> If the basis vectors are orthogonal at every point, the coordinate system is an [[Orthogonal coordinates|orthogonal coordinate system]].
 
An important aspect of a coordinate system is its [[Metric (mathematics)|metric]] ''g''<sub>ik</sub>, which determines the [[arc length]] ''ds'' in the coordinate system in terms of its coordinates:<ref name=Borisenko>{{cite book |title=Vector and Tensor Analysis with Applications |author= A. I. Borisenko, I. E. Tarapov, Richard A. Silverman |page=86 |url=http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q
|isbn=0486638332 |publisher=Courier Dover Publications |year=1979 |pages=pp. 86 ''ff'' |chapter=§2.8.4 Arc length. Metric coefficients |edition=Reprint of Prentice-Hall 1968 ed  }}</ref>
 
:<math>(ds)^2 = g_{ik}\ dx^i\ dx^k \ , </math>
 
where repeated indices are summed over.
 
As is apparent from these remarks, a coordinate system is a mathematical construct, part of an [[axiomatic system]]. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can be used to describe motion by interpreting one coordinate as time. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[coordinate transformation]]s.
 
 
==Notes==
<references/>
[http://books.google.com/books?id=hUWEXphqLo8C&pg=PA111&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEQQ6AEwBA#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Choquet-Bruhat]
[http://books.google.com/books?id=sRaSuentwngC&pg=PA2&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Bishop]
[http://books.google.com/books?id=CGk1eRSjFIIC&pg=PA3&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CE8Q6AEwBg#v=onepage&q=manifold%20%22coordinate%20system%22&f=false O'Neill]
[http://books.google.com/books?id=iaeUqc2yQVQC&pg=PA31&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFgQ6AEwCA#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Warner]

Latest revision as of 12:20, 14 September 2024


The account of this former contributor was not re-activated after the server upgrade of March 2022.


Coordinate system

The coordinates of a point r in an n-dimensional real numerical space ℝn or a complex n-space ℂn are simply an ordered set of n real or complex numbers:[1][2][3]

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[4]

Manifolds

A coordinate system in mathematics is a facet of geometry or of algebra, in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).[5][6] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:[1]

In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.[7] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

   

where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:

   

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e1, e2, …, en} at that point. That is:

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[4] If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

An important aspect of a coordinate system is its metric gik, which determines the arc length ds in the coordinate system in terms of its coordinates:[8]

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can be used to describe motion by interpreting one coordinate as time. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.


Notes

  1. 1.0 1.1 Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478. 
  2. Morita
  3. Fritzche
  4. 4.0 4.1 Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X. 
  5. According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064.  A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.
  6. Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456. 
  7. Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176. 
  8. A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332. 

Choquet-Bruhat Bishop O'Neill Warner