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[[File:Schematic induction regulator.svg|thumb|Schematic of wiring an induction regulator. Power source is connected to R-S-T rotor terminals. Output voltage is N+1-2-3 terminals.]]
{{Probability distribution
[[File:Diagram induction regulator.svg|thumb|Diagram of electrical [[phasor]]s for an induction regulator]]
| name      =Lomax
An '''Induction regulator''' is a device, based on an [[induction motor]], which can provide a continuous ([[Analogue electronics|analogue]]) variable output [[voltage]]. In the past, it was used to control the voltage of electric networks. Nowadays, it has been replaced in this function by the [[Tap (transformer)|tap]] [[transformer]]. Its usage is now mostly confined to electrical laboratories, electrochemical processes and [[arc welding]]. With minor variations, its setup can be used as an isolator [[Quadrature booster|phase-shifting power transformer]].
| type      =density
| pdf_image  =
| cdf_image  =
| parameters =
<math>\lambda >0 </math> [[scale parameter|scale]] (real)<br />
<math>\alpha > 0 </math> [[shape parameter|shape]] (real)
| support    =<math> x \ge 0 </math>
| pdf        =<math> {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}</math>
| cdf        =<math> 1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}</math>
| mean      =<math> {\lambda \over {\alpha -1}} \text{ for } \alpha > 1</math><br /> Otherwise undefined
| median    =<math>\lambda (\sqrt[\alpha]{2} - 1)</math>
| mode      = 0
| variance  =<math> {{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2 </math><br /><math> \infty \text{ for } 1 < \alpha \le 2 </math> <br /> Otherwise undefined
  | skewness  =<math>\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3\,</math>
| kurtosis  =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4\,</math>
| entropy    =
| mgf        =
| char      =
}}


==Construction==
The '''Lomax distribution''', conditionally also called the '''[[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]]''', is a [[heavy tail|heavy-tail]] [[probability distribution]] often used in business, economics, and actuarial modeling.<ref>Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". ''[[Journal of the American Statistical Association]]'', 49, 847–852. {{jstor|2281544}}</ref><ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)</ref>  It is named after K.&nbsp;S.&nbsp;Lomax. It is essentially a [[Pareto distribution]] that has been shifted so that its support begins at zero.<ref>Van Hauwermeiren M and Vose D (2009). [http://www.vosesoftware.com/content/ebook.pdf '' A Compendium of Distributions''] [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com. Accessed 07/07/11</ref>
The induction regulator can be regarded as a wound [[induction motor]]. The rotor is not allowed to turn freely and it can be mechanically shifted by means of a [[worm gear]]. The rest of the regulator's construction follows that of a wound rotor [[induction motor]] with a slotted three-phase stator and a wound three-phase rotor with ends connected to accessible [[slip ring]]s plus common point (neutral)


Depending on the application, the ratio of number of turns on the rotor and the stator can vary.
== Characterization ==
=== Probability density function ===
The [[probability density function]] (pdf) for the Lomax distribution is given by
:<math> p(x) = {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}, \qquad x \geq 0,
</math>
with shape parameter <math>\alpha>0</math> and scale parameter <math>\lambda>0</math>. The density can be rewritten in such a way that more clearly shows the relation to the [[Pareto distribution|Pareto Type I distribution]]. That is:
:<math> p(x) = {{\alpha \lambda^\alpha} \over { (x+\lambda)^{\alpha+1}}}</math>.


==Working==
===Differential equation===
If the rotor terminals are connected to a [[three-phase electric power]] network, a [[rotating magnetic field]] will be driven into the [[magnetic core]].
The pdf of the Lomax distribution is a solution to the following [[differential equation]]:
The resulting flux will produce an [[electromotive force|emf]] on the windings of the stator with the particularity that if rotor and stator are physically shifted by an angle α, then the electric phase shifting of both windings is α too. Considering just the fundamental harmonic, and ignoring the shifting, the following equation rules:
:<math>\left\{\begin{array}{l}
:<math>\frac{U_{stator}}{U_{rotor}}=\frac{\xi_{stator} N_{stator}}{\xi_{rotor} N_{rotor}}
(\gamma +x) p'(x)+(\alpha +1) p(x)=0, \\
p(0)=\frac{\alpha}{\gamma}
\end{array}\right\}
</math>
</math>


Where ξ is the [[winding factor]], a constant related to the construction of the windings.
== Relation to the Pareto distribution ==
The Lomax distribution is a [[Pareto distribution|Pareto Type I distribution]] shifted so that its support begins at zero. Specifically:
:<math>\text{If } Y \sim \mbox{Pareto}(x_m = \lambda, \alpha), \text{ then } Y - x_m \sim \mbox{Lomax}(\lambda,\alpha).</math>
 
The Lomax distribution is a [[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]] with ''x''<sub>m</sub>=λ and μ=0:{{cn|date=October 2012}}
:<math>
\text{If } X \sim \mbox{Lomax}(\lambda,\alpha) \text{ then } X \sim \text{P(II)}(x_m = \lambda, \alpha, \mu=0).</math>


If the stator winding is connected to the primary phase, the total voltage seen from the neutral (N) will be the sum of the voltages at both windings rotor and stator. Translating this to electric [[phasor]]s, both phasors are connected. However, there is an angular shifting of α between them. Since α can be freely chosen between [0, π], both phasors can be added or subtracted, so all the values in between are attainable. The primary and secondary are not isolated. Also, the ratio of transformation between rotor/stator is constant. What makes the regulation possible is the angular shifting.
== Relation to generalized Pareto distribution ==
The Lomax distribution is a special case of the [[generalized Pareto distribution]]. Specifically:


==Advantages==
:<math> \mu = 0,~ \xi = {1 \over \alpha},~ \sigma = {\lambda \over \alpha} .</math>
The output voltage can be continuously regulated within the nominal range. This is a clear benefit against tap transformers where output voltage takes discrete values. Also, the voltage can be easily regulated under working conditions.


==Drawbacks==
== Relation to q-exponential distribution ==
In comparison to tap transformers, induction regulators are expensive, with lower efficiency, high open circuit currents (due to the airgap) and limited in voltage to less than 20kV.
The Lomax distribution is a special case of the [[q-exponential distribution]]. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:


==Applications==
:<math> \alpha = { {2-q} \over {q-1}}, ~ \lambda = {1 \over \lambda_q (q-1)} .</math>
An induction regulator for power networks is usually designed to have a nominal voltage of 14kV and ±(10-15)% of regulation, but this use has declined.  Nowadays, its main uses are in electrical laboratories and [[arc welding]].


==Bibliography==
== Non-central moments ==
{{cite book
  | last = Fraile-Mora
  | first = J. Jesús
  | authorlink = Jesús Fraile-Mora
  | title = Máquinas Eléctricas
  | publisher = [[McGraw-Hill]]
  | series =
  | year = 2003
  | doi =
  | isbn = 84-481-3913-5
  | pages = 358–359
}}


{{cite book
The <math>\nu</math>th non-central moment <math>E[X^\nu]</math> exists only if the shape parameter <math>\alpha</math> strictly exceeds <math>\nu</math>, when the moment has the value
  | last = Serrano-Iribarnegaray
:<math> E(X^\nu) = \frac{ \lambda^\nu \Gamma(\alpha-\nu)\Gamma(1+\nu)}{\Gamma(\alpha)}</math>
  | first = Luis
  | authorlink = Luis Serrano-Iribarnegaray
  | title = Fundamentos de Máquinas Eléctricas Rotativas
  | publisher = [[Marcombo Boixareu]]
  | series =
  | year = 1989
  | doi =
  | isbn = 8426707637
  | pages = 208–210
}}
{{cite book
  | last = Ras
  | first = Enrique
  | authorlink = Enrique Ras
  | title = Transformadores de potencia de medida y de protección
  | publisher = [[Marcombo Boixareu]]
  | series =
  | year = 1991
  | doi =
  | isbn = 84-267-0690-8
  | page = 160
}}


==See also==
== See also ==
*[[Variable frequency transformer]]
*[[Power law]]


[[Category:Articles created via the Article Wizard]]
==References==
[[Category:Transformers (electrical)]]
<references />
[[Category:Energy conversion]]
{{ProbDistributions|continuous-semi-infinite}}
[[Category:Electric motors]]
[[Category:Continuous distributions]]
[[Category:Italian inventions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

Revision as of 14:00, 18 August 2014

Template:Probability distribution

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution often used in business, economics, and actuarial modeling.[1][2] It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.[3]

Characterization

Probability density function

The probability density function (pdf) for the Lomax distribution is given by

p(x)=αλ[1+xλ]−(α+1),x≥0,

with shape parameter α>0 and scale parameter λ>0. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

p(x)=αλα(x+λ)α+1.

Differential equation

The pdf of the Lomax distribution is a solution to the following differential equation:

{(γ+x)p′(x)+(α+1)p(x)=0,p(0)=αγ}

Relation to the Pareto distribution

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

If Y∼Pareto(xm=λ,α), then Y−xm∼Lomax(λ,α).

The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:Template:Cn

If X∼Lomax(λ,α) then X∼P(II)(xm=λ,α,μ=0).

Relation to generalized Pareto distribution

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:

μ=0,ξ=1α,σ=λα.

Relation to q-exponential distribution

The Lomax distribution is a special case of the q-exponential distribution. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:

α=2−qq−1,λ=1λq(q−1).

Non-central moments

The νth non-central moment E[Xν] exists only if the shape parameter α strictly exceeds ν, when the moment has the value

E(Xν)=λνΓ(α−ν)Γ(1+ν)Γ(α)

See also

References

  1. ↑ Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". Journal of the American Statistical Association, 49, 847–852. Template:Jstor
  2. ↑ Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Volume 1, 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)
  3. ↑ Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com. Accessed 07/07/11

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