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2014-11-09T18:01:48Z

103.21.125.76: /* Examples */

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Spectral centroid

2013-08-04T06:20:11Z

103.21.125.79:

In [[complex analysis]], a branch of mathematics, the '''Schwarz integral formula''', named after [[Hermann Schwarz]], allows one to recover a [[holomorphic function]], [[up to]] an imaginary constant, from the boundary values of its real part.

==Unit disc==
Let ''ƒ'' = ''u'' + ''iv'' be a function which is holomorphic on the closed unit disc {''z'' ∈ '''C''' | |''z''| ≤ 1}. Then

: <math> f(z) = \frac{1}{2\pi i} \oint_{|\zeta| = 1} \frac{\zeta + z}{\zeta - z} \text{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta}
+ i\text{Im}(f(0))</math>

for all |''z''| < 1.

==Upper half-plane==
Let ''ƒ'' = ''u'' + ''iv'' be a function that is holomorphic on the closed [[upper half-plane]] {''z'' ∈ '''C''' | Im(''z'') ≥ 0} such that, for some ''α'' > 0, |''z''''α'' ''ƒ''(''z'')| is bounded on the closed upper half-plane. Then

: <math>
f(z)
=
\frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} \, d\zeta
=
\frac{1}{\pi i} \int_{-\infty}^\infty \frac{Re(f)(\zeta+0i)}{\zeta - z} \, d\zeta
</math>

for all Im(''z'') > 0.

Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.

== Corollary of Poisson integral formula ==

The formula follows from [[Poisson integral formula]] applied to ''u'':<ref>
{{cite web
|url=http://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE
|title=Lectures on Entire Functions - Google Book Search
|publisher=books.google.com
|accessdate=2008-06-26
|last=
|first=
}}
</ref><ref>The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html</ref>

: <math>u(z) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} \, d\psi\text{ for }|z| < 1.</math>

By means of conformal maps, the formula can be generalized to any simply connected open set.

== Notes and references ==
<references />

* [[Lars Ahlfors|Ahlfors, Lars V.]] (1979), ''Complex Analysis'', Third Edition, McGraw-Hill, ISBN 0-07-085008-9
* Remmert, Reinhold (1990), ''Theory of Complex Functions'', Second Edition, Springer, ISBN 0-387-97195-5
* Saff, E. B., and A. D. Snider (1993), ''Fundamentals of Complex Analysis for Mathematics, Science, and Engineering'', Second Edition, Prentice Hall, ISBN 0-13-327461-6

[[Category:Complex analysis]]

Semblance analysis

2013-06-01T09:55:45Z

103.21.125.55: /* History */

The '''open-circuit test''', or "no-load test", is one of the methods used in [[electrical engineering]] to determine the [[Electrical impedance|no-load impedance]] in the excitation branch of a [[transformer]].

[[File:Open circuit test.png|right|500px|Circuit diagram for open-circuit test]]

==Method==

The secondary of the transformer is left open-circuited. A [[wattmeter]] is connected to the primary. An [[ammeter]] is connected in series with the primary winding. A [[voltmeter]] is optional since the applied voltage is the same as the voltmeter reading. Rated voltage is applied at primary.

If the applied voltage is normal voltage then normal flux will be set up. Since [[iron loss]] is a function of applied voltage, normal iron loss will occur. Hence the iron loss is maximum at rated voltage. This maximum iron loss is measured using the wattmeter. Since the impedance of the [[Series and parallel circuits|series]] winding of the transformer is very small compared to that of the excitation branch, all of the input voltage is [[voltage drop|dropped]] across the excitation branch. Thus the wattmeter measures only the iron loss. This test only measures the combined iron losses consisting of the [[hysteresis loss]] and the [[eddy current]] loss. Although the hysteresis loss is less than the eddy current loss, it is not negligible. The two losses can be separated by driving the transformer from a variable frequency source since the hysteresis loss varies linearly with supply frequency and the eddy current loss varies with the square.

Since the secondary of the transformer is open, the primary draws only no-load current, which will have some copper loss. This no-load current is very small and because the copper loss in the primary is proportional to the square of this current, it is negligible. There is no copper loss in the secondary because there is no secondary current.

[[Electric current|Current]], [[voltage]] and [[electric power|power]] are measured at the [[primary winding]] to ascertain the [[admittance]] and [[power factor|power-factor angle]].

Another method of determining the series impedance of a real transformer is the [[short circuit test]].

==Calculations==
The current <math>\mathbf{I_0}</math> is very small.

If <math>\mathbf{W}</math> is the wattmeter reading then,

:<math>\mathbf{W} = \mathbf{V_1} \mathbf{I_0} \cos \phi_0 </math>

That equation can be rewritten as,

:<math>\cos \phi_0 = \frac {\mathbf{W}} {\mathbf{V_1} \mathbf{I_0}} </math>

Thus,

:<math>\mathbf{I_m} = \mathbf{I_0} \sin \phi_0 </math>

:<math>\mathbf{I_w} = \mathbf{I_0} \cos \phi_0 </math>

===Impedance===

By using the above equations, <math>\mathbf{X_0}</math> and <math>\mathbf{R_0}</math> can be calculated as,

:<math>\mathbf{X_0} = \frac {\mathbf{V_1}} {\mathbf{I_m}} </math>

:<math>\mathbf{R_0} = \frac {\mathbf{V_1}} {\mathbf{I_w}} </math>

Thus,

:<math>\mathbf{Z_0} = \sqrt {\mathbf{R_0}^2 +\mathbf{X_0}^2} </math>

or

:<math>\mathbf{Z_0} = \mathbf{R_0} + \mathbf{j} \mathbf{X_0} </math>

===Admittance===

The admittance is the inverse of impedance. Therefore,

:<math>\mathbf{Y_0} = \frac {1} {\mathbf{Z_0}} </math>

The conductance <math>\mathbf{G_0}</math> can be calculated as,

:<math>\mathbf{G_0} = \frac {\mathbf{W}} {\mathbf{V_1}^2} </math>

Hence the susceptance,

:<math>\mathbf{B_0} = \sqrt {\mathbf{Y_0}^2 -\mathbf{G_0}^2} </math>

or

:<math>\mathbf{Y_0} = \mathbf{G_0} + \mathbf{j} \mathbf{B_0} </math>

Here, 

<math>\mathbf{W}</math> is the wattmeter reading 

<math>\mathbf{V_1}</math> is the applied rated voltage 

<math>\mathbf{I_0}</math> is the no-load current 

<math>\mathbf{I_m}</math> is the magnetizing component of no-load current 

<math>\mathbf{I_w}</math> is the core loss component of no-load current 

<math>\mathbf{Z_0}</math> is the exciting impedance 

<math>\mathbf{Y_0}</math> is the exciting admittance

==References==
*{{cite book | author=Kosow| title=Electric Machinery and Transformers | publisher=Pearson Education India | year=2007}}
*{{cite book | author=Smarajit Ghosh| title=Fundamentals of Electrical and Electronics Engineering | publisher=PHI Learning Pvt. Ltd. | year=2004}}
*{{cite book | author=Wildi, Wildi Theodore| title=Electrical Machines , Drives And Power Systems, 6th edtn.
| publisher=Pearson | year=2007}}

== See also ==
*[[Short-circuit test]]
*[[Thévenin's theorem]]
*[[Blocked rotor test]]
*[[Circle diagram]]
{{DEFAULTSORT:Open Circuit Test}}
[[Category:Electrical tests]]
[[Category:Transformers (electrical)]]