Kantowski-Sachs metric

2013-07-28T10:15:28Z

86.169.83.19: I deleted the 'and inhomogeneous cosmology' because the Kantowski-Sachs solution is clearly not spatially inhomogeneous. It differs from standard 'Friedmann-Robertson-Walker' solutions in that it is spatially *anisotropic*.

{{disputed|date=May 2012}}
There are three options in a circuit for current. It can be '''leading''', '''lagging''', or in phase with voltage. These can all be seen when one maps current and voltage of alternating (AC) circuits against time. The only time that the voltage and current are in phase together is when the load is resistive. If at some point in the phase shift the current leads the voltage by more than 90 degrees, it can then be stated that the current lags that voltage by 180 degrees minus the phase shift. Ninety degrees phase shift is the determining point if the current is either leading or lagging the voltage.<ref>Gilmore, Besser p. 19</ref>

Each of the main components of a circuit ([[resistor]], [[capacitor]], and [[inductor]]) can be seen as an impedance. All of them produce resistance in either fractional or exponential ways. Here are their complex number forms:

* Resistor, R = R<ref>Bowick, Blyler, Ajluni 2008, pg. 25</ref>
* Capacitor, Z<sub>c</sub> = {{math|1/jωC}}<ref>Bowick, Blyler, Ajluni 2008, pg. 25</ref>
* Inductor, Z<sub>l</sub> = {{math|jωL}}<ref>Bowick, Blyler, Ajluni 2008, pg. 25</ref>
* ω={{math|2πf}}<ref>Bowick, Blyler, Ajluni 2008, pg. 25</ref>

==Angle notation==
[[Angle notation]] can easily determine leading and lagging current:

 <math>A \ang \theta.</math><ref>Nilsson p. 338</ref>

In this equation, the value of theta is the important factor for leading and lagging current. Using [[complex number]]s is a way to simplify analyzing certain components in [[RLC circuits]]. It is also one of the quickest ways to notice right away if the current is leading or lagging in the circuit. For example, it is very easy to convert these between polar and rectangular coordinates. Starting from the polar notation,  <math>\ang \theta </math> can represent either the [[Euclidean vector|vector]]  <math>(\cos \theta, \sin \theta)\,</math>  or the rectangular notation  <math>\cos \theta + j \sin \theta = e^{j\theta},\,</math>  both of which have magnitudes of 1.

==Lagging Current==

 <math>A \ang \theta =A \ang \delta - \beta</math>

Lagging current can be formally defined as “an alternating current that reaches its maximum value up to 90° behind the voltage that produces it”.<ref>"Lagging Current".</ref> This means that current lags the voltage when  <math>\beta</math> is less than  <math>\delta</math>. This can also be stated as the voltage and current being out of phase.

For an inductor, current lags the voltage. This characteristic carries over into circuits with primarily inductive loads, in which current lags the voltage.

==Leading Current==

 <math>A \ang \theta =A \ang \delta - \beta</math>

Leading current can be formally defined as “an alternating current that reaches its maximum value up to 90° ahead of the voltage that produces it”.<ref>"Leading Current".</ref> This means that the current leads the voltage when  <math>\beta</math> is greater than  <math>\delta</math>. They are both out of phase from each other.

In a purely capacitive circuit, current will be at its maximum phase shift and leading the voltage.

==Visualizing Leading and Lagging Current==
A simple [[phasor]] diagram with a two dimensional Cartesian coordinate system and phasors can be used to visualize leading and lagging current.

==Historical Documents Regarding Leading and Lagging Currents==
An early source of data is an article from the 1911 [[American Academy of Arts and Sciences]] by A.E. Kennelly. Kennelly uses traditional methods in solving vector diagrams for oscillating circuits, which can also include alternating current circuits as well. The math goes way beyond the simplification of how people today mathematically solve for different components using vector math for circuits.

For a capacitor, current leads the voltage. This characteristic carries over into circuits with primarily capacitive loads, in which current leads the voltage.

==See also==
* [[Alternating current]]
* [[Angle notation]]
* [[Complex numbers]]
* [[Electrical impedance]]
* [[Ohm's Law]]
* [[Phase (waves)#Phase shift]]
* [[Phasor]]
* [[Power factor]]
* [[RLC circuits]]

== Notes ==
References: {{reflist}}

== References ==
*Bowick, Chris, John Blyler, and Cheryl J. Ajluni. RF Circuit Design. 2nd ed. Amsterdam: Newnes/Elsevier, 2008. Print.

*Gaydecki, Patrick. Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design. 2nd ed. London: Institution of Electrical Engineers, 2004. Print

*Gilmore, Rowan, and Les Besser. Passive Circuits and Systems. Boston [u.a.: Artech House, 2003. Print.

*Hayt, W. H., and J. E. Kemmerly. Engineering Circuit Analysis. 2nd ed. New York: McGraw-Hill, 1971. Print.

*Kennelly, A. E. "Vector-Diagrams of Oscillating-Current Circuits." American Academy of Arts & Sciences 46.17 (1911): 373-421. Jstor. ITHAKA. Web. 1 May 2012. <http://www.jstor.org/stable/20022665>.

*"Lagging Current." TheFreeDictionary.com. Web. 1 May 2012. <http://encyclopedia2.thefreedictionary.com/lagging_current>.

*"Leading Current." TheFreeDictionary.com. Web. 1 May 2012. <http://encyclopedia2.thefreedictionary.com/leading_current>.

*Nilsson, James William; Riedel, Susan A. (2008). Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN 0-13-198925-1., Chapter 9, page 338

*Smith, Ralph J. Circuit Devices and Systems. 3rd ed. New York: John Wiley & Sons, 1976. Print.

[[Category:Electrical parameters]]
[[Category:Electric power]]
[[Category:Electrical engineering]]

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Kantowski-Sachs metric