Editing Norton's theorem

{{About|the theorem in electrical circuits|Norton's theorem for queueing networks|flow-equivalent server method}}
{{Short description|DC circuit analysis technique}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
[[File:NortonEquivalentCircuits.png|thumb|upright=1.6|alt=text|Any [[Black box (systems)|black box]] containing resistances only and ''voltage and current sources can be replaced by an equivalent circuit'' consisting of an equivalent current source in parallel connection with an equivalent resistance.]]
[[File:Edward Lawry Norton.jpg|200px|thumb| [[Edward Lawry Norton]]]]
In [[Direct current|direct-current]] [[circuit theory]], '''Norton's theorem''', also called the '''Mayer–Norton theorem''', is a simplification that can be applied to [[Electrical network|networks]] made of [[Linear time-invariant system|linear time-invariant]] [[Resistor|resistances]], [[Voltage source|voltage sources]], and [[Current source|current sources]]. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel. 

For [[alternating current]] (AC) systems the theorem can be applied to [[Reactive power|reactive]] [[Electrical impedance|impedances]] as well as resistances. The '''Norton equivalent''' circuit is used to represent any network of linear sources and impedances at a given [[frequency]].

Norton's theorem and its dual, [[Thévenin's theorem]], are widely used for circuit analysis simplification and to study circuit's [[Initial condition|initial-condition]] and [[Steady state (electronics)|steady-state]] response.

Norton's theorem was independently derived in 1926 by [[Siemens|Siemens & Halske]] researcher [[Hans Ferdinand Mayer]] (1895–1980) and [[Bell Labs]] engineer [[Edward Lawry Norton]] (1898–1983).<ref name="Mayer_1926"/><ref name="Norton_1926"/><ref name="Johnson_2003a"/><ref name="Johnson_2003b"/><ref name="Brittain_1990"/><ref name="Dorf_2010"/>

To find the Norton equivalent of a linear time-invariant circuit, the Norton current ''I''<sub>no</sub> is calculated as the current flowing at the two terminals ''A'' and ''B'' of the original circuit that is now [[Short circuit|short]] (zero impedance between the terminals). The Norton resistance ''R''<sub>no</sub> is found by calculating the output voltage ''V<sub>o</sub>'' produced at ''A'' and ''B'' with no resistance or load connected to, then ''R''<sub>no</sub> = ''V<sub>o</sub>'' / ''I<sub>no</sub>''; equivalently, this is the resistance between the terminals with all (independent) voltage sources short-circuited and independent current sources [[Open-circuit voltage|open-circuited]] (i.e., each independent source is set to produce zero energy). This is equivalent to calculating the Thevenin resistance.

When there are dependent sources, the more general method must be used. The voltage at the terminals is calculated for an injection of a 1 ampere test current at the terminals. This voltage divided by the 1&nbsp;A current is the Norton impedance ''R''<sub>no</sub> (in ohms). This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.

==Example of a Norton equivalent circuit==
[[File:Norton-example.png|thumb|left|upright=2.9| {{ordered list |1=The original circuit |2=Calculating the equivalent output current |3=Calculating the equivalent resistance |4=Design the Norton equivalent circuit}}]]
{{clear}}

In the example, the total current ''I''<sub>total</sub> is given by:

:[[parallel (operator)|<math>
I_\mathrm{total} = {15\,\mathrm{V} \over 2\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \parallel (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega)} = 5.625\,\mathrm{mA}.
</math>]]

The current through the load is then, using the [[current divider rule]]:

:<math>
\begin{align}
I_\mathrm{no} & = {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega} \cdot I_\mathrm{total} \\[5pt]
& = 2/3 \cdot 5.625\,\mathrm{mA} = 3.75\,\mathrm{mA}.
\end{align}
</math>

And the equivalent resistance looking back into the circuit is:

:[[parallel (operator)|<math>
R_\mathrm{no} = 1\,\mathrm{k}\Omega + (2\,\mathrm{k}\Omega \parallel (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega)) = 2\,\mathrm{k}\Omega.
</math>]]

So the equivalent circuit is a 3.75&nbsp;mA current source in parallel with a 2&nbsp;kΩ resistor.

==Conversion to a Thévenin equivalent==
[[File:norton-to-thevenin.png|thumb|To a Thévenin equivalent]]
A Norton equivalent circuit is related to the [[Thévenin theorem|Thévenin equivalent]] by the equations:
:<math>\begin{align}
& R_{\rm th} = R_{\rm no} \\[8pt]
& V_{\rm th} = I_{\rm no} R_{\rm no} \\[8pt]
& \frac{V_{\rm th}}{R_{\rm th}} = I_{\rm no}.
\end{align}</math>
An original circuit and its Thévenin and Norton equivalents have the same voltage between the two open-circuited terminals, and the same short-circuited current in between.

==Queueing theory==
The passive circuit equivalent of "Norton's theorem" in [[queuing theory]] is called the [[Flow-equivalent server method|Chandy Herzog Woo theorem]].<ref name="Johnson_2003a"/><ref name="Johnson_2003b"/><ref name="Gunther_2004"/> In a [[reversible queueing system]], it is often possible to replace an uninteresting subset of queues by a single ([[FCFS]] or [[Processor sharing|PS]]) queue with an appropriately chosen service rate.<ref name="Chandy_1975"/>

== See also ==
*[[Ohm's law]]
*[[Millman's theorem]]
*[[Source transformation]]
*[[Superposition theorem]]
*[[Thévenin's theorem]]
*[[Maximum power transfer theorem]]
*[[Extra element theorem]]

==References==
{{reflist|refs=
<ref name="Brittain_1990">{{cite journal |author-last=Brittain |author-first=James E. |title=Thevenin's theorem |journal=[[IEEE Spectrum]] |date=March 1990 |volume=27 |issue=3 |page=42 |doi=10.1109/6.48845 |s2cid=2279777 |url=https://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=James+E.+Brittain+Thevenin%27s+theorem&.x=41&.y=17 |access-date=2013-02-01|url-access=subscription }}</ref>
<ref name="Chandy_1975">{{cite journal |author-last1=Chandy |author-first1=Kanianthra Mani |author-link1=Kanianthra Mani Chandy |author-last2=Herzog |author-first2=Ulrich |author-last3=Woo |author-first3=Lin S. |title=Parametric Analysis of Queuing Networks |journal=[[IBM Journal of Research and Development]] |date=January 1975 |volume=19 |issue=1 |pages=36–42 |doi=10.1147/rd.191.0036 |url=https://scholar.google.ca/scholar_url?hl=en&q=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.93.9312%26rep%3Drep1%26type%3Dpdf&sa=X&scisig=AAGBfm1HEBU-rSFYLTIePQWPitczchOopA&oi=scholarr&ei=L3wQUfP9DOHWiwKYtICQAQ&ved=0CC4QgAMoADAA|url-access=subscription }}</ref>
<ref name="Dorf_2010">{{cite conference |author-last1=Dorf |author-first1=Richard C. |author-link1=Richard C. Dorf |author-last2=Svoboda |author-first2=James A. |chapter=Chapter 5: Circuit Theorems |title=Introduction to Electric Circuits |date=2010 |publisher=[[John Wiley & Sons]] |location=Hoboken, NJ, USA |isbn=978-0-470-52157-1 |chapter-url=http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000347.html |edition=8th |pages=162–207 |access-date=2018-12-08 |archive-url=https://web.archive.org/web/20120430052426/http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000347.html |archive-date=2012-04-30 |url-status=dead}}</ref>
<ref name="Gunther_2004">{{cite book |author-last=Gunther |author-first=Neil J. |author-link=Neil J. Gunther |title=Analyzing Computer System Performance with Perl::PDQ |date=2004 |publisher=[[Springer Science+Business Media]] |location=Berlin |isbn=978-3-540-20865-5 |page=281 |edition=Online |url=https://books.google.com/books?id=rp1EZKnr48kC&pg=PA83}}</ref>
<ref name="Johnson_2003a">{{cite journal |author-last=Johnson |author-first=Don H. |title=Origins of the equivalent circuit concept: the voltage-source equivalent |journal=[[Proceedings of the IEEE]] |date=2003 |volume=91 |issue=4 |pages=636–640 |doi=10.1109/JPROC.2003.811716 |url=http://www.ece.rice.edu/~dhj/paper1.pdf |hdl=1911/19968|hdl-access=free }}</ref>
<ref name="Johnson_2003b">{{cite journal |author-last=Johnson |author-first=Don H. |title=Origins of the equivalent circuit concept: the current-source equivalent |journal=[[Proceedings of the IEEE]] |date=2003 |volume=91 |issue=5 |pages=817–821 |doi=10.1109/JPROC.2003.811795 |url=http://www.ece.rice.edu/~dhj/paper2.pdf}}</ref>
<ref name="Mayer_1926">{{cite journal |author-last=Mayer |author-first=Hans Ferdinand |author-link=Hans Ferdinand Mayer |title=Ueber das Ersatzschema der Verstärkerröhre |trans-title=On equivalent circuits for electronic amplifiers |language=de |journal=Telegraphen- und Fernsprech-Technik |date=1926 |volume=15 |pages=335–337}}</ref>
<ref name="Norton_1926">{{cite journal |author-last=Norton |author-first=Edward Lawry |author-link=Edward Lawry Norton |id=Technical Report TM26–0–1680 |title=Design of finite networks for uniform frequency characteristic |date=1926 |publisher=[[Bell Laboratories]]|journal=[[Bell Labs Technical Memo]]|url=https://www.ece.rice.edu/~dhj/norton/Nortonmemo.pdf}}</ref>
}}

==External links==
*{{Commons category inline}}
* [https://www.allaboutcircuits.com/textbook/direct-current/chpt-10/nortons-theorem/ Norton's theorem at allaboutcircuits.com]

[[Category:Circuit theorems]]
[[Category:Eponymous theorems of physics]]
[[Category:Linear electronic circuits]]