Source transformation: Difference between revisions

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{{Short description|Process of simplifying circuit solutions}}

Finding a solution to a circuit can be somewhat difficult without using tricks or methods that make the circuit appear simpler. Circuit solutions are often simplified, especially with mixed sources, by transforming a voltage into a [[Electric current|current]] source, and vice versa.<ref name="Oppenheimer">Oppenheimer, Samuel L. (1984). ''Fundamentals of Electric Circuits''. New Jersey: Prentice Hall.</ref> This process is known as a '''source transformation''', and is an application of [[Thevenin's theorem]] and [[Norton's theorem]].

{{about|electronic circuits|transformation of source code|Program transformation}}

'''Source transformation''' is the process of simplifying a circuit solution, especially with mixed sources, by transforming [[voltage]] sources into [[Electric current|current]] sources, and vice versa, using [[Thévenin's theorem]] and [[Norton's theorem]] respectively.<ref name="CPP">CPP. https://www.cpp.edu/~elab/projects/project_08/index.html.</ref>

== Process ==

Performing a source transformation consists of using [[OhmsOhm's Lawlaw]] to take an existing [[voltage source]] in [[series circuit|series]] with a [[resistor|resistance]], and replacereplacing it with a [[current source]] in [[parallel circuit|parallel]] with the same resistance., Rememberor thatvice Ohmsversa. lawThe statestransformed thatsources aare voltageconsidered inidentical aand materialcan isbe equalsubstituted to the material's resistance times the amount of current through it (V=IR). Since source transformations are bilateral,for one cananother bein deriveda from the othercircuit. <ref name="Nilsson">Nilsson, James W., & Riedel, Susan A. (2002). ''Introductory Circuits for Electrical and Computer Engineering''. New Jersey: Prentice Hall.</ref> Source transformations are not limited to resistive circuits however. They can be performed on a circuit involving [[capacitors]] and [[inductors]], as long as the circuit is first put into the [[frequency ___domain]]. In general, the concept of source transformation is an application of [[Thevenin's theorem]] to a [[current source]], or [[Norton's theorem]] to a [[voltage source]].

Source transformations are not limited to resistive circuits. They can be performed on a circuit involving [[capacitors]] and [[inductors]] as well, by expressing circuit elements as impedances and sources in the [[frequency ___domain]]. In general, the concept of source transformation is an application of [[Thévenin's theorem]] to a [[current source]], or [[Norton's theorem]] to a [[voltage source]]. However, this means that source transformation is bound by the same conditions as Thevenin's theorem and Norton's theorem; namely that the load behaves linearly, and does not contain dependent voltage or current sources.<ref>{{Cite book |last1=Ulaby |first1=Fawwaz T. |title=CIRCUITS-W/ACCESS |last2=Maharbiz |first2=Michel |last3=Furse |first3=Cynthia |author3-link=Cynthia Furse|date=2015-01-01 |publisher=National Technology & Science Press |isbn=978-1-934891-22-3 |edition=3rd |language=English}}</ref>

Specifically, source transformations are used to exploit the equivalence of a real current source and a real voltage source, such as a [[battery (electricity)|battery]]. Application of Thevenin's theorem and Norton's theorem gives the quantities associated with the equivalence. Specifically, suppose we have a real current source I, which is an ideal current source in [[Series and parallel circuits|parallel]] with an [[Electrical impedance|impedance]]. If the ideal current source is rated at I amperes, and the parallel resistor has an impedance Z, then applying a source transformation gives an equivalent real voltage source, which is ideal, and in [[Series and parallel circuits|series]] with the impedance. This new voltage source V, has a value equal to the ideal current source's value times the resistance contained in the real current source <math>V=I\cdot Z</math>. The impedance component of the real voltage source retains its real current source value.

Specifically, sourceSource transformations are used to exploit the equivalence of a real current source and a real voltage source, such as a [[battery (electricity)|battery]]. Application of TheveninThévenin's theorem and Norton's theorem gives the quantities associated with the equivalence. Specifically, suppose we havegiven a real current source I, which is an ideal current source <math>I</math> in [[Series and parallel circuits|parallel]] with an [[Electrical impedance|impedance]]. If the ideal current source is rated at I amperes, and the parallel resistor has an impedance <math>Z</math>, then applying a source transformation gives an equivalent real voltage source, which is an ideal, andvoltage source in [[Series and parallel circuits|series]] with the impedance. The Thisimpedance <math>Z</math> retains its value and the new voltage source <math>V,</math> has a value equal to the ideal current source's value times the resistanceimpedance, containedaccording into theOhm's real current sourceLaw <math>V=I \cdot, Z</math>. In Thethe same way, an ideal voltage source in series with an impedance componentcan ofbe thetransformed realinto voltagean ideal current source retainsin itsparallel realwith the same impedance, where the new ideal current source has value. <math> I = V/Z </math>.

In general, source transformations can be summarized by keeping two things in mind:

* [[Ohm's Law]]

* Impedances remain the same

== Example calculation ==

Source transformations are easy to performcompute as long as there is a familiarity withusing [[OhmsOhm's Lawlaw]]. If there is a voltage source in [[series circuit|series]] with an [[Electrical impedance|impedance]], it is possible to find the value of the equivalent [[current source]] in [[parallel circuit|parallel]] with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also applies hereholds: if a current source in parallel with an impedance is present, multiplying the value of the current source with the value of the impedance will result inprovides the equivalent voltage source in series with the impedance. A visual example of what is being done during a source transformation can be seen in Figure 1.

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'''Remember:'''

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<math>V=I\cdot Z</math>

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<math>I=\cfrac {V}{Z} </math>

:: <math> V = I\cdot Z, \qquad I = \cfrac VZ</math>

[[Image:Sourcetrans.jpg||frame|left|Figure 1. An example of a DC source transformation. Notice that the impedance Z is the same in both configurations.]]

[[Image:Sourcetrans.jpg||frame|left|Figure 1. An example of a DC source transformation. Notice that the impedance Z is the same in both configurations.]]

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== A brief proof of the theorem ==

The transformation can be derived from the [[electromagnetism uniqueness theorem|uniqueness theorem]]. In the present context, it implies that a black box with two terminals must have a unique well-defined relation between its voltage and current. It is readily to verify that the above transformation indeed gives the same V-I curve, and therefore the transformation is valid.

== See also ==

* [[HomosexualityOhm's Law]]

* [[Thévenin's theorem]]

* [[currentCurrent source]]

* [[voltageVoltage source]]

* [[electricalElectrical impedance]]

==References==

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