In electrical engineering, Millman's theorem[1] (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches in parallel.
It is named after Jacob Millman, who proved the theorem.
Let be the generators' voltages. Let be the resistances on the branches with voltage generators . Then Millman states that the voltage at the ends of the circuit is given by:[2]
That is, the sum of the short circuit currents in branch divided by the sum of the conductances in each branch.
It can be proved by considering the circuit as a single supernode.[3] Then, according to Ohm and Kirchhoff, the voltage between the ends of the circuit is equal to the total current entering the supernode divided by the total equivalent conductance of the supernode. The total current is the sum of the currents in each branch. The total equivalent conductance of the supernode is the sum of the conductance of each branch, since all the branches are in parallel.[4]
Branch variationsedit
Current sourcesedit
One method of deriving Millman's theorem starts by converting all the branches to current sources (which can be done using Norton's theorem). A branch that is already a current source is simply not converted. In the expression above, this is equivalent to replacing the term in the numerator of the expression above with the current of the current generator, where the kth branch is the branch with the current generator. The parallel conductance of the current source is added to the denominator as for the series conductance of the voltage sources. An ideal current source has zero conductance (infinite resistance) and so adds nothing to the denominator.[5]
Ideal voltage sourcesedit
If one of the branches is an ideal voltage source, Millman's theorem cannot be used, but in this case the solution is trivial, the voltage at the output is forced to the voltage of the ideal voltage source. The theorem does not work with ideal voltage sources because such sources have zero resistance (infinite conductance) so the summation of both the numerator and denominator are infinite and the result is indeterminate.[6]
Wadhwa, C.L., Network Analysis and Synthesis, New Age International ISBN8122417531'
April 12, 2024
millman, theorem, electrical, engineering, parallel, generator, theorem, method, simplify, solution, circuit, specifically, used, compute, voltage, ends, circuit, made, only, branches, parallel, named, after, jacob, millman, proved, theorem, contents, explanat. In electrical engineering Millman s theorem 1 or the parallel generator theorem is a method to simplify the solution of a circuit Specifically Millman s theorem is used to compute the voltage at the ends of a circuit made up of only branches in parallel It is named after Jacob Millman who proved the theorem Contents 1 Explanation 2 Branch variations 2 1 Current sources 2 2 Ideal voltage sources 3 See also 4 ReferencesExplanation edit nbsp Application of Millman s theoremLet ek displaystyle e k nbsp be the generators voltages Let Rk displaystyle R k nbsp be the resistances on the branches with voltage generators ek displaystyle e k nbsp Then Millman states that the voltage at the ends of the circuit is given by 2 v ekRk 1Rk displaystyle v frac sum frac e k R k sum frac 1 R k nbsp That is the sum of the short circuit currents in branch divided by the sum of the conductances in each branch It can be proved by considering the circuit as a single supernode 3 Then according to Ohm and Kirchhoff the voltage between the ends of the circuit is equal to the total current entering the supernode divided by the total equivalent conductance of the supernode The total current is the sum of the currents in each branch The total equivalent conductance of the supernode is the sum of the conductance of each branch since all the branches are in parallel 4 Branch variations editCurrent sources edit One method of deriving Millman s theorem starts by converting all the branches to current sources which can be done using Norton s theorem A branch that is already a current source is simply not converted In the expression above this is equivalent to replacing the ek Rk displaystyle e k R k nbsp term in the numerator of the expression above with the current of the current generator where the kth branch is the branch with the current generator The parallel conductance of the current source is added to the denominator as for the series conductance of the voltage sources An ideal current source has zero conductance infinite resistance and so adds nothing to the denominator 5 Ideal voltage sources edit If one of the branches is an ideal voltage source Millman s theorem cannot be used but in this case the solution is trivial the voltage at the output is forced to the voltage of the ideal voltage source The theorem does not work with ideal voltage sources because such sources have zero resistance infinite conductance so the summation of both the numerator and denominator are infinite and the result is indeterminate 6 See also editAnalysis of resistive circuitsReferences edit Millman Jacob 1940 A Useful Network Theorem Proceedings of the IRE 28 9 413 417 doi 10 1109 JRPROC 1940 225885 Bakshi amp Bakshi p 7 28 Bakshi amp Bakshi p 3 7 Ghosh amp Chakraborty p 172 Wadhwa p 88 Singh p 64 Bakshi U A Bakshi A V Network Analysis Technical Publications 2009 ISBN 818431731X Ghosh S P Chakraborty A K Network Analysis and Synthesis Tata McGraw Hill 2010 ISBN 0070144788 Singh S N Basic Electrical Engineering PHI Learning 2010 ISBN 8120341880 Wadhwa C L Network Analysis and Synthesis New Age International ISBN 8122417531 Retrieved from https en wikipedia org w index php title Millman 27s theorem amp oldid 1090303996, wikipedia, wiki, book, books, library,
