Parallel Generator Theorem
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In
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, Millman's theorem (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 Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
at the ends of a circuit made up of only branches in parallel. It is named after
Jacob Millman Jacob Millman (1911 in Novohrad-Volynskyi, Russian Empire – May 22, 1991 in Longboat Key, Florida) was a professor of electrical engineering at Columbia University. He immigrated to the United States in 1913 with his mother, Gertrude (Nach ...
, who proved the theorem.


Explanation

Let e_k be the generators' voltages. Let R_k be the resistances on the branches with voltage generators e_k. Then Millman states that the voltage at the ends of the circuit is given by: :v=\frac . That is, the sum of the
short circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the 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. Then, according to
Ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (b ...
and
Kirchhoff Kirchhoff, Kirchoff or Kirchhoffer is a German surname. Notable people with the surname include: * Adolf Kirchhoff (1826–1908), German classical scholar and epigrapher * Alfred Kirchhoff (1838–1907), German geographer and naturalist * Alphonse ...
, 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.


Branch variations


Current sources

One method of deriving Millman's theorem starts by converting all the branches to current sources (which can be done using
Norton's theorem In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of term ...
). A branch that is already a current source is simply not converted. In the expression above, this is equivalent to replacing the e_k/R_k 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.


Ideal voltage sources

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.Singh, p. 64


See also

*
Analysis of resistive circuits A network, in the context of electrical engineering and electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many ...


References

* Bakshi, U.A.; Bakshi, A.V., ''Network Analysis'', Technical Publications, 2009 . * Ghosh, S.P.; Chakraborty, A.K., ''Network Analysis and Synthesis'', Tata McGraw-Hill, 2010 . * Singh, S.N., ''Basic Electrical Engineering'', PHI Learning, 2010 . * Wadhwa, C.L., ''Network Analysis and Synthesis'', New Age International {{ISBN, 8122417531' Electrical engineering Circuit theorems