en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-28T13:49:51Z

References: Fix CS1 deprecated date parameter errors

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	~~Hello~~ and ~~welcome~~. My title ~~is Irwin and~~ I ~~completely dig that~~ title. ~~My working day occupation~~ is a ~~librarian~~. ~~Her spouse~~ and ~~her live~~ in ~~Puerto Rico~~ but ~~she will have~~ to ~~transfer one working~~ day ~~or another~~. To ~~play baseball~~ is the ~~hobby he will never stop performing~~.<br><br>~~My webpage :: [http:~~/~~/youulike~~.~~com~~/~~blogs~~/~~114396~~/~~390615/items~~-to-~~know-~~when-~~confronted-wi std testing at home~~]		'''Harmonic balance''' is a method used to calculate the [[Steady state (electronics)\|steady-state response]] of non-linear differential equations, and is mostly applied to electrical circuits
			<ref>{{cite journal
			\|last = Gilmore \|first=R. J. \|coauthors=Steer, M. B. \|title=Nonlinear circuit analysis using the method of harmonic balance—A review of the art. Part I. Introductory concepts \|journal=Int. J. Microw. Mill.-Wave Comput.-Aided Eng.
			\|volume=1 \|pages=22–37 \|year=1991 \|doi=10.1002/mmce.4570010104}}</ref>
			.<ref>{{cite journal
			\| last = Nakhla
			\| first = Michel S.
			\| authorlink =
			\| coauthors = Vlach, Jiri
			\| title = A piecewise harmonic balance technique for determination of periodic response of nonlinear systems
			\| journal = IEEE Transactions on Circuits and Systems,
			\| volume = CAS-23
			\| pages = 85–91
			\| date = February 1976
			\| issn = 0098-4094 }}
			</ref> It is a ''frequency domain'' method for calculating the steady state, as opposed to the various ''time-domain'' steady state methods. The name "harmonic balance" is descriptive of the method, which starts with Kirchhoff's Current Law written in the frequency domain and a chosen number of harmonics. Effectively the method assumes the solution can be represented by a linear combination of sinusoids, then balances current and voltage sinusoids to satisfy Kirchhoff's law. The method is commonly used to simulate circuits which include [[nonlinear]] elements.<ref>{{cite book
			\| last = Maas
			\| first = Stephen A.
			\| title = Nonlinear microwave and RF circuits
			\| publisher = Artech House
			\| year = 2003
			\| isbn = 1-58053-484-8 }}</ref>

			Microwave circuits were the original application for harmonic balance methods in electrical engineering. Microwave circuits were well-suited because, historically, microwave circuits consist of many linear components which can be directly represented in the frequency domain, plus a few nonlinear components. System sizes were typically small. For more general circuits, the method was considered impractical for all but these very small circuits until the mid-1990s, when Krylov subspace methods were applied to the problem.<ref>{{cite journal
			\|author=Feldmann, P.; Melville, B.; Long, D.
			\|title= Efficient frequency domain analysis of large nonlinear analog circuits
			\|journal=Proceedings of the Custom Integrated Circuits Conference
			\|year=1996
			\|pages=461–464
			\|doi = 10.1109/CICC.1996.510597
			\|isbn=0-7803-3117-6}}</ref><ref>{{cite journal
			\|author=Brachtendorf, H.G.; Welsch, G.; Laur, R.
			\|title=Fast simulation of the steady-state of circuits by the harmonic balance technique
			\|journal=Proceedings, International Symposium on Circuits and Systems
			\|pages=1388
			\|year=1995
			\|doi=10.1109/ISCAS.1995.520406
			\|volume=2
			\|isbn=0-7803-2570-2}}</ref>
			The application of preconditioned Krylov subspace methods allowed much larger systems to be solved, both in size of circuit and in numbers of harmonics. This made practical the present-day use of harmonic balance methods to analyze radio-frequency integrated circuits (RFICs).

			==Algorithm==

			The algorithm starts with Kirchhoff's current law written in the [[frequency-domain]]. To increase the efficiency of the procedure, the circuit may be partitioned into its linear and nonlinear parts, since the linear part is readily described and calculated using [[nodal analysis]] directly in the frequency domain.

			First, an initial guess is made for the solution, then an iterative process continues:

			# Voltages <math>V</math> are used to calculate the currents of the linear part, <math>I_\mbox{linear}</math> in the frequency domain.
			# Voltages <math>V</math> are then used to calculate the currents in the nonlinear part, <math>I_\mbox{nonlinear}</math>. Since nonlinear devices are described in the time domain, the frequency-domain voltages <math>V</math> are transformed into the time domain, typically using inverse fast Fourier transforms. The nonlinear devices are then evaluated using the time-domain voltage waveforms to produce their time-domain currents. The currents are then transformed back into the frequency domain.
			# According to [[Kirchhoff's current law\|Kirchhoff's circuit laws]], the sum of the currents must be zero, <math>\epsilon = I_\mbox{linear} + I_\mbox{nonlinear} = 0</math>. An iterative process, usually [[Newton iteration]], is used to update the network voltages <math>V</math> such that the current residual <math>\epsilon</math> is reduced. This step requires formulation of the [[Jacobian matrix and determinant\|Jacobian]] <math>\tfrac{d\epsilon}{dV}</math>.

			Convergence is reached when <math>\epsilon</math> is acceptably small, at which point all voltages and currents of the steady-state solution are known, most often represented as Fourier coefficients.

			==References==
			<references />

			{{DEFAULTSORT:Harmonic Balance}}
			[[Category:Electronic design]]
			[[Category:Electronic circuits]]
			[[Category:Electrical engineering]]

en>RJHall: Format

2012-08-21T20:31:47Z

Format

New page

Hello and welcome. My title is Irwin and I completely dig that title. My working day occupation is a librarian. Her spouse and her live in Puerto Rico but she will have to transfer one working day or another. To play baseball is the hobby he will never stop performing.<br><br>My webpage :: [http://youulike.com/blogs/114396/390615/items-to-know-when-confronted-wi std testing at home]

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en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

en>RJHall: Format