Editing Network analysis (electrical circuits) (section)

==Equivalent circuits==
[[Image:circuit equivalence.png|200px|right]]
{{main|Equivalent impedance transforms}}
A useful procedure in network analysis is to simplify the network by reducing the number of components.  This can be done by replacing physical components with other notional components that have the same effect.  A particular technique might directly reduce the number of components, for instance by combining impedances in series.  On the other hand, it might merely change the form into one in which the components can be reduced in a later operation.  For instance, one might transform a voltage generator into a current generator using Norton's theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load. 

A [[resistive circuit]] is a circuit containing only [[resistors]], ideal [[current source]]s, and ideal [[voltage source]]s. If the sources are constant ([[Direct current|DC]]) sources, the result is a [[direct current circuit|DC circuit]].  Analysis of a circuit consists of solving for the voltages and currents present in the circuit.  The solution principles outlined here also apply to [[phasor (electronics)|phasor]] analysis of [[#AC circuits|AC circuits]].

Two circuits are said to be '''equivalent''' with respect to a pair of terminals if the [[voltage]] across the terminals and [[Current (electricity)|current]] through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.

If <math>V_2=V_1</math> implies <math>I_2=I_1</math> for all (real) values of {{math|''V''{{sub|1}}}}, then with respect to terminals {{math|ab}} and {{math|xy}}, circuit 1 and circuit 2 are equivalent.

The above is a sufficient definition for a [[one-port]] network.  For more than one port, then it must be defined that the currents and voltages between all pairs of corresponding ports must bear the same relationship. For instance, star and delta networks are effectively three port networks and hence require three simultaneous equations to fully specify their equivalence.

===Impedances in series and in parallel===
{{main|Series and parallel circuits}}
Some two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impedances in series or impedances in parallel.

*Impedances in [[Series and parallel circuits#Series circuits|series]]: <math display="block">Z_\mathrm{eq} = Z_1 + Z_2 + \,\cdots\, + Z_n .</math>
*Impedances in [[Series and parallel circuits#Parallel circuits|parallel]]: <math display="block">\frac{1}{Z_\mathrm{eq}} = \frac{1}{Z_1}  +   \frac{1}{Z_2}  + \,\cdots\, +  \frac{1}{Z_n} .</math>
**The above simplified for only two impedances in parallel: <math display="block">Z_\mathrm{eq} = \frac{Z_1Z_2}{Z_1 + Z_2} .</math>

===Delta-wye transformation===
{{main|Y-Δ transform}}

[[Image:Delta-Star Transformation.svg|right|400px]]

A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit.  An {{mvar|n}}-terminal network can, at best, be reduced to {{mvar|n}} impedances (at worst [[Binomial coefficient|<math>\tbinom{n}{2}</math>]]).  For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or  four node star (Y) network.  These two networks are equivalent and the transformations between them are given below.  A general network with an arbitrary number of nodes cannot be reduced to the minimum number of impedances using only series and parallel combinations.  In general, Y-Δ and Δ-Y transformations must also be used.  For some networks the extension of Y-Δ to [[#General form of network node elimination|star-polygon]] transformations may also be required.

For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations.  The equations below are expressed as resistances but apply equally to the general case with impedances.

====Delta-to-star transformation equations====
:<math>\begin{align}
R_a &=  \frac{R_\mathrm{ac}R_\mathrm{ab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} \\
R_b &=  \frac{R_\mathrm{ab}R_\mathrm{bc}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} \\
R_c &=  \frac{R_\mathrm{bc}R_\mathrm{ac}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} 
\end{align}</math>

====Star-to-delta transformation equations====

:<math>\begin{align}
R_\mathrm{ac} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_b} \\
R_\mathrm{ab} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_c} \\
R_\mathrm{bc} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_a}
\end{align}</math>

===General form of network node elimination===
{{main|Star-mesh transform}}

The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm.  Any node connected by {{mvar|N}} resistors {{math|(''R''{{sub|1}} … ''R{{sub|N}}'')}} to nodes {{math|'''1''' … '''''N'''''}} can be replaced by <math>\tbinom{N}{2}</math> resistors interconnecting the remaining {{mvar|N}} nodes.  The resistance between any two nodes {{mvar|x, y}} is given by:
:<math>R_\mathrm{xy} = R_x R_y\sum_{i=1}^N \frac{1}{R_i}</math>
For a star-to-delta ({{math|1=''N'' = 3}}) this reduces to:
:<math>\begin{align}
R_\mathrm{ab} &= R_a R_b \left(\frac 1 R_a+\frac 1 R_b+\frac 1 R_c\right) = \frac{R_a R_b(R_a R_b + R_a R_c + R_b R_c)}{R_a R_b R_c} \\
&= \frac{R_a R_b + R_b R_c + R_c R_a}{R_c}
\end{align}</math>
For a series reduction ({{math|1=''N'' = 2}}) this reduces to:
:<math>R_\mathrm{ab} = R_a R_b \left(\frac 1 R_a+\frac 1 R_b\right) = \frac{R_a R_b(R_a + R_b)}{R_a R_b} = R_a + R_b</math>
For a dangling resistor ({{math|1=''N'' = 1}}) it results in the elimination of the resistor because <math>\tbinom{1}{2} = 0</math>.

===Source transformation===
[[Image:Sourcetransform.svg|thumb]]

A generator with an internal impedance (i.e. non-ideal generator) can be represented as either an ideal voltage generator or an ideal current generator plus the impedance.  These two forms are equivalent and the transformations are given below.  If the two networks are equivalent with respect to terminals ab, then {{mvar|V}} and {{mvar|I}} must be identical for both networks. Thus,

:<math>V_\mathrm{s} = RI_\mathrm{s}\,\!</math> or <math>I_\mathrm{s} = \frac{V_\mathrm{s}}{R}</math>
* [[Norton's theorem]] states that any two-terminal linear network can be reduced to an ideal current generator and a parallel impedance.
* [[Thévenin's theorem]] states that any two-terminal linear network can be reduced to an ideal voltage generator plus a series impedance.