Network analysis (electrical circuits): Difference between revisions

Revision as of 07:22, 31 January 2023

Revision as of 07:43, 31 January 2023

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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_1</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.

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[[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 {{math|^''n''[[Binomial coefficient|C]]₂}}). 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}

:<math>R_a = \frac{R_\mathrm{ac}R_\mathrm{ab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>

:<math>R_bR_a &= \frac{R_\mathrm{abac}R_\mathrm{bcab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>\\

:<math>R_cR_b &= \frac{R_\mathrm{bcab}R_\mathrm{acbc}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>\\

:<math>R_aR_c &= \frac{R_\mathrm{acbc}R_\mathrm{abac}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>

\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}</math> \\

:<math>R_\mathrm{ab} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_c}</math> \\

:<math>R_\mathrm{bc} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_a}</math>

\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 <math>{{mvar|N</math>}} resistors (<{{math>R_1</math>|(''R''{{sub|1}} ..… <math>R_N</math>''R{{sub|N}}'')}} to nodes {{math|'''1''' ..… '''''N'''''}} can be replaced by <{{math>|{{sup|''N \choose ''}}C{{sub|2}</math>}}} resistors interconnecting the remaining <math>{{mvar|N</math>}} nodes. The resistance between any two nodes <math>{{mvar|x</math>, and <math>y</math>}} 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</math>}}) this reduces to:

:<math>\begin{align}

:<math>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}</math>

For a series reduction (<math>N=2</math>) this reduces to:

&= \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</math>}}) 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</math>}}) it results in the elimination of the resistor because <{{math>|1={{sup|1 \choose }}C{{sub|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>