Talk:Reciprocity (electrical networks)

Draft notes

The reciprocity theorem is a theorem in electrical network analysis.

==Proof==

Reciprocity theorem (pp.148-153)

starting with

$[i_{s}]=[g][e]$

and assuming symmetry of [g] (ie the equilibrium equations refer to the same node-pair voltages as used in the topological development pp.79-81)

Solve for n=2 and setting i_s1=0. Then solve for i_s2=0. Then equate using the symmetry condition yields,

${e_{2} \over i_{s1}}{\bigg |}_{i_{s2}=0}={e_{1} \over i_{s2}}{\bigg |}_{i_{s1}=0}$

To prove generally start with general matrix [G]. Eliminate the node-pair e_n. From the g_nn multiplied sth row subtract the g_nn multiplied last row for s=1, 2, ..., n-1. The resulting matrix is also symmetrical if the original was symmetrical. Repeated application results in a symmetrical 2x2 matrix.

The dual argument applies to loop basis impedance matrix.

Can also be proved with determinants (p.150). If the determinant, A, is symmetrical then the cofactors are also symmetrical and in x_k=(A_sk/A)y_s (p.120) for one source, the s and k can be interchanged.

Cramer's rule applied when there are more than one source the solution for x_k is the sum of these for s=1,2 etc (p.120). Does this detract from the generality of the argument on p.150?

==Bibliography==

• Guillemin, Ernst A., Introductory Circuit Theory, New York: John Wiley & Sons, 1953 OCLC 535111

The above is my draft notes of a potential article at this title, but I then decided to write something elsewhere and make this a redirect. Parking it here on the talk page in case it is of use in the future. SpinningSpark 16:23, 2 August 2020 (UTC)

Mutual inductance and reciprocity

User:Alej27 removed mutual inductance (M) from the list of elements that are guaranteed to form a reciprocal network (R). The edit summary read,

When there are coupled inductors, each one can be modelled as current-dependent voltage sources in series with a non-magnetically coupled inductor (read Charles & Sadiku's or Dorf & Svoboda's textbook.) And, a circuit with dependent sources may or may not have a symmetric impedance or admittance matrix (read K. S. Suresh Kumar's textbook.

It is true that coupled inductors can be modelled with dependent sources (D). It is also true that non-reciprocal networks can be modelled with dependent sources (the gyrator immediately springs to mind). However, putting those two together to conclude that not all mutual inductances are reciprocal is a logical fallacy (Fallacy of the undistributed middle),

All M are D

Some D are ¬R

Some M are ¬R

is a false conclusion. SpinningSpark 13:28, 21 August 2020 (UTC)

@User:SpinningSpark: Brilliant, thanks for pointing out the mistake. Are ideal two-winding transformers (the ones studied in basic circuit analysis textbooks) also elements that guarantee reciprocal networks (or networks with symmetrical impedance/admittance matrices)? I think yes, but I'd appreciate a confirmation. --Alej27 (talk) 20:17, 22 December 2020 (UTC)

@Alej27: Yes, ideal transformers are reciprocal, as is any network with a symmetrical impedance matrix. Two-port parameters lists the reciprocity condition for all the various forms of parameter matrix. Sorry I took so long to reply, I was on a six-month sabbatical from Wikipedia, but your ping to me wouldn't have worked anyway because you added it after committing the post. SpinningSpark 21:23, 27 July 2021 (UTC)

@SpinningSpark: In fact, an ideal transformer has neither an impedance matrix nor an admittance matrix. However, an ideal transformer has a parallel-augmented impedance matrix and a series-augmented admittance matrix, which are both symmetric. This allows us to conclude that the ideal transformer is reciprocal. FreddyOfMaule (talk) 08:32, 28 July 2023 (UTC)

References

Reciprocal relations between some power ratios

@SpinningSpark: Hello. If we consider a reciprocal and passive 2-port, several reciprocal relations exist between some power ratios. For instance: the transducer power gain in the forward direction is equal to the transducer power gain in the reverse direction; and the insertion power gain in the forward direction is equal to the insertion power gain in the reverse direction. In contrast, the operational power gain in the forward direction need not be equal to the operational power gain in the reverse direction. This is explained in . Being one of the authors of this article, I cannot introduce an addition which would cite the article (to avoid a conflict of interest). But, if you think that this is relevant, you could do it, and I can help you. FreddyOfMaule (talk) 09:24, 24 July 2023 (UTC)

References