Reciprocity Theorem

RECIPROCIT RECIPR OCITY Y THEOREM

STATEMENT AND DEFINITION

 Reciprocity theorem states that if an emf E in one branch of a reciprocal network produces a current I in another, then if the emf E is moved from the first to the second branch, it will cause the same current in the first branch, where the emf has been replaced by a short circuit.  Reciprocal network means any network composed of linear, bilateral elements. 

APPLICABILITY

 Applicable to single-source networks.

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Network must have passive, linear and bilateral elements. Network must have absence of independent source.

IMPACT OF THE RECIPROCITY THEOREM

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The theorem by the above circuit suggests that polarity of the voltage source have the same correspondence disection of the branch current in each position.

PROOF OF THE RECIPROCITY THEOREM 

Consider the network below in which values for the elements of fig(a) of the previous slide have been assigned.

To find the current I due to a source time proceed as:  The total resistance is:  R  =R  + R  //( R  + R  ) T 1 2 3 4 = 12Ω+6 Ω //(2 Ω +4 Ω) =12Ω+6 Ω//6 Ω =12Ω+3 Ω =12Ω 

 And

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I5 =I/RT =45V/15Ω =3A With I= 3A/2 = 1.5A

PROOF OF THE RECIPROCITY THEOREM

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Interchanging the location of E and I of previous slide, we have the network as:

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Now

RT =R4 +R3 +R1 //R2 =4 Ω+2 Ω+12

Ω//6 Ω

=10 Ω  And I5 =I/RT =45 V/10 That I=(6

Ω=4.5A

Ω)(4.5A)/12 Ω+6 Ω

=1.5A Which agrees with the previous slides.

RECIPROCITY THEOREM, A THEOREM OR NOT… 

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The reciprocity theorem does not appear in many of the textbooks, such as, Fundamentals of Electric circuit by Alexander and M.Sadika Engineering Circuit Analysis by William H. Hayt & Jack E. Kemmerly

because it is applicable only to single-source networks. It is, therefore, not a theorem used in the analysis of multisource networks.

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But, the theorem is important because:

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It does not hold for all networks

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Utility of the theorem lies in transformers, electromagnetic fields and microwaves.