Y-Δ and Δ-Y Transformations Revisited

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The Y-Δ and Δ-Y transformations form a part of all courses on circuit theory. The relevant formulas are usually derived by equating the resistance between the same pair of nodes in the two networks. It has been pointed out that when the network forms part of a larger closed circuit, this creates confusion among students because when the resistance between a pair of terminals in the Y-network is found out, the third terminal remains open, i.e. the resistance connected to it does not carry any current. The question raised is the following: is the equivalence valid when all resistances carry current? An alternative derivation has recently appeared in the literature where two voltage and current sources are connected to the networks, with their common terminal grounded. This may again give rise to the following question: will the equivalence be valid when all the terminals have arbitrary, non-zero voltages? This is answered in the affirmative in this article by connecting three current sources in each network, and a simple derivation of the well known formulas is presented.