Star Delta Conversion Calculator

The star-delta transformation replaces a three-terminal resistor network with an electrically indistinguishable one. Any circuit attached to the three terminals sees identical terminal voltages and currents before and after the substitution, which is what makes it legal; only the internal node structure changes. Going delta to star, each star arm is the product of the two delta arms touching that terminal divided by the sum of all three. Going the other way, each delta arm is the sum of the three pairwise products of the star arms divided by the opposite star arm. Every star resistance is smaller than either adjacent delta resistance, and in the balanced case the two networks differ by exactly a factor of three, so a 30 ohm delta becomes a 10 ohm star. The technique earns its keep on topologies that no amount of series-parallel reduction will simplify, most famously the bridge network, where converting one triangle into a star leaves a circuit that collapses by inspection. The same algebra applies to complex impedances at a single frequency if R is replaced by Z, though the arithmetic then becomes complex-valued and is only valid at that one frequency. Two cautions apply in practice. The transformation preserves nothing about internal node voltages or branch currents, so it cannot be used to find them directly. And in power apparatus the two configurations are not physically interchangeable, because only the star connection provides a neutral point for grounding and for single-phase loads.

The transformation is generally credited to Arthur Edwin Kennelly, a British-born engineer who worked for Thomas Edison before teaching at Harvard and MIT, and who published the equivalence of triangles and three-pointed stars in conducting networks in Electrical World and Engineer in 1899. Kennelly was working on telephone and telegraph line problems, where multi-terminal networks that resist series-parallel reduction are common, and his short paper gave the substitution rules in essentially the form still used today. The result is sometimes called the Kennelly theorem, and in German-language texts the star-triangle or Stern-Dreieck transformation. The names come from the drawings: the wye or star arrangement resembles the letter Y, and the delta resembles the Greek capital letter of that name. The transformation gained a second life in three-phase power engineering, where generator and load windings are genuinely wired in one configuration or the other, and where the factor-of-three impedance relationship underpins the star-delta motor starter that became standard industrial practice in the early twentieth century. Starting an induction motor in star applies phase voltage reduced by the square root of three, cutting inrush current to roughly a third of its direct-on-line value, after which a contactor reconnects the windings in delta for normal running. The same 1899 algebra generalizes further to the star-mesh conversions used in modern network reduction and in power system fault studies.