A Proof of The Existence and Uniqueness of The Transformation
The feasibility of the transformation can be shown as a consequence of superposition theorem in electric circuit. A short proof, rather than derived as a corollary of the more general star-mesh transform, can be given as follows. The equivalence lies in the statement that for any external voltages (, and ) applying at the three nodes (, and ), the corresponding currents (, and ) are exactly the same for both the Y and Δ circuit, and vice versa. In this proof, we start with given external currents at the nodes. According to superposition theorem, the voltages can be obtained by studying the linear summation of the resulting voltages at the nodes of following three problems: apply at the three nodes with current (1), (2), and (3), . It can be readily shown that due to Kirchhoff's circuit laws, one has . One notes that now each problem is relatively simple, since it only involves one single ideal current source. To obtain exactly the same outcome voltages at the nodes for each problem, the equivalent resistances in two circuits must be the same, this can be easily found by using the basic rules of series and parallel circuits. Considering all three problems, one results in six equations which connect the three resistances in the Y circuit and the three resistance in the Δ circuit, namely,
Though usually six equations are more than enough to express three variables in term of the other three variables, here it is straightforward to show that these equations indeed lead to the above designed expressions. In fact, superposition theorem not only establishes the relation between the values of the resistances, but also guarantees the uniqueness of such solution.
Read more about this topic: Y-Δ Transform
Famous quotes containing the words proof, existence and/or uniqueness:
“The proof of a poet is that his country absorbs him as affectionately as he has absorbed it.”
—Walt Whitman (1819–1892)
“As for types like my own, obscurely motivated by the conviction that our existence was worthless if we didn’t make a turning point of it, we were assigned to the humanities, to poetry, philosophy, painting—the nursery games of humankind, which had to be left behind when the age of science began. The humanities would be called upon to choose a wallpaper for the crypt, as the end drew near.”
—Saul Bellow (b. 1915)
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)