Basic Electrical Engineering Series
Definition
Rosens Theorem
It is always possible to transform a network of n admittance’s which are connected to a star point (node) from n terminals into a corresponding mesh of admittance’s connecting each pair of terminals. It is however only possible to find a unique transform from mesh to star in the case of three elements.
“ The element of the equivalent mesh between any two terminals p q of a set of k terminals is equal to the ratio of the product of the admittance at p and q and the sum of the k admittance’s ”…
So the mesh element between any two terminals p q is given by:
This follows from Millman’s Theorem.
So from this we can immediately state the Star – Delta Transform which is universally used in three-phase circuit theory. In other words we use Rosen’s theorem with just three terminals.
Star- Delta Transform
Hence from Rosen’s Theorem
Delta - Star Transform
Note that this transform is unique to a three-branch mesh.
In general there are more branches in an equivalent mesh than there are elements in the corresponding star. Thus any arbitrary mesh cannot be replaced by a star since there are a greater number of variables in a mesh that a star. The three-branch mesh or delta is unique and hence the inverse transform exists.