# Under what conditions is the star-mesh transform invertible?

We all know and love the Δ-Y (delta-wye) and Y-Δ (wye-delta) transforms for simplifying three-resistor networks: Image from Creative Commons

The Δ-Y and Y-Δ transforms have the nice property that a Δ can always be turned into a Y, and a Y can always be turned into a Δ, no matter the value of the resistances involved.

There's a generalised version of the Y-Δ transform called the star-mesh transform. This converts a "star" of $N$ resistors into a "mesh" of $^{N}C_{2}$resistors. Image from Creative Commons

Wikipedia suggests that the star-to-mesh transform will always exist - but that the inverse transform, mesh-to-star, may not exist. To wit:

The transform replaces N resistors with $^{N}C_{2}$ resistors. For N > 3, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints.

What are the constraints that must be satisfied in order for the inverse to exist?

I am particularly interested in converting a 4-node mesh network into a 4-resistor star network.

Motivation for the question: I have an industrial power systems model (really just a very large network of constant-voltage sources and impedances) containing ~ 2,000 nodes. I am attempting to reduce it to just four nodes of interest.

Edit:

There are some published papers on this topic.

• Versfeld, L., "Remarks on star-mesh transformation of electrical networks," Electronics Letters , vol.6, no.19, pp.597,599, September 17 1970

Two new aspects of the well known star-mesh transformation are studied: (a) the necessary and sufficient conditions for the transformation of a given general mesh network into an equivalent star network; (b) an extension to networks containing sources.

• Bapeswara Rao, V.V.; Aatre, V.K., "Mesh-star transformation," Electronics Letters , vol.10, no.6, pp.73,74, March 21 1974

An equivalent star network exists for a given mesh network if the latter satisfies the Wheatstone relaflonship. Using this fact, it is shown that all the offdiagonal cofactors of the datum-node admittance matrix of such a mesh network are equal. From this property, a simple relationship between the elements of the two networks is derived.

I don't have IEEE Xplore access so I can't read them.

• @user26129: This question is in the same vein as the circuit analysis questions that EE.SE already gets tons of. The only unusual part is that it's not undergraduate coursework, and that it's a general question rather than a specific exercise from a textbook. – Li-aung Yip Sep 30 '13 at 9:48
• @Li-aungYip: I'm not disputing the validity of putting your question in EE.SE, but I do believe you will get more and better responses elsewhere. I'm trying to help you get an answer, not trying to get your question downvoted ;) – user36129 Sep 30 '13 at 9:50
• @user26129: Ah! In any case, the desired answer is in the Electronics Letters papers linked - I am trying to get a copy of them so I can read them and post the pertinent parts as an answer here. – Li-aung Yip Sep 30 '13 at 9:52
• @Li-aungYip well, if that is all you need... efficientelectronics.nl/04245011.pdf – user36129 Sep 30 '13 at 9:57
• I did not really get how to compute the various resistances in the mesh net given the star net resistors, but since the number of resistor increases the additional constraints you are looking for should be arbitrary. Solving the equations for the inverse transform lead to a system of equations that has more variables than equations, so you just choose some resistances and then compute the others. – Vladimir Cravero Oct 3 '13 at 6:53

## 2 Answers

For the mesh-star conversion the problem is that you have more equations than variables, so the number of bond $N_b$ is: $N_b=N_e-N_v$, where $N_e$ is the number of equations, equal also to the number of the resistance in the mesh, $N_v$ is the number of variables, equal to the number of the resistance in the star. In the 4-case, I've demonstrated that the bonds for the transformation are $G_{AB}G_{CD}=G_{AC}G_{BD}=G_{AD}G_{BC}$, in other words the products between the resistance without node in common must be the same.

Ps: The "demonstration" is: The formula for the star-mesh transformation is $G_{XY}=\frac{G_XG_Y}{G_{TOT}}$ with $G_{TOT}=\sum_{i=1}^nG_i$. So, assuming $G_{XY}\ne0$, we can divide two of those equation and obtain $\frac{G_X}{G_Y}=\frac{G_{XZ}}{G_{YZ}}$ for every Z different from X or Y. In the 4-case this means 6 equations, one is the following: $\frac{G_A}{G_B}=\frac{G_{AC}}{G_{BC}}=\frac{G_{AD}}{G_{BD}}\Rightarrow G_{AC}G_{BD}=G_{AD}G_{BC}$. We get the same result from: $\frac{G_C}{G_D}=\frac{G_{AC}}{G_{AD}}=\frac{G_{BC}}{G_{BD}}$. From the last 4 equations we obtain $G_{AB}G_{CD}=G_{AD}G_{BC}$ and $G_{AB}G_{CD}=G_{AC}G_{BD}$ and so we finally have the $G_{AB}G_{CD}=G_{AC}G_{BD}=G_{AD}G_{BC}$ condition. So this is a necessary condition. But if the ratio between any two conductances of the mesh is known ,we can express the $G_{TOT}$ depending on only one of those, like $G_{TOT}=G_{A}+G_B+G_C+G_D=G_A(1+\beta+\gamma+\delta)$, where $\beta=\frac{G_B}{G_A}=\frac{G_{BC}}{G_{AC}}=\frac{G_{BD}}{G_{AD}}$, and so on..$\Rightarrow G_{AB}=\frac{G_AG_B}{G_{TOT}}=\frac{G_AG_B}{G_A(1+\beta+\gamma+\delta)}=\frac{G_B}{(1+\beta+\gamma+\delta)}\Rightarrow G_B=G_{AB}(1+\beta+\gamma+\delta)$. With similar calculations we can find all the 4 conductances(resistances) of the star.

I suppose all of this means the condition is also a sufficient condition.

• Are $G_{AB}G_{CD}=G_{AC}G_{BD}=G_{AD}G_{BC}$ necessary conditions, sufficient conditions, or necessary and sufficient conditions? – Li-aung Yip Nov 16 '13 at 13:09

What this is saying (whether it is true or not) is that there exists more than one way of assigning values to a star network of five resistors such that all the configurations appear indistinguishable according to all external "blackbox" measurements of resistance.

The mesh transformation is a red herring here. If the star networks were uniquely determined, then of course there would always be an inverse of any mapping from that network to any other type, back to that network.