Note on transformation under power invariance

The voltage transformation, corresponding to a current transformation i = C.i¿, is usually taken as v¿=Ct,.v, for invariance of total instantaneous power. It is shown that this solution is not unique, but is one of an infinite family, having the general form v¿=[I+Zs.Z¿0¿1].Ct v where Zs is any skew-symmetric nonreactive impedance matrix, and Z¿0 is the usual transformed impedance matrix. The theory of this matter is outlined, and its topological significance (corresponding to interconnection with gyrators) is discussed. All the usual properties of invariance apply to this more general solution. Whilst this treatment clarifies a mathematical difficulty, the extended solutions are not apparently of great physical importance.