The Quadratic Invariances of a Generalized Network

The Manley-Rowe relation, as applied in the small signal linearized approximation, may be stated as a quadratic form that is invariant under the operation of the system. It is, however, only one of the set of such forms that is invariant through a given type of system. It is shown that the existence of quadratic invariances is a consequence of the eigenvalues of the system operator being either of unit magnitude or else grouped in pairs such that one is the conjugate reciprocal of the other. If this condition applies, then there exists at least n such linearly independent forms, where n is the number of degrees of freedom of the system. Each form then specifies a quantity that is conserved by the system. Methods of determining the quadratic invariant forms from the matrix operation of the system are developed. Application is made to certain simple two-port networks to illustrate the analysis and the significance of the resulting invariances. Parametric circuits are also studied. The Manley-Rowe relation is found, as expected. Other relations, applicable to subclasses of such networks are also found. Finally, application is made to a lossy parametric shunt element, such as an imperfect nonlinear capacity. The quadratic invariances for such a device, for the two-frequency case, are derived.