### General Conditions for Losslessness

The scattering matrices for lossless physical waveguide junctions give an apparently unexplored class of lossless FDN prototypes. However, this is just a subset of all possible lossless feedback matrices. We are therefore interested in the most general conditions for losslessness of an FDN feedback matrix. The results below are adapted from [463,385].Consider the general case in which is allowed to be any scattering matrix,

*i.e.*, it is associated with a not-necessarily-physical junction of physical waveguides. Following the definition of losslessness in classical network theory, we may say that a waveguide scattering matrix is said to be

*lossless*if the

*total complex power*[35] at the junction is scattering invariant,

*i.e.*,

where is any Hermitian, positive-definite

*matrix*(which has an interpretation as a generalized junction admittance). The form is by definition the square of the

*elliptic norm*of induced by , or . Setting , we obtain that must be unitary. This is the case commonly used in current FDN practice. The following theorem gives a general characterization of lossless scattering:

**Theorem**: A scattering matrix (FDN feedback matrix) is lossless if and only if its eigenvalues lie on the unit circle and its eigenvectors are linearly independent.

*Proof*: Since is positive definite, it can be factored (by the Cholesky factorization) into the form , where is an upper triangular matrix, and denotes the Hermitian transpose of ,

*i.e.*, . Since is positive definite, is nonsingular and can be used as a similarity transformation matrix. Applying the Cholesky decomposition in Eq.(C.125) yields

Thus, lossless scattering matrices may be fully parametrized as , where is any unit-modulus diagonal matrix, and is any invertible matrix. In the real case, we have diag and . Note that not all lossless scattering matrices have a simple

*physical*interpretation as a scattering matrix for an intersection of lossless reflectively terminated waveguides. In addition to these cases (generated by all non-negative branch impedances), there are additional cases corresponding to sign flips and branch permutations at the junction. In terms of classical network theory [35], such additional cases can be seen as arising from the use of ``gyrators'' and/or ``circulators'' at the scattering junction [433]).

**Next Section:**

Gyrators

**Previous Section:**

Normalized Scattering