### 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

where , and

Conversely, assume
for each eigenvalue of
, and
that there exists a matrix
of linearly independent
eigenvectors of
. The matrix
diagonalizes
to give
, where
diag. Taking the Hermitian transform of
this equation gives
. Multiplying, we
obtain
. Thus, (C.125) is satisfied for
which is Hermitian and positive
definite.

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