The adaptor equations for a connection of ports, in either the series (2.31) or parallel (2.32) case, may be written as

where and , and where we have

Series adaptor (voltage waves) | (2.38) | |||||

Parallel adaptor (voltage waves) | (2.39) |

Here is an vector containing all ones, is the identity matrix, and and are defined by

The sum of the elements of either or is 2. For power wave variables, we have a similar relationship,

where

Series adaptor (power-normalized waves) | ||||||

Parallel adaptor (power-normalized waves) |

Here the square root sign indicates an entry-by-entry square root of a vector (all entries of and are non-negative).

Defining the Euclidean norm of a column vector as , it is easy to show that a power normalized scattering matrix is norm-preserving in either the series or parallel case, i.e., we have

For voltage waves, we have the preservation of a weighted norm, i.e.,

where ; in this case, is an positive definite diagonal matrix simply given by diag . It should be clear that (2.41) and (2.42) are merely re-statements of power conservation at a memoryless, lossless -port.

Note that multiplying or by a vector requires, in either the series of parallel case, adds and multiplies; in particular, it is cheaper than a full matrix multiply.