General Method for Conversion Between Multimode Network Parameters
Alexander Zhuravlev, Juan D. Baena
TL;DR
This paper addresses the need for general conversion rules between multimode network parameters (e.g., $S$, $T$, $ABCD$, $Z$, $Y$, $h$) by formulating a general algebraic method rooted in a state-vector space, where each conversion is realized as a simple change of basis. The approach defines a $2M$-dimensional state space and six corresponding bases linked to the standard network matrices, yielding a universal conversion formula: $$oldsymbol{N} = igl[ ext{Upper half of }oldsymbol{M} ext{-basis expressed in }oldsymbol{N} ext{-coordinates}igr] imes igl[ ext{Lower half of }oldsymbol{M} ext{-basis expressed in }oldsymbol{N} ext{-coordinates}igr]^{-1}.$$ This framework enables explicit, compact conversion rules (potentially up to ~30) across multimode parameters and highlights how different normalizations of voltages/currents alter the rules, while remaining extensible to new definitions of network matrices. The work provides a practical, uniform methodology for multimode network analysis, especially relevant for densely packed structures with higher-order modes.
Abstract
Different types of network parameters have been used in electronics since long ago. The most typical network parameters, but not the only ones, are $S$, $T$, $ABCD$, $Z$, $Y$ , and $h$ that relate input and output signals in different ways. There exist practical formulas for conversion between them. Due to the development of powerful software tools that can deal efficiently and accurately with higher-order modes in each port, researchers need conversion rules between multimode network parameters. However, the usual way to get each conversion rule is just developing cumbersome algebraic manipulations which, at the end, are useful only for some specific conversion. Here, we propose a general algebraic method to obtain any conversion rule between different multimode network parameters. It is based on the assumption of a state vector space and each conversion rule between network parameters can be interpreted as a simple change of basis. This procedure explains any conversion between multimode network parameters under the same algebraic steps.