Wave-scattering approaches to conservation and causality

Wave-scattering formalisms for multiport energetic systems

“…= ~ is the characteristic impedance, with Z/and I') the per unit length series impedance and shunt admittance, respectively. The scattering relationship is then found by rewriting the hyperbolic functions as equivalent exponential functions, and then transforming the W-line wave matrix, Tw to a scattering matrix (e.g., see (11), p. 300, or (2) In those cases where waves travel at different speeds in each direction, the W-line is described by an asymmetric scattering matrix (2).…”

“…These building blocks are used in both analysis and synthesis, especially for systems where distinct elements having a specified constitutive behavior can be defined. The (normalized) scattering matrix has properties that restrict these defined elements and any system model to be physically realizable (2,12). As not all physical systems lend themselves to reticulation into elements and junctions, a more generally applicable scattering operator forms the basis for our development.…”

“…Equation (2) relates homogeneous (in a unit basis sense) root-power conjugate variables (e, f) to the wave-scattering variables (-ff,'~). This power factorization also reveals that the normalization by v~ commonly used in classical scattering representations (11-13) has a fundamentally physical basis.…”

Flatness and passivity from a bond graph

“…where uitg and ulr3 are the incident and reflected wave variables respectively [20]. Then, tlhe average power absorbed by the n-port is:…”

“…We choose Kamel and Dauphin-Tanguy's method because of its simplicity, it is base on identification of series and :parallel structures (0 and 1 junctions with their related l R,C, and I elements associated) and finally, wave matrices [20] are used to obtain the final scattering matrix of the interconnected structures.…”