Ideal asymmetric junction elements Relax the symmetry assumption and examine the resulting junction structure. For simplicity, consider two-port junction elements. As before, assume instantaneous power transmission between the ports without storage or dissipation of energy. Characterize the power flow in and out of a two-port junction structure using four real-valued wave-scattering variables. Using vector notation: u= ⎣⎢⎡⎦⎥⎤u1u2(A.1) v= ⎣⎢⎡⎦⎥⎤v1v2(A.2) The input and output power flows are the square of the length of these vectors, their inner products. Pin= ∑i=12ui2= utu(A.3) Pout= ∑i=12vi2= vtv(A.4) The constitutive equations of the junction structure may be written as follows. v= f(u) (A.5) Geometrically, the requirement that power in equal power out means that the length of the vector vmust equal the length of the vector u, i.e. their tips must lie on the perimeter of a circle (see figure A.1). For any two particular values of uand v, the algebraic relation f(.) is equivalent to a rotation operator. v= S(u) u(A.6) where the square matrix Sis known as a scattering matrix. Mod. Sim. Dyn. Sys. page 1 Neville Hogan
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