asymmetric_junct

asymmetric_junct - Ideal asymmetric junction elements Relax...

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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 = u 1 u 2 (A.1) v = v 1 v 2 (A.2) The input and output power flows are the square of the length of these vectors, their inner products. P in = i=1 2 u i 2 = u t u (A.3) P out = i=1 2 v i 2 = v t v (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 v must 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 u and v , the algebraic relation f ( . ) is equivalent to a rotation operator . v = S ( u ) u (A.6) where the square matrix S is known as a scattering matrix . Mod. Sim. Dyn. Sys. page 1 Neville Hogan
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v 1 2 v u 1 u 2 u v S need not be a constant matrix, but may in general depend on the power flux through the junction, hence the notation S ( u ). However, S is subject to important restrictions. In particular, v t v = u t S t Su = u t u (A.7) S is an orthogonal matrix: the vectors formed by each of its rows (or columns) are (i) orthogonal and (ii) have unit magnitude; its transpose is its inverse. S t S = 1 (A.8) This constrains the coefficients of the scattering matrix as follows. S = ab cd (A.9) a 2 + c 2 = 1 (A.10) ab + cd = 0 (A.11) b 2 + d 2 = 1 (A.12) Mod. Sim. Dyn. Sys. page 2 Neville Hogan
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As there are only three independent equations and four unknown quantities, we see that this junction is characterized by a single parameter.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.141 taught by Professor Nevillehogan during the Fall '06 term at MIT.

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asymmetric_junct - Ideal asymmetric junction elements Relax...

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