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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