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Ideal asymmetric junction elements
Relax the symmetry assumption and examine the resulting junction structure. For
simplicity, consider twoport 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 realvalued wavescattering 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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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
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.
 Fall '06
 NevilleHogan

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