Discussion
Arrange the following S-parameters of E-plane in increasing order of value :
(a) S12
(b) S13
(c) S23
(d) S33
Answer: D
The increasing order is S23, S33, S12, S13, which is option D: (c), (d), (a), (b).
For the ideal lossless E-plane (series) tee, with port 3 the arm that joins the broad wall, the standard scattering matrix is:
\[ S = \begin{bmatrix} 1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & 1/2 & -1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{bmatrix} \]
So S23 = -0.707 (S13 and S23 are 180 degrees out of phase), S33 = 0 (the side arm is matched), S12 = 0.5 and S13 = 0.707, giving -0.707 < 0 < 0.5 < 0.707.
Options A and C put S33 or S12 below S23 and so ignore its negative sign. Option B puts S23 last, which is its order by magnitude, not by value. The question rests on this ideal-junction matrix, which is the form textbooks give; the key agrees with it.
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