Reciprocity, losslessness and return loss from S-matrix

Why S-parameters are used at high frequencies

At microwave frequencies voltages and currents are hard to measure (and not unique in waveguides), and the open/short terminations needed for Z, Y or h parameters are hard to realise and may make active devices oscillate. S-parameters use incident and reflected waves measured with matched loads, they are measured directly by a network analyzer, and they give reflection, transmission and power flow directly.

Given

 S11 = 0.4 + j0.5    S12 = j0.6
 S21 = j0.6          S22 = 0.4 - j0.5

a) Is the network reciprocal?

A network is reciprocal if [S] is symmetric, S12 = S21. Here S12 = j0.6 = S21, so the network is reciprocal.

b) Is the network lossless?

A lossless network has a unitary S-matrix, so each column must satisfy |S11|² + |S21|² = 1.

 |S11|² = 0.4² + 0.5² = 0.16 + 0.25 = 0.41
 |S21|² = 0.6²        = 0.36
 |S11|² + |S21|²      = 0.77  ≠ 1

(The second column also gives 0.41 + 0.36 = 0.77. The orthogonality condition S11·S12* + S21·S22* = 0 happens to hold, but the column sums fail.) So the network is not lossless: 23% of the power fed at port 1 is absorbed in the network.

c) Return loss at the input

 |S11| = √0.41 = 0.6403
 RL = -20 log|S11| = -20 log(0.6403) = 3.87 dB

Answer: return loss at the input = 3.87 dB

d) Reflected power for 5 W input

 Pref = |S11|² · Pin = 0.41 × 5 = 2.05 W

Answer: reflected power = 2.05 W

(For interest: power transmitted to a matched port 2 = 0.36 × 5 = 1.80 W, and power lost in the network = 5 − 2.05 − 1.80 = 1.15 W.)