Question

A three-port network has the scattering matrix shown below. (a) Is this network lossless? (b) What is the return loss at port 1 when all other ports are matched? (c) What is the scattering matrix between ports 1 and 3, when port 2 is matched? (d) What is the reflection coefficient seen at port 3 if an open circuit is placed at port 1 and port 2 is matched? (e) What is the reflection coefficient seen at port 2 if ports 1 and 3 are connected with an ideal transmission line with an electrical length of 65degree.

0.11∠10°	0.3∠60°	0.2
[ 0.3∠60°	0.12∠ − 40°	0.35∠ − 30°]
0.2	0.35∠ − 30°	0.14∠70°

Answer 1

The given matrix is

a)  

A network is said to be lossless if the unitary property of the scattering matrix is satisfied

Therefore by applying the unitary property in first column,we get

(0.11)² + (0.3)² + (0.2)² = 0.14

As, the unitary property is not satisfied ,hence the network is lossy.

b) Return loss at port 1

When all other ports are matched other than port 1 then no reflection will come from other ports.

Therefore refletction coefficient at port 1 under this condition will be

= S₁₁ = 0.1110⁰

Return loss = -20log| |

= -20 log|0.11|

= 19.17 dB