Question

A four-port network has the scattering matrix shown below. Is this network lossless?

[S] = [0.178∠90° 0.6∠45° 0.4∠45° 0 ]
[ 0.6∠45° 0 0 0.3∠-45° ]
[0.4∠45° 0 0 0.5∠-45° ]
[ 0 0.3∠-45° 0.5∠-45° 0.3∠-45° ]

Answer 1

A four-port network is not lossless if the sum of the diagonal elements in the scattering matrix is not equal to 1.

Step-by-step explanation:

A Lossless network refers to a network in which no power is dissipated or lost as it propagates through the network. To determine whether a four-port network is lossless, we need to check if the total power entering the network is equal to the total power exiting the network. In this case, we can look at the scattering matrix [S] provided and check if the sum of the diagonal elements is equal to 1. If it is, the network is lossless. If it isn't, the network has some loss.

In this case, the sum of the diagonal elements is 0.178∠90° + 0 + 0.5∠-45° + 0.3∠-45° which is approximately equal to 0.478∠90°. Since this value is not equal to 1, the network is not lossless.