A two port network has the following S parameters when the measurements are made at the terminal planes t₁ and t₂.

1. Find whether the network is lossless or not.
2. If the terminal plane of the output port is moved from t₂ to t₃ over a length of b₂d₂=145^o, find the new scattering matrix with regards to the teminal planes t₁ and t₃.

1. Solution I:

   The [S] matrix has to satisfy the unitary condition, that is:

   

   Applying these to the given [S] parameters (first equation)

   

   Also (second equation)

   

   Neither equation is satisfied. Therefore: the network is lossy (First equation would have been sufficient since it did not satisfy the Unitary condition to start with.)

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2. Solution II:

   
   In order to find the new scattering matrix we have to use the transformation matrix (minus sign is for outward displacement of the ports)

   

   For the given problem since only t₂ is moved to t₃, d₁=0, we set e^-jb₁d₁=1. Using the transformation procedure, we find

   

   Note: the arrows show the steps in following the matrix multiplications.
   As expected: S₁₁ did not change. Phases of S₁₂ and S₂₁ have changed by 145^o and the phase of S₂₂ has changed by 2(145^o)