Solved Numerical 8

a) Lossless Network Check

A multiport network is lossless only if the scattering matrix satisfies the unitary condition:

$ [S]^\dagger[S]=[I] $

A quick test is to verify whether the sum of squared magnitudes of any row equals one.

For Row 1:

$ \sum_{j=1}^{4}|S_{1j}|^2 = |S_{11}|^2+|S_{12}|^2+|S_{13}|^2+|S_{14}|^2 $

$ =(0.1)^2+(0.8)^2+(0.3)^2+(0)^2 $

$ =0.01+0.64+0.09+0 $

$ =0.74 $

$ 0.74\neq1 $

Result:

$ \boxed{\text{The network is not lossless}} $

This means some power is dissipated within the network and the structure behaves as a lossy network.

b) Reciprocal Network Check

A network is reciprocal if:

$ S_{ij}=S_{ji} $

Comparing corresponding matrix elements:

$ S_{12}=S_{21}=0.8\angle-45^\circ $

$ S_{13}=S_{31}=0.3\angle-45^\circ $

$ S_{24}=S_{42}=0.6\angle-45^\circ $

$ S_{34}=S_{43}=0.4\angle45^\circ $

Therefore:

$ [S]=[S]^T $

Result:

$ \boxed{\text{The network is reciprocal}} $

c) Return Loss at Port 1

All remaining ports are terminated with matched loads.

For a matched termination:

$ \Gamma_L=0 $

The input reflection coefficient becomes:

$ \Gamma_{in}=S_{11} $

$ \Gamma_{in}=0.1\angle90^\circ $

Return loss is:

$ RL=-20\log_{10}|\Gamma_{in}| $

$ RL=-20\log_{10}(0.1) $

$ RL=20\text{ dB} $

Result:

$ \boxed{RL=20\text{ dB}} $

d) Insertion Loss and Phase Delay Between Port 2 and Port 4

When all unused ports are matched, the transmission coefficient is taken directly from the S-matrix.

$ S_{42}=0.6\angle-45^\circ $

Note: The original solution used 0.4∠45°, which corresponds to S₄₃. For transmission between Port 2 and Port 4, the correct parameter is S₄₂.

Insertion Loss

$ IL=-20\log_{10}|S_{42}| $

$ IL=-20\log_{10}(0.6) $

$ IL=4.44\text{ dB} $

Phase Delay

$ \angle S_{42}=-45^\circ $

Therefore:

$ \text{Phase Delay}=45^\circ $

Result:

$ \boxed{IL=4.44\text{ dB}} $

$ \boxed{\text{Phase Delay}=45^\circ} $

e) Reflection Coefficient at Port 1 When Port 3 is Short-Circuited

Termination conditions:

• Port 2 matched
• Port 4 matched
• Port 3 short-circuited

For a short circuit:

$ \Gamma_L=-1 $

The input reflection coefficient is:

$ \Gamma_{in} = S_{11} + \frac{S_{13}S_{31}\Gamma_L} {1-S_{33}\Gamma_L} $

Substituting values:

$ \Gamma_{in} = 0.1\angle90^\circ + \frac{(0.3\angle-45^\circ)(0.3\angle-45^\circ)(-1)} {1-(0)(-1)} $

Simplifying:

$ (0.3\angle-45^\circ)(0.3\angle-45^\circ) = 0.09\angle-90^\circ $

$ 0.09\angle-90^\circ(-1) = 0.09\angle90^\circ $

Therefore:

$ \Gamma_{in} = 0.1\angle90^\circ + 0.09\angle90^\circ $

$ \Gamma_{in} = 0.19\angle90^\circ $

Rectangular form:

$ \Gamma_{in}=j0.19 $

Result:

$ \boxed{\Gamma_{in}=0.19\angle90^\circ} $

Final Answers

• Lossless: No (row power sum = 0.74 ≠ 1)
• Reciprocal: Yes (S = S^T)
• Return Loss at Port 1: 20 dB
• Insertion Loss (Port 2 → Port 4): 4.44 dB
• Phase Delay (Port 2 → Port 4): 45°
• Reflection Coefficient at Port 1 with Port 3 Shorted: 0.19∠90°

Exam Takeaway

• For a reciprocal multiport network: $ [S]=[S]^T $
• For a lossless network: $ [S]^\dagger[S]=[I] $
• For any row of a lossless S-matrix: $ \sum |S_{ij}|^2=1 $
• If the row magnitude sum is not equal to 1, the network is lossy.
• When all other ports are matched: $ \Gamma_{in}=S_{ii} $
• Return loss is calculated using: $ RL=-20\log_{10}|S_{ii}| $
• Insertion loss is calculated using: $ IL=-20\log_{10}|S_{ij}| $