Solved Numerical 6

Step 1: Check Reciprocity

A two-port network is reciprocal if:

$ S_{12}=S_{21} $

From the given matrix:

$ S_{12}=j0.6 $

$ S_{21}=j0.6 $

Since:

$ S_{12}=S_{21} $

the scattering matrix is symmetric.

$ [S]=[S]^T $

Therefore:

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

Step 2: Check Whether the Network is Lossless

For a lossless two-port network:

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

A quick test is:

$ |S_{11}|^2+|S_{21}|^2=1 $

Calculate |S₁₁|

$ |S_{11}|=\sqrt{(0.4)^2+(0.5)^2} $

$ =\sqrt{0.16+0.25} $

$ =\sqrt{0.41} $

$ =0.6403 $

Therefore:

$ |S_{11}|^2=0.41 $

Calculate |S₂₁|²

$ |S_{21}|^2=(0.6)^2 $

$ =0.36 $

Apply the Lossless Condition

$ |S_{11}|^2+|S_{21}|^2 $

$ =0.41+0.36 $

$ =0.77 $

Since:

$ 0.77\neq1 $

the network does not satisfy the lossless condition.

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

Therefore, it is a lossy network.

Step 3: Calculate Return Loss

Return loss is given by:

$ RL=-20\log_{10}|S_{11}| $

Substituting:

$ RL=-20\log_{10}(0.6403) $

$ RL=3.87\text{ dB} $

Therefore:

$ \boxed{RL=3.87\text{ dB}} $

Step 4: Calculate Reflected Power

Given:

$ P_i=5\text{ W} $

Reflected power is:

$ P_r=|\Gamma|^2P_i $

Since:

$ |\Gamma|=|S_{11}|=0.6403 $

then:

$ P_r=(0.6403)^2(5) $

$ =0.41\times5 $

$ =2.05\text{ W} $

Therefore:

$ \boxed{P_r=2.05\text{ W}} $

Step 5: Calculate Transmitted Power

Using power conservation:

$ P_i=P_r+P_t $

Therefore:

$ P_t=5-2.05 $

$ P_t=2.95\text{ W} $

Hence:

$ \boxed{P_t=2.95\text{ W}} $

Final Answers

• Reciprocal: Yes
• Lossless: No
• Return Loss: 3.87 dB
• Reflected Power: 2.05 W
• Transmitted Power: 2.95 W

Exam Tip

For quick lossless verification in IOE, IIT, B.Tech Electronics, CSIT, and Microwave Engineering examinations, remember:

$ |S_{11}|^2+|S_{21}|^2=1 $

and

$ |S_{22}|^2+|S_{12}|^2=1 $

If either condition is violated, the network cannot be lossless.