Solved Numerical 7

Step 1: Calculate S₁₁

To determine S₁₁, Port 2 is terminated with a matched load equal to the characteristic impedance:

$ Z_0 = 50\,\Omega $

Input Impedance Calculation

The 50 Ω load at Port 2 is connected in parallel with the existing 50 Ω resistor. The equivalent resistance becomes:

$ R_1 = 50\,\Omega \parallel 50\,\Omega = \left( \frac{1}{50} + \frac{1}{50} \right)^{-1} = 25\,\Omega $

This resistance is in series with the 25 Ω resistor:

$ R_2 = 25\,\Omega + 25\,\Omega = 50\,\Omega $

The resulting 50 Ω branch is then in parallel with the 100 Ω resistor:

$ Z_{in} = 100\,\Omega \parallel 50\,\Omega $

$ Z_{in} = \left( \frac{1}{100} + \frac{1}{50} \right)^{-1} $

$ Z_{in} = \left( \frac{3}{100} \right)^{-1} = 33.33\,\Omega $

Reflection Coefficient at Port 1

The input reflection coefficient is:

$ S_{11} = \Gamma_{in} = \frac{Z_{in}-Z_0}{Z_{in}+Z_0} $

Substituting values:

$ S_{11} = \frac{33.33-50}{33.33+50} $

$ S_{11} = -0.2 $

Therefore:

$ \boxed{S_{11}=-0.2} $

Step 2: Calculate S₂₂

To determine S₂₂, Port 1 is terminated with a matched load:

$ Z_0 = 50\,\Omega $

Input Impedance Seen from Port 2

The 100 Ω resistor at Port 1 is in parallel with the 50 Ω matched load:

$ R_1 = 100\,\Omega \parallel 50\,\Omega $

$ R_1 = 33.33\,\Omega $

This resistance is in series with the 25 Ω resistor:

$ R_2 = 25 + 33.33 $

$ R_2 = 58.33\,\Omega $

This branch is then in parallel with the 50 Ω output resistor:

$ Z_{in} = 50\,\Omega \parallel 58.33\,\Omega $

$ Z_{in} = 26.923\,\Omega $

Reflection Coefficient at Port 2

Using the standard reflection coefficient equation:

$ S_{22} = \Gamma_{out} = \frac{Z_{in}-Z_0}{Z_{in}+Z_0} $

$ S_{22} = \frac{26.923-50}{26.923+50} $

$ S_{22} = -0.3 $

Therefore:

$ \boxed{S_{22}=-0.3} $

Step 3: Calculate S₂₁

To determine the forward transmission coefficient, Port 2 remains terminated in a matched 50 Ω load.

Voltage Wave Relationships

The total voltage at Port 1 is:

$ V_1 = V_1^{+} + V_1^{-} $

Since:

$ V_1^{-} = S_{11}V_1^{+} $

Then:

$ V_1 = V_1^{+}(1+S_{11}) $

Substituting:

$ V_1^{+} = \frac{V_1}{1-0.2} = 1.25V_1 $

Current Relationship

Using the input impedance:

$ V_1 = I_1Z_{in} = 33.33I_1 $

The Port 2 branch resistance is:

$ 50\,\Omega \parallel 50\,\Omega = 25\,\Omega $

Thus:

$ I_2 = \frac{V_1}{50} $

$ I_2 = \frac{33.33I_1}{50} = 0.667I_1 $

The reflected wave voltage at Port 2 becomes:

$ V_2^{-} = 25I_2 $

$ V_2^{-} = 25(0.667I_1) $

$ V_2^{-} = 16.67I_1 $

Similarly:

$ V_1^{+} = 1.25(33.33I_1) $

$ V_1^{+} = 41.6625I_1 $

Forward Transmission Coefficient

Using the S-parameter definition:

$ S_{21} = \frac{V_2^{-}}{V_1^{+}} $

$ S_{21} = \frac{16.67I_1}{41.6625I_1} $

$ S_{21} = 0.4 $

Therefore:

$ \boxed{S_{21}=0.4} $

Step 4: Calculate S₁₂

The network contains only passive linear resistors.

Any passive linear resistive network satisfies the reciprocity condition:

$ S_{12}=S_{21} $

Therefore:

$ S_{12}=0.4 $

Hence:

$ \boxed{S_{12}=0.4} $

Final Scattering Matrix

Combining all four S-parameters:

$ [S] = \begin{bmatrix} -0.2 & 0.4\\ 0.4 & -0.3 \end{bmatrix} $

Therefore, the complete two-port scattering matrix is:

$ \boxed{ [S] = \begin{bmatrix} -0.2 & 0.4\\ 0.4 & -0.3 \end{bmatrix} } $

Exam Takeaway

• S₁₁ represents the input reflection coefficient.
• S₂₂ represents the output reflection coefficient.
• S₂₁ represents forward transmission from Port 1 to Port 2.
• S₁₂ represents reverse transmission from Port 2 to Port 1.
• For passive resistive networks, reciprocity gives S₁₂ = S₂₁.
• Always terminate unused ports with matched loads while calculating S-parameters.