Solved Numerical 11

Step 1: Calculate the Reflection Coefficient

The impedance mismatch between the attenuator and the system produces a reflection coefficient at both ports.

$ \Gamma_S=\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0} $

Substituting the values:

$ \Gamma_S=\Gamma_L=\frac{75-50}{75+50} $

$ \Gamma_S=\Gamma_L=\frac{25}{125} $

$ \Gamma_S=\Gamma_L=0.20 $

Step 2: Determine the Original S-Parameters

Assume the attenuator is symmetrical and has zero phase shift.

For a symmetrical attenuator:

$ S_{11}=S_{22} $

$ S_{12}=S_{21} $

Calculate S11 from Return Loss

Return loss is given as:

$ RL=20\text{ dB} $

Using:

$ S_{11}=10^{-\frac{RL}{20}} $

$ S_{11}=10^{-\frac{20}{20}} $

$ S_{11}=0.10 $

Therefore:

$ S_{11}=S_{22}=0.10 $

Calculate S21 from Attenuation

The attenuator has an attenuation of 3 dB.

$ S_{21}=10^{-\frac{IL}{20}} $

$ S_{21}=10^{-\frac{3}{20}} $

$ S_{21}=0.7080 $

Therefore:

$ S_{12}=S_{21}=0.7080 $

Original S-Matrix (50Ω Reference)

$ [S]_{50} = \begin{bmatrix} 0.10 & 0.7080 \\ 0.7080 & 0.10 \end{bmatrix} $

Step 3: Renormalize the S-Matrix to 75Ω

Because the attenuator is now operating in a 75Ω system, the scattering matrix must be renormalized to the new reference impedance.

After renormalization:

$ [S]_{75} = \begin{bmatrix} 0.002379 & 0.7227 \\ 0.7227 & 0.002379 \end{bmatrix} $

Step 4: Calculate the Insertion Loss in the 75Ω System

The new forward transmission coefficient is:

$ S'_{21}=0.7227 $

Insertion loss is calculated from:

$ IL = 10\log_{10} \left( \frac{1}{|S'_{21}|^2} \right) $

Substituting:

$ IL = 10\log_{10} \left( \frac{1}{(0.7227)^2} \right) $

$ IL = 2.82\text{ dB} $

Final Answer

The insertion loss of the 50Ω attenuator when connected to a 75Ω system is:

$ \boxed{IL=2.82\text{ dB}} $

Key Exam Takeaway

Many students assume the insertion loss remains equal to the rated attenuation of the attenuator. That is only true when the device operates in its intended reference impedance. When the reference impedance changes, the S-parameters must be renormalized before calculating insertion loss.

• Original attenuator attenuation = 3 dB
• Return loss = 20 dB
• Reflection coefficient = 0.20
• Renormalized transmission coefficient = 0.7227
• Actual insertion loss in 75Ω system = 2.82 dB

For microwave engineering, RF design, and communication system exams, always check whether the network reference impedance matches the system impedance before using S-parameters directly.