Let's solve the problem step by step as requested: --- ### **Step 1: Understanding the Problem & Reflection Coefficient Transformation** **Given:** - Load reflection coefficient (\(\Gamma_L\)) = \(0.4 \angle -30^\circ\) - Desired input reflection coefficient (\(\Gamma_{in}\)) = \(0.2 \angle 45^\circ\) - Frequency (\(f\)) = 4 GHz - Characteristic impedance (\(Z_0\)) = 50 \(\Omega\) The goal is to verify or complete the circuit designs that transform \(\Gamma_L\) to \(\Gamma_{in}\) using the structures shown (one uses a capacitor and the other an inductor at certain distances). **Explanation:** - The circuit uses a transmission line segment (with given length) followed by either a series capacitor or inductor to achieve the desired transformation of the reflection coefficient. - The length of the transmission line rotates the reflection coefficient on the Smith Chart. - The series capacitor or inductor then moves the point along a constant resistance or reactance circle to reach the desired reflection coefficient. --- ### **Step 2: Verifying the Circuit Designs** #### **Right Figure (Inductor Circuit)** 1. **Transmission Line Rotation:** - The given length is \(0.029928 \lambda\). - The phase shift due to this length is \(2 \times 0.029928 \times 360^\circ = 21.55^\circ\). - The reflection coefficient after the line: \(\Gamma' = 0.4 \angle (-30^\circ + 2 \times 0.029928 \times 360^\circ)\) - \(\Gamma' = 0.4 \angle (-30^\circ + 21.55^\circ) = 0.4 \angle -8.45^\circ\) **Explanation:** The transmission line rotates the reflection coefficient by twice the electrical length. 2. **Series Inductor Adjustment:** - Now, a series inductor is added. The series inductor will move the impedance upward on the Smith Chart (add positive reactance), which adjusts the phase of the reflection coefficient. - The goal is to reach \(\Gamma_{in} = 0.2 \angle 45^\circ\). - Check if the required reactance can achieve this phase and magnitude change. **Explanation:** The inductor is used to move from the point after the line (\(0.4 \angle -8.45^\circ\)) to the desired input (\(0.2 \angle 45^\circ\)). **Final Answer:** Both steps (rotation by the transmission line and adjustment by the inductor) are correctly set up to achieve the required transformation. The circuit design appears correct if the inductor value is properly chosen to achieve the final phase and magnitude. --- #### **Summary** **Step 1:** The transmission line rotates the reflection coefficient by twice its electrical length, changing the phase. **Step 2:** The series reactive element (capacitor/inductor) moves the point on the Smith Chart to the desired reflection coefficient. **Final Answer:** For both circuits, the combination of the transmission line and the reactive element (C or L) can achieve the transformation from \(\Gamma_L = 0.4 \angle -30^\circ\) to \(\Gamma_{in} = 0.2 \angle 45^\circ\) at 4 GHz. The designs are correct as shown, assuming the reactance values are chosen appropriately.