Let's analyze the problem step-by-step. --- ## **Given Data:** - The network consists of three impedances: \( Z_1, Z_2, Z_3 \). - The impedance between the two ports (Port 1 and Port 2) is \( Z_ \). - The impedances are arranged as shown: - \( Z_1 \) between Port 1 and the node. - \( Z_3 \) between the node and Port 2. - \( Z_2 \) between the node and ground (or between the node and the common reference). --- ## **Objective:** Calculate the S-parameters \( S_{11} \) and \( S_{21} \) in terms of the impedances \( Z_1, Z_2, Z_3 \). --- ## **Step 1: Understanding the configuration** The network is a two-port network with the following configuration: - Input port at Port 1. - Output port at Port 2. - The node in the middle connects impedances \( Z_1, Z_2, Z_3 \). The key is to relate the incident and reflected waves at the ports to the impedance network. --- ## **Step 2: Express the S-parameters** Recall the definitions: \[ S_{11} = \frac{\text{Reflected wave at port 1}}{\text{Incident wave at port 1}} \quad \text{(when port 2 is matched)} \] \[ S_{21} = \frac{\text{Transmitted wave at port 2}}{\text{Incident wave at port 1}} \quad \text{(when port 2 is matched)} \] --- ## **Step 3: Write the impedance relations** Assuming the characteristic impedance of the ports is \( Z_ \), and the network is linear, we can express the input and output waves as: \[ a_1, b_1 \text{ at port 1} \] \[ a_2, b_2 \text{ at port 2} \] where: \[ a_1 = \text{incident wave at port 1} \] \[ b_1 = \text{reflected wave at port 1} \] and similarly for port 2. --- ## **Step 4: Find the relation between waves and impedance network** The key is to find the input impedance seen at port 1, \( Z_{in} \), and the transfer impedance for transmission. ### **Input impedance seen at Port 1:** The network between port 1 and port 2 can be viewed as a two-port network with certain impedance relations. --- ## **Step 5: Use the impedance matrix approach** The impedance parameters relate voltages and currents: \[ \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \] But since the problem asks for S-parameters in terms of the impedances \( Z_1, Z_2, Z_3 \), and the network resembles a T-network or Pi-network, the S-parameters can be directly related to the impedance network. --- ## **Step 6: Express \(S_{11}\) and \(S_{21}\) in terms of impedances** ### **Reflection coefficient \( S_{11} \):** \[ S_{11} = \frac{Z_{in} - Z_}{Z_{in} + Z_} \] where \( Z_{in} \) is the input impedance seen at port 1 when port 2 is matched. --- ### **Transmission coefficient \( S_{21} \):** \[ S_{21} = \frac{2 Z_}{Z_{in} + Z_} \] --- ## **Step 7: Find \( Z_{in} \) in terms of \( Z_1, Z_2, Z_3 \)** The impedance looking into port 1, with port 2 terminated in \( Z_ \), can be derived as: \[ Z_{in} = Z_1 + \left( Z_2 \parallel Z_3 \right) \] where: \[ Z_2 \parallel Z_3 = \frac{Z_2 Z_3}{Z_2 + Z_3} \] --- ## **Final expressions:** \[ \boxed{ \begin{aligned} Z_{in} &= Z_1 + \frac{Z_2 Z_3}{Z_2 + Z_3} \end{aligned} } \] --- ## **Step 8: Write the S-parameters** \[ \boxed{ S_{11} = \frac{Z_{in} - Z_}{Z_{in} + Z_} } \] \[ \boxed{ S_{21} = \frac{2 Z_}{Z_{in} + Z_} } \] with \[ Z_{in} = Z_1 + \frac{Z_2 Z_3}{Z_2 + Z_3} \] --- ## **## **Final answers:** \[ \boxed{ \textbf{S}_{11} = \frac{Z_1 + \frac{Z_2 Z_3}{Z_2 + Z_3} - Z_}{Z_1 + \frac{Z_2 Z_3}{Z_2 + Z_3} + Z_} } \] \[ \boxed{ \textbf{S}_{21} = \frac{2 Z_}{Z_1 + \frac{Z_2 Z_3}{Z_2 + Z_3} + Z_} } \] --- **Note:** The derivation assumes the network is linear, reciprocal, and the port characteristic impedance is \( Z_ \).