Question

What does the parameter \( S_{11} \) represent in a two-port network?

• A. Forward transmission coefficient
• B. Reverse transmission coefficient.
• C. Input reflection coefficient
• D. Output reflection coefficient

Hint:

An easy mnemonic for S-parameters is \(S_{\text{destination}, \text{source}}\). Therefore, for \(S_{11}\), the signal originates at Port 1 and is measured at Port 1, which represents reflection at the input. For \(S_{21}\), the signal originates at Port 1 and travels to Port 2, representing forward transmission!

Answer 1

The Correct Option is C

Understanding the Concept: In high-frequency circuit analysis, Scattering parameters (S-parameters) are used to define the behavior of a multi-port electrical network based on traveling voltage waves. For a standard two-port network, the relationship between the incident voltage waves (\(a_1, a_2\)) and reflected voltage waves (\(b_1, b_2\)) at the ports is given by the matrix equation: \[ \begin{bmatrix} b_1\\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \] Expanding this system into simultaneous equations yields: \[ b_1 = S_{11}a_1 + S_{12}a_2 \] \[ b_2 = S_{21}a_1 + S_{22}a_2 \] Each discrete S-parameter has a specific physical interpretation based on terminating the ports in their characteristic impedance to eliminate any reflections arriving back into the network (setting specific incident waves to zero).

Step 1: Setting up the boundary condition for \( S_{11} \).
To isolate the parameter \(S_{11}\) from the first mathematical relationship: \[ b_1 = S_{11}a_1 + S_{12}a_2 \] We must ensure that there is no incident wave entering port 2. This condition is achieved by perfectly terminating port 2 with a load that matches the characteristic impedance of the transmission line, meaning: \[ a_2 = 0 \] Substituting \(a_2 = 0\) directly into our equation gives: \[ b_1 = S_{11}a_1 + S_{12}(0) \implies b_1 = S_{11}a_1 \]

Step 2: Defining the physical meaning of the ratio.
Solving for \(S_{11}\) yields the ratio: \[ S_{11} = \left. \frac{b_1}{a_1} \right|_{a_2 = 0} \] Where:
• \(b_1\) represents the reflected voltage wave propagating out from port 1.
• \(a_1\) represents the incident voltage wave injected directly into port 1.
• \(a_2 = 0\) specifies that port 2 is perfectly matched, preventing any secondary reflections. By definition, the ratio of the reflected wave to the incident wave at the primary port (Port 1) is called the Input Reflection Coefficient.

Step 3: Verification of other parameters for clarity.
To ensure completely robust analysis, let us briefly list the remaining parameters:
• \(S_{21} = \left. \frac{b_2}{a_1} \right|_{a_2 = 0}\) is the Forward Transmission Coefficient.
• \(S_{12} = \left. \frac{b_1}{a_2} \right|_{a_1 = 0}\) is the Reverse Transmission Coefficient.
• \(S_{22} = \left. \frac{b_2}{a_2} \right|_{a_1 = 0}\) is the Output Reflection Coefficient. Thus, \(S_{11}\) specifically corresponds to option (C).