4.1. Scattering (S) Parameters

Microwave junctions with multiple ports are characterized by a Scattering Matrix (S-matrix). The elements of this matrix, known as S-parameters, describe the relationship between incident waves (a_n) and reflected/transmitted waves (b_n) at each port.

• Equation: [b] = [S][a]
• Properties: The S-matrix is a square, symmetric (S_{ij} = S_{ji}), and unitary ([S][S]^* = [I]) matrix for a reciprocal, lossless junction.

4.2. Waveguide Tee Junctions

Tee junctions are three-port devices used to split or combine power in a waveguide system.

• E-Plane Tee (Series Tee): The side arm is attached to the broader dimension of the main waveguide, parallel to the E-field. Outputs from the collinear ports (1 and 2) are 180° out of phase relative to an input at the side arm (port 3).
  • S-Matrix: [S] = \begin{bmatrix} 1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & 1/2 & -1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{bmatrix}
• H-Plane Tee (Shunt Tee): The side arm is attached to the narrower dimension, parallel to the H-field. Outputs from the collinear ports are in phase relative to an input at the side arm.
  • S-Matrix: [S] = \begin{bmatrix} 1/2 & -1/2 & 1/\sqrt{2} \\ -1/2 & 1/2 & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 \end{bmatrix}
• E-H Plane Tee (Magic Tee or Hybrid Tee): A four-port junction combining E-plane and H-plane arms.
  • Ports: Port 1 & 2 (collinear), Port 3 (H-arm, sum port), Port 4 (E-arm, difference port).
  • Characteristics:
    • An input at the H-arm (3) splits equally and in-phase between ports 1 and 2, with no output at port 4.
    • An input at the E-arm (4) splits equally and 180° out-of-phase between ports 1 and 2, with no output at port 3.
    • An input at a collinear port (1 or 2) does not produce an output at the other collinear port.
    • The H-arm and E-arm are isolated from each other.
  • Applications: Used in impedance measurement bridges, duplexers, and mixers.
  • S-Matrix: [S] = \begin{bmatrix} 0 & 0 & 1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 0 & 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 & 0 \end{bmatrix}

4.3. Other Junctions and Couplers

• Rat-race Junction: A four-port circular hybrid junction. An input at port 1 splits equally to ports 2 and 4, while port 3 is isolated. It is used for combining and dividing signals.
• Directional Coupler: A four-port device that samples a small amount of power from a primary waveguide to a secondary waveguide for measurement purposes.
  • Performance Parameters:
    • Coupling Factor (C): Ratio of incident power (P_i) to forward-coupled power (P_f). C = 10 \log_{10}(P_i/P_f).
    • Directivity (D): Ratio of forward-coupled power (P_f) to back power (P_b). D = 10 \log_{10}(P_f/P_b).
    • Isolation (I): Ratio of incident power (P_i) to back power (P_b). I = 10 \log_{10}(P_i/P_b).