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# Microwave Engineering - E-Plane Tee

An E-Plane Tee junction is formed by attaching a simple waveguide to the broader dimension of a rectangular waveguide, which already has two ports. The arms of rectangular waveguides make two ports called **collinear ports** i.e., Port1 and Port2, while the new one, Port3 is called as Side arm or **E-arm**. T his E-plane Tee is also called as **Series Tee**.

As the axis of the side arm is parallel to the electric field, this junction is called E-Plane Tee junction. This is also called as **Voltage** or **Series junction**. The ports 1 and 2 are 180° out of phase with each other. The cross-sectional details of E-plane tee can be understood by the following figure.

The following figure shows the connection made by the sidearm to the bi-directional waveguide to form the parallel port.

## Properties of E-Plane Tee

The properties of E-Plane Tee can be defined by its $[S]_{3x3}$ matrix.

It is a 3×3 matrix as there are 3 possible inputs and 3 possible outputs.

$[S] = \begin{bmatrix}
S_{11}& S_{12}& S_{13}\\
S_{21}& S_{22}& S_{23}\\
S_{31}& S_{32}& S_{33}
\end{bmatrix}$ **........ Equation 1**

Scattering coefficients $S_{13}$ and $S_{23}$ are out of phase by 180° with an input at port 3.

$S_{23} = -S_{13}$**........ Equation 2**

The port is perfectly matched to the junction.

$S_{33} = 0$**........ Equation 3**

From the symmetric property,

$S_{ij} = S_{ji}$

$S_{12} = S_{21} \: \: S_{23} = S_{32} \: \: S_{13} = S_{31}$**........ Equation 4**

Considering equations 3 & 4, the $[S]$ matrix can be written as,

$[S] = \begin{bmatrix}
S_{11}& S_{12}& S_{13}\\
S_{12}& S_{22}& -S_{13}\\
S_{13}& -S_{13}& 0
\end{bmatrix}$**........ Equation 5**

We can say that we have four unknowns, considering the symmetry property.

From the Unitary property

$$[S][S]\ast = [I]$$

$$\begin{bmatrix} S_{11}& S_{12}& S_{13}\\ S_{12}& S_{22}& -S_{13}\\ S_{13}& -S_{13}& 0 \end{bmatrix} \: \begin{bmatrix} S_{11}^{*}& S_{12}^{*}& S_{13}^{*}\\ S_{12}^{*}& S_{22}^{*}& -S_{13}^{*}\\ S_{13}^{*}& -S_{13}^{*}& 0 \end{bmatrix} = \begin{bmatrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{bmatrix}$$

Multiplying we get,

(Noting R as row and C as column)

$R_1C_1 : S_{11}S_{11}^{*} + S_{12}S_{12}^{*} + S_{13}S_{13}^{*} = 1$

$\left | S_{11} \right |^2 + \left | S_{11} \right |^2 + \left | S_{11} \right |^2 = 1$ **........ Equation 6**

$R_2C_2 : \left | S_{12} \right |^2 + \left | S_{22} \right |^2 + \left | S_{13} \right |^2 = 1$ **......... Equation 7**

$R_3C_3 : \left | S_{13} \right |^2 + \left | S_{13} \right |^2 = 1$ **......... Equation 8**

$R_3C_1 : S_{13}S_{11}^{*} - S_{13}S_{12}^{*} = 1$ **......... Equation 9**

Equating the equations 6 & 7, we get

$S_{11} = S_{22} $ **......... Equation 10**

From Equation 8,

$2\left | S_{13} \right |^2 \quad or \quad S_{13} = \frac{1}{\sqrt{2}}$ **......... Equation 11**

From Equation 9,

$S_{13}\left ( S_{11}^{*} - S_{12}^{*} \right )$

Or $S_{11} = S_{12} = S_{22}$ **......... Equation 12**

Using the equations 10, 11, and 12 in the equation 6,

we get,

$\left | S_{11} \right |^2 + \left | S_{11} \right |^2 + \frac{1}{2} = 1$

$2\left | S_{11} \right |^2 = \frac{1}{2}$

Or $S_{11} = \frac{1}{2}$ **......... Equation 13**

Substituting the values from the above equations in $[S]$ matrix,

We get,

$$\left [ S \right ] = \begin{bmatrix} \frac{1}{2}& \frac{1}{2}& \frac{1}{\sqrt{2}}\\ \frac{1}{2}& \frac{1}{2}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0 \end{bmatrix}$$

We know that $[b]$ = $[S] [a]$

$$\begin{bmatrix}b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} \frac{1}{2}& \frac{1}{2}& \frac{1}{\sqrt{2}}\\ \frac{1}{2}& \frac{1}{2}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0 \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix}$$

This is the scattering matrix for E-Plane Tee, which explains its scattering properties.