An E-plane tee is a waveguide tee in which the axis of its side arm is parallel to the E field of the main guide (see Fig.). If the collinear arms are symmetric about the side arm, there are two different transmission characteristics. It can be seen from Fig. that if the E - plane tee is perfectly matched with the aid of screw tuners or inductive or capacitive windows at the junction, the diagonal components of the scattering matrix, $S_{11}, S_{22}, \text{and} S_{33}$, are zero because there will be no reflection. When the waves are fed into the side arm (port 3), the waves appearing at port 1 and port 2 of the collinear arm will be in opposite phase and in the same magnitude. Therefore

$$S_{13}=-S_{23}$$

It should be noted that above Eq. does not mean that $S_{13}$ is always positive and $S_{23}$ is always negative. The negative sign merely means that SB and $S_{23}$ have opposite signs. For a matched junction, the S matrix is given by from the symmetry property of S matrix, the symmetric terms in above Eq. are equal and they are

$S= \begin{bmatrix} 0 & S_{12} & S_{13} \\ S_{21} & 0 & S_{23} \\ S_{31} & S_{32} & 0\end{bmatrix} \dots \text{Eq. a}$

$S_{11}=S_{21} \hspace{1cm} S_{13}=S_{31} \hspace{1cm} S_{23}=S_{32}$ …Eq.1

From the zero property of S matrix, the sum of the products of each term of any column (or row) multiplied by the complex conjugate of the corresponding terms of any other column (or row) is zero and it is

$S_{11}S_{12}^*+S_{21} S_{22}^*+S_{31} S_{32}^*=0$

Hence

$S_{13} S_{32}^*=0$

This means that either $S_{13}$ or $S_{32}^*$, or both, should be zero. However, from the unity property of S matrix, the sum of the products of each term of any one row (or column) multiplied by its complex conjugate is unity; that is,

$S_{21} S_{21}^*+S_{31} S_{31}^*=1 …\text{Eq.2} \\ S_{12} S_{12}^*+S_{32} S_{32}^*=1 …\text{Eq.3} \\ S_{13} S_{13}^*+S_{23} S_{23}^*=1 …\text{Eq.4}$

Substitution of Eq. 1 in Eq. 2 results in

$|S_{12} |^2=1-|S_{13} |^2=1-|S_{23} |^2 …\text{Eq.5}$

Equations 4 and Eq. 5 are contradictory, for if $S_{13} = 0$, then $S_{23}$ is also zero and thus Eq. 4 is false. In a similar fashion, if $S_{23} = 0$, then $S_{13}$ becomes zero and therefore Eq. 5 is not true. This inconsistency proves the statement that the tee junction cannot be matched to the three arms. In other words, the diagonal elements of the S matrix of a tee junction are not all zeros. In general, when an E-plane tee is constructed of an empty waveguide, it is poorly matched at the tee junction. Hence $S_{ij}≠0$ if i = j. However, since the collinear arm is usually symmetric about the side arm, $|S_{13} |=|S_{23} |$ and $S_{11}= S_{22}$. Then the S matrix can be simplified to

$S= \begin{bmatrix} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{11} & -S_{13} \\ S_{31} & -S_{13} & S_{33}\end{bmatrix} \dots \text{Eq. b}$

**H-plane tee (Shunt tee)**

An H-plane tee is a waveguide tee in which the axis of its side arm is "shunting" the E field or parallel to the H field of the main guide as shown in Fig.

It can be seen that if two input waves are fed into port 1 and port 2 of the collinear arm, the output wave at port 3 will be in phase and additive. On the other hand, if the input is fed into port 3, the wave will split equally into port 1 and port 2 in phase and in the same magnitude. Therefore, the S matrix of the H-plane tee is similar to Eq. a and Eq. b except that

$S_{13}=S_{23}$