# S-matrix calculations for E-plane Tee

Updated on 2017/07/27 23:26

## E-plane Tee

We know that,

When TE_{10} mod is made to propagate into port 3 the two outputs at port 1 & port 2 will have a phase shift of 180°

Since it is a three port junction the s-matrix can be derived as follows.

- R1C1: |S11 |
^{2}+ |S12|^{2}+ |S13|^{2 }= 1 -------- (5)

R2C2: |S12|^{2}+ |S22|^{2}+ |S13|^{2}= 1 --------- (6)

R3C3: |S13|^{2}+ |S13|^{2 }= 1 --------- (7)

R3C1: S13.S11^{*}- S13.S12^{*}= 0 ---------- (8)

- Equating (5) & (6), we get, S11 = S22 ---------- (9)
- From equation (7) S13 = --------- (10)
- From equation (8), S13 (S11
^{* }- S12^{*}) = 0

Or S11 = S12 = S22 ----------from (9) ------- (11) - Using equation (9), (10) & (11) in equation (5) |S11|
^{2}+ |S11|^{2}+ = 1

Or 2|S11|^{2 }= or S11 = --------- (12) - ---------- (13)

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

---------- (14)

### Case 1: a1=a2=0, a3≠ 0

Put case 1 values in equation 15, 16, 17:

### Case 2: a1=a2=a; a3=0;

Put case 2 values in equation 15, 16, 17:

### Case 3: a1≠0, a2 = 0, a3 = 0

Put case 3 values in equation 15, 16, 17: