**Magic Tee or Hybrid Tee S Matrix Derivation**

A combination of E plane Tee and M plane Tee is called Hybrid Tee or Magic Tee. It consists of four ports, if two waves of equal magnitude and same phase are fed into port 1 and port 2, the output will be subtractive and zero at port 3 and will be addictive at Port 4. A wave incident at Port 4 divides equally between port 1 and 2, and will not appear at port 3. A wave incident at Port 3 will produce an output of equal magnitude and opposite phase at ports 1 and 2. The magic tee is matched at ports 3 and 4.

The general matrix of the magic tee is given by,

From the property of Symmetry,

S

_{14}= S_{41},
S

_{13}= S_{31},
S

_{23}= S_{32}---------------- (2)
Since port 3 acts as the E plane Tee,

S

_{13}= -S_{23}------------------- (3)
Since port 4 acts as H plane Tee, S

_{14}= S_{24}---------------- (4)
Considering the phase delay in the network,

S

_{34}= -S_{43}= 0 and
S

_{12}= -S_{21}= 0 ---------------- (5)
If port 3 and port 4 are matched,

S

_{33}= S_{44}= 0 ---------------- (6)
Applying equation (2) to (6) in equation (1)

By unitary property, [S][S*] = I

R

_{1}C_{1}=> |S_{11}|^{2}+|S_{13}|^{2}+|S_{14}|^{2}= 1 -------------- (8)
R

_{2}C_{2}=> |S_{22}|^{2}+|S_{13}|^{2}+|S_{14}|^{2}= 1--------------- (9)
R

_{3}C_{3}=> 2|S_{13}|^{2}= 1
S

_{13}= 1/√2 ------------- (10)
R

_{4}C_{4}=> 2|S_{14}|^{2}= 1
S

_{14}= 1/√2 ---------------- (11)
Substitute, Equation (10) and (11) in equation (8)

|S

_{11}|^{2}+ (1/√2)^{2}+ (1/√2)^{2}= 1
|S

_{11}|^{2}= 1 – 1
=> S

_{11}= 0
Equating equation (8) and (9)

We get, S

_{11}= S_{22}
Therefore the s matrix of magic tee is,

**Hybrid Ring S Matrix Derivation:**

Hybrid ring circuits are also known as ‘Rat Race Coupler’. These junctions overcome the power limitations of magic tee. It is constructed by folding rectangular waveguides into circular waveguides. This junction has 4 ports with upper 3 ports separated by λ/4 and lower two ports separated by 3 λ/4. When a wave is fed into port 1, it will not appear at port 3 due to the phase shifts. Similarly wave fed onto port 2 will not appear at port 4 due to phase difference.

The general matrix of hybrid ring is,

If the ports 1,2,3 and 4 are matched then,

S

_{11}= S_{22}= S_{33}= S_{44}= 0 -------------------- (2)
Considering the input – output conditions,

S

_{13}= S_{31}= 0
S

_{24}= S_{42}= 0
S

_{21}= -S_{41}
Therefore, the general matrix can be written as,

**[S][S]* = I**

**R**

_{1}C_{1 }=>

|S

_{12}|^{2 }+ |S_{12}|^{2}= 1
2|S

_{12}|^{2}= 1
S

_{12}= 1/√2**R**

_{2}C_{2 }=>

|S

_{12}|^{2 }+ |S_{23}|^{2}= 1
½ +|S

_{23}|^{2}= 1
S

_{23}= 1/√2**R**

_{3}C_{3 }=>

|S

_{23}|^{2 }+ |S_{34}|^{2}= 1
½ +|S

_{34}|^{2}= 1
S

_{34}= 1/√2**R**

_{4}C_{4 }=>

|S

_{12}|^{2 }+ |S_{34}|^{2}= 1
S

_{12}= 1/√2
Therefore, Matrix of Hybrid Rings is

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