# S-matrix calculation for Directional Coupler

Updated on 2017/08/04 13:05

## Syllabus

- Microwave Network Analysis: S-Matrix calculations for Directional coupler

## Directional Coupler

- A portion of power travelling from Port 1 to Port 2 is coupled to Port 4 but not to Port 3
- A portion of power travelling from Port 2 to Port 1 is coupled to Port 3 but not to Port 4
- Also Port 1 and Port 3 are decoupled as are Port 2 and Port 4

**We will use the above properties of D.C. as follows:**

- Substitute the above values, we get:

-----------(6) - From unity matrix property: [s].[s]*=[I]

- R1c1:- |s12|
^{2}+|s14|^{2}=1 –-------------(7)

R2c2:- |s12|^{2}+|s23|^{2}=1 —-----------(8)

R3c3:- |s23|^{2}+|s34|^{2}=1 --------------(9)

R1c3:-s12. s*23+ S14 s*34=0 --------(10)

- Comparing equation 7 and 8

S14=s23 -------(11)

- Comparing equation 8 and 9

S12=S34 -------(12)

- Let us assume that S12 is real and positive = 'P'

Therfore s12=s34=P=S*34 .........(13)

- Put equation 11 value in equation 10 i.e s14=s23

- From equation 10 & 13

P.s*23+s23.P=0

P[s23+s*23]=0

- Since P0, s23+s*23=0

s23=jy

s*23=-jy

i.e. s23 must be imaginary

- Let s23=jq=s14 .........(14)

Therfore, s12=s34=P and s23=s14=jq also

- Substituting these values, we get:

## Scattering matrix for 3-port Circulator

As it is 3-port device [S] matrix will be of order 3x3:

As we have seen properties of circulator so only s21,s32,s13 parameters exist. So the equation becomes:

---------(1)

The above S-matrix is for perfectly matched, lossless non-reciprocal 3-port circulator:

- output at port 1/input at port 3
- output at port 2/input at port 1
- output at port 3/input at port

## References

- WikiNote Foundation