According to the port numbers in fig. 9.5
the Y-parameters of a coupler write as follows.
![$\displaystyle Y_{11} = Y_{22} = Y_{33} = Y_{44}$](img1014.png) |
![$\displaystyle = \dfrac{A\cdot \left(2-A\right)}{D}$](img1015.png) |
(9.111) |
![$\displaystyle Y_{12} = Y_{21} = Y_{34} = Y_{43}$](img1016.png) |
![$\displaystyle = \dfrac{-A\cdot B}{D}$](img1017.png) |
(9.112) |
![$\displaystyle Y_{13} = Y_{31} = Y_{24} = Y_{42}$](img1018.png) |
![$\displaystyle = \dfrac{C\cdot \left(A-2\right)}{D}$](img1019.png) |
(9.113) |
![$\displaystyle Y_{14} = Y_{41} = Y_{23} = Y_{32}$](img1020.png) |
![$\displaystyle = \dfrac{B\cdot C}{D}$](img1021.png) |
(9.114) |
with
![$\displaystyle A$](img1022.png) |
![$\displaystyle = k^2 \cdot \left( 1+\exp\left(j\cdot 2\phi\right) \right)$](img1023.png) |
(9.115) |
![$\displaystyle B$](img170.png) |
![$\displaystyle = 2 \cdot \sqrt{1-k^2}$](img1024.png) |
(9.116) |
![$\displaystyle C$](img172.png) |
![$\displaystyle = 2 \cdot k \cdot \exp\left(j\cdot\phi\right)$](img1025.png) |
(9.117) |
![$\displaystyle D$](img1026.png) |
![$\displaystyle = Z_{ref}\cdot \left(A^2 - C^2\right)$](img1027.png) |
(9.118) |
whereas
denotes the coupling factor,
the phase shift of
the coupling path and
the reference impedance. The coupler
can also be used as hybrid by setting
. For a 90 degree
hybrid, for example, set
to
. Note that for most
couplers no real DC model exists. Taking the real part of the AC
matrix often leads to non-logical results. Thus, it is better to
model the coupler for DC by making a short between port 1 and port 2
and between port 3 and port 4. The rest should be an open. This
leads to the following MNA matrix.
![$\displaystyle \begin{bmatrix}.&.&.&.& 1 & 0\\ .&.&.&.&-1 & 0\\ .&.&.&.& 0 & 1\\...
...\\ 0\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$](img1031.png) |
(9.119) |
Figure 9.5:
ideal coupler device
|
The scattering parameters of a coupler are:
![$\displaystyle S_{11} = S_{22} = S_{33} = S_{44} = 0$](img1033.png) |
(9.120) |
![$\displaystyle S_{14} = S_{23} = S_{32} = S_{41} = 0$](img1034.png) |
(9.121) |
![$\displaystyle S_{12} = S_{21} = S_{34} = S_{43} = \sqrt{1-k^2}$](img1035.png) |
(9.122) |
![$\displaystyle S_{13} = S_{31} = S_{24} = S_{42} = k\cdot \exp\left(j\phi\right)$](img1036.png) |
(9.123) |
whereas
denotes the coupling factor,
the phase shift of
the coupling path. Extending them for an arbitrary reference
impedance
, they already become quite complex:
![$\displaystyle r$](img1037.png) |
![$\displaystyle = \dfrac{Z_0-Z_{ref}}{Z_0+Z_{ref}}$](img1038.png) |
(9.124) |
![$\displaystyle A$](img1022.png) |
![$\displaystyle = k^2 \cdot \left( \exp\left(j\cdot 2\phi\right)+1 \right)$](img1039.png) |
(9.125) |
![$\displaystyle B$](img170.png) |
![$\displaystyle = r^2 \cdot \left(1-A\right)$](img1040.png) |
(9.126) |
![$\displaystyle C$](img172.png) |
![$\displaystyle = k^2 \cdot \left( \exp\left(j\cdot 2\phi\right)-1 \right)$](img1041.png) |
(9.127) |
![$\displaystyle D$](img1026.png) |
![$\displaystyle = 1 - 2\cdot r^2\cdot \left(1+C\right) + B^2$](img1042.png) |
(9.128) |
![$\displaystyle S_{11} = S_{22} = S_{33} = S_{44} = r\cdot\dfrac{A\cdot B + C + 2\cdot r^2\cdot k^2\cdot\exp\left(j\cdot 2\phi\right)}{D}$](img1043.png) |
(9.129) |
![$\displaystyle S_{12} = S_{21} = S_{34} = S_{43} = \sqrt{1-k^2}\cdot \dfrac{\left(1-r^2\right)\cdot \left(1-B\right)}{D}$](img1044.png) |
(9.130) |
![$\displaystyle S_{13} = S_{31} = S_{24} = S_{42} = k\cdot\exp\left(j\phi\right)\cdot \dfrac{\left(1-r^2\right)\cdot \left(1+B\right)}{D}$](img1045.png) |
(9.131) |
![$\displaystyle S_{14} = S_{23} = S_{32} = S_{41} = 2\cdot\sqrt{1-k^2}\cdot k\cdot\exp\left(j\phi\right)\cdot r\cdot \dfrac{\left(1-r^2\right)}{D}$](img1046.png) |
(9.132) |
An ideal coupler is noise free.
This document was generated by Stefan Jahn on 2007-12-30 using latex2html.