Coupler

According to the port numbers in fig. 9.5 the Y-parameters of a coupler write as follows.

$$Y_{11} = Y_{22} = Y_{33} = Y_{44} = \dfrac{A\cdot \left(2-A\right)}{D}$$ (9.111)

$$Y_{12} = Y_{21} = Y_{34} = Y_{43} = \dfrac{-A\cdot B}{D}$$ (9.112)

$$Y_{13} = Y_{31} = Y_{24} = Y_{42} = \dfrac{C\cdot \left(A-2\right)}{D}$$ (9.113)

$$Y_{14} = Y_{41} = Y_{23} = Y_{32} = \dfrac{B\cdot C}{D}$$ (9.114)

with

$$A = k^2 \cdot \left( 1+\exp\left(j\cdot 2\phi\right) \right)$$ (9.115)

$$B = 2 \cdot \sqrt{1-k^2}$$ (9.116)

$$C = 2 \cdot k \cdot \exp\left(j\cdot\phi\right)$$ (9.117)

$$D = Z_{ref}\cdot \left(A^2 - C^2\right)$$ (9.118)

whereas $0<k<1$ denotes the coupling factor, $\phi$ the phase shift of the coupling path and $Z_{ref}$ the reference impedance. The coupler can also be used as hybrid by setting $k=1/\sqrt{2}$. For a 90 degree hybrid, for example, set $\phi$ to $\pi / 2$. 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.

$$\begin{bmatrix}.&.&.&.& 1 & 0\\ .&.&.&.&-1 & 0\\ .&.&.&.& 0 & 1\\... ...\\ 0\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$$ (9.119)

Figure 9.5: ideal coupler device

The scattering parameters of a coupler are:

$$S_{11} = S_{22} = S_{33} = S_{44} = 0$$ (9.120)

$$S_{14} = S_{23} = S_{32} = S_{41} = 0$$ (9.121)

$$S_{12} = S_{21} = S_{34} = S_{43} = \sqrt{1-k^2}$$ (9.122)

$$S_{13} = S_{31} = S_{24} = S_{42} = k\cdot \exp\left(j\phi\right)$$ (9.123)

whereas $0<k<1$ denotes the coupling factor, $\phi$ the phase shift of the coupling path. Extending them for an arbitrary reference impedance $Z_{ref}$, they already become quite complex:

$$r = \dfrac{Z_0-Z_{ref}}{Z_0+Z_{ref}}$$ (9.124)

$$A = k^2 \cdot \left( \exp\left(j\cdot 2\phi\right)+1 \right)$$ (9.125)

$$B = r^2 \cdot \left(1-A\right)$$ (9.126)

$$C = k^2 \cdot \left( \exp\left(j\cdot 2\phi\right)-1 \right)$$ (9.127)

$$D = 1 - 2\cdot r^2\cdot \left(1+C\right) + B^2$$ (9.128)

$$S_{11} = S_{22} = S_{33} = S_{44} = r\cdot\dfrac{A\cdot B + C + 2\cdot r^2\cdot k^2\cdot\exp\left(j\cdot 2\phi\right)}{D}$$ (9.129)

$$S_{12} = S_{21} = S_{34} = S_{43} = \sqrt{1-k^2}\cdot \dfrac{\left(1-r^2\right)\cdot \left(1-B\right)}{D}$$ (9.130)

$$S_{13} = S_{31} = S_{24} = S_{42} = k\cdot\exp\left(j\phi\right)\cdot \dfrac{\left(1-r^2\right)\cdot \left(1+B\right)}{D}$$ (9.131)

$$S_{14} = S_{23} = S_{32} = S_{41} = 2\cdot\sqrt{1-k^2}\cdot k\cdot\exp\left(j\phi\right)\cdot r\cdot \dfrac{\left(1-r^2\right)}{D}$$ (9.132)

An ideal coupler is noise free.