Circulator

A circulator is a 3-port device, transporting incoming waves lossless from port 1 to port 2, from port 2 to port 3 and from port 3 to port 1. In all other directions, there is no energy flow. The ideal circulator cannot be characterized with Z or Y parameters, because their values are partly infinite. But implementing with S parameters is practical (see equation 9.2).

With the reference impedances $Z_1$, $Z_2$ and $Z_3$ for the ports 1, 2 and 3 the scattering matrix of an ideal circulator writes as follows.

$$denom = 1-r_1\cdot r_2\cdot r_3$$ (9.97)

$$r_1 = \frac{Z_0-Z_1}{Z_0+Z_1} \qquad,\qquad r_2 = \frac{Z_0-Z_2}{Z_0+Z_2} \qquad,\qquad r_3 = \frac{Z_0-Z_3}{Z_0+Z_3}$$ (9.98)

$$S_{11} = \frac{r_2\cdot r_3 - r_1}{denom} \qquad,\qquad S_{22} = ... ...\cdot r_3 - r_2}{denom} \qquad,\qquad S_{33} = \frac{r_1\cdot r_2 - r_3}{denom}$$ (9.99)

$$S_{12} = \sqrt{\frac{Z_2}{Z_1}}\cdot\frac{Z_1+Z_0}{Z_2+Z_0}\cdot\... ... = \sqrt{\frac{Z_3}{Z_1}}\cdot\frac{Z_1+Z_0}{Z_3+Z_0}\cdot\frac{1-r_1^2}{denom}$$ (9.100)

$$S_{21} = \sqrt{\frac{Z_1}{Z_2}}\cdot\frac{Z_2+Z_0}{Z_1+Z_0}\cdot\... ...frac{Z_3}{Z_2}}\cdot\frac{Z_2+Z_0}{Z_3+Z_0}\cdot\frac{r_1\cdot(1-r_2^2)}{denom}$$ (9.101)

$$S_{31} = \sqrt{\frac{Z_1}{Z_3}}\cdot\frac{Z_3+Z_0}{Z_1+Z_0}\cdot\... ... = \sqrt{\frac{Z_2}{Z_3}}\cdot\frac{Z_3+Z_0}{Z_2+Z_0}\cdot\frac{1-r_3^2}{denom}$$ (9.102)

An ideal circulator is noise free.