Question 9.1: Evaluate the scattering matrix of the directional bridge in ......

The scattering matrix of the bridge coupler can be conveniently derived from the resistance matrix, that can be evaluated by inspection as:

R =\left(\begin{array}{ccc} 2 R_0 & 2 R_0 & R_0 \\ 2 R_0 & 3 R_0 & 2 R_0 \\ R_0 & 2 R_0 & 2 R_0 \end{array}\right).

Using:

S =\left( R -R_0 I \right)\left( R +R_0 I \right)^{-1},

we obtain:

S =\left(\begin{array}{ccc} 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \end{array}\right).

The coupling bridge therefore operates like a 3dB coupler between ports 1 and 2, while port 3 is isolated, and a 3dB coupler between ports 2 and 3, port 1 being isolated; the bridge is moreover matched at all ports. In scalar form, we have:

\begin{aligned} b_1 & =\frac{1}{2} a_2 \\ b_2 & =\frac{1}{2} a_1+\frac{1}{2} a_3 \\ b_3 & =\frac{1}{2} a_2 . \end{aligned}

However, the voltmeter is matched and therefore a_3=0 and b_2=a_1 / 2; moreover, a_2=\Gamma_{\text {DUT }} b_2. We therefore have:

\begin{aligned} b_3 & =\frac{1}{2} a_2=\frac{1}{2} \Gamma_{ \text{DUT} } b_2=\frac{1}{4} \Gamma_{ \text{DUT} } a_1 \\ \frac{b_3}{a_1} & =\frac{1}{4} \Gamma_{ \text{DUT} }. \end{aligned}

Thus a measurement of the ratio b_3 / a_1 is proportional to the load reflectance. If the DUT is connected at port 1 and the RF source at port 2, b_1=a_2 / 2=b_3; thus the power wave measured by the voltmeter coincides with the power wave entering the DUT. In conclusion, two cascaded coupling bridges allow the incident and reflected waves to be measured at the DUT port.