RF Analysis of a three port network

Question

I was wondering if anyone could help me understand how to analyze an RF transmission line circuit that I saw in a design used to detect power. It is a three port network that has two transmission lines with a resistor that taps off the center of the transmission line. From what i understand qualitatively is that the 200 Ω resistor attenuates the RF power at port 3, but it messes with the 50 Ω match, so a shunt 50 Ω resistor is used to match port 3. I would like to have an analytical solution to S13 that is similar to how Pozar's Microwave engineering analyzes the resistive power splitter. I have recreated the schematic in QUCS and am having trouble finding an analytical solution for R1.

^{simulate this circuit – Schematic created using CircuitLab}

I'm stuck on finding the input impedance looking into port 1 and how to go about this. From my understanding the voltage at port 1 is the same at the junction of port 2 and 3, because the transmission line does not change impedance, then S13=V1/V3 is a voltage divider V3 = R2/(R1+R2)*V1 so V1/V3=(200+50)/50=5, but that doesn't make sense cause it shouldn't have gain as a passive component, so I'm not sure where I went wrong with my logic. I would like to have a more analytical method of solving this instead of trusting the software.

Thank you

Answer 1

To get the input impedance, you make a few steps. You need to transform everything back to port 1. Starting with you schematic from ADS:

1. Looking into port 2 sees a load of 50 Ω with a 50 Ω line, which gives an input impedance of 50 Ω. Now you have moved port 2 up to the junction point.
2. Then you look towards port 3, here you have 50||50 Ω in series with 200 Ω, which gives 225 Ω.
3. The 50 Ω you found in 1) is in parallel with 225 Ω from 2).
4. Now you can use the standard input impedance formula, with the known ZL= (225*50)/(225+50) and Zo=50 Ω, to give you the input impedance and port 1.