Hello EEs,
I was interested in transmission lines and have been revisiting their theory. They are fascinating.
The reason is that I was trying to design an FM notch filter for a wideband HF amplifier, because I get FM stations all over the place when I connect it to a SDR. I wanted to use a transmission line stub as filter, for its simplicity.
In order to simulate the effects of such a transmission line and verify the measurements with the nanoVNA, I made a GNUplot script that plots several characteristics of this reference circuit:
(https://i.imgur.com/rbbWbjr.png)
The scritpt can be found in https://github.com/nostromo-1/VNA-plots (https://github.com/nostromo-1/VNA-plots) (file vna_plots.plt), if anybody is interested. It simulates a circuit with a certain output impedance Zs (can be a nanoVNA) connected to a transmission line (characteristic impedance Zo, velocity factor k, length l) and connected to a load Zl.
It displays:
- The Smith Chart as seen by the nanoVNA
- The phase of the S11 parameter (reflection coefficient) measured by the nanoVNA
- The Bode plot (frequency response) at the output of the nanoVNA (or circuit): modulus of Vi/Vs
It can also plot the phase of the frequency response Vi/Vs, and modulus/phase of the frequency response at Zl.
As an example, you can connect an RG58 coaxial cable with its end open (open stub) to the nanoVNA (or circuit), and you get the following:
(https://i.imgur.com/FrKSU3dl.png)
It can be seen that the filter works, however it is very very broad: from 50 to 150 MHz. Of course, the Bode plot is not shown in the nanoVNA, but you can see the frequency at which it attenuates most (the length of the coax is \$\lambda \over 4\$ at that frequency) in the phase plot of S11.
You can make the notch filter sharper by placing a 50 ohm resistor at the nanoVNA output (that would be the input impedance of an amplifier, where the coax stub is connected at its input):
(https://i.imgur.com/pE3Dp13l.png)
It is now much narrower: from 70 to 128 MHz. However, the transmission line I used as stub was not a coax, but a twisted pair cable (23 cm long) with an 11 pF capacitor at its output (as load, Zl; I actually used an adjustable capacitor). I measured its characteristic impedance to be 104 ohm, and its velocity factor 0.50, with the nanoVNA.
Changing those data in the script, it plots the following:
(https://i.imgur.com/Gn6nzLMl.png)
It is much shorter than the coax, and it is even narrower due to the higher characteristic impedance: from 83 to 115 MHz.
I implemented it, and it works fine. It reduces the interference from the FM stations. However, it is not as sharp, as the input impedance of my amplifer is higher than 50 ohm.
(https://i.imgur.com/teJuWoZm.jpg)