Vector measurements with the rtl-sdr, 2
In my last post I've talked about a setup for performing vector reflection measurements with the rtl-sdr. I've come up with an idea for a simple time multiplex hardware so that I could receive both the reference and the measured signal with rtl-sdr's single channel ADC. I did some simulations of the setup and I mentioned that I saw some ±5 degree phase errors. I didn't investigate the source of that error at the time.
After spending some more time thinking about it it turned out that the phase errors I've seen in simulations are due to switch cross-talk. It's quite obvious in retrospect. The measured and reference signals get mixed in the switches and this changes their phase a little. It's best to show this with a phasor diagram:
These are the ideal signals. The reference Uref and the measured signal Udut that passed through the device under test. Udut has a different phase and amplitude compared to the reference and I want to measure that difference.
Due to switch cross-talk, what I'm actually measuring is U'ref and U'dut. U'ref is a sum of both the ideal reference and ideal measured signals, but the measured signal has been attenuated by k, which is the switch cross-talk in linear scale. Vice-versa for Udut. εref and εdut are the phase errors due to this addition.
The combined phase error depends on the phase shift α in the signal caused by the device under test and the attenuation of the measured signal. The error is largest when α = 90° and the amplitude of the measured signal is smallest. Some geometry also shows that this maximum phase error ε in radians is, for small errors, roughly the same as the switch cross-talk (CT) minus the attenuation of the DUT (A, ratio between Uref and Udut) in linear scale:
I expect this error to be smaller in practice than what I had in this simulation. First reason is that I made an error and only accounted for one switch in the simulation. I reality there will be two switches and hence, at least in theory, double the attenuation on the unused signal path. The second is that I now plan to use Renesas F2933 switches, which have a much better rated isolation than the F2976 I've considered in my simulation.
Given the limited dynamic range of the rtl-sdr, -80 dB cross-talk or less should probably suffice for a reasonable accuracy over the entire measurable range. I also expect this is the kind of error that I can compensate for, at least to some degree, in software with short-open-load-through (SOLT) calibration. I have to lookup some of my old notes on the math behind that.
Talking about practice, I have the circuit schematic I want to make roughly drawn up on paper. I've decided on all the components I will use. The digital part for driving the switches will be low-voltage 3.3V CMOS logic, since that's compatible with F2933 inputs. For testing purposes I want to also be able to drive the switches from an external signal source and select the signal path manually. Next step is to draw the circuit in some EDA software and design the PCB layout.
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