We are measuring small changes in electrical length in video amplifiers (1 to 30 MHz) and coaxial cables as a function of temperature, all at ten megahertz (with time-mark modulation components to perhaps 30 MHz).

The typical total electrical length is about 300 nanoseconds for cables and 5 nanoseconds for each amplifier, and the changes are in the few picoseconds, so the desired resolution is one picosecond or smaller, all with 10 MHz sinewave signals.

Vector Network Analyzers have absolute phase angle uncertainties of order 0.3 degrees rms, which is far too coarse. However, the trace noise (phase) width is better, with 0.03 degrees rms for an IF bandwidth of 3 KHz being typical. (This from the E5061A data sheet.)

The basic question is how the trace noise (phase) varies with IF bandwidth. First-order theory implies that trace noise in phase will vary with the square root of the ratio of bandwidths, so if we used an IF bandwidth of 10 Hertz, the trace noise (phase) would be (0.03)(Sqrt[10/3000])= 0.0017 degrees rms.

However, a VNA may not be first order, and there will be multiple noise sources, not all being gaussian. Comments?

The typical total electrical length is about 300 nanoseconds for cables and 5 nanoseconds for each amplifier, and the changes are in the few picoseconds, so the desired resolution is one picosecond or smaller, all with 10 MHz sinewave signals.

Vector Network Analyzers have absolute phase angle uncertainties of order 0.3 degrees rms, which is far too coarse. However, the trace noise (phase) width is better, with 0.03 degrees rms for an IF bandwidth of 3 KHz being typical. (This from the E5061A data sheet.)

The basic question is how the trace noise (phase) varies with IF bandwidth. First-order theory implies that trace noise in phase will vary with the square root of the ratio of bandwidths, so if we used an IF bandwidth of 10 Hertz, the trace noise (phase) would be (0.03)(Sqrt[10/3000])= 0.0017 degrees rms.

However, a VNA may not be first order, and there will be multiple noise sources, not all being gaussian. Comments?

The E5061A is probably the worst of our current analyzers with respect to trace noise, so an ENA will be a bit better and the PNA better yet.

For trace noise, you can think that the noise floor will reduce 3X for every 10X in IF BW decrease.

However, a more important effect is cable stability over the measurement time. You will see the trace noise bottom out around 30 Hz or 10 Hz, and then start to increase again, due to long term drift, due to the fact that the measurement time is so long that minute changes in the test cables, VNA stability, and DUT Cable stability will start to affect the trace over very long measurement time spans.

Note, that the absoulte phase accuracy is probably of no concern to you in your measurement. It is the accuracy with which you could repeat a phase measurement on any VNA anywhere in the world with any cal kit, then compare against any other VNA and any other cal kit.

Since you will be using the same VNA with the same cal kit to measure the same cable over different conditions, the only thing you care about is noise in the measurement and stability of the test system. Very likely, the worst aspect of the system is the test lead cable used to test your DUT. It is very difficult to control the phase of these test leads sufficiently if you are not very careful