The Math

[dave]

Before I purchase or make the analyzer, I wish to study the mathematics of the vector measurements. I am especially interested in the method of resolving the phase ambiguity.
Any help or references would be appreciated....... dave

[Bob]

Hi Dave,

As shown by the diagrams in the manual, there are two sine waves proportional to the load voltage and current at 1kHz. The mixing (heterodyning) process involves multiplying two sine waves. Using trig identities it can be shown that the amplitude and the phase relationships are preserved, so all subsequent processing can be done at the lower frequency in the audio range.

These waveforms are both sampled 16 times during one cycle using a 12 bit ADC. From this raw data the Fourier transform is used to find the amplitude and phase.

This procedure gives the true phase (positive or negative) without a dead zone around zero degrees, like the Analog Devices AD8302 that is used in some analyzers. Some analyzers that only measure the magnitude of the impedance take two readings at slightly different frequencies and deduce the sign of the phase from these two readings. This works pretty well for lumped circuits (RLC) but it doesn't always work for distributed circuits like antennas.

The raw data gives the impedance of the load (real and imaginary components) and from this the other parameters are calculated. The equations used can be found in text books or the ARRL Antenna Handbook.

73/ Bob

[dave]

Thanks Bob:
With your help, I have done the Math.
73 David VE3KL

[Deleted]

Bob,

When I use "Refer to Antenna" with Method B, I set loss to 0dB and then expect to see no change in SWR from the "standard" (uncorrected) scan, but only a change in Rs and Xs (i.e. going round a circle of constant radius on a Smith Chart). However, I do notice that the SWR changes. The resultant Rs/Xs values are not exactly what I expect, taking into account the electrical length of the coax. I am doing this just to make sure I understand what is being calculated. The differences are small - no doubt inconsequential - but finite. It seems I am missing something. Can you give me a hint?

Thanks.....John

[dl4raj]

Hi John,

while I'm not Bob I may have the answer for you though.
The Z° of any transmission line,coax or open wire,
has a small imaginary part besides its nominal real part
of the line impedance.
(There is only one single fequency in the GHz range where Z° becomes pure real.)
Given a certain load of any kind on the end of the line you will therefore find periodically small changes of SWR
-over frequency-
on all TL's if your analyzer like the AIM is able to measure it.
The loss of the line is the reason for the small imaginary part
of Z°.

73
Clemens
DL4RAJ

[dave]

Hello John:
Calibration of network analyzers is a very complex and ongoing process. I think that the calibration methods will evolve very nicely for this instrument. In the mean time you might want to see what Agilent do for their instruments. See www.home.agilent.com/agilent/editorial.jspx?cc=US&lc=eng&ckey=895315&id=895315 for a tutorial. We can all help by making precision standards. Measurement errors are a fundamental aspect of these instruments. These errors are determined by the manufacturer and also by its users (You and me).

VY 73 Dave VE3KL

[Bob]

john said:

Bob,

When I use "Refer to Antenna" with Method B, I set loss to 0dB and then expect to see no change in SWR from the "standard" (uncorrected) scan, but only a change in Rs and Xs (i.e. going round a circle of constant radius on a Smith Chart). However, I do notice that the SWR changes. The resultant Rs/Xs values are not exactly what I expect, taking into account the electrical length of the coax. I am doing this just to make sure I understand what is being calculated. The differences are small - no doubt inconsequential - but finite. It seems I am missing something. Can you give me a hint?

Thanks.....John

Hi John,
That's an interesting test. I hadn't thought about it before but when I tried it, the SWR is exactly the same when the loss is set to zero. Of course the SWR is different when loss is taken into account. Make sure you have the correct impedance. The specified length is very critical. I found even 0.1ft makes a noticable difference. By carefully specifying the length and the loss, I got the variation in R to within 5% with a 200ohm terminating resistor.

The Z is affected by the loss. I've uploaded a picture showing this.
w5big.com/ReftoAntennaB.gif.

I noticed the default values intially loaded in this dialog box are not correct. I'll take care of that.

73/ Bob

[Deleted]

Thanks to Clemens, Dave and Bob (especially for new AIM 623) for comments and links to OSL tutorial. I am satisfied that my original question about reproducibility is answered by the statistical fluctuations that can be expected from successive measurements. I want to work offline with data and I use the .scn file which contains all necessary information to calculate everything. [Bob, your manual indicates that 5 pieces of info are provided for each frequency in the .scn file. I see only the two that are really necessary: zmag and angle.]

Not being at all familiar with PCs, I transfer this file to my Mac and then generate all the plots and corrections (Method B only at the moment, but now this is almost indistinguishable from Method A on the PC) that I need, along with tests on the cable. I was going to write about curious values in the .csv files that version 622 can produce when SWR is large; (e.g. negative values of Rs and Xs when the angle is in the positive quadrant. But since using version 623 I have not observed any strange results. Great. (Incidentally, when the angle (in radians) in the .scn file is converted to degrees for the .csv file, I notice that the conversion factor used is 57.3 instead of 180/pi.) I also notice that when calling up methodB, the dialog box always converts to feet and may do this twice if one doesn't notice. It would be nice if it didn't do this as someone else has suggested.

The AIM 4170 is an excellent piece of equipment. Thank you, Bob. By the way, I have solved the BSD problems I mentioned in another thread by using the PL2303 USB/RS232 converter instead of the fancy Belkin items. I have had not one crash since the change. (Much relieved.)

73 John (G4KLA)