Data Analysis, how does it actually work?

[gene]

Hello,
I work in training department and inspired my motto of the article "When SWR isn’t enough — here’s a tool that you can build", we assembled voltage and current sensing schematics from Bob's article, QST, Nov 06.
Instead of discrete Fourier transform (we are not at this level of Math yet), we use oscilloscope to measure Current and Voltage magnitudes and a phase shift between them. And it works fine, see picture in the file.

lecont.net/wp-content/uploads/2015/01/AIM-Measurements_p.pdf

The question is - how to calculate, for example, a value of inductor knowing Current and Voltage magnitudes and a phase shift?
It is known that for good accuracy  the instrument needs to be calibrated.
So, we did 4 measurements: Open, Short, 50 Ohm, and Inductive load, see values in the file. Now, how can we use the data to calculate the actual inductance of the load?

May be the question is very basic for most of the members of this forum, but the answer to it would help me explaining HF fundamentals to our technicians.

I am open for any relevant discussion.
Eugene

[Bob]

Hi Eugene,

For this type of test, be sure to AC couple the scope. That will cancel the DC offset so the sine waves are symmetrical.
For the amplitude of the voltage and current you can use the peak to peak values since only their ratio is needed.
Measure the phase carefully by averaging the shift between the positive slopes and the negative slopes and averaging them. If the signal generator does not have any distortion and you're AC coupled, these shifts will be the same.

The impedance is: Z = R + jX, where X is the inductive reactance = 2*pi*F*L
R is the resistance associated with the inductance and R is usually much smaller than X.

Phase_angle = arctan(X/R) phase is read from the scope, in this case it's approx 80 deg.

Magnitude of Z = Magnitude(voltage)/Magnitude(current) these values are read from the scope

Z is complex, so R = cos(phase)*mag(Z)
X = sin(phase)*mag(Z)
F = 1000 Hz

X = (2.2/1.52)*sin(80.06) = 1.4256 ohms
L = 1.4256/(2*pi*1000) = 226.9 microhenries

I highly recommend using a program like Speqmath (or an equivalent program) to do the complex math calculations. Speqmath is free and it's very user friendly. All the numbers are assumed to be complex so you don't have to worry about declaring variables.
The AIM4170 manual has a tutorial on complex arithmetic in the appendix. This can be downloaded from my website (it's all included in the zip file).

-- 73/Bob - W5BIG

[gene]

Bob, thanks a lot for the detailed explanation. I have some experience with complex numbers and will definately try Speqmath to get more practice.

The main question is - how to use Short, Open and Load measurements, e.g. how to calibrate? Since we use oscilloscope to measure magnitude values, the result is always a voltage drop on 20 Ohm resistors. I believe, calibration would help us to calculate actual current and voltage in the load and to eliminate some uncertainties..

I understand, this may be a heavy subject, but may be you could give general guideline or refer to source if information which is intelligible even to the non specialist.
It is interesting, how did you do it in AIM instrument? I have been using AIM-4170 for years and after calibrating it demonstrates very good accuracy. Unfortunately, nothing mentioned about calibration in the QST article.

Thanks and 73, Evgeny

[Bob]

Hi Eugene,
The calibration procedure used in the AIM involves two port network theory. You can get some info about it here:
cp.literature.agilent.com/litweb/pdf/5950-3000.pdf

Appendix C has the equations that are needed to do the open, short, load calibration. 
Speqmath is very handy for simulating these equations and testing various conditions.

You can get a feel for how this works by calibrating the AIM first with the cal loads at the RF connector. Then attach a piece of coax several feet long and record the data when the cal loads are attached at the far end of this cable. You can feed this raw data into Speqmath and work backward to cancel the effect of the coax.

--73/Bob

[gene]

Great!
This will keep me busy during the rest of my vacation.
Thanks a lot for your kind support.
73, Eugene