How to work Maths is the time sample domain

So far you have only got the magnitude of Z. The actual value is R + jX. From your samples you can recover the peak values of both V and I. You can also count the number of samples between the two waveforms crossing zero in the same direction. This will give you the phase relationship between them. Now do a vector calculation on I to establish the proportion that is in phase, or at ninety degrees to the volt age waveform and do a V/I calculation for each. These will give you R and X for the R+jX calculation.

d Pearce Consulting

Looking for zero crossings only works if you sample really fast.

If you sample V and I several times over a cycle, and demodulate them with inphase and quadrature sine waves:

Vi = (1/N)sum(V(n) * cos(2*pi/M * n)); Vq = (1/N)sum(V(n) * sin(2*pi/M * n)); (and the same for I)

Then you can treat them like complex numbers, so

Z = ((Vi*Ii + Vq*Iq) + j(Vq*Ii - Vi*Iq))/(Ii^2 + Iq^2).

I'm prone to typos so you'll want to check to make sure my complex arithmetic is all OK, and to make yourself understand it.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Alternatively, you average out the v-crossing intervals and i-crossing intervals to get a mean phase shift.

What is the source of your i and q references?

d Pearce Consulting

Hi Tim,

Is this called quadrature dection?

I would like to understand the maths behind this - can you name the algorith you are using.

Regards Joseph

at

8bit V

I read in sci.electronics.design that Tim Wescott wrote (in ) about 'How to work Maths is the time sample domain', on Sat, 17 Jul 2004:

... which would be possible only if you define M, N and n.

--
Regards, John Woodgate, OOO - Own Opinions Only. 
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

Since the waveforms are sampled only with 8..10 bits, my guess is that

10..20 samples/cycle should be enough. Close to the zero crossing of a sine wave, the waveform can be approximated with a straight line, quite accurately within +/-5 degrees, but I guess that even within the +/-30 degree range the approximation would be acceptable in this case.

Take the last sample below zero and first sample above zero and draw a straight line between these two points and calculate were the line crosses the zero line between the sampling point. This will give the _fractional_ sampling period after the first sample.

This way the phase shift between the current and voltage can be measured to within a few degrees even with 10..20 samples/cycle.

Assuming the sampling frequency is not synchronised with the measurement signal, the sampling points around the zero crossing do not always hit the same spots, thus averaging several measurements should improve the resolution.

Calculating the phase shift between the positive going current and voltage transitions and then separately between negative going transitions may help to detect any DC bias in the measurement system.

Paul

In article , Don Pearce wrote: {... i and q method to measure impedance ...]

Since he is just going for the phase difference between the voltage and the current he can just pick a reference at random. Whatever he picks will add the same value to the two values he needs to subtract.

--
--
kensmith@rahul.net   forging knowledge

While this is true, he still needs a constant phase relationship, which means frequency locking to the signal - which implies some sort of decoder to generate the i and q. In hardware, you could have a Costas loop.

d Pearce Consulting

I assume he is creating one of these signals so he already has the frequency. If not then yes he needs to frequency lock. A PLL would do this for him and the error in the phase detector would drop out.

Based on what was implied elsewhere, I think he has a micro that can do the locking if needed.

--
--
kensmith@rahul.net   forging knowledge

You'd have to average a _lot_ of crossings -- my method will get you an answer in one cycle who's accuracy is only dependent on system noise.

The source of the i and q references is the same timebase he's using to generate his V.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Actually it's only possible if _Joseph_ defines M, N and n. _I'm_ just going to give cheap advise.

M is the number of samples in a cycle (conventional wisdom says 3 or more, but could be anything that builds you up a complete picture of the wave).

N is the number of samples in a vector (should end on a cycle, so than N*M is an integer).

n is the particular sample in the vector.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

The easy way to do this is to generate a sine wave (applied voltage) at frequency F, and take adc (current) samples at 4F. Call a set of 4 sequential samples a,b,c, and d. Then let

I = a-c is the angle-zero (in-phase, "I") p-p value

Q = b-d is the 90 degree (quadrature, "Q") p-p value.

You can take a lot of samples and average; this washes out both noise and any DC offset, and extends the effective precision of your ADC.

(Or, equivalently, make the stimulus F the current source and sample resulting voltage at 4F.)

Then just do a little trig.

John