The topic of calculations oddly seems to be a polarizing one around here, with some people discouraging their use entirely, claiming that there is no accurate way to be able to do them at all, and they should be left only to the "professionals". I am pretty sure that is the opposite of what we want if we want anyone to be in this hobby. If you are one of those people, just know that your opinion has already been heard. Let's stay focused on the questions at hand, and not write discouraging comments.

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What we have so far:

This is the equation I have been using so far for many calculations regarding lamps and ballast characteristics:

BallastImpedance=(LampPowerFactor*sqrt(OpenCircuitVoltage^2-LampVoltage^2))/LampCurrent

This is a true equation as far as I am aware, but only when all of the following requirements are met:

1. Ballast is a purely ohmic impedance (usably true for HX or chokes, completely false for CWA)
2. Lamp voltage does not change with current (false, but usable within a small threshold)
3. Lamp distortion power factor does not change with current (false, but usable within a small threshold)

Another inaccuracy of this calculation is my use of average power factor values. I have averaged the power factor values for many different lamps to get average values for each lighting technology, and use those for calculation. But wattage-to-wattage power factor changes exist, I find this is especially problematic with metal halide. This is easy to fix, just calculate it for each lamp (pretty easy), and that is what I will be doing from now on. This still doesn't fix the other issues though.

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What we need:

1. Knowing the characteristics of CWA ballasts will be necessary if we want to be able to use them in our calculations. That would require making the resistance of the ballast into a function dependent on the load presented to the ballast.

2. We need to multiply the nominal lamp voltage by some function of lamp current, to make the voltage change based on current. This requires some knowledge of the thermal characteristics of discharge lamps, which is undoubtedly different for each lamp type.

3. We need to figure out how lamp distortion power factor changes with current. I really doubt there is a way to figure this out without just testing a bunch of lamps.

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If anyone knows anything about how to help with this, I would love to hear it. I know just getting this far is decentle useful, but I would like to go further if possible.

Thank you all! I am curious to see how this goes down...

All calculations we have done so far using those (and related) formulas refer to a steady state condition, or at least an assumed steady state condition. (i.e. they will hold for a warming up lamp too, as long as you use voltage and current measurements made at the same moment)

All processes which are slow (ie. temperature related, as well as any feedback loops going through temperature) are not covered by those calculations, but the opposite is also true : The calculations do hold for any momentary condition, regardless of the long term trends



In steady state, the characteristics of any electrical circuit, which only passes sinewave voltages and currents, can be calculated by linear equations (e.g. Ohms law, vector sum of voltages, etc). When our voltage is not a sinewave, the first thing we can do is disregard this and keep assuming it is sinewave, which is often close enough (and the differences can be packed into empirical factors, such as the lamp power factor)

CWA is not that much different, but there are 2 caveats :

The magnetic saturation mechanism changes the ballast impedance. I.e. we can calculate the impedance from measurements in the running circuit and it will be correct, but not useful because the impedance will be different if the same ballast is powered in another circuit / at different warm up state / etc

With an array of measurements, we can plot a dependence graph which will be correct and useful, even if we don't have the formula for it

CWA Voc is sometimes not sinewave, but sort of a sinewave with warped peaks. However, it is sinewave enough for many purposes, and is clamped to far different waveforms when a lamp is running anyway, and even with them we still use the same calculations....



Go test a bunch of lamps....



If you do accumulate such data, put it in a spreadsheet (e.g. Libreoffice Calc) and you will be able to make useful reference plots with it, interpolate it and more

@Ash
I am aware that CWA is pretty much a fixed impedance when running at a fixed voltage load, and yes impedance this is easy to calculate. But I want to be able to know all of it's possible range of impedances at all of the voltages that could be on it. I want to make it into an equation. Preferably this would not be through load testing, but if that is what must happen then I have no choice.

I was thinking of load testing my two CWA ballasts to see their specs, just as I have done for my HX ballast. When I did it with my HX ballasts, I used the output terminals of an old Variac as a load (variable inductive load). This worked fine, starting at a short circuit and making my way up the windings of the variac. Unsurprisingly when I tried this with CWA, working my way up from a dead short on the Variac actually increased the measured current, because of the capacitor acting as the primary impedance.

Do you have any other suggestions to be able to load test a CWA ballast and get it's characteristics, or ways to calculate instead of load testing? Could I use various different value capacitors as a load (I have countless)? I don't really want to shell out a bunch of money for a Variac-sized wirewound pot just to do this. I only have a lamp for one of my two CWA ballasts, and I really don't want to have to use it because it is very temperamental.

I was not aware that CWA OCV wasn't sinusoidal, that is news to me. I can't even begin to imagine why that is the case.

If I do get these current/voltage points somehow, I will likely plot it on a graph in Desmos so I can have it make the equation of best fit for it. Then I can actually use the data.

Considering the amount of new factors you open up to, experimental results may well have higher precision than theoretical ones (lacking data and equations for many of those factors)

Use automated data logging, whether with "official" instruments connected to a computer, or with something hacked by yourself on a microcontroller. Let it capture entire curves of warm up of lamps at high resolution

@Ash
That might work, and knowing the power factor of the lamp at each of those recorded moments would allow you to plot a usable curve (I am assuming it will change throughout warmup). But how would I measure that change in power factor? All I can know is it's warmed-up power factor from the specs I have on it from datasheets. I can measure RMS voltage and RMS current, but not RMS wattage, which I would need to have to determine PF.

With high enough resolution sampling the voltage and current waveform, you can integrate the power yourself. You don't have to store all this high resolution data, just the final results and at low resolution

Oh my, how would I even do that

Option : Convert the voltage/current signals to insulated / signal ground referenced, connect to an audio recorder (computer line in) and record to a WAV file at stupid sample rate. Process the data from there. It doesn't have to be calibrated to any specific units, use it only for the factors and then apply to the RMS measurements you get from elsewhere

I may have some Perl code lying around to read audio as data if this helps

Oh my, how would I even do that

A cheap and precise way to measure active power + many other useful parameters is to re-purpose a surplus electricity energy meter (real commercial one, not  plug-in toys) . Usually, a DSP chip inside calculates power as an integral of momentary U*I over a few line periods, at sample rate of about 2kHz. This is pretty enough. 24-bit ADCs are typically used, and the precision mainly depends on calibration quality of voltage divider and current shunts or transformers.

Caveats:

On most meters, measuring circuit power supply is feed from voltage input. This power is not high, but will somewhat throw-off  measurements on smaller lamps. May be easily hacked to an aux input, and some more elaborated meters have jumpers for selection from where the power to the supply goes.

Ignition pulses need to be taken into account. A good approach is a SPDT switch which disconnects and shorts meter input from the circuit until the lamp stabilizes.

Input voltage range must be respected. Sometimes, MOVs are placed across the inputs and will burn if overdriven. Also, ADC inputs may start to distort. Feeding lower voltage is usually OK, as ADCs used are very linear across wide range, though the software may start thinking it is a brownout and stop measuring.