Electronics > Beginners
PFC Math
TheDood:
--- Quote from: dietert1 on January 02, 2020, 03:17:59 pm ---When you take the basic situation of 200 V input voltage and 400 V output voltage, then the "charging" voltage of the inductor is 200 V and the "discharging" voltage also (400-200). For one booster cycle those voltages are nearly constant. So is dI/dt, except it's negative during discharge. Input current will have a nice symmetric sawtooth modulation. It will flow from the input during "charge" (increasing) and "discharge" (decreasing). Effective input current will be twice the effective output current into the cap.
If you succeed to model this situation, you will also succeed to model other input voltages. For example at 300 V input voltage "charging" the inductor will be three times faster than "discharging" it. So the symmetric sawtooth becomes more like a triangle. This time the effective booster input current is about 4/3 of its output current.
This example demonstrates how output current of the PFC booster cannot be proportional to input voltage. If you want PFC, you need to control the input current. The booster output current will have a strong 100/120 Hz modulation and the buffer cap does the rest.
Regards, Dieter
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Thanks Dieter,
So if I had a 15Vinput & 31Vload, and it took 1s to charge up an inductor to the point that it was flowing 1A, that that amount of energy stored, is equivalent to enough Joules, such that during discharge, my load would flow 1A down to 0A, and deteriorate its instantaneous current flow at the same rate that the current increased in the inductor during charge up? So 0-1s the inductor charged up to reach a final current of 1A, then discharge begins, and current through load starts at 1A and drops to 0A from time period 1s-2s?
If 1Vinput instead of 15Vinput were used (still 31V load), it would mean that charge up time would be 1/30th of discharge time?
I'm thinking that in order to draw current in a sin wave, as well as keep current constant through my load, that I'm back to a variable frequency switching solution (but all duty cycle pulses same time duration, because L is constant and I'd want to charge L to same % of max with every pulse), varying duty cycle to match desired current. Creating more pulses near zero points while less near peaks. Splitting the waveform up into an arbitrary # of equi-coulomb segments and then determining Hz per segment, or the amount of pulses needed per segment to equate to the average max coloumbs desired to flow. Then use the inductor to squirt the amount of coloumbs I wanted per segment of sin wave onto a capacitor that powers the load? The current waveform would match the proportionality of the voltage waveform due to the constant R during inductor charge durations, but the amount of coulombs transferred or flowed to load would be constant because in times of greater Vinput more coulombs would want to flow so a larger off time would be desired per period, compared to zero points where more pulses would be desired, and less time off per periods. In each pulse, the current rate pulled from mains would match the sine wave proportionality, but the change in pulses per time would create a constant string of coloumbs for my load?
I'd set the R in the inductor charging loop to match Load R (as best as possible so that input I wasn't distorted so much greater than load I if/when R is small), then I'd size the inductor by setting the max pulse duration of the variable switching frequency to 5 Tau (?, maybe 1Tau?) and solving for L, or something there abouts. The smallest variable frequency will also play into the L sizing, and then perhaps the inductor charging R will have to be less than load R because we're trying to accumulate enough coulombs during peak to cover the off portion of switching period, so depending on discharge times maybe half as big?
If not counting the time off and only connecting the dots of pulse current per V, the input waveform would look like its consuming more I than it is, when being billed for meter costs, what would I be billed for? Having a greater "connect the dots" current compared to actual current is what correcting PFC is all about? So I would be back to bad power factor if the "connect the dots" waveform showed higher Irms than actual I?
TheDood:
--- Quote from: T3sl4co1l on January 02, 2020, 04:09:11 pm ---You still don't get anything for power factor, from looking at coulombs or joules. Only thing PF cares about is current averaged over the filter time constant, and that current being proportional to the line voltage.
Easy way to control inductor current to track an average: a hysteretic controller. Switch on when current is below threshold, and off when above. The duty cycle is just whatever turns up, you don't need to know or care what it is. Frequency either.
You do of course care about minimum and maximum times, on and off. Frequency can't go too high (at high voltage and light load) because of switching losses. Frequency can't go too low (at low voltage) because of fixed filter cutoff. On and off times are limited by the gate driver and other logic. So you should probably have some logic to account for that, to limit pulse widths and/or frequency to keep things reasonable. Which of course will screw up the current, you're no longer switching at the expected points; which leads to further logic, like adjusting the hysteresis band, or implementing pulse skipping, or you might go with average current mode control instead, etc.
There's BCM (boundary control mode) PFC, where one threshold is varied, the inductor peak current (switch-off point), and switch-on is timed when inductor current falls to zero. The average of a zero-based triangle wave is peak/2, regardless of the duty cycle. Easy. Frequency isn't too wild, though still gets too fast at low currents, for which some holdoff is needed, and subsequent adjustment of the setpoint.
Tim
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Thanks Tim,
Oh man, didn't see this until now, but I'm going to have to take a break to digest this info for a second (at work). Ill look up the terms BCM, invariance, hysteretic controller ect, and get back. Quickly though, if my "connect the dots" current is greater than my actual current draw, my PF is crap again, or is it a-ok?
EDIT:
Also, if Vpk is 170V or 170× greater than 1V, Ipk is also 170× greater at Vpk than I(v) when V = 1? So maybe 170× more pulses at 1V than 170V? Or time period at 170V is 170× longer than time period when V = 1?
T3sl4co1l:
--- Quote from: TheDood on January 02, 2020, 07:04:07 pm ---Oh man, didn't see this until now, but I'm going to have to take a break to digest this info for a second (at work). Ill look up the terms BCM, invariance, hysteretic controller ect, and get back. Quickly though, if my "connect the dots" current is greater than my actual current draw, my PF is crap again, or is it a-ok?
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Reality doesn't care if you think you can connect dots or not; all that matters is it averages out right. ;)
--- Quote ---EDIT:
Also, if Vpk is 170V or 170× greater than 1V, Ipk is also 170× greater at Vpk than I(v) when V = 1? So maybe 170× more pulses at 1V than 170V? Or time period at 170V is 170× longer than time period when V = 1?
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So, pulse density modulation? Assuming each pulse can charge the inductor to a given peak current? Yeah, that works, you'll find it takes a far higher frequency than is practical though.
Tim
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