Electronics > Beginners
PFC Math
TheDood:
Energy in Capacitor:
J = (1/2)·(F·V^2)
P = W = V·A
W = J/s
I'm thinking if I were to divide each half sine wave into X amount of equi-power sections, let's say 14 sections per half sine wave, that I could then sense the V in the storage cap at every zero cross pulse to determine the nessecary power needed to be flowed per half sine wave, or the necessary duty cycle % for every section. With the wave divided up into equi-power sections or sections of equal area the frequency will be dynamic with it having greater time periods near the 0 points of the curve than in the middle.
Its been awhile since calc days, but I'm trying to integrate the sin function with respect to time to get total relative power. From this total area under the curve I'm trying to divide it into X amount of sections, lets say 14, and then solve for (t) setting integrals equal to 1/14 areas of the total. Then after solving for (t), plug (t) back into the instantaneous equation to find V(t) for each interval. This way I can create equal wattage intervals and sense V to trigger interval duty cycles.
I'm not sure my integration or setup of the sin function representing a 120VACrms 60Hz AC is correct? I'm also not sure if I'm solving in the correct units?? Can someone look over my procedure and give me any confirmation or corrections?
Example:
1000uF 400V Storage Cap = 80J
Zero cross comes and the cap reads 300V, or 45J. I've set the PFC to carry out in (14) individual sections per half sine wave.
35J/0.00833s = 4201W
4201W ÷ 14 sections = 300.07W/section
Each period has a predefined constant average V, so I'd divide 300W by each intervals average V to find the required avgI to provide 300W, then I'd compare this Current nessecary to the possible current flow per interval avgV due to the total ESR of the entire cap charging cct, to then arrive at a % duty cycle calc for each interval. Because the time intervals are dynamic or vary in time duration from interval to internal, once a % "On" duty cycle calc was found for 1 section it should be able to apply to all sections?
unitedatoms:
The power factor is best at non reactive load for mains frequency. Not best for constant power over time. Consider controlling amount of power per interval proportional to what equivalent resistor absorbs for given interval's rms voltage.
TheDood:
--- Quote from: unitedatoms on December 21, 2019, 01:25:57 pm ---The power factor is best at non reactive load for mains frequency. Not best for constant power over time. Consider controlling amount of power per interval proportional to what equivalent resistor absorbs for given interval's rms voltage.
--- End quote ---
Hmmm @unitedatoms, Im not sure Im following.
Also the integration or math I posted wouldn't be power, but rather V*s? For power Id have to find the peak power (pk V * pk I) and use that as my Amplitude in the sine function, I think. I think you're saying to use a resistor to sense when current has flowed sufficiently per interval rather than sensing line V and managing via anticipated/calculated/expected V rather than resistor feedback? Still use the same dynamic frequency, ie longer intervals near zero points ect but multiply the intervals rms V by the calculated I measured across R? So Id conduct until a certain % drop in V occurred? Thanks for the reply.
rstofer:
The integral of sin(x) is -cos(x) plus, perhaps, a constant of integration. If you want the area under some portion, say 0 to PI just solve for -cos(PI) - (-cos(0))
T3sl4co1l:
This is probably an alright, detailed explanation:
https://www.st.com/content/ccc/resource/technical/document/application_note/72/e5/be/2c/74/20/45/bb/CD00195944.pdf/files/CD00195944.pdf/jcr:content/translations/en.CD00195944.pdf
--- Quote from: TheDood on December 21, 2019, 01:00:58 pm ---I'm thinking if I were to divide each half sine wave into X amount of equi-power sections, let's say 14 sections per half sine wave, that I could then sense the V in the storage cap at every zero cross pulse to determine the nessecary power needed to be flowed per half sine wave, or the necessary duty cycle % for every section. With the wave divided up into equi-power sections or sections of equal area the frequency will be dynamic with it having greater time periods near the 0 points of the curve than in the middle.
--- End quote ---
You're missing:
- Definition of power factor
- A measure of distortion
- A load (you're just charging a cap??)
- Variation in supply voltage and load current
All of these are solved by traditional approaches like the above, so it should be quite fruitful to study and understand them.
You hinted elsewhere that you might implement a control in Arduino or something. Judging by your current state of knowledge, this is doomed to fail, in a wide variety of hilarious ways -- the practical result being a lot of magic smoke, and possibly injury.
Consider this: I'm an expert in this subject, I know everything from the finest layout parasitics to the highest control schemes (well, some of them anyway; I get the jist of others but have yet to implement them). And despite all of this, I have yet to implement a fully digital, direct microcontroller-based, control for any kind of power supply. (I did do an FPGA-based one, but that's still easier than software.)
This is very much on the Dunning-Kruger spectrum: you will, at the very first, feel ignorant of the subject; next, you will gain confidence as you gain familiarity with immediate material, but remain ignorant of all the other materials they connect with. This is where one "knows enough to be dangerous", and gets overconfident. You literally don't know what you don't know. If you continue, you will hopefully discover those connections, expanding your perspective massively,
In short: create a lesson plan for yourself. Break it down to the component parts, and understand them. Explode the edge cases, where a given subject hits its limitations, and where it connects to others.
A rough outline for this project might be:
- Basic circuits, KVL, KCL, analysis
- AC circuits, steady-state analysis (phasors, complex numbers), transient analysis
+ Likely prerequisites: calculus, differential equations
- Switching power supplies: types, topologies, operation and waveforms, intro controls
- Controls: feedback, pole-zero analysis, stability, compensation; cascaded and nested loops
- Digital controls: DSP theory, pole-zero equivalence (Z and F transforms), PID controller, implementation
+ Prerequisites: a programming language, on a hardware platform (e.g. ASM or C on AVR); familiarity with the platform (read and understand the datasheets, operation of the CPU and peripherals); hard real-time computing; interrupt-driven IO, and atomic access, race conditions and so on
- And some E&M to understand layout parasitics wouldn't be a bad idea either.
With intensive and directed study, expect to spend a month or two on each of these items. And that's if you learn quickly; the average case where these would be covered in college courses several months long will be more typical, and even longer still if poorly directed.
So, expect your whole project to take at least half a year, more likely several. Budget accordingly, in terms of time, materials, money, whatever. Don't be afraid to pick up some college level courses to cover the more opaque or heavy topics. If you have more budget, you could get a tutor to provide additional direction.
Or if the end product is really your ultimate goal, you can ask others to do the grunt work -- probably at a similar budget to the accelerated plan above. I would expect a project like this to take the better part of 2-6 months (or you can always go higher) by professionals, depending on how much optimization is necessary to push efficiency and such.
Without this, just plodding along -- you can certainly arrive at something that works, but you won't have any of the tools necessary to understand reliability or efficiency, and how to improve them. Let alone real-world problems like manufacturability and production, electromagnetic compatibility (EMC) and safety (UL/ETL).
Case in point: the big project I was working on through highschool was an induction heater. This is essentially a resonant power supply without a rectified DC output; instead it heats metal directly by magnetic field. I started with small scale tests, blew up a few transistors, honed my knowledge of circuitry and theory, and eventually (some years later), demonstrated a supply at 5kW, with frequency, phase and current controls, and overcurrent and desat fault protection. (To this day, most of the home-grown designs out there on the internet, are open loop and unprotected; they're a minefield to operate, not practical to do real work with. It's easy to see why a, say, $3000 Chinese power supply is such a good deal, when you just want to make some metal hot.)
Tim
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