Maths problem
Nov. 17th, 2008 11:25 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
alitalf
I want to solve a tricky maths problem. I really need an engineering solution, and at a pinch I could make adequate approximations, but it would be much better to have a more accurate formula so that I can get very close to the optimum solution before building a magnetic component and doing measurements.
What is happening physically is that there is a magnetic component that transfers a small amount of energy from its input to its output each time the input current is switched on and off, (and rises from zero to a controlled level, then declines back to zero, before immediately doing the same again, in a repeated triangular waveform).
The peak current follows a half sinusoid, and the energy transferred is proportional to the square of the current, so you might think that the power should be:
frequency * some_constant * integral_from_0_to_PI_(sin(x)^2 dx)
The tricky part is that the frequency itself depends on the point on the half sinusoid, and not quite linearly. I could approximate sin(x)^3 as the overall function, but that would not be quite correct - it is probably more like sin(x)^2.7 or something even worse. I need both a way to decide on the correct function for the frequency dependence, and then how to integrate fractional powers of sin(x).
I am more a consumer than a producer of maths, so now I go: "EEK!"
Can anyone help at all?
What is happening physically is that there is a magnetic component that transfers a small amount of energy from its input to its output each time the input current is switched on and off, (and rises from zero to a controlled level, then declines back to zero, before immediately doing the same again, in a repeated triangular waveform).
The peak current follows a half sinusoid, and the energy transferred is proportional to the square of the current, so you might think that the power should be:
frequency * some_constant * integral_from_0_to_PI_(sin(x)^2 dx)
The tricky part is that the frequency itself depends on the point on the half sinusoid, and not quite linearly. I could approximate sin(x)^3 as the overall function, but that would not be quite correct - it is probably more like sin(x)^2.7 or something even worse. I need both a way to decide on the correct function for the frequency dependence, and then how to integrate fractional powers of sin(x).
I am more a consumer than a producer of maths, so now I go: "EEK!"
Can anyone help at all?
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no subject
Date: 2008-11-18 09:31 am (UTC)If you can tell me the function to integrate, then I can do it.
Do you know the current exactly? What is the frequency the frequency of? If it is the frequency of the triangular wave of the current, it doesn't make sense to me to talk about the frequency varying within one cycle, since frequency is number of cycles per time unit.
no subject
Date: 2008-11-18 11:56 am (UTC)The switching frequency, which might be around 100kHz, depends on the time taken for the primary current to reach its programmed level (t1), and the time taken for trhe secondary current to decline to zero (t2).
The system is controlled so that the average current drawn follows the shape of the mains voltage, to a specified accuracy. With the addition of filtering components to keep the current drawn essentially constant with respect to the switching frequency, the whole power supply draws current approximately as if it were a resistor, and adds little in the way of harmonic currents to the mains.
I can't yet see how to determine the duration of a switching cycle at any point in the mains cycle, because, in order to avoid severe inefficiency, the input switch charges the inductor with energy as soon as the output winding has discharged the energy. The current flow is just discontinouous, so that there is no residual current from one switching cycle to affect the control of the next, but this means that the frequency varies over the half sinewave cycle.
What I really want to do is to reach an optimum design that takes into account average power and losses. The loss per switching cycle is not so easy to calculate, because, for typical ferite used for power magnetics, the loss per cycle is seven times as much for a peak flux of 0.2T compared with a peak of 0.1T, but, of course, the energy transferred per cycle is only four times as much.
Whatever the optimum might be, in the end I have to use wire sizes and ferrite parts that of standard sizes, that are normally stocked by distributors. The problem of optimising the design around avaialble parts, as you may observe, is not simple.
It is normally possible to reach a good design, maybe as good as available parts permit, with two or three experiments. Howev er, once I develop a spreadsheet, or some other software approach, to get closer to the optimum design before doing any experiments, much effort will be saved.
Not least, it should be possible to decide which size of magnetic core set will provide the required amount of power while not exceeding whatever temperature is acceptable for the application.
I should be able to figure out what the instantaneouls switching frequency is over the mains half cycle, but even that is not obvious.