I'm highly skeptical there is an exact solution. Can you document the source of this? A numerical solution is a piece of cake by iteration with a modern spreadsheet but as I said, I doubt there is an analytical solution. I spent quite a long time looking for one with no luck.

Hello again,

I had said this is an exact solution, not an analytical solution, which should be apparent by inspection. Thus your skepticism would be either about an exact solution, analytical solution, both, or neither. As to analytical, it's not that although there may be an analytical solution. But if there is, it could come from that equation or one like it whereas it would never be able to come from a purely numerical calculation. The solution comes from a set of analytical equations so there is a chance it may be considered analytical anyway, but i wont claim that at present, and i dont think i need to because the beauty and simplicity of it should stand for itself by it's own right.

A completely numerical solution is different in that it uses general procedures to calculate the solution from the describing ODE's while this one uses analytical solutions from analytic geometry to arrive at the final solution, which may or may not be analytical, but in it's present form it's not as far as i can see right now.

There's also a fairly simple one for the full wave case which i did not post yet in order to give the OP a little more time. We can probably discuss it next though and when you see where these equations come from you'll see why they can be called exact.

The solutions presented here are simpler because certain things about the circuit are just plain simpler, and thus lead to simpler form solutions. Note that these are easily calculable

It would probably be best though if you described what your idea of an analytical solution is.

 

I'm highly skeptical there is an exact solution. Can you document the source of this? A numerical solution is a piece of cake by iteration with a modern spreadsheet but as I said, I doubt there is an analytical solution. I spent quite a long time looking for one with no luck.

Did you ever see this: https://forum.allaboutcircuits.com/threads/capacitor-input-rectifier-filter-analysis.33655/

 

Did you ever see this: https://forum.allaboutcircuits.com/threads/capacitor-input-rectifier-filter-analysis.33655/

Hello there,

I think the graphic to that article is missing, although if anyone wants to see it i can probably dig it up.

However, that article only deals with some non ideal components anyway. A more general solution is shown in the attachment as Equ1, although working that solution into a complete solution for a complex rectifier circuit can take considerable time. It is general enough however to cover every possible case no matter how complicated. For a very complicated rectifier circuit the solution comes out in the form of a set of piecewise analytical expressions, or at least pseudo analytical. The other equations there are for a rectifier circuit that includes capacitor ESR.

The case we were dealing with here though was much simpler, in that the first part of that general solution is already solved, so we can go right into the analytical geometry part, which ends up being just one or two equations that are readily solved unlike the ODE's that would otherwise have to be solved using some method.

I think we may have discussed this once on another site. I had looked for that but could not find it. We had a solution worked out for a simple full wave rectifier.

 

Did you ever see this: https://forum.allaboutcircuits.com/threads/capacitor-input-rectifier-filter-analysis.33655/

I might have. But as soon as he wrote that, "This equation can be solved numerically for ton.", I lost interest and may have dismissed it as just another example of what I still claim, that there is no analytical solution. I'd take another look if the links weren't all dead.

update - I went looking to see if I could find that article in full. No luck except for one place that requires signing up for an account. But in my searching I did find an interesting chart in this article. It could be a handy reference to save time making the calculations yourself.

 

I might have. But as soon as he wrote that, "This equation can be solved numerically for ton.", I lost interest and may have dismissed it as just another example of what I still claim, that there is no analytical solution. I'd take another look if the links weren't all dead.

update - I went looking to see if I could find that article in full. No luck except for one place that requires signing up for an account. But in my searching I did find an interesting chart in this article. It could be a handy reference to save time making the calculations yourself.
View attachment 135455

Hi again,

Numerical solutions have to be used in many practical circuit calculations, it's not a big deal. What is a big deal however is what form the equations are in to begin with. If they are in ODE form then there's a lot of number crunching to be done, but if in an analytic or semi analytic form then there is just a little number crunching to be done and that's considered pretty good.

For example, if we have to use a number solver to solve an equation in one variable, it's much simpler than having to solve a set of ODE's.

Some of the outcome of the geometric analysis comes out analytical. I noticed that in that link the chart included Rs, but i also dont see Rs changing with a given capacitor value. That's one of the drawbacks of charts, although it's still an interesting chart and i didnt think of making one until i saw that so maybe i'll attempt that at some point. However, i would not use 20 pages of calculations and stuff

For an example of how part of the geometric solution is analytic, one of the important calculation points is to be able to calculate the departure point where the capacitor discharge wave leaves the forced sinusoidal part, which i call simple "t1". Since there was Rs in the chart, i wanted to include that here to calculate the departure time but want to review the formula again first, so this is for Rs=0:
t1=atan(1/(w*C*R)/w

and that is completely analytical and is exact. It's simplicity shows that these circuits dont have to be that complicated even for an exact analysis. Once we have an equation like this it is often easy to use for something else too.

I should mention again though that this solution comes from using an ideal diode or diodes where they act as a switch as in that link where there is zero voltage across the diode when conducting. This is often acceptable, however if not, we just have to include the diode equation in the formulation and then we can get something that will be very close to real life. We do need the diode spec's then though that would come from a spice model statement for example.

If i can get to it i'll create a chart and see how it compares, but you can see why i like equations better than charts now, and that is because we can use the equation for any situation while the chart may only show some values.

I am pretty sure the Electrician and myself and another person discussed this on another site and came up with some interesting solutions. He might know where they are as i have not kept the link.

 

I am pretty sure the Electrician and myself and another person discussed this on another site and came up with some interesting solutions. He might know where they are as i have not kept the link.

http://www.electro-tech-online.com/...te-the-value-of-a-smoothing-capacitor.106374/

 

http://www.electro-tech-online.com/...te-the-value-of-a-smoothing-capacitor.106374/

Hi,

Thanks a bunch, i wanted to review that thread myself too and try to remember what we did there.

 

Numerical solutions have to be used in many practical circuit calculations, it's not a big deal.

Oh I completely agree. This is not an especially difficult problem once you accept that you can't solve directly for t1. But with just a little spreadsheet magic you've got it.

 

Oh I completely agree. This is not an especially difficult problem once you accept that you can't solve directly for t1. But with just a little spreadsheet magic you've got it.

Hi,

But just to note, i draw a huge distinction between different numerical forms.
For example the simple formula for t1 as compared to a simulation in a circuit simulator. The circuit simulator uses a completely different numerical approach, which i tend to call "entirely" numerical, while the analytical expressions and even pseudo analytic expressions i place in a different class of solutions because they lend themselves to further study in a more straightforward way, even though obtaining those solutions is not always as straightforward as we would like.
There are also arguments about how different numerical approaches vary from the true solution as time progresses further and further. Some numerical approaches can vary away from the solution while the pseudo analytical type can stay true longer.
I did a quick study on the pendulum a long time ago which in it's most perfect form has a slightly non linear solution. I was surprised to see how the solutions can vary as we use different numerical techniques in different number crunching environments. With the same basic number precision some can be very very accurate, while others are just a little accurate.