Procedure for inverse Laplace transformation to calculate periodic switch-on processes

But they are far more visible and tunable and work a lot better for nonlinear systems, which are the really interesting cases.

There is a legacy of, I think, excess respect for analytical solutions in electronic design, part of the legacy of slide rules and log tables, part in the academic tradition. The really smart people write software for design engineers to use. I couldn't possibly be a math expert for all the different things that I have to do.

Numerical solutions document well too. Just save the .asc file in the project folder.

Sure, there's an equivalence between them if the Laplace transform's ROC contains the imaginary axis.

Practically when solving initial-condition/boundary-value PDEs it can be more appropriate to use one or the other depending on the physical constraints of the problem.

That is to say the transform of your particular solution has to converge and usually you want it to converge to something "nice" like if say your boundary value is a constant at the boundary of some dimension/independent variable the Laplace transform will damp that out while Fourier will return a distribution. Laplace also has the nice property that it transforms a unit step into an algebraic function.

Fourier transform straightforwardly produces the solution to the unconstrained wave equation on the full real line where the general solution has wave packets traveling and dispersing in both directions, or a driven line without loss where the amplitude stays the same to infinity. In steady state amplitude of a driven half-line with loss has to be a sine at one end and zero at infinity, so you'd like the integral transform of your particular solution to converge to something you can work with at boundaries like that.

Anyway it's something like that IIRC, it's been quite a while since I've solved PDEs with pen & paper but point is I have no idea where OP's equation came from, but for solving initial value/boundary value PDEs of that type you have to have to use your brain you can't just transform random shit and generally code-monkey around and hope to come up with anything useful.

Sigh.

Yes you can use a computer to say calculate the magnetic field at a distance r from twisted pair carrying DC current (60Hz is pretty close to DC too) but it doesn't tell you that, blah blah blah several pages of math, modified Bessel functions etc, that for distances greater than about 1/3rd the twist pitch distance, the point of maximum flux density B(max) decreases approximately as 1/[sqrt(pr)]*e^-[(2pi/p)*r] where p is the twist pitch distance in inches, and r is the radius from the centerline in inches.

That's information you could use to actually design something

So the unit step is to turn 'things' on at t=0? Whenever I see the Laplace 's', I just replace with i*omega and it seems to work... There's a bit of confusion in my brain around negative times, negative frequencies (what happens at zero) and getting the prefactors right in Fourier transforms. Does Bracewells book discuss that?

George H.

Come on! it's geek humor. :^) GH

Sigh squared is the probability density.

Cheers

Phil Hobbs

Doesn't matter. You just use analytic continuation, which works fine. Every actual physical process leads to convergent integrals, so there are no worries.

Nah. You can do the hard part with the two-sided transform and then convolve with the transform of the unit step afterwards. (It's a half-strength delta-function plus i/t, probably with a 2 pi in there someplace.)

You say that like it's a bad thing.

When you multiply it by something first, it turns into something you have to look up the solution to. And one-sided Laplace transforms are totally crippled because they don't obey the very powerful Fourier theorems. I do a lot of transforms, but I haven't used vanilla Laplace since I handed in the exam in my undergrad ODE class.

What part of "multiply by the Heaviside unit step" didn't you get?

First year art school, right? ;)

Intelligent messing around is where all the good theorems come from. You're right, it does have to be intelligent though.

Cheers

Phil Hobbs

Sure. He solves the problem by putting the two pi in the exponent and forgetting about it otherwise. That makes all the theorems come out with no ugly multiplicative factors, too.

Bracewell's second edition is selling for cheap on abebooks.com, or you can get a 3rd edition softback from India for $15 plus $10 shipping.

Cheers

Phil Hobbs

You and I have been running this argument for a good dozen years now, and hopefully will keep it up for another dozen. ;)

I couldn't do what I do without a lot of math. Figuring out how good something _could_ be lets you choose from a much wider range of possibilities without excess experimenting (numerical or physical). More important, it lets you know how you're doing--otherwise one risks giving up way too soon or way too late.

Absolutely. And in the git repo.

Cheers

Phil Hobbs

I can always hire a smart guy like you when I need someone to invent a few equations. There's a lot more involved to design and sell a box full of parts.

An idea can be evaluated lots of ways. With math, simulation, testing, or guessing. Most electronic design doesn't need much actual mathematics. None of my EE professors were any good at designing things.

Sure. My math output is mainly what I call 'photon budgets', a particular style of feasibility calculation for various measurement schemes that I've talked about a few times in this very boutique. (I think you have a couple of them someplace, from back when we were working on that EUV litho stuff, so you know what they look like.)

They tend to be full of fun facts, e.g. if you focus a laser beam down to a 1-um diameter spot on some surface, in the steady state the delta-T goes down by half in the first micron of depth. (It's actually a bit faster than that--the input surface has an area of pi r**2 and the spherical shells have area 2 pi r**2.) The local steady state gradient is reached in a nanosecond or so. (The fun fact I talked about last time this came up is that a shiny sphere scatters an incoming collimated beam exactly uniformly in all directions--the full sphere.)

Many of those came from previous photon budgets where I did the math to find them out. I don't have to do that every time. (*)

They also have sections on environmental issues such as the shot noise of sunlight, atmospheric scatter and extinction, temperature coefficients of whatever's relevant, and so forth.

Then a few measurement schemes are evaluated, problems identified, and the best one or two candidates subjected to closer analysis, including circuit issues such as speed vs SNR, power, and cost.

Generally a photon budget for a new-to-me measurement takes about a week all told, including a nice writeup for the files. For things I've done before, it goes faster, but you still have to leave time for the white board and for thought if you want the best results.

Once I get my op amp input capacitance problem solved, I'll ship BEOS3 off and get to work on the next tome, which is largely going to be about photon budgets--I have over 100 of them in stock.

Cheers

Phil Hobbs

(*) The sphere thing I got from my late friend Doug Goodman, who was a pillar of popular optics education as well as a designer of excellent litho lenses. George, you might remember the little kits that the OSA used to sell--they were intended to get kids interested in optics. They had a bunch of moulded lenses, a Fresnel lens, a grating, polarizing film, that sort of stuff. Doug was the mastermind behind that one IIRC.

Something like that. The texts I'm most familiar with are:

and

The first is more geared towards explaining the background towards a goal of understanding numerical techniques. The issue it brings up with respect to the Fourier transform is essentially that the Fourier transform is well-defined, i.e. doesn't return transforms in terms of distributions, in the space of L^2 integrable functions on (-inf, inf), which excludes constants, sines and cosines, polynomials in x, etc.

The Laplace transform damps these on the half-line so the claim is that Laplace can be a more "natural" transform to use in some circumstances for problems on the domain [0, inf) where functions like that appear as boundary conditions or solutions.

If this is a trivial difference that's easy to work around once you know more Fourier/distributional theory then that's the way it is I'm not familiar.

I have a book called "Distribution theory and functional analysis" or somesuch on my shelf which starts with a lot of very theory-heavy expositions on distributions and then goes more in depth into Fourier theorems than you'd get in those books, Greene's functions, Hilbert and Banach spaces and harmonic analysis etc, it's a heavy book and not anything I saw in undergrad as a non-physics major at least, didn't really get a lot out of it the last time I read it like if you were taking a graduate-level quantum mechanics class you'd probably want to know this stuff.

Yes, most private liberal arts colleges do have mathematics and physics and maybe even computer science departments the way many other schools do. Many students aren't there for that stuff they're there to be the next Joss Whedon but the departments do exist, though the quality of the didaction seems to be excellent or meh depending on which of the five faculty members you get there's little in-between.

But no it's not just communism and 2+2 = 5 all day. The point of the liberal arts college originally as I understand it was to produce "well rounded" citizen-scholars well-suited to many types of what we now call "white collar" work, like the Ancient Greeks or some shit.

If you want to learn to be a desk-monkey trained for a purpose you can go to WayHuge U. down the street they specialize in that and have much nicer amenities like jacuzzis in all the lounges I hear; kid's parents like that stuff on tours. also means WayHuge U is often not even that much cheaper nowadays when you factor in potential financial aid for anyone who wasn't a straight-C student, if they're out of state.

That's another case where mathematics proves something that's obviously impossible.

Those same folks are struggling to get nanosecond response from a nanofarad UV photodiode. Without success. I told them we could do it for them. Too many PHBs would lose face now, over the sunk costs.

OK I found the second answer here to be useful. (I never understood the sigma before... how silly of me.)

George H.

We've got a mirrored glass sphere 'gazing ball' in our yard.. (well in the summer time.) I walked around it looking at the sun... and it was the same no matter where I stood. Go to some outdoor garden store on a sunny day and check it out.

George H.

I tried it with a big steel ball bearing and a flashlight and a laser pointer. I still don't believe it.

Ancient history again, unfortunately. It does work, but hasn't been tried recently.

;)

Cheers

Phil Hobbs