Math Problem

Using the multivariate second derivative test to classify the critical points of the function may help:

Are you looking for a closed solution or a "feel" for the domain? Just

so the full term will range over 0 to 1 times V or at most V.

The second term will always be non-negative when F and W are non-negative. The only time f(v) can be negative is if F is negative and the values of F and W (as well as V of course) are such that the magnitude of the second term is larger than the first. I can't pin that down in a closed form but I think this can help you intuitively map the domain a bit more.

Thanks! ...Jim Thompson

You're giving a function, not in terms of time.

Then you're asking a question about it's time derivative.

Either that or that very deliberate-looking dot above f(v) on the bottom line is a typo. (What's a typo called when something is hand-written?)

Or do you mean f'(v), i.e. d/dv f(v)?

d/dv

Picky, picky ;-) ...Jim Thompson

Icky icky.

I flogged it a bit. Partial answers involve transcendental functions, so I suspect there is no closed-form solutions. If there are, they're not available without lots more flogging.

Ah, I'd played with this a little bit before.

Ultimately, I believe Wescott's right. Intuitively, you're emulating a resistor and diode circuit -- a system which SPICE solves iteratively, and a solution which is transcendental (you can only iterate the x[n] = e^x[n-1] form to get an approximate result, there's no closed form analytical solution -- see

Other sigmoid functions come to mind; you can use the Fermi-Dirac statistic (aka Logistic function), 1 / (1 + exp(-x)), to "turn on" the proportional part. But if you unwrap the tanh function, you'll see that's the same thing you're already doing (give or take some constants).

In principle, there exists a function that exactly fills in that little remainder, that you're trying to patch with the Gaussian term. It's most likely transcendental as well... and may not even have a simple description other than to being the remainder to the function in question!

Other sigmoids will have other shapes. Taking the integral of exp(-x^2) might help (erf(x)). There are other kinds out there. Do consider that, since you're SPICEing, you can do some quite excellent numerical solutions, and may not be able to express your desired relation as an equation, but you might be able to with reasonable numerical stability and few additional nodes.

Tim

The derivative of the function is

first we can notice the sum of the first two terms is always positive. second, the third "sech" term is positive whenever kV is positive that last term will be nonnegative with F positive, W positive, V negative, or with V positive, W positive, F negative....

It might be a little bit more amenable to an analytic closed form solution if he was prepared to substitute F/(1+wv^2) for F.exp(-wv^2)

My bad... in my curve-fitting the error looked like (Gaussian) exp(-W*V^2)

Looks the same but easier to deal with: 1-[TANH(K*V)]^2 ...Jim Thompson

-- | James E.Thompson | mens | | Analog Innovations | et | | Analog/Mixed-Signal ASIC's and Discrete Systems | manus | | San Tan Valley, AZ 85142 Skype: skypeanalog | | | Voice:(480)460-2350 Fax: Available upon request | Brass Rat | | E-mail Icon at