Hyperbolic tangent transfer function

It is now known that the tanh function isn't quite the right function.

See:

"Exact analytical solution for current flow through diode with series resistance" by Banwell, T.C. and Jayakumar, A.

This paper appears in Electronics Letters, 17 Feb 2000, Volume 36, Issue 4 on pages 291-292. Abstract: A simple analytical expression is presented for the current flow in a diode driven by a voltage source through a series resistance. The proposed solution is based on the Lambert W-function. The new expression leads to an efficient method for extracting series resistance from measured current-voltage data. Experimental results are presented which validate the proposed solution and extraction method

This paper was published nearly at the time of the thread you referenced, but it must have been submitted at least several months prior to that date. Still, since it is the earliest reference to the subject I found via the IEEE, I am impressed that you were looking at this back then.

What led you to consider the Lambert W function and what did you mean by, "So, sure, I cheated to get the actual solution [...]?"

Regards -- analogspiceman

No it isnt. The tanh function is the correct one.

This has nothing to do with the diff pair having a tanh function.

Secondly, I already have prior priority to using the Lambert function to solve this problem.

In fact, if this TC Banwell is claiming the first credit by his IEEE paper (Nov 2000), it pisses me off no end:-)

I obtained this result in Jan 2000, to wit:

So, sure, I cheated to get the actual solution of x=y.exp(y) in closed form, but it would seem that I beat everyone to the punch in its application to transister circuits.

Kevin Aylward snipped-for-privacy@anasoft.co.uk

I don't see anything in his paper about *claims* one way or another. But at the bottom of the first page of the paper it says that the IEEE received his manuscript on July 15,

If you went to the NG link I posted you will see that I posted to the maths NG.

I asked for the *derivation* of the power series solution to x=y.exp(y), as I was using it to solve said electronics problems, around the 80's, prior to the internet.

I got the answer that it was the Lambert W function. So I cheated in knowing it was named the Lambert W function. However, I was using the function, as is by the PS solution. So, the cheat was in name only.

I first studied this equation around 1983. It was in the shumms advanced calculus book. One of the questions was, show that the solution to that equation was the power series I referenced in my paper. So, I had the PS solution, but I never manged to derive it myself.

I realised later, that the solution was relevant to diode equation circuits. I didn't know until the maths posting that it was already considered as a standard function, and had a name. The schumms book didn't mention that fact.

Kevin Aylward snipped-for-privacy@anasoft.co.uk

Only for an ideal transistor having no parasitic resistances (base spreading resistance, various extrinsic resistances). For example, using a pair of 2N2102 silicon transistors, with the current source in the emitters set to 10 mA, the transfer function looks pretty close to a tanh curve. But, just raise the current source to 100 mA and the curve is plainly no longer a tanh curve. At these higher currents the curve almost becomes a straight line between limits. At 10 mA the same effect can be achieved with 50 ohm resistors in series with the emitters. So, the Lambert W function could be used to provide a better analytical description of the behavior of this circuit when resistances are present, since the tanh function is only an approximation, failing as the voltages across resistances rise past a few tens of millivolts. Fifty millivolts across emitter resistances is enough to cause substantial deviation from a true tanh curve.

resistance,

transistors,

looks pretty

curve is

becomes a

ohm

used to

resistances

voltages

emitter

Well DUH!

Large devices run at small currents come close enough to TANH for government work ;-)

...Jim Thompson

Where did you get the impression that I was referring to a first order model? Your own web site plainly shows that when resistances are taken into account, the Lambert function is appropriate to use. What does Jim say above? "Pretty small range, but can be built out." Then I said "...the tanh function isn't *quite* the right function." The first order model is only approximate, since it *is* just a "first order model". I simply pointed out that if one wants an analysis that goes beyond a first order model, the Lambert W function is an appropriate tool.

resistance,

transistors,

looks pretty

curve is

becomes a

used to

resistances

voltages

across emitter

Do you get the impression that I'm saying something contrary to this? Don't I say above "...with the current source in the emitters set to 10 ma, the transfer function looks pretty close to a tanh curve." I'm discussing what one should do when the currents are large enough that this approximation fails. Is it not permissible to raise this issue? Duh, indeed. :-)

In the context of using the Lambert W function, don't you think it would be interesting to develop an expression for a 2nd order model, possibly using the Lambert W? I've been giving it some effort and the it looks to me like it may not be possible to whip it into a closed form expression, even with the Lambert function. I haven't given up yet, though.

A Lambert W formulation would be making the assumption that the resistances involved are constant, which isn't *quite* true due to conductivity modulation. But I think it would be interesting to see just how close such an analysis could come at moderate to high currents, compared to the straight tanh analysis.

resistance,

transistors,

looks pretty

curve is

becomes a

used to

resistances

voltages

across emitter

I say

function

the currents

this

I think it's unlikely that the typical user would be running a device at such a high current that bulk resistances matter.

...Jim Thompson

Needs to be differential (2x 2N2222), to get TANH.

...Jim Thompson

The statement was in the context of using the Lambert W function. Such a function is not required for the 1st order model.

Ho humm....yes, yes, yes... Ho humm....we all know...but beyond the scope of the discussion as far as I am concerned.

Kevin Aylward snipped-for-privacy@anasoft.co.uk

I suspect there was simply a miscommunication problem on what was being addressed.

Kevin Aylward snipped-for-privacy@anasoft.co.uk

I have been running a short simulation with a 100uA current source in the emitter of 2N2222 and 1k resistors on the opamp. all values are in mV 0000 0000 0020 0036.7 0040 0064.12 0060 0081.1 0080 0090.4 0100 0094.9 0200 0099.31 0500 0099.36 1000 0099.42 There is a certain tolerance as I measured the values with the curser

In Ban's drawing, one must take care with Vcc not to saturate the transistors. A fully-differential input is best (I've edited the drawing to show this), with a common-mode voltage well below Vout. Properly done, the 10k pullup resistors may not be necessary.

Of course it is a differential amp +-----+---o | | +Vcc .-. .-. | | | | 1k_ | |10k| | +--|___|--+ '-' '-' | | | | | |\\ | | +------------+----|-\\ | | | | >-+---o +-----)------------+----|+/ Vout | | | |/ Vin |/ \\| .-. o------| 2222 |--+ | |1k |>

When I was in the USAF (1968 - 1976) there were differential pairs in one TO-5 sized package, but with 6 leads. I'd think they'd temperature- track fairly well, but a very quick web search on "differential transistor pair" didn't seem to turn up much, and I don't really know who has a website where I could look up something like that.

Anybody remember them, and if they're still available? Albeit, I've just looked up LM3080, and it does put more stuff into one package.

Thanks, Rich

Or perhaps an OTA, LM3080 or LM13700?