Re: Logarithmic response of PN junctions

On Wed, 02 Aug 2006 22:31:37 GMT, "Rich Grise, Plainclothes Hippie"
<eatmyshorts@xxxxxxxxxxxxxxx> wrote:

I was just musing about "log amps", where they exploit the
exponential/logarithmic relationship between the current and
voltage of a PN junction.

My question is, is that always "log e", ie., "ln"? My point
being, is it _always_ base "e", rather than, say, log 10, or
log 2, or log something else?

Is that true for silicon, germanium, gallium arsenide, every
semiconductor?

Is that why they're called "natural logs"? If that's true, I
actually find it kind of spooky! :-)

It's all the same, Rich.

log[to base B] of A = ln A / ln B

For example, if B=10, then you have:

log10 A = ln A / ln 10

But since ln 10 is a constant, it's just the same as:

log10 A = k * ln A, where k = 1.0 / ln 10

In other words, only different by a constant factor. Other than that,
it's the same thing. So you can think of it this way: So long as the
relationship between current and voltage in a PN junction maintains
the same fixed logarithmic relationship, it's all the same. Just a
change in the constant factor and that can be adduced in calibration.

It's not spooky.

And the name comes first from Nicolaus Mercator, I think. The value
of 'e' itself was first found, I think, when looking at compound
interest (or when my son asked me the same question he imagined by
himself some years ago when he was learning about limits) -- that is,
in the case of thinking about the limiting case of (1+1/x)^x as x goes
to infinity.

'e^x' is:

1 + x + x^2/2! + x^3/3! + ...

Taking the derivative with respect to x, you get the same series back
again. If you substitute 1.0 for x, you get the value of 'e'. It
also turns out that the series for sine, cosine, hyperbolic sine, and
hyperbolic cosine relate very closely to this series -- especially so
when you include complex numbers and the allow the imaginary value of
i in x (usually using 'z' instead of 'x' for that purpose.) It plays
importantly as an integrating factor for solving linear ordinary
differentials, too.

Jon
.

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