Logarithmic response of PN junctions

[Rich Grise, Plainclothes Hippie]

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!

Thanks!
Rich

 

[Bob Eld]

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!

Thanks!
Rich

It doesn't make any difference. Log, Ln they are both the same curve and
vary only in constants. Remember you can change the base of a logarithm by
simply dividing by the log of the other base, a constant. For example: logA
= lnA/ln10. And, lnB = logB/loge. You can change to any arbitrary base, 2,
hex, octal, whatever, the curve and the math is the same. The word "natural"
has to do with the base e, 2.718....and its relationship to trig functions,
vectors and other relations of higher mathematics, not to PN junctions.
Bob

 

[Jonathan Kirwan]

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

 

[jasen]

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?

log something.

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

natural log is base e.

Bye.
Jasen

 

[Rich Grise]

It's all the same, Rich.

[excellent explanation snipped]

Wow!

Thanks!
Rich

 

[Bill Bowden]

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.

Interesting. I looked up compound interest and the number 2.71828 and
found the formula for continuously compounding interest every
nanosecond or faster.

The formula is just e^(ry) times the principal, where r is the rate and
y is number of years. So, if you invest $100 at 6% for 1 year you get
100* e^(.06*1)= $106.184 which is only 18 cents more than 106 using
simple interest. Not much difference. But as the time increases, the
difference gets greater. $100 at 6% for 100 years compounded anually is
$33930. But compounded every nanosecond, using the formula P* e^(rt)
the return is $40343 for a net gain of $6413. Problem is, you have to
live 100 years to collect the extra 6.4K.

-Bill

 

[Narf]

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!

Thanks!
Rich

e/ln is no more spooky than saying "The earth revolves around the sun.".
The alternatives (other bases) are less elegant. Less convenient.

Why are some bases more convenient than others?
Binary has been very convenient for digital computers so far.
For continuous functions e seems more convenient.

What some people consider 'spooky' is where the
discrete functions meet up with the continuous ones
but then it's more like a mathematician trying to
decide why analog music sounds better than digital music,
or why discrete logs are so hard to compute.

Want to make a quantum computer out of a diode?
Drive it into chaos but then you have the problem
of how to encode your quantum program on the result.
Will it work? I don't think so. It may be non-linear but
it's still classical.

The best you might hope for is some complicated
lossy method of communicating data at super
broadband speeds. Which isn't use-less (and it has already been done),
but it's not a quantum computer. No spooky stuff.