#### Re: Use of Inverse LogLog scales Part 3: Expressions containing e^(-t/T)

Andreas Poschinger

Hi,

the third application is the computation of expressions containing
e^(-t/T), especially the calculation of tables of values for these
functions. For this application the inverse loglog scales are not
necessarily needed, but make it simpler to follow what happens by a more
straight forward recipe.

These e^(-t/T) functions often occur as impulse or step response of
technical system. If one e.g. changes the valve of a radiator in order
to reduce a room temperature from 20 deg C to 15 deg C then the
resulting temperature function will roughly be something like 15 +
5*e^(-t/T). Depending on the room T may be something like 2h, so that we
first need to compute x=e^(-t/2) for all points in the table of values
and then 15+5*x. The other way round, temperature from 15 to 20 would
roughtly follow 20-5*e^(-t/T), so that it is almost the same procedure.

For all values of t in principle we need to repeat the following steps
though the first step only has to be done twice:

Set 2 on C to 1 on D (or 2 on C to 10 on D if out of range)

Cursor to e.g. on 3 of C; on LL03 0.223 for t=3h; on LL02 0.861 for
t=0.3h; on LL01 0.985 for t=0.03h

Then cursor to next position until all t are evaluated.

Next 15+5*x needs to be evaluated. Depending on taste +15 can be done in
head so that only 5*x needs to be calculated:

CI 5 to D1 (or to D10; for general setting of 5 *); at e.g. 0.861 on C
4.3 on D; in head 15+4.3 = 19.3 in total.

Alternatively 15 could be factored out and calculation of 1+5 x /15
could be performed.

Instead of inverse loglog scales also normal LL scales can be used by
inserting this result into the second step by using CI scale:

Set 2 on C to 1 on D (or 2 on C to 10 on D if out of range)

Cursor to e.g. on 3 of C; on LL3 4.48 for t=3h; on LL2 1.162 for t=0.3h;
on LL1 1.015 for t=0.03h

Now we need to look on CI!:

CI 5 to D1 (or to D10; for general setting of 5 * as before); at e.g.
1.162 (for t=0.3h)  on CI 4.3 on D; in head 15+4.3 = 19.3 in total.

This inverse loglog scales only help in order to have a slightly simpler
recipe but not to save work in this use case.

So even with inverse loglog functions table building is quite a lot
work, so that uniform functions  (in the example  e^-x) plot on paper
were used. The actual results were obtained by scaling the x and y axis,
e.g. in the example to write 20 instead of 1 on y and 15 instead of 0 on
y, and 2h instead of 1 on x axis. By use of these uniform function
papers as well as Bode paper and Smith charts paper most likely students
in Germany could "survive" their studies without ever buying a high end
slide rule comprising inverse loglog or even hyperbolic scales.

The next part will be about hyperbolic functions. It is a special case
of reading the inverse numbers as in case 1.

Does anybody know more use cases for the inverse LL scales than those
four mentioned so far?

Best regards

Andreas

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