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

Andreas Poschinger


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
reading at the respective normal LL scale (thus reading e^(t/T)) and
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


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