resistors

[Miguel A. Lerma]

What is the minimum number of 1 Ohm resistors that
you have to combine in order to obtain a given rational
resistance r in one of each of the following kinds of
circuits:

1. A circuit built by starting with one (1 Ohm) resistor and
in each step adding another resistor either in parallel
or in series to the previously built circuit?

2. A circuit built by starting with single (1 Ohm) resistors
and iterating the operations of connecting circuits either
in series or parallel to obtain larger circuits?

3. An arbitrary circuit (built only with 1 Ohm resistors)?

The circuits of the first type are ladder networks, and the
problem can be solved with continued fractions - the number
of 1 Ohm resistors needed is the sum of the terms of the
continued fraction of r. For instance for r = 5/6 Ohms we
need 6 resistors: 5/6 = 1/(1 + 1/5) = [0,1,5], which corresponds
to 1 resistor in parallel with 5 resistors placed in series.
With circuits of the second kind we would need only 5 resistors:
a set of two resistors in parallel placed in series with
a set of three resistors in parallel: 1/2 + 1/3 = 5/6.
It remains to obtain in general the optimal solution
for this kind of circuit. Same for general circuits.

Miguel A. Lerma