# A combinatorial divider

• assume N nominally equal resistors connected as shown in the figure. Also assume that the
relay switch resistance Rsw << R. now if we select K resistors of N at random and turn the corresponding relays OFF and the remaining N-K relays ON.
this is equivalent to the divider shown at the bottom.
now the number of combinations which are possible for a given N and K
(say N=10 and K=7) for a ratio of R/7/R/3 = 3/7 (if used in a non-inverting amp = gain of 10/7)
are N! / (K! (N-K)!) = 120. so there are 120 ways to perform this ratio. if all of these ratios are
averaged the individual variances of the resistors will be averaged.

-zia

• now think of a non-inverting amplifier with the divider as shown above for a nominal gain of
10/7 (= 1 + 3/7) for an LTZ1000 / LM399 type reference (7V approx) to be scaled up to 10V.
if we select 7 resistors from 10 at RANDOM, turn these relays OFF and the remaing relays ON,
wait for some time and repeat the same action.

what would the performance of such a 7-10V converter?

best regards.

-zia

• now as a ratio corresponding to values of K from 1 through (N-1) we will
have ratios like 1/N, 2/N, 3/N ... (N-1)/N

for example N=10, we will have these ratios:-
1/10, 2/10, 3/10 ... 9/10

and the number of combinations which can be averaged for accuracy enhancement would be:-
10, 45, 120, 210, 252, 210, 120, 45 and 10.

this means if you want to have a ratio of 1:2 (5/10), there are 252 ways of making that ratio,
and if ALL of these are averaged, the ratio shall be more accurate.

similarly the ratio of 1:10 or 9:10 would be possible using 10 combinations.

this could be used to make accurate ratio references.