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 NK 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 noninverting amp = gain of 10/7)
are N! / (K! (NK)!) = 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.comments required from learned friends.
zia

now think of a noninverting 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 710V converter?
best regards.
zia

now as a ratio corresponding to values of K from 1 through (N1) we will
have ratios like 1/N, 2/N, 3/N ... (N1)/Nfor example N=10, we will have these ratios:
1/10, 2/10, 3/10 ... 9/10and 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.
comments required.
zia