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Comparison of E12, E24 and E96

Maybe you asked yourself: How good can a voltage divider made up of resistors of a given series approximate even odd ratios? or: Is E96 a must when building an opamp circuit? Here I will give some information on this topic.

Method used: To compare the three standard resistor series, the best pairs (and the relative error) were calculated for one million resistor ratios in the range 1 .. 10 using the code behind the online calculation tool. This data is applicable to any resistor ratio outside of this range, as we can adjust the decade of one of the two resistors to come into the 1 .. 10 ratio range. For example, the relative error for the 1:95 range will be the same as for the 1:9.5 range.

Figure 1: Relative error for resistor ratios R1/R2 in the range 1 .. 10. When it comes to finding a pair of resistors to approximate some given ratio, the E12 series is clearly outperformed by E24 and E96.

The relative errors plotted in figure 1 indicate that E12 is in most cases not suitable for circuits with opamps, where one often has to be close to the required resistor ratio. E96 is (of course) the best of the series here, however most circuit designers don't have a complete E96 at hand. E24 seems to be a good compromise, since most resistor ratios can be approximated quite close. Depending on your application the only range where E24 could be insufficient for your application are ratios between 1:9 and 1:10. For a range of 9.687, the online calculation tool finds the combination 160:1500, while for a range of 9.688 the tool will find 100:1000. In this range you might consider opening up much more possible combinations by using three E24 resistors.

How good is the chance to approximate a given resistor ratio with a series? In figure 2, we sorted all 1000000 data points according to their relative error. One thing you have to know when interpreting this figure, is that I generated the ratios using an exponential function: r(n)=exp(n*ln(10)/1000000), with n being the number of the data point in the set (0<=n<1000000). A large distance between the ratios when coming close to 10 appears to be reasonable since we're discussing the _relative_ error.

If you want to stay beyond 1% relative error, E24 is fine for over 90% of all ratios. E96 is fine for all ratios tested, since the maximum relative error that occurred was about 0.9%. This amplifies that E96 is the best series of the tested ones, but I stick to E24 in most cases.

Note that we neglected the tolerances of the resistors. Finding a resistor pair with a relative error of 1% is fine, but taking 5% tolerance resistors to realize it would be a bad idea.

Figure 2: Relative errors sorted. From the graph you can read how good is the chance to have an relative error less than some threshold. For example with E12, about 87% percent of all possible ratios can be approximated better than 5%.

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