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Resistors are manufactured with various tolerances. We will use the common ± 5%
When resistors are combined either in series or parallel the % tolerance remains unchanged.
Illustration.
Eg Two 1K ± 5 % resistors in series form a combination with a range of
2(1000 ± 50) = 2000 ± 100
= 2000 ± 5 %
Two 1 K ± 5 % resistors in parallel combine,
The reciprocals of numbers have the same percentage uncertainties as the original numbers.
Using the formula and blindly applying the rule, “When multiplying or dividing quantities, add the % uncertainty” suggests a larger error of three times the tolerance ie 15% percent in the case of 5% resistors. However the assumption that tolerances add when quantities are multiplied or divided is only valid for independent quantities.
Clearly (R1 + R2) must have its maximum value when R1 and R2 do so the uncertainty reduces again to 5 % . Percentage uncertainties only add with independent quantities.
The output voltage of a voltage divider with a pull up resistor R 1 and a pull down resistor R2 with the output taken across R2 is given by
· The uncertainty of the voltage divider output is given by the formula
The above formula can be proved using the following standard assumptions
A. The % uncertainty in products and quotients of independent variables is the sum of the individual uncertainties.
B. The % uncertainty in reciprocals is the same as in the original.
This is a special case of the above rule. The % uncertainty in 1/R is the sum of the % uncertainties in the numerator and denominator. There is 0 % uncertainty in the number 1. The percentage uncertainty in R is its % tolerance and this is the total uncertainty.
The % uncertainty in is equal to the % uncertainty in
The right hand side expression has the advantage that R1 and R2 are now a simple quotient of independent variables. The % uncertainty in is twice the % tolerance.
The % tolerance in must be less than that since the number 1 is fixed.
In fact the actual tolerance in = the actual tolerance in