Potential Difference Between Two Points — Question 10

Question 10

A divider output tolerance depends on a resistance ratio rather than its total scale

A divider has positive nominal R1,R2 and ideal input E. Output is taken across R2 without load. Let $k=R_2/(R_1+R_2)$.

1. Derive the first-order fractional output change caused by small fractional changes epsilon1 and epsilon2 in the two resistors.

2. For R1=R2 and independent hard tolerances of one percent, give the first-order worst-case output error and explain why equal tracking errors cancel exactly.

Question 10 - Solution

A divider output tolerance depends on a resistance ratio rather than its total scale

(a) Method and calculation

$$V_o=E\frac{R_2}{R_1+R_2},\quad\delta\ln V_o=\frac{\delta R_2}{R_2}-\frac{\delta R_1+\delta R_2}{R_1+R_2}.$$$$\delta V_o/V_o\simeq(1-k)(\epsilon_2-\epsilon_1).$$

(b) Interpretation and further calculation

$$k=1/2:\quad|\delta V_o|/V_o\lesssim\tfrac12(.01+.01)=.0100.$$$$R_i\prime=(1+\epsilon)R_i\Rightarrow V_o\prime=E\frac{(1+\epsilon)R_2}{(1+\epsilon)(R_1+R_2)}=V_o.$$

Physical check

Matched tracking can preserve a voltage ratio even when the total resistance and source current change.