Calculating the Current in a Single-Loop Circuit — Question 6

Question 6

Worst-case current bounds combine source and resistance tolerances

A source has hard emf bounds $E=12.0\pm.2$ V. Its internal resistance is exactly .500 ohm. Two series resistors have independent hard bounds $R_1=3.00\pm.10$ and $R_2=6.00\pm.20$ ohm.

1. Find the nominal current and exact hard-bound interval.

2. Find the first-order relative uncertainty bound and identify whether it is a substitute for the exact endpoints.

Question 6 - Solution

Worst-case current bounds combine source and resistance tolerances

(a) Method and calculation

$$I_0=12/(.5+3+6)=1.26316\,\mathrm A.$$$$I\in[11.8/(.5+3.1+6.2),\ 12.2/(.5+2.9+5.8)]=[1.20408,1.32609]\,\mathrm A.$$

(b) Interpretation and further calculation

$$\frac{|\delta I|}{I_0}\lesssim\frac{.2}{12}+\frac{.1+.2}{9.5}=.0482456.$$$$I_{\mathrm{linear}}\simeq1.26316\pm.06094\,\mathrm A.$$The first-order estimate is approximate; the ratio bounds are exact under the stated intervals.

Physical check

Maximum current uses maximum emf and minimum total resistance, not all maximum component values.