RC Circuits — Question 10

Question 10

Timing to a fractional threshold separates RC tolerance from source amplitude

An initially uncharged capacitor charges through R from a constant positive source E. A comparator changes state when vC reaches fE, where 0<f<1. Nominal values are R0=100 kilohm, C0=1.00 microfarad and f=.800. Hard component tolerances are plus or minus 2 percent in R and 5 percent in C; f is exact.

1. Derive the trigger time and calculate its nominal value.

2. Find the exact earliest and latest times allowed by the tolerances, and explain whether changing E changes the ideal trigger time.

Question 10 - Solution

Timing to a fractional threshold separates RC tolerance from source amplitude

(a) Method and calculation

$$fE=E(1-e^{-t/(RC)})\Rightarrow t=-RC\ln(1-f).$$$$t_0=(.100\,\mathrm s)\ln5=.160944\,\mathrm s.$$

(b) Interpretation and further calculation

$$t_{\min}=t_0(.98)(.95)=.931t_0=.1498387\,\mathrm s.$$$$t_{\max}=t_0(1.02)(1.05)=1.071t_0=.1723708\,\mathrm s.$$E cancels because the threshold tracks a fixed fraction of the same source voltage.

Physical check

Hard tolerance limits multiply at their extrema; a root-sum-square estimate would require a statistical model.