RC Circuits — Question 3

Question 3

Three equally spaced samples reveal both the time constant and the unknown final voltage

A first-order RC voltage follows $v(t)=V_\infty+A e^{-t/\tau}$. Ideal samples give v(0)=2.00 V, v(.100 s)=6.00 V and v(.200 s)=8.00 V. The driving resistance is 10.0 kilohm.

1. Eliminate the unknown offset to find tau and V infinity.

2. Find C and write the complete voltage function; verify all three samples.

Question 3 - Solution

Three equally spaced samples reveal both the time constant and the unknown final voltage

(a) Method and calculation

$$a=e^{-\Delta t/\tau}=\frac{v_2-v_1}{v_1-v_0}=\frac24=\frac12,\quad\tau=\frac{.100}{\ln2}=.14427\,\mathrm s.$$$$V_\infty=\frac{v_1-av_0}{1-a}=10.0\,\mathrm V.$$

(b) Interpretation and further calculation

$$C=\tau/R=14.427\,\mathrm{\mu F},\quad v(t)=10-8e^{-t/\tau}\ \mathrm V.$$$$v(n\Delta t)=10-8(1/2)^n:\quad v_0=2,\ v_1=6,\ v_2=8\ \mathrm V.$$

Physical check

Successive voltage differences remove the unknown final value, provided one exponential describes the response.