Equivalent Capacitance and Uncertainty of a Circuit

Question

How do you find the equivalent capacitance and the expression for the uncertainty of the equivalent capacitance of this circuit with the following information:

 

Point 1: t(s) = 1.33             t - t₀(s) = .365           V_R(V) = 4.50

Point 2: t(s) = 1.53             t - t0(s) = .565           VR(V) = 4.25

Point 3: t(s) = 1.76             t - t0(s) = .795           VR(V) = 4.00

Point 4: t(s) = 3.64             t - t0(s) = .2.675        VR(V) = 2.51

Point 5: t(s) = 4.7               t - t0(s) = 3.735         VR(V) = 2.00

 

The t₀ = .965         V = 5.0

 

 

Answer 1

The basic physics formula is 

 

V/V0 = exp(-(t-t0)/RC))    where RC is the   RC time constant

 

Taking the ln of both sides and rearranging gives

 

t-to  = RC ln(V0/V) = RC ln(RV0/RV)        (since R is a constant)

 

So if t -t0 is plotted vs.    ln(RV0/RV),   the result should be a straight line with slope RC.

I did this in Excel (electing a linear trend line and choosing to show the trend line and calculated parameters).   I got

RC = 4.0805 sec.   with r^2 = .9985    (very good fit)

Since V at t = t0  is 5,  I conclude that the resistance, R, is 1 ohm.

This means that the capacitance is 4.0805 Farads.     

It is more work to figure the uncertainly (standard error).    Neither TI-84 nor Excel do this and the formula is too complicated to write here (you can Google it).   After some work I got

4.0805 ± 0.039