A capacitor discharges through a resistor

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

A capacitor discharges through a resistor, and its voltage is modelled as

V(t) = V_0 e^{-t/\tau}

. Three measurements are taken, each with the same uncertainty

\sigma = 0.3\,\mathrm{V}

$t$ (s)	0	1	2
$V$ (V)	9.5	5.6	3.4

Starting from the guess

V_0 = 10.0\,\mathrm{V}

\tau = 1.0\,\mathrm{s}

, carry out one Gauss-Newton step. (a) Write the residual

r_i = V_i - V_0 e^{-t_i/\tau}

and the two partial derivatives

\partial f/\partial V_0

and

\partial f/\partial\tau

, and evaluate all of these at the starting guess for the three data points. (b) Assemble the normal equations

J^{\mathsf T}J\,\Delta\mathbf{p} = J^{\mathsf T}\mathbf{r}

, where

J

is the

3\times2

matrix of the derivatives found in (a) (the common weight

1/\sigma^2

cancels because

\sigma

is the same for every point, so it can be dropped from both sides). (c) Solve the

2\times2

system for

\Delta V_0

and

\Delta\tau

, and give the updated parameters

V_0 + \Delta V_0

and

\tau + \Delta\tau

to three significant figures.

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