Quote of the Day

Engineering is achieving function while avoiding failure.

I have been presented with a large amount of experimental data from which I need to determine many exponential time constants. There are so many time constants to calculate that I need to automate the process.

I have data with dozens of exponential curves like shown in Figure 1. I My plan is to estimate three points from every exponential curve: (t0, V0); (t1,V1); and (t2,V2). I will then determine the exponential time constant (τ) and the final voltage (*V _{F}*) by fitting these three points to Equation 1.

Eq. 1 |

where

*V*is the final voltage of reached by the curve._{F}*V*is the initial voltage of the curve._{I}*τ*is the exponential time constant that I need to compute.*v(t)*is the exponential voltage as a function of time.*t*is time.

Because I have two unknowns (*τ*, *V _{F}*), I will need to solve two equations. In my case, I cannot see the point where the exponential curve begins – I will show below that it does not matter. In Figure 1, I select a reference point (

*t0*,

*V*) and measure the two other points relative to

_{0}*V*, i.e., (

_{0}*V1*,

*t1-t0*) and (

*V2*,

*t2-t0*).

To find (*τ*, *V _{F}*), I chose to use a nonlinear solver in Mathcad to solve the problem (Figure 2).

I should mention that it does not matter which point is chosen for the reference. You can prove this as shown below. In this derivation, I show that when you pick a reference *V _{0}*, the point value of the point at

*t1*is driven by the time difference

*t1*-

*t0*.

Thanks to this approach, I was able to determine all the exponential time constants quickly and accurately.