Quote of the Day
Engineering is achieving function while avoiding failure.
Figure 1: Annotated Exponential Curve.
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 (VF) by fitting these three points to Equation 1.
| Eq. 1 |  |
where
- VF is the final voltage of reached by the curve.
- VI is the initial voltage of the curve.
- 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, VF), 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, V0) and measure the two other points relative to V0, i.e., (V1, t1-t0) and (V2, t2-t0).
To find (τ, VF), I chose to use a nonlinear solver in Mathcad to solve the problem (Figure 2).
Figure 2: Mathcad Solution with Test Case.
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 V0, the point value of the point at t1 is driven by the time difference t1-t0.
Figure 3: Derivation Showing that the Value of V1 is related only to the time difference from V0.
Thanks to this approach, I was able to determine all the exponential time constants quickly and accurately.