How to find the time constants of 3 parallel RC networks from curve fitting ?

I am trying to find the time constant tau1,tau2 and tau3.

How can I find the time constant from the plot of time and voltage ? I tried this solution , But i am unable to implement it properly.

[Jon]

What is the exact equation you are trying to fit to this data? Also it looks like you have two step responses in your data set. It would be simpler to fit just the first step response, or just the second.

[SHRISHA BALKUR]

This is a subset of a larger dataset. It is a rising pulse due to three RC parallel network after single step response (relaxation period). I am interested in knowing the time constants of those three RC network.

I am trying to fit f(x) = a*exp(b*x)+c*exp(d*x)+e*exp(f*x). Where b,d and f would be my time constants.

• Here i would like to know how to chose the initial values of a,c and e (Whether it is required or not would also be helpful to know) ?
• How to keep the time constants b,d and f within limits.

Where 1<b<50 , 20<d<1000 and 100<f<3500

[Jon]

As a start, if you hope to match experimental step response data with, theoretical step response values, given certain parameter values, e.g. the three time constants, then you must formulate an expression for the theoretical step response. It will not be of the form that you show, as that expression would produce an ever increasing value of the output with time if any of the exponents are positive (I guess you are calling time, "x". If they are all negative then the output would approach zero. In fact the output should approach a constant, non-zero, steady state value. Also, unless you have some additional information, I would think there would be more unknowns than just the time constants. Wouldn't there also be some unknown gains?

[David Goodmanson]

Hi Shrisha,

There is the issue that the late time response goes to a constant value, whereas the sum of exponentials (assuming the exponentials have negative argument) goes to zero. The data looks more like

C - a*exp(-b*x) -c*exp(-d*x) -e*exp(-f*x)                 b,d,f > 0

where C, the asymtote as t --> inf, appears to be somewhere between 4.164 and 4.165. Even if you determine the value of C (maybe it's known from the experimental conditions) there are other issues. Fitting a,c,e when you already know the time constants b,d,f is one thing. But you should be aware that for a sum of exponentials, fitting the time constants b,d,f is a notoriously ill conditioned problem. Hence whatever answer you get is very likely to be unreliable.

[SHRISHA BALKUR]

Hello Jon and David,

Thank you for your answers. I am sorry for late reply.

I completely agree with your answers. With x being positive or negative my response will have drastic effects. I wanted to have a fairly good enough fit. There are other unknowns.

I intend to use these time constants to simulate and get simulation output. Then use parameter estimation technique to give limits to those time constants.

@David, Thank you for your equation. I did have some values which were negative and hence my simulated values shows drastic behavious and goes negative (This is my understandin,please correct me if I am wrong).

After using parameter estimation technique and giving the proper limits to the time constants, I have all the time constants to be positive. and fairly matches the measured data.

Below is the zoomed image of one of the step response. I still need to process the data to have a good fit. I intend to use linear fit to achieve this. What do you both think of this approach ? Do you think it is reasonable or do you have an alternative suggestion ?