Time constant of a high-order RC circuit

Question

I have an unknown system that can be modelled as a high-order RC circuit:

I performed the impedance spectroscopy measurement of it and fitted the equivalent components. Now I should calculate the time constants of the circuit, three in this case.

The \$\tau_i= R_iC_i\$ approximation doesn’t work for me.

I found some great suggestions online and on this forum but I can't answer to the problem yet.

I thought to use the parallel/series formula: $$Z=R_0+\left(R_1 \parallel \frac{1}{C_1s}\right)+\left(R_2 \parallel \frac{1}{C_2s}\right)+\left(R_3 \parallel \frac{1}{C_3s}\right)$$

Obtaining:

$$Z=R_0+\frac{R_1}{1+R_1C_1s}+\frac{R_2}{1+R_2C_2s}+\frac{R_3}{1+R_3C_3s}$$

But now I'm stuck. Should I find the roots of \$Z\$ (with MATLAB)? Are the \$1/s\$ values the time constants?

With the help of the microcap software I simulated the circuit using the values used by Verbal Kint (blue, red, green). I calculated the time required for each capacitor to reach 63.2% of the maximum voltage or V at infinite time. I also calculated the time constants considering the Thevenin circuit by eliminating the capacitors (cyan, magenta, light green).

I also noticed that the function that I wrote above has zeros for s = -610.5, -89403 and -281440. Calculating I obtain very similar values to the other methods, but I don't know why of this result.

Finally I compared al these "time constants" with the values of a and b obtained by Verbal Kint:

At this point I am wondering which one of these results is the most correct but I guess the definition of time constant it is a bit forced. What do you think? Which one would you pick?

I repeated the same considerations made above but with a circuit having very similar components. In this case it is clear visually that the time constants should be around 0.25 ms. The results are shown in the figure. You can see that in reality with no method you get all three correct values, except the graphical method which, however, is inconvenient.

Answer 1

It is almost impossible to obtain this type of 3rd-order transfer function in a meaningful expression without resorting to the fast analytical techniques or FACTs as described in my book on the subject. By going through a series of sketches without writing a single line of algebra as in this case, you can obtain the transfer function you want.

You start in dc, with \$s=0\$ in which you open all caps. It is easy here and you sum all resistors. Then you have to determine the time constants in various conditions once the stimulus is turned off. In an impedance determination, the stimulus is a test generator \$I_T\$. When turned it to 0 A, it becomes open-circuited and you are left with a simple circuit as shown below:

For the zeroes, an impedance represents an easy case because you have to null the response across the current source to determine the time constants in this mode. A nulled current source (0 V across its terminals) can be conveniently replaced by a short circuit. It is a so-called degenerate case. The new circuit is shown below and what you have to do is determine the resistance \$R\$ in these various conditions:

Finally, you assemble all these time constants to form a low-entropy expression as shown in the below sheet:

And if you plot all expressions, you have the general shape:

The difficulty here is to reorganize the expressions to unveil poles and zeroes by approximating the response. You need to check the time constants and see which ones dominate or can be grouped together. Good luck with this exercise!

Answer 2

It reminded me of a distributed RC delay, where the Elmore Delay approximation seen also here, might be useful for you. The first link provides a tutorial and discusses some limitations and constraints of the approximation.
I tested it on a completely different physical problem (hydraulic transient analysis for flexible pipes submitted to step-pressure response) and could find “reasonable” approximations, even when considered non-linearities of the hydraulic model.
If a quick and dirty solution (+/- 50%) is desired as an order of magnitude estimation, you can even assume certain simplifications. If empirical data is used to fine-tune the parameters of the model, the approximation was even better: +/- 10~20%.

Surely you would need to check if your case is within the restriction of such approximation.

Answer 3

A simple RC circuit is described by a differential equation.When you have multiple capacitor you need to solve a system of differential equations.The differential equations of the system come from doing nodal analysis to each node of the circuit.The resulting system is in most cases not solvable because there exist solutions for only a specific type of system of differential equations.

Answer 4

I guess that you measured "impedance" versus "frequency".
The same simulation can be made if measurements are "impedance" versus "time".

Here is a graphical view of a system where "constants" are very different from each other (simplified case).
The center capacitor varies from the value of the left (/10) to the value of the right (x10).
The resistors are all same values.
One can easily view all choices of components and "determine" graphically all "values" of the model.
Note that, with microcap v12, one can "identify" component values interactively.