What formula would I need to fit my measurements of resistance against frequency in and RLC circuit?

Question

I did some measurements on a system which is currently basically a black box for me. I'm normally a programmer and I'm doing an internship with no prior knowledge on circuitry, or physics in general, so please bear with me.

This is about what I know about the system:

^{simulate this circuit – Schematic created using CircuitLab}

I controlled the input voltage and frequency and measured the output current:

I'm not really sure where that black border comes from, it seems to be caused by matplotlib. It's frequency on x, volt on z, and current on y.

I figured out that on each frequency the voltage and current have the same ratio, so I decided to plot the "measured resistance" against frequency:

(Frequency on x, Resistance on y)

As you see I tried to fit a curve on it. This curve is just an inverted Gaussian curve, which I don't expect to be the right formula. I need to be able to calculate the expected restistance on each frequency (not just the ones I measured, but also in between each frequency measurement.)

Which formula could I use to fit the curve?

Answer 1

This is a standard RLC series resonance with a 2nd order equation with peaking in the transfer function defined by Q with reactance/load impedance ratios and resonance by reactance product values.

The theory was best outlined here.

Finding resonant frequency and cut-off frequency from Bode plot to calculate values for RLC circuit

Since the transformer blocks DC, this implies your output capacitor is redundant and perhaps a problem. If you remove that you will find the low frequency step response to 64% of the target to be T = L/R

Answer 2

I figured out that on each frequency the voltage and current have the same ratio, so I decided to plot the "measured resistance" against frequency:

This is basically saying that the system is a linear system. Twice the voltage means twice the current.

Also: luckily it is linear, because else your problem would have been drastically more difficult.

Which formula could I use to fit the curve?

Fitting, or modeling an impedance in general requires some fairly advanced techniques. Usually, we discuss these things in the Fourier domain (frequency domain). For linear time-invariant systems, you should be able to model it with this equation:

$$\left|Z(j\omega)\right| = \left|\frac{V(j\omega)}{I(j\omega)}\right| = \left|\frac{B(j\omega)}{A(j\omega)}\right| = \left|\frac{\sum_{i=0}^{n_b} b_i\cdot (j\omega)^i}{\sum_{i=0}^{n_a} a_i\cdot (j\omega)^i}\right|$$

While \$a_i\$ and \$b_i\$ are real numbers, these quantities live in the complex plain, representing not only amplitude but also phase. You are trying to solve the problem using only the amplitude which might make it a little bit more tricky.

There are general-purpose algorithms to try and model such curves without knowing the model complexity (\$n_b\$, \$n_a\$) or any of their coefficients (\$a_i\$, \$b_i\$). In your case however, we might try modeling a single resonance peak and see if it matches. It just so happens that we know such resonance can be modeled by a transfer function of order 2.

So you might try to fit this second-order transfer function, normalizing such that \$a_0 = 1\$:

$$\left|Z(j\omega)\right| = \left|\frac{b_0 + b_1\cdot j\omega}{1 + a_1\cdot j\omega - a_2\cdot \omega^2}\right|$$

Or, by squaring the complex amplitudes:

$$\left|Z(j\omega)\right|^2 = \frac{b_0^2 + b_1^2\cdot \omega^2}{(1 - a_2\cdot \omega^2)^2 + a_1^2\cdot \omega^2}$$

I expect \$b_0\$ to be fairly small, because the lowest-frequency amplitude is also very small (if \$\omega = 0\$, then \$|Z(j\omega)|\$ should be small).