by Granger Obliviate
Last Updated October 09, 2019 17:25 PM

So I'm doing experimental work on an unknown impedance. I used a RLC meter for frequencies of 100, 200, 500, 1k, 2k, 5k, 10k, 20k, 50k and 100k Hz. Now, I think the values I got for each frequency are not that relevant (tell me if they are and I can post them). What seems important is how they vary:

- the absolute value always increases (varies between 404 ohm and 2.57k ohm). The increase rate is greater at mid-frequencies.
- the angle starts positive and close to zero (7 degrees), increases until reaching a maximum of 47 degrees at 2k Hz and then it starts decreasing again until like 1 degree at 100k.

Now my attempt. What elements make my impedance and how are they placed?

The way the phase varies tells me that as frequency tends to zero and infinity, the impedance behaves as resistor. The magnitude always increases. The magnitude of the impedance of a resistor is constant The magnitude of the impedance of a capacitor decreases with frequency. Therefore it's theoretically infinity at zero frequency and zero at infinite frequency. The magnitude of the impedance of an inductor increases with frequency. Therefore it's theoretically zero at zero frequency and infinity at infinite frequency.

Because the impedance is low (but not *that* low) at low frequencies, the circuit must be equivalent to a resistor. Same for high frequencies.

I came up with an hypothesis of:

(resistor A in parallel with a capacitor C) in series with (resistor B in parallel with an inductance L). This seems to be the minimum that guarantees the absolute value scheme.

But does it verify with the angle?

Writing the transfer function of this circuit I get.

$$ \frac{R_AR_B + L(R_A+R_B)s + R_AR_BLC s^2}{(1+sCR_A)(1+s\frac{L}{R_2})} $$

Therefore I have 2 real poles and one pair of complex conjugate zeros. Of course without numerical values it is hard to guess if this is correct or not. But I know that the pair of complex conjugate zeros will guarantee me an increase of 180 degrees around the natural frequency and each real pole will guarantee me a -90 degrees fall. I know that the maximum I reach is 47 degrees (so around 45 degrees). Therefore I think my hypothesis can never be correct... To increase the angle the first frequency to appear needs to be the zeros'one. However since the frequency only increases until 45 degrees and then decreases I need to have components that will get me at least -235 degrees of fall... My 2 poles only guarantee me -180 so it would only work if frequency never dropped. Therefore I'm stuck. Do I need more poles in my system? What components can I had to guarantee me that? Thanks in advance!

Your hypothetical model is more complex than the data calls for. It sounds kind of like an RIAA equalization filter, which is just two resistors in series, with a capacitor in parallel with one of them. At very low frequencies, you see just the two resistors in series. At very high frequencies, you see only the resistor that isn't shunted by the capacitor. In between, you see an "S" curve whose exact shape depends on the relative values of the resistors and the capacitor.

Since your circuit has an impedance that rises with frequency, change the capacitor to an inductor.

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