# How to know what this impedance is?

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!

• You should measure the DC resistance. – mkeith Oct 9 '19 at 16:48
• @mkeith you mean at zero frequency? My professor won't let us :/ how would that help? – Granger Obliviate Oct 9 '19 at 16:52
• If the impedance at zero frequency is infinite, you have a series capacitor. Which is a good thing to know. If the impedance is zero, you don't have any series resistors or capacitors, which is also a good thing to know. And measuring DCR is the easiest thing you can do. But I understand that it is not allowed. – mkeith Oct 9 '19 at 16:56
• Yes you are right but I think it might not be the case since the impedance seems pretty much around 404 ohms at low frequencies. – Granger Obliviate Oct 9 '19 at 17:14
• Probably you are right. THEORETICALLY, though, you can always put a really big capacitor in front of a resistor and have the impedance remain unchanged at non-zero frequency. – mkeith Oct 9 '19 at 17:51

• Hi Dave! Thank you very much for your insight. Yes you are right, that circuit you described is enough for my data. Adding the capacitor would just make things more complicated as I would have to add more elements. In terms of angle I also get what I want since I'll have the function $$\arctan (\frac{\omega L (R_1+R_2)R_1}{R_1R_2 + \omega^2L^2(R_1+R_2)})$$ and, if I take the limits to zero and infinity I get the zero degrees I'm looking for. I guess the 47 degrees at mid-frequencies depends on the values of the resistors and the inductor, which need to be adjusted. – Granger Obliviate Oct 9 '19 at 17:36