Measuring variable impedance

I was wondering how to design a measurement system that's able to correctly measure a load impedance that can vary over time and can possibly have very low as well as very high values (see figure below).

If the load impedance is constant, we usually design the measurement system to have a very high input impedance $$\Z_M\$$ compared to load impedance $$\Z_L\$$ (assuming we have an idea of the expected value). Then, knowing $$\v_s\$$ and $$\Z_s\$$, we can deduce $$\Z_L\$$ from the voltage divider formula $$\v_L = v_S \frac{Z_L}{Z_L+Z_S}\$$.

My question is what can we do when $$\Z_L\$$ is no longer negligible compared to $$\Z_M\$$?

• <<< ... can possibly have very low as well as very high values ...>>> Generally, for such values, at least 2 configurations must be used ... or automatic RLC-meter, with more configurations depending on the values to be measured (serial Z or parallel Y or Sxx). Using also "correcting" tables when some "errors" are "well-known". We did so with an "old" measurement system as a Q-meter. See this post electronics.stackexchange.com/questions/268257/… Dec 19, 2022 at 12:29

Assuming this is a "DC" circuit (i.e., we aren't worried about transmission line effects), simply take the Thevenin/Norton equivalent of VS into ZS and ZM, and make this your new VS', ZS'. Compute the divider with VL and ZL and you're done.

The real difficulty is probably that, in general, all three parameters vary with frequency, or maybe other operating conditions, so you have a challenge to calibrate all of it together over the whole frequency range. And if ZL should be variable over a wide range, and your measurement error is comparable to the change in VL (for example, if using an ADC, the change is just a few LSBs), obviously you'll have problems measuring at such extremes.

In that case, having a switchable ZS (and still accounting for ZM as well as you can, including using a different receiver/detector circuit if the ZS || ZM equivalent needs to be that much higher still) is the only remaining option. Which obviously will get problematic if this is an RF context after all.

• This is a (ultra/very) low frequency circuit, working at frequencies ranging from 300 Hz to 30 kHz. The idea of bringing the input impedance of the measurement system to the source side and consider the equivalent Thevenin circuit is not bad. I'm not sure, however, I understand what you mean by 'calibrating all of it' ! If $V_S'$ and $Z_S'$ are know, we're only left with $Z_L$ to determine, right ? Dec 19, 2022 at 19:52
• Exactly -- but you need to know those, and maybe it's easier to know the components of them versus the equivalents. But at low frequencies like that, yeah that should be fine, well within the domain of op-amps (and all the assumptions that make them easy to use). Dec 19, 2022 at 20:05

You can also measure in three steps.

Measure with generator and voltmeter ... e1.
Add Ref = reference and measure ... e2.
You have two equations to solve.

Remove Ref and add Rx, measure ... e3. One more equation to solve.

Here is a Maple sheet for "automating" calculus.

The complete solution will give you the value of the Generator, Voltmeter, and Load resistances.

This should be "complicated" if the phase should also be measured. If you need something already integrated for measuring complex impedance (until 100 kHz), just use AD5933 (100 Ohm -> 10 Meg, 1 Msps) or AD5934 (100 ksps).

• The phase also should be measured as I'm after measuring (complex) impedance. Also, this is more a circuit that needs to be implemented in a pcb. So, I don't unfortunately have the flexibility of measuring through different configurations using generator, volmeter, etc. My measuring system is an ADC. Dec 19, 2022 at 19:56
• No problem. The first part need only be done one time within the measuring system (calibration with one reference resistor switched with a little relay REED -> measuring of rg and rv). Then, just use the last formula zx. Note also that if you need something already integrated for measuring complex impedance, just use AD5933 (100 Ohm -> 10 Meg, 1 Msps) or AD5934 (100 ksps). Dec 19, 2022 at 20:11
• Thanks a lot for the IC references. I didn't know such circuits existed. Dec 19, 2022 at 22:17

One option is to measure both current and voltage. With both VM and ZM known, you know the voltage across the load, but you don't know how much current is consumed by each element. However, if you measure the total current consumed (e.g with a shunt resistor), you can treat the impedance as a current divider. If you know total current and measured current (from VM / ZM), you can always derive load current.

There are more things to consider:

• What frequencies are you expecting?
• What are your operating current and voltage levels?
• What are the input voltage requirements for ZL?
• Application: is this for benchtop/rack-mounted equipment? A PCB? Something else?

Probably the biggest problem is to find a way to balance the shunt resistor's power consumption with its resolution: it must not burn up when ZL is low, and it must provide a useful reading when ZL is high. You may find a shunt resistor that's suitable, or you can choose an appropriate method to switch it in and out, depending on the value of ZL.

• It may not be possible in my case to measure current. However, having more control over source current may be an option instead. Dec 19, 2022 at 19:35