Measuring variable impedance

Question

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\$?

Answer 1

Assuming this is a "DC" circuit (i.e., we aren't worried about transmission line effects), simply take the Thevenin/Norton equivalent of V_S into Z_S and Z_M, and make this your new V_S', Z_S'. Compute the divider with V_L and Z_L 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 Z_L should be variable over a wide range, and your measurement error is comparable to the change in V_L (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 Z_S (and still accounting for Z_M as well as you can, including using a different receiver/detector circuit if the Z_S || Z_M 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.

Answer 2

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).

See also this post if necessary.

Answer 3

One option is to measure both current and voltage. With both V_M and Z_M 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 V_M / Z_M), 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 Z_L?
• 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 Z_L is low, and it must provide a useful reading when Z_L 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 Z_L.