Easy Impedance (LCR) Measurement Method

[Odysseus]

 While researching how to make accurate phase measurements for an impedance analyzer, I stumbled across this document.  It mentions an interesting way to measure phase by making just three measurements with an AC voltmeter and using the law of cosines.  Knowing this, we can devise a way to measure complex impedance with nothing more than a multi-meter, a signal source, and a resistor.

So here's how it works.  First, the measurement:

1. Construct a voltage divider consisting of a resistor of known value and your DUT.
2. Drive the input of the divider with a sine wave at the frequency of interest.
3. Record 3 amplitudes: Input voltage, DUT voltage, and resistor voltage.

Now, the calculations. Excel (or Google Drive, LibreOffice) makes this part easy:

4. Use law of cosines to calculate the angle between the DUT and resistor phasor.
5. Calculate the reactive and real impedance of the DUT. 
6. Convert the reactive into capacitance or inductance using the measurement frequency.
7. If desired, calculate Q or loss tangent.

I made this phasor diagram of a hypothetical lossy inductor to help visualize the math:


We can calculate the magnitude of the DUT impedance simply: |Z| = R*|Vz|/|Vref|
Next, calculate the factor I'll refer to as K: K = ( |Vin|² - |Vref|² - |Vz|²  ) / (2*|Vref|*|Vz|) Note that K=cos(theta).
After this, the real part of the impedance is: R = K*|Z|
And the reactive part of the impedance is: X = sqrt(1-K²)*|Z|
Finally, convert the reactance into either inductance: L = X/(2*pi*f) or capacitance: C = 1/(X*2*pi*f)

Anyway, I've also attached an excel spreadsheet to do all the hard work.  As an example I measured a 1uF ceramic in series with 1 ohm (to simulate a predictable ESR) at roughly 500Hz.  My signal source is way too unstable to get good accuracy on the resistive component (I keep measuring around 6-8 ohms), but the capacitance measurement is solid.

[dannyf]

It is an old "new" approach: I think it was actually covered in one of HP's application notes on lcr measurements.

Its difficulty lines primarily the difficulty and complexity in measuring ac amplitude: you have to make three measurements and each is complex itself.

The modern approach is a computerized v-i approach (=measuring the voltage / phase angles of the input signal + that over the reference load), via FFT (mostly DFT/DCT, or IEEE1057. It is much simpler hardware wise than the syn. detector approach but requires a fairly fast computer.

[Odysseus]

It is an old "new" approach: I think it was actually covered in one of HP's application notes on lcr measurements.

I'd imagine so.  Unfortunately neither the "Agilent Impedance Measurement Handbook" or "LCR Impedance Measurement Basics" by HP make any mention of it.

Its difficulty lines primarily the difficulty and complexity in measuring ac amplitude: you have to make three measurements and each is complex itself.

Do you mean complex as in each of the three measurements returns a complex result, both phase and magnitude? That's not the case here.  Only magnitudes are directly measured.  Maybe you mean complex as in requiring a rectifier/RMS detector.

[DBoulanger]

I really do like that kind of mathematical gymnastic, despite it is somehow old school.

There is nothing wrong at getting back to the roots, even the square ones !   

BTW, aren't the ESR specs evaluated at a higher frequency ?

Sincerely, I never tried measuring ESR on a ceramic cap.  I always was under the impression that ESR was more of a concern for the electrolytic caps.  Oh well, learning something new every day !

Have fun !

[IanB]

I often wonder about an easy way to measure inductance without an LCR meter, and every time the path arrives at the need for a signal generator. You really need to generate an AC signal of a suitable frequency before you can make much progress.

The method described here is somewhat logical, of course. If you have a theoretical circuit with a resistor in series with a reactor, you can calculate what the measured RMS voltages ought to be across each component in the circuit. It is therefore but a short step to measure the voltages on an experimental circuit and reverse the calculations to get the component values.

I suspect that to get the best results you need to vary the source frequency so that the voltage across the reactor is comparable in magnitude to the voltage across the resistor.

[Odysseus]

I suspect that to get the best results you need to vary the source frequency so that the voltage across the reactor is comparable in magnitude to the voltage across the resistor.

Indeed.  For good accuracy the resistor needs to be adjusted to roughly match the impedance of the DUT, which changes with frequency. Thus, doing a frequency sweep with this method gets pretty tedious, unless automated somehow.  But for a single measurement when you just need two or three significant digits, it does the job. 

I figured posting this might save someone, particularly beginners on a low budget, some hassle and second guessing.

[Mechatrommer]

just if you have programming skill to make things easier, if not you can make calculation in your head. based on some sort of math like in the OP, and some real life implementation of the vector mechanic (you can find in google), i made my Goltek Bode Plotter based on that. getting input from FG->DUT->DSO->software, it has impedance analyzer but i believe my math foundation/implementation is not completed making my bode plotter output chaos impedance result, with deeper study i believe it can be refined, but i currently dont have allocated time for it.

1st screen capture is bode plot for the DUT inductor, 2nd screen capture is how the DUT is connected to FG and DSO, assumed model for inductor/capacitor, an the impedance result below (the chaos). of course this is a poor man method, the other option if you dont want to get in the dipsheet of vector mechanic is of course, a proper LCR or impedance analyzer. fwiw.

I often wonder about an easy way to measure inductance without an LCR meter

in the end forgetting about the chaos result, i ended up buying onehunglow LC meter, its a charm for me and giving stable enuff result esp for small value LC.... (wait i find the link)... ah this... http://www.ebay.com.my/itm/1PCS-L-C-Inductance-Capacitance-Multimeter-Meter-LC200A-Tool-/251136559945?pt=LH_DefaultDomain_0&hash=item3a78e7cb49

[dannyf]

Maybe you mean complex as in requiring a rectifier/RMS detector.

Yes. For your approach to work, you will need three separate detectors. And the meter cannot tell the difference  between a capacitor or an inductor.

If you don't need phase detection, you can use simply two detectors.

[G0HZU]

People were measuring impedance >50 years ago using a dual channel scope and a clean signal source and a series sense resistor.

You just need to measure the two voltages either side of the sense resistor and the phase angle between them. Pretty easy on a modern scope especially if you use averaging to remove any trace noise.

I mentioned this on dannyf's ESR meter design thread. The above approach totally outclasses his modern design yet costs about 1 penny for the sense resistor.

i.e. with the above approach you could easily measure a 10uF tant cap with a typical 0.06R ESR. You can also see where it is self resonant (because this is where the phase angle hits zero on the scope)   and see how the ESR (and capacitance) varies with frequency from, say 10kHz to 1MHz and where it starts to look like an inductor. You can then reverse engineer a crude model for it in terms of series LCR. 

The method gives you the complex impedance of the cap at each frequency. This method is ancient and you can do all the vector computation in a simple excel spreadsheet.

With modern soundcards running to 90kHz you might be able to do the whole thing with a soundcard and some basic external hardware and a simple GUI but it limits you to about 90kHz.

[G0HZU]

OK, I just measured a 10uF cap with the above ancient scope + sense resistor method:

I used a 10R sense resistor and the test source was an old function generator with a clean sine wave. I used an old Tek TDS2012 scope.

I measured 222mV and 3.81mV on the scope (using trace averaging to clean up the signals)

The phase angle was -66.5 degrees (so the reactance is capacitive)

With a few simple vector sums this reveals the capacitance to be 9.98uF and the ESR is 0.066 ohms.

I measured this cap a couple of weeks ago on an impedance analyser at work at it showed about 0.06R ESR at 100kHz.

[G0HZU]

Here's a picture of the hastily prepared test setup. I used an old scrap section of PCB to mount the connections. You can see the generator has drifted slightly away from 100kHz since I took the data in the previous post but you can see how easy it is to take the data from the scope.

[G0HZU]

Here's a quick simulation of the jig. It shows the same phase angle (-66.5deg)  and about the same voltage at channel B of approx. 3.75mV.

Some people might argue that this is all a bit fiddly but how often do you need to measure (faulty?) caps for ESR?

NOTE: I have a different system I use for 'in circuit' testing that is very fast and effective that uses a homemade probe so I don't use the above scope method for fautfinding. The probe cost peanuts to make as well and can find dead caps in a circuit board very easily. But the old scope method above is a very simple way to evaluate a capacitor for people who already have a function generator and a reasonable two channel scope.

To give some idea of granularity, if a second cap was tested that had an ESR of 0.1R instead of 0.06R then the results on the simulator are:

V1 222mV
V2 4.08mV
Phase angle -56.6 degrees

So that's a 10 degree change in angle and a change of about 0.3mV on the scope. You have to use averaging on the scope to measure this change well.

[Conrad Hoffman]

It's a perfectly good method for anybody even marginally competent at the test bench. Actually, it's an excellent training exercise for anybody who hasn't done it. It was also common to get values for inductors by resonating them with a known capacitor, often a 1% silver mica.

The excellent free program Visual Analyser has an LCR meter function if you make an interface for your sound card.

It's also very easy to build a traditional capacitance bridge and I'm surprised that more people don't do it.

It's all good!

[Co6aka]

For good accuracy the resistor needs to be adjusted to roughly match the impedance of the DUT...

Without thinking much about it first, how 'bout using a Vactrol as the resistor (or in combination with a resistor) and pre- "calibrating" it (LED current vs resistance, etc) so you've got a controllable VR to use there? 

[dreamcatch16]

Hi to all,

My final year project is about implementation of an impedance meter , I found your approach very interesting since it will save me from using high speed processors and ADCs (and too much headache  since I couldn’t understand how to use the DFT to measure the impedance and why do we need to change it to frequency domain).
My questions are:
Practically is it realizable?
why this approach is not used any more in modern analyzer?
Can I reach 5Mhz frequency, if no what will prevent me from doing so?

Thank you for your help.

[Odysseus]

Hi to all,

My final year project is about implementation of an impedance meter , I found your approach very interesting since it will save me from using high speed processors and ADCs (and too much headache  since I couldn’t understand how to use the DFT to measure the impedance and why do we need to change it to frequency domain).
My questions are:
Practically is it realizable?
why this approach is not used any more in modern analyzer?
Can I reach 5Mhz frequency, if no what will prevent me from doing so?

Thank you for your help.

If you're going to bother designing a proper measurement tool, there are much more accurate ways to do it. See http://cp.literature.agilent.com/litweb/pdf/5950-3000.pdf for a wealth of good information.

[dreamcatch16]

If you're going to bother designing a proper measurement tool, there are much more accurate ways to do it. See http://cp.literature.agilent.com/litweb/pdf/5950-3000.pdf for a wealth of good information.

I have gone through that book and I was imterested by the auto-balancing bridge method, but there ars somethings that I think will limit me from succeeding on it at high frequecies "up to 5Mhz" since the opamp will not behave as Ideal

• The opamp gain will change
• I need high slew rate
• I need to measure simultanusly two voltages at high speed where I think there will be some errors

that is why I thought the technique that you proposed is easier to implement, but I do not know the draw back of it  .

Thank you

[dreamcatch16]

I was thinking about something like the circiut that I draw in the attachement, but I am wondering the following knowing that I want to go up to 5Mhz:

• Will the diod behave like a capacitor at around 5Mhz
• how much slew rate will I need to charge the capacitor for the voltage peak detector
• the precision for the instrument amp and the peak detector
• what type of IA and opamp do I have to use to be able to go up to 5Mhz

I was thinking about the AD8010, do you think it will do the job

thank you

[rbola35618]

Do you need resistors across the caps?  If the impedance is very high, the circuit would not follow the voltage across the DUT if the voltage across is lowered. 

[dreamcatch16]

Do you need resistors across the caps?  If the impedance is very high, the circuit would not follow the voltage across the DUT if the voltage across is lowered.

I will automate the reference resistor to have a somehow equilibrated voltage across both the resistor and the DUT, but will the peak voltage detector reflect the real peak voltage?

[Odysseus]

It is quite the coincidence that you are working on an impedance analyzer project, since I myself am doing more or less the same thing.

You have also run against the same problem that I had when I began the system design for my project: high accuracy and high speed amplitude measurement.  I made previous thread asking for advice about this problem, which contains some useful ideas: https://www.eevblog.com/forum/projects/high-speed-signal-rectification-for-accurate-amplitude-measurement/

Have you attempted the "three voltmeter" technique I described in this thread?  I would try it out using benchtop equipment in order to get an idea of it's limitations. 

Let me address some of your specific concerns:

1) Your front end buffering device(s) need to load or affect the device under test as little as possible.  The relevant opamp specifcations to look for includes input resistnce, input capacitance, which becomes very important at high frequencies, as well as input bias current.  The AD8010, unfortunately,has a relatively low input impedance, 125kohm, and a relatively large input bias current, typically 6uA.  This is because it is a bipolar transistor (BJT) based opamp.  A better option is to use a CMOS or (J)FET input opamp.  For example, I'm using the OPA2354 because it has rail-rail input and output, 3pA input bias current, and input impedance that is really only capacitive, 2pF.

2) A peak-hold detector is a servicable method for amplitude measurement, but the one shown in your schematic will perform poorly at high frequencies. See http://sound.westhost.com/appnotes/an001.htm for information about better circuits. The circuits covered there are rectifiers, but they can be modified to operate as peak-hold circuits.

3) The required slew rate of your opamp needs to be: 2*pi*frequency*amplitude.
I.E. a 5MHz, 2Vpp (1V amplitude) sine wave requires an opamp with a slew rate of 31.4 V/us.

4) Any offset in the opamps will directly affect the accuracy of the resulting measurement. You can use low offset opamps, change the measurement method, or measure and calibrate out the offset.

5) All diodes are not the same.  Suffice it to say that you should look for low capacitance shottky diodes, and use small resistor values in the circuit, a few hundred of ohms at most.  See this post I made for more info: https://www.eevblog.com/forum/projects/high-speed-signal-rectification-for-accurate-amplitude-measurement/msg352834/#msg352834

[dreamcatch16]

2) A peak-hold detector is a servicable method for amplitude measurement, but the one shown in your schematic will perform poorly at high frequencies. See http://sound.westhost.com/appnotes/an001.htm for information about better circuits. The circuits covered there are rectifiers, but they can be modified to operate as peak-hold circuits.

First thank you very much for your precious help, I read the topic of the link and it was very useful for me, but I noticed that almost all the proposed circuit are for rectification only and if you need to hold the peak voltage you need to add a diode in series with the output to prevent the cap from discharging, which will not be anymore a precision peak detector because of the diode, so for my aplication I will use a single supply (0 - 5V) I did a modification for the halfwave rectifier to become a precision fullwave rectifier find it attached, basicaly I will buffer the input because of the low input impedance of the inverting amp, then I will rectify both signals, I put the last stage trying to reduce the effect of the reverce recovry time of the diode, By simulation I got 5mv riple wich is more or less acceptable for me since I am going to use the ADC of PIC18F4550 which is 10bits so for 5v it will be 5mv, also I supose that it's input resistance is High since in the datasheet they say 100pA leak current wich will be around 50Mohm. (all of this is theoritically only I can not try it right now)
my concerne for this design is the high current consuption on the inverting amp since I put small resistance as you suggested to me.

also as you mensionned the offset will be a problem also, I am doinf some reasearch how to get rid of it, but for now I couldn't find a solution with good result.

 I made previous thread asking for advice about this problem, which contains some useful ideas: https://www.eevblog.com/forum/projects/high-speed-signal-rectification-for-accurate-amplitude-measurement/

in that thread I saw the log amp which are new for me, I red about them and I found them interesting (even the AD8302 use that logic for the gain and phase measurement)

did you try them, if so do they give good result.

also do you think that the DC offset will affect the impedance since I am going to feed the DUT with sine wave from 0 - 5V so a DC offset of 2.5V (in the agilent book they speak about it but they say only about the magnetic saturation)

thank you