# LCR meter inplementation and limitations

#### The Electrician

Joined Oct 9, 2007
2,815
LCR meters are a perennial instrument of interest to people involved in electronics. In this thread I'll delve into their modes operation, their capabilities and their limitations.

We see in the previous two posts how users want to know just what can their LCR meters measure and how accurate are they?

LCR meters (and impedance analyzers which usually can act as LCR meters) come a range of capabilities and price, all the way from the "boat anchor" size and US$50,000 for the top-of-the line Keysight units, to$20 kits found on eBay.

It's Friday afternoon and I have to depart, but I'll continue later with a description of operating modes.

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#### narkeleptk

Joined Mar 11, 2019
480
I just picked up a cheap like new keysight u1731c so this thread is right on time, looking forward to following and learning a little something. Thank you.

#### The Electrician

Joined Oct 9, 2007
2,815
So that I can be thorough about the operation of LCR meters, I'm going to give some very basic preliminaries:

I'm going into a fair amount of detail that the EE's won't need--it's here for the beginners.

A long time ago, shortly after the discovery of electricity, experimenters used mostly
static electricity.

Before long, chemical batteries were invented, and experimenters had a good source of
steady electric current. They applied an electrical stimulus to matter in its many forms,
and discovered that some substances conducted electricity better than others. They
noticed that the flow of electricity through matter was opposed in a way that was
analogous to mechanical friction. It was also noticed that when an electric current was
passed through various substances, heat was generated; electric energy was converted
into heat energy.

A way of measuring the opposition to electricity was to apply a known voltage, and
measure the resulting current. If the substance greatly opposed the flow of electricity,
that flow of electricity (current) would be small. If the substance allowed a large current,
the opposition was small. The magnitude of the opposition was given a value by dividing
the applied voltage by the amount of current that resulted.

A word was sought to give a name to this property of matter, a word to convey the notion
of opposition to electric current. A number of words could be used for this: resistance,
reluctance, reactance, impedance, etc.

In those early days when the source of steady electricity was just chemical batteries, the
word chosen was "resistance", and the letter symbol "R" was used. This was the
opposition to a direct current of electricity.

Later, when it was discovered how to produce alternating current, it was found that
capacitors and inductors opposed the flow of an alternating current, but by a different
mechanism. The name chosen for the opposition to alternating electric current exhibited
by capacitors and inductors was "reactance", and the letter "X" was used to stand for it.
It's important to note that whereas the opposition of a resistor to electric current
(resistance) gives rise to heat, the opposition of pure capacitors and inductors
(reactance) does not.

The basis for the opposition to electric current of capacitors and inductors is due to the
nature of the relationship between the AC voltage applied to them, and the current that
results:

For a capacitor, i = C*de/dt. The AC current through a capacitor is equal to the
capacitance times the rate of change of the voltage applied to it.

For an inductor, e = L*di/dt. The AC voltage appearing across an inductor is equal to the
inductance times the rate of change of the current through it.

The expressions de/dt and di/dt may not be familiar to those who haven't studied
calculus; They are called derivatives and refer to the time rate of change of voltage or current.
Just as in the DC case, where we measured resistance by applying a voltage and
measuring the current, then taking the ratio (voltage/current) and calling that
"resistance", in the AC case we can do the same thing. Apply an AC voltage of a
particular frequency to a capacitor or inductor, measure the current and calculate the
ratio (voltage/current) = X (reactance).

(to be continued)

#### The Electrician

Joined Oct 9, 2007
2,815
Considering the 3 fundamental components of circuits, resistors, capacitor and inductors, we have 3 equations relating the voltage across to the current through each. We use upper case for DC voltages, and lower case for AC voltages:

For a resistor, E = I*R

For a capacitor, i = C*de/dt

For an indcutor, e = L*di/dt

Suppose we have a circuit composed of a number of these 3 components, connected in some fashion, and we want to know what the voltages and currents would be if we suddenly applied a DC step of voltage to some part of the circuit. Because the relationships for capacitors and inductors involve derivatives, it is necessary to solve differential equations to get the resulting voltages and currents.

It is known that the differential equations describing circuits composed of only R, L and C are ordinary differential equations with constant coefficients. It is also known, that the solution to such differential equations is always a sum of exponentials, where the arguments to the exponentials may be complex, involving the imaginary number SQRT(-1). Euler's formula:

http://en.wikipedia.org/wiki/Euler's_formula

tells us that the resultant of an exponential with a complex argument involves sines and cosines.

Thus, the solution to a general circuit comprised of only resistors, capacitors and inductors with a suddenly applied DC voltage will be decaying exponentials, sine waves and cosine waves, and combinations of these.

I can tell you that solving a system of differential equations for a complicated circuit is a PITA. Fortunately, C. P. Steinmetz:

http://en.wikipedia.org/wiki/Charles_Proteus_Steinmetz

discovered that ordinary differential equations with constant coefficients can be solved with plain old algebra if you allow the arithmetic to include complex numbers, and those numbers stand for the magnitudes of sine waves of voltage and current. The reason they represent sine waves is because that is what the solutions of the relevant differential equations are comprised of.

It turns out that if the reactance of capacitors and inductors are taken to be imaginary quantities, the algebra of circuits is fairly simple.

#### The Electrician

Joined Oct 9, 2007
2,815
What happens if we connect a resistor and capacitor in series and apply an AC voltage of a particular frequency to the series combination? We could measure the current and calculate the ratio (voltage/current). There would undoubtedly be opposition to the flow of electric current, just as we found with a resistor alone, or a capacitor alone.

When this is done, there is indeed opposition to the current, and this opposition is called "impedance", and the letter "Z" is used to represent it.

A new phenomenon is observed in this case. When an AC voltage of a particular frequency with a sine shape is applied to a resistor, and the voltage and current are displayed on an oscilloscope, it is seen that the voltage wave and the current wave are in phase. But when a capacitor is in series with a resistor, the voltage wave applied to the combination is not in phase with the current wave.

The use of complex numbers deals with this. The impedance of the series combination of a resistor of value R, and a capcitor with reactance X is given by the formula:

Z = R - jX The minus sign is there because the reactance came from a capacitor. If the reactance comes from an inductor, the minus sign becomes a plus sign.

Notice that Z is a complex number, with a real part (R) and an imaginary part (X). This way of representing the impedance is called rectangular notation. A complex number can also be represented with a phase angle like this: |Z|<theta, where |Z| is the magnitude of Z, given by SQRT(R^2 + X^2). This is called polar notation.

If the voltage applied to the series combination is e, then the current i = e/Z (this is sometimes called ohm's law for AC). Since Z is a complex number, i is also a complex number, so the current has a phase angle.

If we want the magnitude of i, which is what we would measure with an ammeter, we would divide e by the magnitude of z thus: |i| = e/|Z|, that is, the magnitude of i is given by the voltage divided by the magnitude of Z.

Here's an app note that goes into much more detail: http://www.ietlabs.com/pdf/application_notes/030122 IET LCR PRIMER 1st Edition.pdf

NOW HERE'S THE IMPORTANT PART OF ALL THIS!

If we have a very complicated circuit of R, C and L, we can pick two nodes in the circuit and measure the impedance at a particular frequency there. The result will be Z = R + jX. This impedance will have a real part R, and an imaginary part X. No matter how complicated the circuit, this will be the form of the impedance. The real part R will most likely not correspond to any single resistor in the complicated circuit. It's a result of the many resistors in parallel, series, bridge, delta or wye, etc., in the complicated circuit. But they all boil down to a single R value, the real part of the measured impedance.

Now, an impedance Z = R + jX also represents the impedance of a series combination of a single resistor and single capacitor (or inductor); R is the resistance of the single resistor and X is the reactance of the single capacitor (or inductor).

Since there will always be a single resistor and capacitor (or inductor) series circuit that has the same impedance as we measured at two nodes of our very complicated circuit, we can say that the EQUIVALENT impedance of our complicated circuit's two nodes is given by the simple series circuit of a resistor and capacitor (or inductor as the case may be).

And the resistance (real part) of our EQUIVALENT series circuit of a resistor and capacitor (or inductor), is the EQUIVALENT SERIES RESISTANCE (ESR) of the impedance measured at the two nodes of our complicated circuit. We could also refer to the reactance of the capacitor (or inductor) of our simple series circuit as the EQUIVALENT SERIES REACTANCE (also ESR; too bad).

(to be continued)

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#### The Electrician

Joined Oct 9, 2007
2,815
Hello, everybody. I am finally able to continue this thread about LCR meters. I will be explaining how they work and discussing their limitations.

For a complete, every detail covered handbook on impedance measurement, this can't be beat: https://assets.testequity.com/te1/Documents/pdf/impedance-measurement-handbook.pdf.

I will distilll it down to a more accessible explanation for the average user in what follows. More to come.

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#### The Electrician

Joined Oct 9, 2007
2,815
Consider the working of a plain old ohmmeter function which is part of nearly every DMM. You have a resistor (DUT; Device Under Test) whose value you want to measure so you connect it to the meter leads. The meter applies a known DC voltage to the resistor and this causes a DC current to flow in the resistor. This current is then measured by the meter and applying ohm's law we calculate the applied voltage divided by the resultant current and this gives us the resistance.

This is what an LCR meter does but it applies an AC voltage to the DUT (which will be a capacitor for this thread) and measures the current which results. The situation is more complicated because we're applying an AC voltage and the DUT is a reactive device. This means that the there will be a phase shift between the applied voltage and resultant current which must be taken into account.

It wouldn't be too far off the mark to say that an LCR meter is just an AC ohmmeter.

In the posts beginning this thread I discuss the concept of impedance and the use of complex numbers to deal with AC impedances. The LCR meter will need to separate the real (resistive) and imaginary (reactive) parts of the measurement. This is done with phase sensitive detectors, an additional complication we didn't have with the plain old DMM ohmmeter function.

Since we want to apply an AC voltage to the DUT one might ask where will we get this voltage? A basic part of an LCR meter is a signal generator. This signal generator must supply a very clean sine wave of selectable frequency, and in some meters, selectable (or at least, known) amplitude. Since the voltage applied to the DUT is generally quite low, 1 volt or less, the signal generator is a low power function. This generator is designed to have a 100 ohm output impedance. This means that when a DUT is connected to the test point of the meter it will result in a circuit that consists of an ideal sine wave generator in series with a 100 ohm resistance and then the DUT. The ideal generator will be generating a "nominal" voltage selected by the user, or fixed by the design such as .6 volts in the DE-5000. The 100 ohm output impedance of the signal generator and the DUT form a voltage divider so that the AC voltage actually applied to the DUT will be less than the "nominal" voltage from the ideal generator.

This leads to a problem right away. Consider that the DUT may be an electrolytic capacitor of a high value, hundreds or thousands of microfarads. Such a capacitor will have a very low value of reactance at the typical frequencies where measurements are made. This low reactance loads down the voltage divider (the 100 ohm output impedance in series with the DUT) and means that the voltage across the DUT is small. This voltage may be so small that it's hard to get a good measurement of it. It will be necessary to amplify it with a gain stage, and the noise associated with that gain stage will be the limiting factor determining the accuracy of the measurement.

More to come.