Impedance, part 2: why do LCR meters exist?
Impedance, part 2: why do LCR meters exist?
Impedance is complex, and a common workbench gadget is not quite what it seems.
In a recent introductory article on electronics, I talked about impedance: the tendency for components and circuits to resist the flow of current. The phenomenon can be divided into three categories: resistance, capacitive reactance, and inductive reactance. All of these can be thought of as the contributors to apparent resistance — and indeed, their effects can be difficult to distinguish at a glance.
Of the three, true resistance is the easiest to grasp: it’s a frequency-independent property most commonly associated with heat dissipation in resistors, semiconductors, and long runs of wire.
Capacitive reactance, in contrast, typically has to do with the reversible storage of energy in electric fields. Its magnitude changes with the frequency of the driving signal. For a given sine wave frequency (f) and a known capacitor value (C), the effect — also measured in ohms — is described by the following formula:
In a similar vein, inductive reactance generally has to do with the creation of magnetic fields inside an inductor. It is likewise frequency-dependent. For a given signal frequency and an inductor value (L), its magnitude in ohms is expressed as:
The last two quantities — XC and XL — produce perfectly opposing effects, and at a given frequency, they can cancel each other out; as a matter of convention, the equation for XC is sometimes written with a minus sign. But the effect of R is another story.
How do digital multimers deal with this?
If the phenomena can’t be easily told apart, it’s worthwhile to ponder how a typical digital multimeter (DMM) might attempt to characterize resistors, capacitors, and inductors — and how much you can depend on the number shown on the screen.
Resistors are a simple special case: the meter can apply a small DC voltage to the terminals of the device under test, indirectly measure the resulting current (usually by observing the voltage drop across a precision shunt placed in series), and then calculate the apparent impedance from Ohm’s law (Z = V/I). Because the frequency of the DC voltage is effectively zero, inductive effects can be disregarded; capacitive effects are also negligible due to the tiny value of C. In other words, when characterizing resistors, Z can be assumed to be equal to R.
The measurement of capacitors also involves measuring impedance — but this time, it’s done with an alternating current. The resulting value is some combination of R, XC, and XL. That said, because capacitors usually have very low internal resistance (ESR), it’s often assumed that R = 0; at low signal frequencies, XL is usually also negligible. Under these circumstances, the meter can roughly assume that Z = XC and then solve the capacitive reactance formula for C to arrive at a ballpark farad figure to put on the screen.
In principle, the same approach could be followed for inductors. That said, relatively few DMMs offer this mode; this is in part because discrete inductors are less common in modern circuits, but also because small inductors often exhibit non-trivial wire resistance — often tens or hundreds of ohms. Under such circumstances, assuming that R = 0 would taint the readings more than it does for most capacitors.
What makes an LCR meter special?
On the face of it, an LCR meter looks like a nerfed DMM, only capable of measuring inductance (L), capacitance (C), and resistance (R). One has to wonder how this piece of gear survived this long, and why it commands a considerable premium for such a limited feature set.
The answer is that the “LCR” moniker is a bit of a misnomer. A better name might be a “|Z|θ meter”. The first thing the meter does is measuring scalar impedance (Z or |Z|) at a user-specified signal frequency. It does so by outputting an AC signal, comparing the observed voltage amplitude and current, and then calculating Z = V/I. Just like in a DMM, the measurement represents the combined effect of resistive, capacitive, and inductive effects; it tells us nothing about the exact values of L, C, or R. That said, the other parameter the meter computes with high accuracy is the phase offset between the voltage and current (θ) — and that extra bit of information changes everything.
Recall from the earlier article that for pure resistive loads, voltage and current stay perfectly in-phase: the momentary current is proportional to the momentary applied voltage. In a capacitor, the situation is different: the charging current is the highest when the capacitor is discharged and the lowest when the device fully charged (and the voltage across the terminals is at its peak). With a repetitive sine wave signal, this gives the appearance of the voltage lagging by 1/4th of a cycle behind current:
Phase shifts in sine waveforms are often measured in degrees; a full wavelength is 360°, so a shift of -1/4th of a cycle can be expressed as θ = -90°.
In an inductor, the shift happens in the opposite direction: the flow of current is impeded at first, allowing the voltage to build up. With repetitive sine waves, this gives the appearance of the current lagging behind by 1/4th of a cycle (θ = +90°):
Because the apparent contributions of inductive or capacitive reactance are shifted ±90° in phase in relation to the effects of resistance, the overall impedance observed for the sine signal isn’t a simple sum of R and XL or XC; instead, it can be visualized as a two-dimensional vector:
A vector associated with a pure resistance (θ = 0°) would point along the x axis; an ideal inductor (θ = 90°) would point up; and an ideal capacitor (θ = -90°) would point down. For the intermediate cases, |Z| is equal to the diagonal of the R-X rectangle:
From basic trigonometry, if we know the length of the impedance vector (|Z|) and the phase angle (θ), we can get back to the values of R and XL or XC:
Did we learn anything useful?
Well, perhaps! For one, we can answer a lingering question about reactance: this polar-coordinate-based analytic approach is precisely why the formulas for for XC and XL contain the “2𝜋” expression. This operation converts “normal” sine wave frequencies (Hz, cycles per second) to angular frequencies (one 360° circle per cycle). This allows the use of standard trigonometry in the two-dimensional plane.
For another piece of trivia, look at the left side of the polar plot: the negative part of the x axis corresponds to negative resistances, which aren’t common in real life — but can be simulated with active circuitry such as op-amps. Feasibility aside, it’s clear that a hypothetical matched “anti-resistor” placed in series with a normal one would result in a net zero resistance. The same idea holds for the y axis: for a single chosen sine wave frequency, it’s possible to match a capacitor with an inductor so that their influences cancel out. Now that we have this new analytical tool, it should be easy to understand why.
What else? Oh, right: now that we know that LCR meters are actually |Z|θ meters, we can talk about the unexpected problems that arise if one tries to use them to measure R, C, and L values without understanding the underlying principles. (The same goes for using the meters to measure more exotic derived parameters, such as the oscillator dampening factors D and Q.)
For example, if you take a 10 µF multilayer ceramic capacitor, connect it to an LCR meter, and set the test frequency to 200 kHz, you might end up with a reading that looks quite a bit off. It might be tempting to blame it on the capacitor, but MLCCs are supposed to behave properly well into tens of megahertz. So, what’s going on?
The answer has to do with the measurement the meter is actually trying to make: scalar impedance. As discussed earlier, for an MLCC operated at 200 kHz, it comprises almost entirely of its capacitive reactance, which is — again — given by the following formula:
In this particular instance, XC works out to about 80 mΩ; that’s too low for many meters to measure accurately. If you lower the frequency to 1 kHz, the math works out to 16 Ω — a much more reasonable result. The issue is evident if you’re looking at the |Z| reading, but it’s easy to miss if you’re staring at the computed C value instead.
There is an inverse problem at low frequencies: a 10 pF capacitor tested at 100 Hz shows up as an impedance of 160 MΩ. Switching to 50 kHz puts the impedance at several hundred kΩ, which is much easier to deal with. Either way, LCR meters are not nearly as plug-and-play as they might seem.
Do I need this on my workbench?
It depends. Unlike an oscilloscope or a DMM, an LCR meter is not essential in most hobby workshops. The device tends to come handy when working with analog circuits, especially in the audio and RF domain; it’s also useful for troubleshooting inductors of all shapes and sizes, and for weeding out electrolytic caps that have gone bad. On the flip side, if you’re working mainly in the digital domain, the meter is likely to end up collecting dust.
It’s perhaps worth noting that for casual experimentation where precise impedance readings are unnecessary, an oscilloscope working in tandem with a signal generator can offer comparable insights — possibly across a wider range of frequencies.
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