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, in some contexts, it can be hard to tell which of the three effects you are looking at.
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 magnitude of this effect — also measured in ohms — is described by the following formula:
In essence, this just says that signals are attenuated in inverse proportion to f and C; as explained in the earlier article, the 2π constant is just a matter of the mathematical convention to define sin(x) over radians, resulting in a natural period of 2π. Because of this, to model a sine wave of a frequency f (in hertz) using a timing variable t (in seconds), we need to calculate sin(2πft).
In contrast to capacitive reactance, 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:
In other words, the attenuation is directly proportional to f and L, with the now-familiar 2π bit thrown in.
In the earlier article, we also explained why the two quantities — XC and XL — introduce opposite phase shifts, exactly 180° (half wavelength) apart.
How do digital multimers deal with this?
If the phenomena can’t be easily told apart, it’s worth pondering how a typical digital multimeter (DMM) might attempt to characterize resistors, capacitors, and inductors — and how much you can trust 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). A real-world capacitor exhibits some parasitic inductance in the conductive path, along with parallel capacitance due to the presence of a less-conductive layer sandwiched between two metal terminals:
That said, at f ≈ 0 Hz, inductive reactance of the component’s conductive path is about zero, while the parallel capacitive reactance is so high that the “side” path via the parasitic capacitor can be ignored. In other words, when characterizing resistors at DC, 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 parasitic resistance (ESR), it’s often assumed that R = 0; at low signal frequencies — below 100 kHz or so — the component’s 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 quite a bit of 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 applied voltage is changing the fastest. 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, there is a similar shift, but it happens in the opposite direction; 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, the behavior can be visualized as a two-dimensional vector in a space where a full waveform cycle corresponds to 360°:
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 approach is precisely why we measure phase shifts in degrees and insist on using angular frequencies for XC and XL. This convention produces a coherent model where circuit dynamics can be analyzed using standard 2D trigonometry.
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.
In general, the resulting impedance magnitude for a pair of inductive and capacitive reactances is simply:
If we plot this formula for some chosen component values across a range of frequencies, we get a result that crops up all over the place in electronics. It shows the behavior of an ideal inductor in series with an ideal capacitor:
You’ll see that shape in the performance plots for real-world capacitors (they have a bit of an inductance that takes over at very high frequencies), in the discussion of signal filters and oscillators, and so on.
We can solve the formula for the crossover frequency where the reactances are equal, and thus |Z| and θ are both zero:
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.
To illustrate, 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. For an MLCC operated at 200 kHz, this comprises almost entirely of its capacitive reactance, which is — again — given by the following formula:
If we plug in f = 200 kHz and C = 10 µF, 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 would be evident when looking at the raw |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 in 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.
👉 To continue reading about capacitors as signal filters, check out this write-up. For more articles about electronics, click here.
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