Fast Impedance Spectroscopy with Multisine Excitation

Whether you need to measure a lot of samples down to low frequencies during high throughput tests, or you need to analyze the difference of successive EIS measurements while your device under test (DUT) is changing (desired, such as charging of a battery - or undesired, such as degradation), the duration of an EIS can be a limiting factor. An impedance spectrum that reaches into the millihertz range is expensive in time, and nearly all of that time is spent on its last few frequency points.

Multisine excitation addresses exactly that part of the spectrum and is able to significantly speed up EIS measurements. Instead of measuring one frequency after another, all frequencies of the low-frequency range are excited at the same time and evaluated from a single recording.

The IM7 offers a multisine implementation for 1 to 5 decades. Thereby, one wave is used to excite all frequencies at once. The length of this wave is the period of the fundamental frequency (the lowest frequency of the multisine measurement). One wave is enough to calculate an impedance spectrum from the excitation and response signals. For example, a measurement from 10 mHz to 1 Hz (2 decades) can be done in 100 s providing 13 frequency points.

The results are pretty impressive. The outstanding accuracy is achieved by careful engineering of the measurement device and software. One crucial factor is the high precision provided by the 24 bit resolution of the measurement channel of the IM7, which allows for significantly higher precision than many other devices.

Amplitude is a challenge because the available excitation amplitude needs to be shared between all excited frequencies. We apply a crest factor optimization (crest factor: ratio between peak and RMS) that increases the amplitude for all frequencies in the multisine excitation significantly, while keeping the total amplitude below a given maximum. This is done by optimizing the phases of the applied Harmonics.

The IM7 provides a quick and automated multisine measurement with selectable parameters to tailor it for your purpose. In the Custom Experiment Builder or through the API, you can design your own multisine, choosing which harmonics are excited and with which amplitude.

Check out this comprehensive application note for practical examples and underlying theory. All measurements shown here were recorded on the same DUT, a commercial 40 Ah LiFePO₄ cell at rest, using a Zahner IM7 in galvanostatic mode with a peak amplitude of 1 A and no DC bias.

Single-Sine and Multisine EIS Compared

In Fig. 1, a direct comparison of a single-sine EIS (solid line) and a multisine EIS (markers) is shown. In this application note, the galvanostatic single-sine EIS sweep is treated as a reference sweep. The reference sweep is recorded from 100 kHz down to 10 mHz, recording 100 points with one pre-wave, and two measured waves per point, these settings are already chosen for speed. The reference sweep consists of 12 points per decade above 66 Hz and 8 points per decade at the lower frequency limit. The second measurement is a Galvanostatic Multisine EIS with identical frequency range and amplitude. It is configured to cover the lowest 3 decades with one multisine excitation, starting at the first harmonic, with no multisine pre-wave and one multisine wave. The crossover frequency is therefore 10 Hz: single-sine mode above, multisine mode below. Both spectra were recorded on the same cell within half an hour of each other.

Fig. 1: Single-sine EIS (line) and multisine EIS (markers) spectra of the 40 Ah LiFePO₄ cell. Comparison of the two measurements in the two common representations. Left: Bode plot, impedance magnitude (blue) and phase (red) against frequency. Right: Nyquist plot, imaginary against real part of the impedance, frequency increasing from the upper right toward the lower left. For the multisine EIS measurement, 21 lowest frequencies between 10 mHz and 10.07 Hz originate from a single multisine recording.

Single-sine and multisine lie almost perfectly on top of each other. In the Bode plot as well as in the Nyquist plot, the results of the multisine cannot be told apart from those measured through the single-sine sweep. How far apart the two actually are, and what that difference should be judged against, is analyzed in detail further down, but we will have a look at the measurement time first.

Where the Time Goes

While there is no big difference in the results between the two measurements, the required measurement time differs significantly.

Fig. 2: Elapsed measurement time at which each frequency point became available. Blue circles: single-sine reference sweep. Red squares: multisine measurement. The vertical column of red squares is the multisine range, whose 21 frequencies are all completed at the same instant.

The runtime of a single-sine sweep is set by the period of each frequency, and the arithmetic is unforgiving: one period at 100 Hz lasts 10 ms, one period at 10 mHz lasts 100 s, and each of those periods buys exactly one point of the spectrum. Both measurements therefore start identically and require less than 10 s to sweep from the start frequency up to 100 kHz and back down to 11.13 Hz. From there the two diverge.

• The reference sweep continues frequency by frequency, and the growing period pulls its points apart along the time axis. The first 71 points were all recorded within the first 9.3 s. The remaining 29 points, the three decades from 9.06 Hz down to 10 mHz, consumed the other 19 min 30 s (1169.7 s), that is 99.2 % of the total runtime. The 10 mHz point alone accounts for 5 min (300 s) of it, exactly the one pre-wave and two measured waves it was asked for.

• The multisine measurement instead applies a single excitation containing 21 frequencies and delivers all of them together after one period of the 10 mHz fundamental.

	Single-sine sweep	Multisine
Recorded points	100	92 (71 swept, 21 multisine)
Below 11.13 Hz	19 min 30 s (1169.7 s), 29 points	1 min 41 s (100.6 s), 21 points
Total runtime	19 min 39 s (1179.0 s)	1 min 49 s (109.4 s)
Speed-up		10.8× overall

Speeding up an impedance measurement therefore is done in its lowest decades. In this example, everything above 10 Hz was measured below 10 s with little potential to save additional time, and that is precisely why the multisine covers only the lowest decades instead of the whole spectrum. Extending it upward would not save significantly more time. Considering the data quality, adding more frequencies to the multisine would decrease the amplitude of every excited frequency because all of them share one peak amplitude. Keeping the multisine confined to the decades where a sweep is genuinely slow is what keeps that price small.

The multisine range here replaces 19 min 30 s of sweeping with 1 min 40 s of measurement. What is traded in is spectral detail and amplitude per point: 21 frequencies (multisine) instead of 29 (single-sine) in the same three decades, and each excited with about one seventh of the total amplitude in multisine mode (see Crest Factor section for more detail), while single-sine mode measures each frequency with the total amplitude.

What One Period Delivers

Two curves lying on top of each other in a Bode or Nyquist plot are a weak statement, because the plot spans four decades of impedance and hides everything below a few percent. Let’s perform a more thorough error analysis and discuss how the obtained residuals should be judged.

Importantly, the comparison of multisine is not against a single-sine optimized for high accuracy but against a rather fast single-sine sweep. It spends one pre-wave and two measured waves per frequency (with a minimum of 20 ms of measurement duration at each frequency point). High-accuracy EIS averages over considerably more waves than two. The reference sweep is meant to be a fast reasonable single-sine sweep since we are comparing methods for fast EIS.

Even set up that way, the three lowest decades cost 19 min 30 s (single-sine, see table above). The multisine lasted one period of the 10 mHz fundamental for all 21 frequencies combined, 100 s in total: no pre-wave, no repetition, no averaging beyond that single period. Fig. 3 shows deviation of multisine from the reference single-sine EIS.

Fig. 3: How does the one-period multisine compare to the reference. Top: magnitude range of the cell impedance; the deviations below are compared against an object ranging between 0.9 and 11 milliohms in impedance magnitude only. Middle and bottom: deviation of the multisine measurement from the reference sweep in magnitude and in phase, respectively. Filled markers are the deviations for the 21 multisine frequencies. The open markers show the deviation of the sweeps above the crossover contained in both measurements and represents the run-to-run reproducibility of cell and instrument rather than anything about the multisine.

Three yardsticks are drawn alongside the residuals to help ranking the residuals.

• The open markers above 10 Hz are not multisine result at all. Above the crossover, both runs contain ordinary single-sine measurements of the very same frequencies, recorded some twenty minutes apart, so their difference is the reproducibility of the instrument and of the cell at this impedance level: 0.43% at worst, 0.20% rms, and in phase 0.24 degree max, 0.08 deg rms. That is the noise floor any method has to be compared against here, multisine or not.

• The filled markers below 10 Hz are multisine result with deviation in impedance of 1.28% max, 0.35% rms and in phase 0.3 degree max, 0.12 degree rms.

• The grey band around the x-axis is the interpolation floor of the comparison. The multisine harmonics do not fall exactly on the frequency grid of the sweep, so the reference has to be interpolated. The band is a leave-one-out test of exactly that interpolation on the reference itself: drop one reference point, interpolate it back from its neighbours, plot the error. It reaches 0.29% and 0.09 degree, so part of the residuals should be judged against the inaccuracy of the comparison itself.

The result speaks for itself. Twenty of the 21 multisine points deviate by less than 0.6% in magnitude, 0.35% rms, and by less than 0.3 degree in phase, 0.12 degree rms. The scatter of the multisine points is thus of the same size as the scatter of two ordinary sweeps of the same cell against each other. One period of the fundamental was not the limiting factor of this measurement: the 19 min 30 s of measurement time on the same three decades in single-sine mode did not result in any accuracy gain above the reproducibility of the cell and instrument.

The single exception is the 10 mHz point, off by 1.3%. It is the frequency at which both grids coincide exactly, so no interpolation is involved, and it is also the pair of values with the largest contrast in time: the reference obtained the 10 mHz point only after 19.7 min of sweeping, while the multisine delivered it after 100 s. Considering that a resting cell may also drift, that drift will be most visible in the point that takes longest to record. That means a drift of the measurement object will be most severe in the slow method, but since that acts as our reference, the error is attributed to the fast method.

What makes this a challenging test is the object. Apart from the drift we just discussed, the whole spectrum of the cell lies between 0.9 and 11 milliohms. With the peak amplitude shared among 21 components each multisine point relies on small response. An impedance obtained from a sub-millivolt response, recorded once, is a considerably bolder claim than on a demonstration object of a few hundred ohms, where the same excitation produces a response several orders of magnitude larger.

A Complete EIS Spectrum in One Second

The crossover frequency between the swept and the multisine part is a free parameter in Zahner Lab, so the swept part can also be removed entirely. Setting the start frequency and the upper frequency limit both equal to the crossover frequency leaves nothing to sweep and measures the whole spectrum in multisine mode.

The EIS measurement (markers) in Fig. 4, performed on the same cell as the measurements before, was configured this way: in the frequency range from 1 Hz to 10 kHz, four decades, 29 (close to) logarithmically spaced frequencies were excited simultaneously on a 1 Hz fundamental, one wave, no pre-wave. The measurement took 1.12 s from start to finish, 1 s of measurement and 0.12 s of post processing. For comparison, the reference sweep of the previous section needed 19.7 s of its runtime to cover the same four decades with 47 single-sine points.

Fig. 4: Impedance spectrum of the 40 Ah LiFePO₄ cell (markers), four decades from 1 Hz to 10 kHz, recorded as one multisine excitation in 1.12 s (1 s excitation and 0.12 s post processing). Solid line is the single-sine reference from before.

Even in this fastest possible configuration, with a single fundamental period and no pre-wave, the spectrum reproduces the features of the slow reference sweep: the impedance minimum near 1.8 kHz, the inductive branch above it, and the rise of the impedance toward low frequencies.

One practical condition applies. The IM7 selects its current range from the response it sees, and on a multisine wave, every ranging attempt costs a full period of the fundamental frequency, which can easily exceed the measurement itself. When the multisine covers only part of the spectrum, the swept part runs first and leaves the instrument on the correct range, so this never becomes visible. When the entire spectrum is one multisine, as in Fig. 4, the range should be established beforehand by a short EIS or Table EIS measurement.

How Multisine Excitation Works

A multisine signal is a superposition of sine waves whose frequencies are all integer multiples, called harmonics, of one common fundamental frequency. That choice is not cosmetic: because every component completes a whole number of periods within one period of the fundamental, a recording of that single period contains each component without spectral leakage, and the individual frequencies can be separated from one another by a discrete Fourier transform.

The consequence for the measurement time is immediate. The fundamental frequency is the lowest frequency of the spectrum, and it alone sets the duration. One period of a 10 mHz fundamental takes 100 s, and every harmonic in the set is measured for those same 100 s.

Exciting a sum of sine waves is an established approach in impedance spectroscopy and goes back to the Fourier-transform-based spectrometers that Popkirov and Schindler described in the early 1990s. What decides whether such an excitation yields usable impedance data is the design of the signal: which harmonics carry energy, how the available amplitude is distributed among them, and with which phases they are added.

Choosing the Harmonics

No electrochemical cell is perfectly linear. Every excitation frequency therefore also produces a small response at its own multiples. If such a distortion product happens to coincide with another excited frequency, it is indistinguishable from the response to that excitation and is recorded as impedance. A consecutive series of harmonics is the worst possible choice for this reason, since every harmonic of the lower frequencies lands on a higher one.

Common practice is therefore to select the harmonics such that distortion products overlap the excited frequencies as little as possible. Restricting the set to odd harmonics is the classical form of this argument.

In the automatic multisine EIS methods of the IM7 this selection is made by the instrument, which generates a harmonic set optimized for the requested frequency range, (closest possible to) logarithmically spaced with up to eight frequencies per decade. The set behind the three decades of the comparison above, for example, consists of the harmonics 1, 3, 5, 7, 11, 13, 17, 23, 31, 43, 59, 73, 101, 131, 179, 239, 317, 421, 563, 751 and 1007: odd throughout, so that the even-order distortion products of every component fall between the excited frequencies rather than on them, and increasingly sparse toward the upper end keeping the points per decade constant. Only the extent of the multisine range and the harmonic it starts on are free parameters, which is why the point density parameters of the sweep have no effect inside the multisine range. Full control over the individual harmonics is available separately, in the Multisine Table EIS block of the Custom Experiment Builder.

Crest Factor

The excitation is limited by its peak value, not by its energy content. This is what makes the phases of the individual sine waves decisive. Added with identical phases, the maximum of the resulting amplitude or peak is almost the sum of the individual amplitudes. With N frequencies in the signal, each one could then carry only little more than 1/N of the permitted amplitude, and the signal-to-noise ratio of every single point in the spectrum would suffer accordingly.

The relevant figure of merit is the crest factor, the ratio of peak value to RMS value.

\[CF = \frac{\hat{I}_{peak}}{I_{rms}} \tag{eq. 1}\]

It is possible to optimize the phases so that the peaks of the components are distributed and the overall amplitude or peak is minimized. Every component can then carry considerably more amplitude within the same peak limit. Optimized phase sets for this purpose have been published by Popkirov and Schindler and are discussed in the standard EIS literature; the IM7 performs this optimization internally for the harmonic set it generates, and applies the optimized phases automatically.

Fig. 5 shows what such a signal looks like: the excitation of the three-decade multisine of the comparison above, over one period of its 10 mHz fundamental.

Fig. 5: The excitation used for the three-decade multisine of the comparison in Figs. 1 to 3, one period of the fundamental, normalized to its peak. All 21 harmonics are present with equal amplitude and phases optimized to minimize the crest factor.

It looks like noise but is nothing of the sort. What matters in the picture is that it fills the band between its limits almost uniformly instead of resting near zero for most of the period and reaching its permitted value in one narrow spike. Its crest factor is 2.16, against \(\sqrt{2} = 1.41\) for a single-sine wave and 6.48 for the same 21 waves added with identical phases. Since the amplitude limit of the instrument sets the peak and nothing else, that difference translates directly into the amplitude available to each single component.

How much amplitude that is follows from two relations. The components of a multisine sit at different frequencies. Their powers add and the RMS value of a signal made of N components of equal amplitude \(\hat{I}_{i}\) is

\[I_{rms} = \sqrt{\sum_{i=1}^{N}\frac{\hat{I}_{i}^{2}}{2}} = \hat{I}_{i}\sqrt{\frac{N}{2}} \tag{eq. 2}\]

The crest factor relates that RMS value to the peak value \(\hat{I}_{peak}\) of the signal, and the peak is what the instrument limits and what the method is parameterized with (1 A here). Inserting eq. 2 into eq. 1 and solving for the amplitude of one component gives

\[\hat{I}_{i} = \frac{\hat{I}_{peak}}{CF}\,\sqrt{\frac{2}{N}} \tag{eq. 3}\]

With the \(N = 21\) harmonics of that measurement and the crest factor of \(CF = 2.16\) the optimized signal achieves

\[\hat{I}_{i} = \frac{\hat{I}_{peak}}{2.16}\,\sqrt{\frac{2}{21}} = 0.143\;\hat{I}_{peak} = \frac{\hat{I}_{peak}}{7.0} \tag{eq. 3a}\]

With the applied 1 A peak amplitude, every of the 21 frequencies is excited with 143 mA, roughly a seventh of the peak amplitude. Their amplitudes add up to \(21 \cdot \hat{I}_{peak}/7 = 3\,\hat{I}_{peak}\), three times the peak the signal is ever allowed to reach.

Eq. 3 also reproduces the two reference values quoted above. A single-sine is the case \(N = 1\) with \(CF = \sqrt{2} = 1.41\), for which eq. 3 returns \(\hat{I}_{i} = \hat{I}_{peak}\), the full amplitude on the one frequency being measured. The same 21 waves added with identical phases instead reach their maxima at the same instant, so the peak becomes \(N \hat{I}_{i}\) and

\[CF_{identical} = \frac{N \hat{I}_{i}}{\hat{I}_{i}\sqrt{N/2}} = \sqrt{2N} = \sqrt{42} = 6.48 \tag{eq. 4}\]

which is the 6.48 above, and eq. 3 then leaves each component with no more than \(\hat{I}_{i} = \hat{I}_{peak}/N = \hat{I}_{peak}/21\). The phase optimization therefore multiplies the amplitude of every single frequency by \(6.48/2.16 = 3\), and only because of that, the penalty by lowering the individual amplitude against a single-sine measurement is a factor of 7 instead of 21.

What the Multisine Trades In

Speed of this magnitude is not free, but only one of the three trade-offs discussed here is a genuine loss of data quality. It is worth keeping the three apart.

The one that costs quality is the shared amplitude. Even with an optimized crest factor, every frequency is excited far weaker than it would be in a single-sine measurement. In the comparison above that was by a factor of about seven, and the signal-to-noise ratio of each point is directly linked to the amplitude it was measured with. Raising the total amplitude is not a solution, because the excitation must stay small enough for the cell to respond linearly. This is the one penalty that grows with every frequency added to the set.

The averaging, contrary to what the simultaneous excitation suggests, is not worse for multisine. All components are evaluated from the full recording, and each of them is evaluated from the whole of it: in a one second record, a 1 kHz component is measured and averaged a thousand times. If the same number of waves is configured, the fundamental frequency of a multisine is averaged exactly as many times as the same point of a sweep. Every frequency above it sees far more periods than a sweep would ever spend there. What is lost is not averaging but the freedom to distribute it. A sweep can ask for many waves where periods are cheap and few where they are expensive, while a multisine has a single wave count for the entire set, and each additional wave costs one more period of the fundamental, the most expensive time in the whole measurement. Averaging can only be bought for all frequencies at once, and always at the price of the slowest one.

The third trade-off is spectral detail. In the multisine range, the frequencies are the harmonic set the instrument generates, up to eight per decade. As such, the comparison above returned 21 points for three decades, whereas in a single-sine sweep the user chooses the number of frequencies freely. On the lower end of the frequency spectrum, the factor between lowest (fundamental) frequency and the next higher frequency is 3 (or 2 if the second Harmonic is applied, which is not the case in automatic multisine).

Only the first of the three becomes more severe the more frequencies a multisine contains, since the peak amplitude is divided among all of them. This is what the split of the spectrum described above addresses: the multisine is only used at frequencies whose periods are long enough to be worth the price. Higher frequencies instead, which a sweep covers in seconds anyway, keep the full amplitude and the freely chosen averaging of a single-sine measurement.

Multisine at Drifting Conditions

As seen above, multisine can dramatically shorten the required measurement time in the low frequency range. That unlocks new opportunities for in operando analysis and online diagnosis. For example, multisine can provide a snapshot of the impedance for a sample that is drifting, whether by design (e.g. charging of a battery) or by accident (e.g. a degrading DUT).

Due to the reduced measurement time, such snapshots are a better representation of a DUT at a specific state while drifting. A single-sine measurement would represent a DUT’s impedance as a function of frequency and time, the time when the respective frequency is measured, and may be inconsistent.

The result is a full spectrum in little time. However, if the drift in DC current or DC voltage in the response signal is substantial, the impedance calculation is affected and yields a systematic error. In that case, the implemented drift correction can be used to mitigate the effect of the DC drift.

Running a Multisine Measurement on the IM7

Multisine excitation is available in Zahner Lab both as a ready-made method and as a building block for the Custom Experiment Builder, and outside Zahner Lab through the API:

• Galvanostatic Multisine EIS (GEIS), the method used for the measurements in this note, and Potentiostatic Multisine EIS (PEIS), which works identically with a voltage excitation. Both sweep the upper frequencies with single-sine waves and measure the lowest decades with one multisine excitation. The parameter descriptions on those pages cover the crossover frequency, the starting harmonic and the number of multisine waves.

• The EIS Multisine block is the same experiment inside a custom experiment, with the multisine pre-waves additionally exposed as a parameter.

• The Multisine Table EIS block gives full control over the excitation: the fundamental frequency, the harmonics themselves, and the weight with which each harmonic shares the total amplitude.

• Without Zahner Lab, the underlying IM7 jobs are started directly through the Zahner Link API: EisMultisineGenerateJob for the automatic multisine and EisMultisineTableJob for a freely defined harmonic set.

References

• G.S. Popkirov, R.N. Schindler; Rev. Sci. Instrum. 63 (1992) 5366

• G.S. Popkirov, R.N. Schindler; Rev. Sci. Instrum. 64 (1993) 3111-3115

• A. Lasia; Electrochemical Impedance Spectroscopy and its Applications, Springer New York (2014), Sect. 3.7.3