# Verification of Low Impedance EIS Using a 1 mω Resistor

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2 Avoid High Frequency Mutual inductance creates a voltage error given by: Vs = M di/dt Vs is the induced voltage on the sense leads, M is the coupling constant (with units of Henries), and di/dt is the rate of change in the cell current. M depends on the degree of coupling and can range from zero up to the value of the inductance in the current-carrying leads. Assuming a constant amplitude waveform in the primary, di/dt is proportional to frequency. The importance of the error voltage depends on its size relative to the true voltage being measured, which in turn is proportional to the cell impedance. Mutual inductance errors appear in the measured EIS spectrum as an inductor of value M in series with the cell s impedance. Minimize the Net Magnetic Field A current passing through a wire creates a magnetic field with the field strength proportional to the current. Fortunately, passing the same current in opposite directions through adjacent wires tends to cancel the external field. Two different wire arrangements are commonly used to minimize inductance and magnetic fields. The first is a coaxial cable; a central conductor is used to carry the current in one direction and a second conductor surrounding the first carries the current in the opposite direction. The second common arrangement is the twisted-pair; two insulated wires carrying current in opposite directions are twisted together. Separate the Pairs The magnetic field produced by a wire loses intensity as the inverse square of the distance away from the wire. Separating the sense wires from the current-carrying wires can dramatically reduce the magnetic coupling. Twisting the sense wires helps in two ways. First, the twisted wires are forced to lie close to each other, minimizing the loop areas. Secondly, adjacent loops pick up opposite polarity voltages, which results in cancellation. Cabling Recommendations Use coaxial cable or twisted pair for each pair of leads. The distance between the pairs should be maximized. Arrange each pair so that they approach the cell from opposite directions as shown in Figure 1. System Current Carrying Leads i Sense Leads Maximize Cell Minimize Figure 1. Recommended Cell Connections. Mutual inductance errors are more significant at lower cell impedances and higher frequencies. For example, on a system with 1 mω of resistance and 1 nh of mutual inductance, EIS phase shift will be 0.4 at 1 khz and 3.6 at 10 khz. If the resistance is lowered to 200 µω without changing the inductance, the phase shifts are 1.8 at 1 khz and 17 at 10 khz. To minimize mutual inductance errors, Gamry Instruments has developed twisted-pair cell cables for our EIS systems. The results below show how these cables can improve the EIS spectrum of a test resistor. i Twist the Sense Wires The concept of a magnetic loop probe is useful in understanding why a twisted sense pair minimizes magnetic pickup. A loop of wire in a changing magnetic field will see a loop voltage proportional to the area of the loop.

4 Why Use Twisted Pair Wiring? Figure 3 shows the importance of wiring in EIS measurement of the 1 mω resistor. There are two Bode plots overlaid in this graph. In all Bode plots, the dark colors are magnitude and the corresponding light colors are phase. Figure 2. Surrogate Connected in Fixture. Why Galvanostatic Mode? Even though Potentiostatic EIS is the most commonly used EIS technique, it is poorly suited to impedance measurements of ECS devices. This is why: current, voltage, and impedance are related through Ohm s Law. A voltage of 1 mv across 1 mω of impedance corresponds to 1 A of current. No commercial potentiostat is specified to control a typical ECS device s potential (0.5 V to 4.5 V) with < 1 mv of error. When a potential with a 1 mv (or larger) error is applied to a low impedance ECS device a very large DC current will flow. This current, given enough time, can alter a battery s state-of-charge. Conversely, a galvanostat can easily control ampere currents to an accuracy of a few milliamps. The voltage on the cell is unaffected when the galvanostat is connected. The DC current is zero or some user defined value. A modern EIS system with AC coupling or offset and gain in the voltage measurement can measure microvolts of AC voltage superimposed on a large DC voltage, as long as that voltage is stable. Galvanostatic EIS is the preferred technique for EIS on ECS devices. This study employed Galvanostatic EIS to better model ECS measurements. Why Use Large Excitation Currents? In a galvanostatic EIS experiment, the voltage signal is proportional to the applied current. Measurement of voltages < 10 µv is difficult, since most measurement systems have a few µv of noise. It is best if the AC excitation current is kept large enough that the AC voltage is >10 µv. For a 1 mω cell, this means the current must be 10 ma or greater. Figure 3. Resistor Spectra with Different Cables. The black and grey data were recorded using the Reference 600 s standard cell cable. Its alligator clips were clipped to the pins on the resistor/header assembly. The red and pink data were recorded with a Low Impedance cable for the Reference 600 connected as described above. Both curves have similar shapes and low frequency impedance close to 1 mω. Both show inductance at higher frequencies. The inductance is much lower with the Low Impedance cable. The Resistor Surrogate s Spectrum Earlier graphs and discussion showed the importance of cabling on the measurement. But, even using a Low Impedance twisted pair cable, one doesn t know how much of the measured impedance is the true resistor impedance and how much to attribute to cable effects. A resistor surrogate allows you to measure the cabling effects. The surrogate is a metal object with the same geometry and connection scheme as the resistor. Ideally it will have zero resistance and inductance. The EIS spectrum of a copper slug used as a surrogate was recorded using the same fixture, wiring, and

5 experimental conditions as the resistor test. Figure 4 shows Bode plots of the surrogate (in black and grey) and resistor spectra (in red and pink). The surrogate spectrum is resistive at low frequencies and becomes inductive at higher frequencies. After removal of the point with an impedance below 1 µω, these data fit fairly well to a series RL model, with R = 4.5 µω and L = 444 ph. Figure 5. Corrected and Uncorrected Resistor Spectra. Figure 5 shows the resistor s spectrum before (in black and grey) and after (in red and pink) a series subtraction of the surrogate s spectrum. The resistive region in the corrected spectrum obviously extends to higher frequencies. Figure 4. Resistor and Surrogate Spectra. Is Spectrum Subtraction Useful? Mutual inductance creates an error that shows up in an EIS spectrum as inductance in series with the cell s real impedance. Similarly, an improper 4-terminal connection, in which current flows through both a current-carrying and a sense lead, can create resistive errors. This error is also in series with the cell s true impedance. In our resistor tests, with a constant fixture and unchanging cell connections, these error should be identical in both the resistor s and resistor surrogate s spectra. A series subtraction of the surrogate s spectrum from the resistor s spectrum can remove these effects. Spectrum Comparisons There were three resistor spectra discussed above. 1. The spectrum using the standard cell cable. 2. The spectrum using the Low Impedance cable. 3. The spectrum using the Low Impedance cable corrected using surrogate data. All of these spectra were fit to a series RL model. In the case of the data recorded with the standard cell cable only the data below 10 khz were used in the fit., since the data above 10 khz showed non-inductive phase shifts. The R and L values resulting from the fits are shown in this table: R error L error Standard 978 µω ± 3.0 µω 53 nh ± 0.4 nh Cable Low Z 972 µω ± 2.6 µω 806 ph ± 65 ph Low Z Corrected 969 µω ± 2.6 µω 390 ph ± 3.8 ph All the resistance values are within the 5% resistor tolerance.

6 Unlike lower value resistors, where the terminal resistance is negligible, the terminal resistance on this 1 mω part must be substantial. If the part were made from solid copper with the same dimensions, its end-toend resistance would be about 50 µω, or 5% of the resistors value. The resistance values from the fits may be low because a significant layer of solder built up on the resistor s terminals, which would lower the component s value, perhaps by as much as µω. In addition, the resistor s manufacturer specified its value assuming a distributed contact onto printed circuit board pads. Our connection geometry is quite different. Notice that the resistance value calculated from the corrected spectrum is lower than the uncorrected data by about 3 µω. This is close to the 4.5 µω resistance value calculated from the surrogate s spectrum. We cannot be certain which resistance value is more accurate. Correction for cable and fixture effects by subtracting a surrogate s spectrum assumes that the surrogate has zero resistance and inductance. If the surrogate has non-zero resistance or inductance, its subtraction can overcorrect the resistor spectrum. Gamry Instruments Accuracy Contour Plot defines two accuracy regions. The first is a region that can be measured with < 1% magnitude error and <2 phase error. The second is a region with < 10% magnitude error and < 10 phase error. What is the highest frequency in our three resistor spectra that lies within these regions? The limits are shown in this table: 1% and 2 10% and 10 Standard cable 79 Hz 505 Hz Low Z cable 5.0 khz 31.6 khz Low Z corrected 12.6 khz khz The frequencies were calculated assuming the resistor s true value is fit value in that spectrum. The resistor is assumed to be perfectly non-inductive. The second assumption led to phase errors calculated as deviations from 0. In practice, the phase error limit was always exceeded at a lower frequency than the magnitude error limit. Conclusions In this Technical Note, Gamry Instruments presents a number of guidelines for accurate EIS measurements on low impedance cells. Galvanostatic cell control, a large AC current, and twisted-pair cell wiring are all important. When these guidelines are followed, an EIS system equipped with a Gamry Instruments Reference 600 can accurately measure the impedance spectrum of a miniature, low inductance 1 mω resistor. When the resistor spectrum is corrected by series subtraction of a low impedance copper surrogate, the accuracy of the measurement is extended to higher frequencies. Assuming the 1 mω resistor that was used in this test is perfectly non-inductive, its impedance can be measured using a Reference 600 based EIS system out to 12.6 khz. Do the inductance values make sense? The inductance calculated from the resistor s spectrum was 806 ph. The inductance calculated from the surrogate s spectrum was 444 ph. The difference between these values (362 ph) is close to the 390 ph inductance calculated from the corrected spectrum. Note that the inductance correction via surrogate subtraction did not involve subtracting one large value from another large value to obtain a small difference. If the resistor inductance and surrogate inductance were much larger than the difference between them, the value of the difference would be suspect. Cabling and fixtures would need to be very, very reproducible to allow this type of correction. Some, or all, inductance remaining after spectrum subtraction may be real inductance in the resistor. Note that the 390 ph inductance value calculated from the corrected spectrum is slightly lower than the Vendor s claim that the WSL resistor has inductance between 0.5 nh and 5.0 nh. In general, EIS measurements that involve inductances in the ph range are difficult and errors in the measurement can be large. Measurement of 1 mω at 1 MHz is virtually impossible. Extension to Real World Devices The miniature resistor used in this validation test is not intended to represent a typical ECS device.

7 Real world ECS devices are generally much larger than this resistor and one cannot easily solder to their terminals. However, the guidelines listed above are still useful. EIS measurements on a packaged battery should always use a connection fixture to insure reproducible cabling. A battery surrogate can often be machined from copper or aluminum. The tinned wires on the Low Impedance cable can be soldered or mechanically connected to the fixture. Fuel cells usually have fixed connection points for the current-carrying and sense leads. These are often bolts or screw terminals. The tinned wires on a Gamry Low Impedance cable can connect directly with these points. Surrogate measurements on a fuel cell assembly can be made with metal foil in place of the fuel cell membrane(s). If you are careful to position your cell leads in a reproducible manner, the surrogate data and fuel cell data should have similar errors. Verification of Low Impedance EIS Using a 1 mω Resistor. Rev /25/2011 Copyright Gamry Instruments, Inc. 734 Louis Drive Warminster PA Tel Fax

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