Verification of Low-impedance EIS Using a 1 mΩ Resistor

Why Use Twisted-pair Wiring? 

Figure 3 shows the importance of wiring in EIS measurement of the 1 mW 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 3. Resistor Spectra with Different Cables.

The black and gray 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 mW. 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, we don’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 has zero resistance and inductance. 

The EIS spectrum of a copper slug used as a surrogate was recorded using the same fixture, wiring, and 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 mW, these data fit fairly well to a series RL model, with R = 4.5 mW and L = 444 pH.  

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 four-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 errors 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.

Figure 5. Corrected (red and pink) and uncorrected resistor spectra (black and gray).

Figure 5 shows the resistor’s spectrum before and after a series-subtraction of the surrogate’s spectrum. The resistive region in the corrected spectrum obviously extends to higher frequencies.

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. For the data recorded with the standard cell cable only the data below 10 kHz were used in the fit, because the data above 10 kHz showed non-inductive phase shifts.

The fitted R and L values are shown in Table 1. All the resistance values are within the 5% resistor tolerance.  

Table 1. Fitted R and L values for the three spectra.

Unlike lower value resistors, where the terminal resistance is negligible, the terminal resistance on this 1 mW part must be substantial. If the part were made from solid copper with the same dimensions, its end-to-end resistance would be about 50 mW, or 5% of the resistor’s 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 10–20 mW. 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 ~ 3 mW. This is close to the 4.5 mW 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 Table 2:

The frequencies were calculated assuming:

• The resistor’s true value is the 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 Application Note, Gamry Instruments presents some 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 the resistor spectrum is corrected by series-subtraction of a low-impedance copper surrogate, the accuracy of the measurement is extended to higher frequencies. 

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. 

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. Cables 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. 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 mW 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. 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 connections. 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. 

You can make surrogate measurements on a fuel-cell assembly 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.

[1] Vishay Intertechnology, Inc., 63 Lancaster Ave., Malvern, PA 19355-2143.