In Impedance track algorithm, How is resistance is related to soc(state of charge)?

Lithium rechargeables have a bathtub ESR curve with a bathtub flat zone between 10 % and 90% SOC and increases rapidly to 3x to 10x minimum ESR at extreme voltages. Depending on chemistry, battery quality and battery aging, ESR roughly starts around 50 mOhm for a 1C 2.4Ah cell . The C rate is controlled by the ESR more than Ah, so you could estimate the min. ESR with a 0.1V (est.)step V change so 1C=0.1V/2.4A=40mOhm so ESR=0.1V/(Ah*C rate). One must be aware of this C rate with cell temp extremes which also increase ESR at cold temps and also high C rates at room temp and low SOC levels so Pd self heating =I^2*ESR which rapidly heats and reaches 10%SOC hopefully quickly otherwise accelerated aging occurs.

Cadex now has an Impedance Spectroscopy tester which can test any batter for which you have a matrix to define its fundamental equivalent circuit and min:max range of values. ( it can store up to 25 matrices) https://www.cadex.com/en/products/spectro

I believe it follows what I have known for 20 years from my past experience in battery testers, rejuvenators and equivalent circuits from a Test Engineering perspective. It is consistent with methods that are similar used in large power transformer insulation (dielectric) defect analysis, except they (Cadex) use 20 to 2kHz, and TFXR testers use 1mHz to 1MHz was on say > 10MVA types.

The reduction of SoC relative to a new battery is like a defect from electrode diffusion into the electrolyte and the effect of Charge voltage on impedance over the range of battery voltages. Cadex indicated they take about 2 million measurements in 15 seconds. I suspect perform pulse current rise vs V drop = “negative resistance” at various incremental pulse widths, then computed impedance , then sinusoidal constant-current decremented log-f burst tests to again compute the V/I vector impedance.

These can be compared to Nyquist Re vs Im plots, with either an estimate of SoC or a Pass/Fail criteria for charged cells or a sorting criteria for perfect cell array matching.

Anecdotal

We used to do this back in late 70’s developing automated Eddy Current instruments with 10 ppm resolution on any impedance change in the xxx kHz range. This was then normalized to samples for each scan for certain defect types in the medium due to microscopic voids or change in conductor thickness by vector impedance plots at different frequencies such that the frequency selected would create a vector result. (After AUTO-zero to null Z changes, two calibration tests with correct gain would give 0 deg=Real blips and 90 deg = Reactive blips of the same amplitude)

Here is Cadex’s patented tester https://www.cadex.com/en/products/spectro

Battery SoC [%] has many dependant variables;

• chemistry ( with multiple impedance RC=T time constants, cell voltage range)
• array geometry Ser-Par
• age derating factors
• specified acceptance criteria capacity from Mfg
• temperature effects
• demand type ( application or C useage rate )

If you have a matrix of parameters for any SOC test which may include most of the above, it is possible to make an accurate % SOC measurement using impedance Spectroscopy methods or in other words a VI “Bode plot” of the battery below 2kHz in order to determine the shift in RC parameters due to the state of charge.

It is well known that the battery is a complex capacitor ( many RC’s in shunt) each with different time constants.

• The imaginary series impedance is the a function of frequency just as it is in caps, ultra caps and batteries, where a small Li-Ion cell can be > 10 kFarads for the largest reactive element which can change slightly DC voltage and drop rapidly below 10% SOC.
• It can be measured by an equivalent RC network in swept frequency plots to determine the asymptotes and thus RC values relative to the type of battery selected for the test.

I could say more about how it works but do not want to guess their copyrights or expose their efforts as I would do the same, but suffice to say ...

Impedance Spectroscopy is used “EVERYWHERE” in medical, biology, geology, industrial, scientific and now consumer SoC BATTERY testers.

The impedance represent the internal serial resistance of the battery.

It is important to know this, because the state of charge greatly depends on it.

Basically, a part of the energy you will input (or draw) on the battery will be lost due to the internal serial resistance.

In other words, the state of charge is somewhat equal to the energy you input, minus the losses of the impedance.

Hence, to know the state of charge of the battery, you also need to know how much energy is being lost because of its impedance.

It also depends a lot on how much current you are injecting or drawing from the battery, as higher current you will have higher losses, as the loss is equal to the square of the current. Ploss = Rser * I ^ 2.

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EDIT

You can have a look at this article it contains valuable information and even refers to an IC able to evaluate this for you, if you are using Li-Ion.