Li-Ion Battery Capacity Evaluation

In the first approximation, we can present a Li-Ion battery as a jar with the following parts:

1. Consumed volume
2. Remaining volume

But why is this not true, and why can't we consume all the charge from the battery?

Each battery, over some period of its life, loses some of its capacity, so even if you charge it to full, it will self-discharge to some extent.

It is a best practice not to fully discharge a Li-Ion battery. The low safe threshold of voltage is around 2500 mV, lower voltage may damage the battery. On one hand, it is better to have a few percent in reserve; on the other hand, the voltage of the battery decreases dramatically when it is a couple of percent from empty. Therefore, it is a good compromise to stop consuming energy from the battery at a voltage of 3100 mV, i.e., reserve approximately 1.8% of charge.

As a result we have four sections of a capacity of the Li-Ion battery:

1. Self-discharged
2. Consumed
3. Remainder
4. Reserved

Obviously, the first and last sections are unavailable to consume.

Charge Capacity

Let's determine different charge capacities, which will be used in the calculations later:

• Q_start: Unavailable amount of charge which is dispensed on self-discharge after the battery is fully charged.
• Q_passed: Amount of charge that has been consumed.
• Q_remaining: Amount of charge that is available to consume at the moment.
• Q_leftover: Amount of charge that is reserved and never consumed, making it unavailable.
• FCC: Amount of charge that can be consumed after the battery is fully charged.
• Q_max: Theoretical maximum capacity of the battery.

Open Circuit Voltage and Depth Of Discharge

Open Circuit Voltage (OCV) - is the voltage of the battery in a relaxed state, i.e., after a period of time without charging or discharging, when the rate of voltage change becomes less than approximately 4 µV/s.

The OCV characteristic is important because, based on it, you can evaluate the depth of discharge (DOD) of the battery.

Depth of Discharge (DOD) - is a value between 0 and 1, or between 0% and 100%, indicating how close the battery is to full discharge. 

• Fully charged = 0
• Fully depleted = 1

Each type of battery has its own OCV vs. DOD dependence. For Li-Ion batteries, it looks like this:

To get the data - see my another article: https://www.linkedin.com/pulse/extracting-li-ion-battery-open-circuit-voltage-ocv-vs-sylichenko-yiygf/

DOD0 - is the absolute depth of discharge relative to Q_max and can be determined using the open-circuit voltage (OCV).

DOD - is a relative depth of discharge relative to FCC and it can only be calculated. There are many ways to do this; here is one of them:

DOD = Q_passed / FCC;

State of Charge (SOC) - is the inverted value of DOD, indicating the percentage of the remaining available charge. This is a relative value, so it is also relative to FCC.

• Fully charged = 1
• Fully depleted = 0

Q_remaining = FCC - Q_passed;

SOC = Q_remaining / FCC =
  = ( FCC - Q_passed ) / FCC =
  = 1 - Q_passed / FCC =
  = 1 - DOD;

Determining Maximum Available Capacity

To evaluate all the capacities of the battery, we first need to process a learning cycle with the battery:

1. Discharge the battery until the voltage drops below 3100 mV and disconnect the load.
2. Wait until rest.
3. Assess the DOD0 - DOD0_fin. This will be our reference point for the maximum achieved discharge of the battery.
4. Charge the battery to full, counting the passed charge - Q_passed.
5. Wait until rest.
6. Assess the DOD0 - DOD0_start. This will be our reference point for the minimum achieved discharge of the battery.

For more details, see my another article: https://www.linkedin.com/pulse/gas-gauging-device-bq28z610-learning-cycle-practical-guide-oleksii-ngk8f/

Calculate Q_max:

Q_max = Q_passed / (DOD0_fin - DOD0_start);

Calculate Q_start:

Q_start = Q_max * DOD0_start;

Calculate Q_leftover:

DOD0(3100) = 0.9812;

Q_leftover = Q_max * ( 1 - DOD0(3100) ) =
  = Q_max * (1 - 0.9812) =
  = Q_max * 0.0188;

Calculate FCC:

FCC = Q_max - Q_start - Q_leftover =
  = Q_max - Q_max * DOD0_start - Q_max * ( 1 - DOD0(3100) ) =
  = Q_max * (1 - DOD0_start - ( 1 - DOD0(3100) ) ) = 
  = Q_max * (1 - DOD0_start - 1 + DOD0(3100) ) = 
  = Q_max * ( DOD0(3100) - DOD0_start );

Example

• DOD0_fin = 0.96;
• DOD0_start = 0.0866;
• Q_passed = 2650;

DOD0(3100) = 0.9812

Q_max = 2650 / (0.96 - 0.0866) = 3034;
Q_start = 3034 * 0.0866 = 263;
Q_leftover = 3034 * 0.0188 = 57;

FCC = 3034 - 263 - 57 = 2714;
FCC = 3034 * ( 0.9812 - 0.0866 ) = 2714;
FCC = 2650 * ( 0.9812 - 0.0866 ) / ( 0.96 - 0.0866 ) = 2714;

State of Charge Calculation

When the battery is being used in the field (under real conditions), you need to determine the state of charge at every moment.

There are two cases for a partially discharged battery:

1. Rest state
2. Under load

Rest state

When the battery is in a rest state, its state of charge can be calculated using the DOD0 value, which can be obtained from tabulated data based on the battery's voltage.

SOC = ( DOD0(3100) - DOD0 ) / ( DOD0(3100) - DOD0_start );

Subcase: without leftover

You may not use Q_leftover in your calculations if your load takes into account the state of charge (SOC) and automatically turns off when the SOC is below a certain threshold. Therefore, it is not necessary to reserve charge on our side. For example, the Bluetooth speaker JBL XTREME 2 turns off when the SOC is below 3%. In that case, you may calculate the SOC for the full charge capacity (FCC) from DOD0_start to DOD0 = 1:

SOC = ( 1 - DOD0 ) / ( 1 - DOD0_start );

Example

• DOD0_start = 0.087;
• OCV = 3655;

DOD0 (3100) = 0.9812;
DOD0 (3655) = 0.6;
SOC = ( 0.9812 - 0.6 ) / ( 0.9812 - 0.087 ) * 100 % = 43 %;

Under Load

When the battery is under load, you cannot use its voltage to determine its DOD. Because of the battery's internal resistance, its voltage drops by an amount that depends on its resistance, which can vary from one battery to another, from one DOD to another, and even over the battery's lifespan.

This case is much more complicated, and I will consider it in my next article.

Stay tuned!

References

• OCV vs. DOD0: https://www.linkedin.com/pulse/extracting-li-ion-battery-open-circuit-voltage-ocv-vs-sylichenko-yiygf/
• Learning Cycle: Practical Guide: https://www.linkedin.com/pulse/gas-gauging-device-bq28z610-learning-cycle-practical-guide-oleksii-ngk8f/
• Texas Instruments “Theory and Implementation of Impedance Track”: https://www.ti.com/lit/an/slua364b/slua364b.pdf
• Drawing equations: https://latex.codecogs.com/eqneditor/editor.php