**Amplifiers/Converters**

# Lithium-ion cell fuel gauging with Dallas Semiconductor devices

**Keywords:power
**

/ARTICLES/2001APR/2001APR10_AMD_POW_AN.PDF |

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INTRODUCTION
Determining the remaining charge of a Lithium-Ion cell accurately under real world conditions requires
much more than just coulomb counting alone. The DS2438's integrated current accumulator (ICA)
provides an accurate measurement of cell capacity under known conditions, however in applications
where temperature and discharge rates vary and the cell's capacity degrades with aging, the DS2438's
ICA needs to be adjusted to achieve the desired accuracy. This document shows how the fuel-gauging
concept of the DS2438 can be expanded to insure greater accuracy under extreme operating conditions.
This is accomplished by characterizing cell capacity over temperature and rate and controlling the
coulomb count in software. This process is not limited to just the DS2438 or a specific type of Lithium-
Ion cell. Any Dallas Semiconductor Battery Management device with a coulomb counter, temperature
converter and 15 bytes of user EEPROM, such as the DS2760 High Precision Li-Ion Battery Monitor is
capable of performing high accuracy fuel gauging on any type of Lithium-Ion cell.
LITHIUM-ION CELL BEHAVIOR
To understand why coulomb counting alone is not sufficient for high accuracy fuel gauging it is helpful to
see how its environment affects a Lithium-Ion cell's charge capacity. The cell in the examples below is a
1200 mAH rated 4/3A cylindrical. It was charged by a two step method, first by a constant 1C rate until
the cell voltage reached 4.2 volts, then by a constant voltage until the charge current fell below C/20 or
60 mA. At this point it was considered fully charged. It was discharged either at a high current rate of 1C
or a low current rate of 0.2C. The cell was considered to be fully discharged when its voltage fell below
2.5 volts.
TEMPERATURE & DISCHARGE RATE
The capacity of the Lithium-Ion cell varies greatly depending on the temperature and discharge rate.
Figure 2 shows its charge capacity in milliamp-hours as temperature and discharge rate are changed. The
"Full" line on the chart is the point at which the cell is considered fully charged by using the above
charging method at the corresponding temperature. The "High Current Empty" line is the point at which
the cell is considered fully discharged by a 1C rate at each temperature. The "Low Current Empty" line
was plotted in the same manner using a discharge rate of 0.2C.
Application Note 131
Lithium-Ion Cell Fuel Gauging with
Dallas Semiconductor Devices
www.dalsemi.com ) 2000 Dallas Semiconductor
HOST
PROCESSOR
DS2438
DS2760
Voltage, Current, and
Temperature Measurements.
Cell Characterization Data.
Accurate Capacity
Estimation.
Figure 1: System Diagram
BATTERY: 59%
APPLICATION NOTE 131
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The capacity of the cell at a given rate and temperature is the difference from the "Full" line and the
corresponding "Empty" line. Because both the empty and full points change over temperature and rate,
every point on the chart is relative to every other point. For example, if a cell was fully charged at a
temperature of 800C and then fully discharged at the low current rate at -200C, the amount of charge able
to be removed would be the difference between the full value at 800C (1340 mAH) and the low current
empty value at -200C (250 mAH) or 1090 mAH. If the cell was then fully recharged at -200C, only the
difference between the full and empty values at -200C or 860 mAH could be returned to the cell.
Only the immediate temperature and rate are needed to determine relative full and empty points. A cell
that is discharged partially at temperature 1 and rate 1, then discharged completely at temperature 2 and
rate 2 will be considered empty at a point based on temperature 2 and rate 2. Similarly, the cell above
could be fully discharged at the high current rate yet is able to be further discharged at the low current
rate by the number of milliamp-hours between the two "Empty" points that correspond to the present cell
temperature. Because of this, it is only necessary to keep track of the present cell temperature and
discharge rate when determining remaining capacity.
CELL AGING
As a Lithium-Ion cell ages it loses its ability to store charge. Figure 3 shows the effects of repeatedly
charging and discharging the cell at room temperature. By maintaining an ongoing coulomb count of the
charge on the cell it can be shown that aging affects the "Full" point only. The "Empty" points remain
unchanged. To account for this, the formulas for calculating remaining capacity must be capable of
dynamically changing over time to remain accurate.
OTHER CONDITIONS
Most other conditions have little or no effect on charge capacity. Lithium-Ion cells are extremely efficient
when charging; very little energy is lost to heat during the cycle unlike other cell chemistries. The self-
discharge of a Lithium-Ion cell is extremely low to the point where it is difficult to even measure. Since
all these conditions combined affect the coulomb count result less than the accuracy of the measurement
device does, they are ignored completely in the fuel gauging equations.
Cell Capacity over Temperature (mAH)
0
200
400
600
800
1000
1200
1400
-20 -10 0 10 20 30 40 50 60 70 80
Temperature (0C)
Capacity(mAH)
Full Line defined
as 4.2V
High Current Emtpy
Line defined as 2.5V
Low Current Empty
Line defined as 2.5V
Figure 2
APPLICATION NOTE 131
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CALCULATING REMAINING CAPACITY
From the charts above it is easy to see how a coulomb counting only method can become very inaccurate
under real world conditions. This section shows how coulomb counting combined with characterization
of the expected "Empty" and "Full" points can maintain an accurate estimation of remaining cell
capacity.
STANDARD ASSUMPTIONS
For the algorithms to function accurately while minimizing computational complexity and parametric
data storage, certain assumptions are made. First it is assumed that similar charging efficiencies and
termination limits are applied universally to the application. It is also assumed in this example that there
are a finite number of repeatable discharge efficiencies encountered, each being well bounded. The cell
must always be considered fully discharged at the same voltage level, 2.5 volts for example. Charge
efficiency and pack self-discharge are assumed negligible in this application and are ignored.
CELL CHARACTERIZATION
The fuel gauging equations work by comparing the DS2438's ICA value with expected "Empty" and
"Full" values for that cell type which are stored in the DS2438's user EEPROM. This data is generated by
characterization of the cell type over the expected temperature range and current consumption of the
application. This information is subsequently stored in a pack resident memory for the algorithms to later
extract and modify. Figure 4 shows the system used to collect the characterization data. Information
should be gathered on several packs so that average or typical values can be stored in every production
pack. For best accuracy, the data should be collected on assembled packs containing the production
circuit as opposed to individual cells.
Cell Capacity over Lifetime (mAH)
0
200
400
600
800
1000
1200
1400
0 100 200 300 400 500 600 700 800 900 1000
Age (Cycles)
Capacity(mAH)
Full Line defined
as 4.2V
High Current Emtpy
Line defined as 2.5V
Low Current Empty
Line defined as 2.5V
Figure 3
APPLICATION NOTE 131
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To collect the data, the cell pack is fully charged at each temperature and fully discharged at each rate at
each temperature. Figure 5 shows the ICA reading of the DS2438 as full charge data and active current
discharge data is collected from 00C to 400C. This process would then be extended to collect the standby
current discharge data over the same temperature range.
All collected data points are arranged in Table 1 below. Since only the difference between points is
important, the absolute values of the data do not matter, they have been normalized to lowest value
(standby current empty at 400C). This reduces the amount of data needing to be stored since Standby
Empty 400C is now always 0.
Table 1: Cell Characterization Data
00C 100C 200C 300C 400C
FULL (mAH) 554 561 578 582 588
STANDBY EMPTY (mAH) 65 42 19 11 0
ACTIVE EMPTY (mAH) 124 90 65 50 44
Cell Pack
Environmental
Chamber
System
Controller
GPIB Bus
Precision Source
Measurement Unit
Figure 4: Cell Characterization System
Capacity Characterization over Temperature
0
100
200
300
400
500
600
700
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
Time (s)
ICA(mAH)
Active Current Empty Points at
each Temperature (mAH).
Full Points at each
Temperature (mAH).
ICA
Reading
Cell
Temperature
Figure 5
APPLICATION NOTE 131
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The characterization data is then stored in two pages of the DS2438's EEPROM memory. Because values
larger than 25510 require more than 1 byte of memory to store, the amount of data is reduced by storing
only the first value and then recording the incremental differences between temperatures. A memory map
of the DS2438 data store is shown in table 2 below.
Table 2: DS2438 Memory Map
Page 3 Page 4
0 STANDBY EMPTY to 200C
1
FULL at 00C
STANDBY EMPTY to 300C
2 FULL to 100C ACTIVE EMPTY to 00C
3 FULL to 200C ACTIVE EMPTY to 100C
4 FULL to 300C ACTIVE EMPTY to 200C
5 FULL to 400C ACTIVE EMPTY to 300C
6 STANDBY EMPTY to 00C ACTIVE EMPTY at 400C
7 STANDBY EMPTY to 100C Unused
The first 6 bytes of page 3 contain the cell's measured FULL point at the different temperatures across the
range. Bytes 0-1 are the capacity of the cell at 00C; the next four bytes are values of the increase in
capacity from the previous temperature. For example if a given cell's capacity was 554 mAH at 00C and
561 mAH at 100C then bytes 0-1 would contain 55410 (0x022Ah) and byte 2 would contain 710 (0x07h).
The next nine bytes hold the STANDBY EMPTY and ACTIVE EMPTY information stored in the same
manner as the FULL values. EMPTY values are incremented in the opposite direction starting at 400C
because it is the lowest value. STANDBY EMPTY at 400C is not included since it is always 010. Table 3
shows the actual information stored in the DS2438 using the characterization data contained in Table 1
above.
Table 3: Memory map of cell data stored in DS2438
0 1 2 3 4 5 6 7
Page 3 0x02h 0x2Ah 0x07h 0x11h 0x04h 0x06h 0x17h 0x17h
FULL STANDBY EMPTY
Page 4 0x08h 0x0Bh 0x22h 0x19h 0x0Fh 0x06h 0x2Ch X
STANDBY EMPTY ACTIVE EMPTY
THE EQUATIONS
After characterization of the cell pack is complete, calculating remaining capacity is very simple. The
characterization data is used to find cell full and empty points based on temperature and discharge rate,
and the DS2438's integrated current accumulator is compared against those values to express remaining
capacity as a percentage. Upon power-up, the characterization data should be read from the DS2438 and
stored in host RAM. When the host decides to update its remaining capacity indicator, it begins the
process by reading the cell temperature and the DS2438's ICA. The present full value for the cell is then
calculated by linearly interpolating between the FULL characterization data points using the cell
temperature. For example, the cell full point at 280C is calculated by:
Full Value (280C) = (FULL 200C) + ((28-20)/10) W (FULL 300C - FULL 200C)
APPLICATION NOTE 131
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The empty point is calculated in exactly the same method, except a determination must be made to use
the ACTIVE or STANDBY characterization data based on the current activation state of the system.
Capacity can then be calculated by determining the location of the ICA between empty and full points as
a percentage. The formulas are summarized below:
Full Value = FULL[Temperature]
Empty Value = STANDBY EMPTY[Temperature] or ACTIVE EMPTY[Temperature]
Capacity = ((ICA - Empty Value) / (Full Value - Empty Value)) W 100%
No estimation of remaining capacity is perfect. To prevent a long-term accumulation of error the ICA
register should be reset to the corresponding EMPTY value each time the cell is fully drained. Likewise,
each time the cell is fully charged, the corresponding FULL value should be changed to match the ICA.
By permanently adjusting the full point based on actual operation, the pack is able to adjust for cells that
are different from the "typical" characterization data and adjusts as the cell ages and deteriorates.
At End of Discharge: ICA = Empty Value
At End of Full Charge: FULL[Temperature] = ICA
DISPLAYING THE INFORMATION
The equations above report remaining capacity as a percentage between calculated empty and full points.
This might not be appropriate for every application. For example, a cell might be discharged to a level
below ACTIVE EMPTY but above STANDBY EMPTY. If the remaining capacity percentage was based
off the STANDBY EMPTY point it would show some capacity left even though the cell would not be
able to support an active current at that time. This could be very confusing for the end user. The way the
host processor presents the capacity data is unique to each application and not covered by standard fuel
gauging equations.
REMAINING ENERGY CALCULATION
For some applications estimating the remaining energy is very important. For example if the circuit had
constant power dissipation where the active current increased and the cell voltage dropped, remaining
time would not be related directly to remaining charge. A good estimation of remaining energy can be
easily calculated with a voltage reading from the DS2438 however. Recall the energy equation:
Energy (J) = Volts W Current W time
Which can be rewritten in terms of remaining energy:
Remaining Energy (J) = 3.6 W Remaining mAH W RAV
Where:
Remaining mAH are the remaining milliamp-hours calculated by the above equations.
3.6 is the conversion factor from milliamp-hours to amp-seconds.
RAV is the remaining average voltage of the cell explained below.
APPLICATION NOTE 131
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The upper plot on Figure 6 shows a typical cell discharge curve. The average remaining cell voltage can
be approximated at any time by finding the average between the cell voltage currently, and the cell
voltage when empty (2.5 volts typically).
RAV = (Voltage + 2.5) / 2
The remaining energy calculation can now be summarized as:
Remaining Energy (J) = 3.6 W Remaining mAH W (Voltage + 2.5) / 2
Where:
Remaining mAH are the remaining milliamp-hours calculated by the fuel gauging equations.
Voltage is present cell voltage measured by the DS2438.
3.6 is the conversion factor from milliamp-hours to amp-seconds.
The second plot on Figure 6 shows the accuracy for this cell when predicting remaining energy with this
method. The more linear a cell's discharge curve is, the more accurate this method will be. The less
linear the cell, the less accurate the calculation. In either case, the calculation becomes more accurate
where it is most important: as the cell voltage approaches the empty point.
EXAMPLE APPLICATION
For the following example a DS2438 demo board was used to monitor a 520 mAH prismatic Li+ cell. For
information on the DS2438 demo board refer to the DS2438K datasheet. A Keithley 2304A DVM/power
supply simulated the charger and load for the cell and a Tenney environmental chamber was used to
control cell temperature. The cell characterization information and actual data stored in the DS2438 are
the same as in the Cell Characterization section above. They are repeated in Tables 4 & 5 below for
convenience. The controlling software was written in Visual Basic and sections of the code dealing
directly with fuel gauging are presented at the end of this paper.
Li+ Discharge Curve & Remaining Energy Calculation Error
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
0 500 1000 1500 2000 2500 3000
Time (s)
Voltage(V)
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
CalculationError(%)
Energy Calculation
Error
Accuracy Improves as
cell voltage drops
Cell Voltage over 1.2C
Discharge Cycle
Figure 6
APPLICATION NOTE 131
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Table 4: Cell Characterization Data
00C 100C 200C 300C 400C
FULL (mAH) 554 561 578 582 588
STANDBY EMPTY (mAH) 65 42 19 11 0
ACTIVE EMPTY (mAH) 124 90 65 50 44
Table 5: Memory map of cell data stored in DS2438
0 1 2 3 4 5 6 7
Page 3 0x02h 0x2Ah 0x07h 0x11h 0x04h 0x06h 0x17h 0x17h
FULL S. EMPTY
Page 4 0x08h 0x0Bh 0x22h 0x19h 0x0Fh 0x06h 0x2Ch X
S. EMPTY ACTIVE EMPTY
The cell was subjected to twenty partial charge-discharge cycles over a variety of temperatures from 00C
to 400C. This test was designed to prove the accuracy of the fuel gauging equations under conditions
which are far more extreme than would generally be encountered in a standard commercial application.
Figure 8 shows the integrated current accumulator's relationship to the dynamically calculated empty and
full points over the duration of the test. The cell temperature is shown at the bottom of the graph. The
X-axis Update Cycle units refer to each time the remaining capacity was updated, approximately every
15 seconds. The worst case error occurred during the first charge (around update cycle 1000) and was
approximately 3%. The fuel gauge algorithms permanently adjusted the full point based on that error and
the second time a charge occurred at the same temperature (around update cycle 5500) the ICA matched
the expected full level almost exactly.
Cell Pack
Tenney Environmental
Chamber
System
Controller
GPIB Bus
Keithly 2304A
Figure 7: Example Application System
DS2438
DemoBoard
520mAHLi+
Prismatic
PAC -
DQ
PAC +
APPLICATION NOTE 131
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The software then calculates the remaining capacity as a percentage of the difference between the empty
and full points. Figure 9 below shows the actual fuel gauge output from the data shown above.
SUMMARY
Considering cell behavior over temperature and discharge rate when calculating remaining cell capacity
provides superior accuracy to coulomb counting alone. Dallas Semiconductor's fuel gauging equations
can be applied to any Lithium-Ion cell type and any Dallas Semiconductor coulomb counting device
while using a minimum of host processor cycles. They also adjust for cell to cell differences and cell
aging, becoming more accurate over time.
Fuel Gauging Example (%)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
110%
0 1000 2000 3000 4000 5000 6000
Update Cycles (~15 seconds each)
RemainingCapcity(%)
Cell Capacity Values
adjusted after a complete
charge
ICA Reset to 0 after a
complete discharge.
Figure 9
Fuel Gauging Example (mAH)
0
100
200
300
400
500
600
700
800
0 1000 2000 3000 4000 5000 6000
Update Cycles (~15 seconds each)
RemainingCapacity(mAH)
Full Point Dynamically Adjusted Based on
Temperature and Previous Cycles.
Empty Point Dynamically Adjusted
Based on Rate and Temperature.
Remaining Capacity
(ICA Register).
Cell Temperature.
Figure 8

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