Circuit Model of 100 Ah Lithium Polymer Battery Cell
Journal of Power and Energy Engineering, 2013, 1, 1-8
Published Online November 2013 (http://www.scirp.org/journal/jpee)
http://dx.doi.org/10.4236/jpee.2013.16001
1
Circuit Model of 100 Ah Lithium Polymer Battery Cell
Bong G. Kim, Dipesh D. Patel, Ziyad M. Salameh
Department of Electrical Engineering, University of Massachusetts Lowell, Lowell, USA.
Email: umass9@hotmail.com, dipesh_patel1@student.uml.edu, ziyad_salameh@uml.edu
Received October 4th, 2013; revised November 4th, 2013; accepted November 15th, 2013
Copyright © 2013 Bong G. Kim et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper presents a circuit model for a 100 Ah Lithium Polymer Battery that takes into account the effect of temperature and discharge rates. This is done by studying the behavior of two advanced 100 Ah Lithium Polymer Battery cells
under different load condition and at different temperatures, to extract the RC parameters needed to develop the equivalent circuit model. This paper presents a methodology to identify the several parameters of the model. The parameters
of the circuit model depend on both, battery cell temperature and discharging current rate. The model is validated by
comparing simulation results with experimental data collected through battery cell tests. The simulation results of the
battery cell model are obtained using MATLAB, and the experimental data are collected through the battery test system
at battery evaluation lab at University of Massachusetts Lowell. This model is useful for optimization of the battery
management system which is needed to run a battery bank safely in an electric car.
Keywords: Lithium Polymer Battery Cell; Battery Circuit Model; Battery Cell Model Simulation
1. Introduction
Battery powered electric vehicles are becoming more and
more attractive with the advancement of new battery
technology that have higher power and energy density,
and it becomes necessary reliable models for design and
simulation of the batteries. 100 Ah Lithium Polymer Battery cells have been tested for cycling, fast chargeability,
realistic load test, self discharge, and life cycle. From
these test results, the battery has high energy density
(373 Wh/L) and specific energy (146 Wh/kg), less self
discharge rate (less than 3% during month), it can be fast
charged, and the battery cell voltage remains within the
limit during the realistic load test. The battery cell has a
long life cycle, more over the battery capacity decreased
very slowly with cycling. The lithium polymer battery
cell is much lighter than other batteries cell and it is one
of the long term criteria for electric vehicle batteries (energy density greater than 300 Wh/L, life cycle over 1000
cycles, and recharge time between 3 to 6 hours) [1].
Two 100 Ah Lithium Battery cells were tested for model.
One is old battery cell which the battery has only 90% of
total capacity and the other one is new battery cell which
has fully 100% of capacity. The old battery cell lost more
than 10% of total capacity because the battery was cycled
more than 700 times. The reason to pick a new and an
old battery cell for the test is to find how the battery SOC
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characteristic changes by battery cell age and capacity.
The circuit model represents the nonlinear relationship
between SOC and the open circuit voltage (Voc) of the
battery cell. The model also demonstrates the constant
and dynamic behavior of the battery cell using a series
resistance and RC circuits. The battery cell model component values vary as a function of SOC. All the models
that have been derived in the past, considered effect of
discharge rate on model components but this model consider the rate of discharge effect as well as the temperature effect. The accurate circuit model will be very helpful for Battery Management System (BMS) design because SOC information makes power sourcing or sinking,
safe and efficient, and it also helps to extend the lifetime
of the traction battery. More detailed study of battery cell
circuit model and simulation will be discussed in following topics.
2. Battery Testing and Circuit Model
2.1. Test Setup
The University of Massachusetts Lowell Battery Evaluation Lab has three complete battery test systems. The
systems are computer controlled. The systems are designed to test batteries ranging from 0.1 mV to 20 volts
at 1 mA to 320 Amps. The current regulators are capable
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
of current sinking or sourcing and can change from
charge to discharge mode instantaneously [2]. Data acquisition and control systems provide a controlled current
and stores data including voltage, current, resting time,
and temperature. Data is taken every 5 sec. so as to limit
the storage space required. Each tester is controlled via
A/D and D/A interfaces. The battery test system also
equipped with temperature chamber to test the battery at
different ambient temperatures between −40˚C to +60˚C.
Temperature of chamber is computer controlled. The
voltage, current, and temperature limit of the battery cell
can set during the initial set time. The test stops automatically when the battery cell reaches the limit [3]. The
MUX connected to the computer, current regulator, and
chamber. The battery cell is inside the chamber and it
connected to the current regulator. The picture of battery
lab at University of Massachusetts Lowell is shown in
Figure 1 and the block diagram of the battery cell test is
shown in Figure 2.
Figure 1. The battery lab at University of Massachusetts
Lowell.
Current Regulator
( Up to 10Volts and 320Amps)
2.2. Battery Cell Circuit Model
Battery cell circuit diagram is shown in Figure 2. Voc is
the open circuit voltage which depends on VSOC. Rs is DC
resistance, Cb is capacitance of battery cell, Rd is resistance of self discharge, R(short) and C(short) are the resistance and capacitance in the shorter time constant RC
circuitand R(long) and C(long) are the resistance and
capacitance in the longer time constant RC circuit. The
voltage controlled by voltage source linking the two circuits is used to represent the nonlinear relationship between the State of SOC and Voc of the battery. This relationship is normalized such that when the voltage across
Cb is 1 V, the battery is at 100% SOC [4]. From Figure 3,
effect of transient components R(short), C(short), R(long),
and C(long)are visible at the start of constant discharge.
After short time of period, decided by R(short) C(short)
and R(long) C(long) time constant, circuit behaves as
resistive circuit. The simulation results of Rs and Voc are
based on polynomial equations and depend on State of
Charge. Values of R(short), C(short), R(long) and C(long)
change abnormally but depend on State of Charge. Equation for Rs and Voc for simulation can be written as:
8
 2

Rs or Voc      l  m  am  k m  SOC n
n0  m0

(1)
where, n is polynomial number, SOC is state of charge,
l is constant of different ambient temperature and discharge current rate, a is constant of quadratic equation
for combine different temperatures, k is constant for constant 0 (for 0˚C @ 1C and for C/10 discharge rate) 2 (for
20˚C @ 1C and for C/4 discharge rate) 4 (for 40˚C @ 1C
and for C/2 discharge rate), and Δ is difference between
different temperature and different discharge rate.
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Li-Poly Battery
Cell
A/D and D/A
Converter
Chamber
(Heating and Cooling)
MUX
Computer
Figure 2. Block diagram of the battery cell test.
C(short)
C(long)
R(short)
R(long)
Vsoc
Rs
Cb
Rd
Voc
Current
Source
Figure 3. Li-Poly battery cell circuit diagram.
The value Δ and constant a at different temperatures
discharge current rates are shown in Tables 1 to 4. The
transient response to a step load current is shown in Figure 4. Vdc is dc resistance voltage, Vs(t) is short time constant voltage, and Vl(t) is long time constant voltage. The
experimental and simulation result of the Voc at 0˚C and
different discharge current rates for new cell is shown in
Figure 5. The experimental and simulation result of the
Voc at C/10 and different temperatures for new cell is
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
Terminal voltage of old cell @ C discharge rate @ 20 deg celcious
old cell (meas.)
Terminal voltage (V)
3.7
3.6
3.5
3.4
3.3
5700
5800
5900
6000 6100
Time (sec.)
6200
6300
6400
Open circuit voltage, Voc (V)
Figure 4. Transient response to a step load current.
4.2
4.1
4
3.9
3.8
3.7
3.6
3.5
3.4
3.3
3.2
Open circuit voltage of new cell @ 0 deg celcious for different discharge rates
C/10 (sim.)
C/10 (meas.)
C/4 (sim.)
C/4 (meas.)
C/2 (sim.)
C/2 (meas.)
C (sim.)
C (meas.)
1
0.9
0.8
0.7
0.6
0.5
0.4
State of charge (SOC)
0.3
0.2
0.1
0
Figure 5. Simulation and experimental result for Voc at various discharge current rate and ambient temperature 0˚C (new
cell).
Table 1. Constant values for open circuit voltage of new battery.
bn
a2
a1
a0
Δ2
Δ1
Δ0
b7
−7.3705
14.741
0
−0.51075
−6.1005
−26.39
b6
−29.075
77.1571
−42.848
−47.19625
123.7646
60.662
b5
48.6737
−167.355
153.79
64.16424125
−186.641083
−17.9
b4
−45.1448
192.389
−219.71
−47.6905
164.7134
−64.28
b3
25.0809
−125.036
159.4
22.174825
−94.42635
77.062
b2
−8.2449
45.3
−61.088
−6.8331875
34.532775
−36.402
b1
1.4449
−8.3407
11.983
1.263825
−6.9768
7.851
b0
−0.1049
0.622
2.6403
−0.1084125
0.623575
−0.6342
Table 2. Constant values for internal DC resistance voltage of new battery.
bn
a2
a1
b7
0.0119488
−0.1033125
b6
−0.0598998
0.4332045
a0
Δ2
Δ1
Δ0
0
0.00246575
−0.0653805
0
−0.075432
−0.06938275
0.4711365
−0.07543
b5
0.1059938
−0.6961025
0.22984
0.164565
−0.96775
0.22984
b4
−0.0790638
0.5003425
−0.20603
−0.19417
1.12784
−0.20603
b3
0.0179244
−0.1190662
0.0082249
0.119324363
−0.82481618
0.008225
b2
0.0065888
−0.035942
0.075254
−0.03827267
0.41748149
0.074943
b1
−0.0050703
0.02809715
−0.035913
0.0065727
−0.1515841
−0.03675
b0
0.000772
−0.0046413
0.0083023
−0.00163701
0.037337975
0.010669
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
Table 3. Constant values for open circuit voltage of old battery.
bn
a2
a1
a0
Δ2
Δ1
Δ0
b7
15.4438
−30.8875
0
26.99555
−71.132
−23.85
b6
−62.5562
125.1125
0
−104.2017
269.6585
106.18
b5
103.4
−213.155
26.862
164.184
−424.648
−167.458
b4
−90.2385
200.6955
−83.366
−136.5385
364.2105
106.674
b3
44.966
−114.8915
100.83
64.7615
−187.6575
−7.43
b2
−12.9533
40.8165
−58.564
−17.671425
59.75125
−22.635
b1
2.0656
−8.4375
16.704
2.633675
−11.147
9.8573
b0
−0.1541
0.814
1.6986
−0.181375
0.9936
−1.273
Table 4. Constant values for internal DC resistance voltage of old battery.
bn
a2
a1
a0
Δ2
Δ1
Δ0
b8
0.3272625
−0.654525
0
0.3272625
−0.654525
0
b7
−2.3734983
8.657718
−7.8229
−1.73876075
4.849293
−2.745
b6
6.915181
−31.568974
35.4135
7.5499185
−35.377399
40.4914
b5
−10.79453
55.20962
−67.017
−13.353155
70.56137
−87.486
b4
9.9834525
−54.46981
68.734
14.25865925
−80.1097235
102.913
b3
−5.63988
31.99052
−41.288
−9.46780248
54.9051424
−71.799
b2
1.9100685
−11.099369
14.544
3.888754375
−22.8884275
30.2023
b1
−0.3561925
2.10245615
−2.7878
−0.94612048
5.58207315
−7.3937
b0
0.0281117
−0.1681538
0.22801
0.122444075
−0.71324865
0.94818
shown in Figure 6. From the figures, the Voc is high at
room temperature and at lower discharge current rate.
The experimental and simulation result of the Rs at 0˚C
and different discharge current rates for old cell is shown
in Figure 7. The experimental and simulation result of
the Rs at C/10 and different temperatures for old cell is
shown in Figure 8. From the figures, the Rs is high at
lower temperature and lower discharge rate. Rs sharply
increase at the lower SOC (10%).
2.3. The Discharge Test Methodology
Each battery cell, new and old, has 12 different test settings which are 3 different ambient temperatures (0˚C,
20˚C, and 40˚C) and each temperature has 4 different
discharge rates (C/10, C/4/, C/2, and C). The battery cell
is first charged to its full capacity through constant current followed by constant voltage (CC/CV) at 1C (100
Amps) and it starts temperature soak until the cell gets to
the set temperature. The battery cell discharged under
different predefined profiles. The cell stops discharging
at decrement of 10% SOC, decrement in the range from
100% to 0% and the cell has resting time for 10 minutes.
Open Access
The resting time is needed for the battery cell voltage to
reach steady state condition [5]. The test completely
stopped when the battery cell reachesat0 capacity. The
open circuit voltage is measured as the steady state open
circuit terminal voltage at various SOC points. The performance of the battery cell during the discharge at 1C
discharge rate and 20˚C is shown in Figure 9.
2.4. Constant Current Discharge Test
The battery cell is first charged to its full capacity
through (CC/CV) and its tarts temperature soak until the
battery cell gets to the set temperature. The battery cell
discharged under different predefined profiles. The battery cell completely stops discharge if it reaches at 0 capacity. The battery cell performance at constant discharge current rate of C/2 and ambient temperature at
0˚C is shown in Figure 10. From the figure, the SOC of
cell drops from a certain value to 0 as the cell output
voltage goes from full capacity voltage to cut off voltage.
The SOC of the full charged battery cell varies with discharge current rate, which reflects the current effect. The
lithium polymer battery cell maintained a relatively flat
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Open circuit voltage (V)
Circuit Model of 100 Ah Lithium Polymer Battery Cell
4.5
4.375
4.25
4.125
4
3.875
3.75
3.625
3.5
3.375
3.25
3.125
3
Open circuit voltage of old cell @ C/10 discharge rate for different temperature
40 deg cel. (sim.)
40 deg cel. (meas.)
20 deg cel. (sim.)
20 deg cel. (meas.)
0 deg cel. (sim.)
0 deg cel. (meas.)
1
0.9
0.8
0.7
0.6
0.5
0.4
State of charge (SOC)
0.3
0.2
0.1
0
DC resistance, Rdc (ohm)
Figure 6. Simulation and experimental result Voc at C/10 and various ambient temperatures (old cell).
12
11
10
9
8
7
6
5
4
3
2
x 10
-3
DC resistance of old cell @ 0 deg celcious for different discharge rate
C/10 (sim.)
C/10 (meas.)
C/4 (sim.)
C/4 (meas.)
C/2 (sim.)
C/2 (meas.)
C (sim.)
C (meas.)
1
0.9
0.8
0.7
0.6
0.5
0.4
State of charge (SOC)
0.3
0.2
0.1
0
DC resistance (ohm)
Figure 7. Simulation and experimental result for Rs at variousdischarge current ratesand ambient temperature 0˚C (old cell).
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
DC resistance of old cell @ C/10 discharge rate for different temperature
0 deg cel. (sim.)
0 deg cel. (meas.)
20 deg cel. (sim.)
20 deg cel. (meas.)
40 deg cel. (sim.)
40 deg cel. (meas.)
1
0.9
0.8
0.7
0.6
0.5
0.4
State of charge (SOC)
0.3
0.2
0.1
0
Figure 8. Simulation and experimental result for Rs at C/10 and various ambient temperatures (old cell).
Terminal voltage different cells @ C discharge rate @ 20 deg celcious
4.2
old (meas.)
old (sim.)
new (meas.)
new (sim.)
Terminal voltage (V)
4
3.8
3.6
3.4
3.2
3
2.8
0
1000
2000
3000
4000 5000 6000
Time (sec.)
7000
8000
9000
10000
Figure 9. Simulation and experimental pulse discharge test result at C and ambient temperature 20˚C.
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
Terminal voltage of different cells @ C/2 discharge rate @ 0 deg celcious
4
new (meas.)
new (sim.)
old (meas.)
old (sim.)
Terminal voltage (V)
3.8
3.6
3.4
3.2
3
2.8
2.6
0
1000
2000
3000
4000
Time (sec.)
5000
6000
7000
Figure 10. Simulation and experimental constant discharge test at C/2 and at ambient temperature 0˚C.
voltage discharge profile with a steep decrease in the
profile near the end of discharge [6].
is Ah difference during the test.
3. Model Simulation Methodology
2.5. Battery Cell Self Discharge Test
The battery cells were tested for self discharge. The cells
are first charged their full capacity through (CC/CV) then
they were set aside for a month in an open circuit condition at +20˚C. The terminal voltage measured once an
every week. The cells were fully discharged after self
discharge test [7].The terminal voltage drop was 0.022
volt (4.13 volt to 4.108 volt) for new battery cell and
0.027 volt (4.113 volt to 4.086 volt) for old battery cell.
The new battery cell lost its capacity 4 Ah, 67 Wh but it
lost 0 Ah, 47 Wh with no self discharge. The old battery
cell lost its capacity 2 Ah, 51 Wh but it lost 0 Ah, 37 Wh
with no self discharge. The cells lost more capacity 4 Ah,
20 Wh for new and 2 Ah, 14 Wh for old due to self discharge. It shows that the lithium polymer battery cell lost
capacity less than 3% per month self discharge. The resistance of self discharge (Rd) can calculated from the
Equation (2).
Rd  Vd Ah   24  30  3600
(2)
where, Vd is voltage difference during the test and Ah
Behavior of the battery in static as well as dynamic response is very non-linear. Several simulation methods
have been approached. Proposed method is circuit-parameter based procedure of equation solving. The equations are written in MATLAB. Simulation is for discharge only as we have done the test for discharge. Values of current for simulation are directly taken from test
results so that accuracy is maintained. For the static response the values of capacitances and resistances depend
on SOC. Value of capacitances at that SOC of battery
measured from dynamic response are taken. The effect of
capacitances, in both the static and dynamic response, die
out after the short and long time constants and resulting
terminal voltage depends on battery internal dc resistance,
short time dc resistance and long time dc resistance. Parameters in the circuit depend on SOC, temperature, rate
of discharge. Circuit’s response can be explained by
equation which is used for the simulation.
m = initial SOC of the battery before discharge;
Ccapacity = capacity of the battery.
T  mCcapacity
t  SOC  t  Ccapacity


  T  t   


 Rs Cs   ,
t T

Vop  t   I  t  Rdc  t   I  t  Rs  t  1  e








 T  s  t   


 R C  
V  t   Vop  t   I  t   Rdc  t   Rs  t    I  t  Rl  t  1  e  l l   , t  T   s






Vop  t   I  t   Rdc  t   Rs  t   Rl  t   ,
t  T  s  l




 s  Rs Cs ,  l  Rl Cl
Open Access
(3)
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
At the start of discharge t = T, terminal voltage is less
than open circuit voltage by voltage drop across internal
DC resistance. Terminal voltage drops due to resistance
of short time dynamic RC branch for t ≤ T. It keeps
dropping till the time constant of short time dynamic RC
branch. After that terminal voltage drops due to the resistance of long time dynamic RC branch, t  T   s . It
keeps dropping till the time constant of long time dynamic RC branch, after that drop terminal voltage is
Terminal voltage different cells @ C discharge rate @ 0 deg celcious
4.5
Terminal voltage (V)
completely resistive, t  T   s   l . Voltage drop across
the internal DC resistance is significantly high than the
latter. The experimental and simulation test result of the
battery cell discharge at different temperatures and C
discharge rate for new and old cell are shown in Figures
11 and 12. The experimental and simulation test result of
the constant discharge at 0˚C and C/4 discharge rate for
new and old is shown in Figure 13. From the figure, it
observed that the model generate voltage response less
old (meas.)
old (sim.)
new (meas.)
new (sim.)
4
3.5
3
2.5
2
0
1000
2000
3000
4000
5000
Time (sec.)
6000
7000
8000
9000
Figure 11. Simulation and experimental pulse discharge test result at C and ambient temperature 0˚C.
Terminal voltage different cells @ C discharge rate @ 40 deg celcious
Terminal voltage (V)
4.5
old (meas.)
old (sim.)
new (meas.)
new (sim.)
4.25
4
3.75
3.5
3.25
3
0
1000
2000
3000
4000 5000 6000
Time (sec.)
7000
8000
9000
10000
Figure 12. Simulation and experimental pulse discharge test result at C and ambient temperature 40˚C.
Terminal voltage of different cells @ C/4 discharge rate @ 0 deg celcious
Terminal voltage (V)
4.5
new (meas.)
new (sim.)
old (meas.)
old (sim.)
4
3.5
3
2.5
0
2000
4000
6000
8000
Time (sec.)
10000
12000
14000
Figure 13. Simulation and experimental constant discharge test result at C/4 and ambient temperature 0˚C
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Circuit Model of 100 Ah Lithium Polymer Battery Cell
than 30 mV, which is less than 3% error margin. Therefore, the simulation results of the battery cell circuit
model match well with the experimental data.
4. Conclusion
A nonlinear battery cell circuit model which is based on
relationship between the battery cell terminal voltage and
battery cell state of discharge (SOC) characteristic under
different discharge current rates and different ambient
temperatures condition has been developed. The nonlinear battery cell circuit model can be used to accurately
model and predict battery cell performance from the experimental and simulation test results. The model will
greatly help research on circuit simulation, multi-cell
battery analysis, battery cell performance prediction and
optimization, and battery cell management. The experimental and simulation of battery circuit cell modeling is
within 3% of error margin. The lithium battery cell has
the best performance when the cell is at room temperature (20˚C) and has C/10 discharge current rate. The
lithium polymer battery cell holds more than 70% of its
capacity between 3.2 and 3.7 volts. Based on this circuit
model, it could be developed with other models like
fuzzy or neuro-fuzzy logic.
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Open Access
JPEE
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