Impedance-Based Simulation Models of Supercapacitors and Li-Ion ...

742 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005 

Impedance-Based Simulation Models of 

Supercapacitors and Li-Ion Batteries 

for Power Electronic Applications 

Stephan Buller, Member, IEEE, Marc Thele, Rik W. A. A. De Doncker, Fellow, IEEE, and 

Eckhard Karden, Member, IEEE 

Abstract—To predict performance of modern power electronic 

systems, simulation-based design methods are used. This work 

employs the method of electrochemical impedance spectroscopy 

to find new equivalent-circuit models for supercapacitors and 

Lithium-ion batteries. 

Index Terms—Lithium-ion (Li-ion) batteries, simulation models, 

supercapacitors (SCs). 

I. INTRODUCTION 

SIMULATION-BASED development methods are increasingly 

employed to cope with the complexity of modern 

power electronic systems. For these methods, suitable submodels 

of all system components are mandatory. However, 

compared to the submodels of most electric and electronic 

components, accurate dynamic models of electrochemical 

energy storage devices are rare. Therefore, this paper employs 

the method of electrochemical impedance spectroscopy (EIS) 

to extent the physics-based, nonlinear equivalent circuit models 

of supercapacitors (SCs) [2] to describe Lithium-ion (Li-ion) 

batteries. 

The following section briefly introduces the method of electrochemical 

impedance spectroscopy and presents measured 

impedance spectra. From these spectra, appropriate equivalent-circuit 

models are deduced. After this, the Matlab/Simulink 

implementation of the new simulation models is discussed and 

simulation results as well as verification measurements are provided. 

Finally, conclusions are drawn and future perspectives 

of the new impedance-based modeling approach are outlined. 

Paper IPCSD-05-006, presented at the 2003 Industry Applications Society 

Annual Meeting, Salt Lake City, UT, October 12–16, and approved for publication 

in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Power 

Electronics Devices and Components Committee of the IEEE Industry Applications 

Society. Manuscript submitted for review July 1, 2003 and released for 

publication March 3, 2005. 

S. Buller was with the Institute for Power Electronics and Electrical Drives 

(ISEA), Aachen University of Technology (RWTH-Aachen), D-52066 Aachen, 

Germany (e-mail: Stephan.Buller@gmx.de). 

M. Thele and R. W. A. A. De Doncker are with the Institute for 

Power Electronics and Electrical Drives (ISEA), Aachen University 

of Technology (RWTH-Aachen), D-52066 Aachen, Germany (e-mail: 

te@isea.rwth-aachen.de; dedoncker@isea.rwth-aachen.de). 

E. Karden is with Energy Management, Ford Research Center Aachen (FFA), 

D-52072 Aachen, Germany (e-mail: ekarden@ford.com). 

Digital Object Identifier 10.1109/TIA.2005.847280 

II. IMPEDANCE SPECTRA OF SCS AND LI-ION BATTERIES 

Electrochemical impedance spectroscopy can be performed 

either in a galvanostatic or in a potentiostatic mode. Following 

the first approach, a small ac current flows through the 

storage device under investigation and its ac voltage response 

is measured. From the ac current and the measured ac voltage 

response, the storage impedance is determined online using 

discrete Fourier transforms (DFTs). Superimposed with the ac 

excitation signal, a dc current (charging or discharging) defines 

the overall working point of the cell. Due to the pronounced 

nonlinearity of most electrochemical storage systems, especially 

of batteries, the differential impedance 

is 

usually not equal to the quotient . In these cases, modeling 

the large-signal behavior of an energy storage device requires 

impedance measurements at several working points followed 

by integration of the differential impedance with respect to 

current, i.e., 

. In addition, the impedance of 

storage devices usually depends on temperature and state of 

charge. Therefore, sets of impedance spectra have to be analyzed 

systematically [3]–[5]. Due to mass transport phenomena, 

dynamic battery performance during continuous discharging 

or charging of batteries differs significantly from that during 

dynamic microcycling with frequent changes between charging 

and discharging. As the latter is typical for many practical 

battery applications (e.g., hybrid-electric vehicles or stop/start 

vehicles), EIS on Li-ion batteries has been performed using 

a specific microcycle technique [5]. 

During the investigation of the SCs, impedance spectra have 

been recorded at four different voltages and five temperatures. 

As an example, Fig. 1 shows the complex-plane representation 

of impedance spectra of a 1400-F SC at 

Vina 

frequency range from 70 Hz down to 160 mHz. This frequency 

range corresponds to typical time constants in most high-power 

applications, e.g., cranking of a vehicle with a combustion 

engine. 

In the high-frequency range Hz , the SCs show inductive 

behavior. Then, at approximately m , 

the impedance plots intersect the real axis. For intermediate frequencies, 

the complex-plane plots form an angle of approximately 

45 with the real axis. This angle is explained by the 

limited current penetration into the porous structure of the electrodes 

(which has been discussed in [2]). For lower frequencies, 

the spectra approach a nearly vertical line in the complex plane, 

which is typical of ideal capacitors. 

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BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES 743 

Fig. 3. 

Equivalent-circuit model of the Li-ion battery. 

Fig. 1. 

Complex-plane diagram of impedance spectra of a 1400-F SC 

manufactured by Montena Components SA, U =1:25 V. 

Fig. 4. Complex-plane impedance diagram of measured and modeled 

impedance data of the Li-ion battery at # =25 C, 80% SOC, I =0A and 

I =1A charge. 

III. EQUIVALENT-CIRCUIT MODELS 

Fig. 2. Impedance spectra of an Li-ion battery (I =0A, # =25 C). 

Fig. 2 shows measured impedance spectra of a Li-ion battery 

(Saft LM 176065, 3.6 V/5 Ah) at room temperature for different 

states of charge (in this case with zero dc current). Impedance 

data have been recorded for eight frequencies per frequency 

decade starting at 6 kHz. For all spectra, some characteristic 

frequencies are given. At approximately , 

the real axis intersection of the impedance spectra is observed. 

For lower frequencies, all spectra show two capacitive semicircles. 

The first semicircle is comparably small and slightly depressed, 

whereas the second one is larger, nearly nondepressed, 

and grows remarkably with decreasing state of charge. Finally, 

at the low-frequency end of the depicted spectra, diffusion becomes 

visible. The diffusion impedance shows a 45 slope, 

which is typical of a so-called Warburg impedance. Due to the 

boundary condition for diffusion of Li ions in the electrodes, 

the diffusion branch of the spectrum approaches a capacitor-like 

impedance spectrum 90 for even lower frequencies. 

Impedance spectra of valve-regulated lead-acid batteries 

(VRLA) with different superimposed dc currents have also 

been measured at several state of changes (SOCs) and temperatures. 

These results are beyond the scope of this paper but can 

be found in [1], and [4]. 

The discussion of the model topology and the general modeling 

principle in this section concentrates on the Li-ion battery 

technology. In the case of SCs, excellent agreement with 

the measured impedance spectra was achieved using a ladder 

network model, consisting of the resistance of the pore electrolyte 

and the nonlinear double-layer capacitance of the phase 

boundary electrode/electrolyte. Detailed results as well as the 

lumped-element representation of the ladder network are reported 

in [1] and [2]. 

To model the recorded impedance spectra, suitable equivalent-circuit 

topologies have to be defined. Based on the underlying 

physical processes, the equivalent circuits should allow 

an optimum representation of the measured spectra with a minimum 

set of model parameters. In a second step, the model 

parameters have to be calculated. To minimize the deviations 

between modeled data and measured spectra, a least-square 

fitting algorithm is employed. 

In Fig. 3, the electric equivalent circuit of an Li-ion battery 

is depicted. This circuit consists of an inductance , an ohmic 

resistance , a so-called element representing a depressed 

semicircle in the complex-plane [1], [3], a nonlinear 

RC circuit ( and ) as well as of a Warburg impedance 

. Using the depicted model topology, the observed ac behavior 

of an Li-ion battery can be described accurately. The 

following adaptation of the model to the measured impedance 

spectra shows that, despite several simplifications, all relevant 

processes including porosity, charge transfer and diffusion are 

modeled with sufficient precision. 

Fig. 4 compares measured and calculated impedance data of 

the Li-ion battery at 25 C and 80% SOC. For all frequencies 

and both depicted dc currents, the corresponding curves show 

nearly perfect agreement. 



Fig. 5. Nonlinearity of the resistance R (# =25 C, 50% SOC). 

Fig. 6. 

Open-circuit voltage as a function of the state of charge (Li-ion). 

Obviously, the diameter of the low-frequency semicircle, i.e., 

parameter , strongly depends on the dc current that is superimposed 

during the impedance measurement. The nonlinearity 

of with dc current is depicted in Fig. 5. Apart from 

the data points, which have been determined from the measured 

impedance spectra, Fig. 5 also shows a calculated curve which 

models the current dependency of . 

The relation between the dc current and the corresponding 

overvoltage at the impedance element can be described 

by a Boltzmann-type equation. In electrochemistry, this type of 

equation is known as Butler–Volmer equation (1). The constants 

are the exchange current , the number of transferred elementary 

charges , the symmetry coefficient , and the 

thermal voltage ( mV if K). 

The nonlinear charge transfer resistance 

as 

(1) 

can be calculated 

Equation (2) can be solved analytically for the special cases 

, (irreversible reactions), and (symmetric 

kinetics). In all other cases, numeric calculation is required. 

Therefore, to determine the Butler–Volmer parameters 

, , and from the data points in Fig. 5, a second fitting algorithm 

is employed. 

The best approximation of the data points is obtained for 

A, , and . For classic redox reactions, 

represents the change in oxidation number of the reaction 

ions. Hence, integer values are expected. However, for 

the Li-ion battery, a noninteger value for was found to be 

best. This result might be explained by the specific nature of the 

Li intercalation reaction or might be due to the simplifications 

that are necessary to allow a parameterization of the simulation 

model without reference electrode measurements. An electrochemical 

investigation of this finding is not required for the further 

development of the dynamic battery model and is therefore 

considered beyond the scope of this work. 

Finally, Fig. 6 shows the open-circuit battery voltage as 

function of the state of charge. The Li-ion battery was partly 

(2) 

Fig. 7. 

Approximation of a ZARC element by RC circuits. 

discharged and the open-circuit voltage was measured after a 

minimum rest period of five hours. The curve in Fig. 6 can be 

stored into a lookup table. 

IV. MODEL IMPLEMENTATION 

So far, the dynamic behavior of the modeled energy storage 

devices is still described in the frequency domain. The time-domain 

behavior of the equivalent-circuit model can be calculated 

by solving a set of ordinary differential equations. For this calculation, 

simulation tools like Matlab/Simulink can be employed. 

However, not all complex impedance elements (e.g., ZARC elements 

and Warburg impedances) can directly be implemented 

in a common circuit simulation tool. For these elements, appropriate 

approximations by means of RC circuits or RC ladder network 

topologies have to be found first [1]. 

As an example, the basic idea for the representation of a 

ZARC element, employed to model a depressed, capacitive 

semicircle in the complex-plane diagram, is depicted in Fig. 7. 

The approximation is based on a series connection of nonlinear 

RC circuits. All RC circuits are fully determined by the parameters 

of the ZARC element. Thus, the number of experimental 

parameters remains constant. With an increasing number of 

RC circuits, the approximation of the ZARC elements becomes 

more and more precise. However, the calculating time increases. 

Thus, an appropriate compromise between simulation 

accuracy and computation effort has to be found. This question 

is thoroughly discussed in [1]. 

The specific model parameters which are needed for the simulation 

of a certain battery are stored in a separate file. To allow 

a linear adaptation of a parameterized battery model to differently 

sized batteries of the same technology, all parameters are 

defined with respect to the battery’s nominal current and the 

number of battery cells connected in series. Furthermore, due to 

the nonlinearity of some impedance elements, the original battery 

current in the simulation model is replaced by the relative 

current . 



Fig. 8. 

Current profile for the verification of the SC model. 

Fig. 10. Comparison of the measured voltage response and the data obtained 

from different simulation models. 

TABLE I 

COMPARISON OF MEASURED AND SIMULATED EFFICIENCY DATA.WORKING 

POINT: 1.5 V, ROOM TEMPERATURE, CYCLE DEPTH 615% Q 

Fig. 9. 

Measured and simulated voltage response to the current profile. 

V. VERIFICATION AND APPLICATION OF THE MODELS 

As a final step, the results of the simulation models are compared 

with measured data in the time domain. Both, the SC 

model as well as the model of the Li-ion battery have been verified 

in detail [1]. In this section, some examples of these verification 

measurements are given. 

For the verification of the SC model, the current profile depicted 

in Fig. 8 has been employed [2]. The imposed charging 

and discharging pulses model a highly dynamic load at the beginning 

as well as deeper charging and discharging periods at 

the end of the evaluation. The corresponding voltage curves are 

depicted in Fig. 9. The measured and the calculated data show 

excellent agreement. 

The influence of the porous structure of the SC electrodes 

can be illustrated by means of a comparison of the full simulation 

model with the voltage response of the simplified model 

which only consists of a series connection of the ohmic resistance 

and the capacitance of the SC. For this comparison, 

Fig. 10 provides an enlarged view of the first current pulses of 

the verification profile. Once more, the excellent agreement of 

the measured and simulated voltage data becomes obvious. In 

addition, remarkable deviations due to the neglect of porosity 

are observed for the simplified model. For low frequencies, i.e., 

for comparably long relaxation times, these deviations could be 

overcome by replacing the ohmic resistance by the larger dc 

resistance with being the resistance of the 

electrolyte in the pores of the electrodes [2]. In this case however, 

the fast voltage transients would not be well represented 

anymore. 

One advantage of SCs used as energy storage devices is their 

good energy efficiency. This efficiency is also influenced by 

the porous structure of the electrodes which means that the increasing 

real part of the impedance with decreasing frequency 

has to be taken into account. 

To compare measured and simulated efficiency data, an SC 

is partly charged and discharged with constant dc currents of 

various amplitudes. The cycle depth is chosen to be 540 A s 

which corresponds approximately to . For each current 

amplitude, the charge/discharge cycle is repeated ten times 

but only the last five cycles, which start and finish at the same 

internal conditions (quasi-stationary), are used for the efficiency 

calculation. In a second step, the same current profile is simulated 

by means of the newly developed capacitor model. Measured 

and simulated efficiency data are compared in Table I. 

Again, very good agreement is observed. 

Next, a simulation example of the Li-ion battery model is 

presented. For the model verification, the dynamic discharge 

current profile depicted in Fig. 11 has been selected. The 

comparison of the simulated and the measured voltage response 

to this current profile at a state of charge of 77.5% and 

room temperature is shown in Fig. 12. Excellent agreement 

of the measured and the simulated voltage curves is found. 

The outstanding accuracy of the simulation model is due to 

the exact representation of the complex battery impedance 

including all important nonlinearities. 

An important precondition for the high quality of the simulation 

results is the nearly perfect reproducibility of the battery 



Fig. 11. 

Current profile for the verification of the Li-ion battery model. 

storage technologies, e.g., NiMH batteries or even fuel-cell 

stacks in the future. This versatility will allow the combined 

simulation of different energy storage devices for the evaluation 

of new storage-hybridization concepts. Furthermore, by means 

of additional submodels, e.g., describing mass transport in 

VRLA batteries, the validity range of the existing models can 

be further enlarged. By this, the simulation of long-lasting 

constant-current charging or discharging periods will become 

possible. 

Another interesting future application of the impedancebased 

simulation models is a detailed thermal battery design. 

All mechanisms of heat generation of SCs or batteries can be 

precisely represented. Combined with the mechanisms of heat 

transport and dissipation, the thermal behavior of batteries can 

be simulated. Consequently, the influence of different future 

cooling concepts, for example, on life-cycle costs, could be 

evaluated. 

ACKNOWLEDGMENT 

The authors are grateful to the Ford Research Center Aachen 

(FFA), especially to Dr. D. Kok and Dr. L. Gaedt for supporting 

this research project. 

Fig. 12. Measured and simulated voltage response of the Li-ion battery (77.5% 

SOC, 25 C). 

behavior during operation as well as the lack of parasitic reactions. 

From this point of view, Li-ion batteries are especially 

suited for any kind of model-based description. Compared to 

Li-ion batteries, the simulation of lead acid batteries turns out 

much more difficult. Nevertheless, the described simulation 

approach could also be successfully adapted to this battery 

technology [1], [4]. 

VI. CONCLUSION AND FUTURE PERSPECTIVES 

This paper has shown that nonlinear, lumped-element equivalent-circuit 

models meet the accuracy requirements for simulation 

models of energy storage devices. To demonstrate the 

power of this modeling concept, Li-ion batteries and SCs were 

selected. 

For the determination of suitable equivalent-circuit topologies 

as well as for the parameterization of these models, the 

method of EIS was employed. After the implementation of the 

models, the simulation results were compared to test-bench 

data. Excellent agreement of simulated and measured voltage 

data was found. 

Due to the versatility the impedance-based modeling approach, 

the described concept can also be employed for other 

REFERENCES 

[1] S. Buller, “Impedance-based simulation models for energy storage devices 

in advanced automotive power systems,” Ph.D. dissertation, ISEA, 

RWTH Aachen, Aachen, Germany, 2003. 

[2] S. Buller, E. Karden, D. Kok, and R. W. De Doncker, “Modeling the 

dynamic behavior of supercapacitors using impedance-spectroskopy,” 

IEEE Trans. Ind. Appl., vol. 38, no. 6, pp. 1622–1626, Nov./Dec. 2002. 

[3] E. Karden, “Using low-frequency impedance spectroscopy for characterization, 

monitoring, and modeling of industrial batteries,” Ph.D. dissertation, 

ISEA, RWTH Aachen, Aachen, Germany, 2001. 

[4] S. Buller, M. Thele, E. Karden, and R. W. De Doncker, “Impedancebased 

nonlinear dynamic battery modeling for automotive applications,” 

J. Power Sources, vol. 113, pp. 422–430, 2003. 

[5] E. Karden, S. Buller, and R. W. De Doncker, “A method for measurement 

and interpretation of impedance spectra for industrial batteries,” J. 

Power Sources, vol. 85, pp. 72–78, 2000. 

Stephan Buller (M’97) received the Ph.D. degree 

from the Institute for Power Electronics and 

Electrical Drives (ISEA), Aachen University of 

Technology (RWTH-Aachen), Aachen, Germany, in 

2002. 

He joined ISEA in 1997, spending five years as a 

Research Associate. From 2002 until 2004, he was 

a Chief Engineer at ISEA. His research activities 

were mainly in the area of batteries and other energy 

storage systems. In January 2005, he joined an 

international consulting company. 

Marc Thele received the Diploma in Electrical 

Engineering from the Institute for Power Electronics 

and Electrical Drives (ISEA), Aachen University of 

Technology (RWTH-Aachen), Aachen, Germany, in 

2002. 

In June 2002, he joined ISEA as a Research Associate. 

His research activities are in the area of battery 

simulation models of different technologies. 



Rik W. A. A. De Doncker (M’87–SM’99–F’01) 

received the Doctor of Electrical Engineering degree 

from the Katholieke Universiteit Leuven, Leuven, 

Belgium, in 1986. 

During 1987, he was appointed as a Visiting 

Associate Professor at the University of Wisconsin, 

Madison. In December 1988, he joined the General 

Electric Company Corporate R&D Center, Schenectady, 

NY. In 1994, he joined Silicon Power 

Corporation as Vice President. In October 1996, 

he became a Professor at Aachen University of 

Technology (RWTH-Aachen), Aachen, Germany, and Head of the Institute for 

Power Electronics and Electrical Drives (ISEA). 

Eckhard Karden (M’00) received the Ph.D. 

degree from Aachen University of Technology 

(RWTH-Aachen), Aachen, Germany. 

He is a Research Engineer for storage systems in 

the Energy Management Group of the Ford Research 

Center Aachen (FFA), Aachen, Germany. Before he 

joined Ford in 2002, he was Chief Engineer at the 

Institute for Power Electronics and Electrical Drives 

(ISEA), RWTH-Aachen.

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