What types of generator excitation are there
Bachelor thesis. Alexander Angold Process for reduced-effort electrochemical impedance spectroscopy for starter batteries
1 Bachelor's thesis Alexander Angold Method for reduced-effort electrochemical impedance spectroscopy for starter batteries Faculty of Technology and Computer Science Department of Information and Electrical Engineering Faculty of Engineering and Computer Science Department of Information and Electrical Engineering
2 Alexander Angold Method for reduced-effort electrochemical impedance spectroscopy for starter batteries Bachelor's thesis submitted as part of the bachelor's examination in the information and electrical engineering course at the Information and Electrical Engineering Department of the Faculty of Technology and Computer Science at the Hamburg University of Applied Sciences Supervising examiner: Prof. Dr.Ing. Karl-Ragmar Riemschneider Second reviewer: Prof. Dr.Ing. Jürgen Vollmer Submitted on
3 Alexander Angold Topic of the bachelor thesis procedure for reduced-effort electrochemical impedance spectroscopy for starter batteries Keywords electrochemical impedance spectroscopy, EIS, starter battery, excitation, DFT Abstract This thesis includes a rough introduction to electrochemical impedance spectroscopy (EIS). Measurement data are analyzed and test circuits are set up and verified from them. The main question is whether and how the measuring systems of the BATSEN research project can use the EIS to measure batteries. In addition to testing the test circuit, a list of guidelines for further investigations is drawn up. Alexander Angold Title of the paper Methods to reduce the effort of an electrochemical impedance spectroscopy with regard to starter batteries. Keywords electrochemical impedance spectroscopy, EIS, starter battery, pertubation, DFT Abstract This thesis is a rough introduction to the electochemical impedance spectroscopy (EIS). It involves the analysis of pertubation parameters, construction of a pertubation circuits, testing and verification of the algorithm used. The prevailing question is whether or not the research project BATSEN is capable of measuring an EIS with its current setup and how this might be achieved. Finally, a rough guideline for further measurements is introduced.
4 Acknowledgments At this point I would like to thank Prof. Dr.-Ing. Karl-Ragmar Riemschneider for making this work possible for me. Also Prof. Dr.-Ing. I would like to thank Jürgen Vollmer for his support with all questions regarding the evaluation. Special thanks go to Dipl.-Ing. Günter Müller and Dipl.-Ing. Matthias Schneider, without whom the work would not have been able to continue. I would also like to thank my brother Eduard. I would like to express my special thanks to the BATSEN team: Matthias Schneider, Valentin Roscher, Armin Hübel, Philipp Schiepel, Johannes Röhn with whom I was able to enjoy discussing over the months.
6 Table of contents Signal form of the excitation Conclusion Summary and evaluation Outlook Bibliography 82 List of tables 86 List of figures 87 A Task 90 B Devices and programs used 93 C Verification of the calculation method 97 D Results of measurements with the power operational amplifier 101 E Estimation errors due to quantization 105 F Source code 113 G List of abbreviations 133 H Extended Appendix 133
7 1 Introduction and motivation Whether starter battery or traction battery in conveyor vehicles, accumulators or batteries are present in almost all vehicles of all sizes. Over the years, the equipping of premium vehicles with electronic comfort and safety systems has increased . On the other hand, there is the battery, the capacity of which is stagnating in the vehicle. The breakdown statistics of the ADAC in Fig. 1.1 name a faulty or discharged battery as reason no. 1 for the failure of a vehicle . Locks) LDWA) LWFS y) 6ü interior) instruments z) tü steering) L brakes) L axles 9) 7ü body O) 9ü power transmission) Lclutch) gearbox) Lcircuit P) 6ü exhaust system) Lcatalyst) particle filter O) yü engineLGMechanics3 7) OüGL general) Lighting3 8) 9ü Fuel system LGPump) Lines) LTank) LFilter3 y) 8ü Engine managementLgeneral 7) 6ü EnginemanagementLotto 8) Pü EnginemanagementLDiesel 8) yü Cooling) LHeating) LClimate T) Pü Battery, generator, starter 39.4% Figure 1.1: ADAC breakdown statistics 2010, modified from . Which part led to the failure? First and foremost the battery. Energy management systems counteract this development. The battery management system (BMS), one of the energy management subsystems, helps ensure the reliability and service life and thus the economic efficiency of the battery . BMS should measure the functionality of the battery on the basis of three loosely defined states with various means. The State of Charge (SoC) is the state of charge, which is typically specified in relation to the nominal capacity of the manufacturer and which fluctuates with the State of Health (SoH). The SoH is that
8 1 Introduction and motivation 8 State of health and includes various aging processes on the electrodes and the discharge and charging history or the type of use over a longer period of time. The high current capability of the lead accumulator is referred to with the State of Function (SoF) and indicates whether the starter battery is able to start the generator of a vehicle. BATSEN research project While conventional BMS are wired and vehicle-bound, the research group Wireless Cell Sensors for Vehicle Batteries (BATSEN) at the Hamburg University of Applied Sciences developed wirelessly communicating sensors that can be built into the individual cells of an accumulator. The aim is to implement cell-by-cell monitoring of the entire battery system in order to be able to determine the SoC, SoH and SoF and to prevent defective individual cells at an early stage. An industrial truck was jacked up in the vehicle technology test bench at the HAW and operated at a constant speed. Fig. 1.2 shows the voltage curve of the 12 cells of the lead traction battery during the test . A significantly larger drop in the voltage at cell 7 can be seen. The cell can therefore be serviced or replaced at an early stage and before the entire battery fails. The sensors measure the cell voltage and transmit this wirelessly to a base station. The base station itself measures the current of the entire battery at the same time. A more detailed description of the BATSEN measuring system can be found in Chapter 4. The SoC in the BATSEN research project has so far been determined via the connection between capacity and open-circuit voltage or via charge balancing. Recently, this method could be improved by optical measurement of the electrolyte density . Compared to this, the SoH is much more complicated to determine and includes many chemical processes and wear and tear in the active material or the electrolyte. Motivation To the previous possibilities for the analysis of the SoC and SoH, the electrochemical impedance spectroscopy (EIS) should be added. The non-destructive process that is used in battery production applies
9 1 Introduction and motivation Switching off the blast cell voltage binbVolt: 14: 00 11:29:45 11:45:30 12:01:15 12:17:00 Time of the critical cell Figure 1.2: Load on a lead traction battery of an industrial truck over about 1 hour. The individual cell voltages are shown over time. A particularly deep voltage drop can be seen in cell 7 after about 45 minutes of loading . to implement experimentally. Particular emphasis is placed on a sensible implementation and optimization of the necessary evaluation algorithms and framework conditions. In addition, test circuits and scripts for the automated measurement of alternating current impedances with special devices and simple measurement technology are to be created. The experimental part of the work is carried out with a battery using conventional lead-acid technology. The lead-acid battery developed more than 100 years ago by Wilhelm Josef Sinsteden can draw a very high current for a short time. This makes it ideal as a starter battery in vehicles, where it is still produced in very large numbers today and installed in various sizes.
10 1 Introduction and motivation 10 Content structure This work is divided into four sections. In the first part, an activation circuit is developed, analyzed and set up, and the necessary calculation algorithm is verified. The second part investigates whether and how an impedance measurement can be carried out with the current status of the BATSEN research project. For this purpose, the necessary accuracy of the cell sensors is determined. In the last part, investigations are made to optimize the excitation parameters. Guidelines for further measurements are being drawn up. Finally, the results of the work are summarized and evaluated. This is followed by an outlook with suggestions for improvement.
11 2 Basics 2.1 Structure and functionality of lead batteries Lead batteries contain chemical energy in their active material, which they convert directly into electrical energy through reduction or oxidation. The lead battery is one of the secondary batteries. These can be recharged by reversing the process . negative electrode lead (alloys) positive electrode lead dioxide porous separator dilute sulfuric acid lead Figure 2.1: Schematic representation of the structure of a lead battery, modified according to . A schematic structure of a lead-acid battery can be seen in Fig. 2.1. It consists of a positive and negative electrode separated by a separator and immersed in an electrolyte (sulfuric acid).
12 2 Basics 12 A starter battery, referred to below as a lead battery, consists of 6 cells connected in series, each with several plus and minus pole plates, which are separated by a separator. Each cell has a nominal voltage of around 2 V. The charging and discharging reactions can be described using Fig. 2.1. According to , when a load is applied to the negative and positive electrodes, the chemical reactions take place on the negative electrode according to equations (2.1) and (2.2). In equation (2.1), lead is oxidized by the load and the current flowing through it. The resulting lead cation reacts with the sulphate of sulfuric acid to form lead sulphate. The excess electrons flow through the conductor (and load) to the positive electrode. At the same time, the hydrogen cations released from the electrolyte move through the porous separator to the positive electrode. negative electrode positive electrode P b P be (2.1) P b 2+ + SO 2 4 P bso 4 (2.2) P bo 2 + 4H + + 2e P b H 2 O (2.3) P b 2+ + SO 2 4 P bso 4 (2.4) At the positive electrode, lead dioxide then reacts to form lead cations and water as in equation (2.3). The lead cations in turn react with the electrolyte to form lead sulfate. In principle, a voltage source can be used to reverse the process, which triggers the chemical reaction in the opposite direction. When the battery is in operation, there are other effects that affect the battery behavior, including the formation of a double layer and diffusion processes of the chemical substances on the electrodes. These can be analyzed using electrochemical impedance spectroscopy. The measurements of the present work are carried out on a wet lead battery of the brand Premium Panther with 100 Ah. The reaction equations and the processes involved in operating a lead battery are described in detail in .
13 2 Basics Chemical processes in a lead battery In electrochemical systems there are measurable effects that arise with the chemical equalization processes in the electrolyte in connection with the electrodes. If a metal is immersed in an electrolyte, ions on the surface of the metal attract ions of opposite charge , as can be seen in Fig. 2.2 on the left. The opposing ions form a layer similar to the dielectric of a capacitor, called an electrochemical double layer. The electrochemical double layer depends, among other things, on the concentration of the reactants and the applied voltage and decreases with distance from the electrode . positively charged electrode electrolyte rigid and diffuse double layer Figure 2.2: The electrochemical double layer on the positively charged electrode in this example, modified according to . The double layer decreases with increasing distance from the electrode. The rigid double layer has dimensions of a few molecules, while the diffuse double layer expands further and fluctuates with concentration and temperature differences in the electrolyte. The mass transfer of a chemical substance on the electrodes and in the electrolyte can be caused by convection, currents (non-electrical) through e.g. Temperature differences, electron migration and diffusion (a concentration gradient) can be described. The equalizing processes of diffusion make up the largest part of the effect . Diffusion effects come to light to varying degrees within an electrochemical system. A rough distinction can be made between diffusion processes inside the porous cathode and anode material and outside the material.
14 2 Basics of Electrochemical Impedance Spectroscopy (EIS) Electrochemical impedance spectroscopy is a widely used method for analyzing chemical processes in an electrochemical system. Among other things, it can be used to analyze the diffusion and transport processes described and the formation of a double layer. The process is used in corrosion research and in the manufacture of batteries  . The dynamic behavior of an electrochemical two-terminal network is determined by an excitation with a small signal of basically any shape (sine, square, pulse). The current response to a voltage excitation is measured or vice versa. The complex impedances for the respective frequencies present in the excitation signal are then calculated from the recorded current and voltage profiles by means of Fourier analysis and Ohm's law. Voltage dropöon theöelectrically conductiveömedien Charge passage / double layer diffusionöwithin semiödesöporousö plate material diffusionöouterhalbö desöporoseö plate material 1 mhz -öimrz) 5 khz 100 Hz 1 Hz 100 µhz ReRZ) Figure 2.3: Locus curve of an electrochemical system, modified according to . Chemical effects that are in the order of magnitude of the period of the excitation frequency can be observed and assigned in the locus curve. Some of the observable effects can be seen in Fig. 2.3. The differing
15 2 Basics 15 to recognize different semicircle-like courses to which physical effects of the battery are assigned. Different battery technologies have different impedance curves depending on SoC, SoH, the temperature, open circuit voltage and the discharge and charging history, but the approaches are very similar to the curve shown in Fig. 2.3 . The results of an EIS can be displayed in a Bode diagram or in a locus / Nyquist plot. Locus curves are easy to interpret visually and for this reason the locus diagram is preferred in the present work. In the case of application, an attempt is made to derive a model from the locus. The simplest possible electrical equivalent circuit (ESB) is created from elementary linear components. Then known physical properties of the system are assigned to these impedances . R i I R s R b U 0 I s I b C Surface C Bulk Figure 2.4: Example of a simple battery model . In this model, the capacity value C Bulk is applied large in order to map the storage behavior of the battery. Fig. 2.4 shows a possible approach . Now the preliminary model has to be tested or the parameters (e.g. resistance value, capacitance value) have to be adjusted so that the behavior of the chemical system matches that of the model as closely as possible. If an optimal choice of parameters results in too great a deviation from the actual behavior, a better ESB must be found. If sufficiently stable and precise parameters are found, long-term measurements (e.g. cycling) can reveal changes in the battery behavior compared to the model. The process of finding parameters is repeated, this time with the knowledge that the battery has been charged in a certain way, which is reflected in the locus.
16 2 Basics 16 With the help of the EIS it is possible to track some changing properties of an electrochemical system by measuring again and revising the equivalent circuit diagram. Among other things, measurements of the SoC, SoH and SoF are conditionally possible. In this thesis, the modeling and fitting of the parameters are not discussed in detail. More information on the methodology of an EIS measurement and the basics of modeling can be found in the literature under . 2.4 Measurement of an impedance by analyzing a system response In principle, the two-pole to be measured is a linear time-invariant system (LTI system), which is generated with an excitation, e.g. is controlled by an impulse. The individual sine components in the pulse are linearly distorted in different amounts and phase by the LTI system. If a voltage pulse is used for excitation, the current response must be measured and vice versa. Both voltage and current are recorded. Calculation and measurement of a linear impedance The linear, complex two-pole Z to be examined is excited with a sinusoidal voltage (2.5). A corresponding current (2.6) is established, the phase of which is related to the zero phase angle of the voltage.i s u s R C L complex two-pole Z Figure 2.5: Excitation of a linear, complex two-pole with sinusoidal voltage of a fixed frequency f.
17 2 Basics 17 Ohm's law yields: us (t) = û s cos (2πf t + φ 0) with φ 0 = 0 (2.5) is (t) = î s cos (2πf t Φ Z) (2.6) Z (f) = us (t) is (t) = Z (f) ejφ Z (2.7) Equation (2.7) can be used to determine the amount by calculating the amount and, using the arctan function, the phase of the impedance for the excitation frequency. The superposition theorem applies, which allows the system to be subjected to several sinusoidal excitations of the form: u s (t) = û s1 cos (2πf 1 t) + û s2 cos (2πf 2 t) (2.8). The system response, analogous to that in equation (2.6), takes the form i s (t) = î s1 cos (2πf 1 t Φ 1) + î s2 cos (2πf 2 t Φ 2) (2.9). The complex impedance spectrum can be calculated using the DFT, Z (f) = DFT (u s) DFT (i s) (2.10) where theoretically the DFT algorithm only delivers a value other than 0 at frequencies f 1 and f 2. Fig. 2.6 shows an example of an RC series circuit that is excited using equation (2.8) and whose complex impedance Z is determined using the DFT. i s R s u s C s Figure 2.6: Excitation of a complex two-terminal pole with sinusoidal voltages of frequencies f 1 and f 2. As an example, R s = 1 kω; C s = 10 µf; f 1 = 10 Hz; f 2 = 100 Hz; u s = 1 V is used.
18 2 Basics 18 Figure 2.7 shows the amplitude and phase spectrum of the impedance Z, which are calculated from the measured current i s and the applied voltage u s from equation 2.10. Z in Ω Φ in degrees voltage / V voltage / V time range amplitude spectrum amplitude response of the impedance spectrum ohm ohms phase response of the impedance spectrum f in Hz 0.5 current / ma current / ma Figure 2.7: Impedance spectrum of the two-pole to be examined from Fig. 2.6 with the associated numerical values. This spectrum only provides meaningful information for the circular frequencies f 1 and f 2. In the top diagram, the superimposition of the two sinusoidal excitations can be seen in the time domain. In the diagram below, the magnitude spectra of the current and the voltage can be seen. The last two diagrams are the magnitude and phase spectrum of the impedance Z (f) according to equation (2.10). The phase and the amount can be read directly from the frequencies f 1 and f 2. The values from Fig. 2.7 determined by means of the DFT agree with the theoretical ones
19 2 Basics 19 Values from equations (2.11) match. Z (f) = 1 R s + j 2 pi fc s (2.11) Z (f 1) = e j57.86 Ω Z (f 2) = e j9.04 Ω
20 2 Fundamentals of DFT and FFT The discrete Fourier transformation (DFT) is an algorithm from Fourier analysis that breaks down a discrete input sequence into its sine and cosine components. The result is a discrete (generally complex) frequency spectrum of the input sequence. The Fast Fourier Transform (FFT) is an algorithm for the efficient calculation of the DFT, whereby the classic variant according to Cooley and Turkey  only allows 2 n -points as input sequences. The FFT (Radix-2) is based on the idea of simplifying complex multi-point DFTs to 2-point DFTs over several stages . In this work, negative frequencies of the complex spectrum are not shown. The spectra shown have been corrected in their amount instead. Computational effort of the DFT and the FFT The calculation of the impedance from a voltage and current consumption is done with the DFT using equation (2.13). The DFT calculates the Fourier coefficients of a signal and supplies a sequence of numbers without any time or frequency reference. This reference must result from the sampling frequency f sampling. X k = N 1 m = 0 mk 2 π jxme N (2.12) Z (f 1) = DFT f 1 (us) DFT f1 (is) X k k-th Fourier coefficient where k = 0 ... N 1 xm m -ter value of the input sequence x N number of points (2.13) The input sequence x is transformed to the output sequence X, where the output sequence X represents the complex spectrum with a frequency spacing f = f sample. X N 4 would therefore be with N = 1000 points and f sampling = 100 Hz. X (4 f) would be the complex spectral line at 0.4 Hz. The computational effort of an N = 2 n = 1024 point DFT is compared with that of a 1024 point FFT according to Cooley and Turkey. M stands for a complex multiplication and A for a complex addition. Multiplications by 1 are ignored. From equation (2.12) it follows that for a DFT M = 2 n and A = 2 n to calculate a spectral line.
21 2 Basics 21 A Cooley Turkey N = 2 n = 1024 points FFT consists of n = 10 stages. A = n 2 n and M = n 2 n 1 apply to the calculation of 2 n spectral lines. Calculation operation Number of DFT Number of FFT Ratio DFT / FFT complex multiplications 2 nn 2 n 1 2 / n complex additions 2 nn 2 n 1 / n Table 2.1: Calculation of a DFT (only one spectral line) and FFT (of the entire spectrum) in comparison for 2 n Points. The computational effort of a DFT in relation to an FFT can be seen in Table 2.1. For the example n = 10, the calculation with the FFT would be five times more complex than the calculation with the DFT. Conversely, if more than five spectral lines have to be calculated, computing time can be saved using the FFT algorithm. This can e.g. be the case when calculating a distortion factor. For this reason, the DFT solution was used to calculate the impedance. The DFT has the advantage that it does not depend on powers of two, so an unorthodox number of points is also possible. Windowing The time-continuous Fourier transform of a periodic cosine function s (t) = cos (2 π f) S (f) = + s (t) ej 2 π t dt. Within the limits from to + would ideally have a spectral line at the point in the one-sided amplitude spectrum +10 Hz result. When setting the integration limits to finite values (which can actually be calculated), the periodic function s (t) is multiplied by a rectangle of a certain width. The limitation of the integration limits is called windowing. A time signal can be seen in Fig. 2.8, which is to be limited with a window. The formal relationship is shown graphically in Fig. 2.9. Fig. 2.8 can be divided into Fig. 2.9 a) and Fig. 2.9 b). In the time domain, the time function is multiplied by a square function of finite width. The amount of the Fourier transform of s (t), S (f) in Fig. 2.9 c) is a δ-function at 10 Hz. Avg
22 a) t t b) 2 Basics 22 c) 10Hz f * 10Hz f d) Figure 2.8: A sinusoidal signal should be limited (windowed). Figure 2.9: a) The signal to be windowed b) the rectangular window c) the ideal one-sided amplitude spectrum measured from to + d) the Fourier transform of the rectangular function. the transformation turns the multiplication in the time domain into a convolution in the image domain with the transform of the rectangular window function, a Si function. The impulse response of the window function can thus be found at 10 Hz. The width of the Si function is inversely proportional to the width of the window used. Leakage The precondition for the exact measurement of a spectral line by means of the DFT is the avoidance of the leakage effect, which occurs due to unsuitable sampling or an unfavorable number of points. The leakage effect is related to the fact that the DFT forcibly periodizes the input sequence.
23 2 Fundamentals 23 The figure shows the quasi-continuous one-sided, windowed amplitude spectrum of s (t) = cos (2 π f) with f = 10 Hz. It is sampled with f sampling = 100 Hz. The signal is multiplied by a rectangular window with a width of 1 s. Conversely, the continuous spectrum S (f) is then sampled with f = f sample n. In Fig. The sampling with f hits exactly the maximum of the continuous Si function and beyond that only all of its zeros. Amplitude in V Fourier transform. DFT frequency in Hz Figure 2.10: The continuous magnitude spectrum of the windowed sinusoidal signal. The DFT amount spectrum is marked with arrows. In Fig. The window was chosen to be slightly larger with 1.01 s, where s (t) remains the same. The main and sidelobes of the Si function of the continuous Fourier transform become a little narrower. By enlarging the time window while maintaining the same sampling frequency f A, the number of samples n and f decreases slightly. S (f) is scanned with the new f and hits, among other things, the secondary lobes of the Si function. The result is a spectrum that does not match that of s (t). The periodic properties of the DFT and IDFT come to light when the discrete spectrum S d [f] is transformed back into a time signal s d [t]. Fig shows the periodic continuation of the sampled signal s (t). At the point 1.01 s, the signal is continued periodically, s (t), but was sampled at the point s (t) t = 0 = 0, so that there is a jump there. In order to avoid the leakage effect, it must first be determined how many sampling periods fit into a searched signal period. A sinusoidal signal f signal = 1 Hz is sampled with f sampling = 100 Hz. N min = 100 sampling periods fit into a signal period of f signal. The signal was sampled leak-free, if multiples
24 2 Basics 24 Amplitude in V Fourier transf. DFT frequency in Hz Figure 2.11: The magnitude spectrum of the sinusoidal signal, windowed with a rectangular window with a width of 1.01 s. It can be seen that the main lobe is minimally narrower due to the larger rectangular window. 1 amplitude in V time in s Figure 2.12: The periodic continuation of the sampled signal s (t) after reverse transformation using the inverse discrete Fourier transformation (IDFT). are recorded by n min points. Appendix F describes the getminimumdatapoints () function, which returns a minimum number of points to be recorded from a given sampling and target frequency. Multiples of this minimum number can be included in order to suppress the noise in the measured values.
25 2 Basics of non-linear systems For a successful analysis of the system response to a stimulus using the simplest of methods, the system must be linear and time-invariant (LTI). The handling and modeling of non-linear systems is a very broad and theoretically demanding field. In this thesis, occurring non-linear effects are limited to those of the non-linear characteristic of a component. Time variance or memory effects of the relevant systems are neglected, but mentioned in the context of the sake of completeness. The subject of non-linearity is discussed in more detail under . Small-signal and large-signal behavior of non-linear systems A diode is used as an example of a non-linear characteristic. In Fig. 2.13a a characteristic curve of the ideal diode can be seen, which results from the Shockley equation I d = I s (e U D n U T 1) (2.14). This characteristic curve has an exponential course and is extremely non-linear, especially at the beginning of the steep slope. In order to show the non-linearity of this system, a circuit as in Fig. 2.13b is calculated with Matlab. Two cases are examined in which the system is controlled with a sine of a fixed frequency f = 100 Hz at a fixed operating point a) I = 10 ma with 2 different amplitudes î = 1 ma and î = 9 ma. The necessary total voltage u g (t) is calculated from the specified current using Schockley's equation (2.14). i s (t) = I + î sin (2π f t) (2.15) Fig shows the spectrum of u g for the î = 1 ma from Fig.2.13a. The spectrum has very small sine components in the harmonics. The time range is the same as the sine that was impressed on the circuit. If the amplitude is increased to î = 9 ma, there is enormous distortion due to the exponential increase in the diode characteristic. Clear spectral lines of the harmonics can be seen. The expected sinusoidal shape of the voltage, as would have been the case with an ohmic resistance, does not occur. u g (t) is strongly distorted nonlinearly.
26 2 Basics 26 I d i d u g R s u g a) id D 0.3 V (a) U d (b) Figure 2.13: (a) An example of an exponential diode characteristic. (b) This circuit was calculated in Matlab, where R s = 207 Ω, I s = 1 na and n U t = mv. U g U g spectrum in dbv t / s f / Hz Figure 2.14: î = 1 ma from Fig.2.13b with i s (t) = 10 ma + 1 ma sin (2πf t) set. The time range can be seen on the left and the spectrum of u g on the right. There is little distortion. Some harmonics can be seen at 200 Hz and 300 Hz, but both with an amplitude less than about 80 dbv = 0.1 mv. Shifting the operating point has a similar effect. The small and large signal behavior of a diode differs from that of an ohmic resistor. In principle, when operating non-linear components, care is taken to work with the smallest possible modulation at the operating point, so that the component behaves quasi-linearly.
27 2 Basics U g U g spectrum in dbv t / s f / Hz Figure 2.15: The circuit from Figure 2.13b is set with i s (t) = 10 ma + 9 ma sin (2πf t). The time range can be seen on the left and the spectrum of u g (t) on the right. There is enormous distortion of u g (t) non-linearity of electrochemical systems. Fig. Shows the characteristic curve of an electrochemical cell with the same resistance converted. The current flowing through the cell is influenced by the ohmic resistance of the cell on the one hand and by the potential necessary for a chemical reaction on the other hand. If the necessary potential is not available, no current flows. The speed of reaction and diffusion processes at the electrodes also influence the current flowing. The resulting characteristic is accordingly non-linear over large parts . Figure 2.16: Characteristic curve of a 1 Ω resistor and an electrochemical solution with an equivalent 1 Ω resistor .
28 2 Basics 28 Especially with batteries, the condition depends on the charging and discharging history . That is, the system is time-variant. An electrochemical system is therefore neither linear nor time-invariant. In order to be able to base the system on a model that is as linear as possible, it must be excited with the smallest possible amplitude. The literature speaks of different limits of around 0.3mV to 10mV excitation amplitude  . As far as possible, the measurement itself must not place any load on the system. Longer measurements with even low currents could unintentionally falsify the status of the system. For this reason, the currents flowing must be as small as possible, especially for measurements in the lower frequency spectrum 0.1 µhz to 1 Hz. For the analysis of non-linear behavior in the large-signal range, such as e.g. is the case while a vehicle is in motion, higher voltage excitations and correspondingly higher currents are required. Nonlinearities are discussed in more detail in Chapter 5. The temperature was kept constant during the measurements with the test systems. The course of the characteristic curve of a battery is significantly influenced by the direct component of the current and the voltage. A higher discharge current during a measurement therefore has an impact on the overall course of the locus.  came to the conclusion that the impedance spectra of batteries that were recorded at higher currents differ mainly in the lower frequencies. Fig shows a measurement with different DC components. It becomes clear that towards the lower frequencies the slope decreases with higher direct currents. The measurements at 0 ma and 100 ma turned out to be the best comparable, with a maximally large impedance.
29 2 Basics Im (Z) / mω -8-6 DC 0 ma DC 100 ma DC 1A DC 10 A Re (Z) / mω 14 Figure 2.17: Influence of the discharge current on impedance spectra for a Hoppecke / 12 V / 95 Ah SoC = 80% battery at different discharge currents, modified according to . No frequency information has been added to this diagram. In the frequency range estimated below 1 Hz, the discharge current has a particularly strong effect.
30 3 Experimental excitation setup Electrochemical impedance spectroscopy is a tool from systems theory. An electrochemical system is excited with a current or voltage signal and its voltage or current response is measured. From this, the impedance of the system can be calculated using Ohm's law. Depending on the type of excitation signal, the impedances can be calculated for one or more frequencies. In general, the prerequisite is that the system to be measured is an LTI system. Suitable signal forms include sine and square wave functions. The aim is to measure the battery impedance versus frequency. There it is advisable to use existing in-vehicle consumers, such as e.g. the heated rear window. Switching the heated rear window on and off would stimulate the system (the battery). Sinusoidal excitations cannot be implemented with this component without major intervention in the on-board electronics . Other approaches use the noise or the residual ripple of the generator during operation in order to achieve an excitation in the range of 1 to 10 khz (depending on the engine speed) . The aim is a reproducible measurement method with sufficiently small errors due to non-linearities in the behavior of the battery. A professional impedance spectrum analyzer is used for better comparability. The focus is on the excitation and measurement of a lead-acid battery with laboratory equipment, which leads to the development of an excitation circuit that is as simple as possible. The calculation and evaluation of the measurement results is carried out with Matlab. An excitation and evaluation circuit in the form of a circuit board with processing of the recorded data by a microcontroller is aimed for. 3.1 Analysis of the excitation of the laboratory device Fig. 3.1 shows the time course of a measurement with an impedance spectrum analyzer from FuelCon (TrueEIS). The TrueEIS is an impedance spectrometer that can excite in the range from 200 µhz to 100 khz and with currents of up to 100 A DC (see Appendix B). The measuring device first lowers the voltage of the battery and then modulates a sinusoidal signal using the load. In front
31 3 Experimental excitation setup 31 the actual measurement, an adjustment is carried out for about 1 s, which does not necessarily correspond to the set excitation frequency. This comparison is possibly used to estimate the expected excitation amplitude at the measurement frequency or for other estimates of the expected non-linearity, possibly via the distortion factor. Adjustment measurement voltage / V time / s Figure 3.1: Time curve of the voltage of a measurement of the TrueEIS. A comparison and then the actual measurement can be clearly seen. The adjustment does not necessarily take place with the measuring frequency. Checking the TrueEIS function The measurement results of the TrueEIS were verified by a test impedance , as can be seen in Fig. 3.2. The equivalent circuit diagram of the test impedance is shown in Fig. 3.3. The RC circuit was connected to the TrueEIS and the impedances from 0.1 Hz to 50 kHz are recorded.
32 3 Experimental excitation setup 32 Figure 3.2: The test circuit or passive test impedance, a 4 mm socket pair for the current-carrying cable (left) and one for the voltage measurement (left) . R 3 R 5 i in u in R 1 R 2 R 7 C 1 u out R 4 R 6 Figure 3.3: ESB of the test circuit. i in and u out are measured. With R 1 = 0.33 Ω, R 2 = 3, 3 Ω, R 3 = R 4 = 127 Ω, R 5 = R 6 = 100 Ω, R 7, C 1 = 330 nf, measuring range 3, modulation current of 750 ma . Im (Z) / mω Hz 1000 Hz Hz TrueEIS 100 Hz LtSpice Sim Re (Z) / mω Figure 3.4: The locus of the measurement of the TrueEIS and an LtSpice simulation of the ESB. The error between measurement and simulation is relatively small and can be explained by measurement noise. The semicircle-like course that is generated by the capacitance in the test circuit can be clearly seen.
33 3 Experimental excitation setup 33 In Fig. 3.4, the measured and simulated locus of the test impedance lie relatively exactly one above the other. The TrueEIS provides sufficiently accurate data. For this reason, the FuelCon impedance spectrum analyzer was used as a reference in the following. A single frequency excitation was used. 3.2 Verification of the calculation method Next, the calculation method was verified. For this purpose, the lead battery was stimulated and measured with the TrueEIS. The excitation by TrueEIS is recorded simultaneously with the oscilloscope. The voltage is measured directly on the battery and the current is recorded using a current clamp. 5 A in measuring range 1 (1 39 mΩ) was selected as the DC current. 50 measurements at the frequencies 1 Hz, 10 Hz, 100 Hz and 1000 Hz are recorded with the oscilloscope. The error criterion for this measurement was defined with Z Calculated, Z Ref CB = B n = Z Calculated Z Ref (3.1) B 100% (3.2) Z Ref φ = (Z Calculated Z Ref) in degrees (3.3), where the mean value from all determined impedances were used for one frequency each. Z (f) = DFT (u s) DFT (i s) (3.4) 50 complete impedance spectra are determined using equation (3.4). A point at the measurement frequency of 1 Hz was taken from each spectrum and entered in Fig. 3.5. The deviations in amount and phase are around 5% and are defined as sufficient for the purposes of the BATSEN project. The other three measurements can be found in Appendix D, which, among other things, indicate a problem with the phase.
34 3 Experimental excitation setup Im (Z) / mω Re (Z) / mω Figure 3.5: Plot of the complex plane with a measurement of 1 Hz excitation. Each circle is an impedance value calculated using the algorithm from equation (3.4). The triangle is the reference value Z Ref measured by the TrueEIS. The square is the mean of the recorded impedance values and lies in the middle of the point cloud. The mean deviation of the phase φ = 1 and the mean deviation in the amount B = mω or B n = 4.8%.
35 3 Experimental set-up of excitation Experimental circuit set-up In this chapter, the function of the TrueEIS and the calculation rule have been demonstrated so far. First of all, the system's excitation is to be modeled on that of the TrueEIS. Among other things, the excitation circuit must: excite with a single sinusoidal signal be able to generate a deflection of about â <5 mV   only act as a load The lead-acid battery to be measured generally has a small alternating current resistance of 2 20 mΩ. Small changes in the voltage, as provided by the excitation of â = 5 mV, can therefore result in relatively large currents. For this reason, the system should be set up to î 4 A. Excitation with audio equipment It was considered to use an audio output stage  for excitation of the lead-acid battery. The frequency range of conventional audio output stages is around 20 Hz to 20 kHz. This area does not quite meet the requirements of impedance spectroscopy. In the literature, µhz excitations are sometimes used. The audio output stage  is also not designed to drive impedances of 2 20 mΩ. For their own protection, many audio amplifiers have an emergency shutdown / current limiter that prevents loads from being driven below the 2 Ω, which is a minimum for loudspeakers. This limitation does not apply to audio output stages in the upper price segment with DC output . For the first attempts to excite the lead battery, a laboratory device was used. Excitation with a power operational amplifier. As a first test, an excitation circuit with a bipolar operational amplifier from Kepco was set up. The structure does not quite meet the specifications for the maximum current. However, it should serve to simulate the measuring process with the simplest of means. The circuit shown in Fig. 3.6 was connected to the battery. It was taken into account that the device is an inverting amplifier, so that a negative DC voltage u Gen is generated accordingly
36 3 Experimental excitation setup 36 R 2 U Batt u R 1 u Gen + u Kep R Labor i Batt U Batt Figure 3.6: Equivalent circuit diagram of the power operational amplifier from Kepco 36-5M, as it can be read on the device, connected to a Lead-acid battery via an adjustable laboratory resistor. The Kepco 36-5M is an inverting DC amplifier and must be fed with a negative DC voltage u Gen for a positive output voltage. must be in order to maintain positive tension. The Kepco was operated as a sink. Operation as a sink (b) U + 36V 12V I -5A -2.8A Figure 3.7: Excerpt from the internal laboratory documents for the Kepco 36-5 M, operated as a sink. Point (b) 1.7 A The specification of the Kepco in Fig. 3.7 shows that the maximum values in operation as a sink, at around 1.7 A + 5 1, 7 3 A = 2.8 A at 12 V (3.5)
37 3 Experimental excitation setup 37 lying. The laboratory resistance of 1 Ω between the battery and the Kepco should therefore limit the current. The load modulation was generated by a voltage difference u between u Kep and u Batt across the laboratory resistor. First, u Gen and the gain of the Kepco were adapted to the voltage u Bat, so that u = 0 V. Then u Kep was lowered by â and a sine of the same size was superimposed, so that u = â sin (2 pi ft) [V] (3.6) The result is a load-modulated excitation of the battery. Matlab generator excitation KepcoG BOP R battery + oscilloscope currentGGGGGGGGCH4 voltageGCH2 - Figure 3.8: Complete measurement system for excitation of the battery and measurement of the battery impedance with the Kepco. Channel 4 of the oscilloscope measures the current flowing through the battery using a current clamp. One of the measurements using the Kepco setup can be seen in Fig. 3.9. 50 measurements were carried out at one frequency, the impedance was determined from current and voltage and drawn in the complex plane. At the same time, the impedance was determined using the TrueEIS for comparison.
38 3 Experimental excitation setup Im (Z) / mω Re (Z) / mω Figure 3.9: 50 measured values at 100 Hz, recorded with the Kepco setup. In this measurement, the measured value (circles) and the reference value (triangle, bottom left in the picture) agree relatively well. The error in the amount is 5%. The mean of all measured values is shown as a square in the diagram. The recordings of the other three signal frequencies can be found in Appendix D. The 4 measurements carried out can be seen in Table 3.1. The error in the amount became relatively large with a maximum of 20%. It is very likely that a stimulus was used that did not match that of the TrueEIS in terms of the DC component. The hardware concept works, a measurement of the TrueEIS could be simulated. The experimental setup, however, is relatively large and by using a clamp meter (whose transfer function was not determined) the phase error is also different and increases proportionally to the frequency. A more compact further development, which is designed for higher currents, is desirable.
39 3 Experimental excitation setup 39 f in Hz B in Ω B n in% Table 3.1: Measurement results at different frequencies. The error in the amount becomes relatively large. This is due to a DC offset in the excitation that differs from the TrueEIS. The phase error is determined by the clamp meter on the battery. The phase was not corrected. Excitation with a regulated load based on a power MOSFET The regulation of the load current by means of a power MOSFET, as is often found with electronic loads, is also useful here for implementing load modulation. Electronic loads can basically be operated in two modes. Either the voltage or the current is kept constant over time. To do this, the resistance of the electronic load is regulated. The circuit in Fig. 3.10a is a current source based on an n-mosfet . This circuit can also be used as a load for a battery, whereby the positive pole of the battery is connected instead of R L as shown in Fig. 3.10b.
40 3 Experimental excitation setup 40 i D + u DS u GS U Batt u Gen u Shunt R (a) (b) Figure 3.10: (a) Current source for large output currents . (b) Power source as a load on a battery. The circuit in Fig. 3.10b is fed back by an operational amplifier and tries to get the virtual zero point at its inverting and non-inverting input. If a voltage u Gen is applied, the difference between u Shunt u Gen is applied to the gate of the n-mosfet. The current i D, which is limited by R, is then set as a function of u DS and u GS. A voltage drops across R through i D, so that U GS changes and a new i D value is set. The load on the battery depends on the input voltage u Gen and can e.g. have a sinus shape, which would excite the system in the form of a sinus. By using a current clamp in the Kepco circuit, the measurement results need a correction of the phase. The MOSFET circuit offers the advantage that the current is determined directly via u Shunt, so that no phase correction is required. Simulation with LtSpice An LtSpice simulation, see Fig. 3.11, was carried out with a generic MOSFET, generic operational amplifier and an R = R shunt = 1 Ω resistor, on which the current can also be calculated at the same time. Under ideal conditions, an input voltage u Gen = 1 V results in an output current i D = R Shunt u Shunt = 1 A (3.7)
41 3 Experimental excitation setup 41 a. In Fig. U Gen = 1 V + 1 sin (2π100) V has been set. As expected, a sinusoidal current i D = 1 A + 1 sin (2π100) A is established. SINE (1O1O100O0O0.001O0O0) Rser = 50 V3 U2 opamp2 M1 NMOS Lead-Acid 12 Rser = 0.005 RShunt 1.tranO0O0.1sO0.0µO1µs Figure 3.11: LtSpice circuit for a load-modulated excitation. In this model, the battery is a voltage source with a very low internal resistance. The operational amplifier can be operated in a unipolar manner and is also rail-to-rail capable, so that control from GND to the supply voltage is possible. u Gen / V i D / A Figure 3.12: u Gen = (1 + 1 sin (2π100)) V was fed in and a sinusoidal current i D = (1 + 1 sin (2π100)) A is established. The current and voltage curves are superimposed. An LM358 was provided as the operational amplifier, which supports unipolar operation. Thanks to the rail-to-rail capability of the IC, the input signal can be amplified from GND to the operating voltage . A power MOSFET 0 is coming
42 3 Experimental excitation setup 42 of the type STE250NS10  from STMicroelectronics, which has already been used in previous work in the BATSEN project . Further simulations were carried out with components that come close to the parameters of the STE250NS10 MOSFET and the LM358 amplifier. These simulations have shown that such a structure is basically functional. Heat dissipation Heat dissipation poses a major problem in practice. The circuit in Fig. 3.10b should be designed for direct currents of up to 4 A in measurement mode and up to 12 A in the event of a fault. The power-carrying part of the circuit consists of an n-MOSFET with the resistor r n Mos and a measuring resistor R Shunt, connected in series to the battery positive terminal against GND. The figure shows the drop in power at the components, depending on the set resistance ratio v, R Shunt = 1 Ω, r n Mos = R Shunt v, where the supply voltage is ideal and U Batt = 12 V. Point (a) in Fig is the fault case in which the MOSFET is fully conductive (r n Mos 5 mω ). 144 W are dropped at R Shunt. The measuring operation up to 4 A ends at point (b) with a maximum of P (r n Mos) = 32 W and P (R shunt) = 16 W and then runs in the direction of the arrows with decreasing current. The structure of the MOSFET is operated with a maximum of 32 W. To calculate the heat dissipation, the excitation is reduced to the direct current component. The necessary heat dissipation is determined by the static thermal resistance of the overall system. A static model for heat dissipation can be formulated analogously to Ohm's law. A series connection of thermal resistances R θtotal, the power loss P v as heat flow and the temperature difference T between component, heat sink and outside temperature as voltage is linked using equation (3.8). P max = T jmax T Ambient R θjunction case + R θuebergang + R θkuehlkoerper = TR θotal (3.8) A heat sink of the type HS MARSTON - 10DN A-200  with R θkuehlkoerper = 2.2 K was used to cool the MOSFET used. The MOSFET is attached to the heat sink W with the thermal conductive paste KP92 Kerafol  and has a negligibly small R θuebergang = 0.1 K. The MOSFET has a thermal resistance of R θjunction Case = 0.25 K and an operating temperature of maximum WWT j = 150 C [ 31]. T Ambient is the ambient operating temperature. According to the International Electrotechnical Commission (IEC), all electrical equipment must
43 3 Experimental excitation setup 43 (a) 10 2 (c) P (rn MOS) P (R Shunt) PowerP / PW (d) 10 1 (b) Ratio Pr n MOS: PR Shunt Figure 3.13: The power P ( R Shunt) and P (rn Mos) depends on the resistance ratio of R Shunt = 1 Ω to rn Mos = R Shunt v. P (R Shunt) decreases inversely quadratically. Point (a) is the fault case in which approximately 150 W would be converted to R Shunt. At the same time, about 0 W would be converted to r n Mos at point (b). From point (c) and (d) in the direction of the arrows, the operating range is r n Mos and R Shunt. The operating range is for direct currents up to a maximum of 4 A. [Flawless] function at air temperatures between +5 C and +40 C . T Ambient is assumed with the permissible upper limit of +40 C. Equation (3.8) gives the MOSFET P max, mosf ET = 110 K 2.45 K W 44.9 W. The MOSFET can therefore radiate the required 32 W via convection with the heat sink. To be on the safe side, a fan was installed, which provides the necessary cooling during operation at 4 A direct current. A practicable approach was chosen for cooling the shunt resistor R Shunt. The emission of P F ehler 150 W power loss with a single shunt is very expensive due to the manufacturing process. That is why 4x1 Ω shunt resistors of the type CGS HSA50  were combined to form a 1 Ω resistor. Fig. 3.14b shows the circuit made up of R = 1 Ω resistors, which is obtained by dividing the
44 3 Experimental excitation setup 44 (1) (2) RRRR (a) (b) Figure 3.14: (a) (1) Power MOSFET on the underside of the cooling plate (2) 100 W, 1 Ω resistor with 4 mm connections (b) Circuit for a 1 Ω resistor composed of 4 R = 1 Ω. Current and division of the voltage can transport 4 times the power. The assembled shunt was mounted on a heat sink of the same type with the addition of cooling paste. Without a heat sink , each of the shunts is able to radiate 20 W power loss at T Ambient = 20 C ambient temperature . The shunt, together with the heat sink, can therefore dissipate an estimated 100 W at T Ambient = 20 C. This does not cover the case of an error. For unattended long-term measurements, a shutdown by means of a temperature measurement must be built in. Another possibility is to install a fuse for i D> 5 A. The fuse must have the lowest possible resistance in order to impair u DS of the MOSFET as little as possible. Problems and possible solutions After the first measurements with the MOSFET structure, significant deviations from the reference impedance values were found. After a hint , the problem was identified and then corrected. The problem becomes clear from the voltage curve at the gate of the MOSFET in Fig. The courses in Fig can be divided into a) and b). In course a) the sine moves towards the lower peak value. It can be clearly seen that the disturbances are increasing. A kind of upswing can be seen. A high
45 3 Experimental excitation setup 45 a) b) a) b) a) 40 Voltage / mv u GS measurement u GS ideal t / ms Figure 3.15: The voltage curve at the gate of the MOSFET. A 200 Hz sine was fed in. The measurement and the expected course are shown.An error in the circuit causes u GS to swing up and causes unforeseen spectral lines and an enormous error in the impedance calculation. Voltage amplitude / mv f / khz Figure 3.16: Amplitude spectrum of the curve at the gate of the MOSFET. As expected, an amplitude of 20 mV can be seen at f signal = 200 Hz. In addition, however, there are high-frequency amplitudes of up to 2 mV in the range from 8 to 16 kHz in the spectrum. Gate capacitance could be found in the data sheet of the MOSFET. Presumably the gate capacitance delays the time it takes to effect a change in the current across the u GS. The swing of the circle increases b) and
46 3 Experimental set-up of excitation 46 the peak value is reached. [edit, mosfetspek] To counteract this, the gate of the MOSFET was loaded with a 10 kΩ resistance to ground. This allows the charge that was brought into the gate to flow away again. The result can be seen in Fig. Voltage / mv t / ms Figure 3.17: The voltage signal from Fig. With a 10 kΩ resistor from the gate of the MOSFET to GND. An upswing can no longer be seen. The problem was solved by the additional resistance. However, the problem could not be simulated by adding a capacitance at the gate within the LtSpice simulation. The corrected activation circuit can be seen in Fig.
47 3 Experimental Excitation Setup u DS u GS U Batt u Gen R Gate u Shunt R Figure 3.18: The corrected excitation circuit with R Gate for the removal of excess charge from the gate. The second operational amplifier is available in the LM358x package and is wired as an impedance converter.
48 3 Experimental excitation setup 48 Evaluation With the corrected circuit, impedance measurements with the measurement setup from Fig. 0.1 to 1000Hz were carried out. The reference curve, the curve from the measured and calculated values, can be seen in Fig. The direct and alternating component of the excitation was modeled on the reference. In this measurement, the excitation was automatically determined by a Matlabcript with successively approximated approximation. Matlab Generator Excitation D G S Battery + R Shunt Oscilloscope Current CH3 Voltage2CH2 - Figure 3.19: Complete measurement system for excitation of the battery and measurement of the battery impedance with the MOSFET structure. Channel 3 of the oscilloscope measures the current through the shunt resistor R Shunt. There is no additional recording of the current by a clamp meter. The measurements of the setup agree for the most part with the reference. A part of a typical semicircle can be seen, which shows the capacitive character of the battery. Table 3.2 shows the approximate deviations of the amount and the phase from the reference. Only at 0.1 Hz and 1000 Hz are there large deviations> 6%. It is assumed that the script for setting the excitation amplitude could not quite set the direct or alternating component of the reference, so that the battery was measured at different operating points. The measurement results with a specially built excitation based on a MOSFET are predominantly positive. The measurement with the Kepco suffered from a large phase error, which can only be explained by the transfer function of the clamp meter. The phase error has been significantly reduced. Amount errors and
49 3 Experimental excitation setup TrueEIS elast setup 8 Im (Z) / mω 6 4 1Hz 2 15Hz 0 200Hz Hz Re (Z) / mω Figure 3.20: Locus curve of the measurement of a lead-acid battery at different frequencies. Reference (TrueEIS) and measurement are largely the same. A typical semicircular shape can be seen. Frequency Z Mosfet structure / mω Z Trueeis / mω absolute value error /% phase error / Table 3.2: Comparison of the measurement of the Mosfet structure to the reference. The absolute value error remains less than 6% in the range from 1 Hz to 500 Hz.
50 3 Experimental excitation setup 50 Phase errors are probably caused by an imprecise setting of the DC and AC excitation compared to the excitation of the reference. According to the measurements, the excitation works through a load modulation based on a MOSFET, up to 500 Hz excitation frequency with an error of about 6% compared to the reference. The error in the phase is slightly higher than 6%, but is considered acceptable for the BATSEN research project. 3.4 Determination of the state-of-charge with electrochemical impedance spectroscopy The next step is a further test of the MOSFET structure. This test circuit had a magnitude error of about 6%. The phase error is negligible for frequencies <100 Hz. The MOSFET design was used to determine the SoC of the starter battery, which is of particular interest in practice. The SoC can be determined by various means. Among other things, the SoC with the complete discharge and charge cycle (discharge, end-of-charge voltage) can be determined with the simplest of means. However, this method is complex and would require a temporary shutdown of driving operations. Other optical measurement methods measure the density of the electrolyte by means of differences in the refractive index  and use this to calculate the SoC. In this context, the EIS can be used to find electrical parameters that are as proportional as possible to the SoC of the system. Individual frequency ranges of the EIS deliver different results in relation to the SoC. Due to their short measurement times, high frequencies are of course particularly attractive for a SoC determination during use. Measurements in these frequency ranges try to determine the internal resistance R i of the battery. This parameter is highly dependent on the discharge and charging history and changes relatively little with a SoC> 40%. If the measurement is carried out at rest, good results below SoC <40%  are possible. A much more pronounced effect can be seen with an EIS in the lower frequency range of 100 MHz to 10 Hz. They are considered more reliable but are time consuming. A measurement in the lower frequency range should also be done in the idle state and is particularly meaningful at SoC <60%.
51 3 Experimental excitation setup 51 Another possibility is shown in  by determining a characteristic frequency. All of the presented methods for the SoC determination using the EIS depend on the temperature and it is recommended to combine them with charge balancing or a learning system . Testing by SoC measurement The MOSFET structure was used to assess the SoC. The battery is first charged as far as possible regardless of the SoH, so that from da SoC =% is assumed. Then it is gradually discharged by 1 of the nominal capacity C over 1 hour. After the gradual unloading, the idle state is awaited. In Fig you can see that the idle state occurs about 2 hours after the end of the discharge. The change in battery voltage u has then dropped to u << 1 mV. Minute voltage / V time / h Figure 3.21: Discharge process of the test battery, after one hour the load is removed and the open-circuit voltage is awaited. The battery was discharged by 110C. Measurements in the range 100 MHz to 20 Hz were carried out after each discharge. The Nyquist plots for the individual SoCs can be seen in Fig. It can be clearly seen how the impedance increases with SoC <50%. In addition, as described in the literature, it can be seen that the impedance at low frequencies changes more strongly with the SoC. With this method, changes in the SoC can be determined, but not quantified absolutely. For this purpose, e.g. a model and load balancing necessary. The condition of the idle state can only rarely be met between the operating phases of a vehicle. Reaching the idle state can take more than 10 hours or more, depending on the load. The SoC estimate via an EIS current / A
52 3 Experimental excitation setup 52 Z in mω C 0.87C 0.77C 0.67C 0.57C 0.47C 0.37C 0.27C f in Hz Figure 3.22: The magnitude spectrum of the alternating current resistance for different (estimated) SoCs. The EIS remains relatively constant above 0.47 C. After reaching 0.47 C, the AC resistance increases significantly, so that the lower states of charge in the spectrum increase slightly. Measurement in the lower frequency range is therefore impractical . Combined with other methods, such as optical density measurement and load balancing, the EIS measurement can provide a relatively accurate SoC.
53 3 Experimental setup phase (Z) in f in Hz 0.97C 0.87C 0.77C 0.67C 0.57C 0.47C 0.37C 0.27C Figure 3.23: Phase response of the alternating current resistance for different (estimated) SoCs. The influence of the SoC is not clearly pronounced on the phase response.
54 4 Estimation of effort: resolution, measuring rate and computing effort The structure of the BATSEN measuring system can be seen in Fig. 4.1 and consists of a base station that measures the current and voltage of the battery and n-cell controllers that record the cell voltages of the battery and send it to the base station via Send on-off keying (OOK) in the 433 MHz or 15.56 MHz band. Base station V A Bidirectional communication Cell sensor 1 Cell sensor 2 Cell sensor 3 Cell sensor n battery Figure 4.1: A BATSEN measurement setup. You can see the base station with measurements of the total voltage and the current flowing through the battery. The cell sensors send the individual cell voltages and temperature to the contacts. The electrochemical impedance spectroscopy works with low alternating voltages of a few mV. An estimate of the necessary resolution, number of samples and reference voltage of the base station and the cell sensors should provide information about whether an impedance measurement can take place. In this part of the thesis, the requirements and characteristics of the cell sensors and the base station are first analyzed in more detail with regard to an impedance measurement.
55 4 Estimation of effort: resolution, measuring rate and computing effort 55 The focus is on the question of whether the current BATSEN measuring system is capable of a successful impedance measurement. In addition, suggestions and practicable solution approaches are examined more closely in order to enable an impedance measurement. In addition, the computational effort involved in an impedance calculation is to be reduced overall. 4.1 The BATSEN measuring system The cell sensors As part of the BATSEN project, three sensor classes were developed and tested. An overview of all sensor classes is given in Table 4.1. Sensor class class 1 class 2 class 3 transmission directions only uplink uplink with reduced uplink and downlink downlink (broadcast wakeup) no receiver passive receiver active receiver receiver in sensor measurement operation and communication required hardware effort autonomous semi-autonomous centrally controlled low complex very complex Table 4.1: Overview of all im BATSEN project developed and available cell sensors. The transmission of data from the cell sensor to the base station is called uplink and vice versa, downlink. To measure the phase of the impedance as precisely as possible, current and voltage must be recorded synchronously. The class 1 sensors are not suitable for synchronous recording because they can only send data without considering other cell sensors, but cannot receive them. The class 2 sensors can send and receive. The class 2 sensor works with 2 bands, a receiver circuit in the 13.56 MHz radio-frequency identification (RFID) band and a transmitter circuit in the 433 MHz band. A wake-up broadcast from the base station can be received, which sends the sensors from the
56 4 Estimation of effort: resolution, measuring rate and computing effort 56 Idle state changed to measurement state. The transmission power is 750 mw and the space required by two antennas finally prompted the development of the class 3 sensor. The class 3 sensor receives and transmits in the 433 MHz band and can initiate synchronous measurements using the wake-up broadcast. The sensor was designed by Durdaut  as part of the BATSEN project. The theoretical ability to be able to measure synchronously was taken up by Sassano  in order to pave the way for an exact impedance measurement. Furthermore, a newer version of the sensor class 3 has been developed, which has a better ADC and more memory. The following considerations with regard to accuracy (sampling frequency and quantization) and memory requirements are based on these class 3 sensors from Sassano. Characteristics of the class 3 sensor according to Sassano The available SRAM memory of the M SP 430x2xx microcontroller is 2 kbytes. The class 3 internal 12 bit ADC requires 2 bytes per measured value . In theory, a maximum of 1000 measured values can be temporarily stored and then downloaded from the base station. However, the MSP430x is a Von Neumann architecture that has to accommodate the program and the data in the SRAM. The number of values that can be saved is now reduced to a maximum of 700 samples. The class 3 sensors were expanded, among other things, by a burst measurement with approximately f sampling = 2 kHz . According to recent studies, f sampling can be increased up to 6 kHz. The MSP 430x2xx has a fix-point hardware multiplier. Characteristics of the base station according to Zimny The current revision of the base station, which was created as part of a thesis by Zimny , was carried out with an LM3S9D92 ARM Cortex M3 from Texas Instruments . The microcontroller clocked at 80 MHz has a dedicated multiply accumulate (MAC) unit, a 512 kb flash, 96 kb RAM. The board has an SD card slot that expands the memory up to 64 GB. The board is equipped with a 16-bit delta-sigma ADC AD7798  which has a reference of U basis full scale = 5V. This results in quantization levels of U LSB = 5 V = µv 216
57 4 Estimation of effort: resolution, measuring rate and computing effort 57 The maximum sampling frequency is 470 Hz . With regard to the EIS, the measurement of the current is particularly important, as this is only done in the base station. A DHAB S / 24 Hall sensor from LEM was used to measure the current. This has 2 channels that are led to the outside. One channel is for currents in the range of ± 500 A and one for ± 75 A. The Hall sensor is based on U basis full scale = 5 V and accordingly has quantization levels of I LSB = 150 A 2 16 = 2.3 ma The base station should measure data with up to can process to a measurement speed of 30 kHz, provided that possible memory bottlenecks due to overhead in the time division multiplex control of the SD card are negligible. 4.2 Requirements for the overall system The overall system consists of the base station, which measures the current, and the cell sensors, which measure and send the cell voltages synchronously with a clock. The base station has sufficient computing power and memory for any further processing or filtering of the data. The Delta-Sigma ADC can record frequencies up to about 100 Hz with an assumed 4- to 5-fold oversampling. An electrochemical system the size of a starter battery generates a relatively high current with just a few mV of excitation. In the worst-case scenario, this is 1 A. 1 A would correspond to around n = 1 A = 434 I 2.3 mA LSB. The recording of cell voltages by class 3 cell sensors is limited by a number of samples of around 500 and the maximum sampling frequency of around 6 kHz. The biggest hurdle, however, is the quantization of a voltage signal of only a few mV. An error is to be expected primarily from the cell voltage measurement. An experimental setup should clarify whether the overall system is suitable for determining impedance. The actual BATSEN hardware was not available for this experiment, so laboratory equipment was used. The recorded data are artificially degraded and evaluated in order to find out how large the additional error is due to quantization and a low number of samples.
58 4 Estimation of effort: resolution, measuring rate and computational effort 58 Procedure Impedance values of a single cell are recorded and the prevailing conditions are used as a requirement for the single cell measuring system. The battery is excited using the MOSFET structure. The current is measured via R Shunt of the MOSFET structure and then quantized with 16 bits, assuming that the Hall sensor is measuring in the ± 75A range. The data recorded with the oscilloscope without artificial deterioration serve as a reference and must be reproduced. The resolution of the oscilloscope far exceeds the required resolution and therefore serves as the basis for the deterioration. Fig. 4.2 shows the recording of the voltage excitation at 1 kHz with the MSO3034. The frequency resolution of the recording is f sampling = 1 MHz and the quantization levels in this data set are 300 nv voltage / m V nv voltage / µv time / msn 0 Figure 4.2: The data set, a noisy 1 kHz sinusoidal signal (left) and the quantization levels that can be seen after differentiating and sorting the data set several times (right). The quantization levels in this data set are 300 n / a. Description of the experimental set-up The entire battery is excited with an amplitude of â 5 mV. The excitation voltage at the single cell is then about â cell = 1 â. The highest frequency that can be measured is set to 1 khz. Three 6 recordings are made with the oscilloscope at the excitation frequency f signal = 1 khz or 10 Hz
59 4 Estimation of effort: resolution, measuring rate and computing effort 59 made. The alternating current impedance is calculated from these recorded curves, in compliance with the sampling theorem and leakage-free sampling, and used as a reference value from now on. In order to do justice to the conditions of the cell sensors and base station, the database is artificially deteriorated. So e.g. the number of samples is reduced to f sample <= f sample max = 6 khz by simply removing every nth sample. This corresponds roughly to the upper limit of a class 3 cell sensor in burst mode. Downsampling occurs without low-pass filtering.This is not necessary, since no signals greater than 500 Hz appear in the measured signal and thus aliasing is limited to a minimum. Minimum number of quantization levels The recording of the voltage u battery cell (sin (2πf signal t) mv is quantized evenly with n = 8 bits and n = 10 bits over the range 0 V to UF ullscale = 2.5 V. The size of a quantization level U LSB is then: U LSB 8Bit = UF ullscale 2 n = 9.8 mv U LSB 10Bit = 2.4 mv The quantization levels U LSB are in both cases about as large as the alternating portion of u battery cell, which is decisive for the measurement Part of the signal power is converted into the noise power. Assuming that with N quantization intervals at full scale up to ± UF ullscale / 2 was quantized uniformly, for an evenly distributed input signal  u 2 quant = U 2 full scale 12 N 2 1 N 2 applies (4.1) P quant = P signal k (4.2) Equation (4.2) describes how the signal power in the quantized signal decreases with decreasing number of stages N. In order to keep the influence of the quantization <1%, N must be 10. On Signal should be represented over at least 10 quantization intervals. 8 and 10 bit quantization is therefore not sufficient without further signal processing. The excitation signal of about â = 0.8 mV is only recorded with about 2 quantization levels even with a 12-bit quantization at U LSB 12-bit = 0.6 mV.
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