Electrochemical Impedance Spectroscopy and Harmonic Distortion Analysis

Transcription

1 Electrochemical Impedance Spectroscopy and Harmonic Distortion Analysis Bernd Eichberger, Institute of Electronic Sensor Systems, University of Technology, Graz, Austria 1

2 Electrochemical Impedance Spectroscopy Table of Content Introduction Impedance and Impedance Spectroscopy Equivalent Circuits EIS Measurement Setup EIS Graphical Presentation Non-Linearity, Distortion Analysis 2

3 Impedance Spectroscopy Measurement Techniques - Introduction Electrochemical Impedance Spectroscopy Non-invasive Method of Measuring Fuel Cell AC Characteristics Physical Model Equivalent Circuit EIS, Nyquist Diagram Physical Model Equivalent Circuit Model Nyquist Diagram 3

4 Impedance Spectroscopy Measurement Techniques - Introduction Electrochemical Impedance Spectroscopy Voltage / Current Characteristics, (complex) Impedance Linearization: Small Signal Sinusoidal Current Excitation Operating Point at the Steady State Curve Voltage vs. Current (typical fuel cell) Discharge Voltage vs. Time (typical NiMH cell) 4

5 Impedance Spectroscopy Measurement Techniques - Impedance Basics Impedance Basics Introduction Electrical Impedance Basics Equivalent Circuits EIS Measurement Setup EIS Graphical Presentation Non-Linearity, Distortion Analysis 5

6 Impedance Spectroscopy Measurement Techniques - Impedance Basics Electrical Impedances: Z - R, C, L Components, Equations Resistor: Z = R Impedance Z = Re + j (Im) Capacitor: Z = 1ΤjωC Magnitude Z = Re 2 + (Im) 2 Inductor: Z = jωl Phase φ = arctan [ Im Τ Re ] Angular Frequency: ω = 2πf 6

7 Impedance Spectroscopy Measurement Techniques - Impedance Basics Impedance: Magnitude and Phase Sinusoidal Voltage, Current v t = vsin(ωt) - magnitude of the complex impedance: ratio of the voltage amplitude to the current amplitude - Phase shift between Voltage v t and Current i t - Phase angle Δφ: phase shift by which the current lags the voltage 7

8 Impedance Spectroscopy Measurement Techniques - Impedance Basics Impedance Z Mathematical Description of Voltage / Current Ratio Impedance Z = v (t) Τi (t) Note: Resistance R = Τ v i, no time dependency (t) - Sinusoidal Voltage v t and Current i t - Time invariant and linear characteristics - Phase angle Δφ does not depend on the amplitude - Z can be any combination of R, L, C : simple complex! - The equivalent circuit of a fuel cell impedance includes elements with special phase characteristic: Warburg Impedance Zw - Nonlinear elements cause distortions harmonics analysis! 8

9 Impedance Spectroscopy Measurement Techniques - Equivalent Circuits Equivalent Circuits Introduction Impedance Basics Equivalent Circuits EIS Measurement Setup EIS Graphical Presentation Non-Linearity, Distortion Analysis 9

10 Impedance Spectroscopy Measurement Techniques - Equivalent Circuits Equivalent Circuit Networks Simplified Model - Resistor and Capacitor in parallel (Voigt Network) - Electrical Layer, Capacitance with Isolation resistance Z = Τ R (1 + jωc) τ = RC Time Constant 10

11 Impedance Spectroscopy Measurement Techniques - Equivalent Circuits Equivalent Circuit Networks Basic Equivalent Circuit Model for Electrochemical Cells Randles Circuit: Mixed Kinetic and Diffusion Model - Electrolyte Resistance Rs - Double Layer Capacitance Cdl - Charge Transfer Resistance Rct - Warburg Impedance Zw: Constant Phase Element, Δφ = - 45 Z w = στ ω j στ ω Z w = 2 στ ω 11

12 Impedance Spectroscopy Measurement Techniques - Equivalent Circuits Warburg Element Warburg Element Zw Model for diffusion processes Significant at (very) low frequencies Porous bounded Warburg model : Z w = Z w = 2 στ στ ω j στ ω ω 12

13 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement Fuel Cell Equivalent Circuit Fuel Cell and Equivalent Circuit Diagram - Ideal Voltage Source and Series Impedance Z -> Vcell (t) Four-Terminal Measurement - Minimize Errors due to wiring resistance / voltage drops - Separate Drive and Sense connections, close to the fuel cell 13

14 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement Principles (1) AC Current Source (a) - Ohm s Law, Precision AC Current Source AC Voltage Source (b) - Voltage Divider, Reference Resistor R for Current Measurement - Ratio metric Measurement 14

15 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement Principles (2) AC Current Source, Bias Load - No Current Flow into Fuel Cell, Bias Load - Full Bandwidth including DC - Fuel Cell Voltage is Offset Voltage for AC Measurement, Limits Dynamic Range! - Best for single (few) Cell Testing Fuel Cell V 15

16 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement Principles (3) AC Current Source, Bias Load, Coupling Capacitors - No Current Flow into Fuel Cell, Bias Load - Limited Lower Bandwidth (τ=rc ) - Suited for Testing Large Stacks (HV) - Requires Sophisticated Protection Fuel Cell V 16

17 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement (1) Practical Measurement Set-up - AC coupling / DC blocking with capacitors 17

18 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement (2) A more sophisticated Measurement Set-up - Direct Digital Synthesizer, Synchronous Rectification - Analogue - Digital Converters, µp Control 18

19 Impedance Spectroscopy Measurement Techniques - EIS Measurement Setup Impedance Measurement (3) Measuring Voltage and Current, Fully Digital (Post) Processing - Analog Filter, High Resolution Analog-to-Digital Conversion - Digital FIR Low Pass Filter with Sample Rate Reduction - Strictly preserves Phase Relation between Voltage and Current - Minimum Adjustments required Top Sensor Plate Bottom Sensor Plate 19

20 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Graphical Presentation Introduction Impedance Basics Equivalent Circuits EIS Measurement Setup EIS Graphical Presentation Non-Linearity, Distortion Analysis. 20

21 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Nyquist Diagram Complex Impedance Z over frequency range (2 dimensional) Zimag and Zreal Missing frequency information High Frequency Low Frequency (DC). 21

22 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Nyquist Diagram (2) Equivalent Circuit Diagram including Warburg Impedance Zimag and Zreal from 1 Hz to 800 Hz Typical Nyquist Plot High Frequency Low Frequency (DC). 22

23 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Nyquist Diagram (3) Series / Parallel Combination of R, C, Zw (Warburg) Double Layer Capacitance (Anode, Cathode), Resistances High Frequency Low Frequency (DC). 23

24 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Nyquist Diagram (4) Example Circuit and calculated impedance vs. frequency RC parallel / series connections, different time constants R1, R2, R3: 10 Ohm C1: 10nF C2: 1µF C3: 100µF. 100 MHz 0 Hz (DC) 24

25 Impedance Spectroscopy Measurement Techniques - Graphical Presentation Bode Diagram Magnitude / Phase vs. (log) Frequency. 25

26 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Enhanced Measurement Techniques Introduction Impedance Basics Equivalent Circuits EIS Measurement Techniques EIS Graphical Presentation Non-Linearity and Distortion Analysis, Intermodulation 26

27 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Harmonic / Intermodulation Distortion Analysis Fuel Cell Voltage vs. Current Density - Partly Non-Linear Transfer Function - Depending on Operating Point and Amplitude / Excitation Level 27

28 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Harmonic Distortion Analysis Exploiting non-linear Voltage/Current Characteristics - Additional information about internal state of a fuel cell (FC) - Total Harmonic Distortion Analysis (THDA) V Nonlinear Voltage / Current Ratio (Impedance) Voltage I Voltage (visible Distortion) Current 28

29 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Harmonic Distortion Analysis Harmonics Analysis (single frequency excitation) - Amplitude Ratio: Fundamental Frequency vs. Harmonics - Fast Fourier Transformation (FFT) Sinusoidal Current Drive Distorted Voltage across Nonlinear Impedance Single Frequency (100 Hz) Fundamental Frequency + Harmonics 29

30 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Harmonic Distortion Analysis Math Example u t = a 0 i t + a 1 i t 2 ω = 2 π f i t = iƹ sin ω t u t = a 0 iƹ sin ω t + a 1 iƹ 2 sin 2 ω t sin 2 x = cos 2x ω u t = a 0 iƹ sin ω t + a 1 Ƹ i2 2 2 ω 1 cos 2 ω t 30

31 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Intermodulation Distortion Analysis Exploiting non-linear Voltage/Current Characteristics - Additional information about internal state of a fuel cell (FC) - Second Order Intermodulation Frequencies f1+f1, f1-f2 V Nonlinear Voltage / Current Ratio (Impedance changes with amplitude) Voltage I Voltage (visible Distortion) Current 31

32 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Intermodulation Distortion Analysis Intermodulation, two (or more) frequencies excitation - Output Spectrum contains f1 + f2, f1 f2, 2 f1, 2 f2, - Fast Fourier Transformation (FFT) or Band pass Filter Two - Tone Sinusoidal Current Drive Distorted Voltage across Nonlinear Impedance f1 f2 f1+f2 f1-f2 2 f1 2 f2 Two Frequencies (1000 Hz, 1050 Hz) Fundamental Frequency + IM f1+f2, f1-f2 32

33 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Intermodulation Analysis A Math Example u t = a 0 i t + a 1 i t 2 i t = iƹ sin ω 1 t + iƹ sin ω 2 t a u t = a 0 i(sin Ƹ ω 1 t + sin ω 2 t ) + a 1 ( iƹ sin ω 1 t + iƹ sin ω 2 t ) 2 u t = a + a 1 iƹ 2 sin 2 ω 1 t + 2 iƹ 2 sin ω 1 t sin ω 2 t + iƹ 2 sin 2 ω 2 t sin α sin β = 1 2 cos α β cos α + β u t = a + a 1 iƹ 2 sin 2 ω 1 t + iƹ 2 (cos ω 1 t ω 2 t cos ω 1 t + ω 2 t ) + iƹ 2 sin 2 ω 2 t ω 1 u t = a 0 iƹ sin ω 1 t ω 2 + sin ω 2 t 2ω 1 + a 1 iƹ 2 (1 1 2 cos 2ω 1t 2ω cos 2ω 2t ω 1 ω 2 + cos ω 1 t ω 2 t ω 1 +ω 2 cos ω 1 t + ω 2 t ) 33

34 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Distortion and Intermodulation Analysis FC Stack I Stack V n V AC C V S tack Load I AC I AC V Cell 2 V 2 Cell 1 V 1 Typical Distortion Analysis Measurement Setup (also suitable for EIS) 34

35 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Multi Channel EIS / Distortion Measurement Top Sensor Plate Bottom Sensor Plate 35

36 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Multi Channel EIS / Distortion Measurement 16 Channel Sensor Plate (courtesy of S++) and 16 Channel Voltage / Current / Temperature Sensor Interface Board 36

37 Impedance Spectroscopy Measurement Techniques - Non-Linearity and Distortion Analysis Multi Channel EIS / Distortion Measurement 2-Dimensional View of Current Density and Impedance 37

38 Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy and Harmonic Distortion Analysis?... 38

39 Electrochemical Impedance Spectroscopy INTRODUCTION Impedance Spectroscopy is a non-invasive electrical method of analysing various frequency dependent phenomena of electrochemical components, especially batteries, fuel cells and electrochemical capacitors. It is useful for single cell testing in the laboratory and for online stack monitoring. Impedance analysis investigates the cell impedance within the frequency range of interest. The (complex) electrical impedance Z of a test object is the result of measurements (and calculations) of voltage and current magnitudes and their phase difference while exciting the system using a sinusoidal voltage or current source. Sophisticated measurement systems can simultaneously apply several different frequencies and so reduce the test time. The equivalent circuit model is an attempt to represent the electrical impedance Z of a fuel cell using capacitive and resistive components and a constant phase element, also called Warburg impedance. It represents anode activation losses, cathode activation losses, barrier layers and mass transfer effects. An electrical equivalent circuit diagram for a single fuel cell or a stack made up of several fuel cells is an ideal voltage source in a series connection with the impedance Z. Only the outer electrical terminals are accessible. The voltage v varies with the load current, as presented in figure x. This defines the operating point for direct current (DC). Impedance measurement means injecting a sinusoidal alternating current (AC) of moderate amplitude, resulting in an equally small variation of the cell/stack voltage. 39

40 Electrochemical Impedance Spectroscopy IMPEDANCE BASICS The electrical impedance Z, also called alternating-current resistance, is the measure of alternating sinusoidal voltage applied to a passive two-pole circuit to the alternating current flowing through it and their phase difference. In mathematical terms it is a complex number in Cartesian form Z=R+jX or in polar form Z= Z e^(jarg[z)). Such a presentation as a complex number is a convenient method of handling amplitude and phase angle in case of harmonic (sine wave) signals. In a linear system, the impedance Z does not depend on the amplitude of voltage or current. This means that the network does not comprise nonlinear voltage/current characteristic such as PNjunctions (Diodes, Transistors). Moreover, it is time invariant and thus does not change its characteristics over time nor has it any kind of memory effect. The basic formulas are Such an impedance Z may be any combination of resistor, capacitor or inductor in various series and parallel connections. Knowing the structure of the network and the values of all its components, the impedance Z can be calculated at any desired frequency. An equivalent circuit diagram of the battery or fuel cell is required for correlating the results of impedance spectroscopy measurements with the assumed chemical and physical processes inside the test object. Such an equivalent circuit mainly consists of mixed series/parallel connections of resistors and capacitors. Inductive behavior is, in most cases, of no relevance at low frequencies. 40

41 Electrochemical Impedance Spectroscopy IMPEDANCE BASICS Ideally, the impedance spectrum of the test object and of the equivalent circuit matches. The degree of matching depends on the accuracy of the equivalent circuit and the fitting of the parts values. The interconnection of the components (R, C) and their values should closely represent the real chemical and physical properties, yet be as simple as possible. Fuel Cell (Stack), Equivalent Circuit including impedance Z and ideal voltage source, Four-Terminal Connections for minimizing measurement errors Z Drive (Force) Sense v ( t ) Z Z Sense Drive (Force) 41

42 Electrochemical Impedance Spectroscopy MEASUREMENT TECHNIQUES A straightforward approach for impedance measurement is by connecting the device under test to an AC current source and measuring voltage, current and phase angle. Another approach is connecting an AC voltage source to a voltage divider consisting of the unknown impedance Z and a precision reference resistor R. In case of a fuel cell or battery, it may be necessary to keep the DC cell voltage of the electrochemical element away from the excitation source and the measuring equipment. A convenient method is using capacitors in a series connection. 42

43 Electrochemical Impedance Spectroscopy MEASUREMENT TECHNIQUES These straightforward setups are good for laboratory measurements using just a few different frequencies. Figure 5 shows a more advanced test arrangement. A microprocessor controls a frequency variable oscillator, which generates a sinusoidal voltage for the voltage controlled current source (VCCS) and in phase / quadrature phase digital signals for two synchronous rectifiers. The current from the VCCS causes a voltage drop across the impedance Z, which is amplified and split up by synchronous rectification into an in phase (real term) and quadrature phase (imaginary term) part. Further, signal processing includes low-pass filtering and analogue-digital conversion. Finally, the microprocessor calculates the magnitude Z and phase angle φ. 43

44 Electrochemical Impedance Spectroscopy HARMONIC DISTORTION ANALYSIS Abnormal operating conditions of FEMFCs typically appear as nonlinearities in the voltage/current characteristics at a specific operating point. Figure 10 shows the typical polarization curve of a PEM fuel cell. Normally it should operate in the region of ohmic polarization, where its equivalent circuit diagram is (in its simplest form) a series connection of a constant voltage source and a resistor. Harmonic distortion analysis requires also the injection of a sinusoidal current into the fuel cell respectively the fuel cell stack. This part of the test setup is similar to the one for electrical impedance spectroscopy. The description for the non-linear transfer function is mathematically a Fourier series and practically an approximation based on a polynomial equation. The output voltage is distorted because it is the sum of the fundamental frequency and harmonic distortions. For harmonic distortion analysis, a low frequency alternating current is impressed on the fuel cell stack by means of a signal generator and a power (current) amplifier. The response of the alternating voltage signal is analyzed for the fundamental frequency and harmonic distortion. A typical approach involves AC-coupling, bandwidth limiting / anti-alias filtering, analog-to-digital conversion and Fourier analysis using digital signal processing. The percentage of (total) harmonic distortions (THD) versus fundamental frequency is an indication for a critical condition of the fuel cell stack. Typical THD values for normal operation are < 0.1% and up to 5% and more in case of abnormal situations. 44