An Interdisciplinary Topic

Transcription

1 : An Interdisciplinary Topic Ahmed Elwakil T S P A H M E D E L W A K I L 1

2 Outline Fractional Calculus History and Definitions Fractional Trigonometric Identities s-domain, Stability and Impulse response Circuit Design The Fractional Capacitor Fractional-order Oscillators Fractional-order Filters Modeling Applications Biological Tissues Energy Devices Mathematical Background Electronic Circuits Bio/electrochemistry 2

3 Part I: Fractional Calculus 3

4 Basic Definitions Differentiations and integrations can be of arbitrary order Integer-order space is a special case! Special Subset Integer-order Fractional-order 4

5 Basic Definitions Riemann-Liouville definition (continuous form) Grünwald-Letnikov definition (discrete form) No physical analogies like slope or area under a curve 5

6 Example: Fractional Derivative of f(x)=x First-order Zero-order derivative-order 6

7 Fractional-order Derivative of f(x)=cos(x) Integer-order derivatives provide phase shifts of (n 90 ) to sine and cosine functions 7

8 Fractional-order Derivative of f(x)=cos(x) Fractional order derivatives increase the range of phase shifts 8

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12 Fractional-order Derivative of f(t)=e t Transient-time behavior Steady-state behavior Space order 12

13 Fractional-order Trigonometry In a fractional-order space (not 2-D or 3-D), time derivatives are functions of the space dimension. 2-D space becomes (2 )-D, 3-D space becomes (3 )-D space.(0< <1) The steady-state value of a time derivative in a space of dimension (n ) is equal to its value in a space of dimension n. The transient of a time derivative in a space of dimension (n ) is NOT the same as a space of dimension n! Transient-Time Fractional Space trigonometry 13

14 Generalized Trigonometry Generalized Euler identity Generalized sine and cosine functions In the transient time: In the steady state Steady-state 14

15 s-domain and Stability Laplace transform is useful because it allows for analysis using algebraic rather than differential equations. Applying the Laplace transform to a fractional derivative with zero initial conditions yields s-plane (cone) 15

16 Stability: Analysis How do we analyze the stability of fractional-order transfer functions? W-plane 1. Define the W-plane such that no corresponding s-domain area 2. The characteristic equation will have the form 3. Solve for all roots in the W-plane 4. The stability criteria is met if for all roots: no corresponding s-domain area A. Radwan, A. Soliman, A. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements, Chaos, Solitons Fractals, vol. 40, no. 5, pp ,

17 Roots in the W-plane of s α +1= 0 Stability: An Example Physical s-domain roots Non-physical s-domain roots A. Radwan, A. Soliman, A. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements, Chaos, Solitons Fractals, vol. 40, no. 5, pp ,

18 Impulse Response 18

19 Part II: Circuit Design The Fractional Capacitor Fractional-order Oscillators Fractional-order Filters 19

20 The Fractance Device A fractance device is a general electrical impedance (V/I) given by M. Nakagawa and K. Sorimachi, Basic characteristics of a fractance device, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. (Japan), vol. E75-A, no. 12, pp , The impedance for the resistor, inductor, and capacitor are Z R = R if α = 0 Z L = (L)s if α = 1 Z C = (1/C)s 1 if α = +1 For 0<α<1, we obtain a fractional capacitor Z = 1/C = ( /2) phase independent of frequency Also called a constant phase element (CPE) Circuit applications Biological applications 20

21 The Fractional Capacitor (1) S. Westerlund, Capacitor theory, IEEE Trans. Dielect. Elect. Insulation, vol. 1,1994 =0.99 Does C have the unit of Farad? Warburg Impedance Pseudo-capacitance =0.5 21

22 The Fractional Capacitor (2) Voltage across a Fractional Capacitor for a triangle-wave exciting current Ideal resistor (no differentiation) Ideal capacitor (integer-order differentiator) Series and parallel connections of Fractional Capacitors don t give a capacitance A. G. Radwan and A. S. Elwakil, "An expression for the voltage response of a current-excited fractance device based on fractional-order trigonometric identities," Int. J. Circuit Theory & Applications, vol. 40, pp ,

23 Approximation of Fractional Capacitor (1) Approximations are based on infinite-tree expansions of In general Increasing accuracy 23

24 Approximation of Fractional Capacitor (2) Fourth-order approximation example Accurate over 4 frequency decades 24

25 Approximation of Fractional Capacitor (3) 0.35u CMOS G. Tsirimokou, C. Psychalinos, A. S. Elwakiland K. N. Salama "Experimental verification of on chip CMOS fractional-order capacitor emulators," Electronics Letters, DOI: /el (in press). 25

26 Approximation of Fractional Capacitor (4) Integrator 26

27 Polymer-based Fractional Capacitors (1) Polymer-based Uses infinite RC trees A. M. Elshurafa, M. N. Almadhoun, K. N. Salama, and H. N. Alshareef, Microscale Electrostatic Fractional Capacitors using Reduced Graphene Oxide Percolated Polymer Composites, Applied Physics letters, 102, (2013). 27

28 Polymer-based Fractional Capacitors (2) Magnitude Phase A. M. Elshurafa, M. N. Almadhoun, K. N. Salama, and H. N. Alshareef, Microscale Electrostatic Fractional Capacitors using Reduced Graphene Oxide Percolated Polymer Composites, Applied Physics letters, 102, (2013). 28

29 Silicon-based Fractional Capacitors (1) uses fractal geometry Photo-lithography Choose fractal shape package fabricate T. Haba, G. Loum, J. Zoueu, G. Ablart Use of a component with fractional impedance in the realization of an analogical regulator of order 1/2, J. Appl. Sciences, vol. 8, no. 1, pp ,

30 Silicon-based Fractional Capacitors (2) 1.6 x 10 6 pseudo-ohm Warburg Impedance T. Haba, G. Loum, J. Zoueu, G. Ablart Use of a component with fractional impedance in the realization of an analogical regulator of order 1/2, J. Appl. Sciences, vol. 8, no. 1, pp ,

31 Liquid-based Fractional Capacitors (1) * Based on a metal-liquid interface * Can realize very low values of * Not practical packaging! Packaging of Single-Component Fractional Order Element, D. Mondal and Karabi Biswas, IEEE Trans. Device & Materials Reliability, vol. 13, pp (2012). 31

32 Liquid-based Fractional Capacitors (2) Equivalent RC tree of the electrode surface Self similar porous structure on the electrode surface 32

33 Part II: Circuit Design The Fractional Capacitor Fractional-order Oscillators Fractional-order Filters 33

34 Fractional capacitors Fractional-order Circuit Design 1) Oscillators let For classical Wien-bridge For a fractional Wienbridge let = Wien-Bridge Can be Independent of RC!! A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples, IEEE Trans. Circuits Syst. Regul. Pap., vol. 55, no. 7 pp ,

35 Experiments with a liquid interface FC with a normal Capacitor with a fractional Capacitor 35

36 2) Multivibrators Classical multivibrator linear Fractional multivibrator Non-linear B. Maundy, A. Elwakil, S. Gift, On a multivibrator that employs a fractional capacitor, Analog Integr. Circ. Sig. Process., vol. 62 pp ,

37 Experiment with fractal geometry FC 37

38 Multivibrator with a fruit (date) Fruits and vegetables show fractional capacitance behavior A.S. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area, IEEE Circuits Syst. Mag., vol. 10, no. 4, pp ,

39 Part II: Circuit Design The Fractional Capacitor Fractional-order Oscillators Fractional-order Filters 39

40 Fractional-order Circuits 3) Analog Filters Expands the frequency responses from integer order steps in the stop-band to fractional steps Fractional Butterworth Low Pass Filter of order (1+ ) (1+.1) (1+.5) The TF is unstable for order 2 att = -20(1+ )db/dec T.J. Freeborn, B. Maundy, A.S. Elwakil, Tow-Thomas fractional-step biquad filters, Nonlinear theory and its Applications (IEICE), vol. 3, no. 3, pp ,

41 Higher-order fractional-step LPF How can we implement stable higher-order fractional-step filters? att = -20(5+ )db/dec Standard Butterworth polynomials B. Maundy, A.S. Elwakil, T.J. Freeborn, On the practical realization of higher-order filters with fractional stepping, Signal Processing, vol. 91, pp ,

42 Fractional-order high-q BPF Both high quality factors and asymmetric characteristics are possible Type-I Type-II asymmetric-slopes P. Ahmadi, B. Maundy, A.S. Elwakil, L. Belostotski, High-quality factor asymmetric-slope band-pass filters: a fractional-order capacitor approach, IET Circuits Devices Syst., vol. 6, pp ,

43 Experimental realizations (1) Field Programmable Analog Array (FPAA) hardware Based on minimum phase-error approximation T.J. Freeborn, B. Maundy, A.S. Elwakil, Field programmable analogue array implementation of fractional step filters IET Circuits Devices Syst., vol. 4, pp ,

44 Experimental realizations (2) Silicon-based fractal geometry FCs Fractional filters have wider bandwidth Fractional response 2nd-order response Classical Tow-Thomas filter T.J. Freeborn, B. Maundy, A.S. Elwakil, Tow-Thomas fractional-step biquad filters, Nonlinear theory and its Applications (IEICE), vol. 3, no. 3, pp ,

45 Experimental realizations (3) CMOS OTA-based emulator 45

46 Part III: Modeling Applications Biological Tissues Energy Storage Devices 46

47 Modeling Applications 1) Biological Tissues Biological tissues exhibit fractional impedances Fruit tissues (apples, apricots, plums, etc...) Human tissues (skull, lungs, breast cancer, etc.) Biologists have been using the Cole-Cole impedance model since1941 to characterize tissue impedance as a function of frequency K.S. Cole and R.H. Cole, Dispersion and absorption in dielectrics: alternating current characteristics, J. Chem. Phys., vol. 9, pp ,

48 Four element Fractional impedance model Single dispersion Cole-Cole model (1) Low-frequency resistor High-frequency resistor Constant Phase Element (fractional capacitor) Tissue characteristic frequency f c = 1/ Finding the Cole parameters is done by Impedance Spectroscopy 48

49 Single dispersion Cole-Cole model (2) Requires an impedance analyzer to measure REAL and IMAGINARY parts (Resistance & Reactance) then fit data to a Nyquist plot. Max reactance at f c Graphically extract the 4 parameters Biochemists have been doing this for over 60 years!!! 49

50 Single dispersion Cole-Cole model (3) Warburg Impedance example 45 degrees Some fruit measurements 50

51 Single dispersion Cole-Cole model (4) New HPF method ideal experimental measure Solve for A. Elwakil and B. Maundy, Extracting the cole-cole impedance model parameters without direct impedance measurement, Electron. Letters, vol. 46, no. 20, pp ,

52 Single dispersion Cole-Cole model (5) New step-response method tissue Apply a step input and measure the Tissue response Find R 2, R 3, C and using an optimization technique Two term Mittag-Leffler function 52

53 Single dispersion Cole-Cole model (5) ideal step-response technique is faster and more accurate experimental 53

54 The AD5933 Analog Devices impedance measurement chip can be used for Impedance Spectroscopy but Post processing is still needed on the data! Several accuracy problems reported! 54

55 Medical Applications: Dentistry three resistors and 3 fractional Capacitors 55

56 Medical Applications: Dentistry 56

57 Medical Applications: Lung Pathology Fractional-order inductor Fractional-order capacitor 57

58 Medical Applications: Lung Pathology 58

59 Medical Applications: Cancer detection T: # of Tumor cells E 1 and E 2 : Immune effectors 59

60 Medical Applications: Body Mass Composition 5 cylinder model 60

61 Body Mass Composition 61

62 Part III: Modeling Applications Biological Tissues Energy Storage Devices 62

63 Super-Capacitors (Ultra-Capacitors) Supercapacitors or Electrochemical Double Layer capacitors exhibit impedances that are modeled very well using fractional transfer functions experimental Fractional model Y. Wang, T.T. Hartley, C.F. Lorenzo, J.L. Adams, J.E. Carletta, R.J. Veillette, Modeling ultracapacitors as fractional-order systems, in New Trends in Nanotechnology and Fractional Calculus Applications, Springer, 2010, pp

64 Super-Capacitors Banks Used in renewable energy sources and hybrid engines T. J. Freeborn, B. Maundy and A. S. Elwakil, "Measurement of supercapacitor fractional-order model parameters from voltage-excited step response," IEEE J. Emerging and Selected Topics in Circuits & Systems, vol. 3, pp , Sept

65 Lithium-Ion Batteries classical model Fractional-order model 65

66 Fuel Cells Double dispersion Cole model has 6 unknowns Used to cover wider frequency range Fuel Cell model 66

67 The Energy Equation Applied step-voltage Not always true M. Fouda, A. S. Elwakil, A. G. Radwan and A. Allagui, "Power and energy analysis of fractional-order electrical energy storage devices," Energy, vol. 111, pp , Sept

68 Some Future Directions 3D modeling of biological tissues using 3D circuit theory Cole model Fruit tissue model (e.g. fruit cell, cancer cell) 3D interconnect network Over-all model (e.g. fruit, tumor) 68

69 Some Future Directions Update Circuit Simulators (Spice, Cadence ) to include Fractional-order devices as standard components Standardize/commercialize fractional-order capacitors and fractional-order inductors Investigate fractional-order properties of new materials Different CPE symbols 69

70 Conclusion Fractional-order systems is an interdisciplinary topic merging Mathematics, Circuits and Biochemistry Applications in Biology and Medicine are immense Fractional-order time-space may revolutionize the understanding of many physical phenomena 70