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Fractional Order Circuits and Systems:An Interdisciplinary Topic
Ahmed Elwakil
1Fractional Order Circuits and Systems
A H M E D E L W A K I L
T S P 2 10
001 6
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Outline
• Fractional Calculus• History and Definitions
• Fractional Trigonometric Identities
• s-domain, Stability and Impulse response
Mathematical
Background
Electronic Circuits
Bio/electrochemistry
• Circuit Design• The Fractional Capacitor
• Fractional-order Oscillators
• Fractional-order Filters
• Modeling Applications• Biological Tissues
• Energy Devices
2Fractional Order Circuits and Systems
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Part I: Fractional Calculus
3Fractional Order Circuits and Systems
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Basic Definitions
Differentiations and integrations can be of arbitrary order
Integer-order space is a special case!
Integer-order Fractional-order
Special Subset
4Fractional Order Circuits and Systems
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Basic Definitions
• Riemann-Liouville definition (continuous form)
• Grünwald-Letnikov definition (discrete form)
• No physical analogies like slope or area under a curve
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Example: Fractional Derivative of f(x)=x
First-order
Zero-order
derivative-order
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Fractional-order Derivative of f(x)=cos(x)
Integer-order derivatives provide phase shifts of (n·90°) to sine and
cosine functions
7Fractional Order Circuits and Systems
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Fractional-order Derivative of f(x)=cos(x)
• Fractional order derivatives increase the range of phase shifts
8Fractional Order Circuits and Systems
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Fractional-order Derivative of f(t)=et
Transient-time
behavior
Steady-state
behavior
Space
order
12Fractional Order Circuits and Systems
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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 NOTthe same as a space of dimension n!
• Transient-Time Fractional Space trigonometry
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Generalized Trigonometry• Generalized Euler identity
• Generalized sine and cosine functions
• In the transient time:
• In the steady state
Steady-state
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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)
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Stability: Analysis• How do we analyze the stability of fractional-order transfer
functions?
1. Define the W-plane such that
3. Solve for all roots in the W-plane
4. The stability criteria is met if for all roots:
2. The characteristic equation will have the form
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. 2317-2328, 2009.
W-plane
no corresponding
s-domain area
no corresponding
s-domain area
16Fractional Order Circuits and Systems
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Stability: An Example
Roots in the W-plane of sα+1= 0
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. 2317-2328, 2009.
Non-physical
s-domain roots
Physical
s-domain roots
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Impulse Response
18Fractional Order Circuits and Systems
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Part II: Circuit Design
19Fractional Order Circuits and Systems
•The Fractional Capacitor
• Fractional-order Oscillators
• Fractional-order Filters
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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. 1814–1819, 1992.
The impedance for the resistor, inductor, and capacitor are
ZR = R if α = 0
ZL = (L)s if α = –1
ZC = (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
20Fractional Order Circuits and Systems
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The Fractional Capacitor (1)
Does C have the unit of Farad? Pseudo-capacitance
S. Westerlund, “Capacitor theory,” IEEE Trans. Dielect. Elect.
Insulation, vol. 1,1994
=0.99
21Fractional Order Circuits and Systems
Warburg Impedance =0.5
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The Fractional Capacitor (2)
Voltage
across a Fractional
Capacitor for a
triangle-wave
exciting current
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. 533-538, 2012
Ideal capacitor
(integer-order differentiator)
Ideal resistor
(no differentiation)
Series and parallel connections
of Fractional Capacitors
don’t give a capacitance
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Approximation of Fractional Capacitor (1)
Approximations are based on infinite-tree expansions of
In general
Increasing
accuracy
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Fourth-order approximation example
Accurate over 4 frequency
decades
24Fractional Order Circuits and Systems
Approximation of Fractional Capacitor (2)
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25Fractional Order Circuits and Systems
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:10.1049/el.2016.1457 (in press).
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26Fractional Order Circuits and Systems
Approximation of Fractional Capacitor (4)
Integrator
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Polymer-based Fractional Capacitors (1)
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, 232901 (2013).
Polymer-based
Uses infinite RC trees
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Polymer-based Fractional Capacitors (2)
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, 232901 (2013).
Magnitude Phase
28Fractional Order Circuits and Systems
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Silicon-based Fractional Capacitors (1)
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. 59-67, 2008.
uses fractal geometry
Choose fractal shapePhoto-lithography
fabricate
package
29Fractional Order Circuits and Systems
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Silicon-based Fractional Capacitors (2)
Fractional Order Circuits and Systems 30
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. 59-67, 2008.
Warburg Impedance
1.6 x 106 pseudo-Ohm
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Liquid-based Fractional Capacitors (1)
Packaging of Single-Component Fractional Order Element, D. Mondal and Karabi
Biswas, IEEE Trans. Device & Materials Reliability, vol. 13, pp. 73-80 (2012).
* Based on a metal-liquid interface
* Can realize very low values of
* Not practical packaging!
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Liquid-based Fractional Capacitors (2)
Self similar porous structure
on the electrode surface
Equivalent RC tree of
the electrode surface
32Fractional Order Circuits and Systems
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Part II: Circuit Design
• The Fractional Capacitor
•Fractional-order Oscillators
• Fractional-order Filters
33Fractional Order Circuits and Systems
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Fractional-order Circuit Design1) Oscillators
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. 2051-2063, 2008.
Fractional
capacitors For classical Wien-bridge
let
For a fractional Wien-
bridge
let =
Can be Independent of RC!!
Wien-Bridge
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Experiments with a liquid interface FC
with a normal
Capacitor
with a fractional
Capacitor
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2) Multivibrators
B. Maundy, A. Elwakil, S. Gift, “On a multivibrator that employs a fractional capacitor,” Analog Integr. Circ. Sig. Process., vol. 62 pp. 99-103, 2010.
Classical multivibrator
linear Fractional multivibrator
Non-linear
36Fractional Order Circuits and Systems
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Experiment with fractal geometry FC
37Fractional Order Circuits and Systems
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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. 40-50, 2010.
38Fractional Order Circuits and Systems
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Part II: Circuit Design
• The Fractional Capacitor
• Fractional-order Oscillators
•Fractional-order Filters
39Fractional Order Circuits and Systems
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Fractional-order Circuits3) Analog Filters
Expands the frequency responses from integer order steps in the
stop-band to fractional stepsFractional Butterworth Low
Pass Filter of order (1+)
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. 357-374, 2012.
The TF is unstable
for order 2
att = -20(1+)dB/dec
(1+.1)
(1+.5)
40Fractional Order Circuits and Systems
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Higher-order fractional-step LPF
How can we implement stable higher-order fractional-step filters?
B. Maundy, A.S. Elwakil, T.J. Freeborn, “On the practical realization of higher-order filters with fractional stepping,” Signal Processing, vol.
91, pp. 484-491, 2011.
Standard Butterworth
polynomials
att = -20(5+)dB/dec
41Fractional Order Circuits and Systems
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Fractional-order high-Q BPFBoth high quality factors and asymmetric characteristics are
possible
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. 187-197,2012.
Type-I
Type-II
asymmetric-slopes
42Fractional Order Circuits and Systems
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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. 514-524, 2010. 43Fractional Order Circuits and Systems
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Experimental realizations (2)
Silicon-based fractal geometry FCs
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. 357-374, 2012.
Classical Tow-Thomas filter
2nd-order
response
Fractional
response
Fractional filters
have
wider bandwidth
44Fractional Order Circuits and Systems
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Experimental realizations (3)
45Fractional Order Circuits and Systems
CMOS OTA-based emulator
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Part III: Modeling Applications
46Fractional Order Circuits and Systems
• Biological Tissues
• Energy Storage Devices
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Modeling Applications1) Biological Tissues
Biological tissues exhibit fractional impedancesFruit 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. 341-351, 1941. 47Fractional Order Circuits and Systems
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Single dispersion Cole-Cole model (1)
Low-frequency resistor
High-frequency resistor
Constant Phase Element (fractional capacitor)
Tissue characteristic frequency fc = 1/
Finding the Cole parameters is done by Impedance Spectroscopy
Four element
Fractional
impedance model
48Fractional Order Circuits and Systems
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Requires an impedance analyzer to measure REAL
and IMAGINARY parts (Resistance & Reactance)
then fit data to a Nyquist plot.
Graphically extract the 4 parameters
Biochemists have been doing this
for over 60 years!!!
Max reactance
at fc
49Fractional Order Circuits and Systems
Single dispersion Cole-Cole model (2)
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Warburg Impedance
example
45 degreesSome fruit
measurements
50Fractional Order Circuits and Systems
Single dispersion Cole-Cole model (3)
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A. Elwakil and B. Maundy, “Extracting the cole-cole impedance model parameters without direct impedance measurement,”
Electron. Letters, vol. 46, no. 20, pp. 1367-1368, 2010.
Solve for
ideal
experimental
measure
51Fractional Order Circuits and Systems
Single dispersion Cole-Cole model (4)New HPF method
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Two term Mittag-Leffler function
tissue
Apply a step input
and measure the
Tissue response
Find R2, R3, C and
using an optimization
technique
52Fractional Order Circuits and Systems
Single dispersion Cole-Cole model (5)New step-response method
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ideal
experimental
step-response technique
is faster and more
accurate
53Fractional Order Circuits and Systems
Single dispersion Cole-Cole model (5)
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The AD5933Analog Devices impedance measurement chip can be used for
Impedance Spectroscopy but Post processing is still needed on the data
! Several accuracy problems reported!
54Fractional Order Circuits and Systems
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Medical Applications: Dentistry
three resistors
and 3 fractional
Capacitors
55Fractional Order Circuits and Systems
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Medical Applications: Dentistry
56Fractional Order Circuits and Systems
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Medical Applications: Lung Pathology
Fractional-order inductor
Fractional-order capacitor
57Fractional Order Circuits and Systems
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Medical Applications: Lung Pathology
58Fractional Order Circuits and Systems
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Medical Applications: Cancer detection
T: # of Tumor cells
E1 and E2: Immune effectors
59Fractional Order Circuits and Systems
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Medical Applications: Body Mass Composition
60Fractional Order Circuits and Systems
5 cylinder model
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Body Mass Composition
61Fractional Order Circuits and Systems
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62Fractional Order Circuits and Systems
• Biological Tissues
•Energy Storage Devices
Part III: Modeling Applications
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Super-Capacitors (Ultra-Capacitors)Supercapacitors or Electrochemical Double Layer capacitors exhibit impedances that
are modeled very well using fractional transfer functions
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. 257–262.
experimental
Fractional model
63Fractional Order Circuits and Systems
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Super-Capacitors BanksUsed in renewable energy sources and hybrid engines
64Fractional Order Circuits and Systems
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. 367-376, Sept. 2013.
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Lithium-Ion Batteries
65 Fractional Order Circuits and Systems
classical model
Fractional-order model
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Fuel Cells
Double dispersion
Cole model has 6 unknowns
Used to cover wider frequency range
Fuel Cell model
66Fractional Order Circuits and Systems
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The Energy Equation
67Fractional Order Circuits and Systems
Applied step-voltage
Not always trueM. 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. 785-792, Sept. 2016
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Some Future Directions
3D modeling of biological tissues using 3D circuit theory
Over-all model
(e.g. fruit, tumor)
Fruit tissue model
(e.g. fruit cell, cancer cell)
3D interconnect network
Cole model
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Some Future Directions
69Fractional Order Circuits and Systems
•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
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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
70Fractional Order Circuits and Systems