Fractional Order Circuits and Systems · 2016-06-30 · • Fractional Calculus • History and...

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

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


Part I: Fractional Calculus


Basic Definitions

Differentiations and integrations can be of arbitrary order

Integer-order space is a special case!

Integer-order Fractional-order

Special Subset


Basic Definitions

• Riemann-Liouville definition (continuous form)

• Grünwald-Letnikov definition (discrete form)

• No physical analogies like slope or area under a curve


Example: Fractional Derivative of f(x)=x

First-order

Zero-order

derivative-order


Fractional-order Derivative of f(x)=cos(x)

Integer-order derivatives provide phase shifts of (n·90°) to sine and

cosine functions


Fractional-order Derivative of f(x)=cos(x)

• Fractional order derivatives increase the range of phase shifts


Fractional-order Derivative of f(t)=et

Transient-time

behavior

Steady-state

behavior

Space

order


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


Generalized Trigonometry• Generalized Euler identity

• Generalized sine and cosine functions

• In the transient time:

• In the steady state

Steady-state


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)


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


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


Impulse Response


Part II: Circuit Design


•The Fractional Capacitor



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


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


Warburg Impedance =0.5

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


Approximation of Fractional Capacitor (1)

Approximations are based on infinite-tree expansions of

In general

Increasing

accuracy


Fourth-order approximation example

Accurate over 4 frequency

decades





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).



Integrator

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


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


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


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

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!


Liquid-based Fractional Capacitors (2)

Self similar porous structure

on the electrode surface

Equivalent RC tree of

the electrode surface



• The Fractional Capacitor

•Fractional-order Oscillators



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


Experiments with a liquid interface FC

with a normal

Capacitor

with a fractional

Capacitor


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


Experiment with fractal geometry FC


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.



• The Fractional Capacitor


•Fractional-order Filters


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)


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


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


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


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




CMOS OTA-based emulator

Part III: Modeling Applications


• Biological Tissues

• Energy Storage Devices

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

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


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



Warburg Impedance

example

45 degreesSome fruit

measurements



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


Single dispersion Cole-Cole model (4)New HPF method

Two term Mittag-Leffler function

tissue

Apply a step input

and measure the

Tissue response

Find R2, R3, C and

using an optimization

technique


Single dispersion Cole-Cole model (5)New step-response method

ideal

experimental

step-response technique

is faster and more

accurate



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!


Medical Applications: Dentistry

three resistors

and 3 fractional

Capacitors


Medical Applications: Dentistry


Medical Applications: Lung Pathology

Fractional-order inductor

Fractional-order capacitor


Medical Applications: Lung Pathology


Medical Applications: Cancer detection

T: # of Tumor cells

E1 and E2: Immune effectors


Medical Applications: Body Mass Composition


5 cylinder model

Body Mass Composition



• Biological Tissues

•Energy Storage Devices

Part III: Modeling Applications

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


Super-Capacitors BanksUsed 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. 367-376, Sept. 2013.

Lithium-Ion Batteries

65 Fractional Order Circuits and Systems

classical model

Fractional-order model

Fuel Cells

Double dispersion

Cole model has 6 unknowns

Used to cover wider frequency range

Fuel Cell model


The Energy Equation


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

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


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

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


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Fractional Order Circuits and Systems · 2016-06-30 · • Fractional Calculus • History and...

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