development of an analytical model for a charge … · memristor and its applications by yojanes...

Development of an analyticalmodel for a charge-controlledmemristor and its applications

by

Yojanes Andrés Rodríguez Velásquez

A dissertation submitted in partial fulfilment ofthe requirements for the degree of

MASTER OF SCIENCES IN THESPECIALTY OF ELECTRONICS

at the

Instituto Nacional de Astrofísica, Óptica yElectrónica

Advisor:

Dr. Librado Arturo Sarmiento ReyesPrincipal Researcher INAOE

August 2017Tonantzintla, Puebla

c©INAOE 2017The author hereby grants to INAOE permission

to reproduce and to distribute copies of thisthesis document in whole or in part.

Abstract

In this work, the development of an analytical model for a memristor is presented. Themodel is based on the solution of the differential equation that governs the physicalbehaviour of the device. The solution has been obtained by resorting to a homotopyperturbation method.

The resulting memristance function is controlled by the electric charge, and fullsymbolic expressions are obtained in function of the main parameters of thememristor.

The obtained model is characterised both for AC and DC sources. The mainfingerprints of the device in AC regime have been verified, as well as the mainparameters in DC (the switching voltages and saturation time).

In order to show the usefulness of the model, two applications that use memristivegrids as an analog processor, are studied. In a first application, the memristive grid isused as a filter for image edge detection and smoothing, obtaining a good performancein comparison with the edge recognition made by humans. In a second application,the memristive grid is used in the maze solving problem, since the analog processorimplements the shortest path method.

Resumen

En este trabajo se presenta el desarrollo de un modelo analítico para un memristor,basado en la solución de la ecuación diferencial que rige el comportamiento físico deldispositivo. La solución se obtiene utilizando el método de homotopía deperturbación.

La memristancia en este modelo es controlada por la carga eléctrica y se obtienenexpresiones puramente simbólicas en función de los principales parámetros delmemristor.

El modelo obtenido es caracterizado tanto para fuentes de AC como DC. Se verificaque el modelo cumple con las principales características en AC, y adicionalmente sedeterminan los principales parámetros DC (los voltajes de conmutación y tiempo desaturación).

Con el fin de mostrar la utilidad del modelo, se estudian dos aplicaciones que utilizanuna red memristiva a modo de procesador analógico. En la primera, la red memristivase utiliza como filtro para suavizado y extracción de bordes en imágenes, mostrando unbuen desempeño al ser comparada con los bordes extraídos por seres humanos. En lasegunda aplicación, la red memristiva se utiliza en la solución de laberintos, ya que elprocesador analógico implementa el método de la trayectoria más corta.

Agradecimientos

Este libro representa la culminación de un ciclo de formación en mi

vida, durante los últimos dos años llegaron y pasaron personas que de

alguna manera contribuyeron a mi formación tanto personal como

académica. Intenté aprender de cada una de ellas y también aportar

algo a sus vidas; profesores, estudiantes, compañeros, amigos,

conocidos de amigos. . . a todos agradezco por permitirme sacar

partida de nuestra interacción, ya que es de la forma en la que estoy

acostumbrado a aprender. También agradezco las oportunidades

presentadas, los momentos de zozobra por alguna situación académica

o personal, y las alegrías vividas en cada viaje; a mi mamá y mis

hermanas por apoyar mi progreso, y contribuir en mi desarrollo como

persona aún estando en la distancia.

Especial agradecimiento a mi asesor Dr. Arturo Sarmiento durante este

proceso, porque me permitió explorar a mi gusto las posibilidades

sobre mi tesis, pero siempre con una guía atenta y adecuada, además de

vii

ser muy correto y atento como ser humano, permitiendo espacios de

esparción y distracción durante este año de tesis.

Gracias México y toda su gente, porque en cada lugar que visité fuí

recibido con un especial calor humano, haciendo que no me sintiera tan

lejos de casa y enamorandome de su cultura. Agredezco el Concejo

Nacional de Ciencia y Tecnología CONACYT, por apoyar

económicamente este proyecto durante el desarrollo del mismo y tiempo

de mi formación como Maestro en ciencias.

Finalmente solo me queda agradecer a ese algo que me ha impulsado a

seguir adelante, a continuar después de cada derrota y fortalecerme en

el proceso.

Contents

Contents ix

1 Introduction 1

2 Fundamentals of memristor and modelling 52.1 The fourth basic circuit element . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Fingerprints of memristor . . . . . . . . . . . . . . . . . . . . 72.1.2 HP memristor . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Types of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Behavioural models . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Approach of modelling in this work . . . . . . . . . . . . . . . 11

3 Development of a charge-controlled model 133.1 Non-linear drift mechanism . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Introduction to homotopy methods . . . . . . . . . . . . . . . . . . . . 163.3 HPM solution to the drift equation . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Memristance equations . . . . . . . . . . . . . . . . . . . . . . 21

4 Characterization of the model 294.1 Dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Fingerprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.2 Comparison with other models . . . . . . . . . . . . . . . . . . 33

4.2 DC tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 Determining the switching voltages . . . . . . . . . . . . . . . 354.2.2 Determining the saturation time . . . . . . . . . . . . . . . . . 384.2.3 Memristance-Charge characteristic . . . . . . . . . . . . . . . 394.2.4 M-q Characteristics of memristor connections . . . . . . . . . . 40

5 Memristive grid for edge detection 455.1 Previous approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Description of the memristive branch . . . . . . . . . . . . . . . . . . . 475.3 Solution of the memristive grid . . . . . . . . . . . . . . . . . . . . . . 49

ix

5.4 Results and comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 515.4.1 Figures of merit for the edge detection procedure . . . . . . . . 525.4.2 Results for a benchmark image . . . . . . . . . . . . . . . . . . 535.4.3 Comparative results on a set of 500 images . . . . . . . . . . . 56

6 Memristive grid for Maze solving 616.1 Previous approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Maze solving: implementing the memristive grid . . . . . . . . . . . . 62

6.2.1 Description of the memristive fuse . . . . . . . . . . . . . . . . 626.3 Simulation flow of the memristive grid . . . . . . . . . . . . . . . . . . 656.4 Mazes under-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.5.1 Single solution mazes . . . . . . . . . . . . . . . . . . . . . . 676.5.2 Multiple solutions mazes . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusions and future work 737.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Fij factors in the memristance equations 75A.1 Factors for η = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 Factors for η = +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B Characterization plots 95B.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.1.1 Ap plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96B.1.2 X0 plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101B.1.3 Ron plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

B.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

List of figures 113

List of tables 117

Bibliography 119

x

Chapter 1

Introduction

Based on a symmetry argument that completes the number of possible relationshipsbetween the four fundamental circuit variables: current, voltage, magnetic flux, andelectric charge, professor Leon O. Chua predicted in 1971 the existence of the fourthbasic circuit element [1]. He called the memristor and defined it as a passive devicewith two terminals, whose branch constitutive function relates magnetic flux andelectric charge. Almost four decades after the theoretical conception of the memristor,in 2008, the R. Stanley Williams work group at Hewlett-Packard Laboratories,presented a device whose behaviour exhibits the memristance phenomenon [2]. Thesemilestones have boosted a series of developments in the field of memristor modellingand applications during the last years.

The memristor is an element that presents properties that can be used both forprocessing and storing information. These properties allow to develop a new paradigmof computation, a paradigm that bases on the behaviour of neurological systems inliving beings [3]. This possibility suggests that the classical computation architecture,where the storing and processing are carried out in different blocks, can be replaced byan architecture where the processing and storing are performed in the same physicalplatform, in fact, this would shorten the processing time because the information isalways available in the processor.

The current architecture of neuro-computing and artificial neural networks isdeveloped with very little connection with neuroscience. Because of this, importantfeatures of biological neuronal systems have been ignored, such as extreme low powerconsumption or their ability to perform robust and efficient computation using massiveparallel arrays of limited precision, or highly variable and unrealistic components [4].This paradigm has been smartly pointed out in [5]: “The idea of taking inspirationfrom the brain’s structure and organization to build advanced systems has attracted alot of interest. These artificial neural networks should be able to carry out tasks thatare not easily or not at all tackled by traditional computer approaches, in particulartasks of a cognitive nature”. A better approach for the operation of biological neuronal

1

2 CHAPTER 1. INTRODUCTION

systems can be achieved by combining CMOS and memristor technology [6], [7].Although the combination of these technologies carries with it a great deal ofvariability in manufacturing processes, this variability and little precision can be takenas a basis for processing in another approach to neural processing, since it has beendemonstrated that biological systems work with stochastic systems and highvariability [8]. This paradigm is called memcomputing [9].

In the last years, studies have been done to carry out logical operations, in digitalsystems, with the exclusive use of memristors [10], or to measure parameters of storeddata without the need for external processing [11]. This way of using the memristorallows to realize digital processing as it is known at the moment, but with theadvantage that the information is stored in the own elements of operation.

This new architecture in computing, also allows to do analog processing within thesame structure, which can be an advance in the hybridization of systems that carry outanalog and digital processing in the same physical platform. Due to the variable natureof the resistance in the memristor, programmable analog processors can be developedwhich allow changing, for example, the characteristics of a particular filter or amplifier[12]. Other systems like memristive grids present properties that make them flexible inthe development of analog applications.

Within INAOE, the CAD group has been focussed during the last years on thedevelopment of memristor models that are suitable for simulation of memristivesystems and its applications. The first model developed by the group is presented in[13], and its implementation in a CAD platform is presented in [14]. This model wasdeveloped to be used in AC regime, i.e. sinusoidal inputs, however, the applicationsthat are studied in this work require a model that can be used in DC circuits and allowsto carry out transient analyses. Consequently, the objective of this work is to developan analytical model of the memristor fabricated by HP, to characterize this model forits application in systems with sources in AC and DC, and to show its usefulness intwo applications that perform analog and parallel processing based on memristivegrids. These applications are: image edge detection and shortest path method as a toolfor solving labyrinths.

To show the work performed in this thesis, this document is organized in six morechapters. In Chapter 2, basic concepts about memristors, such as the ideal memristordefinition and the equations governing the HP memristor, are presented. In Chapter 3,the development of the proposed model is shown and the symbolic equations obtainedas function of the parameters of the memristor are presented. In Chapter 4, a summaryof the characterization of the model is carried out: in the first part, the characterizationis oriented to demonstrate that the model fulfils the main fingerprints of the device; inthe second part, a DC characterization is done in order to determine the main staticparameters of the memristive characteristics. In order to show the potential of the

3

developed model and the memristive grids, in Chapter 5, the memristive grid is usedas a processor for image edge detection. The memristive grid, with little changes, isused in Chapter 6 for computing the shortest path in maze solving, demonstrating theadvantages of the parallel processing that the grid performs. Finally, in Chapter 7 someconclusions are drawn and future lines of research are proposed.

Chapter 2

Fundamentals of memristor andmodelling

As said before, there are two milestones about memristors, the introduction of thememristor as the fourth basic circuit element by Prof. Leon O. Chua in 1971 [1] andthe first memristor fabricated by HP in 2008 [2]; from the latter one many studiesabout memristors has been published in recent years. Models have been developed inorder to find the properties of the new element and to use it in circuit applications.Therefore, in this chapter, the fundamentals about memristors, and the main featuresabout device modelling are presented.

2.1 The fourth basic circuit elementIn circuit theory, four basic electrical variables are used, they are the electric charge q,the magnetic flux φ, the current i and the voltage v. In order to have a relation betweenthe for variables, there are six possibilities of linkage (figure 2.1). Two of these relationsare the well known definition of current

i =dq

dt; q(t) =

∫ t

−∞i(τ)dτ (2.1)

and Faraday’s law

v =dφ

dt; φ(t) =

∫ t

−∞v(τ)dτ (2.2)

Other three relationships are the definition of the basic circuit elements: the resistanceR

v = Ri (2.3)

the capacitance Cq = Cv (2.4)

and the inductance Li = Lφ (2.5)

5

6 CHAPTER 2. FUNDAMENTALS OF MEMRISTOR AND MODELLING

The last relationship, between q and φ, was proposed in 1971 by Leon O. Chua, and isgiven:

dφ = M(q)dq; v(t) = M(q(t))i(t) (2.6)

where M(q) is the ideal memristance and the fourth element is called memristor,because it has the units of a resistor (Ω) and its resistance depends of the complete pasthistory of the memristor current, presenting memory characteristics. The inverseversion of the memristor is called memductor because has units of conductance (Ω−1),its relationship is defined as:

dq = W (φ)dφ; i(t) = W (φ(t))v(t) (2.7)

Figure 2.1: Electrical variables relations.

The symbol of memristor proposed by Chua is shown in figure 2.2.

Figure 2.2: Memristor symbol.

Given the current i(t) or voltage v(t) in the memristor, its behaviour is as a lineartime-varying resistor, or in the same way, the q-φ curve presents a non-linear relation(as shown figure 2.3 a)). In the specific case where the memristor q-φ characteristicis a straight line (as shown figure 2.3 b)), its behaviour is like a linear time-invariantresistor [1].

2.1. THE FOURTH BASIC CIRCUIT ELEMENT 7

a) b)

Figure 2.3: q-φ characteristic of memristor.

2.1.1 Fingerprints of memristorAs reported in [15], the fingerprints are related with the features of the i-v curve in thememristor. A device can be considered a memristor if it fulfils these properties.

Given a memristor controlled by a sinusoidal current source i(t) = Ap sin(ωt), thereare at least three fingerprints for this element:

1) Pinched Hysteresis Loop (PHL)

The i-v characteristic must show a pinched hysteresis loop always, that is, the PHLmust pass through the point v = 0 and i = 0 for any possible input amplitude Ap, andany possible input frequency ω, as shown figure 2.4.

Figure 2.4: Pinched hysteresis loop.


2) Reduction of the lobe area as the frequency increases

Above a critical frequency ωc, the area of the PHL decreases monotonically as thefrequency increases for ω > ωc (as shown figure 2.5).

Extracted from [15]

Figure 2.5: First quadrant lobe area as function of the frequency.

3) Limit behaviour of the PHL at infinite frequency

A memristive device exhibits a PHL that shrinks to a single-valued function when thefrequency ω tends to infinity. This property is maintained no matter what type ofperiodic excitation is used.

2.1.2 HP memristor

The HP memristor is a node in a crossbar structure composed by a nano-scale film oftitanium oxide (TiO2) between two electrodes of platinum (Pt). The TiO2 film hastwo layers: the first one acting as an insulating layer, it has a relation oxygen-titaniumof 2 : 1, the second one acting as an conductor, it has an oxygen decrease of 0.5%(TiO2−x, x = 0.005). The mentioned structure is depicted in figure 2.6.

The total length of the element (doped region and undoped together) is given by thevariable ∆, whereas the length of the doped region is denoted by the variable w. Infunction of the number of dopants, each region has an associated resistance. Theresistance for the doped region is called Ron (resistance of the ON state of the device)and that associated with the undoped region is called Roff (resistance of the OFF stateof the device). The equivalent resistance is the sum of the total resistance in eachregion:

Req = Ronw

∆+Roff

(1− w

∆

)(2.8)

2.2. TYPES OF MODELS 9

Figure 2.6: Structure of HP memristor.

The length w can be normalized in the form x = w∆

, where x is called the state variableof the memristor and it can take a value between 0-1. The variable x can be controlledby the current i(t) that passes through the element, the ratio of change in x(t) is directlyproportional to the current

dx(t)

dt= ηκi(t) (2.9)

where κ = µvRon∆2 , µv is the mobility of the charges in the doped region and it is

measured in m2

V s, and η describes the displacement direction of x(t) (η = −1 or +1)

[16]. Equation 2.9 describes a linear drift mechanism of the HP memristor.

With the aim to model a non-linear drift mechanism, a window function fw(x(t)) isadded to the equation 2.9

dx(t)

dt= ηκi(t)fw(x(t)) (2.10)

This window function must fulfil certain characteristics:• fw(0) = fw(1) = 0 to ensure no drift in the boundaries,• fw(x(t)) is symmetric about x(t) = 1

2, and

• monotonically increasing over the interval 0 ≤ x(t) ≤ 12, 0 ≤ fw(x(t)) ≤ 1.

These properties guarantee that the difference between this model and the linear-driftmodel vanishes in the bulk of the memristor as w → ∆

2[16].

2.2 Types of modelsThe aim of modelling in electronics is to predict the behaviour of electronic circuits,this allows to evaluate a circuit without requiring its implementation [17]. The basicblocks in circuits are the devices or circuit elements, thus, modelling the circuit


implies having a model for each device.

A model seeks to meet the behavioural characteristic of a device, with some precision,in a given region of device operation. As the behaviour can be observed, this knowledgeabout the device can be used as a tool for modelling. These types of models describethe device as a black box without any internal structure. In case the internal operationof the device is known, the black box can be described, and a model that defines theinternal structure of the device can be developed. These two ways of abstracting thecharacteristics of a device are called behavioural and structural modelling respectively.

2.2.1 Behavioural models

The behavioural models are based on the measurement of the response of the device toa determined stimulus. From experimental data, characteristic curves can be generatedthat describe the device, the points generated in these curves are used to obtainmathematical functions that copy the measured behaviour.

2.2.2 Structural models

Structural models try to reproduce the behaviour of a device based on the knowledgeof the physical phenomena that describes it. That is, the internal structure of the deviceis taken into account to describe the black box that represents it. Depending on thelevel of abstraction used, structural models can be classified into physical models andanalytical models.

Physical models

In these types of models, the structure of the model consists of a series of differentialequations that describe the basic physical processes of the device. The model becomesmore specific when the unique properties of the device are taken into account, and theboundary conditions associated with the geometry of the device and its externalcontacts are included in the description.

The validity of these types of models depends on the precision with which they describethe physical phenomena that compose it.

Analytical models

Analytical models can be seen as an abstraction of physical models. In this case, ananalytical solution is found for the differential equations that describe the device, withthis solution, a mathematical expression of the behaviour of the device is obtained.

2.2. TYPES OF MODELS 11

A device can be represented by several analytical models, the precision of them dependson the number of approximations that have been made when solving the differentialequations that describe them.

2.2.3 Approach of modelling in this workIn this work, a memristor model is developed based on the differential equation thatgoverns its physical behaviour.

An analytical model is generated by starting from the approximated solution of thedifferential equation, using a series of mathematical tools. The model is presented asa series of mathematical expressions in function of the principal parameters of the HPmemristor presented in the previous section.

Chapter 3

Development of a charge-controlledmodel

With the aim of developing a symbolic memristor model that can be used for differentforms of current sources,in counter-position to the model reported in [13] which isdefined for a sinusoidal current waveform, a symbolic charge-controlled model isdeveloped in this work that can be used for different input signal waveforms, even DC.It is important to notice that an integration of the current is necessary to obtain thecharge function that control the memristance.

The methodology to obtain a symbolic expression for the memristance as a function ofthe electric charge can be described as: first, the non-linear drift mechanism isexpressed as a function of charge instead of time; then, a homotopic perturbationmethod [18], [19], [20] is used to find a symbolic solution to the non-linear equationfor the normalized state variable x(q); and finally, the x(q) is used to generate thememristance charge-controlled equation. As the method is defined for several ordersof the homotopy formulation, and the window function used in the non-linear equationalso has different exponent values, an extensive treatment of the resulting model ispresented by combining 3 different orders with 5 indices of the window function.

3.1 Non-linear drift mechanismThe non-linear drift mechanism that governs the functioning the HP memristor [2] isdefined by the ODE:

dx(t)

dt= ηκi(t)fw(x(t)) (3.1)

and the solution x(t) is used to determine the memristance:

M(t) = Ronx(t) +Roff (1− x(t)) (3.2)

13

14 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL

In order to obtain a charge-dependent memristance model, equation 3.3 is modified byexpressing the current as the time derivative of the electric charge, i(t) = dq

dt, which

yields:

dx(q)

dq= ηκfw(x(q)) (3.3)

where the state variable x is now a function of charge q, then x(q) can be obtainedsolving the differential equation and after that the memristance as a function of chargeis obtained:

M(q) = Ronx(q) +Roff (1− x(q)) (3.4)

The window function fw must be a bounded function between 0 and 1 in both itsdomain and its rank, also in the boundaries the function must exhibit a strangulation,or tends to 0, in order to model the null displacement of the ON-state resistance andOFF-state resistance interface. Several window functions have been reported in theliterature. All of them are aimed to achieve a normalization of the state variable whilepreserving the physical behaviour of the memristance. In the next paragraphs, a briefintroduction to three functions is given.

In [13] the window function is defined by the polynomial:

fw = ax5 − 2ax3 + ax (3.5)

Some plots can be seen in figure 3.1 for different values of a. It can noted that thefunction presents asymmetry with respect to 0.5 in the x axis, that is reflected in adependence of the drift direction η because the values of the function vary differentlywhen traversing the domain from 0 → 1 than from 1 → 0. Besides if a > 3.49 thenfw > 1.

Another window function, proposed by Biolek, is reported in [21]. This fw is modelledby

fw = 1− (x− stp(−i))2p

stp(i) =

1 if i ≥ 00 if i < 0

(3.6)

where i is the current that passes through the memristor. Some plots for different valuesof p in equation 3.6 are shown in figure 3.2. In this function, when p increases, thedifferential equation approximates to the linear drift case in the domain 0→ 1 of the xvariable, therefore, it can be noticed that the function models the same behaviour whenthe current is positive and negative since in both cases fw goes from one to zero.

3.1. NON-LINEAR DRIFT MECHANISM 15

Figure 3.1: Window function proposed in [13] for different values of a.

Figure 3.2: Window function proposed in [21] for different values of p.

Yogesh N Joglekar presents in [16] the window function modeled by:

fw = 1− (2x− 1)2k (3.7)

where k controls the level of linearity. Similar to the window proposed by Biolek,when k increases, the linearity increases in the range 0 → 1 for x and the function issymmetric in both directions of drift as shown in figure 3.3.


Figure 3.3: Joglekar window for different values of k.

Because the window presents a symmetric bounded function, and it is modelled with asimple algebraic expression, it is used in several works about memristor modelling.The Joglekar window is selected to generate the symbolic expression for thecharge-controlled memristor in this work.

Using the window function from 3.7 in the ODE in 3.3, yields:

dx(q)

dq= ηκ

(1− (2x(q)− 1)2k

)(3.8)

It is possible to find an analytical solution to equation 3.8 for k = 1, however for k > 1 anumerical analysis for the solution of the differential equation becomes necessary [16].In order to obtain a symbolic solution of equation 3.8, for different values of k, as afunction of the parameters of the memristor, the homotopy perturbation method (HPM)reported in [18] is studied. Is important to notice that with this method an approximate,but still analytical, solution is obtained for the differential equation.

3.2 Introduction to homotopy methodsHomotopic methods have been used in circuit theory mainly to solve the problem ofmultiple operating points in DC analysis for non-linear resistive circuits. Homotopy isbased on the fact that the solutions are connected by a so-called solution curve; toobtain this curve an extra parameter is added to the original equation system, whichconverts the static problem into a dynamic problem that contains all the solutions of

3.2. INTRODUCTION TO HOMOTOPY METHODS 17

the static problem, finding in theory all possible solutions.

In topology, if a continuous function f(x) can be deformed into another continuousfunction g(x), then the deformation is called a homotopy between f(x) and g(x) [22]and it can be represented by

H : X × [0, 1]→ YH(x, 0) = f(x) and H(x, 1) = g(x)

(3.9)

where X and Y are the topological spaces where f(x) and g(x) belong respectively.

In this work, homotopy is used to find a symbolic solution to 3.8, by resorting to acomplementary formulation with a perturbation parameter [13]. The homotopyperturbation method (HPM) relies on separating a differential equation into its linearand non-linear part

A(x)− f(r) = 0L(x) +N(x)− f(r) = 0

(3.10)

whereA(x) is a differential operator, and L(x) andN(x) the linear and non-linear partsof the original function. The solution to the linear part, the easy solution, is defined as:

L(v)− L(u0) = 0 (3.11)

Then, the homotopy can find the solution to the complete equation, starting with thesolution to the linear part

H(v, p) = (1− p) [L(v)− L(u0)] + p [L(x)−N(x)− f(r)] = 0 (3.12)

and making a sweep of parameter p from 0 to 1. When p = 0, the initial state of thehomotopy, the solution to the linear part is obtained

H(v, 0) = L(v)− L(u0) = 0 (3.13)

when p = 1 the homotopy find the solution to the original equation

H(v, 1) = L(x) +N(x)− f(r) = 0 (3.14)

The homotopic solution v can be described by a power series of p

v = p0v0 + p1v1 + p2v2 + p3v3 + ... (3.15)

Considering that p tends to 1, in the limit the solution for x (the solution to thedifferential equation) can be described as a sum of the v variables

x = limp→1

(v) = v0 + v1 + v2 + v3 + ... (3.16)


The order of the homotopy is defined by the number of vi terms that are taken toapproximate the solution x(t) in 3.16.

The methodology above is used to solve the ODE given in 3.8. The resulting homotopyformulation is given as:

H(x(q), p) = (1− p)[dx(q)

dq− C1ηκx(q)

]+ p

[dx(q)

dq− ηκfw(x(q))

]= 0 (3.17)

where C1 is a coefficient to complete the linear part of the ODE.

3.3 HPM solution to the drift equationThe homotopy in 3.17 must be solved for x(q). However, the choice of the homotopyorder and the exponent of the Joglekar window function k has be to done. Solutionsfor orders 1,2, and 3 in combination with k = 1, 2, 3, 4, 5 have been obtained, withexception of the case order-3 & k = 5 due to the massive symbolic resultingexpressions. Besides, it must be pointed out that a pair of solutions do indeed exist inevery case because η takes values of +1 and −1 depending on the direction of thecharge displacement.

As example of the x(q) solutions, the equation obtained for order-1, k = 3 and η = −1is given:

xk1,O3,η− = (X40 +X3

0 +X20 +X0)e−4κq − (3X4

0 + 2X30 +X2

0 )e−8κq

+(3X40 +X3

0 )e−12κq −X40e−16κq (3.18)

where X0 corresponds to the initial value of the state variable (when the charge iszero). It can be noted that the model only converges for positive values of q, and thefunction tends to 0 when q →∞.

The solution for η = +1 and positive values of q is given by:

xk1,O3,η+ = 1 + (−X40 + 5X3

0 − 10X20 + 10X0 − 4)e−4κq

+(3X40 − 14X3

0 + 25X20 − 20X0 + 6)e−8κq

+(−3X40 + 13X3

0 − 21X20 + 15X0 − 4)e−12κq

+(X40 − 4X3

0 + 6X20 − 4X0 + 1)e−16κq

(3.19)

In order to have a comparison point for x(q), the numerical solution for the ODE 3.8is calculated with the same values of k and homotopic orders. The numerical solutionis obtained with Backward Euler integration. In figure 3.4 a) and b), a comparisonof the model described by the equations 3.18 and 3.19 with the numerical solution isshown. A better fitting can be observed with the numerical solution for order-3 thanthe other homotopy orders. The same analysis is performed for the other x(q) solutions

3.3. HPM SOLUTION TO THE DRIFT EQUATION 19

and is shown in figures 3.4 c) and d), and figure 3.5. The values that are used for eachparameter are presented in table 3.1.

a) k = 1, η = −1 b) k = 1, η = +1

c) k = 2, η = −1 d) k = 2, η = +1

Figure 3.4: Sweep of the state variable x depending on the electric charge for the Joglekarwindow with k = 1 and 2. In red, numerical solution of the differential equation, in blue, HPMsolution of the differential equation for order 1, in violet for order 2 and cyan for order 3.


a) k = 3, η = −1 b) k = 3, η = +1

c) k = 4, η = −1 d) k = 4, η = +1

e) k = 5, η = −1 f) k = 5, η = +1

Figure 3.5: Sweep of the state variable x depending on the electric charge for the Joglekarwindow with k = 3, 4 and 5. In red, numerical solution of the differential equation, in blue,HPM solution of the differential equation for order 1, in violet for order 2 and cyan for order 3.


µvm2

V s∆ nm κ m

AsX0

1× 10−14 10 10000 0.5

Table 3.1: Parameters for the plots of x(q).

As it is shown in figures 3.4 c) and d), and figures 3.5, the equations for k = 2, 3, 4 and5 have an approximation with notorious variations. However, the approximation is stillbounded for x and presents a behaviour according to the phenomenon of displacementof the dopants.

3.3.1 Memristance equations

Once the solution x(q) has been obtained, this is substituted in equation 3.4 in order todetermine a symbolic expression of the memristance.

The expressions for the k values developed and order-1 are shown in equations3.20-3.29. The notation used to express each equation is of the form Mki,Oj,ηsign ,where index ki determines the value of k for the Joglekar window, index Oj Thehomotopy order and the sign value can be − for η = −1 or + when η = 1. The valueRd denotes Roff −Ron.

Expressions for η = −1

Mk1,O1,η− =

Rd(X0 − 1) [(X0 − 2)e4κq − (X0 − 1)e8κq] +Ron q ≤ 0

RdX0 [X0e−8κq − (X0 + 1)e−4κq] +Roff q > 0

(3.20)

Mk2,O1,η− =

Rd(X0 − 1)

13(2X3

0 + 3X0 − 8)e8κq − 3(X0 − 1)e16κq

−2(X0 − 1)2e24κq − 23(Xo− 1)3e32κq

+Ron q ≤ 0

RdX0

23X3

0e−32κq − 2X2

0e−24κq +X0e

−16κq

−13(2X3

0 − 6X20 + 9X0 + 3)e−8κq

+Roff q > 0

(3.21)


Mk3,O1,η− =

Rd(X0 − 1)

115

(16X5

0 − 20X40 + 20X3

0

+15X0 − 46

)e12κq

−5(X0 − 1)e24κq − 203 (X0 − 1)2e36κq

−203 (Xo− 1)3e48κq − 4(X0 − 1)4e60κq

−1615(X0 − 1)5e72κq

+Ron q ≤ 0

RdX0

1615X

50e−72κq − 4X4

0e−60κq + 20

3 X30e−48κq

−203 X

20e−36κq + 5X0e

−24κq

− 115

(16X5

0 − 60X40 + 100X3

0

−100X20 + 75X0 + 15)e−12κq

)

+Roff q > 0

(3.22)

Mk4,O1,η− =

Rd(X0 − 1)

1105

240X70 − 560X6

0 + 672X50

−420X40 + 210X3

0 + 105X0

−352

e16κq

−7(X0 − 1)e32κq − 14(X0 − 1)2e48κq

− 703 (Xo− 1)3e64κq − 28(X0 − 1)4e80κq

− 1125 (X0 − 1)5e96κq − 32

3 (X0 − 1)6e112κq

− 177 (X0 − 1)7e128κq

+Ron q ≤ 0

RdX0

167 X

70e

−128κq − 323 X

60e

−112κq

+ 1125 X5

0e−96κq − 28X4

0e−80κq

+ 703 X

30e

−64κq − 14X20e

−48κq

+7X0e−32κq

− 1105

240X70 − 1120X6

0 + 2352X50

−2940X40 + 2450X3

0 − 1470X20

+735X0 + 105

e−16κq

+Roff q > 0

(3.23)


Mk5,O1,η− =

Rd(X0 − 1)

1135

1792X9

0 − 6048X80

+9792X70 − 9408X6

0

+6048X50 − 2520X4

0

+840X30 + 315X0 − 1126

e20κq

−9(X0 − 1)e40κq − 24(X0 − 1)2e60κq

−56(Xo− 1)3e80κq − 5045 (X0 − 1)4e100κq

− 6725 (X0 − 1)5e120κq − 128(X0 − 1)6e140κq

− 5767 (X0 − 1)7e160κq − 32(X0 − 1)8e180κq

− 25645 (X0 − 1)9e200κq

+Ron q ≤ 0

RdX0

25645 X

90e

−200κq − 32X80e

−180κq

+ 5767 X7

0e−160κq − 128X6

0e−140κq

+ 6725 X5

0e−120κq − 504

5 X40e

−100κq

+56X30e

−80κq − 24X20e

−60κq

+9X0e−40κq

− 1135

1792X9

0 − 10080X80

+25920X70 − 40320X6

0

+42336X50 − 31752X4

0

+17640X30 − 7560X2

0

+2835X0 + 315

e−20κq

+Roff q > 0

(3.24)

Expressions for η = +1

Mk1,O1,η+ =

RdX0

[X0e

8κq − (X0 + 1)e4κq]

+Roff q ≤ 0

Rd(X0 − 1)[(X0 − 2)e−4κq − (X0 − 1)e−8κq

]+Ron q > 0

(3.25)

Mk2,O1,η+ =

RdX0

23X

30e

32κq − 2X20e

24κq +X0e16κq

− 13 (2X3

0 − 6X20 + 9X0 + 3)e8κq

+Roff q ≤ 0

Rd(X0 − 1)

13 (2X3

0 + 3X0 − 8)e−8κq − 3(X0 − 1)e−16κq

−2(X0 − 1)2e−24κq − 23 (Xo− 1)3e−32κq

+Ron q > 0

(3.26)


Mk3,O1,η+ =

RdX0

1615X

50e

72κq − 4X40e

60κq + 203 X

30e

48κq

− 203 X

20e

36κq + 5X0e24κq

− 115

(165 − 60X4

0 + 100X30

−100X20 + 75X0 + 15

)e12κq

+Roff q ≤ 0

Rd(X0 − 1)

115 (16X5

0 − 20X40 + 20X3

0 + 15X0 − 46)e−12κq

−5(X0 − 1)e−24κq − 203 (X0 − 1)2e−36κq

− 203 (Xo− 1)3e−48κq − 4(X0 − 1)4e−60κq

− 1615 (X0 − 1)5e−72κq

+Ron q > 0

(3.27)

Mk4,O1,η+ =

RdX0

167 X

70e

128κq − 323 X

60e

112κq + 1125 X5

0e96κq

−28X40e

80κq + 703 X

30e

64κq − 14X20e

48κq + 7X0e32κq

− 1105

240X70 − 1120X6

0 + 2352X50

−2940X40 + 2450X3

0 − 1470X20

+735X0 + 105

e16κq

+Roff q ≤ 0

Rd(X0 − 1)

1105

240X70 − 560X6

0 + 672X50

−420X40 + 210X3

0 + 105X0

−352

e−16κq

−7(X0 − 1)e−32κq − 14(X0 − 1)2e−48κq

− 703 (Xo− 1)3e−64κq − 28(X0 − 1)4e−80κq

− 1125 (X0 − 1)5e−96κq − 32

3 (X0 − 1)6e−112κq

− 177 (X0 − 1)7e−128κq

+Ron q > 0

(3.28)


Mk5,O1,η+ =

RdX0

25645 X

90e

200κq − 32X80e

180κq + 5767 X7

0e160κq

−128X60e

140κq + 6725 X5

0e120κq − 504

5 X40e

100κq

+56X30e

80κq − 24X20e

60κq + 9X0e40κq

− 1135

1792X9

0 − 10080X80 + 25920X7

0

−40320X60 + 42336X5

0 − 31752X40

+17640X30 − 7560X2

0 + 2835X0

+315

e20κq

+Roff q ≤ 0

Rd(X0 − 1)

1135

1792X90 − 6048X8

0 + 9792X70

−9408X60 + 6048X5

0 − 2520X40

+840X30 + 315X0 − 1126

e−20κq

−9(X0 − 1)e−40κq − 24(X0 − 1)2e−60κq

−56(Xo− 1)3e−80κq − 5045 (X0 − 1)4e−100κq

− 6725 (X0 − 1)5e−120κq − 128(X0 − 1)6e−140κq

− 5767 (X0 − 1)7e−160κq − 32(X0 − 1)8e−180κq

− 25645 (X0 − 1)9e−200κq

+Ron q > 0

(3.29)

Nested structure of the memristance expressions

Figure 3.6: Nested form of the memristanceequations for each value of k.

Memristance expressions resultingfrom the combinations of homotopyorders and k’s are developed andpresented in nested forms, i.e. a givenmemristance of order i is expressedas function of the memristance of orderi− 1 and so forth, as shown figure 3.6.

To show the nested structure of theexpressions, the memristance for k =1 in the Joglekar window and the threehomotopic orders analysed are shown inequations 3.30, 3.31 and 3.32. It can beseen that for the first homotopic order, the memristance depends on the parameters ofthe memristor and the electric charge. The structure presented in figure 3.6 can bechecked for the second and third order memristance equations.


Mk1,O1,η− =

Rd(X0 − 1) [(X0 − 2)e4κq − (X0 − 1)e8κq] +Ron q ≤ 0

RdX0 [X0e−8κq − (X0 + 1)e−4κq] +Roff q > 0

(3.30)

Mk1,O2,η− = Mk1,O1η− +Rd

(X0 − 1)3 [−e4κq + 2e8κq − e12κq] q ≤ 0

X30 [−e−12κq + 2e−8κq − e−4κq] q > 0

(3.31)

Mk1,O3,η− = Mk1,O2,η− +Rd

(X0 − 1)4 [e4κq − 3e8κq + 3e12κq − e16κq] q ≤ 0

X40 [e−16κq − 3e−12κq + 3e−8κq − e−4κq] q > 0

(3.32)

As the nested structure is preserved for each k value, table 3.2 shows the equationssummarized, where the Fij factors are principally polynomials of X0. The same tablepresent the structure for both negative η and positive η, hence the superscript of η ineach equation is ±. With the aim to reduce the space used in this document the Fijfactors are presented in Appendix A.

O 1 2 3k = 1 Mk1,O1,η± Mk1,O1,η± +RdF12± Mk1,O2,η± +RdF13±

k = 2 Mk2,O1,η± Mk2,O1,η± +RdF22± Mk2,O2,η± +RdF23±



k = 5 Mk5,O1,η± Mk5,O1,η± +RdF52± ———

Table 3.2: Nested memristance equations, structure.

Table 3.3 shows the order of the polynomial of X0 for each homotopic order and k ofmemristance equations. A gradual advance in the value of the order of the polynomialas the order of homotopy grows can be observed, thus it can be deduced the order of thepolynomial that has X0 in the development of the homotopy for the Joglekar windowsas a function of k and the homotopic order (n):

OPX0 = n(2k − 1) + 1 (3.33)


O k = 1 k = 2 k = 3 k = 4 k = 5

1 2 4 6 8 102 3 7 11 15 193 4 10 16 22 28

Table 3.3: Orders of polynomials of X0 for the Joglekar window.

Chapter 4

Characterization of the model

In this chapter, the memristor model, developed in the previous chapter, is extensivelystudied in order to determine the dependence of its main characteristics with respect tothe parameters of the device. This is achieved with the aim of establishing the mainfingerprints of the memristor. For sake of contrasting view, our model is comparedwith the memristor models reported in [2], [21], [23].

A first characterization is achieved in the dynamic domain in order to analyse the mainmemristor characteristics under a sinusoidal test signal. Secondly, a DCcharacterization is done with the purpose of determining the most important staticcharacteristics and parameters of the device, including the meristance vs chargecurves. This is used in the next chapters and applications.

The values of the HP memristor [2] are employed as nominal values in thecharacterization, as shown in table 4.1 (where Ap is the amplitude of the currentsource employed to excite the memristor).

µvm2

V s∆ nm κ m

AsRon Ω Roff Ω X0 Ap µA η

1× 10−14 10 10000 100 16× 103 0.5 40 +1

Table 4.1: Parameters used in the characterization.

4.1 Dynamic tests

In this section, some characteristics of the model under sinusoidal test sources arepresented. First, fingerprints associated to the hysteresis loop are evaluated provingthat the model fulfils them. Then, a comparison with the results reported in [2],Biolek’s model [21] and Affan’s model [23] are performed.

29

30 CHAPTER 4. CHARACTERIZATION OF THE MODEL

4.1.1 FingerprintsThe principal features for memristor are related with the hysteresis loop in the current-voltage characteristics, as was mentioned in Chapter 2, mainly a pinched hysteresisloop (PHL), area lobes decreases when frequency increases, and constant memristancewhen the frequency tends to infinity.

Pinched hysteresis loop (PHL)

In figure 4.1, the hysteresis loop for the memristor model developed is observed. Thecurve for four different amplitudes of excitation current signal and the same frequencyω = 1 for all cases is presented, proving that the characteristic loop of a memristor withpassage through the origin is maintained regardless of the amplitude of the input signal.The PHL is presented for two values of k in the Joglekar window; in the case of figure4.1 b) (k = 5), the saturation point is reached in the memristance since the greater thevalue of k the greater the opening in the lobes, that is why the lobes are deformed whenthe curve approaches the current axis, reaching in this case the minimum memristanceRon = 100 Ω.

a) b)

Figure 4.1: Variation of the hysteresis loop with respect to the amplitude of the input signal. a)Hysteresis loop for a Joglekar window with k = 1. b) Hysteresis loop for a Joglekar windowwith k = 5.

Reduction of the lobe area as the frequency increases

From a critical frequency, the area in the hysteresis loop must decreases monotonicallywith respect to the increases in the frequency of the harmonic input signal [15]. In thisway, as the excitation frequency increases, from a frequency depending on the

4.1. DYNAMIC TESTS 31

characteristics of the device, the hysteresis loop closes continuously.

In figure 4.2, the hysteresis loop is shown for two values of k in the Joglekar windowand four different values of ω in each case. It can be observed how the area lobesdecreases as ω gets larger, however, in the case of k = 5 the area for ω = 1 and 2 cannot be distinguished, in figure 4.2 b), as greater to the area that present the lobes forhigher frequencies; so, figure 4.3 shows a curve of the value of the area as a function ofthe frequency for the same cases of the Joglekar exponent. For a critical frequency, thelobe area decreases as frequency increases, on the contrary, for values smaller than thisfrequency the area increases as frequency increases, this is due to the closeness to thesaturation points in the memristance.

a) b)

Figure 4.2: Variation of the hysteresis loop respect to frequency. a) Joglekar exponent k = 1.b) Joglekar exponent k = 5.

As was shown in figure 4.3, the critical frequency depend of the k Joglekar exponent,in general this frequency increases with k as can be observed in figure 4.4 and table 4.2.

k 1 2 3 4 5ωc 0.947 1.252 1.656 2.013 2.828

Table 4.2: Critical frequencies for different values of k Joglekar exponent.


a) b)

Figure 4.3: Area for the lobe of the hysteresis loop as a function of the frequency, the units ofarea are µm2. a) k = 1, ωc = 0.947 rad

s . b) k = 5, ωc = 2.828.

Figure 4.4: Frequency dependence of the area for different values of Joglekar exponent k.

Limit behaviour of the PHL at infinite frequency

When the excitation frequency tends to infinity, the value of the memristance becomesconstant and the device acts as a linear resistor [15]. As reviewed in the previoussection, as the frequency increases the area in the hysteresis cycle of the memristordecreases, so when the frequency tends to infinity the area is zero, making thememristance in the device constant.

4.1. DYNAMIC TESTS 33

Calculating the limit when the frequency ω →∞ for each memristance equation, it beobtained:

limω→∞

(Mki,Oj) = X0Ron + (1−X0)Roff = Rinit (4.1)

where ki refers to the five k values in the Joglekar window used in the model. Thelimit corresponds to the value of the initial resistance in the device, this is because thestate variable x, which models the position of the doping barrier in the device, can notrespond to such rapid changes in the input signal.

4.1.2 Comparison with other modelsIn order to have a reference with other models, comparisons with the models reportedin [2] (HP), [21] (Biolek) and [23] (Affan) are presented hereafter. For sake ofcomparison, two models are used, namely Mk1,O3 and Mk5,O2.

HP memristor

As can be observed in figure 4.5, both the charge-controlled model (with k = 1) andthe HP results have a similar hysteresis loop but with a difference in the amplitude ofthe current response. The amplitude the charge-controlled model reaches is slightlyhigher than the HP results, where the current and voltage are normalized by a factor of10 mA and 1 V respectively.

a) b) extracted from [2]

c) d) extracted from [2]

Figure 4.5: Comparison of the PHL between k = 1, homotopic order 3 of the model developed,in a), and HP model results, in b). Its response in current for a sinusoidal voltage excitation arepresented in c) and d) respectively.


In figure 4.6, the hysteresis loop and current response of the model using k = 5 inthe Joglekar exponent is presented. It can be seen that the amplitude of the source isenough for the model to reach the ON-state (M(t) = Ron), thus it causes a currentamplitude larger than the model with k = 1.

a) b)

Figure 4.6: PHL for k = 5 homotopic order 2 of the model developed, in a), and its response incurrent for a sinusoidal voltage excitation, in b).

Biolek’s and Affan’s models

As mentioned in Chapter 2, there are in the literature memristor models developed tobe used in different applications. In figure 4.7, the hysteresis loop for Biolek’s model, amacro model implemented in the SPICE platform, and Affan’s model, a mathematicalmodel implemented in MATLAB, are shown and compared with the charge-controlledmodel. The charge-controlled model exhibits a slightly larger current swing that theother models.

−1 0 1−200

−100

0

100

200

V(t) V

I(t)

µA

Charge−controlledBiolekAffan

Figure 4.7: Comparison of the PHL between: k = 1, homotopic order 3 of the model developed(in blue), Biolek’s model (in red) and Affan’s model (in green).

Additionally, in Appendix B a more extensive characterization of the model for

4.2. DC TESTS 35

sinusoidal sources is made. Curves of the memristance-current characteristic,hysteresis loop and passivity are shown for the models with k = 1 and k = 5,sweeping the parameters Ap, Ron and X0. With the help of the passivity curves, it isverified that the model at any moment stops behaving as a passive element for all theranges in which the parameters were varied. This is because it takes into account thesaturation points in memristance, Ron and Roff .

4.2 DC tests

The main features in the DC domain are the voltage thresholds and the saturation timetsat. The voltage necessary to switch, for η = +1, from the OFF-state (Roff ) to theON-state (Ron) is the positive voltage threshold V th+, and the voltage necessary toswitch from the ON-state to the OFF-state is the negative voltage threshold V th−(when η = −1 the switching voltage thresholds are contrary). The saturation time isthe time that the element needs to switch between states.

4.2.1 Determining the switching voltages

In order to find V th± for the models with k = 1 and k = 5 the next steps are applied:• The model is set in the OFF-state (X0 = 0.001, corresponding to an initial

resistance Rinit = 159.841 KΩ).• A voltage source as shown figures 4.8 a) and b) is applied. The description of the

source is given hereafter:– First, an increasing voltage is applied.– When the model passes the V th+ and reaches the ON-state, the voltage

source is stopped to prove that the reached state is maintained.– Then, a decreasing voltage source is applied until the model passes theV th− and reaches the OFF-state.

– Finally, again the voltage source is stopped.

In figure 4.8 c) and d), the current-voltage characteristic for the two models studied areshown. It can be seen that for both, the voltage thresholds are symmetric and measure1.35 V and 0.61 V for k = 1 and k = 5 respectively, consequently a voltage sourcewith these values can switch the initial state of the charge-controlled memristor modelsdeveloped as it is shown in table 4.3. But, what happens if the initial state is OFF and anegative voltage is applied? Or similarly, if the initial state is ON and a positive voltageis applied?


Vth k = 1 k = 5OFF to ON 1.33 V 0.64 VON to OFF −1.33 V −0.64 V

Table 4.3: Switching voltages for k 1 and 5 in the model developed.

As is shown in figure 4.9 a), if the initial state is OFF for the model and an inverseversion of the voltage source is applied, the dynamic of memristance in the thirdquadrant is null because the element can’t change of state. The same happens in theanother case, when the model starts in the ON-state and a positive voltage is applied(presented in figure 4.9 b)), but with the difference that now the non dynamicmemristance is present in the first quadrant. It is important to notice that thesecharacteristics are shown for a model with η = 1, when η = −1 all the issues are theopposite.

a) Mk1,O3 b) Mk5,O2

c) Mk1,O3 d) Mk5,O2

Figure 4.8: c) and d) voltage source in blue, current response of each model in green. c) and d),Current-Voltage characteristics, applying the voltage source shown in a) and b) respectively.

4.2. DC TESTS 37

a) b)

Figure 4.9: Current-Voltage characteristic for the model applying inverted voltage sources. a)Initial state: OFF. b) Initial state: ON.

Charge-Flux characteristic

In addition, the Charge-Flux characteristic is shown in figure 4.10 using the voltagesource present in figure 4.8 a). In the plots, two states of the memristor are easilyidentifiable, and the slope measured in each region corresponds to the inverse of theresistance in the OFF-ON states ( 1

Roff= 6.25 µS and 1

Ron= 1 mS). These simulation

measurements show the direct relationship that the model maintains between electriccharge and magnetic flux. The q-φ characteristic can be easily modelled by a piece-wise-linear equation, and Ron-Roff constitute the principal parameters.

qk1,O3,η+ = −4.22344 + 5.03125φ+ 4.96875|φ− 0.85|mC

qk5,O2,η+ = −1.01859 + 5.03125φ+ 4.96875|φ− 0.205|mC(4.2)

a) qk1,O3,η+ b) qk5,O2,η+

Figure 4.10: Charge-Flux characteristic for the memristor models.


4.2.2 Determining the saturation timeWith the threshold voltage obtained in the previous section, the next step is to apply aVDC = V th to obtain the saturation time tsatOFF−ON , the time that the memristor needsto change from the OFF-state to the ON-state. In order to obtain a theoretical value oftsatOFF−ON , Strukov proposes in [2] the equation:

tsatOFF−ON ≈∆2Roff

2µV+Ron

(4.3)

where V+ is the value of the DC voltage source applied. Using the parameters presentedin table 4.1 and the threshold voltages in table 4.3, the saturation time for both modelsare:

tsatOFF−ON

∣∣V+=1.33

≈ 0.601 s; tsatOFF−ON

∣∣V+=0.64

≈ 1.25 s (4.4)

Figure 4.11 shows the saturation time measured in a simulation of the model, for thecase of k = 1, the value obtained is 0.62 s, closer to the theoretical value, but in thecase of k = 5, the measured value is 0.33 s. The error in the last case may be due to theabrupt changes that the high values of k cause, generating a fast response of the model;but also, on the another side, the equation 4.3 is modelled for the lineal range betweenRon and Roff .

a) b)

c) d)

Figure 4.11: Saturation time measured in the memristance curve (a and b for k = 1 and k = 5)and in the current response (c and b for k = 1 and k = 5 respectively).

4.2. DC TESTS 39

4.2.3 Memristance-Charge characteristicIn order to know the behaviour of the charge-controlled memristor model as a functionof the electric charge, a charge sweep is carried out by applying a DC current source(positive and then negative).

a) Mk1,O3,η− b) Mk1,O3,η+

c) Mk5,O2,η− d) Mk5,O2,η+

Figure 4.12: Memristance-Charge characteristic for the model developed.

Figure 4.12 shows the M-q characteristic of the model for the two values of η. It can beseen that for η = −1 the memristance tends to Roff in the positive range of the charge,and tends to Ron in the negative range of q. In contrary, when η = +1, the memristancetends to Ron in the positive range of the charge, and to Roff in the negative range. Asa function of k, the curves with high k reach first the saturation values of memristance.


4.2.4 M-q Characteristics of memristor connectionsIn order to observe the behaviour of the serial and parallel connections of the memristor,an analysis of the M-q characteristic is then carried out for each type of connection.Because the memristor is a device that has polarity, there are 4 types of connections:series, anti-series, parallel and anti-parallel.

Series connection

The series connection of two memristors, is a circuit configuration with two memristorsconnected in the same polarity (or memristor with the same η). In this case, the M-qcharacteristic for each η, shown in figure 4.12, is accentuated, as shown in figure 4.13.The values of Ron, Roff and Rinit in the connection are:

Rons = Ron1 +Ron2

Roffs = Roff1 +Roff2

Rinits = Rinit1 +Rinit2

(4.5)


Figure 4.13: M-q characteristic for a series connection of two memristors.

Anti-series connection

Figures 4.14 a) and b) show the meristance response for the model with positive andnegative η, starting in ON-state (X0 = 0.999, Rinit = 1159 Ω). Each type of memristorhas a suitable behaviour for a type of excitation, in the case of positive η, it is possible tochange the state by applying a negative current (voltage) source, in contrast, in case ofnegative η, the change of the state is achieved by applying a positive current (voltage)source. When it is necessary a memristive element that can pass from the ON-state

4.2. DC TESTS 41

to the OFF-state, applying a source regardless of its sign, a series connection of eachtype of memristor can be used as can be seen in figure 4.14 c) and d). If the memritiveelements are of the same type, only inverting the polarity of one of the elements, thesame results are obtained, this element is called anti-series connection. The anti-seriesconnection between two memristors has a initial resistance

Rinitas = R+init +R−init (4.6)

where the superscript indicates the sign of η, and the resistance of the of state is given:

Roffas,q−= R−on +R+

off ; Roffas,q+= R+

on +R−off (4.7)

for negative and positive electric charge.

a) Mk1,O3 b) Mk5,O2

c) Mk1,O3 d) Mk5,O2

Figure 4.14: Memristance-Charge characteristic, starting in ON-state, for positive and negativeη. In c) and d), the M-q characteristic for the anti-series connection.

In a comparison for the value of k in the anti-series connection, high values of k implya smaller amount of electric charge to reach the OFF-state.


Parallel connection

The parallel connection of two memristors in the same polarity (or same η), reduce theranges in memristance of the resulting M-q characteristic. This is because theequivalent of parameters Ron, Roff and Rinit for the connection are the parallelbetween the individual memristor parameters:

Ronp = Ron1 ‖ Ron2

Roffp = Roff1 ‖ Roff2

Rinitp = Rinit1 ‖ Rinit2

(4.8)

The plots for the parallel connection are shown in figure 4.15


Figure 4.15: M-q characteristic for a parallel connection of two memristors.

Anti-parallel connection

The anti-series connection can be used for a element that passes from the ON-state tothe OFF-state, but also when it is necessary that it passes from the OFF-state to the ON-state another connection can be used. In this case, each type of memristor starts in theOFF-state. For η = −1, switching to the ON-state is possible by applying a negativeDC current (voltage) source and to the contrary, for η = +1, switching to the ON-stateis possible by applying a positive current (voltage) source, as shown in figures 4.16 a)and b). The M-q characteristic for an anti-parallel memristor connection is shown infigures 4.16 c) and d). For this connection the equivalent resistance parameters are:

Rinitap = R+init ‖ R−init

Ronap,q−= R−on ‖ R+

off ; Ronap,q+= R+

on ‖ R−off(4.9)

4.2. DC TESTS 43

When both memristors are connected in anti-parallel, theRinit equivalent is the parallelresistance between the initial resistances of each type of memristor, and the Ron is R−onfor negative charge and R+

on for positive charge.

a) Mk1,O3 b) Mk5,O2

c) Mk1,O3 d) Mk5,O2

Figure 4.16: a) and b), Memristance-Charge characteristic, starting in ON-state, for positiveand negative η, in c) and d), the M-q characteristic for an anti-parallel connection.

In a comparison for the value of k in the anti-parallel connection, high values of k implya smaller amount of electric charge to reach the ON-state.

Chapter 5

Memristive grid for edge detection

Among the basic types of image signal processing, edge detection constitutes afundamental tool that a reliable image preprocessing algorithm for machine-based orcomputer-based vision has to include. In fact, edge detection can be regarded asdecomposing the original image into a set of topographical curves that are linked to ameasured depth level of intensity. As a clear result, the image contains diminishedinformation which makes it suitable for further and faster treatments.

Non-linear resistive grids have been used in the past for achieving image filtering,focused on smoothing and edge detection by resorting to the non-linear constitutivebranch relationships of the elements in the array in order to carry out in fact aminimization algorithm.

In this chapter a brief description of the advances in resistive and non-linear resistivegrids is presented, then the memristive grid and its components are explained and thesolution to the memristive grid is expounded. Furthermore, using an image databaseextracted from [24], the memristive grid is tested for 500 images and the same imageswith noise. Finally a comparison with the well-known Canny’s method [25] ispresented.

5.1 Previous approaches

As it was mentioned above, edge detection is an important step in image processing.Therefore, it becomes relevant to establish what an edge actually represents in animage. In plain words, an edge is considered when the brightness changes sharply orwhen the image presents physical discontinuities [26].

There are several methods in the literature for image edge detection, they can begrouped in two categories, search-based and zero-crossing-based. In search-basedmethods, edges are detected by first computing a measure of the edge strength as the

45

46 CHAPTER 5. MEMRISTIVE GRID FOR EDGE DETECTION

gradient magnitude and then searching for the local directional maxima for thegradient magnitude. The zero-crossing-based methods search for zero crossings in asecond-order derivative expression computed from the image in order to find edges.As a pre-processing step to edge detection, a smoothing filter, typically Gaussiansmoothing, is widely applied; as a clear consequence, the edge detection methodsdiffer in function of the smoothing filter used [26].

In 1986 John Canny proposed a computational method for image edge detection. Heintroduced the notion of non-maximum suppression, which means that given thepre-smoothing filters, edge points are defined as points where the gradient magnitudeassumes a local maximum in the gradient direction [25]. Although the method wasdeveloped in the beginnings of computer vision, the Canny Edge detector is still in thestate of art.

In 1987 Pietro Perona and Jitendra Malik introduced for the first time the concept ofanisotropic diffusion, also called Perona-Malik diffusion. Anisotropic diffusion is atechnique aiming at reducing the image noise without removing significant parts of theimage content, typically edges, lines or other details that are important for theinterpretation of the image [27]. Based on this technique, through the history ofcomputer vision, many methods for edge detection have been proposed, such as themethod based on non-linear anisotropic diffusion, which consists of considering theoriginal image as an initial state of a parabolic (diffusion-like) process, and extractingfiltered versions from its temporal evolution [28].

Non-linear anisotropic diffusion is a natural phenomenon in non-linear resistive grids.In [29] “the first hardware circuit that explicitly implements either analog or binaryline processes in a controlled fashion” is presented, using a resistive grid as aprocessor for detecting discontinuities in images. The resistive grid and the elementsof this processor are presented in figure 5.1 a), the input voltage sources represent eachpixel of the image to be processed and the output nodes represent each pixel of theimage processed. It is important to note that each branch in the grid is composed of anon-linear resistive element called fuse.

Other applications of the resistive grid as an image filter are presented in [30], butfocused on the smoothing and introducing important figures of merit, such as therelation between the smoothing level L and the branch resistance in the grid, Rbranch.The space constant L measures the smoothing as a number of pixels:

λ = RbranchRin

; ς = cosh−1(1 + λ

2

); L = 1

ς; (5.1)

5.2. DESCRIPTION OF THE MEMRISTIVE BRANCH 47

a) b)

Figure 5.1: Structure and components of the: a) resistive grid, b) memristive grid. In boxes:violet, input pixel, blue, Rin, green, fuse and red, output pixel.

The non-linear resistive grid presents good performance in image smoothing and edgedetection [29] [30]. However, with the development of nano-scale memristors [2],replacing the non-linear resistor in the branches of the grid by memristors yields anetwork that performs a similar discontinuity-preserving image smoothing, but withsignificant advantages, in particular, edges extracted from the outputs of the non-linearmemristive grid more closely match the results of human segmentation [31].Additionally, the dimensions of memristors allow more density of integration in VLSIsystems than resistors. The memristive grid used for image edge detection is shown infigure 5.1 b).

Some additional considerations must be taken into account for processing images witha memristive grid, due to the fact that the memristive grid implements a non-linearanisotropic method. Namely, the method needs a stop criteria to find a solution, sincethey mostly do not converge [32]; the method has good performance in gray-scaleimages with respect to smoothing and edge detection, whereas in color images it givesimproved segmentation results [28]. Because of this, the images processed with thememristive grid, and the model developed in this work, are in gray-scale and a thresholdto stop the process is selected.

5.2 Description of the memristive branchThe memristive fuses of the grid given in figure 5.1 b) must fulfil severalrequirements. First, the memristive fuse is composed by the anti-series connection of


2 memristors. An anti-series connection is defined as the series connection of 2memristors connected back-to-back, as shown in figure 5.2 a). The combined M-qcharacteristic of the memristive fuse has the shape depicted in figure 5.2 b). Ideally,the ON-state memristance is zero and the slope from the ON-state to the OFF-statearound Qt is infinite. In practice, Mon has a very low value, and Moff takes a veryhigh value. It can be said that the larger the values of Qt, the more smoothing on theimage and the longer settling times in the edge detection. The Mth related with Qt isthe memristance threshold; this memristance is selected to define which pixel isidentified as an edge of the original image.

Making a comparison with the M -q characteristic presented in figure 4.14, in theprevious chapter, the anti-series connection of two charge-controlled memristorsfulfils the characteristic behaviour of the required fuse element for the memristivegrid. By using the parameters in table 5.1, figure 5.2 c) is obtained as the fuse elementused in the next simulations, with a Joglekar exponent k = 5 to obtain the smallersaturation times. The Mon, Moff and Rin parameters also are presented in table 5.1.

Parm. µvm2

V s Ron Ω ∆ nm Rinit Ω X0 Roff Ω Mon Ω Moff Ω Rin ΩValue 1× 10−14 1 10 1.1 0.999 1100 2.2 1101 50

Table 5.1: Nominal parameter values.

a)

b) c)

Figure 5.2: Branch element, fuse. a) Anti-series connection of memristors. b) IdealMemristance-Charge characteristics. c) Memristance-Charge characteristics of the modeldeveloped.

5.3. SOLUTION OF THE MEMRISTIVE GRID 49

5.3 Solution of the memristive gridThe functional simulation of the memristive grid is done with MATLAB byimplementing the differential equations describing the circuit as a result of the KCLanalysis. The size of the grid is determined by the number of pixels the image has,each node in the grid is associated with a particular pixel, and the voltage value thateach pixel can represents vary between 0→ 1 V .

Figure 5.3: Current contributions at node Ni,j .

From figure 5.3, KCL analysis of the output node Ni,j yields:

Iin + Ii−1,j + Ii+1,j + Ii,j−1 + Ii,j+1 = 0 (5.2)

This can be established as:

di,j − ui,jRin

+ui−1,j − ui,jMi−1,j

+ui+1,j − ui,jMi+1,j

+ui,j−1 − ui,jMi,j−1

+ui,j+1 − ui,jMi,j+1

= 0 (5.3)

where Mi−1,j,Mi+1,j,Mi,j−1,Mi,j+1 are the memristances incident to the node. Theoutput voltage ui,j is defined as:

ui,j =di,jGin + ui−1,jWi−1,j + ui+1,jWi+1,j + ui,j−1Wi,j−1 + ui,j+1Wi,j+1

Gin +Wi−1,j +Wi+1,j +Wi,j−1 +Wi,j+1

(5.4)

where Gin and W are the inverse of Rin and M , respectively.


The composition of the nodal equations in the boundaries, i.e.in the edges and cornersof the grid, is much simpler due to the fewer number of incident branches. The nodalequations at the corners are given as:

N1,1 : u1,1 = d1,1Gin+u2,1W2,1+u1,2W1,2

Gin+W2,1+Wq1,2

N1,n : u1,n = d1,nGin+u2,nW2,n+u1,n−1W1,n−1

Gin+W2,n+Wq1,n−1

Nm,1 : um,1 = dm,1Gin+um−1,1Wm−1,1+um,2Wm,2

Gin+Wm−1,1+Wm,2

Nm,n : um,n = dm,nGin+um−1,nWm−1,n+um,n−1Wm,n−1

Gin+Wm−1,n+Wm,n−1

(5.5)

The equations at the edges are:

N1,j : u1,j =d1,jGin+u2,jW2,j+u1,j−1W1,j−1+u1,j+1W1,j+1

Gin+W2,j+W1,j−1+W1,j+1

Nm,j : um,j =dm,jGin+um−1,jWm−1,j+um,j−1Wm,j−1+um,j+1Wm,j+1

Gin+Wm−1,j+Wm,j−1+Wm,j+1

Ni,1 : ui,1 =di,1Gin+ui−1,1Wi−1,1+ui+1,1Wi+1,1+ui,2Wi,2

Gin+Wi−1,1+Wi+1,1+Wi,2

Ni,n : ui,n =di,nGin+ui−1,nWi−1,n+ui+1,nWi+1,n+ui,n−1Wi,n−1

Gin+Wi−1,n+Wi+1,n+Wi,n−1

(5.6)

For the whole grid, the resulting KCL analysis yields m × n equations as aboveaccording to the number of pixels of the image. This set of equations is solved for thenodal voltages ui,j and the currents of the memristors are calculated. In a ulterior step,the controlling charges of the memristors are calculated by using a standard numericalintegration algorithm, namely the trapezoidal rule. With the calculated values for theelectric charge, the memristance value is updated by substituting in the memristanceequation.

As mentioned before, in equation 5.1, the level of smoothing depends on the rateMbranch

Rin, i.e. the equivalent of each memristance arriving to the node Ni,j divided by

the input resistance. The initial conditions of the memristive grid yield Mon

Rin= 0.044

which is associated to an L = 4.78 pixels.

As time lapses, the value of Mbranch increases, causing the smoothing to decrease, andthe output image approaches the original image. Therefore, it becomes necessary tostop the smoothing procedure by defining a stop criterion. This criterion is thesmoothing time tsmooth, since it defines when the smoothing level of the output imageis reached. When the process is stopped, in the output pixels a smoothed version of theoriginal image is obtained, and the fuses that reach Mth are selected to determine theedge image.

5.4. RESULTS AND COMPARISONS 51

Each pixel has four adjacent fuses, except the boundary pixels. In order to determineif the pixel belongs to the edge image, the memristance of the right and bottom fuses(Mpi,j andMqi,j respectively in figure 5.3) must be evaluated; then, if any of either fusesreaches Mth the pixel is selected as an edge and included in a binary output image,which is considered the detected edge image. This mechanism is better explained infigure 5.4, where the red branches have a memristance greater than Mth. For the caseof the memristive grid, this threshold must be referred as a fraction of the maximumvalue of the memristance. A percentage of 2 % of Moff has been used, allowing edgesto be detected when the output image still retains a high level of smoothing. As a result,edge detection can be efficiently performed even for images with high levels of noise.

a) b) c)

Figure 5.4: Mechanism to determine the edges in an image of 9× 9 pixels. a) Fuses that reachthe threshold memristance Mth, red. b) Pixels selected as an edge, green. c) Output image.

5.4 Results and comparisons

A benchmark image and its edges drawn by 5 test subjects (human beings) arepresented in figure 5.5 (extracted from the database BSD300 [33]). This image and itsedges are used to evaluate the performance of the memristive grid and to select thebest parameters of Mth, simulation step and integration method for the charge. Theseresults can be observed in table 5.2. After establishing a memristance threshold, theperformance of the grid is evaluated for 500 images (extracted form the databaseBSD500 [24]) and then, the same evaluation is done on the noisy version of theimages with added Gaussian noise with mean 0 and variance 0.01.


a) b)

Figure 5.5: a) Benchmark image. b) Ground-truth for the edges of the benchmark image.

Results of the smoothing are shown in figure 5.6 for several transient values. It can beobserved that as the time increases the smoothing level decreases.

a) t = 1 ms b) t = 5 ms

c) t = 10 ms d) t = 20 ms

Figure 5.6: Output image.

5.4.1 Figures of merit for the edge detection procedureWith the aim of evaluating the efficiency of the memristive grid for the edge-detectionprocedure, we resort to the precision-recall curve and the parameter F as reported in


[24]. The precision (P ) is given as:

P =TP

TP + FP(5.7)

where TP is the number of pixels that belong to the evaluated edge as well as to thereference edge (true positives), and FP is the number of pixels that belong to theevaluated edge but not to the reference edge (false positives). In fact P denotes theprobability related to the quality of the detector.

The recall parameter is defined as:

R =TPTB

(5.8)

where TB is the total number of pixels that belong to the edge in the reference image.Actually the recall-factor indicates the probability for an edge to be detected.

Finally, another commonly used parameter is F :

F =PR

βP + (1− β)R(5.9)

It is the precision-recall cost ratio for a given complementary balance constant β ∈0→ 1. In order to have the same weight for precision and recall, the parameter β usedis 0.5.

5.4.2 Results for a benchmark imageThe edge-detected final output image with the memristive grid is shown in Figure 5.7a). It is compared with the output image from Canny’s method in Figure 5.7 b).

a) b)

Figure 5.7: Edge detection: a) Memristive Grid at t = 20.45 ms , b) using Canny’s method [25]for a threshold 0.422.


Figure 5.8 shows the precision-recall (P -R) curves for the memristive grid and foralgorithm reported in [25]. The maximum F for the memristive grid, is 0.76, wasobtained in a tsmooth = 20.45 ms, while for Canny’s method, the maximum is 0.59 fora threshold of 0.422. In this case, the smoothing time is measured when the maximumof F parameter is reached, and this is the stop criteria of the method; however, whenthere is not ground-truth to compare the detected edge, the stop criterion must betsmooth.

The human average F (for the 5 test subjects) is found to be 0.80, as reported in [24].Therefore, the outcomes of the memristive grid exhibit an excellent agreement withoutcomes made by humans. In all P -R plots, the dotted lines correspond to thecomparison with 5 ground truth databases while the black line represents their mean.Furthermore the green points correspond to the cross-comparisons of the ground truthsubjects, which are close to the human average as reported in [24].

0 0.2 0.4 0.6 0.8 1

recall

0

0.2

0.4

0.6

0.8

1

prec

isio

n

Fmax = 0.75651 t = 20.46 msFmax = 0.75651 t = 20.46 msFmax = 0.75651 t = 20.46 msFmax = 0.75651 t = 20.46 msFmax = 0.77569 t = 20.39 msFmean = 0.76035 t = 20.446 msFh mean = 0.79636

0 0.2 0.4 0.6 0.8 1

recall

0

0.2

0.4

0.6

0.8

1

prec

isio

n

Fmax = 0.57513 th = 0.44Fmax = 0.59548 th = 0.44Fmax = 0.60066 th = 0.44Fmax = 0.5961 th = 0.41Fmax = 0.62154 th = 0.38Fmean = 0.59778 th = 0.422Fh mean = 0.79636

a) b)

Figure 5.8: Precision-recall plots: a) using the memristive grid, b) using Canny’s method [25].

Since the charge update of the memristors in the grid is obtained via numericalintegration of the current, two well-known integration methods were implemented,namely the Backward-Euler and the Trapezoidal Rule methods. The implementationwas aimed at determining the effect that the choice on the integration procedure couldhave on the figures of merit and overall behaviour of the image processing. Table 5.2summarizes the experiments for the first benchmark image. In general, small stepsizes yield small smoothing times and better Fmean. The best Fmean’s are obtained forthe smaller threshold percentages. At the smallest step size, the selection of theintegration method has a negligible impact.


hµs

Backward Euler Trapezoidal TFmsFmean tsmooth ms tsim s Fmean tsmooth ms tsim s

1000 0.67114 54.40 41.6764 0.66531 54.20 42.1378 100100 0.75086 26.42 122.6280 0.75035 26.36 123.7440 3010 0.76033 20.45 1254.0847 0.76035 20.45 1248.5861 30

a) Mth as 2% of Moff .hµs

Backward Euler Trapezoidal TFsFmean tsmooth ms tsim s Fmean tsmooth ms tsim s

1000 0.65868 213.60 1386.3426 0.65826 209.40 1383.6017 3100 0.69418 174.20 2226.3456 0.69403 174.78 3851.7313 0.0510 0.69758 171.90 8730.4390 0.69756 172.78 8739.7100 0.02

b) Mth as 10% of Moff .hµs

Backward Euler Trapezoidal TFsFmean tsmooth ms tsim s Fmean tsmooth ms tsim s

1000 0.58553 1712.60 1386.3426 0.58556 1683.00 1383.6017 3100 0.59075 1671.76 12841.1207 0.59076 1672.10 12839.6735 210 0.60572 1624.65 88654.7841 0.60572 1624.13 88642.6543 2

c) Mth as 50% of Moff .

Table 5.2: Average precision-recall (Fmean), smoothing time and total simulation time forbackward Euler and trapezoidal integration methods for different transient values.

Processing the noisy image

In order to evaluate the performance of the memristive grid in edge detection for imageswith noise, Gaussian noise is added to the benchmark image depicted in figure 5.5. Thenoisy image (figure 5.9) is processed with the grid and the Canny’s method; the edgesdetected are shown in figure 5.10 a) and b) respectively.

Figure 5.9: Benchmark image with Gaussian noise.


a) b)

Figure 5.10: Edge detection for the benchmark image with noise: a) Memristive Grid at t =19.65 ms , b) using Canny’s method [25] for a threshold 0.443.

Figure 5.11 shows P -R curves for the memristive grid and for Canny’s method. Themaximum F for the memristive grid, is 0.75, was obtained in a tsmooth = 19.87 ms,while for Canny’s method, the maximum is 0.59 for a threshold of 0.414. For the imageevaluated, the F measure doesn’t show a significant difference between the noisy andoriginal image.

0 0.2 0.4 0.6 0.8 1

recall

0

0.2

0.4

0.6

0.8

1

prec

isio

n

Fmax = 0.74519 t = 19.89 msFmax = 0.74519 t = 19.89 msFmax = 0.74519 t = 19.89 msFmax = 0.74519 t = 19.89 msFmax = 0.76232 t = 19.78 msFmean = 0.74862 t = 19.868 msFh mean = 0.79636

0 0.2 0.4 0.6 0.8 1

recall

0

0.2

0.4

0.6

0.8

1

prec

isio

n

Fmax = 0.57522 th = 0.43Fmax = 0.59184 th = 0.42Fmax = 0.60003 th = 0.41Fmax = 0.58288 th = 0.4Fmax = 0.61208 th = 0.41Fmean = 0.59241 th = 0.414Fh mean = 0.79636

a) b)

Figure 5.11: Precision-recall plots for the benchmark image with noise: a) using the memristivegrid, b) using Canny’s method [25].

5.4.3 Comparative results on a set of 500 imagesIn order to determine the performance of our memristive grid in a massive task and tosee how its performance is when compared with Canny’s method, a series of testswere carried out on 500 images extracted from [24]. The conditions for these testswere set for a step size of 10 µs, a memristance threshold of 2% of Moff and thetrapezoidal method for the integration of the electric charge.


In figures 5.12 a), b) and c) the histogram of F is shown for the memristive grid results(FMG), the Canny’s method (FC) and the human results (FH) respectively. Also, thehistogram for tsmooth is presented in figure 5.12 d).

As a a way of summarizing the results, it can be noticed that the memristive gridproduces 149 images with the average F , while Canny’s method produces 174. But itshould also be noted that these average images are obtained with better F with thememristive grid. This trend is repeated for the images with the best F ’s. In addition,the human F results from the database show a less spanned distribution centred in theclass 0.6-0.7 for nearly 300 images.

a) b)

c) d)

Figure 5.12: Histogram for the F measures and smoothing time for several images. a) FMG forthe memristive grid. b) FC for the Canny’s method. c) FH for the ground-truth. d) tsmooth forthe memristive grid.

Images with noise

The results of the image edge detection methods compared for the images with noiseare shown in figure 5.13. In this case, a Gaussian noise with mean 0 and variance 0.01


is added to the input images.

a) b)

c) d)

Figure 5.13: Histogram for the F measures and smoothing time for several images withGaussian noise. a) FMG for the memristive grid. b) FC for the Canny’s method. c) FH forthe ground-truth. d) tsmooth for the memristive grid.

The results have a similar behaviour as for the clean images, but show a displacementto the left in the average.

Comparisons

A comparison of the mean in the F measures, the variance σ2 and standard deviationσ is presented in table 5.3, also comparing the results for the images and the sameimages with noise. It can be observed that in both cases the memristive grid has abetter performance than Canny’s method and a good performance comparing with thehuman results. In the cases of the images with and without noise, tsmooth has a standarddeviation around 6 ms; however, because the associated values of the the standarddeviation in F are very close, then the average tsmooth can be taken as a standard stopcriterion for the edge extraction.


FMG tsmooth ms FH FCMean 0.5119 20.2095 0.6396 0.4238σ 0.1332 6.4172 0.0634 0.1091σ2 0.0175 41.1811 0.0040 0.0119

a) Images without noise.

FMG tsmooth ms FH FCMean 0.5017 19.365 0.6396 0.4142σ 0.1324 5.4916 0.0634 0.1085σ2 0.0175 30.1579 0.0040 0.0118

b) Images with noise.

Table 5.3: Statistical results for the performance of the memristive grid and Canny’s method forimage edge detection.

Separated comparisons are given for 3 particular image sets: 3 images with the bestresults, 3 images with the average results and 3 with the worst results. Theaforementioned images and edges are shown in the figure 5.14, where the human edgedepicted for each image is an overlap of the five ground truth of the database. It can beobserved that the edges detected by the memristive grid and Canny’s method havelittle coincidence with those extracted by humans in monotonically coloured images,or cases like the images h) and i), where the object is confused with its surroundings.In these last images, it can be inferred that the success in the extraction of edgedepends on the previous knowledge of the subject (human) of the objects that theimage can contain.

Finally, the F measures of the images in figure 5.14 (the best, average and worst resultsfor the memristive grid in the benchmark images) are presented in table 5.4 includingthe measures for the same images with noise.

Ref. FH FMG FMG Noise FC FC Noisea) 0.7938 0.7772 0.7685 0.7604 0.7644b) 0.7820 0.7682 0.7648 0.7216 0.7218c) 0.6890 0.7610 0.7568 0.6563 0.6628d) 0.6581 0.5127 0.4745 0.3430 0.3457e) 0.6943 0.5126 0.5066 0.4149 0.4217f) 0.6645 0.5117 0.4850 0.3932 0.3746g) 0.5664 0.1506 0.1426 0.1762 0.1640h) 0.6183 0.1309 0.1266 0.1238 0.1137i) 0.5930 0.1220 0.1177 0.1290 0.1242

Table 5.4: Comparative results of F measures in images from figure 5.14.


Image Human MemGrid Canny

a)

b)

c)Best results for memristive grid.

d)

e)

f)Average results for memristive grid.

g)

h)

i)Worst results for memristive grid.

Figure 5.14: Edges extracted by humans, memristive grid and Canny’s method.

Chapter 6

Memristive grid for Maze solving

The memristive grid, regarded as an analog processor [34], has an interestingadvantage, due to the fact that by achieving small changes on the memristive branch,or fuse element, it is possible to use the grid in other analog processing problems. Forexample, the problem of computing the shortest path can be solved with this processorand in the same way the maze solving problem.

In this chapter, the performance of the memristive grid used for maze solving is studied.The developed charge-controlled memristor model is used in this application. The mazesolving application is presented in the following order. First, the fuse used to generatethe memristive branch and to have control of each branch in the circuit is presented;this step is necessary in order to model the maze topology in the circuit. Then, thememristive grid is implemented in HSPICE to simulate the behaviour of the circuit.Subsequently, several mazes and their solutions are presented, including mazes withmultiple solutions, since the grid can process all the solutions in parallel. Finally theresults obtained by the processor (memristive grid) are presented.

6.1 Previous approaches

For thousands of years, mazes have intrigued the human mind [35]. The labyrinthshave been used in research with laboratory animals, in order to study their ability torecognize their environment [36], [37], [38]. In the 90’s, artificial intelligence ofrobots was studied by examining their ability to traverse unfamiliar mazes [39], [40],[41]. Maze exploration algorithms are closely related to graph theory and both havebeen used in mathematics and computer science [42], [43].

There are several algorithms for maze solving in the literature, they can be distributedin two groups: the algorithms used by a traveller in the maze without knowledge of ageneral view of the maze, and the algorithms used for a program that can see thewhole maze. Some examples of the first ones are the wall follower, random mouse,

61

62 CHAPTER 6. MEMRISTIVE GRID FOR MAZE SOLVING

pledge algorithm [44] and Trémaux’s algorithm [45]. In the second group, shortestpath algorithms are useful, because they can find the solution not only for a simpleconnected maze but also for multiple solution mazes.

The main idea resides in the fact that the topology of a maze can be implemented in anon-linear resistive grid. The shortest path approach can be used within non-linearresistive grids [46]. By exploiting the analog computations performed by solvingKirchoft’s Current Laws in a parallel manner, memristive grids have demonstratedtheir ability for computing shortest paths in a given maze, levering on the dynamicadjustment of their intrinsic conductance [47].

6.2 Maze solving: implementing the memristive gridFigure 6.1 a) shows an example of a maze with entrance marked by a green arrow andoutput by a red arrow. The maze is mapped into a non-linear resistive grid (figure 6.1b)) by assigning a high resistance to blocked paths (walls of the labyrinth) and a non-linear resistance to paths that can be walked. The input of the maze is a voltage sourceand the output is the ground node. Each non-linear branch is in fact a memristive fuse.

a) b)

Figure 6.1: Mapping of a maze in the memristive grid.

6.2.1 Description of the memristive fuseIn order to have control in the fuses of the memristive grid, that is, to be able to changeif a fuse represents a path or a wall in the labyrinth, each fuse consists of an anti-seriesconnection of two memristors, connected to a switch [34]. In figure 6.2 the mentioned

6.2. MAZE SOLVING: IMPLEMENTING THE MEMRISTIVE GRID 63

fuse is presented.

Figure 6.2: Fuse configuration in memristive grid for maze solving.

The properties of the anti-series memristive connection have been already expounded inthe previous chapter. However, in figure 6.3 the Memristance-Charge characteristic isgenerated for the parameters in table 6.1. The levels of Roff , Ron and Rinit are selectedaccordingly to the minimum resistance imposed by the implementation of the switch inthe fuse.

µvm2

V s∆ nm Ron Ω Roff Ω Rinit Ω k Order

1× 10−14 10 100 16× 103 1× 103 5 2

Table 6.1: Parameters to generate figure 6.3

Figure 6.3: Memristance-Charge characteristic for anti-series connection.

Switch implementation

In the memristive fuse, for maze solving application, an ideal switch can be used insimulations, however, with the aim to use a more realistic switch, a transmission gateis used. The transmission gate is a switch in CMOS technology, it consists of anNMOS transistor and a PMOS transistor connected in parallel, as in figure 6.4 a). Thetwo devices in combination can fully transmit any signal value between Vdd (thesupply voltage of the transistors) and ground. In order to switch, each transistor


requires a complementary control input. Therefore, it is necessary to add an inverterconnected between the control input and the PMOS gate [48].

If the control input is Vdd then the switch is closed, the transistors can pass the inputsignal to output presenting a low resistance. On the contrary, if the control input isgrounded then the switch is opened and the transistors present a high resistance.

a) Configuration. b) Symbol.

Figure 6.4: Transmission gate.

To implement the transmission gate in the grid (in simulation), a CMOS 180 nmtechnology is used. The parameters of the two complementary transistors are in table6.2 and the medium resistance in ON and OFF state. The resistance values areextracted making a sweep of the input voltage and measure the equivalent resistanceof the transistors in the ON-state (switch closed) figure 6.5 a) and the OFF-state(switch opened) figure 6.5 b).

CMOS TG W µm L µmPMOS 1.44 0.18NMOS 0.48 0.18

RTGon Ω RTGoff Ω2.504× 103 10.854× 109

Table 6.2: Transmission gate parameters.

a) ON-state. b) OFF-state.

Figure 6.5: Resistance in the transmission gate with parameters from table 6.4.

6.3. SIMULATION FLOW OF THE MEMRISTIVE GRID 65

6.3 Simulation flow of the memristive gridThe electrical behaviour of the memristive grid is simulated in HSPICE by modellingthe memristor as a non-linear resistor, the behaviour of this model is described by theequations presented in Chapter 3.

The simulation flow is shown in figure 6.6 and is described as follows:

Maze Generator A script in MATLAB generates an aleatory maze and it is shown asa plot. Then, the labyrinth topology is described in a .sp file with the net-listcorresponding to that topology. All fuses that represent walls have the switchin the OFF-state, whereas, those that represent paths have the switch in the ON-sate. The inputs in the maze are represented by step voltage sources and theoutput nodes are grounded.

Simulation The net-list obtained by the maze generator is simulated in HSPICE. Here,a transient analysis for 20 s is carried out, this time is enough to find the solutionsof the mazes under-test, however, the exact time when the solutions are founddepends on the maze dimensions (grid). The results are saved in a .tr0 output file.

Graphic display of the results In order to visualize the results, a scrip in MATLABimport the output simulation signals obtained in HSPICE. The ∆R parameter iscalculated for each simulation time and then the grid is represented by a plot,where the color in each fuse indicates the changes in the resistance (∆R) at agiven time, being white for the minimum change and black for the maximumchange. The unconnected fuses that represent the walls in the maze are shown ingreen color (perpendicular to the fuse trajectory) in the plot.

Figure 6.6: Simulation flow.

In order to measure ∆R, first, the minimum resistance of each branch is set as

Rmin = 2 ·Rinit +RTGon = 2 · 1000 + 2504 = 4504 Ω (6.1)


whereRinit is the initial resistance in each memristor (of the anti-series connection) andRTGon is the average resistance of the transmission gate in conduction mode. At eachinstant, the equivalent resistance is monitored in the fuses, then, the difference betweenthe measured resistance and the minimum resistance in equation 6.1 is calculated, and∆R is obtained. Thus, the fuses that first reach the highest ∆R indicate the solutionpath of the labyrinth. In mazes with multiple solutions, fuses that belong to the shortestpath first reach high values of ∆R; as time lapses, other solution paths are revealedreaching high values of ∆R. For a given time, all fuses within the solution paths reachthe maximum resistance

Rmax = Roff +RTGon (6.2)

corresponding to the sum of the OFF-state in the memristors and the ON-state of thetransmission gate.

6.4 Mazes under-testTo prove the behaviour of the memristive grid in maze solving, six different mazes areused (figure 6.7), three single solution mazes and three mazes with multiple solutions.In each case, a maze that can be represented in a 5× 5 nodes and a maze with 10× 10nodes is used, the third single solution maze has 15 × 15 nodes and the third multiplesolution maze has 10× 10 nodes but two different inputs and outputs.

a) b) c)

d) e) f)

Figure 6.7: Mazes under test. Single solution: a) 5 × 5, b) 10 × 10 and c) 15 × 15. Multiplesolution: d) 5 × 5 with two solutions, e) 10 × 10 with four solutions, and f) 10 × 10 with twoinputs and two outputs.

6.5. RESULTS 67

6.5 ResultsThe solutions found for the mazes in figure 6.7 by the memristive grid are shownhereafter.

6.5.1 Single solution mazesThe first labyrinth used to evaluate the memristive grid is the shown in figure 6.7 a).

Figure 6.8 a) shows the evolution in time of the resistance of the fuses in the pathways.As time lapses, the fuses in the solution path exhibit a significant increment in theresistance value compared with the paths that do not constitute a solution of the maze.A zoom for smaller values in t is shown in Figure 6.8 b), where the mechanism ofdifferentiation among the paths with regard to the resistance value is highlighted. Infact, the memrsitive grid achieves the finding of the solution of the maze in a parallelprocessing by calculating the resistance of the fuses at the same time. The solution ofthe maze is represented by a color scale that shows how the resistance of the fusesevolve until the finish point is reached (figure 6.9). At t = 0 s all fuses of the pathwayhave the same resistance (figure 6.9 a)), as time lapses, only the fuses in the solutionpaths present significant changes in their resistance, as shown in figure 6.8 b) fort = 0.197 s. This changes are represented for the maze in figure 6.9 b) for the sametime.

a) 20 s. b) 0.2 s.

Figure 6.8: Resistance measures of the maze in figure 6.7 a). In blue lines, fuses outside thesolution path. In red lines, fuses that belong to solution path.

Finally, the solution, in the representation of the whole maze, can be easily identifiablein red color as shown figure 6.9 c). The solutions for mazes in figure 6.7 b) and c) areshown in figure 6.10 a) and b) respectively.


a) t = 0 s. b) t = 0.197 s.

c) t = 0.638 s. d) t = 20 s.

Figure 6.9: Results for the maze from figure 6.7 a) showing the evolution in ∆R.

a) t = 1.3929 s. b) t = 3.7886 s.

Figure 6.10: Results of the memristive grid for the mazes in figure 6.7 b) and c).

6.5. RESULTS 69

6.5.2 Multiple solutions mazes

In order to evaluate the performance of the memristive grid in solving mazes withmultiple solutions, the results for three different labyrinths are presented bellow.

First, the solution for the maze in figure 6.7 d) is analysed. Similarly to the mazes witha single solution, a measure of the resistance response in the fuses is carried out;however, in this case only the resistance in the fuses belonging to the solution pathsare presented in figure 6.11. The resistance in fuses that belong to the shortest pathhave a fastest increase than the fuses in the other solution paths, as shown figure 6.11a) in red color lines. By zooming in the plot (figure 6.11 b)), it can be seen that theshortest path are easily differentiable at a time of 0.6 s. The solutions found by thememristive grid are presented in figure 6.12 for four different times.

All the solutions for the maze in figure 6.7 e) are shown in figure 6.13 a). Additionally,due to the difference in the evolution of the rate of change in the resistance in thefuses, the shortest path is distinguished in red color, followed by a lighter colouredpath for the other solutions. For a long time, all the solutions reach the same resistancevalue as shown in figure 6.13 b).

A maze with two entrances and two outputs is shown in figure 6.7 f). In this case,the memristive grid finds first the shortest path between an entrance and an output, asshown in figure 6.14 a). Afterwards, the connection between the two solutions paths isfound, as shown in figure 6.14 b).

a) 20 s. b) 0.7 s.

Figure 6.11: Resistance measures of the fuses in the solution paths. In red lines, fuses in theshortest path and in blue lines the fuses in the another solution path.


a) t = 0 s. b) t = 0.2204 s.

c) t = 0.638 s. d) t = 20 s.

Figure 6.12: Results of the memristive grid for the maze in figure 6.7 d).

a) t = 1.901 s. b) t = 20 s.

Figure 6.13: Results of the memristive grid for the maze in figure 6.7 d).

6.5. RESULTS 71

a) t = 1.0276 s. b) t = 8.0716 s.

Figure 6.14: Results of the memristive grid for the maze in figure 6.7 f).

Chapter 7

Conclusions and future work

A symbolic model for a charge-controlled memristor has been developed. Theresulting model can be used in memristive systems where the stimuli may be anycurrent waveform. The model can be used in any simulation tool of electronic systemsdue to its symbolic nature.

A first characterization of the model was achieved in AC. The results of thecharacterization demonstrated that the model fulfils the fingerprints for the i-v pinchedhysteresis loop. Sweeps of several parameters were done on order to determine thepassivity of the memristor.

A second characterization was achieved in DC with the aim of establishing the mainparameters of the static characteristics, such as the switching voltages and saturationtimes. Furthermore, the charge-flux characteristic was studied, finding that each slopecorresponds to a state in the memristor (Ron in ON-state and Roff ) in OFF-state).

Special attention was devoted to the Memristance-Charge characteristic of theanti-series and anti-parallel connection in DC. It can be observed that the anti-seriesconnection allows to have a memristive element that can pass from the ON-state to theOFF-state applying a DC source regardless of its sign. Similarly, the anti-parallelconnection allows to pass from the OFF-state to the ON-state.

The developed model has been used in applications of memristive grids. Memristivegrids can be used as an analog processor for different applications. In this work, twoof these possible applications were presented. First, the memristive grid as a filter forimage smoothing and edge detection, and second, as a processor to determine theshortest path in maze solving problems.

The methods for image edge detection usually use a smoothing filter as the first step toimprove edge detection. In the memristive grid, the smoothing filter is naturallyimplemented by the same circuit, which allows to have a processor that implements

73

74 CHAPTER 7. CONCLUSIONS AND FUTURE WORK

both functions, image smoothing and image edge detection. Additionally, in the testsperformed to measure the behaviour of the memristive grid in this application, the gridpresents a good performance in edge detection in comparison with the human results,and its performance is better than the obtained with the Canny’s method.

In a second application, the performance of memristive grid was evaluated whensolving mazes for the case of single and multiple solutions. In the case of a mazehaving a single solution, the solution trajectory is found by sensing the high variationsin the resistance of the fuses that belong to the path, whereas the other fuses aremaintained in the initial resistance value. For a multiple solution maze, the memristivegrid is able to asses firstly the shortest path by sensing the higher resistance-variantfuses. The rest of the solution paths are naturally determined later by the analogprocessor.

7.1 Future workMemristive grids can be used in other types of problems involving trajectories andobstacles, mapping the space that needs to be analysed in the grid and finding thesolution paths. In particular, if a guided search of the solution is devised, i.e. such asthe case of assigning priority to a given path, then the assignment of the initialresistance of those fuses in the path should be done. This procedures is similar toassigning weight in graphs, so that the desired paths are followed.

Moreover, seeking for additional applications of the developed model (in DC and AC)can also be the matter of future research, such of chaos.

Appendix A

Fij factors in the memristanceequations

Complementary factors for the memristance equations in table 3.2 are presentedhereafter. In the equations A = X0 − 1.

A.1 Factors for η = −1

F12− =

A3[−e4κq + 2e8κq − e12κq

]q ≤ 0

X30

[−e−12κq + 2e−8κq − e−4κq

]q > 0

(A.1)

F13− =

A4[e4κq − 3e8κq + 3e12κq − e16κq

]q ≤ 0

X40

[e−16κq − 3e−12κq + 3e−8κq − e−4κq

]q > 0

(A.2)

F22− =

A3

− 145 ( 40X4

0 + 44X30 + 123X2

0 + 92X0 + 106 ) e8κq

+2 ( 2X20 + 2X0 + 5 ) e16κq

+ ( 4X30 + 6X0 − 19 ) e24κq

+ 89A ( 2X3

0 + 3X0 − 23 ) e32κq

−13A2e40κq − 245 A

3e48κq − 89A

4e56κq

q ≤ 0

X30

− 89X

40e

−56κq + 245 X

30e

−48κq − 13X20e

−40κq

+ 89X0 ( 2X3

0 − 6X20 + 9X0 + 18 ) e−32κq

− ( 4X30 − 12X2

0 + 18X0 + 9 ) e−24κq

+2 ( 2X20 − 6X0 + 9 ) e−16κq

− 145 ( 40X4

0 − 204X30 + 495X2

0 − 630X0 + 405 ) e−8κq

q > 0

(A.3)

75

76 APPENDIX A. FIJ FACTORS IN THE MEMRISTANCE EQUATIONS

F23− =

A4

− 12835

(3920X6

0 + 4578X50 + 14484X4

0 + 13310X30

+19767X20 + 11805X0 + 8681

)e8κq

− 115 ( 100X4

0 + 128X30 + 366X2

0 + 284X0 + 337 ) e16κq

− 15 ( 40X5

0 + 16X40 + 106X3

0 − 124X20 − 64X0 − 379 ) e24κq

− 1135

(560X6

0 − 288X50 + 1320X4

0 − 7840X30

+900X20 − 11400X0 + 20393

)e32κq

+ 13A ( 130X3

0 + 195X0 − 586 ) e40κq

+ 65A

2 ( 16X30 + 24X0 − 143 ) e48κq

+ 2135A

3 ( 280X30 + 420X0 − 7009 ) e56κq

− 4444105 A

4e64κq − 545 A

5e72κq − 11281 A

6e80κq

q ≤ 0

X40

11281 X

60e

−80κq − 545 X

50e

−72κq + 4444105 X

40e

−64κq

− 2135X

30 ( 280X3

0 − 840X20 + 1260X0 + 6309 ) e−56κq

+ 65X

20 ( 16X3

0 − 48X20 + 72X0 + 103 ) e−48κq

− 13X0 ( 130X3

0 − 390X20 + 585X0 + 261 ) e−40κq

+ 1135

(560X6

0 − 3072X50 + 8280X4

0 − 5760X30

−9180X20 + 25920X0 + 3645

)e−32κq

− 15

(40X5

0 − 216X40 + 570X3

0 − 690X20

+270X0 + 405

)e−24κq

+ 115

(100X4

0 − 528X30 + 1350X2

0−1800X0 + 1215

)e−16κq

− 12835

(3920X6

0 − 28098X50 + 96174X4

0 − 195426X30

+251181X20 − 195615X0 + 76545

)e−8κq

q > 0

(A.4)

A.1. FACTORS FOR η = −1 77

F32− =

A3

− 14725

(16128X8

0 − 19264X70 + 31864X6

0 − 2208X50

+27560X40 + 5332X3

0 + 24474X20 + 17996X0 + 16243

)e12κq

+ 23 ( 16X4

0 − 4X30 + 16X2

0 + 16X0 + 31 ) e24κq

+ 13 ( 64X5

0 − 80X40 + 80X3

0 + 60X0 − 199 ) e36κq

+ 169 A ( 16X5

0 − 20X40 + 20X3

0 + 15X0 − 81 ) e48κq

+ 23A

2 ( 32X50 − 40X4

0 + 40X30 + 30X0 − 337 ) e60κq

+ 875A

3 ( 64X50 − 80X4

0 + 80X30 + 60X0 − 2449 ) e72κq

− 20969 A4e84κq − 3328

21 A5e96κq

− 2323 A6e108κq − 640

27 A7e120κq − 256

75 A8e132κq

q ≤ 0

X30

− 25675 X

80e

−132κq + 64027 X

70e

−120κq − 2323 X6

0e−108κq

+ 332821 X5

0e−96κq − 2096

9 X40e

−84κq

+ 875X

30

(64X5

0 − 240X40 + 400X3

0 − 400X20

+300X0 + 2325

)e−72κq

− 23X

20

(32X5

0 − 120X40 + 200X3

0 − 200X20

+150X0 + 275

)e−60κq

+ 169 X0

(16X5

0 − 60X40 + 100X3

0−100X2

0 + 75X0 + 50

)e−48κq

− 13

(64X5

0 − 240X40 + 400X3

0−400X2

0 + 300X0 + 75

)e−36κq

+ 23 ( 16X4

0 − 60X30 + 100X2

0 − 100X0 + 75 ) e−24κq

− 14725

(16128X8

0 − 109760X70 + 348600X6

0 − 687600X50

+949200X40 − 959700X3

0 + 708750X20

−367500X0 + 118125

)e−12κq

q > 0

(A.5)


F33− factor for q ≤ 0

Fq−

33− = A4

110135125

135303168X12

0 − 291838976X110 + 524242048X10

0−394171360X9

0 + 470277760X80 − 194812256X7

0+378083352X6

0 + 24470824X50 + 222941560X4

0+94997940X3

0 + 155371038X20 + 90038574X0

+51986953

e12κq

− 1945

(37632X8

0 − 41216X70 + 74816X6

0 + 3648X50 + 78640X4

0+16208X3

0 + 75156X20 + 56824X0 + 52667

)e24κq

− 1945

(86016X9

0 − 173824X80 + 259616X7

0 − 148384X60

+180896X50 − 236416X4

0 + 178124X30

−123016X20 − 85216X0 − 292171

)e36κq

− 12835

387072X100 − 1146880X9

0 + 1908480X80

−1766400X70 + 1464960X6

0 − 2709504X50

+2933280X40 − 2378880X3

0 + 189000X20

−1708560X0 + 3181807

e48κq

− 2189A

10752X100 − 31360X9

0 + 52080X80

−47520X70 + 40320X6

0 − 139944X50

+162960X40 − 146580X3

0 + 5670X20

−108045X0 + 277792

e60κq

− 423625A

2

236544X100 − 680960X9

0 + 1128960X80

−1017600X70 + 880320X6

0 − 10459008X50

+12821760X40 − 12443760X3

0 + 132300X20

−9295020X0 + 34774589

e72κq

+ 4135A

3(

58688X50 − 73360X4

0 + 73360X30

+55020X0 − 316283

)e84κq

+ 64315A

4(

6656X50 − 8320X4

0 + 8320X30

+6240X0 − 59671

)e96κq

+ 8105A

5(

9744X50 − 12180X4

0 + 12180X30

+9135X0 − 169004

)e108κq

+ 320567A

6(

448X50 − 560X4

0 + 560X30

+420X0 − 19931

)e120κq

+ 3223625A

7(

29568X50 − 36960X4

0 + 36960X30

+27720X0 − 5933713

)e132κq

− 4796595210395 A8e144κq − 5811808

2835 A9e156κq

− 11560961755 A10e168κq − 25600

189 A11e180κq − 450563375 A

12e192κq

(A.6)


F33− factor for q > 0

Fq+

33− = X40

+ 450563375 X

120 e−192κq − 25600

189 X110 e−180κq + 1156096

1755 X100 e−168κq

− 58118082835 X9

0e−156κq + 47965952

10395 X80e

−144κq

− 3223625X

70

(29568X5

0 − 110880X40 + 184800X3

0 − 184800X20

+138600X0 + 5876425

)e−132κq

+ 320567X

60 ( 448X5

0 − 1680X40 + 2800X3

0 − 2800X20 + 2100X0 + 19063 ) e−120κq

− 8105X

50

(9744X5

0 − 36540X40 + 60900X3

0 − 60900X20

+45675X0 + 150125

)e−108κq

+ 64315X

40

(6656X5

0 − 24960X40 + 41600X3

0 − 41600X20

+31200X0 + 46775

)e−96κq

− 4135X

30

(58688X5

0 − 220080X40 + 366800X3

0−366800X2

0 + 275100Xo+ 202575

)e−84κq

+ 423625X

20

236544X100 − 1684480X9

0 + 5644800X80

−11884800X70 + 17841600X6

0 − 10483200X50

−18984000X40 + 49350000X3

0 − 55282500X20

+43942500X0 + 16078125

e−72κq

− 2189X0

(10752X10

0 − 76160X90 + 253680X8

0 − 530400X70

+789600X60 − 778680X5

0 + 357000X40 + 178500X3

0−435750X2

0 + 433125X0 + 76125

)e−60κq

+ 12835

(387072X10

0 − 2723840X90 + 9004800X8

0 − 18662400X70

+27484800X60 − 28896000X5

0 + 19908000X40 − 6720000X3

0−1995000X2

0 + 5040000X0 + 354375

)e−48κq

− 1945

(86016X9

0 − 600320X80 + 1965600X7

0 − 4027200X60

+5846400X50 − 6199200X4

0 + 4651500X30 − 2257500X2

0+472500X0 + 354375

)e−36κq

+ 1945

(37632X8

0 − 259840X70 + 840000X6

0 − 1694400X50

+2410800X40 − 2528400X3

0 + 1942500X20

−1050000X0 + 354375

)e−24κq

− 110135125

135303168X12

0 − 1331799040X110 + 6244022400X10

0−18563802400X9

0 + 39335264800X80 − 63139533600X7

0+79353131000X6

0 − 79311375000X50 + 63211005000X4

0−39722182500X3

0 + 19050281250X20 − 6475218750X0

+1266890625

e−12κq

(A.7)


F42− =

A3

− 11576575

32947200X12

0 − 108556800X110 + 199027840X10

0−222057280X9

0 + 186742656X80 − 100672928X7

0+58217348X6

0 − 21399216X50 + 25550660X4

0+836944X3

0 + 11453523X20 + 8346392X0

+6815836

e16κq

+ 215

(40X6

0 − 320X50 + 352X4

0 − 68X30

+142X20 + 142X0 + 247

)e32κq

+ 15

(480X7

0 − 1120X60 + 1344X5

0 − 840X40

+420X30 + 210X0 − 739

)e48κq

+ 89A(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 541

)e64κq

+ 13A2(

960X70 − 2240X6

0 + 2688X50 − 1680X4

0+840X3

0 + 420X0 − 3585

)e80κq

+ 1625A3(

480X70 − 1120X6

0 + 1344X50 − 840X4

0+420X3

0 + 210X0 − 3679

)e96κq

+ 3245A4(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 5343

)e112κq

+ 128735

A5(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 29353

)e128κq

−5716A6e144κq − 467849

A7e160κq − 9299225

A8e176κq

− 11033655

A9e192κq − 3443245

A10e208κq − 716839

A11e224κq − 102449

A12e240κq

q ≤ 0

X30

− 102449

X120 e−240κq + 7168

39X11

0 e−224κq − 3443245

X100 e−208κq + 110336

55X9

0e−192κq

− 9299225

X80e

−176κq + 467849

X70e

−160κq − 5716X60e

−144κq

+ 128735

X50

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 29106

)e−128κq

− 3245X4

0

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 5096

)e−112κq

+ 1625X3

0

(480X7

0 − 2240X60 + 4704X5

0 − 5880X40

+4900X30 − 2940X2

0 + 1470X0 + 3185

)e−96κq

− 13X2

0

(960X7

0 − 4480X60 + 9408X5

0 − 11760X40

+9800X30 − 5880X2

0 + 2940X0 + 2597

)e−80κq

+ 89X0

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 294

)e−64κq

− 15

(480X7

0 − 2240X60 + 4704X5

0 − 5880X40

+4900X30 − 2940X2

0 + 1470X0 + 245

)e−48κq

+ 215

(240X09

6 − 1120X50 + 2352X4

0 − 2940X30

+2450X20 − 1470X0 + 735

)e−32κq

− 11576575

32947200X12

0 − 286809600X110 + 1179418240X10

0−3045981120X9

0 + 5541471936X80 − 7552985440X7

0+8015307300X6

0 − 6788101320X50 + 4641997360X4

0−2554471920X3

0 + 1107281175X20 − 360510150X0

+77252175

e−16κq

q > 0

(A.8)


F43− for q ≤ 0, where B1 = 1160408623375

, B2 = 1225225

, B3 = 125025

, B4 = 1135135

,B5 = 1

9009, B6 = 2

125125, B7 = 2

96525and B8 = 4

11036025.

Fq−

43−= A4

B1

38311004160000X18

0 − 201859698585600X170 + 556256047564800X16

0−1001651202918400X15

0 + 1321167468456960X140 − 1328148147600960X13

0+1067173367461376X12

0 − 701242505786112X110 + 413565804177216X10

0−220688447545360X9

0 + 123615165067560X80 − 52622661333672X7

0 + 28135641118004X60

−4736605328742X50 + 8738475447000X4

0 + 1960566325250X30 + 3768059229321X2

0+2150563919883X0 + 1127263989101

e16κq

−B2

(74131200X12

0 − 239078400X110 + 436860160X10

0 − 480997120X90 + 407173824X8

0−211440512X7

0 + 135349592X60 − 53870064X5

0 + 76089140X40

+2637136X30 + 35821662X2

0 + 26723948X0 + 22355959

)e32κq

−B3

(27456000X13

0 − 114470400X120 + 245569280X11

0 − 331181440X100 + 319580672X9

0−220799168X8

0 + 125266552X70 − 81477128X6

0 + 66550752X50 − 44824512X4

0+17427678X3

0 − 8968692X20 − 6866592X0 − 19013727

)e48κq

−B4

(362419200X14

0 − 1856870400X130 + 4684359680X12

0 − 7466833920X110 + 8394177792X10

0−6941975040X9

0 + 4443959520X80 − 2938890240X7

0 + 2558475920X60 − 2198388192X5

0+1279398120X4

0 − 570329760X30 + 31531500X2

0 − 272552280X0 + 537869405

)e64κq

−B5A

(39536640X14

0 − 201062400X130 + 504824320X12

0 − 801265920X110 + 897921024X1

00

−739939200X90 + 473513040X8

0 − 374328240X70 + 414894480X6

0 − 403026624X50

+241801560X40 − 1127025900Xo

3 + 3783780X20 − 55090035X0 + 141011006

)e80κq

−B6A2

285542400X140 − 1442918400X13

0 + 3608084480X120 − 5705656320X11

0+6376161792X10

0 − 5237872640X90 + 3350867520X8

0 − 3807598080X70

+5636510880X60 − 6080378304X5

0 + 3728765040X40 − 1801319520X3

0+29429400X2

0 − 892251360X0 + 3059914287

e96κq

−B7A3

131788800X140 − 662323200X13

0 + 1650288640X120 − 2601231360X11

0+2899792896X10

0 − 2375493120X90 + 1519277760X8

0 − 3748018560X70

+7271584320X60 − 8411090688X5

0 + 5224499280X40 − 2581619040X3

0+14414400X2

0 − 1287205920X0 + 6094717037

e112κq

−B8A4

1976832000X140 − 9887539200X13

0 + 24559575040X120 − 38600540160X11

0+42937230336X10

0 − 35086571520X90 + 22434572160X8

0 − 269962767360X70

+608134246720X60 − 725047010688X5

0 + 452675102880X40 − 225858272640X3

0+227026800X2

0 − 112878685920X0 + 762685947887

e128κq

+ 635A5(

685920X70 − 1600480X6

0 + 1920576X50 − 1200360X4

0+600180X3

0 + 300090X0 − 3039649

)e144κq

+ 8567

A6(

8421120X70 − 19649280X6

0 + 23579136X50 − 14736960X4

0+7368480X3

0 + 3684240X0 − 60492517

)e160κq

+ 87875

A7(

92062080X70 − 214811520X6

0 + 257773824X50 − 161108640X4

0+80554320X3

0 + 40277160X0 − 1201895119

)e176κq

+ 321925

A8(

3310080X70 − 7723520X6

0 + 9268224X50 − 5792640X4

0+2896320X3

0 + 1448160X0 − 91924095

)e192κq

+ 20851975

A9(

5681280X70 − 13256320X6

0 + 15907584X50 − 9942240X4

0+4971120X3

0 + 2485560X0 − 417110305

)e208κq

+ 8966435

A10(

42240X70 − 98560X6

0 + 118272X50 − 73920X4

0+36960X3

0 + 18480X0 − 11349341

)e224κq

+ 12849049

A11(

274560X70 − 640640X6

0 + 768768X50 − 480480X4

0+240240X3

0 + 120120X0 − 490916677

)e240κq

− 29552380674563378375

A12e256κq − 110876614208225225

A13e272κq − 94401020928425425

A14e288κq

− 31183015936405405

A15e304κq − 29933158415561

A16e320κq − 20070465

A17e336κq

− 81920343

A18e352κq

(A.9)


F43− for q > 0, C1 = 411036025

, C2 = 296525

, C3 = 2125125

, C4 = 19009

, C5 = 1135135

,C6 = 1

25025, C7 = 1

225225and C8 = 1

160408623375.

Fq+

43−= X4

0

+ 81920343

X180 e−352κq − 200704

65X17

0 e−336κq + 29933158415561

X160 e−320κq − 31183015936

405405X15

0 e−304κq

+ 94401020928425425

X140 e−288κq − 110876614208

225225X13

0 e−272κq + 29552380674563378375

X120 e−256κq

− 12849049

X110

(274560X7

0 − 1281280X60 + 2690688X5

0 − 3363360X40

+2802800X30 − 1681680X2

0 + 840840X0 + 490634109

)e−240κq

+ 8966435

X100

(42240X7

0 − 197120X60 + 413952X5

0 − 517440X40

+431200X30 − 258720X2

0 + 129360X0 + 11305869

)e−224κq

− 20851975

X90

(5681280X7

0 − 26512640X60 + 55676544X5

0 − 69595680X40

+57996400X30 − 34797840X2

0 + 17398920X0 + 411263321

)e−208κq

+ 321925

X80

(3310080X7

0 − 15447040X60 + 32438784X5

0 − 40548480X40

+33790400X30 − 20274240X2

0 + 10137120X0 + 88517471

)e−192κq

− 87875

X70

(92062080X7

0 − 429623040X60 + 902208384X5

0 − 1127760480X40

+939800400X30 − 563880240X2

0 + 281940120X0 + 1107147895

)e−176κq

+ 8567

X60

(8421120X7

0 − 39298560X60 + 82526976X5

0 − 103158720X40

+85965600X30 − 51579360X2

0 + 25789680X0 + 51825781

)e−160κq

− 635X5

0

(685920X7

0 − 3200960X60 + 6722016X5

0 − 8402520X40

+7002100X30 − 4201260X2

0 + 2100630X0 + 2333723

)e−144κq

+C1X40

1976832000X140 − 17788108800X13

0 + 75913277440X120 − 204453150720X11

0+390235862016X10

0 − 562389667840X90 + 637894857600X8

0 − 329340211200X70

−754883729600X60 + 2239760979456X5

0 − 3004883001120X40

+2564360198400X30 − 1555178985360X2

0 + 783151649280X0 + 498309146335

e−128κq

−C2X30

131788800X140 − 1182720000X13

0 + 5032867840X120 − 13512145920X11

0+25701451776X10

0 − 36898301440X90 + 41672030400X8

0 − 35238403200X70

+15303288000X60 + 10283809536X5

0 − 26097991920X40 + 25623197600X3

0−16433376960X2

0 + 8569841280X0 + 3139381245

e−112κq

+C3X20

285542400X140 − 2554675200X13

0 + 10834503680X120 − 28981155840X11

0+54900797952X10

0 − 78460462080X90 + 88153665600X8

0 − 78245207040X70

+51112981920X60 − 18458119680X5

0 − 4532127600X40 + 11810999200X3

0−9270261000X2

0 + 5356150800X0 + 1107281175

e−96κq

−C4X0

39536640X140 − 352450560X13

0 + 1488847360X120 − 3965095680X11

0+7474731264X10

0 − 10623733120X90 + 11860889040X8

0 − 10596265680X70

+7439752320X60 − 3806987184X5

0 + 1112431320X40 + 127036910X3

0−360510150X2

0 + 272957685X0 + 29870841

e−80κq

+C5

362419200X140 − 3216998400X13

0 + 13525191680X120 − 35830179840X11

0+67143428352X10

0 − 94783969280X90 + 104987282400X8

0 − 93393780480X70

+66694868240X60 − 37048083072X5

0 + 14791216440X40 − 3296092800X3

0−432612180X2

0 + 988827840X0 + 46351305

e−64κq

−C6

(27456000X13

0 − 242457600X120 + 1013492480X11

0 − 2667450240X100 + 4961628672X9

0−6944217280X8

0 + 7613806200X70 − 6706539840X6

0 + 4781296520X50 − 2719276560X4

0+1184533350X3

0 − 360510150X20 + 51501450X0 + 25750725

)e−48κq

+C7

(74131200X12

0 − 650496000X110 + 2699656960X10

0 − 7047156480X90

+12983914944X80 − 17969311360X7

0 + 19436016600X60 − 16860523680X5

0+11872800940X4

0 − 6756990240X30 + 3038585550X2

0 − 1030029000X0 + 231756525

)e−32κq

−C8

38311004160000X18

0 − 487738376294400X170 + 2986224808089600X16

0−11707305945036800X15

0 + 33014202823848960X140 − 71320807749804480X13

0+122759862696843456X12

0 − 172880531381093760X110 + 202956974680017600X10

0−201252782111624880X9

0 + 170026414374038040X80 − 122940893286371160X7

0+76113841094391100X6

0 − 40179886298001450X50 + 17901638015010750X4

0−6599973597767550X3

0 + 1940377565701575X20 − 421821209935125X0

+55020157817625

e−16κq

(A.10)


F52− for q ≤ 0

Fq−

52− = A3

− 1241215975

39032913920X160 − 208653885440X15

0+559517038080X14

0 − 965042670592X130

+1193227729408X120 − 1109879485440X11

0+805011916544X10

0 − 458411498240X90

+212176270080X80 − 81859173152X7

0+34876432304X6

0 − 13249902912X50

+7945302160X40 − 150922936X3

0 + 2188697388X20

+1585738072X0 + 1223994731

e20κq

+ 235

(1792X8

0 − 4256X70 + 5536X6

0 − 3872X50

+2176X40 − 344X3

0 + 496X20 + 496X0 + 811

)e40κq

+ 135

(14336X9

0 − 48384X80 + 78336X7

0 − 75264X60

+48384X50 − 20160X4

0 + 6720X30 + 2520X0 − 9323

)e60κq

+ 3245A

(1792X9

0 − 6048X80 + 9792X7

0 − 9408X60

+6048X50 − 2520X4

0 + 840X30 + 315X0 − 1621

)e80κq

+ 45A

2(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 4907

)e100κq

+ 1625A

3(

7168X90 − 24192X8

0 + 39168X70 − 37632X6

0+24192X5

0 − 10080X40 + 3360X3

0 + 1260X0 − 17041

)e120κq

+ 6445A

4(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 17939

)e140κq

+ 512245A

5(

1792X90 − 6048X8

0 + 9792X70 − 9408X6

0+6048X5

0 − 2520X40 + 840X3

0 + 315X0 − 24562

)e160κq

+ 1635A

6(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 195689

)e180κq

+ 642835A

7(

14336X90 − 48384X8

0 + 78336X70 − 75264X6

0+48384X5

0 − 20160X40 + 6720X3

0 + 2520X0 − 5953103

)e200κq

− 8599045 A8e220κq − 2032640

11 A9e240κq − 14763529 A10e260κq

− 153395213 A11e280κq − 16412672

245 A12e300κq − 454819841575 A13e320κq

− 31078435 A14e340κq − 147456

85 A15e360κq − 65536405 A16e380κq

(A.11)


F52− for q > 0

Fq+

52− = X30

− 65536405 X16

0 e−380κq + 14745685 X15

0 e−360κq − 31078435 X14

0 e−340κq

+ 454819841575 X13

0 e−320κq − 16412672245 X12

0 e−300κq + 153395213 X11

0 e−280κq

− 14763529 X10

0 e−260κq + 203264011 X9

0e−240κq − 859904

5 X80e

−220κq

+ 642835X

70

(14336X9

0 − 80640X80 + 207360X7

0 − 322560X60

+338688X50 − 254016X4

0 + 141120X30

−60480X20 + 22680X0 + 5946615

)e−200κq

− 1635X

60

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30

−15120X20 + 5670X0 + 194067

)e−180κq

+ 512245X

50

(1792X9

0 − 10080X80 + 25920X7

0 − 40320X60

+42336X50 − 31752X4

0 + 17640X30

−7560X20 + 2835X0 + 23751

)e−160κq

− 6445X

40

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30

−15120X20 + 5670X0 + 16317

)e−140κq

+ 1625X

30

(7168X9

0 − 40320X80 + 103680X7

0 − 161280X60

+169344X50 − 127008X4

0 + 70560X30

−30240X20 + 11340X0 + 13797

)e−120κq

− 45X

20

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30

−15120X20 + 5670X0 + 3285

)e−100κq

+ 3245X0

(1792X9

0 − 10080X80 + 25920X7

0 − 40320X60

+42336X50 − 31752X4

0 + 17640X30

−7560X20 + 2835X0 + 810

)e−80κq

− 135

(14336X9

0 − 80640X80 + 207360X7

0 − 322560X60

+338688X50 − 254016X4

0 + 141120X30

−60480X20 + 22680X0 + 2835

)e−60κq

+ 235

(1792X8

0 − 10080X70 + 25920X6

0 − 40320X50 + 42336X4

0−31752X3

0 + 17640X20 − 7560X0 + 2835

)e−40κq

− 1241215975

39032913920X160 − 415872737280X15

0+2113658426880X14

0 − 6817969686528X130

+15666108936192X120 − 27282941199360X11

0+37411675741440X10

0 − 41429875818240X90

+37715217636096X80 − 28588332925152X7

0+18206154766800X6

0 − 9785373998400X50

+4430655028800X40 − 1671221463912X3

0+512342730900X2

0 − 121572851400Xo+ 19538493975

e−20κq

(A.12)

A.2. FACTORS FOR η = +1 85

A.2 Factors for η = +1

F12+ =

X30

[−e12κq + 2e8κq − e4κq

]q ≤ 0

A3[−e−4κq + 2e−8κq − e−12κq

]q > 0

(A.13)

F13+ =

X40

[e16κq − 3e12κq + 3e8κq − e4κq

]q ≤ 0

A4[e−4κq − 3e−8κq + 3e−12κq − e−16κq

]q > 0

(A.14)

F22+ =

X30

− 89X

40e

56κq + 245 X

30e

48κq − 13X20e

40κq

+ 89X0 ( 2X3

0 − 6X20 + 9X0 + 18 ) e32κq

− ( 4X30 − 12X2

0 + 18X0 + 9 ) e24κq

+2 ( 2X20 − 6X0 + 9 ) e16κq

− 145 ( 40X4

0 − 204X30 + 495X2

0 − 630X0 + 405 ) e8κq

q ≤ 0

(X0 − 1)3

− 145 ( 40X4

0 + 44X30 + 123X2

0 + 92X0 + 106 ) e−8κq

+2 ( 2X20 + 2X0 + 5 ) e−16κq

+ ( 4X30 + 6X0 − 19 ) e−24κq

+ 89 (X0 − 1) ( 2X3

0 + 3X0 − 23 ) e−32κq

−13(X0 − 1)2e−40κq − −245 (X0 − 1)3e−48κq

− 89 (X0 − 1)4e−56κq

q > 0

(A.15)


F23+ =

X40

11281 X

60e

80κq − 545 X

50e

72κq + 4444105 X

40e

64κq

− 2135X

30 ( 280X3

0 − 840X20 + 1260X0 + 6309 ) e56κq

+ 65X

20 ( 16X3

0 − 48X20 + 72X0 + 103 ) e48κq

− 13X0 ( 130X3

0 − 390X20 + 585X0 + 261 ) e40κq

+ 1135

(560X6

0 − 3072X50 + 8280X4

0 − 5760X30 − 9180X2

0+25920X0 + 3645

)e32κq

− 15 ( 40X5

0 − 216X40 + 570X3

0 − 690X20 + 270X0 + 405 ) e24κq

+ 115 ( 100X4

0 − 528X30 + 1350X2

0 − 1800X0 + 1215 ) e16κq

− 12835

(3920X6

0 − 28098X50 + 96174X4

0 − 195426X30 + 251181X2

0−195615X0 + 76545

)e8κq

q ≤ 0

A4

− 12835

(3920X6

0 + 4578X50 + 14484X4

0 + 13310X30 + 19767X2

0+11805X0 + 8681

)e−8κq

− 115 ( 100X4

0 + 128X30 + 366X2

0 + 284X0 + 337 ) e−16κq

− 15 ( 40X5

0 + 16X40 + 106X3

0 − 124X20 − 64X0 − 379 ) e−24κq

− 1135

(560X6

0 − 288X50 + 1320X4

0 − 7840X30 + 900X2

0−11400X0 + 20393

)e−32κq

+ 13A ( 130X3

0 + 195X0 − 586 ) e−40κq

+ 65A

2 ( 16X30 + 24X0 − 143 ) e−48κq

+ 2135A

3 ( 280X30 + 420X0 − 7009 ) e−56κq

− 4444105 A

4e−64κq − 545 A

5e−72κq − 11281 A

6e−80κq

q > 0

(A.16)


F32+ =

X30

− 25675 X

80e

132κq + 64027 X

70e

120κq − 2323 X6

0e108κq

+ 332821 X5

0e96κq − 2096

9 X40e

84κq

+ 875X

30 ( 64X5

0 − 240X40 + 400X3

0 − 400X20 + 300X0 + 2325 ) e72κq

− 23X

20 ( 32X5

0 − 120X40 + 200X3

0 − 200X20 + 150X0 + 275 ) e60κq

+ 169 X0 ( 16X5

0 − 60X40 + 100X3

0 − 100X20 + 75X0 + 50 ) e48κq

− 13 ( 64X5

0 − 240X40 + 400X3

0 − 400X20 + 300X0 + 75 ) e36κq

+ 23 ( 16X4

0 − 60X30 + 100X2

0 − 100X0 + 75 ) e24κq

− 14725

(16128X8

0 − 109760X70 + 348600X6

0 − 687600X50

+949200X40 − 959700X3

0 + 708750X20

−367500X0 + 118125

)e12κq

q ≤ 0

A3

− 14725

(16128X8

0 − 19264X70 + 31864X6

0 − 2208X50

+27560X40 + 5332X3

0 + 24474X20 + 17996X0 + 16243

)e−12κq

+ 23 ( 16X4

0 − 4X30 + 16X2

0 + 16X0 + 31 ) e−24κq

+ 13 ( 64X5

0 − 80X40 + 80X3

0 + 60X0 − 199 ) e−36κq

+ 169 A ( 16X5

0 − 20X40 + 20X3

0 + 15X0 − 81 ) e−48κq

+ 23A

2 ( 32X50 − 40X4

0 + 40X30 + 30X0 − 337 ) e−60κq

+ 875A

3 ( 64X50 − 80X4

0 + 80X30 + 60X0 − 2449 ) e−72κq

− 20969 A4e−84κq − 3328

21 A5e−96κq − 2323 A6e−108κq

− 64027 A

7e−120κq − 25675 A

8e−132κq

q > 0

(A.17)


F33+ for q ≤ 0

Fq−

33+ = X40

+ 450563375 X

120 e192κq − 25600

189 X110 e180κq + 1156096

1755 X100 e168κq

− 58118082835 X9

0e156κq + 47965952

10395 X80e

144κq

− 3223625X

70

(29568X5

0 − 110880X40 + 184800X3

0−184800X2

0 + 138600X0 + 5876425

)e132κq

+ 320567X

60 ( 448X5

0 − 1680X40 + 2800X3

0 − 2800X20 + 2100X0 + 19063 ) e120κq

− 8105X

50

(9744X5

0 − 36540X40 + 60900X3

0 − 60900X20

+45675X0 + 150125

)e108κq

+ 64315X

40

(6656X5

0 − 24960X40 + 41600X3

0 − 41600X20

+31200X0 + 46775

)e96κq

− 4135X

30

(58688X5

0 − 220080X40 + 366800X3

0−366800X2

0 + 275100Xo+ 202575

)e84κq

+ 423625X

20

(236544X10

0 − 1684480X90 + 5644800X8

0 − 11884800X70

+17841600X60 − 10483200X5

0 − 18984000X40 + 49350000X3

0−55282500X2

0 + 43942500X0 + 16078125

)e72κq

− 2189X0

(10752X10

0 − 76160X90 + 253680X8

0 − 530400X70

+789600X60 − 778680X5

0 + 357000X40 + 178500X3

0−435750X2

0 + 433125X0 + 76125

)e60κq

+ 12835

(387072X10

0 − 2723840X90 + 9004800X8

0 − 18662400X70

+27484800X60 − 28896000X5

0 + 19908000X40 − 6720000X3

0−1995000X2

0 + 5040000X0 + 354375

)e48κq

− 1945

(86016X9

0 − 600320X80 + 1965600X7

0 − 4027200X60

+5846400X50 − 6199200X4

0 + 4651500X30 − 2257500X2

0+472500X0 + 354375

)e36κq

+ 1945

(37632X8

0 − 259840X70 + 840000X6

0 − 1694400X50 + 2410800X4

0−2528400X3

0 + 1942500X20 − 1050000X0 + 354375

)e24κq

− 110135125

135303168X12

0 − 1331799040X110 + 6244022400X10

0−18563802400X9

0 + 39335264800X80 − 63139533600X7

0+79353131000X6

0 − 79311375000X50 + 63211005000X4

0−39722182500X3

0 + 19050281250X20 − 6475218750X0

+1266890625

e12κq

(A.18)


F33+ for q > 0

Fq+

33+ = A4

110135125

135303168X120 − 291838976X11

0 + 524242048X100

−394171360X90 + 470277760X8

0 − 194812256X70

+378083352X60 + 24470824X5

0 + 222941560X40

+94997940X30 + 155371038X2

0 + 90038574X0 + 51986953

e−12κq

− 1945

(37632X8

0 − 41216X70 + 74816X6

0 + 3648X50 + 78640X4

0+16208X3

0 + 75156X20 + 56824X0 + 52667

)e−24κq

− 1945

(86016X9

0 − 173824X80 + 259616X7

0 − 148384X60 + 180896X5

0−236416X4

0 + 178124X30 − 123016X2

0 − 85216X0 − 292171

)e−36κq

− 12835

(387072X10

0 − 1146880X90 + 1908480X8

0 − 1766400X70

+1464960X60 − 2709504X5

0 + 2933280X40 − 2378880X3

0+189000X2

0 − 1708560X0 + 3181807

)e−48κq

− 2189A

(10752X10

0 − 31360X90 + 52080X8

0 − 47520X70

+40320X60 − 139944X5

0 + 162960X40 − 146580X3

0+5670X2

0 − 108045X0 + 277792

)e−60κq

− 423625A

2

(236544X10

0 − 680960X90 + 1128960X8

0 − 1017600X70

+880320X60 − 10459008X5

0 + 12821760X40 − 12443760X3

0+132300X2

0 − 9295020X0 + 34774589

)e−72κq

+ 4135A

3 ( 58688X50 − 73360X4

0 + 73360X30 + 55020X0 − 316283 ) e−84κq

+ 64315A

4 ( 6656X50 − 8320X4

0 + 8320X30 + 6240X0 − 59671 ) e−96κq

+ 8105A

5 ( 9744X50 − 12180X4

0 + 12180X30 + 9135X0 − 169004 ) e−108κq

+ 320567A

6 ( 448X50 − 560X4

0 + 560X30 + 420X0 − 19931 ) e−120κq

+ 3223625A

7 ( 29568X50 − 36960X4

0 + 36960X30 + 27720X0 − 5933713 ) e−132κq

− 4796595210395 A8e−144κq − 5811808

2835 A9e−156κq − 11560961755 A10e−168κq

− 25600189 A11e−180κq − 45056

3375 A12e−192κq

(A.19)


F42+ =

X30

− 102449

X120 e240κq + 7168

39X11

0 e224κq − 3443245

X100 e208κq + 110336

55X9

0e192κq

− 9299225

X80e

176κq + 467849

X70e

160κq − 5716X60e

144κq

+ 128735

X50

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 29106

)e128κq

− 3245X4

0

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 5096

)e112κq

+ 1625X3

0

(480X7

0 − 2240X60 + 4704X5

0 − 5880X40

+4900X30 − 2940X2

0 + 1470X0 + 3185

)e96κq

− 13X2

0

(960X7

0 − 4480X60 + 9408X5

0 − 11760X40

+9800X30 − 5880X2

0 + 2940X0 + 2597

)e80κq

+ 89X0

(240X7

0 − 1120X60 + 2352X5

0 − 2940X40

+2450X30 − 1470X2

0 + 735X0 + 294

)e64κq

− 15

(480X7

0 − 2240X60 + 4704X5

0 − 5880X40

+4900X30 − 2940X2

0 + 1470X0 + 245

)e48κq

+ 215

(240X09

6 − 1120X50 + 2352X4

0 − 2940X30

+2450X20 − 1470X0 + 735

)e32κq

− 11576575

32947200X120 − 286809600X11

0 + 1179418240X100

−3045981120X90 + 5541471936X8

0 − 7552985440X70

+8015307300X60 − 6788101320X5

0 + 4641997360X40

−2554471920X30 + 1107281175X2

0 − 360510150X0 + 77252175

e16κq

q ≤ 0

A3

− 11576575

32947200X120 − 108556800X11

0 + 199027840X100

−222057280X90 + 186742656X8

0 − 100672928X70

+58217348X60 − 21399216X5

0 + 25550660X40

+836944X30 + 11453523X2

0 + 8346392X0 + 6815836

e−16κq

+ 215

(40X6

0 − 320X50 + 352X4

0 − 68X30

+142X20 + 142X0 + 247

)e−32κq

+ 15

(480X7

0 − 1120X60 + 1344X5

0 − 840X40

+420X30 + 210X0 − 739

)e−48κq

+ 89A(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 541

)e−64κq

+ 13A2(

960X70 − 2240X6

0 + 2688X50 − 1680X4

0+840X3

0 + 420X0 − 3585

)e−80κq

+ 1625A3(

480X70 − 1120X6

0 + 1344X50 − 840X4

0+420X3

0 + 210X0 − 3679

)e−96κq

+ 3245A4(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 5343

)e−112κq

+ 128735

A5(

240X70 − 560X6

0 + 672X50 − 420X4

0+210X3

0 + 105X0 − 29353

)e−128κq

−5716A6e−144κq − 467849

A7e−160κq − 9299225

A8e−176κq

− 11033655

A9e−192κq − 3443245

A10e−208κq − 716839

A11e−224κq − 102449

A12e−240κq

q > 0

(A.20)


F43+ for q ≤ 0, where D1 = 411036025

, D2 = 296525

, D3 = 2125125

, D4 = 19009

, D5 =1

135135, D6 = 1

25025, D7 = 1

225225and D8 = 1

160408623375.

Fq−

43+= X4

0

+ 81920343

X180 e352κq − 200704

65X17

0 e336κq + 29933158415561

X160 e320κq − 31183015936

405405X15

0 e304κq

+ 94401020928425425

X140 e288κq − 110876614208

225225X13

0 e272κq + 29552380674563378375

X120 e256κq

− 12849049

X110

(274560X7

0 − 1281280X60 + 2690688X5

0 − 3363360X40

+2802800X30 − 1681680X2

0 + 840840X0 + 490634109

)e240κq

+ 8966435

X100

(42240X7

0 − 197120X60 + 413952X5

0 − 517440X40

+431200X30 − 258720X2

0 + 129360X0 + 11305869

)e224κq

− 20851975

X90

(5681280X7

0 − 26512640X60 + 55676544X5

0 − 69595680X40

+57996400X30 − 34797840X2

0 + 17398920X0 + 411263321

)e208κq

+ 321925

X80

(3310080X7

0 − 15447040X60 + 32438784X5

0 − 40548480X40

+33790400X30 − 20274240X2

0 + 10137120X0 + 88517471

)e192κq

− 87875

X70

(92062080X7

0 − 429623040X60 + 902208384X5

0 − 1127760480X40

+939800400X30 − 563880240X2

0 + 281940120X0 + 1107147895

)e176κq

+ 8567

X60

(8421120X7

0 − 39298560X60 + 82526976X5

0 − 103158720X40

+85965600X30 − 51579360X2

0 + 25789680X0 + 51825781

)e160κq

− 635X5

0

(685920X7

0 − 3200960X60 + 6722016X5

0 − 8402520X40

+7002100X30 − 4201260X2

0 + 2100630X0 + 2333723

)e144κq

+D1X40

1976832000X140 − 17788108800X13

0 + 75913277440X120 − 204453150720X11

0+390235862016X10

0 − 562389667840X90 + 637894857600X8

0 − 329340211200X70

−754883729600X60 + 2239760979456X5

0 − 3004883001120X40

+2564360198400X30 − 1555178985360X2

0 + 783151649280X0 + 498309146335

e128κq

−D2X30

131788800X140 − 1182720000X13

0 + 5032867840X120 − 13512145920X11

0+25701451776X10

0 − 36898301440X90 + 41672030400X8

0 − 35238403200X70

+15303288000X60 + 10283809536X5

0 − 26097991920X40 + 25623197600X3

0−16433376960X2

0 + 8569841280X0 + 3139381245

e112κq

+D3X20

285542400X140 − 2554675200X13

0 + 10834503680X120 − 28981155840X11

0+54900797952X10

0 − 78460462080X90 + 88153665600X8

0 − 78245207040X70

+51112981920X60 − 18458119680X5

0 − 4532127600X40 + 11810999200X3

0−9270261000X2

0 + 5356150800X0 + 1107281175

e96κq

−D4X0

39536640X140 − 352450560X13

0 + 1488847360X120 − 3965095680X11

0+7474731264X10

0 − 10623733120X90 + 11860889040X8

0 − 10596265680X70

+7439752320X60 − 3806987184X5

0 + 1112431320X40 + 127036910X3

0−360510150X2

0 + 272957685X0 + 29870841

e80κq

+D5

362419200X140 − 3216998400X13

0 + 13525191680X120 − 35830179840X11

0+67143428352X10

0 − 94783969280X90 + 104987282400X8

0 − 93393780480X70

+66694868240X60 − 37048083072X5

0 + 14791216440X40 − 3296092800X3

0−432612180X2

0 + 988827840X0 + 46351305

e64κq

−D6

(27456000X13

0 − 242457600X120 + 1013492480X11

0 − 2667450240X100 + 4961628672X9

0−6944217280X8

0 + 7613806200X70 − 6706539840X6

0 + 4781296520X50 − 2719276560X4

0+1184533350X3

0 − 360510150X20 + 51501450X0 + 25750725

)e48κq

+D7

(74131200X12

0 − 650496000X110 + 2699656960X10

0 − 7047156480X90

+12983914944X80 − 17969311360X7

0 + 19436016600X60 − 16860523680X5

0+11872800940X4

0 − 6756990240X30 + 3038585550X2

0 − 1030029000X0 + 231756525

)e32κq

−D8

38311004160000X18

0 − 487738376294400X170 + 2986224808089600X16

0−11707305945036800X15

0 + 33014202823848960X140 − 71320807749804480X13

0+122759862696843456X12

0 − 172880531381093760X110 + 202956974680017600X10

0−201252782111624880X9

0 + 170026414374038040X80 − 122940893286371160X7

0+76113841094391100X6

0 − 40179886298001450X50 + 17901638015010750X4

0−6599973597767550X3

0 + 1940377565701575X20 − 421821209935125X0 + 55020157817625

e16κq

(A.21)


F43+ for q > 0, where E1 = 1160408623375

, E2 = 1225225

, E3 = 125025

, E4 = 1135135

,E5 = 1

9009, E6 = 2

125125, E7 = 2

96525and E8 = 4

11036025.

Fq+

43+= A4

E1

38311004160000X18

0 − 201859698585600X170 + 556256047564800X16

0−1001651202918400X15

0 + 1321167468456960X140 − 1328148147600960X13

0+1067173367461376X12

0 − 701242505786112X110 + 413565804177216X10

0−220688447545360X9

0 + 123615165067560X80 − 52622661333672X7

0+28135641118004X6

0 − 4736605328742X50 + 8738475447000X4

0+1960566325250X3

0 + 3768059229321X20 + 2150563919883X0 + 1127263989101

e−16κq

−E2

(74131200X12

0 − 239078400X110 + 436860160X10

0 − 480997120X90

+407173824X80 − 211440512X7

0 + 135349592X60 − 53870064X5

0+76089140X4

0 + 2637136X30 + 35821662X2

0 + 26723948X0 + 22355959

)e−32κq

−E3

(27456000X13

0 − 114470400X120 + 245569280X11

0 − 331181440X100

+319580672X90 − 220799168X8

0 + 125266552X70 − 81477128X6

0 + 66550752X50

−44824512X40 + 17427678X3

0 − 8968692X20 − 6866592X0 − 19013727

)e−48κq

−E4

362419200X140 − 1856870400X13

0 + 4684359680X120 − 7466833920X11

0+8394177792X10

0 − 6941975040X90 + 4443959520X8

0 − 2938890240X70

+2558475920X60 − 2198388192X5

0 + 1279398120X40 − 570329760X3

0+31531500X2

0 − 272552280X0 + 537869405

e−64κq

−E5A

39536640X140 − 201062400X13

0 + 504824320X120 − 801265920X11

0+897921024X1

00 − 739939200X90 + 473513040X8

0 − 374328240X70

+414894480X60 − 403026624X5

0 + 241801560X40 − 1127025900Xo

3

+3783780X20 − 55090035X0 + 141011006

e−80κq

−E6A2

285542400X140 − 1442918400X13

0 + 3608084480X120 − 5705656320X11

0+6376161792X10

0 − 5237872640X90 + 3350867520X8

0 − 3807598080X70

+5636510880X60 − 6080378304X5

0 + 3728765040X40 − 1801319520X3

0+29429400X2

0 − 892251360X0 + 3059914287

e−96κq

−E7A3

131788800X140 − 662323200X13

0 + 1650288640X120 − 2601231360X11

0+2899792896X10

0 − 2375493120X90 + 1519277760X8

0 − 3748018560X70

+7271584320X60 − 8411090688X5

0 + 5224499280X40 − 2581619040X3

0+14414400X2

0 − 1287205920X0 + 6094717037

e−112κq

−E8A4

1976832000X14

0 − 9887539200X130 + 24559575040X12

0−38600540160X11

0 + 42937230336X100 − 35086571520X9

0+22434572160X8

0 − 269962767360X70 + 608134246720X6

0−725047010688X5

0 + 452675102880X40 − 225858272640X3

0+227026800X2

0 − 112878685920X0 + 762685947887

e−128κq

+ 635A5(

685920X70 − 1600480X6

0 + 1920576X50 − 1200360X4

0+600180X3

0 + 300090X0 − 3039649

)e−144κq

+ 8567

A6(

8421120X70 − 19649280X6

0 + 23579136X50 − 14736960X4

0+7368480X3

0 + 3684240X0 − 60492517

)e−160κq

+ 87875

A7(

92062080X70 − 214811520X6

0 + 257773824X50 − 161108640X4

0+80554320X3

0 + 40277160X0 − 1201895119

)e−176κq

+ 321925

A8(

3310080X70 − 7723520X6

0 + 9268224X50 − 5792640X4

0+2896320X3

0 + 1448160X0 − 91924095

)e−192κq

+ 20851975

A9(

5681280X70 − 13256320X6

0 + 15907584X50 − 9942240X4

0+4971120X3

0 + 2485560X0 − 417110305

)e−208κq

+ 8966435

A10(

42240X70 − 98560X6

0 + 118272X50 − 73920X4

0+36960X3

0 + 18480X0 − 11349341

)e−224κq

+ 12849049

A11(

274560X70 − 640640X6

0 + 768768X50 − 480480X4

0+240240X3

0 + 120120X0 − 490916677

)e−240κq

− 29552380674563378375

A12e−256κq − 110876614208225225

A13e−272κq − 94401020928425425

A14e−288κq

− 31183015936405405

A15e−304κq − 29933158415561

A16e−320κq − 20070465

A17e−336κq

− 81920343

A18e−352κq

(A.22)


F52+ for q ≤ 0

Fq−

52+ = X30

− 65536405 X16

0 e380κq + 14745685 X15

0 e360κq − 31078435 X14

0 e340κq

+ 454819841575 X13

0 e320κq − 16412672245 X12

0 e300κq + 153395213 X11

0 e280κq

− 14763529 X10

0 e260κq + 203264011 X9

0e240κq − 859904

5 X80e

220κq

+ 642835X

70

(14336X9

0 − 80640X80 + 207360X7

0 − 322560X60

+338688X50 − 254016X4

0 + 141120X30 − 60480X2

0+22680X0 + 5946615

)e200κq

− 1635X

60

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30 − 15120X2

0+5670X0 + 194067

)e180κq

+ 512245X

50

(1792X9

0 − 10080X80 + 25920X7

0 − 40320X60

+42336X50 − 31752X4

0 + 17640X30 − 7560X2

0+2835X0 + 23751

)e160κq

− 6445X

40

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30 − 15120X2

0+5670X0 + 16317

)e140κq

+ 1625X

30

(7168X9

0 − 40320X80 + 103680X7

0 − 161280X60

+169344X50 − 127008X4

0 + 70560X30 − 30240X2

0+11340X0 + 13797

)e120κq

− 45X

20

(3584X9

0 − 20160X80 + 51840X7

0 − 80640X60

+84672X50 − 63504X4

0 + 35280X30 − 15120X2

0+5670X0 + 3285

)e100κq

+ 3245X0

(1792X9

0 − 10080X80 + 25920X7

0 − 40320X60

+42336X50 − 31752X4

0 + 17640X30 − 7560X2

0+2835X0 + 810

)e80κq

− 135

(14336X9

0 − 80640X80 + 207360X7

0 − 322560X60

+338688X50 − 254016X4

0 + 141120X30

−60480X20 + 22680X0 + 2835

)e60κq

+ 235

(1792X8

0 − 10080X70 + 25920X6

0 − 40320X50 + 42336X4

0−31752X3

0 + 17640X20 − 7560X0 + 2835

)e40κq

− 1241215975

39032913920X160 − 415872737280X15

0+2113658426880X14

0 − 6817969686528X130

+15666108936192X120 − 27282941199360X11

0+37411675741440X10

0 − 41429875818240X90

+37715217636096X80 − 28588332925152X7

0+18206154766800X6

0 − 9785373998400X50

+4430655028800X40 − 1671221463912X3

0+512342730900X2

0 − 121572851400X0 + 19538493975

e20κq

(A.23)


F52+ for q > 0

Fq+

52+ = A3

− 1241215975

39032913920X160 − 208653885440X15

0+559517038080X14

0 − 965042670592X130

+1193227729408X120 − 1109879485440X11

0+805011916544X10

0 − 458411498240X90

+212176270080X80 − 81859173152X7

0+34876432304X6

0 − 13249902912X50

+7945302160X40 − 150922936X3

0+2188697388X2

0 + 1585738072X0 + 1223994731

e−20κq

+ 235

(1792X8

0 − 4256X70 + 5536X6

0 − 3872X50

+2176X40 − 344X3

0 + 496X20 + 496X0 + 811

)e−40κq

+ 135

(14336X9

0 − 48384X80 + 78336X7

0 − 75264X60

+48384X50 − 20160X4

0 + 6720X30 + 2520X0 − 9323

)e−60κq

+ 3245A

(1792X9

0 − 6048X80 + 9792X7

0 − 9408X60

+6048X50 − 2520X4

0 + 840X30 + 315X0 − 1621

)e−80κq

+ 45A

2(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 4907

)e−100κq

+ 1625A

3(

7168X90 − 24192X8

0 + 39168X70 − 37632X6

0+24192X5

0 − 10080X40 + 3360X3

0 + 1260X0 − 17041

)e−120κq

+ 6445A

4(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 17939

)e−140κq

+ 512245A

5(

1792X90 − 6048X8

0 + 9792X70 − 9408X6

0+6048X5

0 − 2520X40 + 840X3

0 + 315X0 − 24562

)e−160κq

+ 1635A

6(

3584X90 − 12096X8

0 + 19584X70 − 18816X6

0+12096X5

0 − 5040X40 + 1680X3

0 + 630X0 − 195689

)e−180κq

+ 642835A

7(

14336X90 − 48384X8

0 + 78336X70 − 75264X6

0+48384X5

0 − 20160X40 + 6720X3

0 + 2520X0 − 5953103

)e−200κq

− 8599045 A8e−220κq − 2032640

11 A9e−240κq − 14763529 A10e−260κq

− 153395213 A11e−280κq − 16412672

245 A12e−300κq − 454819841575 A13e−320κq

− 31078435 A14e−340κq − 147456

85 A15e−360κq − 65536405 A16e−380κq

(A.24)

Appendix B

Characterization plots

In this appendix, the curves for the AC analysis for two of the models generated arepresented. The parameters used are shown in table B.1.

A variation in some parameters is carried out, first for Ap, the stimulus currentamplitude, second for X0, the initial state of the model, and third for Ron, theresistance of the ON-state of the model.

µvm2

V s∆ nm κ m

AsRon Ω Roff Ω X0 Ap µA

1× 10−14 10 10000 100 16× 103 0.5 40

Table B.1: Parameters used in the characterization.

B.1 Figures

The plots are presented for the two values of η and four values of frequency ω (1, 2, 5and 10 rad

s). The structure of the figures for each parameter is:

• The pinched hysteresis loop for the model Mk1O3η± and after for Mk5O2η± .• The M -i characteristic for the same models.• A plot of the minimum (for η = +1) memristance in function of the parameter

evaluated, with the aim to evaluate the passivity of the models (Mk1O3η+ andMk5O2η+) with respect that parameter. In case of η = −1 the minimummemristance is always Rinit, because the memristance vary from initial state toOFF-state, then the memristance increases.

Additionally, each parameter is evaluated in the curves for the values shown hereafter:• Ap = 10, 20, 40, 50, 80, 100 and 120 µA.• X0 = 0.1, 0.2, 0.3, 0.5, 0.7 and 0.9.• Ron = 1, 3, 10, 30, 100 and 300 Ω.

95

96 APPENDIX B. CHARACTERIZATION PLOTS

B.1.1 Ap plots

η = +1 η = −1ω

=1

ω=

2ω

=5

ω=

10

Figure B.1: Hysteresis loop for the memristance equation Mk1O3η± , varying Ap.

B.1. FIGURES 97

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.2: Hysteresis loop for the memristance equation Mk5O2η± , varying Ap.


η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.3: M -i characteristic for the memristance equation Mk1O3η± , varying Ap.

B.1. FIGURES 99

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.4: M -i characteristic for the memristance equation Mk5O2η± , varying Ap.


Mk1O3η+ Mk5O2η+

ω=

1ω

=2

ω=

5ω

=10

Figure B.5: Passivity characteristic for η = +1, varying Ap.

B.1. FIGURES 101

B.1.2 X0 plots

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.6: Hysteresis loop for the memristance equation Mk1O3η± , varying X0.


η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.7: Hysteresis loop for the memristance equation Mk5O2η± , varying X0.

B.1. FIGURES 103

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.8: M -i characteristic for the memristance equation Mk1O3η± , varying X0.


η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.9: M -i characteristic for the memristance equation Mk5O2η± , varying X0.

B.1. FIGURES 105

Mk1O3η+ Mk5O2η+

ω=

1ω

=2

ω=

5ω

=10

Figure B.10: Passivity characteristic for η = +1, varying X0.


B.1.3 Ron plots

η = +1 η = −1ω

=1

ω=

2ω

=5

ω=

10

Figure B.11: Hysteresis loop for the memristance equation Mk1O3η± , varying Ron.

B.1. FIGURES 107

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.12: Hysteresis loop for the memristance equation Mk5O2η± , varying Ron.


η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.13: M -i characteristic for the memristance equation Mk1O3η± , varying Ron.

B.1. FIGURES 109

η = +1 η = −1

ω=

1ω

=2

ω=

5ω

=10

Figure B.14: M -i characteristic for the memristance equation Mk5O2η± , varying Ron.


Mk1O3η+ Mk5O2η+

ω=

1ω

=2

ω=

5ω

=10

Figure B.15: Passivity characteristic for η = +1, varying Ron.

B.2. DISCUSSION 111

B.2 DiscussionThe voltage swing is greater for η = −1 in all curves as compared to η = +1. This isbecause for positive η the equivalent resistance tends to Ron, and to Roff for negativeη. In this way, a higher resistive range is presented in the case η = −1, making thePHL has wider lobes and a greater amplitude in voltage for the same current values.

In the M -i characteristic, the plots have a bell shaped at low frequencies, that isbecause the model reaches the saturation resistance, Ron for η = +1 and Roff forη = −1. This feature is accentuated as k increases.

For the parameters used, the model developed is maintained as a passive element. Itcan be observed in all cases that the minimum memristance tends to Ron, because ofthis, in the figures where a sweep of the parameter Ron is made, the trend of the curveis a straight line from a given Ron.

List of figures

2.1 Electrical variables relations. . . . . . . . . . . . . . . . . . . . . . . . 62.2 Memristor symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 q-φ characteristic of memristor. . . . . . . . . . . . . . . . . . . . . . . 72.4 Pinched hysteresis loop. . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 First quadrant lobe area as function of the frequency. . . . . . . . . . . 82.6 Structure of HP memristor. . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Window function proposed in [13] for different values of a. . . . . . . . 153.2 Window function proposed in [21] for different values of p. . . . . . . . 153.3 Joglekar window for different values of k. . . . . . . . . . . . . . . . . 163.4 Sweep of the state variable x depending on the electric charge for the

Joglekar window with k = 1 and 2. In red, numerical solution of thedifferential equation, in blue, HPM solution of the differential equationfor order 1, in violet for order 2 and cyan for order 3. . . . . . . . . . . 19

3.5 Sweep of the state variable x depending on the electric charge for theJoglekar window with k = 3, 4 and 5. In red, numerical solution of thedifferential equation, in blue, HPM solution of the differential equationfor order 1, in violet for order 2 and cyan for order 3. . . . . . . . . . . 20

3.6 Nested form of the memristance equations for each value of k. . . . . . 25

4.1 Variation of the hysteresis loop with respect to the amplitude of theinput signal. a) Hysteresis loop for a Joglekar window with k = 1. b)Hysteresis loop for a Joglekar window with k = 5. . . . . . . . . . . . 30

4.2 Variation of the hysteresis loop respect to frequency. a) Joglekarexponent k = 1. b) Joglekar exponent k = 5. . . . . . . . . . . . . . . 31

4.3 Area for the lobe of the hysteresis loop as a function of the frequency,the units of area are µm2. a) k = 1, ωc = 0.947 rad

s. b) k = 5, ωc = 2.828. 32

4.4 Frequency dependence of the area for different values of Joglekarexponent k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Comparison of the PHL between k = 1, homotopic order 3 of themodel developed, in a), and HP model results, in b). Its response incurrent for a sinusoidal voltage excitation are presented in c) and d)respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

113

114 LIST OF FIGURES

4.6 PHL for k = 5 homotopic order 2 of the model developed, in a), andits response in current for a sinusoidal voltage excitation, in b). . . . . . 34

4.7 Comparison of the PHL between: k = 1, homotopic order 3 of themodel developed (in blue), Biolek’s model (in red) and Affan’s model(in green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.8 c) and d) voltage source in blue, current response of each model ingreen. c) and d), Current-Voltage characteristics, applying the voltagesource shown in a) and b) respectively. . . . . . . . . . . . . . . . . . . 36

4.9 Current-Voltage characteristic for the model applying inverted voltagesources. a) Initial state: OFF. b) Initial state: ON. . . . . . . . . . . . . 37

4.10 Charge-Flux characteristic for the memristor models. . . . . . . . . . . 374.11 Saturation time measured in the memristance curve (a and b for k = 1

and k = 5) and in the current response (c and b for k = 1 and k = 5respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.12 Memristance-Charge characteristic for the model developed. . . . . . . 394.13 M-q characteristic for a series connection of two memristors. . . . . . . 404.14 Memristance-Charge characteristic, starting in ON-state, for positive

and negative η. In c) and d), the M-q characteristic for the anti-seriesconnection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.15 M-q characteristic for a parallel connection of two memristors. . . . . . 424.16 a) and b), Memristance-Charge characteristic, starting in ON-state, for

positive and negative η, in c) and d), the M-q characteristic for an anti-parallel connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 Structure and components of the: a) resistive grid, b) memristive grid.In boxes: violet, input pixel, blue, Rin, green, fuse and red, output pixel. 47

5.2 Branch element, fuse. a) Anti-series connection of memristors. b)Ideal Memristance-Charge characteristics. c) Memristance-Chargecharacteristics of the model developed. . . . . . . . . . . . . . . . . . . 48

5.3 Current contributions at node Ni,j . . . . . . . . . . . . . . . . . . . . . 495.4 Mechanism to determine the edges in an image of 9×9 pixels. a) Fuses

that reach the threshold memristance Mth, red. b) Pixels selected as anedge, green. c) Output image. . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 a) Benchmark image. b) Ground-truth for the edges of the benchmarkimage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.6 Output image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.7 Edge detection: a) Memristive Grid at t = 20.45 ms , b) using Canny’s

method [25] for a threshold 0.422. . . . . . . . . . . . . . . . . . . . . 535.8 Precision-recall plots: a) using the memristive grid, b) using Canny’s

method [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.9 Benchmark image with Gaussian noise. . . . . . . . . . . . . . . . . . 55

LIST OF FIGURES 115

5.10 Edge detection for the benchmark image with noise: a) MemristiveGrid at t = 19.65 ms , b) using Canny’s method [25] for a threshold0.443. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.11 Precision-recall plots for the benchmark image with noise: a) using thememristive grid, b) using Canny’s method [25]. . . . . . . . . . . . . . 56

5.12 Histogram for the F measures and smoothing time for several images.a) FMG for the memristive grid. b) FC for the Canny’s method. c) FHfor the ground-truth. d) tsmooth for the memristive grid. . . . . . . . . . 57

5.13 Histogram for the F measures and smoothing time for several imageswith Gaussian noise. a) FMG for the memristive grid. b) FC for theCanny’s method. c) FH for the ground-truth. d) tsmooth for thememristive grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.14 Edges extracted by humans, memristive grid and Canny’s method. . . . 60

6.1 Mapping of a maze in the memristive grid. . . . . . . . . . . . . . . . . 626.2 Fuse configuration in memristive grid for maze solving. . . . . . . . . . 636.3 Memristance-Charge characteristic for anti-series connection. . . . . . . 636.4 Transmission gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.5 Resistance in the transmission gate with parameters from table 6.4. . . . 646.6 Simulation flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.7 Mazes under test. Single solution: a) 5× 5, b) 10× 10 and c) 15× 15.

Multiple solution: d) 5 × 5 with two solutions, e) 10 × 10 with foursolutions, and f) 10× 10 with two inputs and two outputs. . . . . . . . 66

6.8 Resistance measures of the maze in figure 6.7 a). In blue lines, fusesoutside the solution path. In red lines, fuses that belong to solution path. 67

6.9 Results for the maze from figure 6.7 a) showing the evolution in ∆R. . . 686.10 Results of the memristive grid for the mazes in figure 6.7 b) and c). . . . 686.11 Resistance measures of the fuses in the solution paths. In red lines,

fuses in the shortest path and in blue lines the fuses in the anothersolution path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.12 Results of the memristive grid for the maze in figure 6.7 d). . . . . . . . 706.13 Results of the memristive grid for the maze in figure 6.7 d). . . . . . . . 706.14 Results of the memristive grid for the maze in figure 6.7 f). . . . . . . . 71

B.1 Hysteresis loop for the memristance equation Mk1O3η± , varying Ap. . . 96B.2 Hysteresis loop for the memristance equation Mk5O2η± , varying Ap. . . 97B.3 M -i characteristic for the memristance equation Mk1O3η± , varying Ap. . 98B.4 M -i characteristic for the memristance equation Mk5O2η± , varying Ap. . 99B.5 Passivity characteristic for η = +1, varying Ap. . . . . . . . . . . . . . 100B.6 Hysteresis loop for the memristance equation Mk1O3η± , varying X0. . . 101B.7 Hysteresis loop for the memristance equation Mk5O2η± , varying X0. . . 102B.8 M -i characteristic for the memristance equation Mk1O3η± , varying X0. . 103B.9 M -i characteristic for the memristance equation Mk5O2η± , varying X0. . 104

116 LIST OF FIGURES

B.10 Passivity characteristic for η = +1, varying X0. . . . . . . . . . . . . . 105B.11 Hysteresis loop for the memristance equation Mk1O3η± , varying Ron. . . 106B.12 Hysteresis loop for the memristance equation Mk5O2η± , varying Ron. . . 107B.13 M -i characteristic for the memristance equation Mk1O3η± , varying Ron. 108B.14 M -i characteristic for the memristance equation Mk5O2η± , varying Ron. 109B.15 Passivity characteristic for η = +1, varying Ron. . . . . . . . . . . . . 110

List of tables

3.1 Parameters for the plots of x(q). . . . . . . . . . . . . . . . . . . . . . 213.2 Nested memristance equations, structure. . . . . . . . . . . . . . . . . 263.3 Orders of polynomials of X0 for the Joglekar window. . . . . . . . . . . 27

4.1 Parameters used in the characterization. . . . . . . . . . . . . . . . . . 294.2 Critical frequencies for different values of k Joglekar exponent. . . . . . 314.3 Switching voltages for k 1 and 5 in the model developed. . . . . . . . . 36

5.1 Nominal parameter values. . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Average precision-recall (Fmean), smoothing time and total simulation

time for backward Euler and trapezoidal integration methods fordifferent transient values. . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Statistical results for the performance of the memristive grid andCanny’s method for image edge detection. . . . . . . . . . . . . . . . . 59

5.4 Comparative results of F measures in images from figure 5.14. . . . . . 59

6.1 Parameters to generate figure 6.3 . . . . . . . . . . . . . . . . . . . . . 636.2 Transmission gate parameters. . . . . . . . . . . . . . . . . . . . . . . 64

B.1 Parameters used in the characterization. . . . . . . . . . . . . . . . . . 95

117

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