model for a charge-controlled
memristor and its applications
by
Yojanes Andrés Rodríguez Velásquez
A dissertation submitted in partial fulfilment of
the requirements for the degree of
MASTER OF SCIENCES IN THE
SPECIALTY OF ELECTRONICS
at the
Instituto Nacional de Astrofísica, Óptica y
Electrónica
Advisor:
Dr. Librado Arturo Sarmiento Reyes
Principal Researcher INAOE
August 2017
Tonantzintla, Puebla
c
INAOE 2017
The author hereby grants to INAOE permission
to reproduce and to distribute copies of this
In this work, the development of an analytical model for a memristor is presented. The model is based on the solution of the differential equation that governs the physical behaviour of the device. The solution has been obtained by resorting to a homotopy perturbation method.
The resulting memristance function is controlled by the electric charge, and full symbolic expressions are obtained in function of the main parameters of the memristor.
The obtained model is characterised both for AC and DC sources. The main fingerprints of the device in AC regime have been verified, as well as the main parameters in DC (the switching voltages and saturation time).
In order to show the usefulness of the model, two applications that use memristive grids as an analog processor, are studied. In a first application, the memristive grid is used as a filter for image edge detection and smoothing, obtaining a good performance in 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 processor implements the shortest path method.
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 del dispositivo. La solución se obtiene utilizando el método de homotopía de perturbación.
La memristancia en este modelo es controlada por la carga eléctrica y se obtienen expresiones puramente simbólicas en función de los principales parámetros del memristor.
El modelo obtenido es caracterizado tanto para fuentes de AC como DC. Se verifica que el modelo cumple con las principales características en AC, y adicionalmente se determinan los principales parámetros DC (los voltajes de conmutación y tiempo de saturación).
Con el fin de mostrar la utilidad del modelo, se estudian dos aplicaciones que utilizan una red memristiva a modo de procesador analógico. En la primera, la red memristiva se utiliza como filtro para suavizado y extracción de bordes en imágenes, mostrando un buen desempeño al ser comparada con los bordes extraídos por seres humanos. En la segunda aplicación, la red memristiva se utiliza en la solución de laberintos, ya que el procesador analógico implementa el método de la trayectoria más corta.
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
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
Contents ix
1 Introduction 1
2 Fundamentals of memristor and modelling 5
2.1 The fourth basic circuit element . . . 5
2.1.1 Fingerprints of memristor . . . 7
2.1.2 HP memristor . . . 8
2.2 Types of models . . . 9
2.2.1 Behavioural models . . . 10
2.2.2 Structural models . . . 10
2.2.3 Approach of modelling in this work . . . 11
3 Development of a charge-controlled model 13 3.1 Non-linear drift mechanism . . . 13
3.2 Introduction to homotopy methods . . . 16
3.3 HPM solution to the drift equation . . . 18
3.3.1 Memristance equations . . . 21
4 Characterization of the model 29 4.1 Dynamic tests . . . 29
4.1.1 Fingerprints . . . 30
4.1.2 Comparison with other models . . . 33
4.2 DC tests . . . 35
4.2.1 Determining the switching voltages . . . 35
4.2.2 Determining the saturation time . . . 38
4.2.3 Memristance-Charge characteristic . . . 39
4.2.4 M-q Characteristics of memristor connections . . . 40
5 Memristive grid for edge detection 45 5.1 Previous approaches . . . 45
5.2 Description of the memristive branch. . . 47
5.3 Solution of the memristive grid . . . 49
5.4 Results and comparisons . . . 51
5.4.1 Figures of merit for the edge detection procedure . . . 52
5.4.2 Results for a benchmark image . . . 53
5.4.3 Comparative results on a set of 500 images . . . 56
6 Memristive grid for Maze solving 61 6.1 Previous approaches . . . 61
6.2 Maze solving: implementing the memristive grid . . . 62
6.2.1 Description of the memristive fuse . . . 62
6.3 Simulation flow of the memristive grid . . . 65
6.4 Mazes under-test . . . 66
6.5 Results. . . 67
6.5.1 Single solution mazes . . . 67
6.5.2 Multiple solutions mazes . . . 69
7 Conclusions and future work 73 7.1 Future work . . . 74
A Fij factors in the memristance equations 75 A.1 Factors forη =−1 . . . 75
A.2 Factors forη = +1 . . . 85
B Characterization plots 95 B.1 Figures . . . 95
B.1.1 Applots . . . 96
B.1.2 X0 plots . . . 101
B.1.3 Ronplots . . . 106
B.2 Discussion. . . 111
List of figures 113
List of tables 117
Bibliography 119
Introduction
Based on a symmetry argument that completes the number of possible relationships between the four fundamental circuit variables: current, voltage, magnetic flux, and electric charge, professor Leon O. Chua predicted in 1971 the existence of the fourth basic circuit element [1]. He called the memristor and defined it as a passive device with two terminals, whose branch constitutive function relates magnetic flux and electric 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]. These milestones have boosted a series of developments in the field of memristor modelling and applications during the last years.
The memristor is an element that presents properties that can be used both for processing and storing information. These properties allow to develop a new paradigm of computation, a paradigm that bases on the behaviour of neurological systems in living beings [3]. This possibility suggests that the classical computation architecture, where the storing and processing are carried out in different blocks, can be replaced by an architecture where the processing and storing are performed in the same physical platform, in fact, this would shorten the processing time because the information is always available in the processor.
The current architecture of neuro-computing and artificial neural networks is developed with very little connection with neuroscience. Because of this, important features of biological neuronal systems have been ignored, such as extreme low power consumption or their ability to perform robust and efficient computation using massive parallel arrays of limited precision, or highly variable and unrealistic components [4]. This paradigm has been smartly pointed out in [5]: “The idea of taking inspiration from the brain’s structure and organization to build advanced systems has attracted a lot of interest. These artificial neural networks should be able to carry out tasks that are not easily or not at all tackled by traditional computer approaches, in particular tasks of a cognitive nature”. A better approach for the operation of biological neuronal
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 of variability in manufacturing processes, this variability and little precision can be taken as a basis for processing in another approach to neural processing, since it has been demonstrated that biological systems work with stochastic systems and high variability [8]. This paradigm is called memcomputing [9].
In the last years, studies have been done to carry out logical operations, in digital systems, with the exclusive use of memristors [10], or to measure parameters of stored data without the need for external processing [11]. This way of using the memristor allows to realize digital processing as it is known at the moment, but with the advantage that the information is stored in the own elements of operation.
This new architecture in computing, also allows to do analog processing within the same structure, which can be an advance in the hybridization of systems that carry out analog and digital processing in the same physical platform. Due to the variable nature of the resistance in the memristor, programmable analog processors can be developed which allow changing, for example, the characteristics of a particular filter or amplifier [12]. Other systems like memristive grids present properties that make them flexible in the development of analog applications.
Within INAOE, the CAD group has been focussed during the last years on the development of memristor models that are suitable for simulation of memristive systems 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 was developed to be used in AC regime, i.e. sinusoidal inputs, however, the applications that are studied in this work require a model that can be used in DC circuits and allows to carry out transient analyses. Consequently, the objective of this work is to develop an analytical model of the memristor fabricated by HP, to characterize this model for its application in systems with sources in AC and DC, and to show its usefulness in two applications that perform analog and parallel processing based on memristive grids. These applications are: image edge detection and shortest path method as a tool for solving labyrinths.
To show the work performed in this thesis, this document is organized in six more chapters. In Chapter 2, basic concepts about memristors, such as the ideal memristor definition and the equations governing the HP memristor, are presented. In Chapter 3, the development of the proposed model is shown and the symbolic equations obtained as function of the parameters of the memristor are presented. In Chapter 4, a summary of the characterization of the model is carried out: in the first part, the characterization is oriented to demonstrate that the model fulfils the main fingerprints of the device; in the second part, a DC characterization is done in order to determine the main static parameters of the memristive characteristics. In order to show the potential of the
developed model and the memristive grids, in Chapter 5, the memristive grid is used as a processor for image edge detection. The memristive grid, with little changes, is used in Chapter 6 for computing the shortest path in maze solving, demonstrating the advantages of the parallel processing that the grid performs. Finally, in Chapter 7 some conclusions are drawn and future lines of research are proposed.
Fundamentals of memristor and
modelling
As said before, there are two milestones about memristors, the introduction of the memristor as the fourth basic circuit element by Prof. Leon O. Chua in 1971 [1] and the first memristor fabricated by HP in 2008 [2]; from the latter one many studies about memristors has been published in recent years. Models have been developed in order 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 features about device modelling are presented.
2.1
The fourth basic circuit element
In circuit theory, four basic electrical variables are used, they are the electric chargeq, the magnetic fluxφ, the currentiand the voltagev. In order to have a relation between the for variables, there are six possibilities of linkage (figure2.1). Two of these relations are the well known definition of current
i= dq
dt; q(t) =
Z t
−∞
i(τ)dτ (2.1)
and Faraday’s law
v = dφ
dt; φ(t) =
Z t
−∞
v(τ)dτ (2.2)
Other three relationships are the definition of the basic circuit elements: the resistance R
v =Ri (2.3)
the capacitanceC
q =Cv (2.4)
and the inductanceL
i=Lφ (2.5)
6 CHAPTER 2. FUNDAMENTALS OF MEMRISTOR AND MODELLING
The last relationship, betweenqandφ, was proposed in 1971 by Leon O. Chua, and is given:
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 past history of the memristor current, presenting memory characteristics. The inverse version 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 figure2.2.
Figure 2.2: Memristor symbol.
Given the current i(t) or voltage v(t) in the memristor, its behaviour is as a linear time-varying resistor, or in the same way, theq-φ curve presents a non-linear relation (as shown figure 2.3 a)). In the specific case where the memristorq-φ characteristic is a straight line (as shown figure 2.3b)), its behaviour is like a linear time-invariant resistor[1].
a) b)
Figure 2.3:q-φcharacteristic of memristor.
2.1.1
Fingerprints of memristor
As reported in [15], the fingerprints are related with the features of thei-v curve in the memristor. A device can be considered a memristor if it fulfils these properties.
Given a memristor controlled by a sinusoidal current source i(t) = Apsin(ωt), there
are 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 PHL must pass through the pointv = 0andi = 0for any possible input amplitudeAp, and
any possible input frequencyω, as shown figure2.4.
8 CHAPTER 2. FUNDAMENTALS OF MEMRISTOR AND MODELLING
2) Reduction of the lobe area as the frequency increases
Above a critical frequency ωc, the area of the PHL decreases monotonically as the
frequency increases forω > ωc(as shown figure2.5).
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 the frequency ω tends to infinity. This property is maintained no matter what type of periodic excitation is used.
2.1.2
HP memristor
The HP memristor is a node in a crossbar structure composed by a nano-scale film of titanium oxide (TiO2) between two electrodes of platinum (P t). The TiO2 film has
two layers: the first one acting as an insulating layer, it has a relation oxygen-titanium of 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 figure2.6.
The total length of the element (doped region and undoped together) is given by the variable ∆, whereas the length of the doped region is denoted by the variable w. In function of the number of dopants, each region has an associated resistance. The resistance for the doped region is calledRon (resistance of the ON state of the device)
and that associated with the undoped region is calledRof f (resistance of the OFF state
of the device). The equivalent resistance is the sum of the total resistance in each region:
Req=Ron
w
∆+Rof f
1− w ∆
Figure 2.6: Structure of HP memristor.
The lengthwcan be normalized in the formx= w∆, wherexis called the state variable of the memristor and it can take a value between0-1. The variablexcan be controlled by the currenti(t)that passes through the element, the ratio of change inx(t)is directly proportional 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 mV s2, and η describes the displacement direction of x(t) (η = −1or +1) [16]. Equation2.9describes a linear drift mechanism of the HP memristor.
With the aim to model a non-linear drift mechanism, a window function fw(x(t))is
added to the equation2.9
dx(t)
dt =ηκi(t)fw(x(t)) (2.10)
This window function must fulfil certain characteristics: • fw(0) =fw(1) = 0to ensure no drift in the boundaries,
• fw(x(t))is symmetric aboutx(t) = 12, and
• monotonically increasing over the interval0≤x(t)≤ 1
2,0≤fw(x(t))≤1.
These properties guarantee that the difference between this model and the linear-drift model vanishes in the bulk of the memristor asw→ ∆
2 [16].
2.2
Types of models
The 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 basic blocks in circuits are the devices or circuit elements, thus, modelling the circuit
10 CHAPTER 2. FUNDAMENTALS OF MEMRISTOR AND MODELLING
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 knowledge about the device can be used as a tool for modelling. These types of models describe the device as a black box without any internal structure. In case the internal operation of the device is known, the black box can be described, and a model that defines the internal structure of the device can be developed. These two ways of abstracting the characteristics 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 to a determined stimulus. From experimental data, characteristic curves can be generated that describe the device, the points generated in these curves are used to obtain mathematical functions that copy the measured behaviour.
2.2.2
Structural models
Structural models try to reproduce the behaviour of a device based on the knowledge of the physical phenomena that describes it. That is, the internal structure of the device is taken into account to describe the black box that represents it. Depending on the level of abstraction used, structural models can be classified into physical models and analytical models.
Physical models
In these types of models, the structure of the model consists of a series of differential equations that describe the basic physical processes of the device. The model becomes more specific when the unique properties of the device are taken into account, and the boundary conditions associated with the geometry of the device and its external contacts are included in the description.
The validity of these types of models depends on the precision with which they describe the physical phenomena that compose it.
Analytical models
Analytical models can be seen as an abstraction of physical models. In this case, an analytical solution is found for the differential equations that describe the device, with this solution, a mathematical expression of the behaviour of the device is obtained.
A device can be represented by several analytical models, the precision of them depends on the number of approximations that have been made when solving the differential equations that describe them.
2.2.3
Approach of modelling in this work
In this work, a memristor model is developed based on the differential equation that governs its physical behaviour.
An analytical model is generated by starting from the approximated solution of the differential equation, using a series of mathematical tools. The model is presented as a series of mathematical expressions in function of the principal parameters of the HP memristor presented in the previous section.
Development of a charge-controlled
model
With the aim of developing a symbolic memristor model that can be used for different forms of current sources,in counter-position to the model reported in [13] which is defined for a sinusoidal current waveform, a symbolic charge-controlled model is developed 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 the charge function that control the memristance.
The methodology to obtain a symbolic expression for the memristance as a function of the electric charge can be described as: first, the non-linear drift mechanism is expressed as a function of charge instead of time; then, a homotopic perturbation method [18], [19], [20] is used to find a symbolic solution to the non-linear equation for the normalized state variable x(q); and finally, the x(q) is used to generate the memristance charge-controlled equation. As the method is defined for several orders of the homotopy formulation, and the window function used in the non-linear equation also has different exponent values, an extensive treatment of the resulting model is presented by combining 3 different orders with 5 indices of the window function.
3.1
Non-linear drift mechanism
The non-linear drift mechanism that governs the functioning the HP memristor [2] is defined by the ODE:
dx(t)
dt =ηκi(t)fw(x(t)) (3.1)
and the solutionx(t)is used to determine the memristance:
M(t) = Ronx(t) +Rof f(1−x(t)) (3.2)
14 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
In order to obtain a charge-dependent memristance model, equation3.3is modified by expressing the current as the time derivative of the electric charge, i(t) = dqdt, 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 obtained solving the differential equation and after that the memristance as a function of charge is obtained:
M(q) =Ronx(q) +Rof f(1−x(q)) (3.4)
The window function fw must be a bounded function between 0 and 1 in both its
domain 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 and OFF-state resistance interface. Several window functions have been reported in the literature. All of them are aimed to achieve a normalization of the state variable while preserving the physical behaviour of the memristance. In the next paragraphs, a brief introduction 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 the function presents asymmetry with respect to 0.5 in the x axis, that is reflected in a dependence of the drift directionη because the values of the function vary differently when traversing the domain from0 → 1 than from1 → 0. Besides ifa > 3.49then fw >1.
Another window function, proposed by Biolek, is reported in [21]. Thisfwis modelled
by
fw = 1−(x−stp(−i))2p
stp(i) =
1 if i≥0 0 if i < 0
(3.6)
whereiis the current that passes through the memristor. Some plots for different values of p in equation 3.6 are shown in figure 3.2. In this function, when p increases, the differential equation approximates to the linear drift case in the domain0→1of thex variable, therefore, it can be noticed that the function models the same behaviour when the current is positive and negative since in both casesfw goes from one to zero.
Figure 3.1: Window function proposed in [13] for different values ofa.
Figure 3.2: Window function proposed in [21] for different values ofp.
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 range0 → 1forx and the function is symmetric in both directions of drift as shown in figure3.3.
16 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
Figure 3.3: Joglekar window for different values ofk.
Because the window presents a symmetric bounded function, and it is modelled with a simple algebraic expression, it is used in several works about memristor modelling. The Joglekar window is selected to generate the symbolic expression for the charge-controlled memristor in this work.
Using the window function from3.7in the ODE in3.3, yields:
dx(q)
dq =ηκ 1−(2x(q)−1)
2k
(3.8)
It is possible to find an analytical solution to equation3.8fork = 1, however fork > 1a numerical analysis for the solution of the differential equation becomes necessary [16]. In order to obtain a symbolic solution of equation3.8, for different values of k, as a function 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 methods
Homotopic methods have been used in circuit theory mainly to solve the problem of multiple operating points in DC analysis for non-linear resistive circuits. Homotopy is based on the fact that the solutions are connected by a so-called solution curve; to obtain this curve an extra parameter is added to the original equation system, which converts the static problem into a dynamic problem that contains all the solutions of
the static problem, finding in theory all possible solutions.
In topology, if a continuous function f(x) can be deformed into another continuous functiong(x), then the deformation is called a homotopy betweenf(x)andg(x)[22] and it can be represented by
H :X×[0,1]→Y
H(x,0) = f(x) and H(x,1) = g(x) (3.9)
whereXandY are the topological spaces wheref(x)andg(x)belong respectively.
In this work, homotopy is used to find a symbolic solution to 3.8, by resorting to a complementary formulation with a perturbation parameter [13]. The homotopy perturbation method (HPM) relies on separating a differential equation into its linear and non-linear part
A(x)−f(r) = 0
L(x) +N(x)−f(r) = 0 (3.10)
whereA(x)is a differential operator, andL(x)andN(x)the linear and non-linear parts of 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 the solution 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 from0to 1. When p = 0, the initial state of the homotopy, the solution to the linear part is obtained
H(v,0) =L(v)−L(u0) = 0 (3.13)
whenp= 1the homotopy find the solution to the original equation
H(v,1) =L(x) +N(x)−f(r) = 0 (3.14)
The homotopic solutionv can be described by a power series ofp
v =p0v0+p1v1+p2v2+p3v3 +... (3.15)
Considering that p tends to 1, in the limit the solution for x (the solution to the differential equation) can be described as a sum of thev variables
x= lim
18 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
The order of the homotopy is defined by the number of vi terms that are taken to
approximate the solutionx(t)in3.16.
The methodology above is used to solve the ODE given in3.8. The resulting homotopy formulation is given as:
H(x(q), p) = (1−p)
dx(q)
dq −C1ηκx(q)
+p
dx(q)
dq −ηκfw(x(q))
= 0 (3.17)
whereC1is a coefficient to complete the linear part of the ODE.
3.3
HPM solution to the drift equation
The homotopy in3.17 must be solved forx(q). However, the choice of the homotopy order and the exponent of the Joglekar window function k has be to done. Solutions for orders 1,2, and 3 in combination with k = 1,2,3,4,5 have been obtained, with exception of the case order-3 & k = 5 due to the massive symbolic resulting expressions. Besides, it must be pointed out that a pair of solutions do indeed exist in every case because η takes values of +1 and −1 depending on the direction of the charge displacement.
As example of thex(q)solutions, the equation obtained for order-1,k= 3andη=−1 is given:
xk1,O3,η− = (X4
0 +X03+X02+X0)e−4κq−(3X04+ 2X03+X02)e
−8κq
+(3X04+X03)e−12κq−X04e−16κq (3.18)
where X0 corresponds to the initial value of the state variable (when the charge is
zero). It can be noted that the model only converges for positive values of q, and the function tends to0whenq→ ∞.
The solution forη= +1and positive values ofqis given by:
xk1,O3,η+ = 1 + (−X04+ 5X03−10X02+ 10X0−4)e−4κq
+(3X4
0 −14X03+ 25X02−20X0+ 6)e−8κq
+(−3X4
0 + 13X03−21X02+ 15X0 −4)e−12κq
+(X04−4X03+ 6X02−4X0+ 1)e−16κq
(3.19)
In order to have a comparison point for x(q), the numerical solution for the ODE3.8
is calculated with the same values ofk and homotopic orders. The numerical solution is obtained with Backward Euler integration. In figure 3.4 a) and b), a comparison of the model described by the equations 3.18 and3.19 with the numerical solution is shown. A better fitting can be observed with the numerical solution for order-3 than the other homotopy orders. The same analysis is performed for the otherx(q)solutions
and is shown in figures 3.4c) and d), and figure3.5. The values that are used for each parameter are presented in table3.1.
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 Joglekar window withk= 1and 2. 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.
20 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
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 Joglekar window withk = 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.
µv V s ∆nm κ As X0
1×10−14 10 10000 0.5
Table 3.1: Parameters for the plots ofx(q).
As it is shown in figures3.4c) and d), and figures3.5, the equations fork = 2,3,4and 5have an approximation with notorious variations. However, the approximation is still bounded forxand presents a behaviour according to the phenomenon of displacement of the dopants.
3.3.1
Memristance equations
Once the solutionx(q)has been obtained, this is substituted in equation3.4in order to determine a symbolic expression of the memristance.
The expressions for the k values developed and order-1 are shown in equations
3.20-3.29. 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 The homotopy order and thesignvalue can be−forη =−1or+whenη = 1. The value RddenotesRof f −Ron.
Expressions forη=−1
Mk1,O1,η− =
Rd(X0−1) [(X0−2)e4κq−(X0−1)e8κq] +Ron q ≤0
RdX0[X0e−8κq −(X0+ 1)e−4κq] +Rof f q >0
(3.20)
Mk2,O1,η− =
Rd(X0−1)
1 3(2X
3
0 + 3X0−8)e8κq−3(X0−1)e16κq
−2(X0−1)2e24κq −23(Xo−1)3e32κq
+Ron q≤0
RdX0
2 3X
3 0e
−32κq−2X2 0e
−24κq+X
0e−16κq
−1 3(2X
3
0 −6X02+ 9X0+ 3)e−8κq
+Rof f q >0
22 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
Mk3,O1,η− =
Rd(X0−1)
1 15
16X05−20X04+ 20X03
+15X0−46
e12κq
−5(X0−1)e24κq−203(X0−1)2e36κq
−203 (Xo−1)3e48κq−4(X
0−1)4e60κq
−16
15(X0−1) 5e72κq
+Ron q≤0
RdX0
16
15X05e−72κq−4X04e−60κq+203X03e−48κq
−20
3X02e−36κq+ 5X0e−24κq
−1 15
16X05−60X04+ 100X03
−100X02+ 75X0+ 15)e−12κq
+Rof f q >0
(3.22)
Mk4,O1,η−=
Rd(X0−1)
1 105
240X07−560X06+ 672X05
−420X04+ 210X03+ 105X0
−352
e16κq
−7(X0−1)e32κq−14(X0−1)2e48κq
−70
3(Xo−1)
3e64κq−28(X
0−1)4e80κq
−112
5 (X0−1)
5e96κq−32
3(X0−1) 6e112κq
−17
7(X0−1) 7e128κq
+Ron q≤0
RdX0
16 7X 7
0e−128κq−323X 6 0e−112κq
+1125 X5
0e−96κq−28X04e−80κq
+703X3
0e−64κq−14X02e−48κq
+7X0e−32κq
− 1
105
240X7
0−1120X06+ 2352X05
−2940X4
0+ 2450X03−1470X02
+735X0+ 105
e−16κq
+Rof f q >0
Mk5,O1,η− =
Rd(X0−1)
1 135
1792X9
0−6048X08
+9792X7
0−9408X06
+6048X5
0−2520X04
+840X3
0+ 315X0−1126
e20κq
−9(X0−1)e40κq−24(X0−1)2e60κq
−56(Xo−1)3e80κq−504
5 (X0−1) 4e100κq
−672
5 (X0−1)
5e120κq−128(X
0−1)6e140κq
−576
7 (X0−1)
7e160κq−32(X
0−1)8e180κq
−256
45(X0−1) 9e200κq
+Ron q≤0
RdX0
256 45X 9
0e−200κq−32X08e−180κq
+576 7 X
7
0e−160κq−128X06e−140κq
+672 5 X
5
0e−120κq−5045 X 4 0e−100κq
+56X3
0e−80κq−24X02e−60κq
+9X0e−40κq
− 1 135
1792X9
0−10080X08
+25920X7
0−40320X06
+42336X5
0−31752X04
+17640X03−7560X02
+2835X0+ 315
e−20κq
+Rof f q >0
(3.24)
Expressions forη= +1
Mk1,O1,η+ =
RdX0X0e8κq−(X0+ 1)e4κq+Rof f q≤0
Rd(X0−1)(X0−2)e−4κq−(X0−1)e−8κq+Ron q >0
(3.25)
Mk2,O1,η+=
RdX0
2 3X
3
0e32κq−2X02e24κq+X0e16κq
−1
3(2X 3
0−6X02+ 9X0+ 3)e8κq
+Rof f q≤0
Rd(X0−1)
1 3(2X
3
0+ 3X0−8)e−8κq−3(X0−1)e−16κq
−2(X0−1)2e−24κq−23(Xo−1)3e−32κq
+Ron q >0
24 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
Mk3,O1,η+=
RdX0
16 15X 5
0e72κq−4X04e60κq+ 20
3X 3 0e48κq
−20
3X 2
0e36κq+ 5X0e24κq
−1
15
165−60X4
0+ 100X03
−100X02+ 75X0+ 15
e12κq
+Rof f q≤0
Rd(X0−1)
1 15(16X
5 0−20X
4 0 + 20X
3
0+ 15X0−46)e−12κq
−5(X0−1)e−24κq−203(X0−1)2e−36κq
−20
3(Xo−1)
3e−48κq−4(X
0−1)4e−60κq
−16
15(X0−1) 5e−72κq
+Ron q >0
(3.27)
Mk4,O1,η+=
RdX0
16 7X 7 0e
128κq−32 3X
6 0e
112κq+112 5 X
5 0e
96κq
−28X04e80κq+703X
3
0e64κq−14X02e48κq+ 7X0e32κq
− 1
105
240X7
0−1120X06+ 2352X05
−2940X4
0+ 2450X03−1470X02
+735X0+ 105
e16κq
+Rof f q≤0
Rd(X0−1)
1 105
240X7
0−560X06+ 672X05
−420X4
0 + 210X03+ 105X0
−352
e−16κq
−7(X0−1)e−32κq−14(X0−1)2e−48κq
−70
3(Xo−1)
3e−64κq−28(X
0−1)4e−80κq
−112
5 (X0−1)
5e−96κq−32
3(X0−1)
6e−112κq
−17
7(X0−1)
7e−128κq
+Ron q >0
Mk5,O1,η+ =
RdX0
256 45X 9
0e200κq−32X08e180κq+ 576
7 X 7 0e160κq
−128X06e140κq+6725 X05e120κq−504
5 X 4 0e
100κq
+56X03e80κq−24X02e60κq+ 9X0e40κq
− 1 135
1792X9
0−10080X08+ 25920X07
−40320X6
0+ 42336X05−31752X04
+17640X3
0−7560X02+ 2835X0
+315
e20κq
+Rof f q≤0
Rd(X0−1)
1 135
1792X9
0−6048X08+ 9792X07
−9408X6
0 + 6048X05−2520X04
+840X3
0+ 315X0−1126
e−20κq
−9(X0−1)e−40κq−24(X0−1)2e−60κq
−56(Xo−1)3e−80κq−504
5 (X0−1)
4e−100κq
−672
5 (X0−1)
5e−120κq−128(X
0−1)6e−140κq
−576
7 (X0−1)
7e−160κq−32(X
0−1)8e−180κq
−256
45(X0−1)
9e−200κq
+Ron q >0
(3.29)
Nested structure of the memristance expressions
Figure 3.6: Nested form of the memristance equations for each value ofk.
Memristance expressions resulting from the combinations of homotopy orders and k’s are developed and presented in nested forms, i.e. a given memristance of order i is expressed as function of the memristance of order i−1and so forth, as shown figure3.6.
To show the nested structure of the expressions, the memristance for k = 1 in the Joglekar window and the three homotopic orders analysed are shown in equations 3.30, 3.31 and 3.32. It can be
seen that for the first homotopic order, the memristance depends on the parameters of the memristor and the electric charge. The structure presented in figure 3.6 can be checked for the second and third order memristance equations.
26 CHAPTER 3. DEVELOPMENT OF A CHARGE-CONTROLLED MODEL
Mk1,O1,η− =
Rd(X0−1) [(X0−2)e4κq−(X0−1)e8κq] +Ron q≤0
RdX0[X0e−8κq−(X0 + 1)e−4κq] +Rof f q >0
(3.30)
Mk1,O2,η− =Mk1,O1η−+Rd
(X0−1)3[−e4κq+ 2e8κq−e12κq] q ≤0
X03[−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
X04[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 equations summarized, where theFij factors are principally polynomials ofX0. The same table
present the structure for both negativeη and positive η, hence the superscript of η in each equation is ±. With the aim to reduce the space used in this document the Fij
factors are presented in Appendix A.
O 1 2 3
k= 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= 3 Mk3,O1,η± Mk3,O1,η±+RdF32± Mk3,O2,η±+RdF33±
k= 4 Mk4,O1,η± Mk4,O1,η±+RdF42± Mk4,O2,η±+RdF43±
k= 5 Mk5,O1,η± Mk5,O1,η±+RdF52± ———
Table 3.2: Nested memristance equations, structure.
Table3.3 shows the order of the polynomial ofX0 for each homotopic order andk of
memristance equations. A gradual advance in the value of the order of the polynomial as the order of homotopy grows can be observed, thus it can be deduced the order of the polynomial that hasX0 in the development of the homotopy for the Joglekar windows
as a function ofk and the homotopic order (n):
O k = 1 k= 2 k = 3 k = 4 k= 5
1 2 4 6 8 10
2 3 7 11 15 19
3 4 10 16 22 28
Characterization of the model
In this chapter, the memristor model, developed in the previous chapter, is extensively studied in order to determine the dependence of its main characteristics with respect to the parameters of the device. This is achieved with the aim of establishing the main fingerprints of the memristor. For sake of contrasting view, our model is compared with the memristor models reported in [2], [21], [23].
A first characterization is achieved in the dynamic domain in order to analyse the main memristor characteristics under a sinusoidal test signal. Secondly, a DC characterization is done with the purpose of determining the most important static characteristics and parameters of the device, including the meristance vs charge curves. This is used in the next chapters and applications.
The values of the HP memristor [2] are employed as nominal values in the characterization, as shown in table 4.1 (where Ap is the amplitude of the current
source employed to excite the memristor).
µv m
2
V s ∆nm κ
m
As RonΩ Rof f 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 are presented. First, fingerprints associated to the hysteresis loop are evaluated proving that the model fulfils them. Then, a comparison with the results reported in [2], Biolek’s model [21] and Affan’s model [23] are performed.
30 CHAPTER 4. CHARACTERIZATION OF THE MODEL
4.1.1
Fingerprints
The principal features for memristor are related with the hysteresis loop in the current-voltage characteristics, as was mentioned in Chapter 2, mainly a pinched hysteresis loop (PHL), area lobes decreases when frequency increases, and constant memristance when the frequency tends to infinity.
Pinched hysteresis loop (PHL)
In figure4.1, the hysteresis loop for the memristor model developed is observed. The curve for four different amplitudes of excitation current signal and the same frequency ω= 1for all cases is presented, proving that the characteristic loop of a memristor with passage through the origin is maintained regardless of the amplitude of the input signal. The PHL is presented for two values ofkin the Joglekar window; in the case of figure
4.1b) (k = 5), the saturation point is reached in the memristance since the greater the value ofkthe greater the opening in the lobes, that is why the lobes are deformed when the curve approaches the current axis, reaching in this case the minimum memristance Ron = 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 withk = 1. b) Hysteresis loop for a Joglekar window withk= 5.
Reduction of the lobe area as the frequency increases
From a critical frequency, the area in the hysteresis loop must decreases monotonically with respect to the increases in the frequency of the harmonic input signal [15]. In this way, as the excitation frequency increases, from a frequency depending on the
characteristics of the device, the hysteresis loop closes continuously.
In figure 4.2, the hysteresis loop is shown for two values ofk in the Joglekar window and four different values of ω in each case. It can be observed how the area lobes decreases asωgets larger, however, in the case ofk = 5the area forω = 1and 2 can not be distinguished, in figure 4.2 b), as greater to the area that present the lobes for higher frequencies; so, figure4.3shows a curve of the value of the area as a function of the frequency for the same cases of the Joglekar exponent. For a critical frequency, the lobe area decreases as frequency increases, on the contrary, for values smaller than this frequency the area increases as frequency increases, this is due to the closeness to the saturation points in the memristance.
a) b)
Figure 4.2: Variation of the hysteresis loop respect to frequency. a) Joglekar exponentk = 1. b) Joglekar exponentk= 5.
As was shown in figure 4.3, the critical frequency depend of the kJoglekar exponent, in general this frequency increases withkas can be observed in figure4.4and table4.2.
k 1 2 3 4 5
ωc 0.947 1.252 1.656 2.013 2.828
32 CHAPTER 4. CHARACTERIZATION OF THE MODEL
a) b)
Figure 4.3: Area for the lobe of the hysteresis loop as a function of the frequency, the units of area areµm2. a)k= 1,ωc= 0.947rads . b)k= 5,ωc= 2.828.
Figure 4.4: Frequency dependence of the area for different values of Joglekar exponentk.
Limit behaviour of the PHL at infinite frequency
When the excitation frequency tends to infinity, the value of the memristance becomes constant and the device acts as a linear resistor [15]. As reviewed in the previous section, as the frequency increases the area in the hysteresis cycle of the memristor decreases, so when the frequency tends to infinity the area is zero, making the memristance in the device constant.
Calculating the limit when the frequencyω → ∞for each memristance equation, it be obtained:
lim
ω→∞(Mki,Oj) =X0Ron+ (1−X0)Rof f =Rinit (4.1)
where ki refers to the five k values in the Joglekar window used in the model. The
limit corresponds to the value of the initial resistance in the device, this is because the state variablex, which models the position of the doping barrier in the device, can not respond to such rapid changes in the input signal.
4.1.2
Comparison with other models
In order to have a reference with other models, comparisons with the models reported in [2] (HP), [21] (Biolek) and [23] (Affan) are presented hereafter. For sake of comparison, two models are used, namelyMk1,O3andMk5,O2.
HP memristor
As can be observed in figure 4.5, both the charge-controlled model (withk = 1) and the HP results have a similar hysteresis loop but with a difference in the amplitude of the current response. The amplitude the charge-controlled model reaches is slightly higher than the HP results, where the current and voltage are normalized by a factor of 10mAand1V respectively.
a) b) extracted from [2]
c) d) extracted from [2]
Figure 4.5: Comparison of the PHL betweenk= 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 are presented in c) and d) respectively.
34 CHAPTER 4. CHARACTERIZATION OF THE MODEL
In figure 4.6, the hysteresis loop and current response of the model using k = 5 in the Joglekar exponent is presented. It can be seen that the amplitude of the source is enough for the model to reach the ON-state (M(t) = Ron), thus it causes a current
amplitude larger than the model withk= 1.
a) b)
Figure 4.6: PHL fork= 5homotopic order 2 of the model developed, in a), and its response in current 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 to be used in different applications. In figure4.7, the hysteresis loop for Biolek’s model, a macro model implemented in the SPICE platform, and Affan’s model, a mathematical model implemented in MATLAB, are shown and compared with the charge-controlled model. The charge-controlled model exhibits a slightly larger current swing that the other models.
−1 0 1
−200 −100 0 100 200
V(t) V
I(t)
µ
A
Charge−controlled Biolek
Affan
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).
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 is
verified that the model at any moment stops behaving as a passive element for all the ranges in which the parameters were varied. This is because it takes into account the saturation points in memristance,RonandRof f.
4.2
DC tests
The main features in the DC domain are the voltage thresholds and the saturation time tsat. The voltage necessary to switch, for η = +1, from the OFF-state (Rof f) to the
ON-state (Ron) is the positive voltage threshold V th+, and the voltage necessary to
switch from the ON-state to the OFF-state is the negative voltage threshold V th−
(when η = −1the switching voltage thresholds are contrary). The saturation time is the time that the element needs to switch between states.
4.2.1
Determining the switching voltages
In order to findV th±for the models withk = 1andk = 5the next steps are applied:
• The model is set in the OFF-state (X0 = 0.001, corresponding to an initial
resistanceRinit = 159.841KΩ).
• A voltage source as shown figures4.8a) 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 the V th−and reaches the OFF-state.
– Finally, again the voltage source is stopped.
In figure4.8c) and d), the current-voltage characteristic for the two models studied are shown. It can be seen that for both, the voltage thresholds are symmetric and measure 1.35V and 0.61V for k = 1andk = 5respectively, consequently a voltage source with these values can switch the initial state of the charge-controlled memristor models developed as it is shown in table4.3. But, what happens if the initial state is OFF and a negative voltage is applied? Or similarly, if the initial state is ON and a positive voltage is applied?
36 CHAPTER 4. CHARACTERIZATION OF THE MODEL
Vth k = 1 k= 5
OFF to ON 1.33V 0.64V ON to OFF −1.33V −0.64V
Table 4.3: Switching voltages fork1 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 inverse version of the voltage source is applied, the dynamic of memristance in the third quadrant is null because the element can’t change of state. The same happens in the another 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 dynamic memristance is present in the first quadrant. It is important to notice that these characteristics are shown for a model withη = 1, whenη = −1all the issues are the opposite.
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.
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 voltage source present in figure 4.8 a). In the plots, two states of the memristor are easily identifiable, and the slope measured in each region corresponds to the inverse of the resistance in the OFF-ON states (R1
of f = 6.25µS and 1
Ron = 1mS). These simulation
measurements show the direct relationship that the model maintains between electric charge and magnetic flux. The q-φ characteristic can be easily modelled by a piece-wise-linear equation, andRon-Rof f 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+
38 CHAPTER 4. CHARACTERIZATION OF THE MODEL
4.2.2
Determining the saturation time
With the threshold voltage obtained in the previous section, the next step is to apply a VDC =V thto obtain the saturation timetsatOF F−ON, the time that the memristor needs
to change from the OFF-state to the ON-state. In order to obtain a theoretical value of tsatOF F−ON, Strukov proposes in [2] the equation:
tsatOF F−ON ≈
∆2Rof f
2µV+Ron
(4.3)
whereV+is the value of the DC voltage source applied. Using the parameters presented
in table4.1and the threshold voltages in table4.3, the saturation time for both models are:
tsatOF F−ON
V+=1.33≈0.601s; tsatOF F−ON
V+=0.64 ≈1.25s (4.4)
Figure4.11 shows the saturation time measured in a simulation of the model, for the case of k = 1, the value obtained is 0.62s, closer to the theoretical value, but in the case ofk = 5, the measured value is0.33s. The error in the last case may be due to the abrupt changes that the high values ofkcause, generating a fast response of the model; but also, on the another side, the equation4.3is modelled for the lineal range between RonandRof f.
a) b)
c) d)
Figure 4.11: Saturation time measured in the memristance curve (a and b fork= 1andk= 5) and in the current response (c and b fork= 1andk= 5respectively).
4.2.3
Memristance-Charge characteristic
In order to know the behaviour of the charge-controlled memristor model as a function of 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.
Figure4.12shows the M-q characteristic of the model for the two values ofη. It can be seen that forη=−1the memristance tends toRof f in the positive range of the charge,
and tends toRonin the negative range ofq. In contrary, whenη= +1, the memristance
tends toRon in the positive range of the charge, and toRof f in the negative range. As
40 CHAPTER 4. CHARACTERIZATION OF THE MODEL
4.2.4
M-q Characteristics of memristor connections
In 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 memristors connected in the same polarity (or memristor with the sameη). In this case, the M-q characteristic for eachη, shown in figure4.12, is accentuated, as shown in figure4.13. The values ofRon,Rof f andRinitin the connection are:
Rons =Ron1 +Ron2
Rof fs =Rof f1 +Rof f2
Rinits =Rinit1 +Rinit2
(4.5)
a)Mk1,O3,η− b)Mk1,O3,η+
Figure 4.13: M-q characteristic for a series connection of two memristors.
Anti-series connection
Figures4.14 a) and b) show the meristance response for the model with positive and negativeη, starting in ON-state (X0 = 0.999,Rinit = 1159 Ω). Each type of memristor
has a suitable behaviour for a type of excitation, in the case of positiveη, it is possible to change the state by applying a negative current (voltage) source, in contrast, in case of negativeη, 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
to the OFF-state, applying a source regardless of its sign, a series connection of each type of memristor can be used as can be seen in figure4.14c) and d). If the memritive elements are of the same type, only inverting the polarity of one of the elements, the same results are obtained, this element is called anti-series connection. The anti-series connection between two memristors has a initial resistance
Rinitas =R +
init+R
−
init (4.6)
where the superscript indicates the sign ofη, and the resistance of the of state is given:
Rof fas,q− =R
−
on+R
+
of f; Rof fas,q+ =R
+
on+R
−
of f (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 negative
η. In c) and d), the M-q characteristic for the anti-series connection.
In a comparison for the value ofkin the anti-series connection, high values ofkimply a smaller amount of electric charge to reach the OFF-state.
42 CHAPTER 4. CHARACTERIZATION OF THE MODEL
Parallel connection
The parallel connection of two memristors in the same polarity (or sameη), reduce the ranges in memristance of the resulting M-q characteristic. This is because the equivalent of parameters Ron, Rof f and Rinit for the connection are the parallel
between the individual memristor parameters:
Ronp =Ron1 kRon2
Rof fp =Rof f1 kRof f2
Rinitp =Rinit1 kRinit2
(4.8)
The plots for the parallel connection are shown in figure4.15
a)Mk1,O3,η− b)Mk1,O3,η+
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 to the 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 the OFF-state. Forη = −1, switching to the ON-state is possible by applying a negative DC current (voltage) source and to the contrary, forη= +1, switching to the ON-state is possible by applying a positive current (voltage) source, as shown in figures4.16 a) and b). The M-q characteristic for an anti-parallel memristor connection is shown in figures4.16c) and d). For this connection the equivalent resistance parameters are:
Rinitap =R +
initkR
−
init
Ronap,q− =R
−
on kR
+
of f; Ronap,q+ =R
+
onkR
−
of f
When both memristors are connected in anti-parallel, theRinitequivalent is the parallel
resistance between the initial resistances of each type of memristor, and theRonisR−on
for negative charge andR+onfor 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 positive and negativeη, in c) and d), the M-q characteristic for an anti-parallel connection.
In a comparison for the value ofkin the anti-parallel connection, high values ofkimply a smaller amount of electric charge to reach the ON-state.
Memristive grid for edge detection
Among the basic types of image signal processing, edge detection constitutes a fundamental tool that a reliable image preprocessing algorithm for machine-based or computer-based vision has to include. In fact, edge detection can be regarded as decomposing the original image into a set of topographical curves that are linked to a measured depth level of intensity. As a clear result, the image contains diminished information 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 constitutive branch relationships of the elements in the array in order to carry out in fact a minimization algorithm.
In this chapter a brief description of the advances in resistive and non-linear resistive grids is presented, then the memristive grid and its components are explained and the solution to the memristive grid is expounded. Furthermore, using an image database extracted from [24], the memristive grid is tested for 500 images and the same images with noise. Finally a comparison with the well-known Canny’s method [25] is presented.
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 an image. In plain words, an edge is considered when the brightness changes sharply or when the image presents physical discontinuities [26].
There are several methods in the literature for image edge detection, they can be grouped in two categories, search-based and zero-crossing-based. In search-based methods, edges are detected by first computing a measure of the edge strength as the
