Modeling of Memristive Systems and Its Applications

Modeling of memristive systems and its applications

Jose´ Balaam Alarcon´ Angulo

Thesis submitted as a partial requirement for the degree of

Master in Science with specialty in Electronics at the

Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ San Andres´ Cholula, Puebla

Adviser:

Dr. Librado Arturo Sarmiento Reyes, INAOE

c INAOE 2017

The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole or in part

Modeling of memristive systems and its applications

Master’s Thesis

By: Jos´eBalaam Alarc´onAngulo

Adviser: Dr. Librado Arturo Sarmiento Reyes

Instituto Nacional de Astrof´ısica Optica´ y Electr´onica Electronics Department

San Andres´ Cholula, Puebla. January 16, 2018

Agradezco a mis padres Guadalupe Alarcóny Ma. Teresa y a mi hermana Dayanira Alarcónpor su incondicional apoyo a lo largo de toda mi vida. Al doctor Arturo Sarmiento por su orientacióndurante el desarrollo de esta tesis.Y a todos mis amigos que siempre estánah´ıe hicieron invaluables aportes para el desarrollo de este trabajo.

Modeling of memristive systems and its applications

iii

To my future readers. I hope you will enjoy reading this work as much as I did when writing it and that it will serves as a guide for future work.

Modeling of memristive systems and its applications

Resumen

Desde el advenimiento del memristor como elemento básicode circuito real, ha habido un significativo impulso en la investigaciónorientada al desarrollo de aplicaciones del memristor en diseñode circuitos y procesamiento de señales.Amplificadores con retroalimentacónnegativa basados en nullores se encuentran entre las aplicaciones más factibles debido a que la propiedad de memoria del memristor puede ser incorporada a la transferencia de amplificaciónal colocar el memristor directamente en el lazo de retroalimentación. Esta tesis introduce un modelo del memristor que resulta ad hoc para el análisis de amplificadores de retroalimentaciónnegativa basados en nullores. El modelo ha sido desarrollado utilizando el método de homotop´ıamodificado que resulta en una expresióntotalmente simbólicade la memristancia con una estructura de funciones armónicas.El modelo desarrollado puede ser codificado en Verilog-A para fines de simulacióneléctricade los amplificadores. Se ha realizado el análisisde distorsión armónicay el análisisde ruido para los cuatro tipos de amplificadores basados en nullores, de voltaje, de transmemristancia, de transmemductancia y de corriente. En un paso posterior, el nullor se implementa con un memistor, el cual consiste en la conexiónanti-serie de dos memristores. Obteniéndosede esta manera, un amplificador puramente memristivo. Finalmente un amplificador de transmemristencia se analiza como caso de estudio.

[v]

Abstract

Since the advent of the memristor as an actual basic element, a significant thrust of the research has been oriented to develop applications of the memristor in circuit design and signal processing. Nullor-based negative-feedback amplifiers are among the most directly feasible applications due to the fact that the memory property of the memristor can be incorporated to the overall transfer gain by placing a memristive feedback. This thesis introduces a tailored memristor model for the analysis of nullor-based negative-feedback memristive amplifiers. The model has been developed by using a modified homotopy method that yields a fully symbolic harmonic expression of the memristance. This model can be recast in a piece of Verilog-A code for electric simulation of the amplifiers. Harmonic distortion and noise analyses are carried out for the four types of nullor-based memristive amplifiers, namely, voltage, transmemristance, transmemductance and current amplifiers. In a further step, the nullor is implemented by a memistor which consists in the back-to-back connection of two memristors. As a result, it yields a full memristive amplifier. Finally, a transmemristance memistor- based configuration is tackled as a case study.

[vii]

Contents

Resumenv

Abstract vii

List of Figures xiii

List of Tables xv

1 Introduction1 1.1 Objective...... 2 1.2 Hypothesis...... 2 1.3 Methodology of the research...... 2

2 Fundamentals5 2.1 Memristor and memristive systems...... 5 2.2 The Nullor...... 9 2.3 Nullor-based ampliﬁers...... 10

3 Memristor model generation 13 3.1 Modelling methodology...... 13 3.2 Model characterization...... 16

4 The memistor 23 4.1 Introduction...... 24 4.2 Memistor with HPM memristor model...... 24 4.2.1 Characterization...... 25

[ix] x CONTENTS

5 Memristive nullor-based amplifiers 31 5.1 Basic topologies...... 32 5.1.1 Memristive voltage amplifier...... 32 5.1.2 Transmemristance amplifier...... 33 5.1.3 Transmemductance amplifier...... 35 5.1.4 Memristive current amplifier...... 36 5.1.5 Summary...... 37

6 Harmonic analysis 39 6.1 Introduction...... 39 6.2 Basic amplifiers & symbolic harmonic analysis...... 40 6.2.1 Memristive voltage amplifier...... 40 6.2.2 Transmemristance amplifier...... 41 6.2.3 Transmemductance amplifier...... 42 6.2.4 Memristive current amplifier...... 43 6.3 Simulation...... 44 6.3.1 Memristive voltage amplifier...... 44 6.3.2 Transmemristance amplifier...... 44 6.3.3 Transmemductance amplifier...... 45

7 Noise Analysis 47 7.1 Introduction...... 47 7.1.1 Thermal noise...... 47 7.2 Basic Ampliﬁers...... 49

8 Case of study 53 8.1 Noise in memistor...... 53 8.2 Transmemristance ampliﬁer...... 55 8.2.1 Noise simulations...... 55 8.2.2 Harmonic analysis...... 56

9 Conclusions and future work 59

Appendices 62

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ CONTENTS xi

A Appendix A 65

A.1 Code for Order-1k=7 ...... 65

A.2 Code for Order-3k=1 ...... 66

B Appendix B 67 B.1 Positive-gain transmemristance ampliﬁer...... 67 B.2 Positive-gain transmemductance ampliﬁer...... 69

Bibliography 71

Modeling of memristive systems and its applications xii CONTENTS

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ List of Figures

2.1 Basic circuit elements...... 6 2.2 Struture of HP memristor...... 7 2.3 Basic fingerprint for memristors...... 7 2.4 Structure of a nullor...... 9 2.5 Single-loop configuration of nullor-based negative-feedback amplifiers. 10

3.1 Joglekar window with some discrete values to k...... 14 3.2 Equivalent circuit of the coupled ohmic-tunnelin variable-resistor cir-

cuit model, where Ron is de ohmic resistance and Roff is the tunnelling resistance...... 15 3.3 Pinched Hysteresis Loop for both models and diﬀerents values of ω.. 17 3.4 M-I characteristics for several values of ω...... 18 3.5 Memristance vs frequency...... 19 3.6 Current & voltage of the memristor for ω = 1...... 20

3.7 Characterization of memristance vs Xo...... 20

3.8 Passivity vs Xo ...... 21

3.9 Monotonicity vs Xo ...... 21

4.1 The Widrow memistor...... 23 4.2 Memristor-based memistor...... 24 4.3 Circuit for the characterization of the memistor...... 25

4.4 Ix for several cases...... 26

4.5 Ix for several cases...... 27 π 4.6 Ix-VD characteristic for t = 0, 2 , π...... 28 π 4.7 Ix-VG characteristic for t = 0, 2 , π...... 29

[xiii] xiv LIST OF FIGURES

5.1 Basic topologies for the nullor as ideal amplifier...... 31 5.2 Pinched hysteresis transfer loop for the voltage amplifier at ω = 1.. 32 5.3 Voltage gain for ω = 1...... 33 5.4 Pinched hysteresis transfer loop for the transmemristance amplifier at ω = 1...... 34 5.5 Gain for ω = 1...... 34 5.6 Pinched hysteresis transfer loop for the transmemductance amplifier at ω = 1...... 35 5.7 Gain for ω = 1...... 36 5.8 Pinched hysteresis transfer loop for the voltage amplifier at ω = 1.. 36 5.9 Current gain for ω = 1...... 37

6.1 Voltage amplifier with input output voltage for ω = 1...... 40 6.2 Transmemristance amplifier with input current and output voltage for ω = 1...... 42 6.3 Transmemductance amplifier with input voltage and output current for ω = 1...... 43 6.4 Memristive current amplifier with input and output current for ω = 1. 43

7.1 Noise model of a resistor...... 48 7.2 Noise model of a memristor...... 49 7.3 Basice memristive ampliﬁers: noisy conﬁgurations...... 50

8.1 Nullor implementation with memistor...... 53 8.2 Memistor with noise contributions...... 54 8.3 Transmemristance ampliﬁer with memistor...... 55 8.4 Noise contributions in the transmemristance ampliﬁer with memistor. 56

8.5 Vo and Ii of the transmemristance ampliﬁer with memistor: simulated results...... 57

B.1 Transmemristance nullor-based amplifier...... 67 B.2 Transmemristance amplifier with input current and output voltage for ω = 1...... 68 B.3 Transmemductance amplifier with input voltage and output current for ω = 1...... 69

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ List of Tables

3.1 HP-memristor parameters...... 16 3.2 Memristance values for both models and for some values of ω..... 18

4.1 Cases for the characterization of the memistor...... 25

6.1 Normalised symbolic harmonic analysis for the memristive voltage amplifier...... 44 6.2 Normalised symbolic harmonic analysis for the transmemristance amplifier...... 44 6.3 Harmonic analysis for the transmemductance amplifier: simulations results...... 45

7.1 Noise analysis of the memristive ampliﬁers...... 51

8.1 Noise equivalent at the input of the memristor from evaluations of equation (8.2)...... 54 8.2 Total noise present at the input of the transmemristance ampliﬁer using memistor...... 55 8.3 Normalised symbolic harmonic analysis for the transmemristance am- pliﬁer...... 56

B.1 Normalised symbolic harmonic analysis for the positive-gain transmemristance ampliﬁer...... 68 B.2 Harmonic analysis for the transmemductance ampliﬁer: simulations results...... 69

[xv] xvi LIST OF TABLES

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ Chapter 1 Introduction

The history of memristor development has two milestones. The seminal paper of Prof. L.O. Chua from 1971 [1] is being unanimously hailed as the first milestone in memristor development. Herein, the memristor was introduced as the fourth basic circuit element that closes the loop around the main electrical variables, namely, voltage, current, electric charge and flux-linkage. The actual fabrication of a memristor t the Hewleet-Packard Laboratories in 2008 constitutes the second milestone in memristor development [2]. Since the advent of the memristor as an actual device, a significant thrust of the research has been oriented to develop applications of the memristor in circuit design and signal processing. In this thesis, we focus on the analysis of nullor-based negative-feedback memristive amplifiers which result from combining both the memristor and the nullor, which is a pathologic circuit element. The nullor was firstly employed by Carlin and Youla in 1961 [3], where the term “pathologic” was coined for both the nullator and the norator. However, the nullor was formally introduced in 1964 by H.J. Carlin [4]. In his work, Prof. Carlin wisely pointed out that the nullor exhibits a more natural (and understandable) behaviour when it appears interconnected with other circuit elements. However, it was Prof. Bernard D.H. Tellegen who stated the concept of the four ideal amplifiers in 1954 [5].

[1] 2 1. Introduction

1.1 Objective

The main objective of this thesis is to develop of a memristor model that can be used for the analysis of a special class of memristive systems, namely nullor-based negative-feedback memristive ampliﬁers. Key features are listed as follow:

Develop and characterize a fully symbolic harmonic memristor model. • Generate the basic conﬁgurations of single-loop nullor-based memristive conﬁg- • urations.

Determine the behavior for distortion and noise. • Generate a memistor suitable for the nullor synthesis. •

1.2 Hypothesis

Given that the nullor-based negative-feedback ampliﬁers features the nullor as the plant to be controlled and a feed-back network constituted by a passive component. The research focusses on the strength to obtain memristive transfer gains by placing a memristor in the feed-back network and to achieve harmonic distortion as well as noise analyses to the resulting memristive ampliﬁers. In order to perform such analyses, a specially tailored memristor model must be developed.

1.3 Methodology of the research

In order to conduct our research, a series of main tasks are listed as follow:

Determine the key characteristics for memristive ampliﬁers are the milstone for • this Thesis.

From the mechanism that describes the physical behavior of the memristor, a • model should be developed.

A full characterization of the memristor model must be done in order to verify • its robustness.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 1.3 Methodology of the research 3

Synthesize the nullor by using a memistor, which in turn is composed by an • anti-series connection of two memristors.

Achieve the complete characterization of the memistor. • Obtain the harmonic distortion analyses of the memristive ampliﬁers. • Get the noise analyses. •

Modeling of memristive systems and its applications

Chapter 2 Fundamentals

This Chapter focuses on the basic concepts regarding nullor-based memristive amplifiers. Section 2.1 introduces the basic concepts concerning the memristor and memristive systems, with emphasis on the main features of the memristor. Section 2.2 explains the concept of nullor and finally Section 2.3 gives an overview on the basic configurations of negative-feedback nullor-based amplifiers.

2.1 Memristor and memristive systems

The memristor is the fourth basic electric element that closes the loop around the fundamental electric variables of circuit theory, namely electric charge (q), voltage (v), current (i), and flux-linkage (ϕ), as sketched in Figure 2.1. Chua noted that only four different mathematical relations connecting pairs of the principal electric variables. The branch relationship for the resistor has a dependence of the voltage an current (v(t) = fR(i) or i(t) = fR(v)), the branch function of the capacitor is defined by the relationship between charge and voltage (q(t) = fC (v) or v(t) = fC (q)), and the inductor is delimited by the correlation between flux and current (ϕ(t) = fL(i) or i(t) = fL(ϕ)). In addition, the current is defined as the time derivate of the charge and the voltage is defined as the time derivate of the flux [6].

dq(t) Z t i(t) = or q(t) = i(τ)dτ (2.1) dt −∞ and

dϕ(t) Z t v(t) = or ϕ = v(τ)dτ (2.2) dt −∞

[5] 6 2. Fundamentals

which are depicted as the diagonal lines in Figure 2.1.

Figure 2.1: Basic circuit elements.

Due to the fact that electric charge and ﬂux-linkage are the variables involved in the memristor branch relationship, a memristor can either ﬂux-controled or charge- controlled:

q(t) = gM (ϕ) ϕ(t) = fM (q) (2.3) | {z } | {z } flux−controlled charge−controlled The voltage across a charge-controlled memristor can be found according to [7] and [8]:

df v(t) = M(q(t))i(t) M(q(t)) = M(q) (2.4) dq

Similarly, the current of a ﬂux-controlled memristor is given by:

dg (ϕ) i(t) = W (ϕ(t))v(t) W (ϕ(t)) = M (2.5) dϕ

where M(q) and W (ϕ) are de memristance and memductance respectively.

The HP memristor

The theoretical aspects of the memristor stayed in latency for nearly 30 years, until a nanometric memristor was actually fabricated at HP Labs [2,9]. The so-called HP memristor has a crossbar structure, which consists in a mesh of perpendicular

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 2.1 Memristor and memristive systems 7 wires. At the crossing points a sandwiched layered structure arises. The cross-section is fabricated by two TiO2 nanometric layers between two Pt electrodes. The ﬁrst layer is formed by TiO2 with a perfect 2:1 oxygen titanium ratio which works as an insulator. The second layer TiO2−x has a 5% deﬀect of vacancies, which mimics as a conductor. This structure is depicted in Figure 2.2.

Figure 2.2: Struture of HP memristor.

Memristor ﬁngerprints

The fingerprints are properties that the element must exhibit. In this work, we pay attention to three of them [10, 11, 12]. The first fingerprint is shown in Figure 2.3a, the current-voltage characteristic is a self-crossing by zero parametric curve denoted as the Pinched Hysteresis Loop (PHL). The second fingerprint establishes that the area of the PHL diminishes with the frequency, as shown in Figure 2.3b. Consequently, in the limit (as ω ) the PHL reduced itself to a rectline, i.e. the memristor behaves → ∞ as a linear resistor – as shown in Figure 2.3c.

(a) Pinched Hysteresis Loop. (b) Deterioration of the PHL (c) Memristance when ω → ∞. when ω → ∞.

Figure 2.3: Basic ﬁngerprint for memristors.

Modeling of memristive systems and its applications 8 2. Fundamentals

Memristive systems

Memristive systems are deﬁned as a dynamic system that contain an input (u), an output (y) and a state variable (x). The term was introduced by Chua in 1976 [13]. A memristive system can be expressed as:

y = g(x, u, t)u x˙ = f(x, u, t) (2.6) where f is a continuous n-dimensional function and g is a continuous scalar function. A memristive system is current-controlled when:

v = R(x, i, t)i x˙ = f(x, i, t) (2.7)

A memristive system is voltage-controlled when:

i = G(x, v, t)v x˙ = f(x, v, t) (2.8)

In [13], the properties of memristive systems were introduced. A list of them is hereafter given:

Passivity criterion. •

No storage of electricity. •

The operating DC is equivalent to time-invariant lineal resistor. •

Double value in Lissajous ﬁgure. •

Linearization of the Pinched Histeresis Loop. •

Small signal equivalent does exist if the memristive system is assimptotically • stable.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 2.2 The Nullor 9

2.2 The Nullor

The nullor was introduced as a two-port element deﬁned by a nullator at port 1 and a norator at port 2 by Carlin in [4] – as depicted in Figure 2.4. Later on, Carlin and Youla pointed that the nullor exhibits a more natural behaviour when it appears interconnected with other circuit elements [3]. It is important to mention that from a theoretical point of view, the nullor has been an excellent auxiliar in achieving circuit transformations not only in the context of circuit theory but also in network synthesis [5,7,8, 14]. The nullator at the input port handles:

vi = 0

ii = 0 (2.9) on the other side, the norator at the output port handles:

vo = × io = (2.10) × where indicates that both, voltage and current have arbitrary values that are de- × termined by the connected load, i.e. the devices that are connected at the output of the nullor.

Figure 2.4: Structure of a nullor.

If the equations 2.9 and 2.10 are combined, the resulting chain matrix K of the nullor is expressed as: " # " # AD 0 0 K = = (2.11) CD 0 0

Modeling of memristive systems and its applications 10 2. Fundamentals

with: A = 1 = vi = 0 B = 1 = vi = 0 µ vo γ io io=0 uo=0

C = 1 = ii = 0 D = 1 = ii = 0 ζ vo βF io io=0 uo=0 Therefore, the nullor models the four ideal inﬁnite gain ampliﬁers, as stated by Prof. Tellegen in his paper in 1954 [15]:

µ = γ = ∞ ∞ (2.12)

ζ = βF = ∞ ∞ This is the reason why the nullor has been used as an important building block in the design of negative feedback ﬁnite-gain ampliﬁers [16, 17].

2.3 Nullor-based ampliﬁers

(a) Voltage Ampliﬁer. (b) Transconductance Ampliﬁer.

Figure 2.5: Single-loop conﬁguration of nullor-based negative-feedback ampliﬁers.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 2.3 Nullor-based amplifiers 11

In this section, we introduce the basic topologies of negative feedback amplifiers that are constituted by a passive feedback network and the nullor as the active plant [16]. The resulting nullor-based single-loop amplifiers are shown in 2.5. The corresponding ideal closed-loop gains for the four amplifiers are given as:

R2 Av = 1 + Agm = G R1 − (2.13)

R2 Arm = RAI = 1 + − R1 where Av, Agm, Arm and AI are the voltage, transconductance, transresistance and current gains , respectively.

Modeling of memristive systems and its applications

Chapter 3 Memristor model generation

In order to obtain the adequate design for the memristive ampliﬁers introduced in the previous chapter, it clearly becomes necessary to develop a suitable model of the memristor. In Section 3.1, two memristor models are generated by resorting to a homotopy perturbation method (HPM). Furthermore, in Section 3.2, the generated models are fully characterized in order to verify the ﬁngerprints of the device.

3.1 Modelling methodology

The starting point of the methodology is the diﬀerential equation that describes the physical behavior of the device. The non-linear drift mechanism is deﬁne by [18]:

dx(t) µR = on i(t)f (x) (3.1) dt ∆2 w where ∆ stands for the full length of the semiconductor material and x(t) is the normalized state-variable (x = w/∆). Besides, µ is the mobility of the charges, Ron is the ON-state resistance and fw(x) is a window function that bounds the state- variable x i.e. fw(0) = fw(1) = 0 to guarantee no drift at the boundaries. The current is the stimuli function given as: i(t) = Ap sin(ωt) where Ap is the amplitude, and ω is the angular frequency. There are several window functions reported in the literature [19, 20]. In order to solve the nonlinear drift equation above, we resort to the Joglekar function [21]:

2k fw(x) = 1 (2x 1) (3.2) − −

[13] 14 3. Memristor model generation

where k aﬀects ﬂatness of the window curve as shown in Figure 3.1.

Figure 3.1: Joglekar window with some discrete values to k.

Finding a numerical solution to equation (3.1) has several shortcomings regarding not only the accuracy and the stability of the numerical algorithms, but also the lack of insight in the solution, and thus in the memristance behavior. Therefore, a symbolic solution is foreseen. A fully symbolic solution to the ordinary diﬀerential equation in (3.1) has been found by using the HPM from [22, 23, 24]. The HPM introduces a homotopy parameter p that takes values ranging from 0 up to 1. The nonlinear diﬀerential equation, for the case of HPM, can be expressed as:

L(v) + N(v) f(r) = 0 (3.3) − where L and N are the linear and nonlinear operators, and f(r) is a known analytic function. Therefore the homotopy formulation can be established as:

H(v, p) = (1 p)[L(v) L(uo)] + p[L(v) + N(v) f(r)] = 0 (3.4) − − −

where uo is the initial approximation for the solution of (3.4), and p is known as the perturbation homotopy parameter. Assuming that the solution (3.4) can be represented as a power series of p.

0 1 n v = p v0 + P v1 + + p vn (3.5) ···

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 3.1 Modelling methodology 15

Then, the approximate solution takes the form:

v = v0 + v1 + + vn (3.6) ··· An important parameter that aﬀects the complexity of the symbolic solution is n which is deﬁned as the homotopy order.

Memristance expression

Regarding the scheme that describes the memristance, an electrical equivalent of the memristance has been introduced in [25]. It consists of a series connexion of two coupled resistors, the ON-state resistance Ron and the OFF-state resistance Roff , as shown in Figure 3.2. The memristance can be expressed as:

M(t) = Ronx(t) + Roff (1 x(t)) (3.7) −

Figure 3.2: Equivalent circuit of the coupled ohmic-tunnelin variable-resistor circuit model, where Ron is de ohmic resistance and Roff is the tunnelling resistance.

The equation (3.1) has been solved for two homotopy orders, namely Order-1 and Order-3. The memristance expressions for Order-1 is:

2 MO1k=n = Ronfwk γ1(α 1)[ 1 + cos(ωt)] + Rinit − − 2k fw = fw(Xo) k=n = 1 (2Xo 1) k | − − (3.8) Rinit = [Xo + α(1 Xo)]Ron µAp − γ = ∆2ω where Rinit is the initial memristor resistance, with Xo as the initial condition of the state variable.

Modeling of memristive systems and its applications 16 3. Memristor model generation

The memristance expressions for Order-3 is:

2 MO3 = R fw γ1(α 1) [ 1 + cos(ωt)] + Rinit k=n on k − − +Ron3f f 0 γ2(α 1) 3 + cos(ωt) ( 1 ) cos(2ωt) wk wk 1 4 4 4 00 3 − − − +Ron fw f γ Px (α 1) [a0 + a1 cos(ωt) + a2 cos(2ωt) + a3 cos(3ωt)] k wk 1 o3kn − (3.9)

f 0 = 4k(2Xo 1)2k−1 wk − − f 00 = 8k(2k 1)(2Xo 1)2(k−1) wk − − −

The coeﬃcients ai and the polynomial Pxo3kn are function of the particular k that is selected. The expressions (3.8) and (3.9) constitute indeed a fully symbolic harmonic memristor model as function of the physical parameters. It clearly results that the homotopy order can be regarded as the number of harmonics in the obtained memristance expressions.

Parameter Value µ 10−10cm2/sV

Ron 100Ω ∆ 10nm

Ap 40µA α 160

Table 3.1: HP-memristor parameters

3.2 Model characterization

According to the memristance expressions shown elsewhere ((3.8) and (3.9)), two models are characterized, the models for Order-1 with k = 7 and Order-3 with k = 1.

The expression for Order-1k=7 is:

2 MO1k=7 = Ronfw7 γ1(α 1)[ 1 + cos(ωt)] + Rinit − − 14 fw = fw(Xo) k=7 = 1 (2Xo 1) 7 | − − (3.10) Rinit = [Xo + α(1 Xo)]Ron µAp − γ = ∆2ω

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 3.2 Model characterization 17

The memristance expression for Order-3k=1 is:

2 MO3k=1 = Ronfw1 γ1(α 1) [ 1 + cos(ωt)] + Rinit 3 0− 2 − 3 1 +Ron fw f γ (α 1) + cos(ωt) ( ) cos(2ωt) 1 w1 1 − − 4 − 4 +Ron4f f 00 γ3P (α 1) 5 5 cos(ωt) + 1 cos(2ωt) 1 cos(3ωt) w1 w1 1 xo3k1 6 4 2 12 − 2 − − fw1 = fw(Xo) k=1 = 1 (2Xo 1) 0 | − − fw1 = 4(2Xo 1) 00 − − fw1 = 8(2Xo 1) − 2 − Px = (6X 6Xo + 1) O3kn o − (3.11) Equations (3.10) and (3.11) have been recast in Verilog-A modules, that can be used for electric simulation of memristive circuits. Verilog-A codes are displayed in AppendixA. Further numeric evaluations of (3.10) and (3.11) are obtained in a very straightfor- ward form in order to verify the fingerprints of the memristor models. The evaluations are done using the nominal values corresponding to the well-known HP-memristor as listed in Table 3.1. The i(t)-v(t) pinched hysteresis loop (PHL) for several values of the angular frequency are shown in the family of parametric curves of Figure 3.3 for both models. It can be noticed that the area of these characteristics decreases as the frequency increases, which verifies the fingerprint.

(a) Order-1k=7. (b) Order-3k=1.

Figure 3.3: Pinched Hysteresis Loop for both models and diﬀerents values of ω.

Modeling of memristive systems and its applications 18 3. Memristor model generation

The memristance-current plots for the same set of ω values in both models are depicted in Figure 3.4 for both models. The memristance has an excursion that spans from a maximum to a minimum value. The diﬀerence between both values decreases with the frequency. The minimum and maximum values of the memristance are hereafter explained.

(a) Order-1k=7. (b) Order-3k=1.

Figure 3.4: M-I characteristics for several values of ω.

On one side, the maximum value of the memristance is the same for any frequency, and it is deﬁned as:

Mmax = M(t) t=0 = [Xo + α(1 Xo)]Ron (3.12) | − i.e. Mmax = Rinit and it depends only on the initial condition of the state variable, the on-state resistance and the ratio with the oﬀ-state.

O1 O3 ω k=7 k=1 Max Min Max Min 1 14.41KΩ 2.25KΩ 14.41KΩ 374Ω 2 14.41KΩ 8.33KΩ 14.41KΩ 1.02KΩ 5 14.41KΩ 11.98KΩ 14.41KΩ 13.23KΩ 10 14.41KΩ 13.19KΩ 14.41KΩ 13.89KΩ

Table 3.2: Memristance values for both models and for some values of ω.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 3.2 Model characterization 19

On the other side, the minimum value of the memristance occurs when the memristance-current plot takes the half of its excursion in time, and it is expressed as:

Mmin = M(t) t= π (3.13) | ω

Evaluations of expressions (3.12) and (3.13) with the nominal parameters are recast in Table 3.2 for several discrete values of the angular frequency ω.

(a) Order-1k=7. (b) Order-3k=1.

Figure 3.5: Memristance vs frequency.

Continuous plots of the memristance vs the frequency are shown in Figure 3.5. It is evident that the Order-3k=1 model shows a sharper behavior when asymptotically approaching to the maximum value.

Additionaly, the voltage and current waveforms of the memristor for ω = 1 are shown in Figure 3.6 for both models. In here, the plots exhibits the nullor condition (v = 0, i = 0) of the variables which veriﬁes another ﬁngerprint of the memristor.

Modeling of memristive systems and its applications 20 3. Memristor model generation

(a) Order-1k=7. (b) Order-3k=1.

Figure 3.6: Current & voltage of the memristor for ω = 1.

In order to illustrate the behavior of the pinched hysteresis loop for both models

against variations of the initial condition of the state variable Xo, 3D plots are shown in Figure 3.7.

(a) HPM Order-1k=7. (b) HPM Order-3k=1.

Figure 3.7: Characterization of memristance vs Xo.

Passivity

Passivity is analyzed against variations of the initial conditions Xo by obtaining the plots of the memristance for several ω values. These plots are shown in Figure 3.8 for both models. It can be clearly seen that passivity of the model Order-1k=7 is

guaranteed for Xo 0.2. For the other model, passivity is guaranteed for 0.35 ≤ ≤ Xo 0.88. ≤

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 3.2 Model characterization 21

(a) Passivity for Order-1k=7. (b) Passivity for model Order-3k=1.

Figure 3.8: Passivity vs Xo

Monotonicity criterion

Another important feature of the models is the shape of a certain curve against a parameter. The monotonicity of the memristance is analyzed vs the initial condition

Xo, which is done by obtaining the derivatives of the curves 3.8 with respect to Xo – as shown of Figure 3.9.

(a) Monotonicity for model Order-1k=7.

(b) Monotonicity for model Order-3k=1.

Figure 3.9: Monotonicity vs Xo

Modeling of memristive systems and its applications

Chapter 4 The memistor

This chapter is devoted to the memistor which is a three-terminal device that behaves as a potentiometer [26]. In fact, memistor and memristor are physically diﬀerent elements and their names are some how misleading. In Section 4.1, the memistor is introduced as a three-terminal device. Finally, in Section 4.2, the diﬀerence between the memistor and the memristor is explained.

Sensing current

Control current

(a) Pencil-lead memistor element (from [27]). (b) Electrical symbol of memistor.

Figure 4.1: The Widrow memistor.

[23] 24 4. The memistor

4.1 Introduction

The first time that the memistor was mentioned like a three-terminal device was by Widrow and Hoff in his technical laboratory report from 1960 [27], in which they were exploring the artificial neural networks through the development of an adaptive classification machine of patterns called Adelines (Adaptive Linear). They develop a first chemical memistor (a resistor with memory), which was fabricated with a thin graphite rod, sealed inside a glass tube that had a control electrode and a solution of sulphuric acid of copper and copper sulphate. This structure and the electrical symbol of the memistor are given in Figure 4.1. By applying a voltage between the graphite rod and the control electrode, the rod was covered by copper thus reducing its resistance. If a voltage with reverse polarity was applied, then resistance was increased. Later, a solid-state realization of the memistor was achieved by Thakoor et al [28, 29]. More recently, the memistor has been regarded as a transistor after an analysis from the memristive systems point of view [30].

Figure 4.2: Memristor-based memistor.

A memristor-based memistor was introduced in [31], consisting in a two back-to- back series connection of HP memristors. Figure 4.2 shows the scheme of the resulting memistor.

4.2 Memistor with HPM memristor model

In this section the implementation of the memristor-based memistor is carried out by using the memristance expressions obtained with the HPM method from the previous chapter.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 4.2 Memistor with HPM memristor model 25

The resulting memistor is further characterized by applying a methodology similar to the methodology used for BJT’s. [30]

4.2.1 Characterization

In order to obtain the transfer characteristic of the memistor, we resort to the characterization circuitry given in Figure 4.3, where VD and VG denote the sensing voltage and the control voltage respectively.

In this scheme, the memristors M1 and M2 are modeled by the equation of Order-

3k=1 (3.11).

Figure 4.3: Circuit for the characterization of the memistor.

The characterization of the memistor is carried out by considering seven cases that are listed in Table 4.1, which describes the ratio between the memristances M1 and M2.

Case 1 2 3 4 5 6 7 → 1 1 1 M1 = 2 M2 5 M2 10 M2 M2 M2 M2 M2 × × × 2 5 10 Table 4.1: Cases for the characterization of the memistor.

The plots in Figure 4.4 show the resulting currents IMx (t) when VG = 0.8 and VD has three values (1, 1.5 and 2 volts).

Modeling of memristive systems and its applications 26 4. The memistor

(a) Case 1. (b) Case 2. (c) Case 3.

(d) Case 4. (e) Case 5. (f) Case 6.

(g) Case 7.

Figure 4.4: Ix for several cases

The time-dependance of the current in Figure 4.4 comes from the fact that the expressions for the memristances are also time-dependent. The maximum current coincides when both Mmin’s occur, i.e. at the half of the period.

Another characterization is done for the complementary condition with VD = 0.8 and VG takes three values (0.8, 1.5 and 2 volts).

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 4.2 Memistor with HPM memristor model 27

(a) Case 1. (b) Case 2. (c) Case 3.

(d) Case 4. (e) Case 5. (f) Case 6.

(g) Case 7.

Figure 4.5: Ix for several cases

When both memristors are identical, VG has not eﬀect on the resulting current, which reaches a maximum peak of 2.2mA at the half of the period, which coincides again with the minimum memristances. For the rest of the cases, the voltage VG can be seen as a modulator voltage of the current.

A set of fully static characteristics (literally) taken at three diﬀerent time points are obtained in the following scheme: VG remains constant (0.8 volts), with a sweep π in VD within a range of 0 to 2 volts. The memristances are evaluated for t = 0, 2 , π seconds.

Modeling of memristive systems and its applications 28 4. The memistor

(a) Case 1. (b) Case 2. (c) Case 3.

(d) Case 4. (e) Case 5. (f) Case 6.

(g) Case 7.

π Figure 4.6: Ix-VD characteristic for t = 0, 2 , π.

The instantaneous Ix-VD characteristics from Figure 4.6 veriﬁes that the responses of the memistor at t = 0 and t = π are nearly identical. The maximum values of Ix for the VD sweep occur at the half period, i.e. when the instantaneous equivalent of the series memristance reaches its minimum.

Another set of static characteristics is generated by the following scheme: VD remains constant (2 volts), with a sweep in VG from 0 to 2 volts. Here again, the π memristances are evaluated for t = 0, 2 , π seconds.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 4.2 Memistor with HPM memristor model 29

(a) Case 1. (b) Case 2. (c) Case 3.

(d) Case 4. (e) Case 5. (f) Case 6.

(g) Case 7.

π Figure 4.7: Ix-VG characteristic for t = 0, 2 , π.

The instantaneous Ix-VG characteristics from Figure 4.7 shows horizontal lines for case 7. For the rest of the cases the slope of the rectlines is positive (negative) when the ratio is greater (lower) than one. Again, the characteristic with higher current is π for t = 2 .

Modeling of memristive systems and its applications

Chapter 5 Memristive nullor-based ampliﬁers

A particular case of memristive circuit application arises when the memristor is combined with the nullor, in order to achieve a memristive output-input transfer function.

(a) Voltage Ampliﬁer (b) Transconductance Ampliﬁer

Figure 5.1: Basic topologies for the nullor as ideal ampliﬁer

[31] 32 5. Memristive nullor-based amplifiers

5.1 Basic topologies

The memristive versions of the sigle-loop nullor-based negative-feedback ampliﬁers are shown in Figure 5.1. Under the assumption of instantaneous linearity [32], the gain of the ampliﬁers depicted in Figure 5.1 can be expressed as:

V M(t) V I I M(t) o (t) = 1 + o (t) = M(t) o (t) = W (t) o (t) = 1 + (5.1) Vi R Ii − Vi − Ii R

5.1.1 Memristive voltage ampliﬁer

Figure 5.1a shows the topology for the memristive voltage ampliﬁer and the gain is given as:

V M(t) o (t) = 1 + (5.2) Vi R

(a) Numeric evaluation with (b) Simulation in AnalogInsydes (c) Simulation in HSpice Maple

Figure 5.2: Pinched hysteresis transfer loop for the voltage ampliﬁer at ω = 1

By using the memristor model Order-3k=1, and choosing a value of 300Ω for R, the Vo Vi transfer characteristic of the amplifier is obtained. The resulting transfer − loops are shown in Figure 5.2. An usual way of report the behavior of the overal gain of the amplifier is by plotting the time-dependent gain functions for a given frequency [33]. For this amplifier, the time-varying voltage gain of the amplifier is shown in Figure 5.3 for ω = 1. This curve can be better understood by looking at the values of Mmax and Mmin in Table 3.2,

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 5.1 Basic topologies 33 which yield maximun and minimum gains, respectively: Vo M(t) 14410) max = 1 + = 1 + = 49.033 V R 300 i M(t)=Mmax Vo M(t) 374) min = 1 + = 1 + = 2.246 (5.3) V R 300 i M(t)=Mmin

The maximum gain occurs at t = 0, 2π, i.e. at the begin and the end of the PHL, which can be seen as the cross at the origin with maximum slope. On the other hand, the minimum gain occurs at the half of the period, i.e. when crossing the origin with minimum slope.

Figure 5.3: Voltage gain for ω = 1

5.1.2 Transmemristance ampliﬁer

Figure 5.1c shows the topology for the ampliﬁer, a similar treatment is done. The gain of this conﬁguration is given as:

V o (t) = M(t) (5.4) Ii − i.e. the gain is purely deﬁned by the memristance. The Vo Ii transfer characteristic − of the ampliﬁer is obtained and show in Figure 5.4.

Modeling of memristive systems and its applications 34 5. Memristive nullor-based amplifiers

(a) Numeric evaluation with (b) Simulation in AnalogInsyde (c) Transresistance Ampliﬁer maple

Figure 5.4: Pinched hysteresis transfer loop for the transmemristance ampliﬁer at ω = 1

The maximum and minimum gains at ω = 1 are in fact determined by the limit values of the memristance at ω = 1. Vo max = = Mmax = 14410Ω Ii − − Vo min = = Mmin = 374Ω (5.5) Ii − −

Notice that the maximum and minimum correspond to the absolute values of the gain, because the negative sign complies with the fact that the PHL transfer loop is on the second and fourth quadrants. Figure 5.5 shows the time-varying gain.

Figure 5.5: Gain for ω = 1

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 5.1 Basic topologies 35

5.1.3 Transmemductance ampliﬁer

Figure 5.1b shows the conﬁguration of this ampliﬁer, the gain is given as:

I 1 o (t) = = W (t) (5.6) Vi −M(t) −

The Io Vi transfer characteristics of the ampliﬁer is shown in Figure 5.6. −

(a) Numeric evaluation with (b) Simulation in AnalogInsyde (c) Transresistance Ampliﬁer maple

Figure 5.6: Pinched hysteresis transfer loop for the transmemductance ampliﬁer at ω = 1

The maximum and minimum inverting gains at ω = 1 are in fact determined by the limit values of the inverse of the memristance at ω = 1: Io 1 1 max = = = = 2.67mf Vi −Mmin −374 − Io 1 1 min = = = = 69.39µf (5.7) Vi −Mmax −14410 −

The PHL of the transfer, that Figure 5.6 depicts, has a steep slope in zero crossing when t = π. The plot of the gain as a function of time given in Figure 5.7 shows how the gain drops very fast in the vicinity of this time value. In general, the sharpness of the gain and the PHL indicate a highly nonlinear behaviour of this ampliﬁer.

Modeling of memristive systems and its applications 36 5. Memristive nullor-based amplifiers

Figure 5.7: Gain for ω = 1

5.1.4 Memristive current ampliﬁer

Figure 5.1d shows the connection of this ampliﬁer, its gain is given as:

I M(t) o (t) = 1 + (5.8) Ii R

The Io Ii transfer characteristic of the ampliﬁer is shown in Figure 5.8. The − time-dependent gain is shown in Figure 5.9. The maximum and minimum current gains are 49.03 and 2.24 respectively.

(a) Numeric evaluation with (b) Simulation in AnalogInsyde (c) Transresistance Ampliﬁer maple

Figure 5.8: Pinched hysteresis transfer loop for the voltage ampliﬁer at ω = 1

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 5.1 Basic topologies 37

Figure 5.9: Current gain for ω = 1

5.1.5 Summary

The substitution of the feedback network by a memristor in the basic topologies of nullor-based ampliﬁers leads to obtain overall transfer characteristics that mimic the memristor pinched hysteresis loop. Furthermore the constant gains of the original resistive ampliﬁers are transformed into time-varying transfer gains. All presented results have been obtained with Maple evaluations, AnalogInsydes and HSpice.

Modeling of memristive systems and its applications

Chapter 6 Harmonic analysis

Harmonics or harmonic frequencies of a periodic voltage or current are frequency components in the signal that are at integer multiplies of the frequency of the main signal. This is the basic outcome that Fourier analysis of a periodic signal shows. Harmonic distortion is the distortion of the signal due to these harmonics, that are introduced when a circuit has nonlinear elements and it is necessary to analyze its behavior. The use of the total harmonic distortion (THD) is perhaps the most widespread power quality index, and many electric utilities have adopted a THD-based measure of the limits of customer load currents. Distortion factor, a closely related term, is sometimes used as a synonym.

6.1 Introduction

The total harmonic distortion (THD) is a measurement of the harmonic distortion present in a signal that passes through a nonlinear device. It is deﬁned as the ratio of the sum of the power of all harmonic components to the power of the fundamental frequency [34, 35]. In the case of nullor-based memristive ampliﬁers the nonlinearty is introduced by the memristor, so the THD is the ratio of the sum of all harmonic components of the output voltage to the fundamental component of the output voltage. THD can be expressed as:

pv2 + v2 + + v2 THD = 2 3 ··· n (6.1) v1 where n is the maximum harmonic component, so vn is the output root mean square

[39] 40 6. Harmonic analysis

(RMS) voltage of the n-th harmonic and n = 1 is the output voltage of the fundamental frequency. The result of the equation 6.1 is a comparison of the output voltages of all harmonic components and the output voltage of the fundamental component.

6.2 Basic ampliﬁers & symbolic harmonic analysis

In this section, the memristor model described by equations (3.8) and (3.9) is used to obtain the symbolic expressions for the harmonic components of the output signal for the memristive conﬁgurations introduced in Chapter5.

6.2.1 Memristive voltage ampliﬁer

The voltage ampliﬁer and the input and output signals obtained from the symbolic simulation are shown in Figure 6.1 for ω = 1. We have chosen this value for ω since the memristor exhibits its higher nonlinearity at this frequency. Therfore, the transfer function is given as:

V M o = 1 + (6.2) Vi R

with R = 300Ω.

(a) Input and output waveforms (b) Voltage memristive nullor- (c) Input and output waveforms with model Order1k=7. based ampliﬁer. with model Order3k=1.

Figure 6.1: Voltage ampliﬁer with input output voltage for ω = 1.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 6.2 Basic amplifiers & symbolic harmonic analysis 41

The resulting symbolic expressions of the harmonic components for the model Order1k=7 are given as:

2 R+Rinit Ronfwnγ1(α−1) Fv = Ain O1 R − R (6.3) R2 f γ (α−1) H = A on wn 1 2vO1 in 2R

and for Order3k=1:

2 3 0 2 4 ” 3 R+Rinit Ronfwnγ1(α−1) 5 Ronfwnfwnγ1 (α−1) 7 Ronfwnfwnγ1 P (α−1) Fv = Ain + O3 R − R − 8 R 12 R

2 3 0 2 4 ” 3 Ronfwnγ1(α−1) Ronfwnfwnγ1 (α−1) 7 Ronfwnfwnγ1 P (α−1) H2 = Ain + vO3 2R 2R − 12 R (6.4)

4 ” 3 3 0 2 1 Ronfwnfwnγ1 P (α−1) 1 Ronfwnfwnγ1 (α−1) H3 = Ain vO3 4 R − 8 R

4 ” 3 1 Ronfwnfwnγ1 P (α−1) H4 = Ain vO3 − 24 R

where Fv and Hn stand for the fundamental, and n-th harmonic components of the output voltage, respectively. Equations (6.3) and (6.4) constitutes indeed the symbolic harmonic model for the memristive voltage ampliﬁer.

6.2.2 Transmemristance ampliﬁer

The transmemristance ampliﬁer and the input output signals obtained are shown in Figure 6.2 for ω = 1.The transfer function at this frecquency is given as:

V o = M (6.5) Ii −

The symbolic harmonic model yields the expressions of the harmonic component for the model Order1k=7 are given as:

2 Rinit Ronfwnγ1(α−1) Fm = Ain + O1 − R R (6.6) 2 Ronfwnγ1(α−1) H2 = Ain mO1 − 2R

Modeling of memristive systems and its applications 42 6. Harmonic analysis

(a) Input and output waveforms (b) Transmemristance nullor- (c) Input and output waveforms with model Order1k=7. based ampliﬁer. with model Order3k=1.

Figure 6.2: Transmemristance ampliﬁer with input current and output voltage for ω = 1.

and for Order3k=1:

2 3 0 2 4 ” 3 Rinit Ronfwnγ1(α−1) 5 Ronfwnfwnγ1 (α−1) 7 Ronfwnfwnγ1 P (α−1) Fm = Ain + + O3 − R R 8 R − 12 R

2 3 0 2 4 ” 3 Ronfwnγ1(α−1) Ronfwnfwnγ1 (α−1) 7 Ronfwnfwnγ1 P (α−1) H2 = Ain + mO3 − 2R − 2R 12 R (6.7)

4 ” 3 3 0 2 1 Ronfwnfwnγ1 P (α−1) 1 Ronfwnfwnγ1 (α−1) H3 = Ain + mO3 − 4 R 8 R

R4 f f ” γ3P (α−1) H = A 1 on wn wn 1 4mO3 in 24 R

In a similar way, Equations (6.6) and (6.7) are the symbolic harmonic model for the transmemristance ampliﬁer.

6.2.3 Transmemductance ampliﬁer

The transmemductance ampliﬁer and the input and output signals are shown in Figure 6.3 for ω = 1. The transfer function is given as:

I 1 o = = W (6.8) Vi −M −

From the waveform of the output current, it follows that this ampliﬁer exhibits the most nonlinear behaviour, which is a direct result of obtaining the transmemductance transfer function by inverting the memristance expression as denoted in equation (6.8). In fact it conveys to a sec-like function, which is well known for hav-

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 6.2 Basic amplifiers & symbolic harmonic analysis 43

(a) Input and output waveforms (b) Transmemductance nullor- (c) Input and output waveforms with model Order1k=7. based ampliﬁer. with model Order3k=1.

Figure 6.3: Transmemductance ampliﬁer with input voltage and output current for ω = 1. ing discontinuities.

6.2.4 Memristive current ampliﬁer

For this ampliﬁer, diagram as well as the input and output currents are shown in Figure 6.4 for ω = 1. The memristive transfer function is:

I M o = 1 + (6.9) Ii R

which results identical to the transfer function of the memristive voltage ampliﬁer. As

(a) Input and output waveforms (b) Memristive current nullor- (c) Input and output waveforms with model Order1k=7. based ampliﬁer. with model Order3k=1.

Figure 6.4: Memristive current ampliﬁer with input and output current for ω = 1.

a clear result, the harmonic analysis yields the same symbolic and numerical results.

Modeling of memristive systems and its applications 44 6. Harmonic analysis

6.3 Simulation

6.3.1 Memristive voltage ampliﬁer

The simulation was made in Hspice using a sinusoidal voltage source with value of 0.25 sin ωt and a resistor with value of 300Ω.

Memristive Order1 Order3 Voltage Amp. Evaluated Simulated Evaluated Simulated F 1 1 1 1 H2 332.416m 331.745m 346.504m 345.7994m H3 - 59.333m 71.755m 72.202m H4 - 1.954m 5.730m 5.505m H5 - 175.097µ - 189.061µ THD% 29.85 29.91 35.39 35.33

Table 6.1: Normalised symbolic harmonic analysis for the memristive voltage ampliﬁer

The results above are normalised with respect to the fundamental, and they appear in Table 6.1. Besides, columns 2 and 4 contains the evaluated items from the nominal values gives in Table 3.1. The simulation results (columns 3 and 5) are obtained by using HSPICE with the memristor model recast as a Verilog-A module.

6.3.2 Transmemristance ampliﬁer

From equations (6.6) and (6.7), the normalised harmonic components can be obtained, as shown in Table 6.2.

Transmemristance Order1 Order3 Amp. Evaluated Simulated Evaluated Simulated F 1 1 1 1 H2 345.914m 342.146m 357.44m 355.746m H3 - 55.387m 74.020m 73.959m H4 - 1.281m 5.911m 5.848m H5 - 161.702µ - 174.77µ THD% 31.967 31.180 36.507 36.34

Table 6.2: Normalised symbolic harmonic analysis for the transmemristance ampliﬁer

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 6.3 Simulation 45

6.3.3 Transmemductance ampliﬁer

The simulations results allow to verify that the transmemductance ampliﬁer is the most nonlinear, as shown in Table 6.3.

Transmemductance Order1 Order3 Simulation Simulation Ampliﬁer F 1 1 H2 631.456m 650.989m H3 549.848m 574.594m H4 439.281m 462-649m H5 195.384m 211.17m THD 100.942 105.654

Table 6.3: Harmonic analysis for the transmemductance ampliﬁer: simulations results.

Modeling of memristive systems and its applications

Chapter 7 Noise Analysis

The problems caused by electrical noise are visible in the output device of an electrical system, but the sources of noise are unique to the low-signal-level portions of the system. As an example, the snow that may be observed on a television receiver display is the result of internally generated noise in the first stages of signal amplification. This chapter explains the thermal noise that is generated by the devices in a circuit. Section 7.1 defines the fundamentals types of noise present in electronic systems. Section 7.2 expound the noise generated by the feedback and a comparison is made between the numerical and simulation results for the nullor-based memristivie amplifiers.

7.1 Introduction

Noise in the broadest sense, can be deﬁned as any unwanted disturbance that obscures or interferes with a desired signal [36]. In electronic the noise is a totally random signal which consists of frequency components that are random in both amplitude and phase. Although the long-term rms value can be measured, the exact amplitude at any instant of time cannot be predicted. The word noise is used to represent basic random-noise generators or spontaneous ﬂuctuations that result from the physics of the devices and materials that compose an electrical circuit. Most of the noise has a Gaussian or normal distribution of instantaneous amplitudes in time [37].

7.1.1 Thermal noise

Due to the mechanism that causes the noise, noise can be classiﬁed as: thermal noise, shot noise, and low-frequency (f −1) noise. In the analysis of nullor-based memristive

[47] 48 7. Noise Analysis amplifiers, thermal noise constitutes the main origin of noise. Thermal noise is caused by the random thermally excited vibration of the charge carriers in a conductor. It was first observed by J. B. Johnson of Bell Telephone Laboratories in 1927 [38], and a theoretical analysis was provided by H. Nyquist in 1928 [39]. Because of their work thermal noise also is called Johnson noise or Nyquist noise. In every resistor, the electrons are in random motion, and this vibration is dependent on temperature. Since each electron carries a charge of 1.602 10−19C, there × are many current peaks as electrons randomly move about in the material [40]. Al- though the average current in the conductor resulting from these movements is zero, instantaneously there is a current fluctuation that gives rise to a voltage across the terminals of the resistor. The noise can be represented as a series noise voltage source in series with the noiseless resistor as shown in Figure 7.1. The power density can be expressed as:

2 2 v¯n = 4kT R (V /Hz) (7.1) where k is Boltzmann’s constant, T is the absolute operating temperature of the resistor in kelvins [K], and R is the value of the resistor [Ω]. Norton’s equivalent allows us to express the noise current as:

4kT ¯i2 = (A2/Hz) (7.2) n R

+ vn − R in

(a) Voltage noise source (b) Current noise source

Figure 7.1: Noise model of a resistor

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 7.2 Basic Amplifiers 49

In order to determine how the noise of the memristor can be represented, it is useful to point out that the memristor models given by equations (3.10) and (3.11) imply an ohmic relationship.

v(t) = M(t)i(t) (7.3)

Under the assumption of instantaneous linearity [32], the noise of the memristor is regarded in a way similar to the noise in a linear resistor. Herein, the power density can be expressed as:

v¯2 = 4kT M (V 2/Hz) (7.4) nM and the noise current is given by:

4kT ¯i2 = (A2/Hz) (7.5) nM M where M denotes de instant memristance. The equivalent circuit noise for the memristor are given in Figure 7.2.

+ vnM − M inM M

(a) Voltage noise source (b) Current noise source

Figure 7.2: Noise model of a memristor

7.2 Basic Ampliﬁers

In the basic schemes of the nullor-based memristive amplifiers, the noise comes from the resistors and memristor. Figure 7.3 shows the diagram of the memristive amplifiers with the sources of noise that contribute to the overall noise. As mentioned, the noise contributions arise from the source resistor Rs and the feedback network. For the case of the voltage and current amplifiers R and M are the noise contributiors, while for the transmemristance and transmemductance amplifiers, M is the only contributor.

Modeling of memristive systems and its applications 50 7. Noise Analysis

Besides, even though the nullor is noiseless, two noise contributions are set apart for the nullor when it is synthesised by an active device in forthcoming step, namely a

noise voltage vnn and noise current inn [16].

Rs Rs Un Un N - + N - + Un Un Rs I Rs I nN nN + - + - + + U U - s - s

Un M R1 M RL RL

Un Un M R1 M

(a) Voltage Ampliﬁer. (b) Transmemductance Ampliﬁer. U U nM M nM M

U U nN nN

- + - + I I nN nN + - + - I R I I R I S S nS S S nS

U nR

RL RL R1

Figure 7.3: Basice memristive ampliﬁers: noisy conﬁgurations.

The equivalent input noise of the voltage ampliﬁer is given by:

RM RM RM R+M R+M vn = vn + vn + Rs + in + vn + vn (7.6) eqin s n R + M n R R R M for the transmemristance ampliﬁer is:

1 1 1 i = i + i + + v + v (7.7) neqin ns nn nn nM Rs M M

for transmenductance ampliﬁer is:

v = v + v + (R + M)i + Mi (7.8) neqin ns nn s nn nM

and ﬁnally for the current ampliﬁer:

1 1 R M i = i + i + + v + i + i (7.9) neqin ns nn nn nR nM Rs M + R R + M R + M

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 7.2 Basic Amplifiers 51

In fact, equations (7.6), (7.7), (7.8) and (7.9) represent the equivalent noise at the input of the ampliﬁers. Evaluations of these equations and simulation with Hspice are summarized in Table 7.1.

Voltage Current Transmemristance Transmemductance Amplifier Amplifier t Amplifier Amplifier Rs = 100, R = 300 Rs = 100, R = 300 Evaluated Simulated Evaluated Simulated Evaluated Simulated Evaluated Simulated 0 3.7853nV 2.5466nV 1.1127pA 1.0697pA 16.7344nV 15.4605nV 1.2423pA 1.0588pA π 3.6319nV 2.0946nV 6.6909pA 6.6352pA 3.7770nV 2.7937nV 7.0376pA 4.9428pA

Table 7.1: Noise analysis of the memristive ampliﬁers.

The deviations between evaluated and simulated results arise from the fact that the nullor has been approximated by high-gain controlled sources because of the lack of the nullor as an available component in the simulator.

Modeling of memristive systems and its applications

Chapter 8 Case of study

The synthesis of the nullor with active devices has been already matter of study by several researchers and scholars [41, 17, 16, 42, 43, 44, 45]. The synthesis methodology is carried out by attending a series of design guidelines with the aim of fulﬁlling the user speciﬁcations. In particular, the research just mentioned has been focussed on tackling noise, distortion and bandwidth. The nullor is implemented by the memistor explained in Chapter4, in order to establish a two-port network. The nullor implementation results from selecting one of the end terminals of the back-to-back connection as the common terminal of the two-port, as depicted in Figure 8.1. The memristor model given by the equation (3.9) is used to describe the memristors that appear in the scheme.

- +

+ -

Figure 8.1: Nullor implementation with memistor

8.1 Noise in memistor

As established before, the memristor contributes with noise in the same form a resistor does, and as result of this, all possible sources of noise were treated in Chapter7. In

[53] 54 8. Case of study

addition, the noise contributions for the nullor were temporarily set aside because of the ideal properties of the nullor. However, equations (7.6)-(7.9) already considered the forthcoming noise contributions of the nullor when it is synthesyzed with devices. In our nullor implementation with a memistor, it clearly results that they contribute with noise currents as depicted in Figure 8.2. According to equation (7.5), the noise currents for the memristors are given by:

4kT 4kT ¯i2 = ¯i2 = (8.1) nM nM S MS D MD

in MD - +

MD M in S MS + -

Figure 8.2: Memistor with noise contributions

After manipulating the ¯i and ¯i , the amount of equivalent noise present at nMS nMD the input of the memistor two port can be expressed as: MS +MD in = in + 1 + in n MS MS MD (8.2) v = M i nn D nMD

Both equations are incorporated to equations (7.6)-(7.9) in order to calculate the total noise present at the input of the ampliﬁers. The equivalent noise contributions at the input of the memistor from equation (8.2) are avaluated at t = 0, π and the values are summarized in Table 8.1.

t inn vnn 0 4.288033476 pA 15.44764060 nV π 26.60082721 pA 24.90148125 nV

Table 8.1: Noise equivalent at the input of the memristor from evaluations of equation (8.2).

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 8.2 Transmemristance amplifier 55

8.2 Transmemristance ampliﬁer

As an example of the use of the memistor for determining a transmemristive transfer in a nullor-based amplifier, the transmemristance amplifier is designed by resorting to the memistor configuration. This is shcematically shown in Figure 8.3

- + + MD vo Rs RL i MS s - + -

Figure 8.3: Transmemristance ampliﬁer with memistor.

8.2.1 Noise simulations

Figure 8.4 depicts the transmemristance ampliﬁer with all components having at- tached the noise contributions. In fact the contributions of the nullor, namale vnn and inn , are result of equation (8.2). The memristor of the feedback network M, and the memristors of the memistor are modelled by the equation (3.9). It is possible to determine that the total noise at the input of the ampliﬁer can be expressed as:

i = i + i (8.3) neqin ns Ms

Evaluations of equations (8.3) and results from Hspice simulations are recast in Table 8.2 for t = 0, π.

t Evaluated Simulated 0 1.1127 pA 1.3118 nA π 6.691 pA 8.1268 nA

Table 8.2: Total noise present at the input of the transmemristance ampliﬁer using memistor.

Modeling of memristive systems and its applications 56 8. Case of study i nM

M v i nn nM + - D - + + MD vo Rs RL i MS s in in in - S n MS + -

Figure 8.4: Noise contributions in the transmemristance ampliﬁer with memistor.

8.2.2 Harmonic analysis

Table 8.3 shows the simulation results of the ampliﬁer for the harmonic analysis. The columns corresponding to the memristive ampliﬁer when using an ideal nullor are repeated from Tables 6.2 for sake of comparsion.

Transmemristance Order1 Order3 Memistor Ampliﬁer Evaluated Simulated Evaluated Simulated Simulated F 1 1 1 1 1

H2 345.914m 342.146m 357.44m 355746m 356.112m H3 - 55.387m 74.020m 73.959m 73.8547m H4 - 1.281m 5.9113m 5.848m 5.777m H5 - 161.702µ - 174.770µ 170.0354µ THD 31.967 31.180 36.507 36.34 36.3736

Table 8.3: Normalised symbolic harmonic analysis for the transmemristance ampliﬁer

Finally, Figure 8.5 shows the simulated waveforms of the input current and output voltage of the transmemristance amplifier. It can be noticed that these waveforms are very similar. Those are close to the waveforms for the case of the nullor-based amplifier from Figure 6.2c, which justifies that the nullor canb be synthesized by a memistor.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 8.2 Transmemristance amplifier 57

Figure 8.5: Vo and Ii of the transmemristance ampliﬁer with memistor: simulated results.

Modeling of memristive systems and its applications

Chapter 9 Conclusions and future work

In this thesis, we have demostrated that memristive transfer gains can be achieved by nullor-based memristive amplifiers. In order to do this, a fully-symbolic harmonic- structured model for a memristor has been developed in first instance. The model has been obtained from the differential equation that describes the nonlinear drift conduction mechanism of the memristor. The model is fully characterized in order to verify that the memristor fingerprints are fulfilled. Furthermore, the model has been applied to the analysis of nullor-based memristive amplifiers. These amplifiers are constituted by the nullor, as the plant to be controlled, and the feedback network, where the memristor is placed. The four basic single-loop memristive configurations have been analyzed, namely the voltage, transmemductance, transmemristance and current gains. An important feature of the memristive amplifiers is that the typical pinched-hysteresis loop of the memristor is mimicked by the resulting overall transfer gain for all configurations. Besides, by taking advantage of the harmonic structure of the developed memristor model, a fully symbolic harmonic analysis of the four configurations is carried out. In addition, noise analysis of the four amplifiers has been done under the assumption of instantaneous linearity. In an ulterior step, the synthesis of the nullor is done by implementing a memistor with two back-to-back series connection of memristors. The resulting 3-terminal element is fully characterized when the developed model is employed. It clearly results that including the memistor converts the amplifier into a full-memristor circuit. Harmonic distortion and noise analyses are applied in a case study consisting in a transmemristance amplifier. Finally, it is worthy of mentioning that the developed symbolic models have been recast in Verilog-A in order to carry out electric simulations. The results from electric

[59] 60 9. Conclusions and future work simulation agree totally with the numerical evaluations of the symbolic expressions which veriﬁes the correctness of the memristor model.

Future work

There do exist several lines for future developments due to the novelty of the topic. An obvious thread is to extend this work to double-loop nullor-based configurations. The use of memcapacitors and meminductors in the feedback network in order to study a wider class of possible mem-transfer characteristics is also another line of future work. In the field of mathematical modeling of the memristor, the development of other kinds of models, such as charge-controlled models, and their applications to memristive amplifiers is also a topic to be developed in short-term. The aforemen- tioned topics are intended to end-up in the development of a structured synthesis methodology for memristive nullor-based amplifiers.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ Appendices

[61]

Appendix Appendix A

In the following, the memristance expressions given in (3.10)(3.11) are recast in Verilog-A modules. They represent indeed the behavioral models of the memristor that can be used in electrical simulation.

A.1 Code for Order-1k=7

[63] 64 A. Appendix A

mem = Ron*Ron*(fw)*(Gamma)*(alpha-1)*(-1+cos(omega*($abstime)))+Rinit; I(in,out) <+ (V(in,out))/mem; end endmodule

A.2 Code for Order-3k=1

‘include "const.va" ‘include "std.va" ‘include "disciplines.vams" module Memristor(in,out); inout in,out; electrical in,out; parameter real Delta = 10.0e-9; parameter real Pi = 3.1416; parameter real Ron = 100; parameter real mu = 1.0e-14; parameter real alpha = 160; parameter real Ap = 40.0e-6; parameter real omega = 1; parameter real Xo = 0.1; real mem, gamma1; analog begin gamma1 = ((mu*Ap)/(Delta*Delta*omega)); mem = Ron*Ron*(fw)*(Gamma)*(alpha-1)*(-1+cos(omega*($abstime)))+Rinit + pow(Ron,3)*fw*dfw*(Gamma*Gamma)*(alpha-1)*(cos(omega*($abstime))-3/4* cos(2*omega*($abstime))) +pow(Ron,4)*fw*ddfw*(pow(Gamma,3))*(alpha-1)*(5/4*cos(omega*($abstime))-1/2* cos(2*omega*($abstime))-1/12*cos(3*omega*($abstime)); I(in,out) <+ (V(in,out))/mem; end endmodule

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ Appendix Appendix B

Due to the fact that the memristor has polarity, the transfer functions of the nullor- based memristive ampliﬁers change sign depending on the memristor polarity. In this appendix, analyses for the transmemristance and transmemductance ampliﬁers are presented when the memristor is placed in reverse polarization.

B.1 Positive-gain transmemristance ampliﬁer

The conﬁguration for the transmemristance ampliﬁer with the memristor in reverse polarization is shown in Figure B.1, in counterposition to Figure 5.1c. The transfer function – here again under the assumption of instantaneous linearity, is given as:

V o = M (B.1) Ii which in fact is the unsigned version of equation (5.6).

Figure B.1: Transmemristance nullor-based ampliﬁer

[65] 66 B. Appendix B

Evaluation of the expression in equation (B.1), yields the waveforms for the output voltage and input current as shown in Figure B.2 for two diﬀerent orders of the memristor model.

(a) Input and output waveforms with model (b) Input and output waveforms with model Order1k=7. Order3k=1.

Figure B.2: Transmemristance ampliﬁer with input current and output voltage for ω = 1.

Regarding the harmonic analysis of this conﬁguration, the evaluated and simulated results an identical to the normalized values for the inverting conﬁguration as shown in Table B.1.

Table B.1: Normalised symbolic harmonic analysis for the positive-gain transmemristance ampliﬁer.

Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ B.2 Positive-gain transmemductance amplifier 67

B.2 Positive-gain transmemductance ampliﬁer

Figure B.3 depicts the topology when the memristor is in reverse polaritation. The transfer function change by:

I o = W (B.2) Vi this diﬀerence in the sign of function is due to the polarity of memristor. The nor-

(a) Io − Vi ratio with model (b) Transmemductance nullor- (c) Io − Ii ratio with model Order1k=7 based ampliﬁer Order3k=1

Figure B.3: Transmemductance ampliﬁer with input voltage and output current for ω = 1 malized values for the harmonic are show in Table B.2

Transmemductance Order1 Order3 Simulation Simulation Ampliﬁer F 1 1 H2 631.456m 650.989m H3 549.848m 574.594m H4 439.281m 462-649m H5 195.384m 211.17m THD 100.942 105.654

Table B.2: Harmonic analysis for the transmemductance ampliﬁer: simulations results.

Modeling of memristive systems and its applications

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Modeling of memristive systems and its applications