Modeling of memristive systems and its applicationsThe main objective of this thesis is to develop...

Modeling of memristive systemsand its applications

By

Jose Balaam Alarcon Angulo

Thesis submitted as a partial requirement for thedegree of

Master in Science with specialty in Electronics

at the

Instituto Nacional de Astrofısica, Optica yElectronica

San Andres Cholula, Puebla

Adviser:

Dr. Librado Arturo Sarmiento Reyes, INAOE

c©INAOE 2017The author hereby grants to INAOE permission

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

Modeling of memristive systems and itsapplications

Master’s Thesis

By:

Jose Balaam Alarcon Angulo

Adviser:

Dr. Librado Arturo Sarmiento Reyes

Instituto Nacional de Astrofısica Optica y Electronica

Electronics Department

San Andres Cholula, Puebla. January 16, 2018

i

Agradezco a mis padres Guadalupe Alarcon y Ma. Teresa y a mi hermana Dayanira

Alarcon por su incondicional apoyo a lo largo de toda mi vida. Al doctor Arturo

Sarmiento por su orientacion durante el desarrollo de esta tesis.Y a todos mis

amigos que siempre estan ahı e 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.


Resumen

Desde el advenimiento del memristor como elemento basico de circuito real, ha habido

un significativo impulso en la investigacion orientada al desarrollo de aplicaciones

del memristor en diseno de circuitos y procesamiento de senales. Amplificadores con

retroalimentacon negativa basados en nullores se encuentran entre las aplicaciones mas

factibles debido a que la propiedad de memoria del memristor puede ser incorporada

a la transferencia de amplificacion al colocar el memristor directamente en el lazo de

retroalimentacion.

Esta tesis introduce un modelo del memristor que resulta ad hoc para el analisis

de amplificadores de retroalimentacion negativa basados en nullores. El modelo ha

sido desarrollado utilizando el metodo de homotopıa modificado que resulta en una

expresion totalmente simbolica de la memristancia con una estructura de funciones

armonicas. El modelo desarrollado puede ser codificado en Verilog-A para fines de

simulacion electrica de los amplificadores. Se ha realizado el analisis de distorsion

armonica y el analisis de 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

conexion anti-serie de dos memristores. Obteniendose de 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 simu-

lation 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

Resumen v

Abstract vii

List of Figures xiii

List of Tables xv

1 Introduction 1

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Methodology of the research . . . . . . . . . . . . . . . . . . . . . . . 2

2 Fundamentals 5

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




6.2.4 Memristive current amplifier . . . . . . . . . . . . . . . . . . . 43

6.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44




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 Astrofısica, 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


xii CONTENTS


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

4.6 Ix-VD characteristic for t = 0, π2, π. . . . . . . . . . . . . . . . . . . . . 28

4.7 Ix-VG characteristic for t = 0, π2, 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


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 am-

plifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2 Normalised symbolic harmonic analysis for the transmemristance am-

plifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.3 Harmonic analysis for the transmemductance amplifier: simulations re-

sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 trans-

memristance amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.2 Harmonic analysis for the transmemductance amplifier: simulations re-

sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

[xv]

xvi LIST OF TABLES


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 mem-

ristive 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.


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.


Chapter 2

Fundamentals

This Chapter focuses on the basic concepts regarding nullor-based memristive ampli-

fiers. Section 2.1 introduces the basic concepts concerning the memristor and mem-

ristive 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].

i(t) =dq(t)

dtor q(t) =

∫ t

−∞i(τ)dτ (2.1)

and

v(t) =dϕ(t)

dtor ϕ =

∫ t

−∞v(τ)dτ (2.2)

[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)︸︷︷︸flux−controlled

ϕ(t) = fM(q)︸︷︷︸charge−controlled

(2.3)

The voltage across a charge-controlled memristor can be found according to [7]

and [8]:

v(t) = M(q(t))i(t) M(q(t)) =dfM(q)

dq(2.4)

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

i(t) = W (ϕ(t))v(t) W (ϕ(t)) =dgM(ϕ)

dϕ(2.5)

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


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 PHLwhen ω →∞.

(c) Memristance when ω →∞.

Figure 2.3: Basic fingerprint for memristors.


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.


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:

K =

[A D

C D

]=

[0 0

0 0

](2.11)


10 2. Fundamentals

with:A = 1

µ= vi

vo

∣∣∣io=0

= 0 B = 1γ

= viio

∣∣∣uo=0

= 0

C = 1ζ

= iivo

∣∣∣io=0

= 0 D = 1βF

= iiio

∣∣∣uo=0

= 0

Therefore, the nullor models the four ideal infinite gain amplifiers, as stated by

Prof. Tellegen in his paper in 1954 [15]:

µ =∞ γ =∞

ζ =∞ βF =∞(2.12)

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.

(c) Transresistance Amplifier. (d) Current Amplifier.

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


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:

Av = 1 + R2

R1Agm = −G

Arm = −R AI = 1 + R2

R1

(2.13)

where Av, Agm, Arm and AI are the voltage, transconductance, transresistance and

current gains , respectively.


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)

dt=µRon

∆2i(t)fw(x) (3.1)

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]:

fw(x) = 1− (2x− 1)2k (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 param-

eter 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.

v = p0v0 + P 1v1 + · · ·+ pnvn (3.5)


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 ohmicresistance 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:

MO1k=n = R2onfwkγ1(α− 1)[−1 + cos(ωt)] +Rinit

fwk = fw(Xo)|k=n = 1− (2Xo − 1)2k

Rinit = [Xo + α(1−Xo)]Ron

γ = µAp∆2ω

(3.8)

where Rinit is the initial memristor resistance, with Xo as the initial condition of the

state variable.



The memristance expressions for Order-3 is:

MO3k=n= R2

onfwkγ1(α− 1) [−1 + cos(ωt)] +Rinit

+Ron3fwkf ′wk

γ21(α− 1)[− 3

4 + cos(ωt)− ( 14 ) cos(2ωt)]

+Ron4fwkf ′′wk

γ31Pxo3kn(α− 1) [a0 + a1 cos(ωt) + a2 cos(2ωt) + a3 cos(3ωt)]

f ′wk= −4k(2Xo− 1)2k−1

f ′′wk= −8k(2k − 1)(2Xo− 1)

2(k−1)

(3.9)

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:

MO1k=7= R2

onfw7γ1(α− 1)[−1 + cos(ωt)] +Rinit

fw7 = fw(Xo)|k=7 = 1− (2Xo − 1)14

Rinit = [Xo + α(1−Xo)]Ron

γ = µAp∆2ω

(3.10)


3.2 Model characterization 17

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

MO3k=1= R2

onfw1γ1(α− 1) [−1 + cos(ωt)] +Rinit

+Ron3fw1f′w1γ2

1(α− 1)[−3

4+ cos(ωt)− (1

4) cos(2ωt)

]+Ron4fw1f

′′w1γ3

1Pxo3k1(α− 1)[

56− 5

4cos(ωt) + 1

2cos(2ωt)− 1

12cos(3ωt)

]fw1 = fw(Xo)|k=1 = 1− (2Xo − 1)2

f ′w1= −4(2Xo− 1)

f ′′w1= −8(2Xo− 1)

PxO3kn= (6X2

o − 6Xo + 1)

(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

Appendix A.

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 fre-

quency 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.



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 here-

after explained.


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.

ωO1k=7 O3k=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.



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= πω

(3.13)

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.


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.




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.



(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


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.

(a) Pencil-lead memistor element (from [27]).

Control current

Sensing current

(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.


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 charac-

terization 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

M1 = 2×M2 5×M2 10×M212M2

15M2

110M2 M2

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).


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).





(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

in VD within a range of 0 to 2 volts. The memristances are evaluated for t = 0, π2, π

seconds.


28 4. The memistor



(g) Case 7.

Figure 4.6: Ix-VD characteristic for t = 0, π2, 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

memristances are evaluated for t = 0, π2, π seconds.





(g) Case 7.

Figure 4.7: Ix-VG characteristic for t = 0, π2, 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

for t = π2.


Chapter 5

Memristive nullor-based amplifiers

A particular case of memristive circuit application arises when the memristor is com-

bined with the nullor, in order to achieve a memristive output-input transfer function.

(a) Voltage Amplifier (b) Transconductance Amplifier

(c) Transresistance Amplifier (d) Current 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 amplifiersare shown in Figure 5.1. Under the assumption of instantaneous linearity [32], thegain of the amplifiers depicted in Figure 5.1 can be expressed as:

VoVi

(t) = 1 +M(t)

R

VoIi

(t) = −M(t)IoVi

(t) = −W (t)IoIi(t) = 1 +

M(t)

R(5.1)

5.1.1 Memristive voltage amplifier

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

given as:

VoVi

(t) = 1 +M(t)

R(5.2)

(a) Numeric evaluation withMaple

(b) Simulation in AnalogInsydes (c) Simulation in HSpice

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,


5.1 Basic topologies 33

which yield maximun and minimum gains, respectively:

max

(VoVi

)=

(1 +

M(t)

R

)∣∣∣∣M(t)=Mmax

= 1 +14410)

300= 49.033

min

(VoVi

)=

(1 +

M(t)

R

)∣∣∣∣M(t)=Mmin

= 1 +374)

300= 2.246 (5.3)

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:

VoIi

(t) = −M(t) (5.4)

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.



(a) Numeric evaluation withmaple

(b) Simulation in AnalogInsyde (c) Transresistance Amplifier

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.

max =

(VoIi

)= −Mmax = −14410Ω

min =

(VoIi

)= −Mmin = −374Ω (5.5)

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



5.1.3 Transmemductance amplifier

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

IoVi

(t) = − 1

M(t)= −W (t) (5.6)

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



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:

max =

(IoVi

)= − 1

Mmin

= − 1

374= −2.67mf

min =

(IoVi

)= − 1

Mmax

= − 1

14410= −69.39µf (5.7)

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.



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:

IoIi

(t) = 1 +M(t)

R(5.8)

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.



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



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.


Chapter 6

Harmonic analysis

Harmonics or harmonic frequencies of a periodic voltage or current are frequency com-

ponents 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 volt-

age.

THD can be expressed as:

THD =

√v2

2 + v23 + · · ·+ v2

n

v1

(6.1)

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 funda-

mental frequency.

The result of the equation 6.1 is a comparison of the output voltages of all har-

monic 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 Chapter 5.


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:

VoVi

= 1 +M

R(6.2)

with R = 300Ω.

(a) Input and output waveformswith model Order1k=7.

(b) Voltage memristive nullor-based amplifier.

(c) Input and output waveformswith model Order3k=1.

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


6.2 Basic amplifiers & symbolic harmonic analysis 41

The resulting symbolic expressions of the harmonic components for the modelOrder1k=7 are given as:

FvO1= Ain

(R+Rinit

R − R2onfwnγ1(α−1)

R

)H2vO1

= Ain

(R2

onfwnγ1(α−1)2R

) (6.3)

and for Order3k=1:

FvO3= Ain

(R+Rinit

R − R2onfwnγ1(α−1)

R − 58R3

onfwnf′wnγ

21(α−1)

R + 712R4

onfwnf”wnγ

31P (α−1)

R

)

H2vO3= Ain

(R2

onfwnγ1(α−1)2R +

R3onfwnf

′wnγ

21(α−1)

2R − 712R4

onfwnf”wnγ

31P (α−1)

R

)

H3vO3= Ain

(14R4

onfwnf”wnγ

31P (α−1)

R − 18R3

onfwnf′wnγ

21(α−1)

R

)

H4vO3= Ain

(− 1

24R4

onfwnf”wnγ

31P (α−1)

R

)

(6.4)

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.


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:

VoIi

= −M (6.5)

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

FmO1= Ain

(−Rinit

R +R2

onfwnγ1(α−1)R

)H2mO1

= Ain

(−R

2onfwnγ1(α−1)

2R

) (6.6)




(b) Transmemristance nullor-based amplifier.


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

and for Order3k=1:

FmO3= Ain

(−Rinit

R +R2

onfwnγ1(α−1)R + 5

8R3

onfwnf′wnγ

21(α−1)

R − 712R4

onfwnf”wnγ

31P (α−1)

R

)

H2mO3= Ain

(−R

2onfwnγ1(α−1)

2R − R3onfwnf

′wnγ

21(α−1)

2R + 712R4

onfwnf”wnγ

31P (α−1)

R

)

H3mO3= Ain

(− 1

4R4

onfwnf”wnγ

31P (α−1)

R + 18R3

onfwnf′wnγ

21(α−1)

R

)

H4mO3= Ain

(124R4

onfwnf”wnγ

31P (α−1)

R

)

(6.7)

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

the transmemristance 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:

IoVi

= − 1

M= −W (6.8)

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 transmem-

ductance 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-


6.2 Basic amplifiers & symbolic harmonic analysis 43


(b) Transmemductance nullor-based amplifier.


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:

IoIi

= 1 +M

R(6.9)

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


(b) Memristive current nullor-based amplifier.


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.



6.3 Simulation


The simulation was made in Hspice using a sinusoidal voltage source with value of

0.25 sinωt and a resistor with value of 300Ω.

MemristiveVoltage Amp.

Order1 Order3Evaluated 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.


From equations (6.6) and (6.7), the normalised harmonic components can be obtained,

as shown in Table 6.2.

TransmemristanceAmp.


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


6.3 Simulation 45


The simulations results allow to verify that the transmemductance amplifier is the

most nonlinear, as shown in Table 6.3.

Transmemductance

Amplifier

Order1Simulation

Order3Simulation

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.


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 in-

stantaneous 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 depen-

dent 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:

v2n = 4kTR (V 2/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:

i2n =4kT

R(A2/Hz) (7.2)

R in

(a) Voltage noise source

R

n

+

−

v

(b) Current noise source

Figure 7.1: Noise model of a resistor


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:

v2nM

= 4kTM (V 2/Hz) (7.4)

and the noise current is given by:

i2nM =4kT

M(A2/Hz) (7.5)

where M denotes de instant memristance. The equivalent circuit noise for the mem-

ristor are given in Figure 7.2.

inMM

(a) Voltage noise source

−

nM

M

+

v

(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.


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].

+

- +

-

R1 RL

Rs

Us

UnRs

UnR1

MUnM

UnN

InN

+

-

(a) Voltage Amplifier.

+

- +

-

RL

Rs

Us

UnRs

MUnM

UnN

InN

+

-

(b) Transmemductance Amplifier.

+

- +

-

M

RL

RSIS InS

InN

UnN

UnM

(c) Transmemristance Amplifier.

+

- +

-

R1

M

RL

RSIS InS

InN

UnN

UnM

UnR

(d) Current Amplifier.

Figure 7.3: Basice memristive amplifiers: noisy configurations.

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

vneqin = vns + vnn +

(Rs +

RM

R +M

)inn +

RMR+M

RvnR +

RMR+M

RvnM (7.6)

for the transmemristance amplifier is:

ineqin = ins + inn +

(1

Rs

+1

M

)vnn +

1

MvnM (7.7)

for transmenductance amplifier is:

vneqin = vns + vnn + (Rs +M)inn +MinM (7.8)

and finally for the current amplifier:

ineqin = ins + inn +

(1

Rs

+1

M +R

)vnn +

R

R +MinR +

M

R +MinM (7.9)


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.

t

VoltageAmplifier

Rs = 100, R = 300

TransmemristanceAmplifier

TransmemductanceAmplifier

CurrentAmplifier

Rs = 100, R = 300Evaluated 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.


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 Chapter 4, 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.

+

- +

-

MD

MS

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 Chapter 7. 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 con-

tribute with noise currents as depicted in Figure 8.2. According to equation (7.5), the

noise currents for the memristors are given by:

i2nMS=

4kT

MS

i2nMD=

4kT

MD

(8.1)

+

- +

-

MD

MSinMS

inMD

Figure 8.2: Memistor with noise contributions

After manipulating the inMS and inMD , the amount of equivalent noise present at

the input of the memistor two port can be expressed as:

inn = inMS +(

1 + MS+MD

MS

)inMD

vnn = MDinMD

(8.2)

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 vnn0 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).


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

MSRL

+

-

vo

M

is

Rs

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:

ineqin = ins + iMs (8.3)

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.


56 8. Case of study

+

- +

-

MD

MSRL

+

-

vo

M

is

Rs

+ -

inMD

inMS

inn

vnn

inM

inS

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.

TransmemristanceAmplifier

Order1 Order3 MemistorEvaluated 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.


8.2 Transmemristance amplifier 57

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


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 mem-

ristive 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 ba-

sic 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 memis-

tor 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 re-

sults 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 fu-

ture 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.


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

‘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 Ron = 100;

parameter real mu = 1.0e-14;

parameter real alpha = 160;

parameter real Ap=400.0e-7;

parameter real omega=1;

parameter real Xo=0.1;

real mem, im, Rinit, fw, Gamma;

analog begin

Gamma = ((mu*Ap)/(Delta*Delta*omega));

fw = (-4*Xo*(Xo-1));

Rinit = (Xo+alpha*(1-Xo))*Ron;

[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


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:

VoIi

= M (B.1)

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 modelOrder1k=7.

(b) Input and output waveforms with modelOrder3k=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.

TransmemristanceAmp.


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


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:

IoVi

= W (B.2)

this difference in the sign of function is due to the polarity of memristor. The nor-

(a) Io − Vi ratio with modelOrder1k=7

(b) Transmemductance nullor-based amplifier

(c) Io − Ii ratio with modelOrder3k=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

Amplifier

Order1Simulation

Order3Simulation

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.


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