Development of a modeling methodology for circuits with ...Development of a modeling methodology for...
Transcript of Development of a modeling methodology for circuits with ...Development of a modeling methodology for...
Development of a modelingmethodology for circuits with
memristors
by
Carlos Manuel Hernandez Mejıa
A dissertation submitted to the ElectronicsDepartment in partial fulfillment of the
requirements for the degree of
D.Sc. in Electronics
at the
National Institute for Astrophysics, Opticsand ElectronicsFebruary 2015
Tonantzintla, Puebla
Advisor:
Dr. Librado Arturo Sarmiento Reyes -INAOEDr. H ector Vazquez Leal -UV
c©INAOE 2015All rights reserved
The author hereby grants to INAOE permissionto reproduce and to distribute copies of this
thesis document in whole or in part
Resumen
A partir de la publicacion revolucionaria del Profesor Chua en los anos 70’s, el concepto
del memristor ha sido introducido. Sin embargo, tuvo que transcurrir un periodo aproxi-
madamente de 40 anos para que la concepcion fısica del memristor se llevara a cabo por
los laboratorios HP en el 2008. Hoy en dıa, este dispositivo emergente ha sido empleado
en nuevas aplicaciones; las cuales abarcan desde el desarrollo de memorias modernas hasta
circuitos analogicos en combinacion con elementos convencionales para obtener funcional-
idades novedosas. Lo anterior ha provocado la necesidad de modelos apropiados del mem-
ristor para llevar a cabo simulaciones de circuitos con el proposito de ayudar en el flujo
de diseno de circuitos y sistemas memristivos. La presente tesis introduce una novedosa
metodologıa de modelado para generar expresiones semi-simbolicas del memristor en forma
de modelos orientados a la simulacion de circuitos en los dominios de DC y tiempo. Estos
modelos son expresados mediante relaciones de rama voltaje-corriente; es decir, modelos
funcionales del memristor. Ademas, una atencion especial es dedicada durante el proceso
de generacion para verificar que las principales senas de identidad del memristor sean
cumplidas; lo cual significa que los modelos propuestos poseen las propiedades para ser
considerados verdaderos memristores. Los modelos han sido caracterizados detalladamente
con respecto a varios parametros del dispositivo - asumiendo los valores del laboratorio de
HP. En el ultimo paso de la metodologıa, los modelos son establecidos mediante bloques
funcionales; los cuales se pueden utilizar por las herramientas industriales de simulacion
de circuitos integrados, tales como SPICE. Los modelos del memristor han sido probados
mediante una serie de circuitos con resultados satisfactorios.
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Abstract
Since the seminal paper of Prof. Chua in the early 70’s when the concept of the memristor
was introduced, a period of nearly 40 years has passed in order to have a real memristor
device in 2008 at the HP Labs. Nowadays, this emerging device has been already used in
novel applications that span from the development of modern memories to analog circuit
applications in combination with traditional electronics to obtain novel functionalities.
As a direct consequence, especially suited models for the memristor are needed that can
be used in circuit simulation applications to help the design flow of memristive circuits
and systems. This work introduces a novel modeling methodology in order to generate
semi-symbolical expressions in the form of circuit simulation-oriented memristor models
in the DC and time domains. The generated models are expressed in the form of an
i-v constitutive branch relationship, i.e. a functional memristor model. Special attention
during the generation process has been devoted to verify that the fingerprints of the
memristor are fulfilled, which means that the proposed models posses the main properties
stipulated for a device to be considered a memristor. The models have been thoroughly
characterized versus several parameters of the device - assuming the HP Lab values. In
the last step of the methodology, the models are recast in functional blocks that can be
used within industrial IC simulation frameworks such as SPICE. The generated memristor
models have been tested in a series of bench-mark circuits with excellent results.
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iv
Contents
Resumen ii
Abstract iv
1 Introduction 3
1.1 Original concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Finding the missing memristor . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Fingerprints of the memristor . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Other mem-elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Memcapacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Meminductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.3 Memcapacitive and meminductive systems . . . . . . . . . . . . . . 17
1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 State of the art of memristor models 27
2.1 Why is modeling necessary? . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 The memristor needs to be modeled . . . . . . . . . . . . . . . . . . 28
2.2 Analysis domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 DC-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Transient-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Frequency-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Hierarchy of design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Circuital model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Behavioral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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3 Development of DC memristor models 41
3.1 DC memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 PWL approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.2 Voltage-controlled memristive system . . . . . . . . . . . . . . . . . 43
3.1.3 Current-controlled memristive system . . . . . . . . . . . . . . . . . 45
3.1.4 Variations of the PWL parameters . . . . . . . . . . . . . . . . . . 47
3.2 Generation of the v-i branch relationship . . . . . . . . . . . . . . . . . . . 49
3.3 A glimpse to DC analysis of memristive circuits . . . . . . . . . . . . . . . 52
3.4 Cases of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 A single-memristor circuit . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.2 A two-memristor circuit . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Transient memristor model 63
4.1 Polynomial-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Mathematical description . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Characteristic of the memristor model . . . . . . . . . . . . . . . . 65
4.2 Homotopy-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Homotopy perturbation method . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Memristor model using HPM . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.4 Symbolical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.5 Parameter variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.6 Characteristic of the memristor model . . . . . . . . . . . . . . . . 82
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Circuit applications 87
5.1 Nullor-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.1 Trans-memresistance amplifier . . . . . . . . . . . . . . . . . . . . . 92
5.1.2 Trans-memconductance amplifier . . . . . . . . . . . . . . . . . . . 108
5.2 Opamp-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.1 Memristor-based inverter amplifier . . . . . . . . . . . . . . . . . . 124
5.2.2 Memristor-based noninverter amplifier . . . . . . . . . . . . . . . . 136
5.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3 Mem-elements emulator circuits . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3.1 Memcapacitor and meminductor grounded . . . . . . . . . . . . . . 145
5.3.2 Mutator using current conveyor and OTA . . . . . . . . . . . . . . 148
5.3.3 Emulator using single-output current conveyor . . . . . . . . . . . . 150
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Contents
5.3.4 Meminductor emulator . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Conclusions 157
A Nonlinear boundary of HP memristor 159
B Transient memristor model 161
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Contents
viii
List of Figures
1.1 The four fundamental two-terminal electric elements . . . . . . . . . . . . . 4
1.2 The general structure of HP memristor . . . . . . . . . . . . . . . . . . . . 6
1.3 V -I characteristic curve of the unipolar switching . . . . . . . . . . . . . . 7
1.4 The positive instantaneous power areas . . . . . . . . . . . . . . . . . . . . 10
1.5 Lissajous figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Memristor fingerprint: Pinched hysteresis loop . . . . . . . . . . . . . . . . 11
1.7 Memristor fingerprint: Hysteresis lobe area . . . . . . . . . . . . . . . . . . 12
1.8 Frequency response of Lissajous figures . . . . . . . . . . . . . . . . . . . . 12
1.9 Memristor fingerprint: Shrunken hysteresis loop . . . . . . . . . . . . . . . 13
1.10 Circuit elements with memory and conventional elements . . . . . . . . . . 14
1.11 Characteristic curves of the memcapacitive system . . . . . . . . . . . . . . 18
3.1 PWL representation of the memristive system . . . . . . . . . . . . . . . . 42
3.2 PWL functions for the voltage-controlled memristive system . . . . . . . . 43
3.3 Circuit with voltage-controlled memristive system . . . . . . . . . . . . . . 44
3.4 DC-responses of the voltage-controlled memristive systems . . . . . . . . . 44
3.5 PWL functions for the current-controlled memristive system . . . . . . . . 45
3.6 Circuit with current-controlled memristive system . . . . . . . . . . . . . . 46
3.7 DC-responses of the current-controlled memristive systems . . . . . . . . . 46
3.8 Scheme of variations for memristive slopes . . . . . . . . . . . . . . . . . . 47
3.9 Simulation results using the variations of M1 coefficient . . . . . . . . . . . 48
3.10 Simulation results using the variations of M2 coefficient . . . . . . . . . . . 48
3.11 Scheme of variations for hyperplane position . . . . . . . . . . . . . . . . . 49
3.12 Simulation results using the variations of the hyperplane VT . . . . . . . . 49
3.13 Methodology for DC analysis of memristors . . . . . . . . . . . . . . . . . 50
3.14 DC memristive circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.15 DC-response of different memristor characteristics . . . . . . . . . . . . . . 55
3.16 DC two-memristor circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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List of Figures
3.17 DC-response of the M1 and M2 memristors . . . . . . . . . . . . . . . . . . 57
4.1 Hysteresis loops of the memristor model . . . . . . . . . . . . . . . . . . . 65
4.2 Memristance of the memristor model . . . . . . . . . . . . . . . . . . . . . 65
4.3 General procedure of the approximate solution using HPM . . . . . . . . . 67
4.4 RMSE using different homotopy orders . . . . . . . . . . . . . . . . . . . . 69
4.5 HPM and RK4 curves of the solution x(t) using the homotopy order 6. . . 70
4.6 RMSE using different frequency values. The HPM order is 6. . . . . . . . . 70
4.7 Symbolical state variable x(t) using different ω values . . . . . . . . . . . . 74
4.8 Symbolical memristance M using different ω values . . . . . . . . . . . . . 75
4.9 Analysis of x(t) using different values of current amplitude. . . . . . . . . . 77
4.10 M-i and v-i characteristic using not possible value of parameter A. . . . . 78
4.11 Analysis of window function coefficient using different values. . . . . . . . . 79
4.12 RMSE using different window function coefficient values . . . . . . . . . . 80
4.13 M-i and v-i characteristic curves using different values of parameter a. . . 80
4.14 Hysteresis loops of the memristor model . . . . . . . . . . . . . . . . . . . 82
4.15 Memristance of the memristor model . . . . . . . . . . . . . . . . . . . . . 82
5.1 The nullor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The asymptotic-gain model . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Negative-feedback amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Negative-feedback trans-mem amplifiers . . . . . . . . . . . . . . . . . . . . 90
5.5 Nullor synthesis by MOS-equivalent . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Nullor synthesis by Memistor realization . . . . . . . . . . . . . . . . . . . 91
5.7 Trans-memresistance amplifier with nullor . . . . . . . . . . . . . . . . . . 92
5.8 uo(t) curves for the trans-memresistance amplifier with nullor using poly-
nomial memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.9 Transmem-resistance ii-uo hysteresis with nullor using polynomial memris-
tor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.10 uo(t) curves for the trans-memresistance amplifier with nullor using HPM
memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.11 Transmem-resistance ii-uo hysteresis with nullor using HPM memristor model 95
5.12 Trans-memresistance amplifier with MOS-equivalent . . . . . . . . . . . . . 96
5.13 uo(t) curves for the trans-memresistance amplifier with MOS-equivalent us-
ing polynomial memristor model . . . . . . . . . . . . . . . . . . . . . . . . 96
5.14 Transmem-resistance ii-uo hysteresis with MOS-equivalent using polyno-
mial memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
x
List of Figures
5.15 Transmission curves for Analysis I. . . . . . . . . . . . . . . . . . . . . . . 98
5.16 Transmission curves for Analysis II. . . . . . . . . . . . . . . . . . . . . . . 99
5.17 uo(t) curves for the trans-memresistance amplifier with MOS-equivalent us-
ing HPM memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.18 Transmem-resistance ii-uo hysteresis with MOS-equivalent using HPMmem-
ristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.19 Transmission curves for Analysis I. . . . . . . . . . . . . . . . . . . . . . . 101
5.20 Transmission curves for Analysis II. . . . . . . . . . . . . . . . . . . . . . . 102
5.21 Trans-memresistance amplifier with Memistor realization . . . . . . . . . . 103
5.22 uo(t) curves for the trans-memresistance amplifier with Memistor realization
using polynomial memristor model . . . . . . . . . . . . . . . . . . . . . . 103
5.23 Transmem-resistance ii-uo hysteresis (black) and memristor hysteresis (red)
for Memistor realization using polynomial memristor model . . . . . . . . . 104
5.24 uo(t) curves for the trans-memresistance amplifier with Memistor realization
using HPM memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.25 Transmem-resistance ii-uo hysteresis (black) and memristor hysteresis (blue)
for Memistor realization using HPM memristor model . . . . . . . . . . . . 105
5.26 Trans-memconductance amplifier with nullor . . . . . . . . . . . . . . . . . 108
5.27 io(t) curves for the trans-memconductance amplifier with nullor using poly-
nomial memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.28 Transmem-conductance ui-io hysteresis with nullor using polynomial mem-
ristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.29 io(t) curves for the trans-memconductance amplifier with nullor using HPM
memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.30 Transmem-conductance ui-io hysteresis with nullor using HPM memristor
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.31 Trans-memconductance amplifier with MOS-equivalent . . . . . . . . . . . 112
5.32 io(t) curves for the trans-memconductance amplifier with MOS-equivalent
using polynomial memristor model . . . . . . . . . . . . . . . . . . . . . . 112
5.33 Transmem-conductance ui-io hysteresis with MOS-equivalent using polyno-
mial memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.34 Transmission curves for Analysis I. . . . . . . . . . . . . . . . . . . . . . . 114
5.35 Transmission curves for Analysis II. . . . . . . . . . . . . . . . . . . . . . . 115
5.36 io(t) curves for the trans-memconductance amplifier with MOS-equivalent
using HPM memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . 116
xi
List of Figures
5.37 Transmem-conductance ui-io hysteresis with MOS-equivalent using HPM
memristor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.38 Transmission curves for Analysis I. . . . . . . . . . . . . . . . . . . . . . . 117
5.39 Transmission curves for Analysis II. . . . . . . . . . . . . . . . . . . . . . . 118
5.40 Trans-memconductance amplifier with Memistor realization . . . . . . . . . 119
5.41 io(t) curves for the trans-memconductance amplifier with Memistor realiza-
tion using polynomial memristor model . . . . . . . . . . . . . . . . . . . . 119
5.42 Transmem-conductance ui-io hysteresis (black) and memristor hysteresis
(red) for Memistor realization using polynomial memristor model . . . . . 120
5.43 io(t) curves for the trans-memconductance amplifier with Memistor realiza-
tion using HPM memristor model . . . . . . . . . . . . . . . . . . . . . . . 121
5.44 Transmem-conductance ui-io hysteresis (black) and memristor hysteresis
(blue) for Memistor realization using HPM memristor model . . . . . . . . 121
5.45 Memristor-based inverter amplifier . . . . . . . . . . . . . . . . . . . . . . 124
5.46 t-Av characteristic curves using different memristor models . . . . . . . . . 125
5.47 Transmission characteristic curves using different memristor models . . . . 126
5.48 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 126
5.49 Memristor-based inverter amplifier . . . . . . . . . . . . . . . . . . . . . . 127
5.50 t-Av characteristic curves using different memristor models . . . . . . . . . 127
5.51 Transmission characteristic curves using different memristor models . . . . 128
5.52 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 128
5.53 Gain-THD curves using the HPM memristor model . . . . . . . . . . . . . 131
5.54 Gain-THD curves using the polynomial memristor model . . . . . . . . . . 132
5.55 Memristor-based inverter amplifier . . . . . . . . . . . . . . . . . . . . . . 133
5.56 t-Av characteristic curves using different memristor models . . . . . . . . . 134
5.57 Transmission characteristic curves using different memristor models . . . . 134
5.58 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 135
5.59 Memristor-based noninverter amplifier . . . . . . . . . . . . . . . . . . . . 136
5.60 t-Av characteristic curves using different memristor models . . . . . . . . . 136
5.61 Transmission characteristic curves using different memristor models . . . . 137
5.62 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 137
5.63 Memristor-based noninverter amplifier . . . . . . . . . . . . . . . . . . . . 138
5.64 t-Av characteristic curves using different memristor models . . . . . . . . . 138
5.65 Transmission characteristic curves using different memristor models . . . . 139
5.66 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 139
5.67 Gain-THD curves using the HPM memristor model . . . . . . . . . . . . . 140
xii
List of Figures
5.68 Gain-THD curves using the polynomial memristor model . . . . . . . . . . 141
5.69 Memristor-based noninverter amplifier . . . . . . . . . . . . . . . . . . . . 142
5.70 t-Av characteristic curves using different memristor models . . . . . . . . . 142
5.71 Transmission characteristic curves using different memristor models . . . . 143
5.72 HSPICE and AnalogInsydes simulation results . . . . . . . . . . . . . . . . 143
5.73 Memcapacitor and meminductor emulators . . . . . . . . . . . . . . . . . . 145
5.74 Vin and Vout curves for the memcapacitor emulator. . . . . . . . . . . . . . 146
5.75 Vin and Vout curves for the meminductor emulator. . . . . . . . . . . . . . . 146
5.76 The capacitance C(t) using different memristor models. . . . . . . . . . . . 147
5.77 The inductance L(t) using different memristor models. . . . . . . . . . . . 147
5.78 Mutator MR to MC using CCII+ and OTA . . . . . . . . . . . . . . . . . . 148
5.79 q and IMccurves for the memcapacitor using polynomial memristor model 148
5.80 Vin-Capacitance characteristic curves using polynomial memristor model . 149
5.81 q and IMccurves for the memcapacitor using HPM memristor model . . . . 149
5.82 Vin-Capacitance characteristic curves using HPM memristor model . . . . . 149
5.83 Floating memcapacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.84 Memcapacitor characteristic-curves using polynomial memristor model . . 150
5.85 Memcapacitor characteristic-curves using HPM memristor model . . . . . . 151
5.86 Floating meminductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.87 Meminductor characteristic-curves using polynomial memristor model . . . 152
5.88 Meminductor characteristic-curves using HPM memristor model . . . . . . 153
A.1 The HP memristor structure using nonlinear boundary . . . . . . . . . . . 159
xiii
List of Figures
xiv
List of Figures
1
List of Figures
2
Chapter 1
Introduction
The idea of the memristor, the missing circuit element that relates the electrical variables
of charge and flux, has represented a revolution in electronics and a key piece of the
grandest technology challenges: mimicking the functions of the brain. The history of the
memristor has two remarkable milestones [1, 2]: the first one [1] in 1971 when Professor L.
O. Chua introduced the mathematical existence of the device as the fourth basic electric
circuit element, and the second one [2] in 2008 when researches of HP Labs were able to
fabricate the memristor, i.e. the memristor as a real device. This introductory chapter
presents, firstly, an overview of the fundamental concepts introduced by Chua with special
emphasis in pointing out the main properties of the memristor. Naturally, attention is
also devoted to the discovery of the device and to the open possibilities for electronic
design incorporating the memristor. In addition, the most important fingerprints of the
device are addressed and discussed. Furthermore, the expansion of the original concept
of the memristor to circuit elements with memory is presented in order to define the new
family of memory passive elements. Hereafter, the motivation for tackling the problem of
memristor modeling is justified and the global objectives of this thesis are introduced.
1.1 Original concept
At the end of 1971, Chua [1] introduced a new two-terminal circuit element that closed the
loop around the four main electrical variables of Circuit Theory, namely, electric charge
q, voltage v, current i, and flux linkage ϕ. This new fundamental circuit element was
called memristor (a contraction for memory resistor) because it behaves as a resistor
with memory. The memristor constituted the missing electric element that established
a direct branch relationship between the electric charge and the flux linkage. Figure 1.1
shows the concept explained above.
3
Chapter 1. Introduction
ϕ
i
q
v
Capacitor Inductor
Resistor
Memristor
Figure 1.1: The four fundamental two-terminal electric elements
In the Figure 1.1, the crossed lines denote the time-dependent derivative of charge and
flux, the so-called charge-conservation and flux-conservation laws, which are expressed as:
dq(t)
dt= i(t) or q(t) =
∫ t
−∞
i(τ)dτ
and
dϕ(t)
dt= v(t) or ϕ(t) =
∫ t
−∞
v(τ)dτ
respectively.
Chua mathematically predicted that there is a solid-state device, which defines the
missing relationship between four basic variables [1]. This relationship establishes that
the memristor can be either flux-controlled or charge-controlled:
q(t) = gM(ϕ) ϕ(t) = fM(q)
flux-controlled charge-controlled(1.1)
After differentiating with respect to t, both terms of (1.1), it yields
i(t) ,dq(t)
dt=
dgM(ϕ)
dtv(t) ,
dϕ(t)
dt=
dfM(q)
dt
By applying the chain rule to the expressions above, it is possible to obtain
i(t) =dgM(ϕ)
dϕ
dϕ
dtv(t) =
dfM(q)
dq
dq
dt
Therefore, the current of a flux-controlled memristor and the voltage of a charge-
controlled memristor are given by
4
1.1. Original concept
i(t) =dgM(ϕ)
dϕv(t) v(t) =
dfM(q)
dqi(t)
The expressions can be defined as
i(t) = W (ϕ)v(t) v(t) = M(q)i(t) (1.2)
where W (ϕ) is called memconductance and M(q) is called memresistance.
In 1976, Chua and Kang [3] considered the memristor as a particular case of a general
class of dynamical systems, called memristive systems and defined by
x = f (x, ς, t)
y = g(x, ς, t)ς (1.3)
where x represents the state of the system, and the output and input of the system are
represented by y and ς respectively.
The equation system of (1.3) opens the possibility for two different types of memristive
system: voltage and current controlled. The current-controlled and the voltage-controlled
memristive systems are represented by
x = f(x, i, t)
v = R(x, i, t)i
current-controlled
x = f(x, v, t)
i = G(x, v, t)v
voltage-controlled
(1.4)
where v and i denote the voltage and current port, respectively.
The original conception of the memristor was demonstrated by building operational
laboratory models with the help of active circuits [1]. The M-R Mutator was used to
transform the v-i characteristic curve of the nonlinear resistor into the corresponding ϕ-q
characteristic curve of the memristor. The absence of a physical device was the major
drawback for demonstrating the properties and potential applications of memristors.
The first conceptual application of the memristor was introduced by Chua and Tseng
[4] in 1974. The dynamical behavior of the real diode under reverse, forward and sinusoidal
operating modes was simulated using a circuital model, called memristive diode model.
This model was compared with existing models for diodes as: the Two-Capacitor model
[5], the Barna-Horelick model [6], the Linvill’s multi-lumped model [7] and the Wang-
Branin model [8]. The simulation results indicated that the memristors are useful for
modeling devices with charge-storage and conductivity modulation effects.
5
Chapter 1. Introduction
The memristor also was considered as a new bond graph element [9] in order to simulate
mechanical and electrochemical systems with memristors.
The theory of the memristor carried out by Chua had to wait over thirty years until
the scale of electronic devices would allow the observation of the memristive behavior.
1.2 Finding the missing memristor
The memristive behavior had been previously reported [10] to the original conception
of the memristor. These anomalous behaviors in the v-i charateristic curves of micro
and nano-scale devices have been observed in different materials and by various types of
physical mechanisms [11, 12, 13, 14, 15, 16, 17, 18].
In 2008, HP Labs presented a physical model of a two-terminal electrical device which
behaves like a memristor [2, 19]. The HP memristor is a nanometric scale thin film of
TiO2 which can be divided by two main regions: the undoped region with high resistance
ROFF and the doped region with low resistance RON .
D
wTiO2−x
TiO2
Pt
Pt
D
wRON
ROFF
Figure 1.2: The general structure of HP memristor
Figure 1.2 shows the general structure of the HP memristor, where D is the total
length of the device and w is the length of doped region. In this thesis, the boundary of
the two regions of the memristor is considered as an one-dimensional problem; i.e. the
boundary is linear. A simple proposal in order to tackled the nonlinear boundary is shown
in Appendix A.
The application of an external bias v(t) across the memristor has the effect of displacing
the boundary between the two regions. The length of doped region is a charge-dependent
function and is responsible for the memristive behavior of the device.
The above explanation has been modeled by the linear dopant drift model. This model
is based on the following equation
6
1.2. Finding the missing memristor
dw(t)
dt= µV
RON
Di(t) (1.5)
where µV is the average ion mobility.
Integrating the above expression with respect to t, it yields
w(t) = µV
RON
Dq(t) (1.6)
By inserting the expression 1.6 into the following v-i relationship,
v(t) =
(
RON
w(t)
D+ROFF
(
1−w(t)
D
))
i(t) (1.7)
It is possible to obtain
M(q) = ROFF
(
1−µVRON
D2q(t)
)
(1.8)
where M(q) is the memresistance.
The q-dependent term indicates that the HP memristor is a charge-controlled memris-
tor. According to the charge-conservation law, shown in equation 1.1, the HP memristor
can also be considered as a current-controlled memristor.
The HP memristor is considered a bipolar switching memristive system because of
both voltage polarities are required to switch a device from a low resistance state (ON)
to a high resistance state (OFF) and back. However, there is another type of switching
memristive system reported in the literature: the unipolar switching memristive systems.
The unipolar switching is defined by two voltages: the set (driving OFF→ON transition)
and the reset (driving ON→OFF transition) voltages, as shown in Figure 1.3.
OFF
ON
V
I
Figure 1.3: V -I characteristic curve of the unipolar switching
7
Chapter 1. Introduction
This resistance switching is considered by some authors [20] to be based on a thermal
effect. Physically, a weak conducting filament with a controlled resistance is formed in
the electroforming process. During the reset operation, this filament is partially destroyed
because of the large amount of heat released, similarly to a traditional house-hold fuse
[20]. In the set operation, the filament is reconstructed again. The nanoscale filament
formation has been observed experimentally in such material as TiO2 [21] and NiO [22].
In this thesis, the bipolar switching memristive system is resorted in order to carry out
the study of the memristive systems.
The physical realization of the memristor by HP Labs opens the possibility for includ-
ing memristors and memristive systems into ICs. This possibility has the advantage of
extending the functionality of the circuits. The conception of a hybrid chip composed of
transistors and memristors can improve the performance of circuits without the need for
reducing the size of the devices [23].
Among others, memristors are promising devices for a wide range of potential uses
[24], such as
Non-volatile memory The memristors can retain memory states and data without re-
quiring external biasing. Nonvolatile characteristic, nano-scale geometry and com-
patibility with CMOS technology are properties of the memristors that increase the
density of memory cells and reduce the power dissipation. Besides, the memristors
provide new approaches for managing and reducing the power by disabling of CAM
(Content Addressable Memory) cell blocks without loss of stored data [25].
Dynamic load Recent advances in the development of new devices have allowed the
implementation of the memristor as a dynamic load, e.g. the programmable gain
amplifier using the memristor as a variable resistor [26]. The memristors fulfill
the need of programmable resistors in high-frequency circuits for the purpose of
adapting to particular applications or compensating PVT variations. Programmable
comparators, Schmitt triggers and Oscillators [27] have also been developed using
memristors.
Neuromorphic∗ and biological systems The physical conception of the memristor has
created excitement in the community of neuromorphic researchers. The dynamics of
the memristor is correlated with the behavior of the chemical synapses in neuromor-
phic systems [28]. Electronic circuits based on LC and memristors networks have
recently been constructed and used to model the adaptive behavior of unicellular
organisms [29]. Moreover, the memristor can be used for modeling the electrical
properties of the human skin [30].
8
1.3. Fingerprints of the memristor
Crossbar latches The high power consumption of transistors has been a major barrier
to miniaturization and development of microprocessors. Solid-state memristors can
be combined into devices called Crossbar latches [31], which can replace transistors
in future computing systems using a much smaller area.
1.3 Fingerprints of the memristor
The anomalous behavior in the v-i charateristic curves of micron and nano-scale devices
and the advent of the HP memristor has contributed to a misunderstanding about the
interpretation of the general characteristics of the memristor.
The analysis of the results reported as memristive behavior reveals that some devices
or systems are not memristors in the sense of the original definition [1] but are memristive
systems according to the more general definition [3]. Moreover, there are systems or
devices that behave as memristors or memristive systems solely under special conditions.
The concept of fingerprints has been introduced [32, 33, 34] for the purpose of iden-
tifying the memristive nature of the systems and therefore to determine the violation of
the principle of operation for memristors and memristive systems. The fingerprints can
be classified as: DC-domain fingerprints and time-domain fingerprints.
Time-domain fingerprints
The fingerprints in the time-domain that a device or system should exhibit in order to be
classified as a memristor, are the following
Pinched hysteresis loop
Hysteresis lobe area
Shrunken hysteresis loop
Pinched hysteresis loop
The pinched hysteresis loop fingerprint implies that the hysteresis loop must pass through
the intersection (0,0) of the v-i characteristic curve for any possible amplitude A and
frequency ω values of the sinusoidal input-signal. Besides, this signature must be fulfilled
for any value of the state variable x. This fingerprint is based on three fundamental
∗This is a term used to describe VLSI circuits containing analog systems which mimic neurobiological
architectures present in the nervous system.
9
Chapter 1. Introduction
properties of the memristors [3], called: passivity criterion, no energy discharge and double-
valued Lissajous figure.
The passivity criterion establishes that the memristors are passive elements when
G(x, v) ≥ 0 for any input voltage v(t) or R(x, i) ≥ 0 for any input current i(t), for
all t ≥ t0; i.e. the output (v or i) is zero whenever the input (i or v) is zero, regardless
of the state variable x. Therefore, this property is depicted as a zero-crossing in the v-i
characteristic curve.
The property of no energy discharge implies that because G(x, v) ≥ 0 (o R(x, i) ≥ 0)
the instantaneous power (p(t) = v(t)i(t)) is always positive for memristors, i.e. the v-i
characteristic curve is located in the first and third quadrant, as shown in Figure 1.4, in
order to satisfy the principle of a positive instantaneous power.
I
III
i
v
Figure 1.4: The positive instantaneous power areas
The property of double-valued Lissajous figure establishes that, under periodic opera-
tion, the memristors always exhibit a v-i Lissajous figure whose the current i (or voltage
v) is at most a double-valued function of v (or i).
i
v
i
v
(a) (b)
Figure 1.5: Lissajous figures
10
1.3. Fingerprints of the memristor
The Lissajous figure 1.5a corresponds to a memristor because any value of one axis
belongs to at most two values of the other axis. Figure 1.5b shows an impossible Lissajous
figure for a memristor because the dashed-line intersects more than two points.
The waveform of Figure 1.6 is related to a double-valued function. This type of func-
tion implies that the output variable and the state variable show no additional transient
phenomena to the input-signal; and therefore ensures that the device is a pure memristor.
i(t)
v(t)
i-v characteristic curve
v(t)
M
v-M characteristic curve
Figure 1.6: Memristor fingerprint: Pinched hysteresis loop
The memristive systems and the pure memristors can be distinguished by two different
types of pinched hysteresis loops, called self-crossing and not self-crossing [35]. These types
of crossings are related to the path of the pinched hysteresis loop. The pure memristors
generate pinched hysteresis loop of type self-crossing (black arrows in Figure 1.6). The
not self-crossing type (red arrows in Figure 1.6) is associated to the memristive systems.
Moreover, the v(t)-M characteristic curve is depicted in Figure 1.6(right-side). The
pinched hysteresis loop causes the v(t)-M characteristic curve is located in the first and
second quadrant.
Hysteresis lobe area
The hysteresis lobe area fingerprint implies that the area of the pinched hysteresis lobe
decreases monotonically as the frequency ω of the input-signal tends to infinity.
Figure 1.7(left-side) shows two hysteresis lobe areas using different frequency values.
The solid-line represents the hysteresis lobe area for ω1 and the dashed-line displays the
hysteresis lobe area for ω2=10ω1.
11
Chapter 1. Introduction
ω1
ω2
i(t)
v(t)
i-v characteristic curve
ω1
ω2
v(t)
M
v-M characteristic curve
Figure 1.7: Memristor fingerprint: Hysteresis lobe area
The area of the lobe in the i-v characteristic curve is related to the corresponding part
of memristive potential in the q-ϕ plane. The lobe area of the pinched hysteresis loop has
been widely analyzed [36, 37, 38] in order to understand the hysteresis effects in circuits
employing memristors. This fingerprint causes that the range of memristance is reduced
as the area of the pinched hysteresis lobe decreases monotonically, as shown in Figure
1.7(right-side).
Shrunken hysteresis loop
The shrunken hysteresis loop fingerprint implies that the shape of the pinched hysteresis
loop tends to a single-valued function as the frequency ω approaches infinity. This finger-
print is based on one fundamental property of the memristors [3], called: limiting linear
characteristic.
i
v
ω3>ω2>ω1
Figure 1.8: Frequency response of Lissajous figures
12
1.3. Fingerprints of the memristor
The limiting linear characteristic establishes that the memristors, under periodic oper-
ation, degenerate into a linear time-invariant resistor as the excitation frequency increases
toward infinity. This property is illustrated in Figure 1.8.
Figure 1.9 shows the pinched and shrunken hysteresis loops. The solid-line represents
the pinched hysteresis loop for ω1 and the dashed-line displays the shrunken hysteresis
loop for ω2=∞.
ω1
ω2
i(t)
v(t)
i-v characteristic curve
ω1
ω2
v(t)
M
v-M characteristic curve
Figure 1.9: Memristor fingerprint: Shrunken hysteresis loop
The waveforms of Figure 1.9 indicate that as the frequency ω of the input-signal in-
creases to infinity, the state variable converges to a constant value and establishes a linear
relationship between the input variable and the output variable. Therefore, the effect of
the pinched hysteresis loop disappears.
Furthermore, the v(t)-M characteristic curve is depicted in Figure 1.9(right-side). The
shrunken hysteresis loop causes the v(t)-M characteristic curve is shrunken with a fixed
value of memristance.
The physical interpretation of this phenomenon [39] is that the system possesses certain
inertia and can not respond as rapidly as the fast variation in the excitation waveform
and therefore must settle to some equilibrium state. This implies that the hysteresis effect
of the memristive system decreases as the frequency increases and hence it eventually
degenerates into a purely resistive system.
DC-domain fingerprint
The fingerprint in DC-domain is based on the fundamental property of the memristors
[3], called: DC characteristic. For dc operation, the v-i characteristic curves behave as
13
Chapter 1. Introduction
time-independent no linear resistors; i.e., the derivative of the state variable, shown in
(1.3), is used in the time-independent version.
This property implies that the state variable x is considered not time dependent and
causes that the value of the derivative is equal to zero. Then, the output variable depends
only on the input variable. Therefore, a value of the input variable gives a value of the
output variable.
In DC-domain, the nullor has been considered as equivalent element of the behavior
of the memristors [33].
1.4 Other mem-elements
The idea of the memristor has been extended to capacitive and inductive elements, called
memcapacitor and meminductor [40]. The coexistence between the passive non-memory
elements and the mem-elements is shown in Figure 1.10.
ϕ
i
q
v
σρ
Capacitor Inductor
Resistor
MemristorMemcapacitorMeminductor
Figure 1.10: Circuit elements with memory and conventional elements
These new circuit elements with memory show pinched hysteresis loops in the two
constitutive variables that define them: charge-voltage for the memcapacitor, and current-
flux for the meminductor. Hereafter, the concepts of memcapacitor and meminductor are
developed mathematically for the purpose of determining the relationship between the
constitutive variables and comparing with the conventional circuit elements.
14
1.4. Other mem-elements
1.4.1 Memcapacitor
The capacitor with memory or memcapacitor is a two-terminals element which constitutive
relationship is represented by
∫ t
−∞
q(τ)dτ = σ† = f(ϕ) (1.9)
Differentiating with respect to time t, both terms of (1.9), it yields
q(t) ,dσ
dt=
df(ϕ)
dt
By applying the chain rule to the expression above, it is possible to obtain
q(t) =df(ϕ)
dϕ
dϕ
dt(1.10)
The equation (1.10) can be rearranged in two different forms
q(t) =dσ
dϕ
dϕ
dt
dϕ
dt=
dϕ
dσq(t)
The above expressions can be simplified as
q(t) , C(ϕ)v(t)
voltage-controlled
v(t) , D(σ)q(t)
charge-controlled
where
C(ϕ) =dσ
dϕD(σ) =
dϕ
dσ
are called memcapacitance and memelastance, respectively.
1.4.2 Meminductor
The meminductor or inductor with memory is a two-terminals element which constitutive
variables are related by
∫ t
−∞
ϕ(τ)dτ = ρ‡ = f(q) (1.11)
†This term has been introduced by Chua [41] in order to assign a symbol to the time-domain integralof charge
‡This term has been introduced by Chua [41] in order to assign a symbol to the time-domain integralof flux
15
Chapter 1. Introduction
After applying some derivative steps with respect to time t, both terms of (1.11), it
yields
ϕ(t) ,dρ
dt=
df(q)
dt
The chain rule is used into the expression above in order to obtain
ϕ(t) =df(q)
dq
dq
dt(1.12)
The equation (1.12) is realigned in two different forms
ϕ(t) =dρ
dq
dq
dt
dq
dt=
dq
dρϕ(t)
These expressions are abbreviated as
ϕ(t) , L(q)i(t)
current-controlled
i(t) , Γ(ρ)ϕ(t)
flux-controlled
where
L(q) =dρ
dqΓ(ρ) =
dq
dρ
are called meminductance and memreluctance, respectively.
In order to compare the circuit elements with memory, the conventional circuit ele-
ments, namely, resistor, capacitor and inductor can be represented by
Resistor
v(t) = Ri(t)
current-controlled
i(t) = Gv(t)
voltage-controlled(1.13)
Capacitor
q(t) = Cv(t)
voltage-controlled
v(t) = Dq(t)
charge-controlled(1.14)
16
1.4. Other mem-elements
Inductor
ϕ(t) = Li(t)
current-controlled
i(t) = Γϕ(t)
flux-controlled(1.15)
Table 1.1 resumes the circuit elements: resistor (R), capacitor (C), inductor (I), mem-
ristor (MR), memcapacitor (MC) and meminductor (MI) with the respective controlled
variable: voltage-controlled, current-controlled, charge-controlled and flux-controlled.
Table 1.1: Duality between conventional and mem-elements
conventional elements mem-elements
Rcurrent-controlled v = Ri v = M(q)i current-controlled
MRvoltage-controlled i = Gv i = W (ϕ)v voltage-controlled
Cvoltage-controlled q = Cv q = C(ϕ)v voltage-controlled
MCcharge-controlled v = Dq v = D(σ)q charge-controlled
Lcurrent-controlled ϕ = Li ϕ = L(q)i current-controlled
MLflux-controlled i = Γϕ i = Γ(ρ)ϕ flux-controlled
1.4.3 Memcapacitive and meminductive systems
Similarly that the memristor is considered as a particular case of a general class of dynami-
cal systems, called memristive systems; the memcapacitor and meminductor are particular
instances of more extensive dynamical systems, called memcapacitive and meminductive
systems.
The general class of nth-order u-controlled memory devices [42] can be described by
x = f(x, u, t) y(t) = g(x, u, t)u(t) (1.16)
where x is a vector representing n internal state variables, y(t) and u(t) are output/input
variables (i.e., current, charge, voltage, or flux), g is a generalized response and f is a
continuous n-dimensional vector function.
Memcapacitive and meminductive systems are particular cases of 1.16, where the con-
stitutive variables are charge and voltage for the memcapacitance and flux and current for
the meminductance.
1.4.3.1 Memcapacitive system
The nth-order memcapacitive system charge and voltage-controlled can be defined by
17
Chapter 1. Introduction
x = f (x, q, t)
VC(t) = D(x, q, t)q(t)
charge-controlled
x = f (x, VC, t)
q(t) = C(x, VC , t)VC(t)
voltage-controlled
(1.17)
where D y C are called memelastance and memcapacitance, respectively. VC(t) y q(t) are
the voltage and the charge on the memcapacitive system at time t.
ω1
ω2
vC(t)
q(t)
vC-q characteristic curve
ω1
ω2
vC(t)
C
vC-C characteristic curve
Figure 1.11: Characteristic curves of the memcapacitive system
Figure 1.11 shows the vC(t)-q(t) and vC(t)-C characteristic curves of the memcapacitive
system. The memcapacitive system behaves as a linear capacitor in the limit of infinite
frequency and displays the pinched hysteresis loop in the vC(t)-q(t) characteristic curve.
1.4.3.2 Meminductive system
The nth-order meminductive system flux and current-controlled can be defined by
x = f (x, ϕ, t)
IL(t) = Γ(x, ϕ, t)ϕ(t)
flux-controlled
x = f (x, IL, t)
ϕ(t) = L(x, IL, t)IL(t)
current-controlled
(1.18)
where Γ y L are called memreluctance and meminductance, respectively. IL(t) y ϕ(t) are
the current and the flux on the meminductive system at time t.
The meminductive system can be modeled similarly to the memcapacitive system pre-
sented above. The meminductive system presents the pinched hysteresis loop in the iL(t)-
ϕ(t) characteristic curve and the iL(t)-L characteristic curve tends to a fixed value as the
18
1.5. Motivation
frequency approaches infinity.
Table 1.2 resumes the systems with memory: memristive system (MRS), memcapaci-
tive system (MCS) and meminductive system (MLS) with the respective controlled vari-
able: voltage-controlled (VC), current-controlled (CC), charge-controlled (QC) and flux-
controlled (FC).
Table 1.2: Systems with memory using state variables
System Port equation State equation
CCMRS v = M(x, i, t)i x = f(x, i, t)VCMRS i = W (x, v, t)v x = f(x, v, t)
VCMCS q = C(x, v, t)v x = f(x, v, t)QCMCS v = D(x, q, t)q x = f(x, q, t)
CCMLS ϕ = L(x, i, t)i x = f(x, i, t)FCMLS i = Γ(x, ϕ, t)ϕ x = f(x, ϕ, t)
The circuit elements with memory can be used for non-volatile memory and promise to
allow simulation of learning, adaptation and spontaneous behavior. These kind of devices
are common at nano-scale [43, 44, 45, 46, 47] where the dynamical properties of electrons
and ions depend on the history of the system during certain time scales [48].
1.5 Motivation
The theoretical conception of the memristor opens the possibility of simulating hybrid
circuits, i.e. circuits with conventional and memory elements. The design of hybrid
circuits and systems based on conventional and memory elements require rethinking of
fundamental circuit analysis, i.e. conventional circuit analysis with modified nodal analysis
(MNA) cannot consider mem-elements such as memristor together with the traditional
devices. Therefore, the incorporation of the memristor in a simulation scheme represents
a major part in order to improve the properties of the new circuit element with the
conventional devices.
The physical conception of the memristor by HP Labs has motivated a great interest
for modeling the memristive behavior of the device. A lot of work has been done with the
purpose of obtaining a SPICE model or macro-model of the memristor [49, 50, 51, 52, 53,
54]. Macro-models or SPICE models have the major drawback of using a large number of
nodes or circuit elements to mimic the behavior of the memristor in a particular analysis
domain, instead of using a two-terminal model that accomplishes the principal fingerprints
of the memristor.
19
Chapter 1. Introduction
In addition, the system of equations that represents the memristive behavior must be
solved by a numerical integration method. This type of numerical method has the major
disadvantage of employing a long CPU time to obtain the numerical values at each instant
of time.
1.6 Objective of the thesis
The research area of this thesis is focused on modeling the memristor in order to be
included in a simulation procedure for circuits with memristors. Therefore, the objective
of this thesis is
Generating memristor models for different domain analysis (DC, time and frequency
domain) in order to be included in a simulation procedure for circuits with memris-
tors.
20
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[3] L. Chua and S. M. Kang, “Memristive devices and systems,” Proceedings of the IEEE,
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26
Chapter 2
State of the art of memristor models
The discovery of the physical memristor has caused widespread interest for modeling the
memristive behavior in order to include the device in a simulation procedure. The principal
goal of the memristive models is showing the eight-shape or the bow tie-shape in the v-i
characteristic curve. However, the pinched hysteresis loop is not sufficient for validating the
memristive model. The behavior of the memristance respect to current, voltage, charge,
flux or state variable must be reviewed in order to ensure the reliability of the memristive
model. This chapter presents a brief introduction about the need for modeling in the
electronic circuit design and introduces different types of memristive models by classifying
three principal areas. Firstly, the analysis-domain classification is presented. In addition,
the hierarchy of design is used as classification of the memristor models. Furthermore, the
memristor models are organized based on the mathematical expressions. Hereafter, the
advantages and disadvantages of the memristive models are discussed with the purpose of
obtaining the necessary features for modeling these kind of devices.
2.1 Why is modeling necessary?
Modeling is a major step in the electronic circuit design. The principal purpose consists of
generating equivalent approximations that allow to carry out the simulation tasks of anal-
ysis and synthesis. The process of conceiving and selecting suitable models is a problem
of central importance in the design and simulation of electronic circuits [1]. Besides, the
assignment of proper numerical values to the model parameters are keys to the successful
computer-aided analysis of a circuit [1].
The adequacy of the models in the simulation tasks involves three principal commit-
ments [2]: agreement between the simulation results and the actual performance of the
physically implemented circuit, compromise between computational complexity and in-
27
Chapter 2. State of the art of memristor models
herent inaccuracy; and choosing which physically phenomena should be incorporated in
the model and which are to be neglected.
2.1.1 The memristor needs to be modeled
The design of hybrid circuits and systems based on conventional elements and memristors
is not possible without the generation of models for memristors. The modeling of the new
fundamental circuit element allow to carry out the simulation of novel electronic circuits
that contain memory elements and traditional devices.
2.2 Analysis domain
The electrical circuits are broken down in three different analysis domains [2, 1, 3]: DC,
transient and frequency. DC-analysis is based on the response of the circuit for time-
invariant excitation sources and is referred to the determination of operating points.
The time-variable responses established in circuits under periodic operation of excita-
tion sources are treated in the transient-analysis. The transfer characteristics of electrical
circuits, e.g. amplitude, phase and delay, as functions of frequency are examined in the
frequency-analysis.
2.2.1 DC-model
DC-analysis is the starting point for circuit simulators. The fundamental problem consists
of finding DC operating points, which are used as initial states in other types of analysis.
The behavior of the memristor in the DC-domain has been poorly studied in the
literature. In [4], the DC-model of the memristor is developed by considering current
contributions arising from different conduction mechanisms. The proposed circuit model
is obtained by fitting the filament-assisted tunneling currents from different regions of the
v-i characteristic curve of the memristor. The parametric representation of the currents
is performed by polynomial functions which depend on the voltage applied to the device.
In [5], the DC-model of the memristor is based on movement of the dopant boundary
w. The memristor is considered a constant resistance which tends either RON or ROFF
in a time required, called resistance saturation time. The time required for resistance
saturation depends on the sign and the value of the voltage in the device.
In [6], the DC-response of the memristor is represented by the waveform of the cur-
rent, the time after which the current reaches the equilibrium value and the decaying
28
2.2. Analysis domain
curve as time increases. The behavior of the memristor in DC-regime is related to the
decomposition of the hysteresis characteristic curve as the frequency tends to decrease.
The fundamental properties of the memristive systems [7] establish that under DC-
operation the memristive systems are equivalent to nonlinear resistors. However, this
property has not been tackled in the literature.
2.2.2 Transient-model
The transient-domain is the most widely used to model the behavior of the memristor. In
this domain, the three most important features of the memristive systems can be displayed:
the pinched hysteresis loop in the i-v characteristic curve, the behavior of the memristance
and the waveform of the state variable.
In [8], the memristor is based on a single internal state variable. The rate of system
resistance change (R changes between two limiting values) is characterized by a PWL func-
tion with two segments and one hyperplane. Besides, the piece-wise linear approximation
has been used for establishing the q-ϕ relationship [9, 10], the state variable of memristive
devices [8] and the pinched hysteresis loop in the v-i characteristic curve of the memristor
[11]. The numerical calculations of this kind of models are performed using a numerical
integration method in order to show the characteristic curves of the memristor.
In [12], the memristor is based on the movement of the boundary between two regions
with different dopant concentration. This model is called linear drift model. The linear
drift velocity of the memristor is described by a normalized state variable. The state
variable is proportional to the charge that passes through the device. However, this model
is not bounded to the physical limits of the device and the nonlinear behavior of the
transport mechanism is not incorporated. Therefore, the linear drift model has been
extended to the nonlinear drift model. The system of equations for the nonlinear model
is similar to the linear model except by the adding of a term to the state equation, called
window function [12, 13, 14, 15]. This function bounds the normalized state variable
within zero and unity, i.e. the velocity changes based on the position of the boundary.
The concept of the window function has been carried out with propose of incorporating
the nonlinear behavior into the state variable equation.
In [16, 17, 18], the memristor has been modeled using emulators or SPICE models.
These models are composed of individual elements which together can display the principal
characteristics of the memristors. Moreover, the incorporation of new elements in these
kind of models contributes to represent additional phenomena in the device.
The principal properties [19, 7] and fingerprints [20, 21, 22, 23] of the memristors have
been established in the time-domain in order to ensure the reliability of the models.
29
Chapter 2. State of the art of memristor models
2.2.3 Frequency-model
The modeling of the memristor in the frequency-domain is closely related to the behavior
of the memristor in the time-domain, e.g. the collapse of the pinched hysteresis loop
is observed in the time-domain despite of being an effect caused by the variation of the
frequency. This correlation of analysis domain has led to low interest for making memristor
models in the frequency-domain.
In [24], the memristor is considered a device that generates passive second and higher
harmonics signals. This feature is incorporated into the memristor model for harmonic
generation in combination with active circuits.
In [25], the memristor is represented by the Fourier response of the current in the
device. The Fourier series of the memristor current contains sine-type components and
no DC component. These characteristics are related with the pinched hysteresis loop and
the odd-symmetry of the memristor.
In [26], the discrete Fourier transform is used to explain the properties of the memristor
current in terms of time evolution of the width and the memconductance in the device.
The Fourier analysis shows that the shape of the hysteresis curve is related to the weights
of the harmonics.
2.3 Hierarchy of design
The automated design of analog systems has been structured over three principal levels of
hierarchy [27]: behavioral, circuital and physical. Behavioral-level is used to represent the
general idea of the system. The generation of architectures composed of circuit elements
in order to mimic the behavior of the system is treated in the circuital-level. Physical-level
is related to the behavior of real devices.
2.3.1 Physical model
The physical models reflect the physical phenomena taking place into memristor devices
[28, 29, 30]. The goal of this kind of models is to obtain physical insight into the trans-
port process responsible for memristive behavior. These models are validated with the
experimental data from real devices.
In [29], the memristor model is based on numerical solutions of coupled drift-diffusion
equations for electrons and ions with appropriate boundary conditions. The dynamics
of the memristive device is simulated by a semiconductor thin film with mobile dopants
that are partially compensated using a small amount of immobile acceptors. This model
30
2.3. Hierarchy of design
presents different conclusions about the behavior of a thin film semiconductor memristive
device. First, the ON and nearby low-resistance states have ohmic v-i characteristics, but
as the zero-bias resistance of the device increases toward the OFF state, the v-i curves
become increasingly nonlinear because of the npn potential barrier that develops in the
semiconductor. Second, for the same magnitude but opposite polarity of applied voltage,
the speed of switching ON will be significantly faster than switching OFF. This is related
to the electric field and the dopant concentration gradients. Finally, it is possible to invert
the switching polarity of a device; for example, applying an ON-switching voltage for a
long enough time will eventually cause ions from the grounded side of the device to drift
toward the center of the film, thus turning the device OFF.
In [28], the behavior of the memristor device is modeled from classical macroscopic
viewpoint, which is represented by the electron density distribution inside the device. The
memristor model is based on classic ion transportation theory and composed of three
regions, namely, the conductive region, the transition region and the insulating region.
The conductive region corresponds to the region of filaments while the transition region
connects the filaments and the remaining insulating region. The formation or dissolution
of the filament is simulated as the ion surface motion within the transition region. The
parameters w, λ and D are used in order to denote the lengths of the conductive region,
the transition region, and the entire device, respectively.
In [30], a general model is proposed in order to explain the memristive switching
behavior of nanodevices. This model suggests that the switching involves changes in the
electronic barrier at the Pt/TiO2 interface induced by drift of oxygen vacancies under an
applied electric field. When vacancies drift towards the interface, they create conducting
channels that short-circuit the electronic barrier. When vacancies drift away from the
interface they eliminate these channels, restoring the original electronic barrier. This
model fits the experimentally measure I-V characteristic curve of the memristor using an
equation comprising of two terms. The first term is used to approximate the ON-state
behavior of the memristor due to the electron tunneling. The second term represents the
approximation of the OFF-state behavior of the memristor similar to a P-N junction.
2.3.2 Circuital model
The circuital models allow the optimization of the memristive behavior of the devices
by adjusting circuit elements and bias conditions [31]. These type of models are used for
building macro-models or SPICE models. The macro and SPICE models have been widely
used in order to mimic the memristive behavior of the device built by HP Labs.
On one hand, the SPICE memristor models have been developed for different appli-
31
Chapter 2. State of the art of memristor models
cations, such as: chaos and oscillation [32, 33], arithmetic and logic [33, 34, 35, 36, 37],
amplifier [38, 39] and filter [40]; and different areas of knowledge, such as: neuromorphic
systems [41], biological systems [42] and electronics [42]. On the other hand, the SPICE
memristor models have been implemented with three [43], four [44, 45, 46], five [47, 48],
six [46], eight [49] and ten [50] internal nodes using passive elements and eight [46] internal
nodes using active elements.
2.3.3 Behavioral model
The behavioral models provide an overview of the expected memristive characteristic [51].
The classical procedure in order to build behavioral models consist of using mathematical
expressions to represent the general behavior of the device.
In [12], the behavior of the memristor has been modeled using the state variable equa-
tion of the boundary between two regions (doped and undoped) of the HP memristor.
Moreover, this mathematical expression has been modified by adding one term to the
state variable equation, called window function [52].
In [53], the piece-wise linear approximation has been used for modeling the behavior
of the memristor in biological applications.
2.4 Discussion
The memristor models classified by analysis domain and hierarchy of design can be located
in more than one category. In [54], the memristor model is performed using behavioral
approaches and then it is transformed into a SPICE model. In [4], the dc memristor
model is based on the behavioral approximation of the i-v characteristic curve and it is
represented by the polynomial function of the current response across the device. In [55],
the fundamental concept of memristor is used to develop behavioral models of non-volatile
resistance switching memory devices. Therefore, the memristor models have been built
to cover different design levels in different analysis domains. However, in the literature
there is not memristor model or memristive model that can be used in the three domains
of analysis: DC, transient and frequency.
The possible causes of the lack of memristor models using the three domains of anal-
ysis include: the principal fingerprints [20, 21, 22, 23] are displayed in the time-domain,
the absence of analytical expressions to represent the behavior of the memristor and the
misunderstanding of the dc response of the memristor.
The macro and SPICE models have been constantly proposed in order to incorporate
32
2.4. Discussion
the memristor in a simulation procedure. However, these models present two major draw-
backs: the number of nodes employed and the validity of the model by using the pinched
hysteresis loop in the v-i characteristic curve as the unique fingerprint of the device.
In [12], the memristor model is described by using the state variable equation and
represents the linear diffusion of dopants through the device. However, this model has no
physical boundaries in the frontiers; i.e., the normalized state variable can be increased
above the unity, which means that the boundary of the doped region has exceeded the
length of the device. The memristor models in [13, 14, 15] solve the problem of the
physical boundaries by adding a term to the equation of the state variable, called window
function. The window function restricts the diffusion of dopants into the physical limits of
the device; i.e., the normalized state variable can not exceed the boundaries of the device.
This model provides an adequate approximation of the memristive behavior. However,
the principal drawback is the need for a numerical integration method to solve the system
of equations.
The memristor model of [30] is considered a closer approach to the real behavior of the
memristor device due to the use of mathematical expressions related to physical phenom-
ena present in the device. However, there are many parameters that can be adjusted and
do not represent realistic values of the memristive characteristic.
The PWL memristor models [8, 9, 10, 11] are used to represent the rate of memris-
tance, state variable and resistance change. These models are useful to represent switching
mechanism because of the piece-wise linear representation is composed of slopes and hy-
perplanes. However, the necessity of numerical integration methods and the inability to
include nonlinear effects represent two principal disadvantages of these models.
33
Chapter 2. State of the art of memristor models
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40
Chapter 3
Development of DC memristor
models
In DC-domain, the calculation of the operating points has been regarded as the most im-
portant simulation stage because it represents the stating point for other types of analysis.
The incorporation of the memristor opens the possibility of studying the influence of the
memristive behavior in the operating points of circuits with memristors. This chapter
presents, firstly, a simple model of the memristor based on the piece-wise linear (PWL)
approximation. In addition, the influence of the parameters of the PWL approximation in
the v-i characteristic curve is studied. Furthermore, the methodology for obtaining the v-i
characteristic is explained. Hereafter, the existence of multiple operation points is treated
by two cases of study.
3.1 DC memristor model
In this section, a DC model is developed from the original dynamic formulation of a
memristive system.
3.1.1 PWL approximation
The memristive systems are represented by
x = f(x, i)
v = R(x, i)i
current-controlled
x = f(x, v)
i = G(x, v)v
voltage-controlled
(3.1)
The PWL approximation can be used to establish the branch-characteristic of the
41
Chapter 3. Development of DC memristor models
memristive elements. The general representation of the memristive system using the first
and the third quadrant is depicted in Figure 3.1.
VM
f(VM)
VT1
M1
M2M4
M3
VT2
Figure 3.1: PWL representation of the memristive system
This representation is composed of the slopes M1, M2, M3 and the break points VT1,
VT2. Additionally, the blue dashed-line is incorporated into the PWL representation to
complete a possible path between the first and third quadrant and therefore establish a
pinched hysteresis loop.
According to the literature, this representation can be carried out using the general
explicit piece-wise linear function of Chua and Kang [1]. The formal definition of this
canonical PWL function is expressed as [1]:
f(x) = a0 + a1x+
n∑
j=1
bj |x− xj |+ cjsgn(x− xj) (3.2)
The characteristic of the blue dashed-line in Figure 3.1 may cause, under certain con-
ditions, problems with the canonical PWL function of (3.2). In order to overcome that
problem, the decomposed formulation based on the hyperplane unbending has been pro-
posed [2].
In this part of the thesis, the particular explicit piece-wise linear function of Chua and
Kang [3] has been resorted in order to represent the memristive behavior. Furthermore,
the analysis is focused to the first quadrant of the PWL representation in Figure 3.1.
However, the proposed methodology can be applied to the third quadrant.
The formal definition of the particular PWL function is expressed as [4]:
42
3.1. DC memristor model
f(x) = a +Bx+σ
∑
i=1
ci|(αi, x)− βi| (3.3)
This function is capable of modeling the change of slopes and the break point which
are essential conditions for carrying out the memristive effect.
3.1.2 Voltage-controlled memristive system
The voltage-controlled memristive systems can be divided in two different cases: memris-
tive state and memconductive state. These cases are described by
x = f(VM)
IM = VM
x(VM )
memristive-state
x = f(VM)
IM = x(VM)VM
memconductive-state
(3.4)
where the state variable x is established as the function that governs the change in mem-
ristive or memconductive value of the system and VM denotes the voltage across the
memristive system.
The state equations are defined by the canonical formulation of (3.3), compounded of
two lineal-segments and one hyperplane, as shown in Figure 3.2.
VM
f(VM)
VT
M1
M2
VM
f(VM)
VT
W1
W2
Figure 3.2: PWL functions for the voltage-controlled memristive system
The state equations are expressed as
x = a0 + a1VM + b1|VM − VT |
a0 = −12(M2 −M1)|VT |
a1 = M1 +12(M2 −M1)
b1 =12(M2 −M1)
memristive-state
x = a0 + a1VM + b1|VM − VT |
a0 = −12(W2 −W1)|VT |
a1 = W1 +12(W2 −W1)
b1 =12(W2 −W1)
memconductive-state
(3.5)
43
Chapter 3. Development of DC memristor models
where M1 and M2 are the memristive slopes, W1 and W2 are the memconductive slopes
and VT represents the position of the hyperplane.
The voltage-controlled memristive systems established in (3.4) and (3.5) are used to
simulate the circuit shown in Figure 3.3.
+−V VM
+
–
IM
Figure 3.3: Circuit with voltage-controlled memristive system
where V=2V. The hyperplane position is VT=0.5 and the values of M1, M2, W1 and
W2 are listed in Table 3.1.
Table 3.1: Parameter values
Parameter Value
M1 0.5 kΩM2 1 kΩW1 2 m
W2 1 m
The values of the parameters M1, M2, W1 and W2 have been established to fulfill the
PWL function for the voltage-controlled memristive system, as shown in Figure 3.2. In
the case of memristive-state, the value of M2 is the double of the M1 value in order to
ensure the change of slope in the memristor. For the case of memconductive-state, the
value of W2 is the half of W1 value.
non-monotonic monotonic
0.0 0.5 1.0 1.5 2.00
2
4
6
8
VoltageHVL
Cur
rentHm
AL
Figure 3.4: DC-responses of the voltage-controlled memristive systems
Figure 3.4 shows the dc-responses of the voltage-controlled memristive systems. The
44
3.1. DC memristor model
simulation results have been obtained by sweeping the power supply V of the electri-
cal circuit shown in Figure 3.3. In each value of voltage, the system of equations of
the voltage-controlled memristive system has been resolved using a numerical integration
method. On one hand, the dashed-line represents the monotonically increasing curve of the
voltage-controlled memristive system using the memconductive-state. On the other hand,
the solid-line represents the non-monotonically increasing curve of the voltage-controlled
memristive system using the memristive-state.
3.1.3 Current-controlled memristive system
In the case of the current-controlled memristive systems, the memristive-state and the
memconductive-state are described by
x = f(IM)
VM = x(IM)IM
memristive-state
x = f(IM)
VM = IMx(IM )
memconductive-state
(3.6)
where x is the state variable and IM denotes the current across the memristive system.
The state equations are defined by the canonical formulation of (3.3), as shown in
Figure 3.5.
IM
f(IM)
IT
M1
M2
IM
f(IM)
IT
W1
W2
Figure 3.5: PWL functions for the current-controlled memristive system
The state equations are expressed as
x = a0 + a1IM + b1|IM − IT |
a0 = −12(M2 −M1)|IT |
a1 = M1 +12(M2 −M1)
b1 =12(M2 −M1)
memristive-state
x = a0 + a1IM + b1|IM − IT |
a0 = −12(W2 −W1)|IT |
a1 = W1 +12(W2 −W1)
b1 =12(W2 −W1)
memconductive-state
(3.7)
45
Chapter 3. Development of DC memristor models
where M1 and M2 are the memristive slopes, W1 and W2 are the memconductive slopes
and IT represents the position of the hyperplane.
The current-controlled memristive systems established in (3.6) and (3.7) are used to
simulate the circuit shown in Figure 3.6.
I VM
+
–
IM
Figure 3.6: Circuit with current-controlled memristive system
where I=2mA. The hyperplane position is IT=0.5mA and the values of M1, M2, W1
and W2 are listed in Table 3.1. The values of the parameters M1, M2, W1 and W2 have
been established according to the criteria for the case of voltage-controlled PWL function.
monotonic non-monotonic
0.0 0.5 1.0 1.5 2.00.0
0.1
0.2
0.3
0.4
0.5
CurrentHmAL
Vol
tageHVL
Figure 3.7: DC-responses of the current-controlled memristive systems
Figure 3.7 shows the dc-responses of the current-controlled memristive systems. The
procedure for obtaining the simulation results is similar to the case of voltage-controlled
memristive systems with the difference that the power supply in Figure 3.6 is current-
mode. On one hand, the dashed-line represents the monotonically increasing curve of
the current-controlled memristive system using the memristive-state. On the other hand,
the solid-line represents the non-monotonically increasing curve of the current-controlled
memristive system using the memconductive-state.
In summary, the Figures 3.4 and 3.7 show that the DC-response curve of the memris-
tive system, under certain conditions, exhibits a non-monotonically increasing curve and
therefore it is possible to cause multiple operating points.
46
3.1. DC memristor model
3.1.4 Variations of the PWL parameters
The parameters M1, M2 and VT of the PWL memristor model can be related with the
memristive behavior of the device. In the bipolar switching memristive systems, the
hysteresis process often is of a threshold type: while a high applied voltage is needed
to change the device state, at low applied voltages the resistance of the device remains
unchanged. This is very often associated [5] with ionic transport and electrochemical
reactions. The experimental work of [6] assumes that the rate of the resistance change is
small below a threshold voltage VT and fast above VT .
The location, concentration and distribution of oxygen vacancies in the TiO2 film
control the conductance, rectification and switching polarity of the device [7]. Moreover,
the ratio M2
M1
is dependent of the thickness of the TiOx/TiO2 layers [8].
Hereafter, the effects in the voltage-current characteristic of the memristive systems at
DC-regime caused by the parameters M1, M2 and VT of the PWL memristor model are
determined using the systematically variations of the parameter values. The variations
of parameters are principal focused on two issues: variations of memristive slopes and
variations of hyperplane position.
Variations of memristive slopes
In order to assesses the effects of the memristive slopes M1 and M2 in the voltage-current
characteristic, the scheme of variations in Figure 3.8 is proposed.
VM
f(VM)
VT
M1
M2
VM
f(VM)
VT
M1
M2
Figure 3.8: Scheme of variations for memristive slopes
The previous scheme causes two different cases. The first case consists of varying the
memristive slope M1 for the following values (in KΩ): 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4
and 5; while M2=1KΩ.
The resulting family of V -I curves are depicted in Figure 3.9. These simulation results
have been obtained using the numerical integration method of Runge-Kutta for each value
47
Chapter 3. Development of DC memristor models
of the memristive slope M1. In plain words, it can be established that M1 affects the
asymptotic V -I characteristics by flattening the curve and locating the peak closer to zero
as M1 increases.
á
ç
á 0.1ç 5
0.0 0.5 1.0 1.5 2.00
5
10
15
VoltageHVL
Cur
rentHm
AL
Figure 3.9: Simulation results using the variations of M1 coefficient
In the second case, the memristive slope M2 is varied for the following values (in KΩ):
0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4 and 5; while M1=1KΩ.
In similar way, the resulting V -I DC branch relationships are depicted in the fam-
ily of curves of Figure 3.10. It can be established that M2 affects the asymptotic V -I
characteristics by lowering the curve at high voltages (for V > VT approximately) as M2
increases. Besides M1 has no further effect at low voltages and the peak remains basically
unchanged.
á
ç
á 0.1ç 5
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
VoltageHVL
Cur
rentHm
AL
Figure 3.10: Simulation results using the variations of M2 coefficient
48
3.2. Generation of the v-i branch relationship
Variations of hyperplane position
For assessing the effect that the hyperplane position VT has on the voltage-current char-
acteristic of the memristive system, the scheme of variations in Figure 3.11 is proposed.
VM
f(VM)
VT
M1
M2
Figure 3.11: Scheme of variations for hyperplane position
In this case, the hyperplane position is varied for the following values (in V): 0.1, 0.2,
0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1; while M1=500Ω and M2=1KΩ.
The resulting family of characteristic curves V -I are depicted in Figure 3.12. It can be
established that the position of hyperplane VT affects the asymptotic characteristic curve
V -I by lowering the curve at voltages above VT .
á 0.1ç 1
çá
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
6
7
VoltageHVL
Cur
rentHm
AL
Figure 3.12: Simulation results using the variations of the hyperplane VT
In summary, it could be noticed that the variations on the PWL parameters affect the
shape of the resulting v-i curves.
3.2 Generation of the v-i branch relationship
The main goal of the methodology is to obtain a v-i branch relationship that constitutes
the memristor model in DC. This methodology is shown in Figure 3.13.
49
Chapter 3. Development of DC memristor models
Input parameters
PWL description
Expression ofDC characteristic
Simulationprocedure
Figure 3.13: Methodology for DC analysis of memristors
Input parameters
In this initial stage, the voltage-controlled characteristic is given by a PWL function that
consists of two lines segments separated by a hyperplane at VT . The slopes of the segments
are M1 and M2.
PWL description
The PWL function is based on the well-known canonical representation of (3.3). In the
case of the voltage-controlled memristive system, the PWL description is established as
(3.4) and (3.5).
Expression of DC characteristic
The state equation of (3.5) is used for obtaining the current-voltage characteristic of the
memristive system. In order to do this, the numerical integration formula of the fourth-
order Runge-Kutta method is resorted. The formula of RK4 [9] is expressed as
xn+1 = xn +h
6(k1 + 2k2 + 2k3 + k4)
using
50
3.2. Generation of the v-i branch relationship
k1 = f(VM)
k2 = f(VM + 12k1h)
k3 = f(VM + 12k2h)
k4 = f(VM + k3h)
This method approaches the solution of the ordinary differential equation using an
iterative process where the numerical solution xn+1 is determined by the present value xn
plus the weighted average of the products of the size of the interval h and an estimated
slope. Besides, the RK4 method is a fourth-order method meaning that the local truncal
error is on the order of O(h5) and the total accumulated error is on order of O(h4). Runge-
Kutta methods have local truncation error of the same order of Taylor methods, but do
not use the calculation and evaluation of the derivatives of the function [9]. Because
of these characteristics and the use only of monotonically increasing curves to represent
the memristive system in the DC-domain, the RK4 method is resorted to approach the
solution of the ODE. The results of the integration procedure are recast in the output
equation of (3.4).
At this point, the process of constructing a mathematical function that fit the series
of data points from the output variable of the memristive system is performed using the
command Fit from Wolfram Mathematica [10]. This curve fitting procedure is applied
in order to approximate the current-voltage characteristic to a polynomial function of the
form:
i(VM) = anVnM + an−1V
n−1M + ...+ a2V
2M + a1VM + a0 (3.8)
The expression of (3.8) represents the constitutive branch relationship of the memris-
tive system that will be used for DC analysis in the simulation procedure. In particular,
special attention is devoted to assess the existence of multiple DC operating points because
the expression in (3.8) may be a non monotonically increasing characteristic.
Simulation procedure
In the simulation procedure an input netlist is used to establish the equilibrium equation
that will be placed in a homotopy solver for obtaining the multiple DC solutions. The
input netlist is oriented to input the circuit description containing components as volt-
age and current independent-sources, linear resistors and memristors. The equilibrium
equation is formulated by using a slightly changed version of the modified nodal analysis
51
Chapter 3. Development of DC memristor models
formulation that incorporates the current-voltage characteristics of the memristive systems
as a separated nonlinear current vector, as follows:
f(x) = YmnaXmna + i(VM)− S = 0
where, Xmna represents the unknowns as the MNA variables, S is the stimuli vector and
i(VM) is the vector of the voltage-controlled memristor branch relationships.
Because Newton-Rhapson methods face several convergence problems especially when
dealing with equations emanating from memristive systems, the homotopy methods are
resorted. Homotopy methods formulate an associated equation to the original system
of non-linear equations by adding an extra parameter, the homotopy parameter λ. The
homotopy formula is represented by
H(f(x), λ) = f(x) + (λ− 1)f(x0) = 0
The solutions are found on the homotopy path.
3.3 A glimpse to DC analysis of memristive circuits
In general, the problem of finding the DC operating point is important because it is the
starting point for other kinds of analysis such as the small signal and transient analysis.
The work of finding the DC operating consists in solving the nonlinear algebraic equations
(NAEs) emanating from circuits.
The complete solution to the general DC problem of nonlinear circuits focusses on
the next aspects [11]: determining whether or not the uniqueness of the dc solution is
guaranteed, establishing an upper bond on the number of dc solutions, and as last step
calculating all of them, as well as determining their conditions for stability.
As for nonlinear resistive circuits, the resulting equilibrium equation for memristive
circuits at dc regime has the form of system of NAEs:
f(x) = 0 (3.9)
where x is the set of unknowns node voltages and currents of the non-NA compatible
elements.
For memristor circuits, the highly nonlinear resulting curves impose serious convergence
problems when using the well-known Newton-Raphson method for solving the system of
NAEs in (3.9). Besides, Newton-Raphson methods are locally convergent, it means that
they usually fail to converge to a solution when the starting point is not close enough to
52
3.3. A glimpse to DC analysis of memristive circuits
the solution.
Homotopy methods have proved its usefulness not only to overcome this drawback but
also to find more than one DC solutions. They are based on establishing an auxiliary
scheme in order to convert the problem of finding the roots of the system in (3.9) a
static problem into a problem of finding the solution of an associated ordinary differential
equation (a dynamic problem) [12, 13].
The homotopy equation is formed by adding a parameter to the original equation:
H(f(x), λ) = 0
where λ is the homotopy parameter, and H is the resulting homotopy formula.
However, the homotopy methods exhibit three principal drawbacks, such as: the initial
point where the homotopy starts to trace the solution, the stop criterion to decide when
to stop seeking for more solutions and the path tracking to ensure the trajectory of the
solution curve.
In [13], a homotopy scheme that overcomes the problem of the stop criterion for these
kinds of methods has been introduced and it will be applied to solve the equilibrium
equations of memristive circuits. This homotopy method is called the double-bounded
homotopy (DBH) and is defined by
H(f(x), λ) = CQ + eQ ln(Df 2(x) + 1)
where f(x) is the function to solve, λ is the homotopy parameter, a and b are the values
of the double bounding lines, C and D are positive constants of the homotopy, and Q is
given by:
(λ− a)(λ− b)
Hereafter, the DBH method is employed to find the solutions in the memristive circuits
at dc-regimen.
53
Chapter 3. Development of DC memristor models
3.4 Cases of study
3.4.1 A single-memristor circuit
The first example consists of the memristive circuit shown in Figure 3.14.
M
+−V
R
Figure 3.14: DC memristive circuit
where V=2V and R=400Ω. The voltage-controlled memristor M is given by three
different I-V expressions with the purpose of inspecting the impact of parameter variation
on the amount of solutions for memristive circuits. These expressions are obtained using
the methodology for generating the v-i characteristics as shown in Figure 3.13 and are
defined as follow
IM = 0.0505VM + 0.00933V 2M − 1.045V 3
M + 4.1928V 4M − 8.6192V 5
M + 10.9289V 6M
− 9.0108V 7M + 4.8515V 8
M − 1.647V 9M + 0.320V 10
M − 0.0272V 11M
(3.10)
IM = 0.0547VM + 0.099V 2M − 0.6338V 3
M + 3.5221V 4M − 8.2V 5
M + 11.1881V 6M
− 9.6988V 7M + 5.419V 8
M − 1.8937V 9M + 0.3769V 10
M − 0.0326V 11M
(3.11)
IM = 0.057VM − 0.1689V 2M − 0.2551V 3
M + 2.462V 4M − 6.351V 5
M + 9.054V 6M
− 8.0449V 7M + 4.5671V 8
M − 1.6142V 9M + 0.324V 10
M − 0.0282V 11M
(3.12)
As it can be seen, the order of the I-V expressions is set to a value of 11 in order
to obtain greater accuracy with respect to the numerical solution from the process of
integration. In addition, a lower order of the expression compromises the tolerance with
respect to the original curve of the memristive system.
According to the expression of the memristor, three different cases of DC-response
can emerge. These cases are the result of the intersection between the asymptotic I-V
memristor characteristic and the load line -defined by the values of V and R. DC-responses
54
3.4. Cases of study
are depicted in Figure 3.15.
The first case is shown in Figure 3.15a withM1=500 andM2=500. In this circumstance,
a Newton-Rapshon scheme can find the solution provided the Jacobian of the system
remains non-singular. This case uses the I-V expression of (3.10).
The second case uses the I-V expression of (3.11) with M1=810 and M2=500. In this
case, Newton-like methods are able to reach only the first solution (at low voltages) but
they experience serious convergence problems when assessing the second solution (when
the load line is tangent to the I-V characteristic). Figure 3.15b shows the DC-response of
this case.
The third case is shown in Figure 3.15c with M1=1000 and M2=100. In this case,
Newton-Raphson methods are able to find only one solution depending on the selection
of the starting guess of the method. The I-V expression of (3.12) is used for this case.
S
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
6
7
VoltageHVL
Cur
rentHm
AL
(a) One solution
S1
S2
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
VoltageHVL
Cur
rentHm
AL
(b) Two solutions
S1
S2
S3
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
VoltageHVL
Cur
rentHm
AL
(c) Three solutions
Figure 3.15: DC-response of different memristor characteristics
Table 3.2 shows the operating points of the three cases explained above. The cases of
multiple solutions, Figures 3.15b and 3.15c, have been addressed by resorting to the DBH
method.
55
Chapter 3. Development of DC memristor models
In the following, the stability of the DC solutions for each case is addressed.
First case Herein, the system possesses a single DC operating point. This can be
straightforwardly solved by using the traditional Newton method. The solution
lies on a positive -slope region of the v-i memristor DC characteristic, which means
that the solution is stable.
Second case Here, two solutions are found. S1 is stable while S2 is unstable.
Third case In this case three solutions appear. S1 and S3 are stable while S2 is unstable.
Table 3.2: Summary of the operating points of the three cases
Sol. VM (V) IM (A)
S 0.108692 0.00472827S1 0.124707 0.00468823S2 0.860166 0.00284958S1 0.140453 0.00464887S2 0.393065 0.00401734S3 1.03058 0.00242355
Additional comparative information of the three cases are summarized in Table 3.3.
Table 3.3: Analysis of the operating points of the DC memristive circuit
Sol. Vv Iv VR IR VM IM Pv PR PM
∑
VI
S1 2 -0.00472827 1.89131 0.00472827 0.108692 0.00472827 -0.00945654 0.00894261 0.000513927 2.81893x10−18
S1 2 -0.00468823 1.87529 0.00468823 0.124707 0.00468823 -0.00937646 0.00879181 0.000584655 -4.33681x10−19S2 2 -0.00284958 1.13983 0.00284958 0.860166 0.00284958 -0.00569917 0.00324805 0.00245112 -4.57273x10−15
S1 2 -0.00464887 1.85955 0.00464887 0.140453 0.00464887 -0.00929774 0.00864479 0.000652947 -1.6263x10−18S2 2 -0.00401734 1.60693 0.00401734 0.393065 0.00401734 -0.00803467 0.0064556 0.00157907 -8.0231x10−18S3 2 -0.00242355 0.96942 0.00242355 1.03058 0.00242355 -0.0048471 0.00234944 0.00249766 -7.23076x10−15
56
3.4. Cases of study
3.4.2 A two-memristor circuit
The second example consists of a two-memristor circuit [14], as shown in Figure 3.16.
M1
M2
+−V
RVb
Vc
Figure 3.16: DC two-memristor circuit
where the linear resistor R=400 and V=2V. Besides, the voltage-controlled memristors
(M1 and M2) are given by the following I-V expressions
IM1= 0.05417(Vb − Vc)− 0.0822(Vb − Vc)
2 − 0.7455(Vb − Vc)3 + 3.9447(Vb − Vc)
4
− 9.1982(Vb − Vc)5 + 12.6736(Vb − Vc)
6 − 11.11(Vb − Vc)7 + 6.273(Vb − Vc)
8
− 2.2124(Vb − Vc)9 + 0.4438(Vb − Vc)
10 − 0.0386(Vb − Vc)11
IM2= 0.057Vc − 0.1689V 2
c − 0.2551V 3c + 2.462V 4
c − 6.351V 5c + 9.0546V 6
c
− 8.0449V 7c + 4.5671V 8
c − 1.6142V 9c + 0.324V 10
c − 0.0282V 11c
(3.13)
S1
S2
S3
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
VoltageHVL
Cur
rentHm
AL
(a) M1
S1
S2
S3
0.0 0.5 1.0 1.5 2.00
2
4
6
8
10
VoltageHVL
Cur
rentHm
AL
(b) M2
Figure 3.17: DC-response of the M1 and M2 memristors
Figure 3.17 shows the DC-responses of the M1 and M2. The memristive characteristics
of M1 and M2 present three intersections with the load line. Table 3.4 shows the operating
points of the memristors M1 and M2.
57
Chapter 3. Development of DC memristor models
Table 3.4: Summary of the operating points of M1 and M2
Sol. VM(V ) IM(mA)
S1 0.1220 4.6949S2 0.7397 3.1507S3 1.0811 2.2970S4 0.1410 4.6472S5 0.3880 4.0297S6 1.0661 2.3345
The branch electrical-variables of M1 and M2 are shown in Table 3.5.
Table 3.5: Analysis of the operating points of M1 y M2
Sol. VM(V ) IM(mA) I ′M(VM) = GM(S) RM(Ω) = G−1M PM(mW )
S1 0.12202298599799923 4.694942535005002 0.0211003 47.39269 0.572891S2 0.739714187344127 3.1507145316396826 -0.00407294 -245.52289 2.33063S3 1.0811685470000758 2.2970786324998105 -0.000197111 -5.073x103 2.48353S4 0.14109890901914046 4.647252727452149 0.0118107 84.66898 0.655722S5 0.3880811511346674 4.029797122163331 -0.00660631 -151.37043 1.56389S6 1.0661612891168126 2.3345967772079685 -0.000998819 -1.001x103 2.48906
With the purpose of obtaining the solutions of circuit in Figure 3.16, the DBH method
is resorted. Table 3.6 shows the operating points of the circuit compounded of M1 and
M2 memristors. Besides, the branch electrical-variables of the operation points are shown
in Table 3.7.
Table 3.6: Solutions using the DBH method
Soluciones Vb(V ) Vc(V ) Iv(mA)
S1 1.2082259570272 1.4078452267629 -1.9794351074322S2 0.041500014945818 0.25237891076774 -4.8962499626355S3 0.23403114962084 0.12430622762259 -4.4149221259480S4 0.92068199991821 0.057622615964098 -2.6982950002045S5 1.0567275225081 0.048702503521150 -2.3581811937297S6 0.52990240876302 0.44637855425860 -3.6752439780925S7 1.0334780064944 0.98384546118236 -2.4163049837640
58
3.4.Cases
ofstu
dy
Tab
le3.7:
Analy
sisof
thesolu
tionsusin
gDBH
meth
od
Sol. VE IE VR IR VM1IM1
VM2IM2
PE PR PM1PM2
∑
V I
S1 2 -0.00197944 0.791774 0.00197944 -0.199619 0.00197944 1.40785 0.00197944 -0.00395887 0.00156727 -0.000395133 0.00278674 -5.24448x10−11S2 2 -0.00489625 1.9585 0.00489625 -0.210879 0.00489625 0.252379 0.00489625 -0.0097925 0.00958931 -0.00103252 0.00123571 -1.37078x10−10S3 2 -0.00441492 1.76597 0.00441492 0.109725 0.00441492 0.124306 0.00441492 -0.00882984 0.00779661 0.000484427 0.000548802 -1.11586x10−14S4 2 -0.0026983 1.07932 0.0026983 0.863059 0.0026983 0.0576226 0.0026983 -0.00539659 0.00291232 0.00232879 0.000155483 2.15945x10−13S5 2 -0.00235818 0.943272 0.00235818 1.00803 0.00235818 0.0487025 0.00235818 -0.00471636 0.00222441 0.00237711 0.000114849 1.30692x10−9S6 2 -0.00367524 1.4701 0.00367524 0.0835239 0.00367524 0.446379 0.00367524 -0.00735049 0.00540297 0.000306971 0.00164055 1.02618x10−11S7 2 -0.0024163 0.966522 0.0024163 0.0496325 0.0024163 0.983845 0.0024163 -0.00483261 0.00233541 0.000119927 0.00237727 4.26559x10−13
59
Chapter 3. Development of DC memristor models
60
Bibliography
[1] L. Chua and S. M. Kang, “Section-wise piecewise-linear functions: Canonical repre-
sentation, properties, and applications,” Proceedings of the IEEE, vol. 65, pp. 915–
929, June 1977.
[2] V. Jimenez-Fernandez, L. Hernandez-Martinez, and A. Sarmiento-Reyes, “Decom-
posed piecewise-linear models by hyperplanes unbending,” in Circuits and Systems,
2006. ISCAS 2006. Proceedings. 2006 IEEE International Symposium on, pp. 4 pp.–,
May 2006.
[3] L. Chua and S. M. Kang, “Section-wise piecewise-linear functions: Canonical repre-
sentation, properties, and applications,” Proceedings of the IEEE, vol. 65, pp. 915–
929, June 1977.
[4] D. M. W. Leenaerts andW. M. V. Bokhoven, Piecewise Linear Modeling and Analysis.
Kluwer Academic Publishers, 1998.
[5] M. Waser, R.; Aono, “Nanoionics-based resistive switching memories,” Nature Mate-
rials, vol. 6, no. 11, pp. 833–840, 2007.
[6] G. S. S. D. R. W. R. S. Strukov, Dmitri B.; Snider, “The missing memristor found,”
Nature, vol. 453, no. 7191, pp. 80–83, 2008.
[7] M. D. L. X. O. A. A. S. D. R. W. R. S. Yang, J. Joshua; Pickett, “Memristive switching
mechanism for metal/oxide/metal nanodevices,” Nature, vol. 3, no. 7, pp. 429–433,
2008.
[8] A. Emelyanov, V. Demin, I. Antropov, G. Tselikov, Z. Lavrukhina, and P. Kashkarov,
“Effect of the thickness of the tio x /tio2 layers on their memristor properties,”
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[9] R. Burden and J. Faires, Numerical Analysis. Cengage Learning, 2004.
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[10] S. Wolfram, The MATHEMATICA R© Book, Version 4. Cambridge University Press,
1999.
[11] A. Sarmiento-Reyes, A partition method for the determination of multiple DC oper-
ating points. Ph.D. Thesis, Delft University, 1994.
[12] J. Ogrodzki, Circuit Simulation Methods and Algorithms. Electronic Engineering
Systems, Taylor & Francis, 1994.
[13] H. Vazquez-Leal, L. Hernandez-Martinez, and A. Sarmiento-Reyes, “Double-bounded
homotopy for analysing nonlinear resistive circuits,” in Circuits and Systems, 2005.
ISCAS 2005. IEEE International Symposium on, pp. 3203–3206 Vol. 4, May 2005.
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solutions of nonlinear resistive circuits,” International Journal of Circuit Theory and
Applications, vol. 4, no. 3, pp. 215–239, 1976.
62
Chapter 4
Transient memristor model
The principal properties and fingerprints of the memristors have been established in the
time-domain. The models of the memristors have been generated using SPICE-models,
mathematical-models and behavioral-models. However, these kind of models do not fulfill
the fundamental characteristics of the memristive systems.
From the point of view of CAD, the modeling of the memristor must comply with the
following requirements: two-terminal model, full-symbolical expression and the capacity
of showing the principal variables of the memristive systems.
This chapter presents two memristor models using two different methods in order to
generate the v-i expressions that satisfy the properties and fingerprints of the memristors.
The first one is based on the polynomial approximation of the constitutive variables of the
memristor. The second one is based on the solution of the memristive system from the
HP memristor using the homotopy perturbation method.
4.1 Polynomial-based model
In this part, the memristor model is obtained using the polynomial expression. The
mathematical description is presented in order to establish the connection between the
q-ϕ curve and the v-i expression of the model. Besides, the memristor model is simulated
with the purpose of showing the behavior in the time-domain.
4.1.1 Mathematical description
The memristor can be either q-controlled or ϕ-controlled. Therefore, the memristor can
be described by the following function
63
Chapter 4. Transient memristor model
ϕ(t) = M1q(t) +M2q(t)2 (4.1)
where M1 and M2 are the memristance values defined as
M1 = ROFF M2 =RON −ROFF
4
where ROFF and RON are the OFF-state and ON-state resistance, respectively. The
order of the polynomial function shown in (4.1) is set to a value of two for the purpose
of maintaining simplicity in the formulation without performing complex calculations.
In addition, a higher order of the polynomial expression involves a faster switching time
between the values of ROFF and RON of the memristor. This area of study is not addressed
in this thesis and therefore the order selected in (4.1) is useful.
The current across the memristor is defined by
i(t) = A sin(ωt) (4.2)
where A is the current amplitude and ω represents the frequency. By integrating (4.2),
the electric charge q(t) is obtained
q(t) =
∫ t
0
A sin(ωτ)dτ =A
ω[1− cos(ωt)] (4.3)
A new expression for the flux is obtained, after substituting (4.3) in (4.1)
ϕ(t) =A
ω(1− cos(ωt))
[
M1 +M2A
ω(1− cos(ωt))
]
Because the time-derivative of the flux is the voltage, it is possible obtain an expression
for the voltage
v(t) =[
M1 + 2M2A
ω(1− cos(ωt))
]
i(t) (4.4)
The expression of (4.4) constitutes in fact an v-i branch relationship that models the
memristor, and it can be recast as a functional model for simulation purposes. It clearly
results that the memristance is given as:
M =[
M1 + 2M2A
ω(1− cos(ωt))
]
(4.5)
It is important to notice, that the limit value of the memristance at high frequency is:
M∞ = limω→∞
(M) = M1 = ROFF
64
4.1. Polynomial-based model
i.e. the value of the OFF-state resistance
4.1.2 Characteristic of the memristor model
The polynomial memristor model is established using the expression of (4.4). This mem-
ristor model is evaluated for discrete values of the frequency at ω = 1, 2, 10, and the
corresponding hysteresis loops are shown in Figure 4.1. It possesses the main fingerprints
of a passive memristor, such as: pinched hysteresis loop, hysteresis lobe area and shrunken
hysteresis loop. The curves have been obtained for ROFF=14.41 KΩ, RON=1.4 KΩ and
A=1.
é éé é
é éé ééééééééééééé
ééé
é
é
é
é
é
éééé
ééééééééééé éé é
é éé é
éí íí í
í íí íííííííííí
ííííí
íí
í
í
í
íí
ííííííí
ííííííííí í
í íí í
íá áá á
á áá ááááááááááááááááááááááááááááááááááá
ááá á
á áá á
á
á 10Ωí 2Ωé Ω
-1.0 -0.5 0.0 0.5 1.0
-200
-100
0
100
200
VoltageHVL
Cur
rentHΜ
AL
Figure 4.1: Hysteresis loops of the memristor model
Besides, the waveform of the memristance given by (4.5) is depicted in Figure 4.2. The
loci traced out by (M, v(t)) is a hysteresis loop, but it is not pinched since M > 0 for all
times.
é é é ééééééééé
é
é
éééééé
éé
ééééééé
éé
éééééé
é
é
ééééééééé é
é éí í í í í í í íííííííííííííííííííííí
íííí
ííííííííííí íí í
í í í íá á á á á á á ááááááááááááááááááááááááááááááááááááá
á á á á á á á
é Ωí 2Ωá 10Ω
-1.0 -0.5 0.0 0.5 1.0
2
4
6
8
10
12
14
VoltageHVL
Mem
rist
anceHKWL
Figure 4.2: Memristance of the memristor model
65
Chapter 4. Transient memristor model
4.2 Homotopy-based model
In this part, the memristor model is obtained using the homotopy perturbation method
(HPM). Firstly, a brief glimpse of the HPM procedure is introduced. Then, the relation
between the system of equations of the HP memristor and the homotopy equation is
presented. Hereafter, the numerical and symbolical analysis of the solutions from the HPM
procedure is realized in order to establish the behavior of the principal characteristics of
the memristor model. Moreover, the variation of the parameter is addressed with the
purpose of finding the range of usefulness of the memristor model. Finally, the memristor
model is simulated with the purpose of showing the behavior in the time-domain.
4.2.1 Homotopy perturbation method
The homotopy perturbation method (HPM) is employed to obtain an approximate solution
for the nonlinear differential equations [1, 2, 3]. HPM introduces a homotopy parameter
p, which 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
where L and N are the linear and nonlinear operators, and f(r) is a known analytic
function.
The homotopy formulation can be established as
H(v, p) = (1− p)[
L(v)− L(uo)]
+ p[
L(v) +N(v)− f(r)]
= 0 (4.6)
where u0 is the initial approximation for the solution of (4.6), and p is known as the
perturbation homotopy parameter.
Assuming that the solution of (4.6) can be represented as a power series of p
v = p0v0 + p1v1 + p2v2 + · · · .
Therefore, the approximate solution of (4.6) can be of the form
v = v0 + v1 + v2 + · · · . (4.7)
The general procedure for the approximate solution of the nonlinear differential equa-
tion using HPM is shown in Figure 4.3. This procedure consists of formulating the homo-
topy equation with the nonlinear differential equation. Then, the order of approximation
66
4.2. Homotopy-based model
n∗ should be set to generate the power series of p. This power series is substituted in
the homotopy equation with the purpose of obtaining terms in function of the homotopy
parameter. The coefficient of the power n homotopy parameter is used as the nonlin-
ear differential equation and the solution is considered the particular case of the power
n homotopy parameter. Hereafter, the particular solutions of the nonlinear differential
equations using the coefficient of the power n homotopy parameter are obtained until the
power n equals to the order of approximation n∗. Finally, the general solution is composed
of the sum of all the particular solutions.
Homotopy equation
order of approximation n∗
and power series of p
Homotopy equation withpower series of p
coefficient of the power nhomotopy parameter
Nonlinear differential equation
Solution xn(t)xG(t) = xn(t) + xG(t)
If n=n∗
SolutionxG(t)
n=0; xG(t)=0
No
Yes
n = n + 1
Figure 4.3: General procedure of the approximate solution using HPM
67
Chapter 4. Transient memristor model
4.2.2 Memristor model using HPM
The HP memristor can be represented by the following system of equations
dw(t)
dt= µV
RON
Di(t)f(w(t))
f(w(t)) = aw(t)5 − 2aw(t)3 + aw(t)
v(t) =
(
RON
w(t)
D+ROFF
(
1−w(t)
D
))
i(t)
(4.8)
where µV is the average ion mobility, D is the full length of the device, RON and ROFF are
the ON and OFF state resistances, w(t) is the state variable that represents the length of
the doped region, f(w(t)) is the window function and a is the window function coefficient.
The current across the memristor i(t) is represented by the following expression
i(t) = A sin(ωt)
where A is the current amplitude and ω is the angular frequency.
The state variable equation of (4.8) can be re-written as
dx(t)
dt= µV
RON
D2i(t)f(x(t)) (4.9)
where x(t) = w(t)D
is the normalized state variable.
The nonlinear equation of (4.9) is substituted in the homotopy formulation of (4.6)
H(x(t), p) = (1− p)
[
dx(t)
dt
]
+ p
[
dx(t)
dt− µV
RON
D2i(t)f(x(t))
]
= 0 (4.10)
The expression of (4.10) is developed using the general procedure of HPM shown
in Figure 4.3. Hereafter, the approximate solutions of the homotopy formulation are
evaluated using two principal types of analysis: numerical and symbolical analysis.
4.2.3 Numerical analysis
The numerical analysis is based on the choice of the homotopy order by comparing the
margin of error between HPM procedure and the numerical integration method. Moreover,
the behavior of the margin of error as the frequency tends to infinity is studied. The
numerical analysis is carried out using the parameter values of x0=0.1, a=3, A=400×10−7,
µ=1×10−14, RON=100, α=160 and D=10×10−9.
68
4.2. Homotopy-based model
Homotopy order
The principal objective is to find the most suitable homotopy order in order to carry out
the symbolical analysis. The way to do this is comparing the solution x(t) from HPM
and the solution x∗(t) from the four-order Runge-Kutta integration method (RK4). The
comparison procedure is performed by assessing the root-mean-square error (RMSE).
The RMSE can be determined by
RMSE =
√
√
√
√
1
n
n∑
i=1
(x∗(t)− x(t))2 (4.11)
where n is the number of samples.
5 10 15 20 25 30 350.001
0.002
0.005
0.010
0.020
0.050
0.100
0.200
Order
RM
SE
Figure 4.4: RMSE using different homotopy orders
Figure 4.4 shows the RMSE values for different homotopy orders. The lowest RMSE
values are located in the 19 and 32 homotopy orders. However, these homotopy orders
generate solutions x(t) with significant amount of terms. Therefore, the homotopy order
should be less than 19.
The values of homotopy order under 19 present wide dispersion of RMSE values, as
shown in Figure 4.4. However, the homotopy orders of 6 and 15 remain with low values of
RMSE. Besides, the number of terms in the solution x(t) using the homotopy order of 6 is
negligible compared to the 15, 19 and 32 homotopy orders values. This feature represents
an advantage, from the point of view of modeling, because with a minor amount of terms;
the behavior of the HP memristor can be modeled.
Assuming that, the difference between the curve of the solution x(t) from HPM pro-
cedure and the curve of the solution x∗(t) from the RK4 procedure is not significant, as
shown in Figure 4.5, the homotopy order 6 is chosen to carry out the symbolical analysis.
69
Chapter 4. Transient memristor model
- - - HPMRK4
___
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
TimeHsL
Stat
eva
riab
leFigure 4.5: HPM and RK4 curves of the solution x(t) using the homotopy order 6.
Frequency
The purpose of the frequency variation is to analyze the RMSE values as the frequency
tends to infinity. This part of the numerical analysis is carried out using the mathematical
expression of (4.11) and the homotopy order of 6. Moreover, the frequency value is varied
from 1 to 100.
Table 4.1: RMSE values using different frequency values. The HPM order is 6.
Frequency RMSE Frequency RMSE Frequency RMSE
1 0.008607500 35 0.000853501 70 0.0008182595 0.001047520 40 0.000841938 75 0.00081825910 0.000931147 45 0.000844512 80 0.00081825915 0.000895380 50 0.000827569 85 0.00081825920 0.000870859 55 0.000818259 90 0.00081825925 0.000853746 60 0.000818259 95 0.00081825930 0.000859321 65 0.000818259 100 0.000771825
1 2 5 10 20 50 100
0.0010
0.0050
0.0020
0.0030
0.0015
0.0070
Frequency
RM
SE
Figure 4.6: RMSE using different frequency values. The HPM order is 6.
Table 4.1 show the RMSE values between the solutions x(t) and x∗(t) as the frequency
tends to infinity. As it can be seen, the RMSE value decreases as the frequency increases.
70
4.2. Homotopy-based model
Besides, the behavior of the RMSE values is depicted in Figure 4.6 from 1 to 100 frequency
values.
The simulation results show two principal behaviors. Firstly, the RMSE value decreases
rapidly as the frequency increases the double of the value. After that, the RMSE value
decreases gradually as the frequency continuously increasing. The reduction of the RMSE
is related to the form of the solution x(t) as the frequency approaches infinity. The
waveform of the solution x(t) is an asymmetrical and periodical curve which has a margin
of error with respect to the solution x∗(t) from the numerical integration method. This
difference is reduced as the frequency increases because the solutions of the x(t) and x∗(t)
curves become symmetrical and periodical waveforms and therefore the error tends to zero.
4.2.4 Symbolical analysis
The symbolical analysis is based on the composition of the symbolical solution x(t) using
the homotopy order established in the numerical analysis. Moreover, the symbolical mem-
ristance M(x(t)) is studied in order to establish the mathematical expression that models
the memristor. Finally, the behavior of the x(t) and M(x(t)) symbolical expressions as
the frequency increases is studied in order to establish the dependency of the frequency.
Symbolical solution x(t)
The symbolical solution x(t) is obtained from HPM procedure using the mathematical
expression of (4.10) and the homotopy order of 6. This symbolical expression is represented
by
x(t) = x0 +ChaAµRON
ωD2+
Cha2A2µ2R2
ON
ω2D4+
Cha3A3µ3R3
ON
ω3D6+
Cha4A4µ4R4
ON
ω4D8
−Cha
5A5µ5R5ON
ω5D10−
Cha6A6µ6R6
ON
ω6D12
+ cos(ωt)[
−ChaAµRON
ωD2−
Cha2A2µ2R2
ON
ω2D4−
Cha3A3µ3R3
ON
ω3D6
−Cha
4A4µ4R4ON
ω4D8+
Cha5A5µ5R5
ON
ω5D10+
Cha6A6µ6R6
ON
ω6D12
]
+ cos(2ωt)[
+Cha
2A2µ2R2ON
ω2D4+
Cha3A3µ3R3
ON
ω3D6+
Cha4A4µ4R4
ON
ω4D8
−Cha
5A5µ5R5ON
ω5D10−
Cha6A6µ6R6
ON
ω6D12
]
+ cos(3ωt)[
−Cha
3A3µ3R3ON
ω3D6−
Cha4A4µ4R4
ON
ω4D8+
Cha5A5µ5R5
ON
ω5D10
71
Chapter 4. Transient memristor model
+Cha
6A6µ6R6ON
ω6D12
]
+ cos(4ωt)[
+Cha
4A4µ4R4ON
ω4D8−
Cha5A5µ5R5
ON
ω5D10−
Cha6A6µ6R6
ON
ω6D12
]
+ cos(5ωt)[
+Cha
5A5µ5R5ON
ω5D10+
Cha6A6µ6R6
ON
ω6D12
]
+ cos(6ωt)[
−Cha
6A6µ6R6ON
ω6D12
]
(4.12)
where x0 is the initial width and Ch is a constant of the homotopy.
The solution x(t) from HPM is composed of (p + 1) terms, where p is the homotopy
order. The first term is called DC. The second term is called the fundamental. The
following terms are called the harmonics. The total number of harmonics depends on the
value of p.
In the case of the homotopy order of 6, the solution x(t) is composed of 7 terms and
the total number of harmonics is 6, as shown in (4.12).
Symbolical memristance M(x(t))
The symbolical memristance M(x(t)) is represented by
M(x(t)) = RONx(t) + αRON(1− x(t)) (4.13)
Substituting the symbolical solution x(t) in (4.13), the symbolical memristanceM(x(t))
can be re-written as
M(x(t)) = x0RON +RONα(1− x0) +ChaAµR
2ON
ωD2+
Cha2A2µ2R3
ON
ω2D4+
Cha3A3µ3R4
ON
ω3D6
+Cha
4A4µ4R5ON
ω4D8−
Cha5A5µ5R6
ON
ω5D10−
Cha6A6µ6R7
ON
ω6D12−
ChaAµR2ONα
ωD2
−Cha
2A2µ2R3ONα
ω2D4−
Cha3A3µ3R4
ONα
ω3D6−
Cha4A4µ4R5
ONα
ω4D8+
Cha5A5µ5R6
ONα
ω5D10
+Cha
6A6µ6R7ONα
ω6D12
+ cos(ωt)[
−ChaAµR
2ON
ωD2−
Cha2A2µ2R3
ON
ω2D4−
Cha3A3µ3R4
ON
ω3D6
−Cha
4A4µ4R4ON
ω4D8+
Cha5A5µ5R5
ON
ω5D10+
Cha6A6µ6R6
ON
ω6D12
+ChaAµR
2ONα
ωD2+
Cha2A2µ2R3
ONα
ω2D4+
Cha3A3µ3R4
ONα
ω3D6
+Cha
4A4µ4R5ONα
ω4D8−
Cha5A5µ5R6
ONα
ω5D10−
Cha6A6µ6R7
ONα
ω6D12
]
72
4.2. Homotopy-based model
+ cos(2ωt)[
+Cha
2A2µ2R2ON
ω2D4+
Cha3A3µ3R3
ON
ω3D6+
Cha4A4µ4R4
ON
ω4D8
−Cha
5A5µ5R5ON
ω5D10−
Cha6A6µ6R6
ON
ω6D12−
Cha2A2µ2R3
ONα
ω2D4
−Cha
3A3µ3R4ONα
ω3D6−
Cha4A4µ4R5
ONα
ω4D8+
Cha5A5µ5R6
ONα
ω5D10
+Cha
6A6µ6R7ONα
ω6D12
]
+ cos(3ωt)[
−Cha
3A3µ3R3ON
ω3D6−
Cha4A4µ4R4
ON
ω4D8+
Cha5A5µ5R5
ON
ω5D10
+Cha
6A6µ6R6ON
ω6D12+
Cha3A3µ3R4
ONα
ω3D6+
Cha4A4µ4R5
ONα
ω4D8
−Cha
5A5µ5R6ONα
ω5D10−
Cha6A6µ6R7
ONα
ω6D12
]
+ cos(4ωt)[
+Cha
4A4µ4R4ON
ω4D8−
Cha5A5µ5R5
ON
ω5D10−
Cha6A6µ6R6
ON
ω6D12
−Cha
4A4µ4R5ONα
ω4D8+
Cha5A5µ5R6
ONα
ω5D10+
Cha6A6µ6R7
ONα
ω6D12
]
+ cos(5ωt)[
+Cha
5A5µ5R5ON
ω5D10+
Cha6A6µ6R6
ON
ω6D12−
Cha5A5µ5R6
ONα
ω5D10
−Cha
6A6µ6R7ONα
ω6D12
]
+ cos(6ωt)[
−Cha
6A6µ6R6ON
ω6D12+
Cha6A6µ6R7
ONα
ω6D12
]
(4.14)
The symbolical memristance M(x(t)) from HPM is composed of (p+ 1) terms. These
terms are called DC and the harmonics. In the case of the homotopy order of 6, the
symbolical memristance is composed of 7 terms and the total number of harmonics is 6.
The symbolical expression of (4.14) constitutes in fact an v-i branch relationship that
models the memristor, and it can be recast as a functional model for simulation purposes.
x(t) and M(x(t)) frequency dependency
In order to determine the influence of the frequency in the solution x(t) and memristance
M(x(t)) symbolical expressions, the parameters are replaced with the following values:
x0=0.1, a=3, A=400×10−7, µ=1×10−14, RON=100, α=160 and D=10×10−9.
The symbolical solution x(t) of (4.12) can be simplified as
x(t) =0.1−0.0241758
ω6−
0.015989
ω5+
0.0106446
ω4+
0.0542583
ω3+
0.0995527
ω2+
0.117612
ω1
+ cos(ωt)
[
+0.0414442
ω6+
0.0266483
ω5−
0.0170313
ω4−
0.0813875
ω3−
0.132737
ω2
73
Chapter 4. Transient memristor model
−0.117612
ω
]
+ cos(2ωt)
[
−0.0259026
ω6−
0.0152276
ω5+
0.00851566
ω4+
0.032555
ω3+
0.0331842
ω2
]
+ cos(3ωt)
[
+0.0115123
ω6+
0.00571034
ω5−
0.00243304
ω4−
0.00542583
ω3
]
+ cos(4ωt)
[
−0.00345368
ω6−
0.00126897
ω5+
0.000304131
ω4
]
+ cos(5ωt)
[
+0.000627942
ω6+
0.000126897
ω5
]
+ cos(6ωt)
[
−0.0000523285
ω6
]
(4.15)
The expression of (4.15) is dependent of the frequency and establishes that as ω tends
to infinity the solution x(t) approaches to the value of x0. This behavior can be observed
by simulating, separately, each one of the terms that constitute the symbolical solution
x(t).
DCΩ
2Ω
0 20 40 60 80 100-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
Ω
Stat
eva
riab
lexH
tL
(a)
3Ω4Ω5Ω6Ω
2 4 6 8 10-4
-2
0
2
4
6
8
Ω
Stat
eva
riab
lexH
tLH1
0-3 L
(b)
Figure 4.7: Symbolical state variable x(t) using different ω values
Figure 4.7 shows the behavior of the DC, the fundamental and the harmonics terms
as ω tends to infinity. The DC term tends to the value of 0.1 as ω tends to infinity, as
shown in Figure 4.7a. The fundamental and the harmonic terms tends to zero as ω tends
to infinity, as shown in Figures 4.7a and 4.7b. Therefore, the DC term is the dominant
term in accordance ω tends to infinity. In general can be concluded that as ω tends to
infinity, the symbolical solution x(t) approaches to the value of x0.
The symbolical memristance M(x(t)) of (4.14) can be simplified as
74
4.2. Homotopy-based model
M(x(t)) =14410 +384.395
ω6+
254.225
ω5−
169.249
ω4−
862.707
ω3−
1582.89
ω2−
1870.03
ω
+ cos(ωt)
[
−658.963
ω6−
423.708
ω5+
270.798
ω4+
1294.06
ω3+
2110.52
ω2
+1870.03
ω
]
+ cos(2ωt)
[
+411.852
ω6+
242.119
ω5−
135.399
ω4−
517.624
ω3−
527.629
ω2
]
+ cos(3ωt)
[
−183.045
ω6−
90.7945
ω5+
38.6854
ω4+
86.2707
ω3
]
+ cos(4ωt)
[
+54.9136
ω6+
20.1765
ω5−
4.83568
ω4
]
+ cos(5ωt)
[
−9.98428
ω6−
2.01765
ω5
]
+ cos(6ωt)
[
+0.832024
ω6
]
(4.16)
The expression of (4.16) establishes that as ω tends to infinity the memristanceM(x(t))
approaches to the following expression
M(x(t))ω→∞ = RONx0 + αRON(1− x0) (4.17)
This behavior can be observed by simulating, separately, each one of the terms that
constitute the symbolical memristance M(x(t)).
DCΩ
2Ω
0 20 40 60 80 100
0
2
4
6
8
10
12
14
Ω
Mem
rist
anceHkWL
(a)
4Ω3Ω
5Ω6Ω
2 4 6 8 10-150
-100
-50
0
50
Ω
Mem
rist
anceHWL
(b)
Figure 4.8: Symbolical memristance M using different ω values
Figure 4.8 shows the behavior of the DC, the fundamental and the harmonics terms as
ω tends to infinity. The DC term tends to the value of 14.41 kΩ as ω tends to infinity, as
75
Chapter 4. Transient memristor model
shown in Figure 4.8a. The fundamental and the harmonic terms tends to zero as ω tends
to infinity, as shown in the figures 4.8a and 4.8b. Therefore, the DC term is the dominant
term in accordance ω tends to infinity. In general can be concluded that as ω tends to
infinity, the symbolic memristance M(x(t)) is dependent on the variables α, RON and x0.
4.2.5 Parameter variation
In this part of the analysis, the impact of the variation of the parameters: current ampli-
tude A, the window function coefficient a, the resistive factor α and the initial width x0
is studied with the purpose of finding the solution x(t) from HPM that approaches to the
solution x∗(t) using the RK4 integration method in order to achieve the widest memris-
tance range. The RMSE value is used to measure the differences between the solutions
x(t) and x∗(t).
The parameters are replaced with the following values: x0=0.1, a=3, A=400×10−7,
µ=1×10−14, RON=100, α=160 and D=10×10−9. In the case of the parameters are not
varied remain with the starting values.
Current Amplitude (A)
The current amplitude (A) is varied for the following values (in µA):
10 20 30 40 50 60 70 80 90 100
The current amplitude A values with the corresponding RMSE values are shown in
Table 4.2. The behavior of the data shows that the maximum value of the current ampli-
tude, before the RMSE value increases considerably, is located in the range from 40µA to
50µA.
Table 4.2: RMSE values for the variation of the current amplitude A
A RMSE A RMSE10 0.000278065 60 0.25024800020 0.000787793 70 0.89694500030 0.003571840 80 2.38112000040 0.008607500 90 5.31549000050 0.034329900 100 10.57850000
The value of the parameter A impacts, significantly, the range of memristance in the
memristor because it modifies the value of the state variable x(t); i.e., as the current
amplitude increases the range of the state variable increases and therefore the range of
memristance increases too.
76
4.2. Homotopy-based model
0 1 2 3 4 5 60.10
0.12
0.14
0.16
0.18
TimeHsL
Stat
eva
riab
lewHtL
(a) A=10µA
0 1 2 3 4 5 60.10
0.15
0.20
0.25
0.30
TimeHsL
Stat
eva
riab
lewHtL
(b) A=20µA
0 1 2 3 4 5 60.1
0.2
0.3
0.4
TimeHsL
Stat
eva
riab
lewHtL
(c) A=30µA
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
TimeHsL
Stat
eva
riab
lewHtL
(d) A=40µA
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
0.7
TimeHsL
Stat
eva
riab
lewHtL
(e) A=50µA
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
TimeHsL
Stat
eva
riab
lewHtL
(f) A=60µA
0 1 2 3 4 5 6-1.5
-1.0
-0.5
0.0
0.5
TimeHsL
Stat
eva
riab
lewHtL
(g) A=70µA
0 1 2 3 4 5 6
-5
-4
-3
-2
-1
0
1
TimeHsL
Stat
eva
riab
lewHtL
(h) A=80µA
0 1 2 3 4 5 6
-12
-10
-8
-6
-4
-2
0
TimeHsL
Stat
eva
riab
lewHtL
(i) A=90µA
0 1 2 3 4 5 6
-25
-20
-15
-10
-5
0
TimeHsL
Stat
eva
riab
lewHtL
(j) A=100µA
Figure 4.9: Analysis of x(t) using different values of current amplitude.
Figure 4.9 shows the differences between the solutions x(t) from the HPM procedure
(blue line) and the RK4 integration method (red line). These series of frames confirm
the range of acceptable current amplitude and also establish that as the value of the
parameter A increases above 50µA the curve of the variable x(t) is deformed to achieve
physical values not allowed.
The problems caused by the use of physical values not allowed are shown in Figure
4.10. The principal drawback is located in the v-i characteristic curve. According to the
77
Chapter 4. Transient memristor model
fundamental properties of the memristive systems is impossible to obtain the performance
of the curve in the enclosed area -black arrows-, as shown in Figure 4.10. Moreover, the
M-i characteristic curve reaches the maximum values of memristance above the physical
limits allowed.
-2 -1 0 1 2
-50
0
50
VoltageHVL
Cur
rentHΜ
AL
(a)
-50 0 50
20
40
60
80
100
CurrentHΜAL
Mem
rist
anceHkWL
(b)
Figure 4.10: M-i and v-i characteristic using not possible value of parameter A.
As it can be seen, the pinched hysteresis loop and the range of memristance suffer
disturbances in particular areas. These areas correspond to phenomena unrelated to the
behavior of the memristor. Therefore, the curves shown in Figures 4.10a and 4.10b do not
represent the performance of the memristive system.
Window function coefficient (a)
The equation of the window function has the form
f(x(t)) = a[
x(t)]5
− 2a[
x(t)]3
+ ax(t) (4.18)
where a is called the window function coefficient.
The window function coefficient is used for modifying proportionally the behavior of
the window function. The range of operation for the window function can be established
as
1 ≥ f(x(t)) ≥ 0
1 ≥ x(t) ≥ 0(4.19)
According to (4.18) and (4.19) the maximum value for the window function coefficient
a is 3.55.
The window function coefficient (a) is varied for the following values:
78
4.2. Homotopy-based model
1 1.5 2 2.5 3 3.5
The values of the window function coefficient a with the corresponding RMSE values
are shown in Table 4.3. The behavior of the data establishes that as the window function
coefficient increases, the RMSE increases too.
Table 4.3: RMSE values for the variation of the window function coefficient a
a RMSE1 0.0004091241.5 0.0007877932 0.0020628002.5 0.0056483603 0.0086075003.5 0.012122600
According to (4.8), the window function coefficient adjusts the amount of current across
the memristor and therefore modifies the value of the state variable x(t).
0 1 2 3 4 5 60.10
0.12
0.14
0.16
0.18
0.20
TimeHsL
Stat
eva
riab
lexH
tL
(a) a=1
0 1 2 3 4 5 60.10
0.15
0.20
0.25
0.30
TimeHsL
Stat
eva
riab
lexH
tL
(b) a=1.5
0 1 2 3 4 5 60.10
0.15
0.20
0.25
0.30
0.35
0.40
TimeHsL
Stat
eva
riab
lexH
tL
(c) a=2
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
TimeHsL
Stat
eva
riab
lexH
tL
(d) a=2.5
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
TimeHsL
Stat
eva
riab
lexH
tL
(e) a=3
0 1 2 3 4 5 60.1
0.2
0.3
0.4
0.5
0.6
0.7
TimeHsL
Stat
eva
riab
lexH
tL
(f) a=3.5
Figure 4.11: Analysis of window function coefficient using different values.
Figure 4.11 shows the differences between the solutions x(t) from the HPM procedure
(blue line) and the RK4 integration method (red line). The maximum value of the state
variable x(t) is located for a window function coefficient a of 3.5. However, the value of
3.5 presents the maximum RMSE value, as shown in Figure 4.12.
The reason that the value of 3.5 causes the maximum RMSE is the presence of the
disturbance at the top of the solution x(t) from the HPM procedure, as shown in Figure
79
Chapter 4. Transient memristor model
4.11f. The appropriate window function coefficient is one that approaches the solution
x(t) to the unity value for achieving the widest memristance range.
According to Figures 4.11 and 4.12 the widest memristance range is caused by a window
function coefficient value of 3.3. The value of 3.3 has the same amount of RMSE that a
value of 2.7 as shown in Figure 4.12.
1.0 1.5 2.0 2.5 3.0 3.51´10-4
5´10-40.001
0.0050.010
0.050
Window function coefficient
RM
SE
Figure 4.12: RMSE using different window function coefficient values
The consequences of varying the parameter a can be observed in the v-i and M-
i characteristic curves. The lobe area of the pinched hysteresis loop and the range of
memristance increase as the value of the window function coefficient increases, as shown
in Figure 4.13.
a=3.5
a=1
-0.4 -0.2 0.0 0.2 0.4-40
-20
0
20
40
VoltageHVL
Cur
rentHΜ
AL
(a)
a=1
a=3.5
-40 -20 0 20 40
6
8
10
12
14
CurrentHΜAL
Mem
rist
anceHkWL
(b)
Figure 4.13: M-i and v-i characteristic curves using different values of parameter a.
80
4.2. Homotopy-based model
Resistive factor (α)
The resistive factor α establishes the resistive ratio.
α =ROFF
RON
(4.20)
The fixed value of RON causes that ROFF assumes the unique value according to (4.20).
The resistive factor α is varied for the following values:
100 120 140 160 180 200 220
The values of the resistive factor α with the corresponding MMax, MMin and Mω→∞
values are shown in Table 4.4. The variations of the parameter α does not lead to changes
for the solution x(t).
Table 4.4: Analysis of the memristance using different α values
α MMax MMin Mω→∞
100 9010 3653 9010120 10810 4370 10810140 12610 5088 12610160 14410 5806 14410180 16210 6524 16210200 18010 7241 18010220 19810 7959 19810
This behavior is caused because the resistive factor is not related to the state variable
of (4.8). The mathematical expression of (4.17) establishes that the resistive factor affects
the maximum and minimum values of memristance.
Initial width (x0)
The initial width (x0) is varied for the following values:
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
The behavior of the data establishes that as the parameter x0 increases, the low value
of the state variable increases and therefore the range of the solution x(t) is reduced. This
reduction in the range of the solution x(t) causes that the range of memristance is reduced
considerably.
81
Chapter 4. Transient memristor model
4.2.6 Characteristic of the memristor model
The HPM memristor model is established using the expression of (4.14). This memristor
model is evaluated for discrete values of the frequency at ω = 1, 2, 10, and the corre-
sponding hysteresis loops are shown in Figure 4.14. It possesses the main fingerprints of a
passive memristor, such as: pinched hysteresis loop, hysteresis lobe area and shrunken hys-
teresis loop. The curves have been obtained for x0=0.1, a=3, A=400×10−7, µ=1×10−14,
RON=100, ROFF=16KΩ and D=10×10−9.
éééééééééééééééééééééééééééééééééééééééééééééééééééééé
éééééééééééééééééééééééééééééééééééé
ééééééééééééééééééééééééééééééééééééííííííííííííííííííííííííí
ííííííí
íííííííííííííííííí
íííííííííííííááááááááááááááááááááááááá
ááááááá
ááááááááááááááááááá
áááááááááááá
é Ωí 2Ωá 10Ω
-1.0 -0.5 0.0 0.5 1.0-100
-50
0
50
100
VoltageHVL
Cur
rentHΜ
AL
Figure 4.14: Hysteresis loops of the memristor model
Moreover, the waveform of the memristance is shown in Figure 4.15.
é é é é é é éééééé
é
é
é
é
é
é
éééééééééé
ééééé
é
é
é
é
é
é
éééééé é
é é éé éí í í í
íííí
íí
ííííí
íí
íííííí í í í í í í
íííí
íí
ííííí
íí
íííííí í í íá á áááááááá á á áááááááá á á áááááááá á á áááááááá á á áááááááá á
é Ωí 2Ωá 10Ω
-0.4 -0.2 0.0 0.2 0.46
8
10
12
14
VoltageHVL
Mem
rist
anceHKWL
Figure 4.15: Memristance of the memristor model
82
4.3. Discussion
4.3 Discussion
The conventional memristor models [4, 5] use mathematical expressions in order to simu-
late the behavior of the memristive systems. These models present the principal drawback
of requiring a numerical integration method to solve the system of equations. The neces-
sity of using a large amount of memristors causes problems in the execution speed of
the computer programs. Therefore, this part of the thesis proposes the use of behavioral
modeling of the memristor as alternative path for obtaining the memristive characteristic.
The polynomial model of the memristor is based on the approximation of the q-ϕ curve
by using the polynomial function. This function uses the RON and ROFF parameters to
bound the range of the memristance in the device. However, this model uses the theoretical
conception of the memristive behavior in order to include the memristor in the simulation
procedure; i.e. the polynomial approximation of the memristive characteristic does not
use the physical parameters to adjust the behavior and the boundary conditions of the
device. Thus, the polynomial model is useful to carry out new studies in circuits with
conventional elements in order to investigate the response of the systems with the insertion
of the memristor.
The novel HPM memristor model is based on the full-symbolical expression of the
memristive behavior using the physical parameters of the HP memristor. This feature
represents the principal advantage with respect to the numerical integration methods.
The numerical integration methods present errors as the time-variable increases, called
local truncation error and total accumulated error [6]. These errors do not occur in the
new memristor model because the numerical integration method is not necessary to display
the characteristic curves of the memristive behavior. Moreover, the expression of the HPM
memristor model is composed of terms classified as fundamental and harmonic elements.
This feature provides a particular vision to understand the collapse of the v-i characteristic,
the state variable and the memristance curve as frequency approaches infinity.
The summary of the characteristics of the polynomial memristor model and the HPM
memristor model is exposed in Table 4.5. This summary is based on the following charac-
teristics: memristive values, parameters of control, number of terms and the characteristic
curves of the v-i, memristance and state-variable. Firstly, the memristive values consist
of establishing the values of the parameters in the memristor model in order to display
memristive behavior. Then, parameters of control mention the parameters used in the
mathematical expression of the model. After that, number of terms specifies the amount
of elements that constitutes the mathematical approximation of the model. Finally, the
characteristic curves of the memristive behavior are incorporated into the summary of the
83
Chapter 4. Transient memristor model
memristor models.
Table 4.5: Summary of the characteristics of the memristor models
Characteristics Polynomial memristor model HPM memristor model
memristive behavior X RON=1.4KΩ, ROFF=14.41KΩ, A=1 X x0=0.1, a=3, A=400×10−7, α=160, RON=100Ωparameters of control RON , ROFF , A RON , α, A, a, x0
number of terms 2 7v-i characteristic curve X X
M characteristic curve X X
The classical figure of performance for memristor models shows the pinched hysteresis
loop in the v-i characteristic curve. The novel HPM memristor model presents the clas-
sical figure of performance and also shows the state variable w and the memristance M
characteristic curves in order to ensure the memristive behavior of the device. Besides, the
memristive characteristic curves tends to be a straight line as the frequency approaches
infinity.
However, the HPM memristor model operates in a bounded range. The range of
usefulness is established by comparing the response of the HPM memristor model and
the response from the RK4 integration method. This range is set out in Table B.6 and
ensures that the RMSE value is less than 1 % with respect to the numerical integration
method. Finally, the HPM memristor model presents two principal drawback: the range of
usefulnes and the inability of incorporate the phenomenon of the state variable’s drift speed
as w approaches to the boundaries. These disadvantages are caused by HPM procedure
problems.
84
Bibliography
[1] J.-H. He, “A coupling method of a homotopy technique and a perturbation technique
for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35,
no. 1, pp. 37 – 43, 2000.
[2] J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechan-
ics and Engineering, vol. 178, no. 34, pp. 257 – 262, 1999.
[3] F.-N. U, “Hpm applied to solve nonlinear circuits: A study case,” Applied Mathematical
Sciences, vol. 6, no. 87, pp. 4331 – 4344, 2012.
[4] M. Di Ventra, Y. V. Pershin, and L. O. Chua, “Circuit elements with memory: mem-
ristors, memcapacitors and meminductors,” ArXiv e-prints, Jan. 2009.
[5] Y. V. Pershin, S. La Fontaine, and M. di Ventra, “Memristive model of amoeba learn-
ing,” Physical Review E, vol. 80, p. 021926, Aug. 2009.
[6] L. O. Chua and P. Y. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms
and Computational Techniques. Prentice Hall Professional Technical Reference, 1975.
85
Bibliography
86
Chapter 5
Circuit applications
The incorporation of the memristor in the simulation procedure represents a new vision
for generating novel types of circuits and systems. Moreover, the new fundamental circuit
element breaks the arbitrary restriction of electronic circuit theory to linear systems. This
chapter describes the use of the HPM and polynomial memristor models in different types
of applications, such as: negative feedback amplifiers, opamp-based circuits and mem-
elements emulator circuits.
5.1 Nullor-based examples
The nullor is a two-port singular element used in the structured design methodology for
the design of amplifier circuits [1, 2]. This electric component is composed of a nullator
(input port) and a norator (output port), as shown in the Figure 5.1.
uo
+
–
ui
+
–
ii io
Figure 5.1: The nullor
The current and voltage values of the nullator are by definition zero
ii = 0 vi = 0
The values of current and voltage of the norator can be arbitrary
87
Chapter 5. Circuit applications
io = × vo = ×
There are many forms to describe a nullor, but the most convenient one is the chain
matrix K, because we are focussed on its amplification properties. This matrix has the
form
K =
[
A B
C D
]
=
[
0 0
0 0
]
which establishes the following definitions of transfer functions:
A =ui
uo
∣
∣
∣
∣
i0=0
= 0 B =ui
io
∣
∣
∣
∣
u0=0
= 0
(5.1)
C =ii
uo
∣
∣
∣
∣
i0=0
= 0 D =ii
io
∣
∣
∣
∣
u0=0
= 0
Therefore, the nullor possesses infinite-gains for all four basic amplifiers, i.e. the nullor
represents the ideal infinite gain plant to be controlled by the feedback network which is
usually constituted by passive components. This description yields the asymptotic-gain
model of negative-feedback amplifiers [2, 3], as shown in Figure 5.2.
+
–
The nullor
A
β
The resistive network
Figure 5.2: The asymptotic-gain model
In this way, the four types of amplifiers can be obtained when a passive feedback
network is connected with the nullor in a single negative-feedback loop [1, 4] as shown in
Figure 5.3.
The ideal closed-loop gains for the negative-feedback amplifiers above are given as
88
5.1. Nullor-based examples
R2R1
Voltage Amplifier
+
– +
–
+
–
+
–
ui
uo
G
Transconductance Amplifier
+
– +
–
+
–
ui
io
R
Transresistance Amplifier
+
– +
–
+
–
uo
ii
R2
R1
Current Amplifier
+
– +
–
iiio
Figure 5.3: Negative-feedback amplifiers
AV = 1 +R2
R1
AG = −G
AR = −R AI = 1 +R2
R1
(5.2)
where AV , AG, AR and AI are the voltage, transconductance, transresistance and current
gains, respectively.
The existence of the memristor opens the possibility for introducing a novel family of
memristive transfer functions that are obtained by substituting the resistors of the passive
feedback network by memristors. The resulting configurations are shown in Figure 5.4.
Besides, depending on the selection made of the components of the feedback network,
a specific member of the family emerges, as listed in Table 5.1.
In order to demonstrate the feasibility of achieving a memristive transfer function, the
nullor part of the amplifiers is implemented by appropriated active devices. In this thesis,
two different nullor implementations are carried out: MOS-realization and Memistor∗-
realization.
∗The attent reader should notice the word memistor. It is not a mistake, it is a 3-terminal devicedifferent from the memristor.
89
Chapter 5. Circuit applications
M2M1
Voltage Amplifier
+
– +
–
+
–
+
–
ui
uo
W
Trans-memconductance Amplifier
+
– +
–
+
–
ui
io
M
Trans-memresistance Amplifier
+
– +
–
+
–
uo
ii
M2
M1
Current Amplifier
+
– +
–
iiio
Figure 5.4: Negative-feedback trans-mem amplifiers
Table 5.1: Family of Trans-memristive Amplifiers
Configuration Feedback network Member
M1 −M2 pure transmem-voltage amplifiervoltage M1 −R2 transmem-voltage amplifier type I
R1 −M2 transmem-voltage amplifier type IItransconductance W transmem-conductance amplifiertransresistance M transmem-resistance amplifier
M1 −M2 pure transmem-current amplifiercurrent M1 −R2 transmem-current amplifier type I
R1 −M2 transmem-current amplifier type II
MOS implementation
The MOS in common-source (CS) configuration is used for implementing the nullor since
this configuration has the best approximation to small entries in the chain matrix K. The
nullor implemented by a first-order small-signal model of the CS-stage is shown in Figure
5.5.
The chain matrix K is given as
KCS =
[
A B
C D
]
=
(
− gmgds
)−1
(−gm)−1
0 0
(5.3)
where gm is the transconductance and gds is the drain-source conductance.
Because no capacitors are included in the model, the maximum possible stage gain is
90
5.1. Nullor-based examples
uo
+
–
ui
+
–
vgs
+
–
gmvgs
gds
ii io
Figure 5.5: Nullor synthesis by MOS-equivalent
given as gmgds
[5], which is also refereed as the DC-gain expressed usually as:
DC-gain(dBs) = 20 log(gm
gds
)
(5.4)
In this thesis, the single MOS-stage is resorted in order to carry out the methodology
of the structured electronic design. This method [1] assumes that it is possible to design
analog circuits in a well-defined way and therefore the best performing circuits are found.
Memistor characterization
Another way of implementing the nullor with an active device is by using a memistor. The
memistor is a three-terminal device in which the current entering in the control terminal
modulates the resistance between the other two terminals [6, 7]. The memistor can be
obtained by a back-to-back series connection of two memristors, as shown in Figure 5.6.
uo
+
–
ui
+
–
ii io
Figure 5.6: Nullor synthesis by Memistor realization
In fact, the memristor is the ’fundamental element’ while the memistor is the ’derived
circuit element’. By implementing the nullor with the memistor, fully-memristor versions
of the trans-memristive amplifiers can be obtained.
Hereafter, the HPM and polynomial memristor models are used to carry out the circuit-
level simulation of the trans-memresistance and trans-memconductance amplifiers.
91
Chapter 5. Circuit applications
5.1.1 Trans-memresistance amplifier
5.1.1.1 Nullor-based design
The circuit shown in Figure 5.7 represents the trans-memresistance amplifier using the
nullor as the plant to be controlled. This negative-feedback amplifier is simulated by
using the polynomial and HPM memristor models.
ii RL
M
uo
+
–
Figure 5.7: Trans-memresistance amplifier with nullor
where ii=A sinωt, A=40µA, RL=1MΩ andM is characterized by the parameter values
of the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.2. The memristor model is incorporated in the feedback loop of the amplifier, as shown
in Figure 5.7. This circuit is simulated in order to examine the behavior of the output
variable and the transfer ratio of the amplifier.
Table 5.2: M parameter values using polynomial memristor model
M(Ω) Mmin Mmax α M(ω → ∞)
M 16000-100 100Ω 16000Ω 160 16000Ω
Results
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.8. Besides, in order to gain insight on the
dynamics of the amplifier, simulations were also carried out by substituting the memristor
92
5.1. Nullor-based examples
M for linear resistors having the minimum and maximum memristance values that the
memristor can reach, namely Mmin and Mmax respectively.
@Mmin
Mmax @×10-2
M
0 2 4 6 8 10 12
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
TimeHsL
Vol
tageHVL
(a) ω=1
@×10-2M
@
Mmax
Mmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.8: uo(t) curves for the trans-memresistance amplifier with nullor using polynomialmemristor model
The corresponding curves shown in Figure 5.8 denote uo with the memristor by the
red lines, and the black lines are associated to the simulations with Mmin and Mmax.
Clearly it can be noticed that the uo curve jumps from the lower to the upper limits at
low-frequency and it sticks to the upper limit at high frequencies, i.e. the amplifier reaches
the maximum trans-memresistance value. The label @x10−2 means that the curve of Mmin
has been increased one hundred times of the original value in order to be displayed.
Ω=1
Ω=10
-40 -20 0 20 40-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
CurrentHΜAL
Vol
tageHVL
Figure 5.9: Transmem-resistance ii-uo hysteresis with nullor using polynomial memristormodel
Another result of the simulations is the transmission curves (uo
ii) of the amplifier for the
same frequency values that are shown in Figure 5.8. The memristive transresistance ratio
(uo
ii) shows the typical eight-shape of the memristor hysteresis, tending to a straight-line at
high frequencies, as shown in Figure 5.9. Table 5.3 shows the breakdown of the memristive
transresistance ratio by reducing the memristive range as the frequency increases.
93
Chapter 5. Circuit applications
Table 5.3: Memristive transresistance ratio using different frequency values
ω uo
ii(KΩ) ω uo
ii(KΩ)
1 16-0.010 6 16-13.352 16-8.050 7 16-13.723 16-10.70 8 16-14.014 16-12.02 9 16-14.235 16-12.82 10 16-14.41
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.4.
Now, the circuit shown in Figure 5.7 is performed by using the HPM memristor model.
This circuit is simulated in order to examine the behavior of the output variable and the
transfer ratio of the amplifier.
Table 5.4: M parameter values using HPM memristor model
M(Ω) x0 a A α Mmin Mmax M(ω → ∞)
M 14410-5806 0.1 3 40µA 160 5806Ω 14410Ω 14410Ω
Results
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.10.
Mmax
M
Mmin
0 2 4 6 8 10 12-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
TimeHsL
Vol
tageHVL
(a) ω=1
MmaxM
Mmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.10: uo(t) curves for the trans-memresistance amplifier with nullor using HPMmemristor model
In this case, the corresponding curves shown in Figure 5.10 denote uo with the memris-
tor by the blue lines, and the black lines are associated to the simulations with the Mmin
and the Mmax reached by the memristor.
94
5.1. Nullor-based examples
Ω=1
Ω=10
-40 -20 0 20 40-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
CurrentHΜALV
olta
geHVL
Figure 5.11: Transmem-resistance ii-uo hysteresis with nullor using HPM memristor model
The transmission curves (uo
ii) of the amplifier are shown in Figure 5.11. Table 5.5 shows
the behavior of the trans-memresistance values as the frequency increases.
Table 5.5: Memristive transresistance ratio using different frequency values
ω uo
ii(KΩ) ω uo
ii(KΩ)
1 14.41-5.806 6 14.41-13.652 14.41-11.15 7 14.41-13.783 14.41-12.59 8 14.41-13.874 14.41-13.16 9 14.41-13.935 14.41-13.47 10 14.41-13.99
95
Chapter 5. Circuit applications
5.1.1.2 MOS-based design
The circuit shown in Figure 5.12 represents the trans-memresistance amplifier using the
nullor implementation of the MOS in common-source configuration. This implementation
is simulated by using the polynomial and HPM memristor models.
ii RL
M
uo
+
–
vgs
+
–
gmvgs
gds
Figure 5.12: Trans-memresistance amplifier with MOS-equivalent
The values of the amplifier are: ii=A sinωt, A=40µA,RL=1MΩ, gm=320µA
V, gds=320nS
and M is characterized by the parameter values of the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.2. By following the same procedure as in the previous implementation, the circuit of
Figure 5.12 is simulated.
Results
Mmin
Mmax
M
0 2 4 6 8 10 12
-0.4
-0.2
0.0
0.2
0.4
TimeHsL
Vol
tageHVL
(a) ω=1
Mmax
M
Mmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.4
-0.2
0.0
0.2
0.4
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.13: uo(t) curves for the trans-memresistance amplifier with MOS-equivalent usingpolynomial memristor model
96
5.1. Nullor-based examples
The simulation result of the output voltage uo(t) are shown in Figure 5.13. In low-
frequency, the trans-memresistance amplifier displays a negative trans-memresistance re-
gion. This region is related to the performance of the equation that governs the trans-
memresistance amplifier. Note that, the negative trans-memresistance values disappear
as the frequency tends to infinity because of the range of memristance is reduced.
M
-40 -20 0 20 40-0.4
-0.2
0.0
0.2
0.4
CurrentHΜAL
Vol
tageHVL
(a) ω=1
M
-40 -20 0 20 40-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
CurrentHΜALV
olta
geHVL
(b) ω=10
Figure 5.14: Transmem-resistance ii-uo hysteresis with MOS-equivalent using polynomialmemristor model
For this implementation, the resulting ii-uo hysteresis loops (in black) and the hysteresis
loops of the memristor (in red) are shown in Figure 5.14.
It clearly result that the trans-memresistance is smaller than the memresistance at the
feedback, which can be explained by the fact that the implemented nullor does not have
zeroes in its chain matrix. Table 5.6 shows the memristive transresistance ratio with the
MOS-equivalent as the frequency increases.
Table 5.6: Memristive transresistance ratio using different frequency values
ω M(KΩ) uo
ii(KΩ) ω M(KΩ) uo
ii(KΩ)
1 16-0.010 12.86-3.021 6 16-13.35 12.86-10.212 16-8.050 12.86-4.920 7 16-13.72 12.86-10.593 16-10.70 12.86-7.567 8 16-14.01 12.86-10.874 16-12.02 12.86-8.891 9 16-14.23 12.86-11.095 16-12.82 12.86-9.685 10 16-14.41 12.86-11.27
Effect of the DC-gain
In this part, further analysis of the amplifier is achieved for several values of the DC-gain.
This important parameter expresses how closer the MOS-model is to the nullor. Because
the DC-gain depends on gds and gm, separate analysis are carried out by varying gds and
gm.
97
Chapter 5. Circuit applications
Analysis I: Effect of gds
In this analysis, the output conductance gds is varied while the transconductance gm
remains fixed. The trans-memresistance ii–uo hystereses are shown in Figure 5.15 for
DC-gain values of 60dB, 40dB and 20dB.
é
é
é
é
ééé ééééééé
é
ééééééééé
éééé
éééééééé
é
éééééé é
ééé
é
é
é
éí
í
í
í
ííí íííí
ííííííííííííí
íííí
íííííííííííííííí í
íí
í
í
í
í
é 60dB & 40dBí 20dB
-40 -20 0 20 40-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
CurrentHΜAL
Vol
tageHVL
(a) ω=1
é
é
ééééééééééééééééééé
éé
éé
éé
éé
éééééééééééééééééééé
é
éííííííííííííííííí
íííí
íí
íí
íí
íí
íííííííííííííííííííííí
é 60dB & 40dBí 20dB
-40 -20 0 20 40
-0.4
-0.2
0.0
0.2
0.4
CurrentHΜAL
Vol
tageHVL
(b) ω=10
Figure 5.15: Transmission curves for Analysis I.
Another form of looking at the simulation results is recast in Table 5.7. Herein, the
values associated to gds in order to produce the DC-gain values above yield a limit trans-
memresistance value at ω=∞, that is to say high frequencies. Decreasing the DC-gain
(increasing gds) causes a small reduction in this limit.
Table 5.7: Summary of Analysis I
DC-gain (dB) gm (10−6)AV
gds()uo
ii(kΩ) for ω=∞
60 320 320×10−9 12.8640 320 320×10−8 12.7420 320 320×10−7 11.70
98
5.1. Nullor-based examples
Analysis II: Effect of gm
In this analysis, gm is varied while gds is fixed. The resulting trans-memresistance ii-uo
hystereses are shown in Figure 5.16 for the same DC-gain values of the previous analysis.
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ííííííííííííííííí í í í í í í íá
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á 20dB
é 60dBí 40dB
-40 -20 0 20 40
-10
-5
0
5
10
CurrentHΜAL
Vol
tageHVL
(a) ω=1
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é 60dBí 40dBá 20dB
-40 -20 0 20 40
-10
-5
0
5
10
CurrentHΜAL
Vol
tageHVL
(b) ω=10
Figure 5.16: Transmission curves for Analysis II.
The summary of the effect of gm is shown in Table 5.8. Now, the values associated to
gm produce a limit trans-memresistance value at ω=∞. Note that, the variation of gm
causes negative trans-memresistance values even though the frequency tends to infinity.
Table 5.8: Summary of Analysis II
DC-gain (dB) gmAV
gds(10−9) uo
ii(kΩ) for ω = ∞
60 320×10−6 320 12.8640 320×10−7 320 -15.0920 320×10−8 320 -269.5
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.4.
By following the same procedure as in the previous implementation, the circuit of Figure
5.12 is simulated in order to examine the behavior of the output variable and the transfer
ratio of the amplifier.
Results
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.17. For this implementation, the resulting
ii-uo hysteresis loops (in black) and the hysteresis loops of the memristor (in blue) are
shown in Figure 5.18.
99
Chapter 5. Circuit applications
Mmax
M
Mmin
0 2 4 6 8 10 12
-0.4
-0.2
0.0
0.2
0.4
TimeHsL
Vol
tageHVL
(a) ω=1
Mmax
M
Mmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.4
-0.2
0.0
0.2
0.4
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.17: uo(t) curves for the trans-memresistance amplifier with MOS-equivalent usingHPM memristor model
M
-40 -20 0 20 40
-0.4
-0.2
0.0
0.2
0.4
CurrentHΜAL
Vol
tageHVL
(a) ω=1
M
-40 -20 0 20 40-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
CurrentHΜAL
Vol
tageHVL
(b) ω=10
Figure 5.18: Transmem-resistance ii-uo hysteresis with MOS-equivalent using HPM mem-ristor model
Table 5.9: Memristive transresistance ratio using different frequency values
ω M(KΩ) uo
ii(KΩ) ω M(KΩ) uo
ii(KΩ)
1 14.41-5.806 11.27-2.678 6 14.41-13.65 11.27-10.522 14.41-11.15 11.27-8.026 7 14.41-13.78 11.27-10.643 14.41-12.59 11.27-9.456 8 14.41-13.87 11.27-10.734 14.41-13.16 11.27-10.03 9 14.41-13.93 11.27-10.805 14.41-13.47 11.27-10.33 10 14.41-13.99 11.27-10.85
Note that, the negative trans-memresistance values are not present by using the HPM
memristor. Table 5.9 shows the memristive transresistance ratio with the MOS-equivalent
as the frequency increases.
100
5.1. Nullor-based examples
Effect of the DC-gain
The analysis of the amplifier is achieved for several values of the DC-gain. Because the
DC-gain depends on gds and gm, separate analysis are carried out by varying gds and gm.
Analysis I: Effect of gds
The conductance gds is varied while the transconductance gm remains fixed.
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-40 -20 0 20 40
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
CurrentHΜAL
Vol
tageHVL
(a) ω=1
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-40 -20 0 20 40
-0.4
-0.2
0.0
0.2
0.4
CurrentHΜAL
Vol
tageHVL
(b) ω=10
Figure 5.19: Transmission curves for Analysis I.
The trans-memresistance ii–uo hystereses are shown in Figure 5.19 for DC-gain values
of 60dB, 40dB and 20dB.
Table 5.10: Summary of Analysis I
DC-gain (dB) gmAV
gds(10−9) uo
ii(kΩ) for ω = ∞
60 320×10−6 320 11.2740 320×10−7 320 11.1720 320×10−8 320 10.25
In Table 5.10, the values associated to gds in order to produce the DC-gain values
above yield a limit trans-memresistance value at ω=∞, that is to say high frequencies.
Decreasing the DC-gain (increasing gds) causes a small reduction in this limit.
101
Chapter 5. Circuit applications
Analysis II: Effect of gm
In this analysis, gm is varied while gds is fixed.
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é 60dBí 40dBá 20dB
-40 -20 0 20 40
-10
-5
0
5
10
CurrentHΜAL
Vol
tageHVL
(a) ω=1
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é 60dBí 40dBá 20dB
-40 -20 0 20 40
-10
-5
0
5
10
CurrentHΜAL
Vol
tageHVL
(b) ω=10
Figure 5.20: Transmission curves for Analysis II.
The resulting trans-memresistance ii-uo hystereses are shown in Figure 5.20 for the
same DC-gain values of the previous analysis. The summary of the effect of gm is shown
in Table 5.11. Now, the values associated to gm produce a limit trans-memresistance value
at ω=∞.
Table 5.11: Summary of Analysis II
DC-gain (dB) gmAV
gds(10−9) uo
ii(kΩ) for ω = ∞
60 320×10−6 320 11.2740 320×10−7 320 -16.6720 320×10−8 320 -270.9
Note that, in spite of using the HPM memristor model, the variation of gm causes
negative trans-memresistance values even though the frequency tends to infinity.
102
5.1. Nullor-based examples
5.1.1.3 Memistor-based design
The circuit shown in Figure 5.21 represents the trans-memresistance amplifier using the
nullor implementation of the memistor. This implementation is simulated by using the
polynomial and HPM memristor models.
ii RL
M
uo
+
–
Figure 5.21: Trans-memresistance amplifier with Memistor realization
where ii=A sinωt, A=40µA and RL=1MΩ. The memristor M and the memistor are
characterized by the parameter values of the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.2. By following the same procedure as in the previous implementation, the circuit of
Figure 5.21 is simulated.
Results
Mmin
M
Mmax
0 2 4 6 8 10 12-0.2
-0.1
0.0
0.1
0.2
TimeHsL
Vol
tageHVL
(a) ω=1
MminM
Mmax
0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.22: uo(t) curves for the trans-memresistance amplifier with Memistor realizationusing polynomial memristor model
103
Chapter 5. Circuit applications
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.22. For this implementation, the resulting
ii-uo hysteresis loops (in black) and the hysteresis loops of the memristor (in red) are
shown in Figure 5.23.
M
-40 -20 0 20 40
-100
-50
0
50
100
CurrentHΜAL
Vol
tageHm
VL
(a) ω=1
M
-40 -20 0 20 40-200
-100
0
100
200
CurrentHΜAL
Vol
tageHm
VL
(b) ω=10
Figure 5.23: Transmem-resistance ii-uo hysteresis (black) and memristor hysteresis (red)for Memistor realization using polynomial memristor model
Table 5.12: Memristive transresistance ratio using different frequency values
ω M(KΩ) uo
ii(KΩ) ω M(KΩ) uo
ii(KΩ)
1 16-0.010 5.333-0.033 6 16-13.35 5.333-4.4502 16-8.050 5.333-2.683 7 16-13.72 5.333-4.5763 16-10.70 5.333-3.566 8 16-14.01 5.333-4.6704 16-12.02 5.333-4.008 9 16-14.23 5.333-4.7445 16-12.82 5.333-4.273 10 16-14.41 5.333-4.803
Table 5.12 shows the memristive transresistance ratio with the Memistor-realization
as the frequency increases.
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.4.
By following the same procedure as in the previous implementation, the circuit of Figure
5.21 is simulated in order to examine the behavior of the output variable and the transfer
ratio of the amplifier.
Results
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.24.
104
5.1. Nullor-based examples
Mmin
MmaxM
0 2 4 6 8 10 12-0.2
-0.1
0.0
0.1
0.2
TimeHsL
Vol
tageHVL
(a) ω=1
Mmin
Mmax
M
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.2
-0.1
0.0
0.1
0.2
TimeHsL
Vol
tageHVL
(b) ω=10
Figure 5.24: uo(t) curves for the trans-memresistance amplifier with Memistor realizationusing HPM memristor model
For this implementation, the resulting ii-uo hysteresis loops (in black) and the hysteresis
loops of the memristor (in blue) are shown in Figure 5.25. The trans-memresistance and
the memresistance at the feedback remain the same behavior as the previous analysis using
the polynomial memristor model.
M
-40 -20 0 20 40
-150
-100
-50
0
50
100
150
CurrentHΜAL
Vol
tageHm
VL
(a) ω=1
M
-40 -20 0 20 40-200
-100
0
100
200
CurrentHΜAL
Vol
tageHm
VL
(b) ω=10
Figure 5.25: Transmem-resistance ii-uo hysteresis (black) and memristor hysteresis (blue)for Memistor realization using HPM memristor model
Table 5.13 shows the memristive transresistance ratio with the Memistor-realization
as the frequency increases.
105
Chapter 5. Circuit applications
Table 5.13: Memristive transresistance ratio using different frequency values
ω M(KΩ) uo
ii(KΩ) ω M(KΩ) uo
ii(KΩ)
1 14.41-5.806 4.803-1.935 6 14.41-13.65 4.803-4.5522 14.41-11.15 4.803-3.719 7 14.41-13.78 4.803-4.5933 14.41-12.59 4.803-4.197 8 14.41-13.87 4.803-4.6234 14.41-13.16 4.803-4.389 9 14.41-13.93 4.803-4.6465 14.41-13.47 4.803-4.490 10 14.41-13.99 4.803-4.663
5.1.1.4 Conclusions
The feasibility of input-output characteristics that exhibit the typical memristive hysteresis
has been proved by the analysis of the trans-memresistance amplifier amplifiers using
the polynomial and the HPM memristor models. The simulation results show that the
pinched hysteresis loop of the memristor has been transmitted to the input-output transfer
characteristics which indicates a memristive behavior of the amplifier.
The nullor has been implemented in two forms, namely a MOS-realization and a novel
Memistor-realization. On the one hand, the first implementation allow us to foresee that
hybrid (MOS/memristor) circuits can be used for this type of application. On the other
hand, the second implementation opens the possibility to obtain full-memristor realizations
for these circuits. Besides, the results on the analysis of the effect of the DC-gain when
the nullor is implemented by a MOS-stage are in concordance with the basic concepts of
the design of nullor-based negative feedback amplifiers.
Table 5.14: Summary of data from the trans-memresistance amplifier using the polynomialmemristor model
Trans-memresistance amplifier M(KΩ) for ω=∞ uM(V ) uo
ii(KΩ) for ω=1 uo
ii(KΩ) for ω=∞
Nullor-based 16 0.61 16–0.100 16MOS-equivalent 16 0.48 12.86–3.021 12.86
Memistor-implementation 16 0.20 5.333–0.033 5.333
Table 5.15: Summary of data from the trans-memresistance amplifier using the HPMmemristor model
Trans-memresistance amplifier M(KΩ) for ω=∞ uM(V ) uo
ii(KΩ) for ω=1 uo
ii(KΩ) for ω=∞
Nullor-based 14.41 0.57 14.41–5.806 14.41MOS-equivalent 14.41 0.44 11.27–2.678 11.27
Memistor-implementation 14.41 0.19 4.803–1.935 4.803
The differences on the simulation results in the case of using Nullor-based, MOS-
equivalent and Memistor-implementation in the trans-memresistance amplifier are recast
in Table 5.14 and Table 5.15.
106
5.1. Nullor-based examples
The information establishes that the active part of the trans-memresistance amplifier
contributes significantly to the performance of the amplifier. The use of non-ideal blocks in
the active part causes the value of the memristive transmission decreases as the frequency
tends to infinity, i.e. the gain of the amplifier is reduced. Moreover, the memristive
behavior of the transfer ratios has not disappeared in the case of MOS-implementation
and Memistor-implementation which suggests the usefulness of the memristor as feedback
element.
107
Chapter 5. Circuit applications
5.1.2 Trans-memconductance amplifier
5.1.2.1 Nullor-based design
The circuit shown in Figure 5.26 represents the trans-memconductance amplifier using the
nullor as the plant to be controlled. This negative-feedback amplifier is simulated by using
the polynomial and HPM memristor models.
W
+−ui RL
io
Figure 5.26: Trans-memconductance amplifier with nullor
where ui=A sinωt, A=1V, RL=1Ω and W is characterized by the parameter values of
the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.16.
Table 5.16: W parameter values using polynomial memristor model
W (10−5S) Wmax(S) Wmin(S) α W (ω → ∞)(S)
W 6.25-1000 1×10−2 6.25×10−5 160 6.25×10−5
The memristor model is incorporated in the feedback loop of the amplifier, as shown
in Figure 5.26. This circuit is simulated in order to examine the behavior of the output
variable and the transfer ratio of the amplifier.
Results
The simulation result of the output current io(t) of the amplifier for two discrete values of
the frequency ω = 1, 10 is shown in Figure 5.27.
108
5.1. Nullor-based examples
Wmin
Wmax W
@×10@
0 2 4 6 8 10 12-1.0
-0.5
0.0
0.5
1.0
TimeHsL
Cur
rentHm
AL
(a) ω=1
@×10
W
@
Wmin
Wmax
0.0 0.2 0.4 0.6 0.8 1.0 1.2-1.0
-0.5
0.0
0.5
1.0
TimeHsL
Cur
rentHm
AL
(b) ω=10
Figure 5.27: io(t) curves for the trans-memconductance amplifier with nullor using poly-nomial memristor model
Besides, in order to gain insight on the dynamics of the amplifier, simulations were also
carried out by substituting the memristor W for linear conductances having the minimum
and maximum memconductances values that the memristor can reach, namely Wmin and
Wmax respectively.
The corresponding curves shown in Figure 5.27 denote io with the memristor by the red
lines, and the black lines are associated to the simulations with Wmin and Wmax. Clearly it
can be noticed that the io curve jumps from the lower to the upper limits at low-frequency
and it sticks to the lower limit at high frequencies, i.e. the amplifier reaches the minimum
trans-memconductance value. The label @x10 means that the curve of Wmax has been
reduced to one tenth of the original value in order to be displayed.
Another results of the simulations are the transmission curves ( ioui
) of the amplifier,
as shown in Figure 5.28. These curves show the typical eight-shape of the memristor
hysteresis, tending to a straight-line at high frequencies.
Ω=1
Ω=10
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
VoltageHVL
Cur
rentHm
AL
Figure 5.28: Transmem-conductance ui-io hysteresis with nullor using polynomial mem-ristor model
109
Chapter 5. Circuit applications
Table 5.17 shows the breakdown of the memristive transconductance ratio by reducing
the memristive range as the frequency increases.
Table 5.17: Memristive transconductance ratio using different frequency values
ω ioui
(x10−5S) ω ioui
(x10−5S)
1 6.250-1000 6 6.250-7.4902 6.250-12.42 7 6.250-7.2843 6.250-9.345 8 6.250-7.1364 6.250-8.316 9 6.250-7.0255 6.250-7.800 10 6.250-6.939
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.18.
Table 5.18: W parameter values using HPM memristor model
W (10−5S) x0 a A α Wmax(S) Wmin(S) W (ω → ∞)(S)
W 6.939-17.22 0.1 3 40µA 160 1.722×10−4 6.939×10−5 6.939×10−5
Now, the circuit shown in Figure 5.26 is performed by using the HPM memristor model.
Results
The simulation result of the output voltage io(t) of the amplifier for two discrete values of
the frequency ω = 1, 10 is shown in Figure 5.29.
Wmax
W
Wmin
0 2 4 6 8 10 12
-150
-100
-50
0
50
100
150
TimeHsL
Cur
rentHΜ
AL
(a) ω=1
Wmax
W
Wmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-150
-100
-50
0
50
100
150
TimeHsL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.29: io(t) curves for the trans-memconductance amplifier with nullor using HPMmemristor model
110
5.1. Nullor-based examples
In this case, the corresponding curves shown in Figure 5.29 denote io with the memris-
tor by the blue lines, and the black lines are associated to the simulations with the Wmin
and the Wmax reached by the memristor.
Ω=1
Ω=10
-1.0 -0.5 0.0 0.5 1.0-100
-50
0
50
100
VoltageHVL
Cur
rentHΜ
AL
Figure 5.30: Transmem-conductance ui-io hysteresis with nullor using HPM memristormodel
The transmission curves ( ioui
) of the amplifier are shown in Figure 5.30. The memristive
transconductance ratio ( ioui
) shows the typical eight-shape of the memristor hysteresis,
tending to a straight-line at high frequencies, as shown in Figure 5.30. Table 5.19 shows
the behavior of the memristive range as the frequency increases.
Table 5.19: Memristive transconductance ratio using different frequency values
ω ioui
(x10−5S) ω ioui
(x10−5S)
1 6.939-17.22 6 6.939-7.3222 6.939-8.960 7 6.939-7.2563 6.939-7.942 8 6.939-7.2094 6.939-7.594 9 6.939-7.1745 6.939-7.423 10 6.939-7.147
111
Chapter 5. Circuit applications
5.1.2.2 MOS-based design
The circuit shown in Figure 5.31 represents the trans-memconductance amplifier using the
nullor implementation of the MOS in common-source configuration. This implementation
is simulated by using the polynomial and HPM memristor models.
W
+−ui RL
iovgs
+
–
gmvgs
gds
Figure 5.31: Trans-memconductance amplifier with MOS-equivalent
where ui=A sinωt, A=1V, RL=1Ω, gm=320µA
V, gds=320nS and W is characterized by
the parameter values of the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.16. By following the same procedure as in the previous implementation, the circuit of
Figure 5.31 is simulated.
Results
Wmax
Wmin
W
0 2 4 6 8 10 12-300
-200
-100
0
100
200
300
TimeHsL
Cur
rentHΜ
AL
(a) ω=1
Wmax
W
Wmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2-300
-200
-100
0
100
200
300
TimeHsL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.32: io(t) curves for the trans-memconductance amplifier with MOS-equivalentusing polynomial memristor model
112
5.1. Nullor-based examples
By following the same procedure as in the previous implementation, the curves io(t)
are obtained and shown in Figure 5.32.
W
-1.0 -0.5 0.0 0.5 1.0
-100
-50
0
50
100
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
W
-1.0 -0.5 0.0 0.5 1.0
-40
-20
0
20
40
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.33: Transmem-conductance ui-io hysteresis with MOS-equivalent using polyno-mial memristor model
For this implementation, the Figure 5.33 shows the resulting ui-io hysteresis loops (in
black) and the hysteresis loops of the memristor are also shown (in red).
It clearly result that the trans-memconductance is smaller than the memconductance
at the feedback, which can be explained by the fact that the implemented nullor does not
have zeroes in its chain matrix. Table 5.20 shows the memristive transconductance ratio
with the MOS-equivalent as the frequency increases.
Table 5.20: Memristive transconductance ratio using different frequency values
ω W (x10−5S) ioui
(x10−5S) ω W (x10−5S) uo
ii(x10−5S)
1 6.250-1000 5.224-31.00 6 6.250-7.490 5.224-6.0642 6.250-12.42 5.224-8.942 7 6.250-7.284 5.224-5.9283 6.250-9.345 5.224-7.227 8 6.250-7.136 5.224-5.8304 6.250-8.316 5.224-6.595 9 6.250-7.025 5.224-5.7565 6.250-7.800 5.224-6.266 10 6.250-6.939 5.224-5.698
Effect of the DC-gain
In this part, further analysis of the amplifier is achieved for several values of the DC-gain.
This important parameter expresses how closer the MOS-model is to the nullor. Because
the DC-gain depends on gds and gm, separate analysis are carried out by varying gds and
gm.
113
Chapter 5. Circuit applications
Analysis I: Effect of gds
In this analysis, the output conductance gds is varied while the transconductance gm
remains fixed.
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í íí í
é 60dB & 40dBí 20dB
-1.0 -0.5 0.0 0.5 1.0
-100
-50
0
50
100
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
ééééééééééééééééé
éééé
éé
é
é
é
é
éé
ééééééééééééééééééééééííííííííííííííííí
íííí
íí
íí
íí
íí
íííííííííííííííííííííí
é 60dB & 40dBí 20dB
-1.0 -0.5 0.0 0.5 1.0
-40
-20
0
20
40
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.34: Transmission curves for Analysis I.
The trans-memconductance ui–io hystereses are shown in Figure 5.34 for DC-gain
values of 60dB, 40dB and 20dB.
Another form of looking at the simulation results is recast in Table 5.21. Herein, the
values associated to gds in order to produce the DC-gain values above yield a limit trans-
memconductance value at ω=∞, that is to say high frequencies. Decreasing the DC-gain
(increasing gds) causes a small reduction in this limit.
Table 5.21: Summary of Analysis I
DC-gain (dB) gm (10−6)AV
gds()ioui
() for ω = ∞
60 320 320×10−9 5.224×10−5
40 320 320×10−8 5.185×10−5
20 320 320×10−7 4.825×10−5
114
5.1. Nullor-based examples
Analysis II: Effect of gm
In this analysis, gm is varied while gds is fixed.
é ééééééééééééééééé
éééé
é
é
é
é
é
é
é
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éí í íí í í
í íííííííííííííííííííííííííííííííííííí
í í íí í í
í íá á á á á á ááááááááááááááááááááááááááááááááááááááá á á á á á á
é 60dBí 40dBá 20dB
-1.0 -0.5 0.0 0.5 1.0
-100
-50
0
50
100
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
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éééé
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é
é
é
é
éé
ééééééééééééééééééééééí í
í íí í
í ííííííííííííííííííííííííííííííííííí
ííí í
í íí í
íá á á á á á á ááááááááááááááááááááááááááááááááááááá á á á á á á
á
é 60dBí 40dBá 20dB
-1.0 -0.5 0.0 0.5 1.0
-40
-20
0
20
40
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.35: Transmission curves for Analysis II.
The resulting trans-memconductance ui-io hystereses are shown in Figure 5.35 for the
same DC-gain values of the previous analysis.
The summary of the effect of gm is shown in Table 5.22. Now, the values associated to
gm produce a limit trans-memresistance value at ω=∞.
Table 5.22: Summary of Analysis II
DC-gain (dB) gmAV
gds(10−9) io
ui
() for ω = ∞
60 320×10−6 320 5.224×10−5
40 320×10−7 320 2.109×10−5
20 320×10−8 320 3.029×10−6
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.18.
By following the same procedure as in the previous implementation, the circuit of Figure
5.31 is simulated in order to examine the behavior of the output variable and the transfer
ratio of the amplifier.
Results
The simulation result of the output current io(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.36. The resulting ui-io hysteresis loops (in
black) and the hysteresis loops of the memristor (in blue) are shown in Figure 5.37.
115
Chapter 5. Circuit applications
Wmax
W
Wmin
0 2 4 6 8 10 12
-100
-50
0
50
100
TimeHsL
Cur
rentHΜ
AL
(a) ω=1
Wmax
W
Wmin
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-100
-50
0
50
100
TimeHsL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.36: io(t) curves for the trans-memconductance amplifier with MOS-equivalentusing HPM memristor model
W
-1.0 -0.5 0.0 0.5 1.0
-50
0
50
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
W
-1.0 -0.5 0.0 0.5 1.0-60
-40
-20
0
20
40
60
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.37: Transmem-conductance ui-io hysteresis with MOS-equivalent using HPMmemristor model
The trans-memconductance and the memconductance at the feedback remain the same
behavior as the previous analysis using the polynomial memristor model. Table 5.23
shows the memristive transconductance ratio with the MOS-equivalent as the frequency
increases.
Table 5.23: Memristive transconductance ratio using different frequency values
ω W (x10−5S) ioui
(x10−5S) ω W (x10−5S) uo
ii(x10−5S)
1 6.939-17.22 5.698-11.18 6 6.939-7.322 5.698-5.9542 6.939-8.960 5.698-6.994 7 6.939-7.256 5.698-5.9103 6.939-7.942 5.698-6.357 8 6.939-7.209 5.698-5.8784 6.939-7.594 5.698-6.133 9 6.939-7.174 5.698-5.8555 6.939-7.423 5.698-6.020 10 6.939-7.147 5.698-5.837
116
5.1. Nullor-based examples
Effect of the DC-gain
The analysis of the amplifier is achieved for several values of the DC-gain. Because the
DC-gain depends on gds and gm, separate analysis are carried out by varying gds and gm.
Analysis I: Effect of gds
The conductance gds is varied while the transconductance gm remains fixed.
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éé
é
é
é
é
é
é
é
é
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íí
í
í
í
í
í
í
í
í
íííííííí
íííííííííííííí
é 60dB & 40dBí 20dB
-1.0 -0.5 0.0 0.5 1.0
-50
0
50
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
ééééééééééééééééé
éééé
éé
éé
éé
éé
ééééééééééééééééééééééííííííííííííííííí
íííí
íí
íí
íí
íí
íííííííííííííííííííííí
é 60dB & 40dBí 20dB
-1.0 -0.5 0.0 0.5 1.0-60
-40
-20
0
20
40
60
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.38: Transmission curves for Analysis I.
The trans-memconductance ui-io hystereses are shown in Figure 5.38 for DC-gain values
of 60dB, 40dB and 20dB.
In Table 5.24, the values associated to gds in order to produce the DC-gain values
above yield a limit trans-memconductance value at ω=∞, that is to say high frequencies.
Decreasing the DC-gain (increasing gds) causes a small reduction in this limit.
Table 5.24: Summary of Analysis I
DC-gain (dB) gm (10−6)AV
gds()ioui
() for ω=100
60 320 320×10−9 5.698×10−5
40 320 320×10−8 5.656×10−5
20 320 320×10−7 5.269×10−5
117
Chapter 5. Circuit applications
Analysis II: Effect of gm
In this analysis, gm is varied while gds is fixed.
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éé
é
é
é
é
é
é
é
é
éééééééé
ééééééééééééééí í
í íí í
í ííííííííííííííííííííííííííííííííííí
íí íí í
í íí íá á á á
á á á ááááááááááááááááááááááááááááááááááááá á á á á á á á
é 60dBí 40dBá 20dB
-1.0 -0.5 0.0 0.5 1.0
-50
0
50
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
ééééééééééééééééé
éééé
éé
éé
éé
éé
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í íí í
í ííííííííííííííííííííííííííííííííííí
ííí í
í íí í
íá á á á á á á ááááááááááááááááááááááááááááááááááááá á á á á á á
á
é 60dBí 40dBá 20dB
-1.0 -0.5 0.0 0.5 1.0-60
-40
-20
0
20
40
60
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.39: Transmission curves for Analysis II.
The resulting trans-memconductance ui-io hystereses are shown in Figure 5.39 for the
same DC-gain values of the previous analysis.
The summary of the effect of gm is shown in Table 5.25. Now, the values associated to
gm produce a limit trans-memconductance value at ω=∞.
Table 5.25: Summary of Analysis II
DC-gain (dB) gmAV
gds(10−9) io
ui
() for ω = ∞
60 320×10−6 320 5.698×10−5
40 320×10−7 320 2.183×10−5
20 320×10−8 320 3.045×10−6
118
5.1. Nullor-based examples
5.1.2.3 Memistor-based design
The circuit shown in Figure 5.40 represents the trans-memconductance amplifier using the
nullor implementation of the memistor. This implementation is simulated by using the
polynomial and HPM memristor models.
W
+−ui RL
io
Figure 5.40: Trans-memconductance amplifier with Memistor realization
where ui=A sinωt, A=1V and RL=1Ω. The memristor W and the memristors that
constitutes the memistor are characterized by the parameter of the memristor model.
Polynomial model
The polynomial memristor model is characterized by the parameter values shown in Table
5.16. By following the same procedure as in the previous implementation, the circuit of
Figure 5.40 is simulated.
Results
W
Wmin
Wmax
0 2 4 6 8 10 12-400
-200
0
200
400
TimeHsL
Cur
rentHΜ
AL
(a) ω=1
@
@×10-1Wmin
W
Wmax
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-20
-10
0
10
20
TimeHsL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.41: io(t) curves for the trans-memconductance amplifier with Memistor realiza-tion using polynomial memristor model
119
Chapter 5. Circuit applications
The simulation result of the output voltage uo(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.41. The label @x10−1 means that the curve
of Wmax has been increased ten times of the original value in order to be displayed.
M
-1.0 -0.5 0.0 0.5 1.0
-200
-100
0
100
200
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
M
-1.0 -0.5 0.0 0.5 1.0
-20
-10
0
10
20
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.42: Transmem-conductance ui-io hysteresis (black) and memristor hysteresis (red)for Memistor realization using polynomial memristor model
For this implementation, the resulting ui-io hysteresis loops (in black) and the hysteresis
loops of the memristor (in red) are shown in Figure 5.42. Table 5.26 shows the memristive
transconductance ratio with the Memistor-implementation as the frequency increases.
Table 5.26: Memristive transconductance ratio using different frequency values
ω W (x10−5S) ioui
(x10−5S) ω W (x10−5S) uo
ii(x10−5S)
1 6.250-1000 2.083-333.33 6 6.250-7.490 2.083-2.4962 6.250-12.42 2.083-4.140 7 6.250-7.284 2.083-2.4283 6.250-9.345 2.083-3.115 8 6.250-7.136 2.083-2.3784 6.250-8.316 2.083-2.772 9 6.250-7.025 2.083-2.3415 6.250-7.800 2.083-2.600 10 6.250-6.939 2.083-2.313
HPM model
The HPM memristor model is characterized by the parameter values shown in Table 5.18.
By following the same procedure as in the previous implementation, the circuit of Figure
5.40 is simulated.
Results
The simulation result of the output current io(t) of the amplifier for two discrete values
of the frequency ω = 1, 10 is shown in Figure 5.43. For this implementation, the resulting
120
5.1. Nullor-based examples
ui-io hysteresis loops (in black) and the hysteresis loops of the memristor (in blue) are
shown in Figure 5.44.
Wmin
W
Wmax
0 2 4 6 8 10 12-40
-20
0
20
40
TimeHsL
Cur
rentHΜ
AL
(a) ω=1
WminW
Wmax
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-20
-10
0
10
20
TimeHsL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.43: io(t) curves for the trans-memconductance amplifier with Memistor realiza-tion using HPM memristor model
M
-1.0 -0.5 0.0 0.5 1.0
-30
-20
-10
0
10
20
30
VoltageHVL
Cur
rentHΜ
AL
(a) ω=1
M
-1.0 -0.5 0.0 0.5 1.0
-20
-10
0
10
20
VoltageHVL
Cur
rentHΜ
AL
(b) ω=10
Figure 5.44: Transmem-conductance ui-io hysteresis (black) and memristor hysteresis(blue) for Memistor realization using HPM memristor model
Table 5.27: Memristive transconductance ratio using different frequency values
ω W (x10−5S) ioui
(x10−5S) ω W (x10−5S) uo
ii(x10−5S)
1 6.939-17.22 2.313-5.740 6 6.939-7.322 2.313-2.4402 6.939-8.960 2.313-2.986 7 6.939-7.256 2.313-2.4183 6.939-7.942 2.313-2.647 8 6.939-7.209 2.313-2.4034 6.939-7.594 2.313-2.531 9 6.939-7.174 2.313-2.3915 6.939-7.423 2.313-2.474 10 6.939-7.147 2.313-2.382
The trans-memconductance and the memconductance at the feedback remain the same
behavior as the previous analysisis using the polynomial memristor model. Table 5.27
121
Chapter 5. Circuit applications
shows the memristive transconductance ratio with the Memistor-implementation as the
frequency increases.
5.1.2.4 Conclusions
The feasibility of input-output characteristics that exhibit the typical memristive hysteresis
has been proved by the analysis of the trans-memconductance amplifiers. The simulation
results show that the pinched hysteresis loop of the memristor has been transmitted to
the input-output transfer characteristics which indicates a memristive behavior of the
amplifier.
The nullor has been implemented in two forms, namely a MOS-realization and a novel
Memistor-realization. On the one hand, the first implementation allow us to foresee that
hybrid (MOS/memristor) circuits can be used for this type of application. On the other
hand, the second implementation opens the possibility to obtain full-memristor realizations
for these circuits. Besides, the results on the analysis of the effect of the DC-gain when
the nullor is implemented by a MOS-stage are in concordance with the basic concepts of
the design of nullor-based negative feedback amplifiers.
The differences on the simulation results in the case of using Nullor-based, MOS-
equivalent and Memistor-implementation in the trans-memconductance amplifier are re-
cast in Table 5.28 and Table 5.29.
Table 5.28: Summary of data from the trans-memconductance amplifier using the poly-nomial memristor model
Trans-memconductance amplifier W (µS) for ω=∞ uW (V ) ioui
(µ) for ω=1 ioui
(µ) for ω=∞
Nullor-based 62.5 1 10x103–62.5 62.5MOS-equivalent 62.5 0.83 310–54.6 54.6
Memistor-implementation 62.5 0.33 3.333x103–20.8 20.8
Table 5.29: Summary of data from the trans-memconductance amplifier using the HPMmemristor model
Trans-memconductance amplifier W (µS) for ω=∞ uW (V ) ioui
(µ) for ω=1 ioui
(µ) for ω=∞
Nullor-based 69.39 1 172.2–69.39 69.39MOS-equivalent 69.39 0.83 111.8–56.98 56.98
Memistor-implementation 69.39 0.33 57.40–23.13 23.13
The information establishes that the active part of the trans-memconductance amplifier
contributes significantly to the performance of the amplifier. The use of non-ideal blocks in
the active part causes the value of the memristive transmission decreases as the frequency
122
5.1. Nullor-based examples
tends to infinity, i.e. the gain of the amplifier is reduced. Moreover, the memristive
behavior of the transfer ratios has not disappeared in the case of MOS-implementation
and Memistor-implementation which suggests the usefulness of the memristor as feedback
element.
123
Chapter 5. Circuit applications
5.2 Opamp-based examples
The op-amp is a differential amplifier that exhibits a large voltage gain, low output
impedance and high input impedance. In the conventional inverter and noninverter am-
plifiers, the op-amp is connected using two resistors R1 and R2.
The existence of the memristor provides the opportunity to use memristors instead of
resistors. Besides, depending of the resistor replaced; a new configuration of the memristor-
based amplifier emerges. These new configurations are listed in Table 5.30.
Table 5.30: New configurations of memristor-based amplifiers
Amplifier Feedback network Configuration
M −M MM
configurationinverter M − R M
Rconfiguration
R −M RM
configuration
M −M MM
configurationnoninverter M − R M
Rconfiguration
R −M RM
configuration
In this thesis, the ideal functional-model of the operational amplifier is resorted in order
to accomplish the point of view of the structured electronic design. Moreover, the analysis
of the composition of fundamental and harmonic terms in the case of MR
configuration
inverter and noninverter amplifiers in order to determine the total harmonic distortion
(THD).
Hereafter, the HPM and polynomial memristor models are used to carry out the circuit-
level simulation of the inverter and noninverter amplifiers.
5.2.1 Memristor-based inverter amplifier
M1
M2
configuration
−
+
M1
Vout
M2
Vin
Figure 5.45: Memristor-based inverter amplifier
124
5.2. Opamp-based examples
The transmission characteristic of the circuit shown in the Figure 5.45, can be estab-
lished as
Av =Vout
Vin
= −M1
M2(5.5)
The memristors M1 and M2 are characterized with the parameter values shown in
Table 5.32 and Table 5.31 for the polynomial and HPM memristor model, respectively.
Table 5.31: M1 and M2 parameter values using polynomial memristor model
M RON ROFF α M(ω → ∞)
M1 38000-100 100 38000 380 38000M2 16000-100 100 16000 160 16000
Table 5.32: M1 and M2 parameter values using HPM memristor model
M x0 a A α M(ω → ∞)
M1 14410-9385 0.1 2 40µA 160 14410M2 14410-5806 0.1 3 40µA 160 14410
Ω=1
Ω=100
0 1 2 3 4 5 6-2.4
-2.2
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
TimeHsL
Av
(a) Polynomial memristor model
Ω=1
Ω=100
0 1 2 3 4 5 6
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
TimeHsL
Av
(b) HPM memristor model
Figure 5.46: t-Av characteristic curves using different memristor models
The Figure 5.46 shows the t-Av curve using the M1
M2
configuration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
The Figure 5.47 shows the transmission curve using the M1
M2
configuration. The trans-
mission curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends
to be a straight line according ω approaches to infinity.
125
Chapter 5. Circuit applications
Ω=1 Ω=100
-1.0 -0.5 0.0 0.5 1.0
-2
-1
0
1
2
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.47: Transmission characteristic curves using different memristor models
-HSPICEAnalogInsydes
0 1 2 3 4 5 6-2.4
-2.2
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
TimeHsL
Av
(a) Polynomial memristor model
AnalogInsydes-HSPICE
0 1 2 3 4 5 6
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
TimeHsL
Av
(b) HPM memristor model
Figure 5.48: HSPICE and AnalogInsydes simulation results
The memristors M1 and M2 have been incorporated to Verilog-A modules in order to
be employed in a HSPICE simulation procedure. A comparison of HSPICE and Analo-
gInsydes simulation results is depicted in Figure 5.48.
126
5.2. Opamp-based examples
5.2.1.1 M2
Rconfiguration
−
+
M2
Vout
RVin
Figure 5.49: Memristor-based inverter amplifier
The transmission characteristic of the circuit shown in the Figure 5.49, can be estab-
lished as
Av =Vout
Vin
= −M2
R(5.6)
where R=5KΩ
Ω=1
Ω=100
0 1 2 3 4 5 6
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
TimeHsL
Av
(a) Polynomial memristor model
Ω=1
Ω=100
0 1 2 3 4 5 6
-2.5
-2.0
-1.5
TimeHsL
Av
(b) HPM memristor model
Figure 5.50: t-Av characteristic curves using different memristor models
The Figure 5.50 shows the t-Av curve using the M2
Rconfiguration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
The Figure 5.51 shows the transmission curve using the M2
Rconfiguration. The trans-
mission curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends
to be a straight line according ω approaches to infinity.
127
Chapter 5. Circuit applications
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0
-3
-2
-1
0
1
2
3
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=100
Ω=1
-1.0 -0.5 0.0 0.5 1.0-3
-2
-1
0
1
2
3
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.51: Transmission characteristic curves using different memristor models
-HSPICEAnalogInsydes
0 1 2 3 4 5 6
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
TimeHsL
Av
(a) Polynomial memristor model
-HSPICEAnalogInsydes
0 1 2 3 4 5 6
-2.5
-2.0
-1.5
TimeHsL
Av
(b) HPM memristor model
Figure 5.52: HSPICE and AnalogInsydes simulation results
The memristor M2 has been incorporated to Verilog-A module in order to be em-
ployed in a HSPICE simulation procedure. A comparison of HSPICE and AnalogInsydes
simulation results is depicted in Figure 5.52.
THD
The total harmonic distortion, or THD, is the sum of all harmonic components compared
against the fundamental component. In the case of memristor-based inverter amplifiers,
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
THD =
√
v22 + v23 + v24 + . . .+ v2nv1
(5.7)
128
5.2. Opamp-based examples
where n is the maximum harmonic component, vk is the output voltage of the k-th har-
monic and k=1 is the output voltage of the fundamental frequency.
The result of the equation 5.7 is a comparison of the output voltages of all harmonic
components and the output voltage of the fundamental component.
The output voltage of the memristor-based inverter amplifier in the M2
Rconfiguration,
are given by
vout = −M
Rvin (5.8)
where vin = A sin(ωt).
In the case of HPM memristor model, the memristance M is defined by
M = C1 + C2 cos(ωt) + C3 cos(2ωt) + C4 cos(3ωt)
+ C5 cos(4ωt) + C6 cos(5ωt) + C7 cos(6ωt)(5.9)
where C1, C2, C3, C4, C5, C6 and C7 are constants from the physical parameters and the
homotopy perturbation method.
Substituting the above expression in the equation 5.8
vout = −
[
C1
Rsin(ωt) +
C2
Rcos(ωt) sin(ωt) +
C3
Rcos(2ωt) sin(ωt)
+C4
Rcos(3ωt) sin(ωt) +
C5
Rcos(4ωt) sin(ωt)
+C6
Rcos(5ωt) sin(ωt) +
C7
Rcos(6ωt) sin(ωt)
]
(5.10)
Applying some trigonometric identities
vout = −
[(
C1
R−
C3
2R
)
sin(ωt) +
(
C2
2R−
C4
2R
)
sin(2ωt)
+
(
C3
2R−
C5
2R
)
sin(3ωt) +
(
C4
2R−
C6
2R
)
sin(4ωt)
+
(
C5
2R−
C7
2R
)
sin(5ωt) +C6
2Rsin(6ωt) +
C7
2Rsin(7ωt)
]
(5.11)
The above expression establishes the vout voltage using the frequency components:
fundamental, second, third, fourth, fifth, sixth and seventh harmonic.
The equation 5.7 can be rewritten as
129
Chapter 5. Circuit applications
THD =
√
v22 + v23 + v24 + v25 + v26 + v25v1
(5.12)
where the v1, v2, v3, v4, v5, v6 and v7 expression of voltage are described as
v1 =
(
C1
R−
C3
2R
)
sin(ωt)
v2 =
(
C2
2R−
C4
2R
)
sin(2ωt)
v3 =
(
C3
2R−
C5
2R
)
sin(3ωt)
v4 =
(
C4
2R−
C6
2R
)
sin(4ωt)
v5 =
(
C5
2R−
C7
2R
)
sin(5ωt)
v6 =
(
C6
2R
)
sin(6ωt)
v7 =
(
C7
2R
)
sin(7ωt)
THD can be calculated by substituting the vout voltage of each frequency component
in the equation 5.12.
In the case of polynomial memristor model, the memristance M is defined by
M = B1 +B2 cos(ωt) (5.13)
where B1 and B2 are constants from the physical parameters and the polynomial memristor
model.
Substituting the above expression in the equation 5.8
vout = −
[
B1
Rsin(ωt) +
B2
Rcos(ωt) sin(ωt)
]
(5.14)
Applying some trigonometric identities
vout = −
[(
B1
R
)
sin(ωt) +
(
B2
2R
)
sin(2ωt)
]
(5.15)
130
5.2. Opamp-based examples
The above expression establishes the vout voltage using the frequency components:
fundamental and second harmonic.
The equation 5.7 can be rewritten as
THD =
√
v22v1
(5.16)
where the v1 and v2 expression of voltage are described as
v1 =
(
B1
R
)
sin(ωt)
v2 =
(
B2
2R
)
sin(2ωt)
THD can be calculated by substituting the vout voltage of each frequency component
in the equation 5.16.
2 4 6 8
10
20
30
40
GainHdBsL
TH
D%
(a) ω=1
7.0 7.5 8.0 8.5 9.0
2
4
6
8
10
12
GainHdBsL
TH
D%
(b) ω=2
8.6 8.7 8.8 8.9 9.0 9.1 9.20.0
0.5
1.0
1.5
2.0
2.5
3.0
GainHdBsL
TH
D%
(c) ω=5
8.95 9.00 9.05 9.10 9.150.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
GainHdBsL
TH
D%
(d) ω=10
Figure 5.53: Gain-THD curves using the HPM memristor model
Figure 5.53 shows the Gain-THD curves using different angular frequency values. The
131
Chapter 5. Circuit applications
behavior of the Gain-THD curves establishes that the THD value decreases as the angular
frequency tends to infinity. Furthermore, the gain value for the minimum value of THD
tends to a fixed point as the angular frequency approaches infinity. The gain value at
ω=∞ is a fixed point with a THD value of zero.
Table 5.33: Parameter values using different frequency values
ω Mmax(kΩ) Mmin(kΩ) Gainmax(dBs) Gainmin(dBs) THDmax(%) THDmin(%)
1 16 0.1 10.1 -33.97 98.75 0.022 16 8.05 10.1 4.13 33.05 0.0925 16 12.82 10.1 8.17 11.03 0.008731510 16 14.41 10.1 9.19 5.22 0.0041113
Table 5.33 shows the maximum memristance, the minimum memristance, the maxi-
mum gain, the minimum gain, the maximum THD and the minimum THD values as the
frequency increases.
-30 -20 -10 0 100
20
40
60
80
100
GainHdBsL
TH
D%
(a) ω=1
5 6 7 8 9 100
5
10
15
20
25
30
GainHdBsL
TH
D%
(b) ω=2
8.5 9.0 9.5 10.00
2
4
6
8
10
GainHdBsL
TH
D%
(c) ω=5
9.2 9.4 9.6 9.8 10.00
1
2
3
4
5
GainHdBsL
TH
D%
(d) ω=10
Figure 5.54: Gain-THD curves using the polynomial memristor model
Figure 5.54 shows the Gain-THD curves using different angular frequency values. The
behavior of the Gain-THD curves establishes that the THD value decreases as the angular
frequency tends to infinity. Furthermore, the gain value for the minimum value of THD
132
5.2. Opamp-based examples
tends to a fixed point as the angular frequency approaches infinity. The gain value at
ω=∞ is a fixed point with a THD value of zero.
Table 5.34: Parameter values using different frequency values
ω Mmax(kΩ) Mmin(kΩ) Gainmax(dBs) Gainmin(dBs) THDmax(%) THDmin(%)
1 14.41 5.80 9.19 1.29 43.49 2.772 14.41 11.15 9.19 6.97 12.51 0.735 14.41 13.47 9.19 8.60 3.36 0.0907510 14.41 13.99 9.19 8.93 1.47 0.02045
Table 5.34 shows the maximum memristance, the minimum memristance, the maxi-
mum gain, the minimum gain, the maximum THD and the minimum THD values as the
frequency increases.
5.2.1.2 RM2
configuration
−
+
R
Vout
M2
Vin
Figure 5.55: Memristor-based inverter amplifier
The transmission characteristic of the circuit shown in the Figure 5.55, can be estab-
lished as
Av =Vout
Vin
= −R
M2(5.17)
where R=5KΩ
133
Chapter 5. Circuit applications
Ω=1
Ω=100
0 1 2 3 4 5 6-50
-40
-30
-20
-10
0
TimeHsL
Av
(a) Polynomial memristor model
Ω=1
Ω=100
0 1 2 3 4 5 6
-0.8
-0.7
-0.6
-0.5
-0.4
TimeHsL
Av
(b) HPM memristor model
Figure 5.56: t-Av characteristic curves using different memristor models
Figure 5.56 shows the t-Av curve using the RM2
configuration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0
-0.4
-0.2
0.0
0.2
0.4
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.57: Transmission characteristic curves using different memristor models
Figure 5.57 shows the transmission curve using the RM2
configuration. The transmission
curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends to be a
straight line according ω approaches to infinity.
134
5.2. Opamp-based examples
-HSPICEAnalogInsydes
0 1 2 3 4 5 6-50
-40
-30
-20
-10
0
TimeHsL
Av
(a) Polynomial memristor model
-HSPICEAnalogInsydes
0 1 2 3 4 5 6
-0.8
-0.7
-0.6
-0.5
-0.4
TimeHsL
Av
(b) HPM memristor model
Figure 5.58: HSPICE and AnalogInsydes simulation results
The memristor M2 has been incorporated to Verilog-A module in order to be em-
ployed in a HSPICE simulation procedure. A comparison of HSPICE and AnalogInsydes
simulation results is depicted in Figure 5.58.
135
Chapter 5. Circuit applications
5.2.2 Memristor-based noninverter amplifier
M1
M2
configuration
−
+
M1
Vout
Vin
M2
Figure 5.59: Memristor-based noninverter amplifier
The transmission characteristic of the circuit shown in the Figure 5.59, can be estab-
lished as
Av =Vout
Vin
= 1 +M1
M2(5.18)
The memristors M1 and M2 are characterized with the parameter values shown in
Table 5.32 and Table 5.31 for the polynomial and HPM memristor model, respectively.
Figure 5.60 shows the t-Av curve using the M1
M2
configuration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
Ω=1
Ω=100
0 1 2 3 4 5 62.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
TimeHsL
Av
(a) Polynomial memristor model
Ω=1
Ω=100
0 1 2 3 4 5 62.0
2.1
2.2
2.3
2.4
2.5
2.6
TimeHsL
Av
(b) HPM memristor model
Figure 5.60: t-Av characteristic curves using different memristor models
Figure 5.61 shows the transmission curve using the M1
M2
configuration. The transmission
136
5.2. Opamp-based examples
curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends to be a
straight line according ω approaches to infinity.
Ω=1Ω=100
-1.0 -0.5 0.0 0.5 1.0
-3
-2
-1
0
1
2
3
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=100
Ω=1
-1.0 -0.5 0.0 0.5 1.0
-2
-1
0
1
2
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.61: Transmission characteristic curves using different memristor models
The memristors M1 and M2 have been incorporated to Verilog-A modules in order to
be employed in a HSPICE simulation procedure. A comparison of HSPICE and Analo-
gInsydes simulation results is depicted in Figure 5.62.
-HSPICEAnalogInsydes
0 1 2 3 4 5 62.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
TimeHsL
Av
(a) Polynomial memristor model
-HSPICEAnalogInsydes
0 1 2 3 4 5 62.0
2.1
2.2
2.3
2.4
2.5
2.6
TimeHsL
Av
(b) HPM memristor model
Figure 5.62: HSPICE and AnalogInsydes simulation results
137
Chapter 5. Circuit applications
5.2.2.1 M2
Rconfiguration
−
+
M2
Vout
Vin
R
Figure 5.63: Memristor-based noninverter amplifier
The transmission characteristic of the circuit shown in the Figure 5.63, can be estab-
lished as
Av =Vout
Vin
= 1 +M2
R(5.19)
where R=5KΩ
Ω=1
Ω=100
0 1 2 3 4 5 61.0
1.5
2.0
2.5
3.0
3.5
4.0
TimeHsL
Av
(a) Polynomial memristor model
Ω=100
Ω=1
0 1 2 3 4 5 6
2.5
3.0
3.5
TimeHsL
Av
(b) HPM memristor model
Figure 5.64: t-Av characteristic curves using different memristor models
Figure 5.64 shows the t-Av curve using the M2
Rconfiguration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
138
5.2. Opamp-based examples
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0
-4
-2
0
2
4
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=100
Ω=1
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.65: Transmission characteristic curves using different memristor models
The Figure 5.65 shows the transmission curve using the M2
Rconfiguration. The trans-
mission curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends
to be a straight line according ω approaches to infinity.
-HSPICEAnalogInsydes
0 1 2 3 4 5 61.0
1.5
2.0
2.5
3.0
3.5
4.0
TimeHsL
Av
(a) Polynomial memristor model
-HSPICEAnalogInsydes
0 1 2 3 4 5 6
2.5
3.0
3.5
TimeHsL
Av
(b) HPM memristor model
Figure 5.66: HSPICE and AnalogInsydes simulation results
The memristor M2 has been incorporated to Verilog-A module in order to be em-
ployed in a HSPICE simulation procedure. A comparison of HSPICE and AnalogInsydes
simulation results is depicted in Figure 5.66.
139
Chapter 5. Circuit applications
THD
By following the same procedure of the memristor-based inverter amplifier, the THD
analysis is realized.
7 8 9 10 11
5
10
15
20
25
30
GainHdBsL
TH
D%
(a) ω=1
10.5 11.0 11.5
2
4
6
8
GainHdBsL
TH
D%
(b) ω=2
11.4 11.5 11.6 11.70.0
0.5
1.0
1.5
2.0
2.5
GainHdBsL
TH
D%
(c) ω=5
11.60 11.65 11.70 11.750.0
0.2
0.4
0.6
0.8
1.0
GainHdBsL
TH
D%
(d) ω=10
Figure 5.67: Gain-THD curves using the HPM memristor model
Figure 5.67 shows the Gain-THD curves using different angular frequency values. The
behavior of the Gain-THD curves establishes that the THD value decreases as the angular
frequency tends to infinity. Furthermore, the gain value for the minimum value of THD
tends to a fixed point as the angular frequency approaches infinity. The gain value at
ω=∞ is a fixed point with a THD value of zero.
Table 5.35: Parameter values using different frequency values
ω Mmax(kΩ) Mmin(kΩ) Gainmax(dBs) Gainmin(dBs) THDmax(%) THDmin(%)
1 14.41 5.80 11.78 6.69 29.75 1.892 14.41 11.15 11.78 10.18 9.05 0.535 14.41 13.47 11.78 11.35 2.47 0.06610 14.41 13.99 11.78 11.59 1.09 0.015
Table 5.35 shows the maximum memristance, the minimum memristance, the maxi-
140
5.2. Opamp-based examples
mum gain, the minimum gain, the maximum THD and the minimum THD values as the
frequency increases.
0 2 4 6 8 10 120
10
20
30
40
50
60
GainHdBsL
TH
D%
(a) ω=1
9 10 11 120
5
10
15
20
GainHdBsL
TH
D%
(b) ω=2
11.2 11.4 11.6 11.8 12.0 12.2 12.40
2
4
6
8
GainHdBsL
TH
D%
(c) ω=5
11.8 11.9 12.0 12.1 12.2 12.3 12.40
1
2
3
4
GainHdBsL
TH
D%
(d) ω=10
Figure 5.68: Gain-THD curves using the polynomial memristor model
Figure 5.68 shows the Gain-THD curves using different angular frequency values. The
behavior of the Gain-THD curves establishes that the THD value decreases as the angular
frequency tends to infinity. Furthermore, the gain value for the minimum value of THD
tends to a fixed point as the angular frequency approaches infinity. The gain value at
ω=∞ is a fixed point with a THD value of zero.
Table 5.36: Parameter values using different frequency values
ω Mmax(kΩ) Mmin(kΩ) Gainmax(dBs) Gainmin(dBs) THDmax(%) THDmin(%)
1 16 0.1 12.46 0.1720 60.91 0.01242 16 8.05 12.46 8.3328 23.34 0.01855 16 12.82 12.46 11.0388 8.19 0.006410 16 14.41 12.46 11.7811 3.93 0.0030
Table 5.36 shows the maximum memristance, the minimum memristance, the maxi-
mum gain, the minimum gain, the maximum THD and the minimum THD values as the
frequency increases.
141
Chapter 5. Circuit applications
5.2.2.2 RM2
configuration
−
+
R
Vout
Vin
M2
Figure 5.69: Memristor-based noninverter amplifier
The transmission characteristic of the circuit shown in the Figure 5.69, can be estab-
lished as
Av =Vout
Vin
= 1 +R
M2(5.20)
where R=5KΩ
Ω=1
Ω=100
0 1 2 3 4 5 60
10
20
30
40
50
TimeHsL
Av
(a) Polynomial memristor model
Ω=1
Ω=100
0 1 2 3 4 5 6
1.4
1.5
1.6
1.7
1.8
TimeHsL
Av
(b) HPM memristor model
Figure 5.70: t-Av characteristic curves using different memristor models
Figure 5.70 shows the t-Av curve using the RM2
configuration. The gain Av exhibits
a periodical variation in different instants of time. Besides, the gain Av tends to a fixed
value according ω approaches to infinity.
142
5.2. Opamp-based examples
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
VinHVL
Vo
utHVL
(a) Polynomial memristor model
Ω=1
Ω=100
-1.0 -0.5 0.0 0.5 1.0-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
VinHVL
Vo
utHVL
(b) HPM memristor model
Figure 5.71: Transmission characteristic curves using different memristor models
Figure 5.71 shows the transmission curve using the RM2
configuration. The transmission
curve exhibits a pinched hysteresis loop. Moreover, the transmission curve tends to be a
straight line according ω approaches to infinity.
-HSPICEAnalogInsydes
0 1 2 3 4 5 60
10
20
30
40
50
TimeHsL
Av
(a) Polynomial memristor model
-HSPICEAnalogInsydes
0 1 2 3 4 5 6
1.4
1.5
1.6
1.7
1.8
TimeHsL
Av
(b) HPM memristor model
Figure 5.72: HSPICE and AnalogInsydes simulation results
The memristor M2 has been incorporated to Verilog-A module in order to be em-
ployed in a HSPICE simulation procedure. A comparison of HSPICE and AnalogInsydes
simulation results is depicted in Figure 5.72.
5.2.3 Conclusions
The simulation results show that the pinched hysteresis loop of the memristor has been
transmitted to the amplifier causing a memristive behavior in the gain characteristic curve.
Further, the simulation results have been compared with a HSPICE simulation procedure
using Verilog-A modules.
143
Chapter 5. Circuit applications
The advantage of using the novel polynomial and HPM memristor models in amplifiers
is the ability to locate the gain value over the time, which is not achieved with a PWL
function-based memristor model [8].
The composition of fundamental and harmonic terms in the novel polynomial and HPM
memristor models facilitates the mathematical analysis of the total harmonic distortion.
The THD in the memristor-based inverter and noninverter amplifiers has been discussed
in [9]. The novel memristor models confirms that the THD is worse in the lower gain
values [9].
144
5.3. Mem-elements emulator circuits
5.3 Mem-elements emulator circuits
The concept of memristance, i.e. the property of an element whose resistance depends
on the memory of the system state, has been extended to inductance and capacitance
[10], which has brought us to the concept of meminductance and memcapacitance. The
associated memory elements are called meminductor and memcapacitor, respectively.
In this part of the thesis, the polynomial and HPM memristor models are used to carry
out the circuit-level simulation of these emulation circuits.
5.3.1 Memcapacitor and meminductor grounded
The memcapacitor emulator [11] consists of a memristor M , a capacitor C and a resistor
R connected to an operational amplifier as shown in Figure 5.73. The expression of the
capacitance is determined by
C(t) =M(t)C
R(5.21)
On the other hand, the meminductor emulator [11] is similar to the design of a gyrator
with a memristor replacing a resistor as shown in Figure 5.73. The equivalent inductance
is determined by
L(t) = RM(t)C (5.22)
−
+
Vout
R
MVin
C
−
+
Vout
R
C
Vin
M
Figure 5.73: Memcapacitor and meminductor emulators
The parameter values for the memcapacitor and meminductor emulators are: R=480Ω,
C=10µA and M is characterized by the parameter values shown in Table 5.37 and Table
5.38. The input signal is a square function.
The simulation results are shown in the waveforms of Figures 5.74 and 5.75. It shows
the input voltage (solid line) and the voltage at the output of the operational amplifier
145
Chapter 5. Circuit applications
Table 5.37: M parameter values using HPM memristor model
M x0 a A α Mmin Mmax M(ω → ∞)
M 14410-9385 0.1 2 40µA 160 9385 14410 14410
Table 5.38: M parameter values using polynomial memristor model
M Mmin Rmax α M(ω → ∞)
M 16000-100 100 16000 160 16000
(dashed line) which represents the voltage of the emulate memcapacitor and meminductor.
The waveforms agree with those reported in [11].
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
TimeHsL
Vol
tageHVL
(a) Polynomial memristor model
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
TimeHsL
Vol
tageHVL
(b) HPM memristor model
Figure 5.74: Vin and Vout curves for the memcapacitor emulator.
0.0 0.2 0.4 0.6 0.8 1.0-3
-2
-1
0
1
2
3
TimeHsL
Vol
tageHVL
(a) Polynomial memristor model
0.0 0.2 0.4 0.6 0.8 1.0-3
-2
-1
0
1
2
3
TimeHsL
Vol
tageHVL
(b) HPM memristor model
Figure 5.75: Vin and Vout curves for the meminductor emulator.
Besides, the equivalent capacitance C(t) and inductance L(t) are displayed in Figures
5.76 and 5.77. The simulation results confirm the presence of hysteresis loops in the
146
5.3. Mem-elements emulator circuits
capacitance and inductance as a function of voltage. The behavior of the typical property
in memory circuit elements is observed: the hysteresis loop is much smaller as the frequency
tends to infinity.
Ω=1
Ω=5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
60
65
70
75
80
85
90
Vout
Cap
acit
anceHΜ
FL
(a) HPM memristor model
Ω=1
Ω=5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50
20
40
60
80
100
Vout
Cap
acit
anceHΜ
FL
(b) Polynomial memristor model
Figure 5.76: The capacitance C(t) using different memristor models.
Ω=1
Ω=5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
14
15
16
17
18
19
20
VinHVL
Indu
ctan
ceHHL
(a) HPM memristor model
Ω=1
Ω=5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50
5
10
15
20
VinHVL
Indu
ctan
ceHHL
(b) Polynomial memristor model
Figure 5.77: The inductance L(t) using different memristor models.
147
Chapter 5. Circuit applications
5.3.2 Mutator using current conveyor and OTA
The mutator consists of a memristor M , a capacitor C, an operational transconductance
amplifier (OTA) and a current conveyor (CCII+). The implementation of the mutator
[12] is shown in Figure 5.78.
C
v2v1
M
x y
zCCII+
–
+
gm
i1
iM
Figure 5.78: Mutator MR to MC using CCII+ and OTA
According to [12], the CCII+ curent conveyor transforms voltage v2 into voltage v1;
i.e. v1=v2. The current i1 is conveyed to the z terminal causing a voltage drop on the
capacitor C which is proportional to the time-domain integral of this current. This voltage
is then transformed using the OTA with the transconductance gm into current iM .
The parameter values of the memristor to memcapacitor mutator are: C=10µF, gm=100µAV
and M is characterized by the parameter values shown in Table 5.37 and Table 5.38. The
input port is exited by the voltage source vin of sinusoidal waveform with the amplitude
of 2V and the frequency of 1Hz.
qMc
IMc
0 2 4 6 8 10 12
-1500
-1000
-500
0
500
1000
1500
TimeHsL
qHΜ
CL-
I McHΜ
AL
Figure 5.79: q and IMccurves for the memcapacitor using polynomial memristor model
The simulation results are shown in the waveforms of Figures 5.79 and 5.80. Figure
5.79 shows the behavior of the charge and the current across the memcapacitor mutator.
Moreover, Figure 5.80 shows the behavior of the Vin-Capacitance characteristic curve as
frequency tends to infinity.
148
5.3. Mem-elements emulator circuits
Ω=1
-2 -1 0 1 20
200
400
600
800
1000
vinHVL
Cap
acit
anceHΜ
FL
(a)
Ω=2
Ω=5
Ω=10
-2 -1 0 1 2
7
8
9
10
11
12
vinHVL
Cap
acit
anceHΜ
FL
(b)
Figure 5.80: Vin-Capacitance characteristic curves using polynomial memristor model
In the case of using HPM memristor model, the simulation results are shown in the
waveforms of Figures 5.81 and 5.82. Figure 5.81 shows the behavior of the charge and the
current across the memcapacitor mutator.
IMc
qMc
0 2 4 6 8 10 12
-150
-100
-50
0
50
100
150
200
TimeHsL
qHΜ
CL-
I McH
10-
7 AL
Figure 5.81: q and IMccurves for the memcapacitor using HPM memristor model
Moreover, Figure 5.82 shows the behavior of the Vin-Capacitance characteristic curve
as frequency tends to infinity.
Ω=1
Ω=2
Ω=5-10
-2 -1 0 1 27.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
vinHVL
Cap
acit
anceHΜ
FL
Figure 5.82: Vin-Capacitance characteristic curves using HPM memristor model
149
Chapter 5. Circuit applications
5.3.3 Emulator using single-output current conveyor
The implementation of the memcapacitor emulator [13] is shown in Figure 5.83. This
emulator consists of a memristor M , a inductor L, a resistor R and four current conveyors
(CCII+).
y x
zCCII+
x y
zCCII+
z
xyCCII+
y
zxCCII+
M
R
L
v1 v2
Figure 5.83: Floating memcapacitor
The parameter values of the memcapacitor emulator are: L=10mH, R=480Ω and M
is characterized by the parameter values shown in Table 5.37 and Table 5.38. The input
port is exited by the voltage source vin of sinusoidal waveform with the amplitude of 1V
and the frequency of 1Hz.
Ω=2
Ω=5
Ω=10
-1.0 -0.5 0.0 0.5 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
VCHVL
qH10-
9CL
(a) VC -q characteristic curves
Ω=2
Ω=10
Ω=5
-1.0 -0.5 0.0 0.5 1.0
1.4
1.6
1.8
2.0
2.2
2.4
2.6
VCHVL
Cap
acit
anceHn
FL
(b) VC -C characteristic curves
Figure 5.84: Memcapacitor characteristic-curves using polynomial memristor model
The simulation results are shown in the waveforms of Figure 5.84. Figure 5.84a shows
the parametric graph of the charge and the voltage across the memcapacitor emulator.
Moreover, Figure 5.84b shows the behavior of the VC-C characteristic curve of the mem-
capacitor emulator.
150
5.3. Mem-elements emulator circuits
Ω=1
Ω=2-10
-1.0 -0.5 0.0 0.5 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
VCHVL
qH10-
9CL
(a) VC -q characteristic curves
Ω=1
Ω=2
Ω=5
Ω=10
-1.0 -0.5 0.0 0.5 1.0
1.4
1.6
1.8
2.0
2.2
VCHVL
Cap
acit
anceHn
FL
(b) VC -C characteristic curves
Figure 5.85: Memcapacitor characteristic-curves using HPM memristor model
In the case of using HPM memristor model, the simulation results are shown in the
waveforms of Figure 5.85. Figure 5.85a shows the parametric graph of the charge and the
voltage across the memcapacitor emulator. Moreover, Figure 5.85b shows the behavior of
the VC-C characteristic curve of the memcapacitor emulator.
151
Chapter 5. Circuit applications
5.3.4 Meminductor emulator
The implementation of the meminductor emulator [13] is shown in Figure 5.86. This
emulator consists of a memristor M , a capacitor C, a resistor R and four current conveyors
(CCII+).
y x
zCCII+
x y
zCCII+
z
xyCCII+
y
zxCCII+
R
M
C
v1 v2
Figure 5.86: Floating meminductor
The parameter values of the meminductor emulator are: C=1nF, R=480Ω and M is
characterized by the parameter values shown in Table 5.37 and Table 5.38. The input
port is exited by the voltage source vin of sinusoidal waveform with the amplitude of 1V
and the frequency of 1Hz.
Ω=1
Ω=2
Ω=5-10
-2 -1 0 1 2-1.5
-1.0
-0.5
0.0
0.5
1.0
ILHΜAL
jH1
0-8
WL
(a) IL-ϕ characteristic curves
Ω=1
Ω=2
Ω=5
Ω=10
-2 -1 0 1 20
2
4
6
ILHΜAL
Indu
ctan
ceHm
HL
(b) IL-L characteristic curves
Figure 5.87: Meminductor characteristic-curves using polynomial memristor model
The simulation results are shown in the waveforms of Figure 5.87. Figure 5.87a shows
the parametric graph of the flux and the current across the meminductor emulator. More-
over, Figure 5.87b shows the behavior of the IL-L characteristic curve of the meminductor
emulator.
152
5.3. Mem-elements emulator circuits
Ω=1
Ω=2-10
-2 -1 0 1 2-1.5
-1.0
-0.5
0.0
0.5
1.0
ILHΜAL
jH1
0-8
WL
(a) IL-ϕ characteristic curves
Ω=1
Ω=2
Ω=5Ω=10
-2 -1 0 1 24.5
5.0
5.5
6.0
6.5
7.0
ILHΜAL
Indu
ctan
ceHm
HL
(b) IL-L characteristic curves
Figure 5.88: Meminductor characteristic-curves using HPM memristor model
In the case of using HPM memristor model, the simulation results are shown in the
waveforms of Figure 5.88. Figure 5.88a shows the parametric graph of the flux and the
current across the meminductor emulator. Moreover, Figure 5.88b shows the behavior of
the IL-L characteristic curve of the meminductor emulator.
153
Chapter 5. Circuit applications
154
Bibliography
[1] C. Verhoeven and G. Monna, Structured Electronic Design: Negative-Feedback Am-
plifiers. Springer, 2003.
[2] G. Palumbo and S. Pennisi, Feedback Amplifiers: Theory and Design. Springer, 2002.
[3] J. Stoffels, Automation in High-performance Negative Feedback Amplifier Design.
Technische Universiteit Delft, 1988.
[4] E. Nordholt, Design of high-performance negative-feedback amplifiers. Studies in elec-
trical and electronic engineering, Elsevier Scientific Pub. Co., 1983.
[5] F. Maloberti, Analog Design for CMOS VLSI Systems. The Kluwer international
series in engineering and computer science.VLSI, computer architecture and digital
signal processing, Springer, 2001.
[6] S. P. Adhikari and H. Kim, “Why Are Memristor and Memistor Different Devices?,”
IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 11,
pp. 2611–2618.
[7] B. Mouttet, “Memristive systems analysis of 3-terminal devices,” in Electronics, Cir-
cuits, and Systems (ICECS), 2010 17th IEEE International Conference on, pp. 930–
933, 2010.
[8] Q. Yu, Z. Qin, J. Yu, and Y. Mao, “Transmission characteristics study of memristors
based op-amp circuits,” in Communications, Circuits and Systems, 2009. ICCCAS
2009. International Conference on, pp. 974–977, 2009.
[9] T. Wey and W. Jemison, “Variable gain amplifier circuit using titanium dioxide mem-
ristors,” Circuits, Devices Systems, IET, vol. 5, no. 1, pp. 59–65, 2011.
[10] M. Di Ventra, Y. Pershin, and L. Chua, “Circuit elements with memory: Memristors,
memcapacitors, and meminductors,” Proceedings of the IEEE, vol. 97, pp. 1717–1724,
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[11] Y. Pershin and M. Di Ventra, “Memristive circuits simulate memcapacitors and me-
minductors,” Electronics Letters, vol. 46, no. 7, pp. 517–518, 2010.
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156
Chapter 6
Conclusions
A methodology for the generation of circuit simulation-oriented memristor models in the
DC and time domains has been developed. In both cases the generated models are recast
in the form of current-voltage branch relationships.
The methodology for the generation of the DC models starts from the state variable
formulation of the memristive system given through a piece-wise linear (PWL) definition.
A numerical integration method yields the solution of the ODE which is treated by a curve
fitting procedure in order to obtain a polynomial i-v characteristic. The dependence of the
characteristics on the parameters of the PWL definition has been studied. As a result, the
problem of the occurrence of multiple DC operating points in memristive DC circuits has
been established. The multiple solutions of the DC memristive equilibrium equations have
been assessed by resorting to a homotopy method for two cases study, namely a single-
and two-memristor circuit.
The methodology for the generation of the time-domain models has been focused on
generating to two different models. Firstly, a polynomial model has been developed from
the ϕ-q branch relationship of a generic memristor. In second place, a semi-symbolical
memristor model has been developed from the ODE that expresses the linear drift mech-
anism of the device with the parameters of the HP memristor. This model has been
generated by using the homotopy perturbation method (HPM). The impact of the pa-
rameters of the HPM memristor model has been determined by symbolical and numerical
analyzes. Both, the polynomial and the HPM models comply with the properties and
fingerprints that a memristor must posses.
Both models have been used in circuit applications. When the memristor is used in
negative-feedback amplifier schemes, a new family of trans-mem amplification ratios has
been established. The results are presented in the form of pinched hysteresis loops for the
amplifier gains and the analysis of the composition of fundamental and harmonic terms
157
Chapter 6. Conclusions
in the case of using the HP memristor model in order to determine the total harmonic
distortion (THD). Another usage of these models has been in circuits for emulating mem-
capacitors and meminductors.
Future work
The methodology can be extended to memristors with different physical parameters be-
cause the starting definition in this work is based on the physical behavior of the device
by using the linear drift mechanism.
The memristor models should be able to unify the analysis domain of DC and time in
order to provide continuity in the simulation procedure with traditional elements. This
problem has been caused by using functional models which are approximations that depend
on the analysis domain and the validity of the variable range. However, from the computer-
aided circuit analysis point of view, the insertion of the memristor requires a reformulating
of the normal form equations by using the computationally efficient tableau approach,
where each memristor can be represented simply by three equations; namely, φ=v, q=i,
and φ=φ(q).
The insertion of the fundamental variables of the memristor, namely the electric charge
and the flux-linkage in the matrix representation employed in circuit simulators involves
a dramatic change in the paradigm of circuital simulation, and it clearly constitutes a
different approach from the current work.
158
Appendix A
Nonlinear boundary of HP
memristor
This appendix is focused on carrying out a simple proposal in order to tackle the problem
of the nonlinear boundary between the two regions of the HP memristor. This problem is
depicted in Figure A.1.
L
D
wTiO2−x
TiO2
Pt
Pt
D
wRON
ROFF
Figure A.1: The HP memristor structure using nonlinear boundary
The nonlinear boundary can be included into the following v-i relationship as a two-
dimensional problem
v(t) =
(
RON
(
wL+L
2
)
+ROFF
(
DL−
(
wL+L
2
)))
i(t) (A.1)
According to the expression of (A.1), the RON and ROFF parameters must be expressed
in units of area.
159
Appendix A. Nonlinear boundary of HP memristor
160
Appendix
B
Tra
nsie
ntmemristo
rmodel
Tab
leB.1:
RMSEvalu
esfor
thevariation
ofx0an
da.
x0 = 0.1 x0 = 0.2 x0 = 0.3a RMSE a RMSE a RMSE1 0.000409124 1 0.000683479 1 0.0007908611.5 0.000787793 1.5 0.00199432 1.5 0.001373672 0.0020628 2 0.0129016 2 0.001787372.5 0.00564836 2.5 0.0575805 2.5 0.01452663 0.0086075 3 0.18392 3 0.0879223.5 0.0121226 3.5 0.469092 3.5 0.319383
x0 = 0.4 x0 = 0.5 x0 = 0.6a RMSE a RMSE a RMSE1 0.00082034 1 0.000693107 1 0.000573981.5 0.00464389 1.5 0.00141201 1.5 0.002870392 0.0312958 2 0.00379775 2 0.01907882.5 0.133853 2.5 0.00430202 2.5 0.08261453 0.425554 3 0.0251001 3 0.268813.5 1.10884 3.5 0.13394 3.5 0.719861
x0 = 0.7 x0 = 0.8 x0 = 0.9a RMSE a RMSE a RMSE1 0.000386982 1 0.000221111 1 0.0000745421.5 0.000564692 1.5 0.000382429 1.5 0.0001022932 0.00121035 2 0.00170416 2 0.0001289812.5 0.00253369 2.5 0.00727705 2.5 0.0001997693 0.00276473 3 0.0239203 3 0.0005017763.5 0.0086793 3.5 0.0649502 3.5 0.00136137
161
Appendix
B.Tran
sientmem
ristormodel
Tab
leB.2:
RMSEvalu
esfor
thevariation
ofx0an
dA.
x0 = 0.1 x0 = 0.2 x0 = 0.3 x0 = 0.4 x0 = 0.5 x0 = 0.6 x0 = 0.7 x0 = 0.8 x0 = 0.9A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000278065 10 0.00048208 10 0.000584091 10 0.000596038 10 0.000538745 10 0.000437633 10 0.000310316 10 0.00017654 10 0.00005866815 0.000484012 15 0.000806414 15 0.00090117 15 0.00105733 15 0.000782525 15 0.000713037 15 0.00042297 15 0.000243087 15 0.000081934920 0.000787793 20 0.00199432 20 0.00137367 20 0.00464389 20 0.00141201 20 0.00287039 20 0.000564692 20 0.000382429 20 0.00010229325 0.00155968 25 0.00829826 25 0.00183512 25 0.020411 25 0.00309802 25 0.0124417 25 0.000969213 25 0.00113793 25 0.00012154630 0.00357184 30 0.0286619 30 0.00396581 30 0.0677394 30 0.00479793 30 0.0414709 30 0.00183941 30 0.0036525 30 0.00015129135 0.0067454 35 0.0790299 35 0.024592 35 0.182904 35 0.00427864 35 0.113467 35 0.00281501 35 0.010013 35 0.00024294540 0.0086075 40 0.18392 40 0.087922 40 0.425554 40 0.0251001 40 0.26881 40 0.00276473 40 0.0239203 40 0.00050177645 0.00780889 45 0.37745 45 0.238474 45 0.885969 45 0.0938475 45 0.571153 45 0.00502304 45 0.0513479 45 0.0010696250 0.0343299 50 0.704616 50 0.547012 50 1.69313 50 0.252615 50 1.11479 50 0.0215133 50 0.101321 50 0.0021584155 0.107208 55 1.22294 55 1.11777 55 3.02368 55 0.570556 55 2.03295 55 0.0607852 55 0.186816 55 0.0040889560 0.250248 60 2.00446 60 2.09674 60 5.11178 60 1.14941 60 3.50706 60 0.139217 60 0.325783 60 0.00732365 0.498526 65 3.1379 65 3.68084 65 8.25984 65 2.13184 65 5.77693 65 0.28126 65 0.542273 65 0.012499470 0.896945 70 4.73114 70 6.128 70 12.8502 70 3.71077 70 9.15191 70 0.521191 70 0.867698 70 0.020475375 1.50154 75 6.91371 75 9.7681 75 19.3577 75 6.13963 75 14.0229 75 0.90537 75 1.34219 75 0.032372880 2.38112 80 9.83958 80 15.0149 80 28.3632 80 9.74365 80 20.8755 80 1.49475 80 2.01607 80 0.049630485 3.61912 85 13.69 85 22.3787 85 40.5679 85 14.9319 85 30.3038 85 2.36763 85 2.95148 85 0.07405990 5.31549 90 18.6765 90 32.48 90 56.8086 90 22.2104 90 43.0254 90 3.62267 90 4.22404 90 0.10790495 7.58882 95 25.0441 95 46.0643 95 78.0741 95 32.196 95 59.8974 95 5.38207 95 5.92473 95 0.15391100 10.5785 100 33.0747 100 64.017 100 105.522 100 45.6316 100 81.9327 100 7.79511 100 8.16177 100 0.215395
162
Tab
leB.3:
RMSEvalu
esfor
thevariation
ofx0 ,
Aan
da.ω=1.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.0000761997 10 0.000168005 10 0.000278065 10 0.000340764 10 0.000140454 10 0.000301515 10 0.00048208 10 0.000579293 10 0.000183746 10 0.000380563 10 0.000584091 10 0.00068669720 0.000168005 20 0.000409124 20 0.000787793 20 0.00120177 20 0.000301515 20 0.000683479 20 0.00199432 20 0.00519635 20 0.000380563 20 0.000790861 20 0.00137367 20 0.0017644730 0.000278065 30 0.000787793 30 0.00357184 30 0.0067454 30 0.00048208 30 0.00199432 30 0.0286619 30 0.0790299 30 0.000584091 30 0.00137367 30 0.00396581 30 0.02459240 0.000409124 40 0.0020628 40 0.0086075 40 0.0121226 40 0.000683479 40 0.0129016 40 0.18392 40 0.469092 40 0.000790861 40 0.00178737 40 0.087922 40 0.31938350 0.000567576 50 0.00564836 50 0.0343299 50 0.1928 50 0.000988288 50 0.0575805 50 0.704616 50 1.7096 50 0.00102741 50 0.0145266 50 0.547012 50 1.7140660 0.000787793 60 0.0086075 60 0.250248 60 0.896945 60 0.00199432 60 0.18392 60 2.00446 60 4.73114 60 0.00137367 60 0.087922 60 2.0967 60 6.12870 0.00120177 70 0.0121226 70 0.896945 70 2.74939 70 0.00519635 70 0.469092 70 4.73114 70 11.0105 70 0.00176447 70 0.319383 70 6.128 70 17.20680 0.0020628 80 0.0765817 80 2.38112 80 6.75879 80 0.0129016 80 1.0248 80 9.83958 80 22.7515 80 0.00178737 80 0.890327 80 15.0149 80 41.097690 0.00357184 90 0.250248 90 5.31549 90 14.447 90 0.0286619 90 2.00446 90 18.6765 90 43.0904 90 0.00396581 90 2.09674 90 32.48 90 87.3805100 0.00564836 100 0.612127 100 10.5785 100 27.9996 100 0.0575805 100 3.61197 100 33.0747 100 76.3266 100 0.0145266 100 4.38596 100 64.0171 100 170.129
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000201575 10 0.00040228 10 0.000596038 10 0.000694675 10 0.000193944 10 0.000374608 10 0.000538745 10 0.00061553 10 0.000164469 10 0.000309973 10 0.000437633 10 0.00049882120 0.00040228 20 0.00082034 20 0.00464389 20 0.0128944 20 0.000374608 20 0.000693107 20 0.00141201 20 0.00242716 20 0.000309973 20 0.00057398 20 0.00287039 20 0.0078731530 0.000596038 30 0.00464389 30 0.0677394 30 0.182904 30 0.000538745 30 0.00141201 30 0.00479793 30 0.00427864 30 0.000437633 30 0.00287039 30 0.0414709 30 0.11346740 0.00082034 40 0.0312958 40 0.425554 40 1.10884 40 0.000693107 40 0.00379775 40 0.0251001 40 0.13394 40 0.00057398 40 0.0190788 40 0.26881 40 0.71986150 0.00159178 50 0.133853 50 1.69313 50 4.31444 50 0.000907294 50 0.00430202 50 0.252615 50 0.919605 50 0.00102931 50 0.0826145 50 1.11479 50 2.9404360 0.00464389 60 0.425554 60 5.11178 60 12.8502 60 0.00141201 60 0.0251001 60 1.14941 60 3.71077 60 0.00287039 60 0.26881 60 3.50706 60 9.1519170 0.0128944 70 1.10884 70 12.8502 70 32.0376 70 0.00242716 70 0.13394 70 3.71077 70 11.2742 70 0.00787315 70 0.719861 70 9.15191 70 23.699380 0.0312958 80 2.50885 80 28.3632 80 70.3595 80 0.00379775 80 0.441367 80 9.74365 80 28.5236 80 0.0190788 80 1.67518 80 20.8755 80 53.753190 0.0677394 90 5.11178 90 56.8086 90 140.497 90 0.00479793 90 1.14941 90 22.2104 90 63.4014 90 0.0414709 90 3.50706 90 43.0254 90 110.318100 0.133853 100 9.60829 100 105.522 100 260.512 100 0.00430202 100 2.58071 100 45.6316 100 127.909 100 0.0826145 100 6.76163 100 81.9327 100 209.398
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000119491 10 0.000222123 10 0.000310316 10 0.000349828 10 0.0000678299 10 0.000126096 10 0.00017654 10 0.000199359 10 0.0000217025 10 0.0000411466 10 0.000058668 10 0.000066795620 0.000222123 20 0.000386982 20 0.000564692 20 0.000784894 20 0.000126096 20 0.000221111 20 0.000382429 20 0.000759171 20 0.0000411466 20 0.000074542 20 0.000102293 20 0.00011495630 0.000310316 30 0.000564692 30 0.00183941 30 0.00281501 30 0.00017654 30 0.000382429 30 0.0036525 30 0.010013 30 0.000058668 30 0.000102293 30 0.000151291 30 0.00024294540 0.000386982 40 0.00121035 40 0.00276473 40 0.0086793 40 0.000221111 40 0.00170416 40 0.0239203 40 0.0649502 40 0.000074542 40 0.000128981 40 0.000501776 40 0.0013613750 0.000460415 50 0.00253369 50 0.0215133 50 0.107395 50 0.000269078 50 0.00727705 50 0.101321 50 0.272173 50 0.0000890015 50 0.000199769 50 0.00215841 50 0.0060647960 0.000564692 60 0.00276473 60 0.139217 60 0.521191 60 0.000382429 60 0.0239203 60 0.325783 60 0.867698 60 0.000102293 60 0.000501776 60 0.007323 60 0.020475370 0.000784894 70 0.0086793 70 0.521191 70 1.74979 70 0.000759171 70 0.0649502 70 0.867698 70 2.29537 70 0.000114956 70 0.00136137 70 0.0204753 70 0.056875180 0.00121035 80 0.0443066 80 1.49475 80 4.73095 80 0.00170416 80 0.153377 80 2.01607 80 5.30396 80 0.000128981 80 0.00332668 80 0.049630 80 0.13703290 0.00183941 90 0.139217 90 3.62267 90 11.0418 90 0.0036525 90 0.325783 90 4.22404 90 11.0627 90 0.000151291 90 0.007323 90 0.107904 90 0.296322100 0.00253369 100 0.348343 100 7.79511 100 23.1384 100 0.00727705 100 0.636856 100 8.16177 100 21.2956 100 0.000199769 100 0.0147975 100 0.215395 100 0.588657
163
Appendix
B.Tran
sientmem
ristormodel
Tab
leB.4:
RMSEvalu
esfor
thevariation
ofx0 ,A
anda.ω=10.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.0000698368 10 0.000141 10 0.000213513 10 0.000250285 10 0.000131107 10 0.000264226 10 0.00039937 10 0.000467705 10 0.000176231 10 0.000354186 10 0.00053383 10 0.00062427420 0.000141 20 0.000287403 20 0.000439412 20 0.000517584 20 0.000264226 20 0.00053655 20 0.000817057 20 0.000960399 20 0.000354186 20 0.000715127 20 0.00108252 20 0.0012685330 0.000213513 30 0.000439412 30 0.000678388 30 0.000803 30 0.00039937 30 0.000817057 30 0.00125329 30 0.0014784 30 0.00053383 30 0.00108252 30 0.00164494 30 0.0019308940 0.000287403 40 0.000597233 40 0.000931147 40 0.00110766 40 0.00053655 40 0.00110581 40 0.00170817 40 0.00202177 40 0.000715127 40 0.00145602 40 0.00221976 40 0.0026091150 0.000362694 50 0.000761075 50 0.0011984 50 0.00143271 50 0.000675776 50 0.00140284 50 0.00218166 50 0.0025903 50 0.000898038 50 0.00183524 50 0.00280545 50 0.0033005960 0.000439412 60 0.000931147 60 0.00148088 60 0.00177929 60 0.000817057 60 0.00170817 60 0.00267354 60 0.00318341 60 0.00108252 60 0.00221976 60 0.00340029 60 0.0040024170 0.000517584 70 0.00110766 70 0.00177929 70 0.0021485 70 0.000960399 70 0.00202177 70 0.00318341 70 0.00380017 70 0.00126853 70 0.00260911 70 0.00400241 70 0.0047114380 0.000597233 80 0.00129084 80 0.00209435 80 0.00254151 80 0.00110581 80 0.00234359 80 0.00371066 80 0.00443926 80 0.00145602 80 0.0030028 80 0.00460983 80 0.005424390 0.000678388 90 0.00148088 90 0.00242675 90 0.00295928 90 0.00125329 90 0.00267354 90 0.00425448 90 0.00509888 90 0.00164494 90 0.00340029 90 0.00522043 90 0.00613761100 0.000761075 100 0.00167801 100 0.00277714 100 0.00340277 100 0.00140284 100 0.00301148 100 0.0048138 100 0.00577685 100 0.00183524 100 0.00380102 100 0.00583207 100 0.00684795
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000199547 10 0.00039969 10 0.000600357 10 0.000700863 10 0.000198141 10 0.000395469 10 0.000591915 10 0.000689787 10 0.000172582 10 0.000343363 10 0.000512321 10 0.00059611120 0.00039969 20 0.000801472 20 0.00120475 20 0.00140675 20 0.000395469 20 0.000787414 20 0.00117532 20 0.0013676 20 0.000343363 20 0.000679439 20 0.00100809 20 0.0011695930 0.000600357 30 0.00120475 30 0.00181111 30 0.00211438 30 0.000591915 30 0.00117532 30 0.00174854 30 0.00203087 30 0.000512321 30 0.00100809 30 0.0014869 30 0.0017198840 0.000801472 40 0.0016089 40 0.00241736 40 0.00282044 40 0.000787414 40 0.00155869 40 0.00231011 40 0.00267736 40 0.000679439 40 0.0013292 40 0.00194857 40 0.0022467950 0.00100296 50 0.00201331 50 0.00302142 50 0.00352168 50 0.000981902 50 0.00193709 50 0.00285873 50 0.00330525 50 0.000844698 50 0.0016427 50 0.00239304 50 0.0027503660 0.00120475 60 0.00241736 60 0.00362128 60 0.00421501 60 0.00117532 60 0.00231011 60 0.00339334 60 0.00391308 60 0.00100809 60 0.00194857 60 0.00282041 60 0.0032309270 0.00140675 70 0.00282044 70 0.00421501 70 0.00489761 70 0.0013676 70 0.00267736 70 0.00391308 70 0.00449982 70 0.00116959 70 0.00224679 70 0.00323092 70 0.0036889280 0.0016089 80 0.00322194 80 0.00480085 80 0.0055670 80 0.00155869 80 0.00303854 80 0.00441732 80 0.00506482 80 0.0013292 80 0.00253739 80 0.00362485 80 0.0041250190 0.00181111 90 0.00362128 90 0.00537725 90 0.00622139 90 0.00174854 90 0.00339334 90 0.00490563 90 0.0056078 90 0.0014869 90 0.00282041 90 0.00400261 90 0.00453989100 0.00201331 100 0.00401789 100 0.00594291 100 0.00685943 100 0.00193709 100 0.00374153 100 0.00537779 100 0.00612886 100 0.0016427 100 0.00309594 100 0.00436463 100 0.00493447
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.00012757 10 0.000253258 10 0.000377082 10 0.000438303 10 0.0000726084 10 0.00014408 10 0.000214436 10 0.000249203 10 0.0000228015 10 0.0000453416 10 0.0000676248 10 0.000078671320 0.000253258 20 0.000499065 20 0.000737585 20 0.000854166 20 0.00014408 20 0.000283699 20 0.000419029 20 0.000485138 20 0.0000453416 20 0.0000896553 20 0.000132975 20 0.00015427330 0.000377082 30 0.000737585 30 0.00108208 30 0.00124852 30 0.000214436 30 0.000419029 30 0.000614342 30 0.000708687 30 0.0000676248 30 0.000132975 30 0.000196162 30 0.00022697840 0.000499065 40 0.000968992 40 0.0014112 40 0.00162237 40 0.000283699 40 0.000550236 40 0.000800911 40 0.000920682 40 0.0000896553 40 0.000175334 40 0.000257291 40 0.00029694850 0.000619225 50 0.00119347 50 0.00172556 50 0.00197675 50 0.00035189 50 0.000677478 50 0.000979239 50 0.0011219 50 0.000111437 50 0.000216762 50 0.000316459 50 0.00036433660 0.000737585 60 0.0014112 60 0.00202584 60 0.00231269 60 0.000419029 60 0.000800911 60 0.00114981 60 0.00131306 60 0.000132975 60 0.000257291 60 0.000373761 60 0.00042928270 0.000854166 70 0.00162237 70 0.00231269 70 0.00263124 70 0.000485138 70 0.000920682 70 0.00131306 70 0.00149485 70 0.000154273 70 0.000296948 70 0.000429282 70 0.00049191780 0.000968992 80 0.00182719 80 0.00258676 80 0.00293342 80 0.000550236 80 0.00103693 80 0.00146943 80 0.0016679 80 0.000175334 80 0.000335763 80 0.000483106 80 0.00055236390 0.00108208 90 0.00202584 90 0.0028487 90 0.00322021 90 0.000614342 90 0.00114981 90 0.00161932 90 0.00183281 90 0.000196162 90 0.000373761 90 0.000535309 90 0.000610735100 0.00119347 100 0.00221853 100 0.00309912 100 0.00349254 100 0.000677478 100 0.00125943 100 0.0017631 100 0.00199013 100 0.000216762 100 0.000410967 100 0.000585965 100 0.000667141
164
Tab
leB.5:
RMSEvalu
esfor
thevariation
ofx0 ,
Aan
da.ω=100.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000063667 10 0.000127452 10 0.000191354 10 0.00022335 10 0.000119713 10 0.000239607 10 0.000359682 10 0.000419787 10 0.000161309 10 0.000322777 10 0.000484403 10 0.00056527520 0.000127452 20 0.000255375 20 0.000383773 20 0.00044815 20 0.000239607 20 0.000479938 20 0.000720992 20 0.000841792 20 0.000322777 20 0.000646187 20 0.000970226 20 0.0011324830 0.000191354 30 0.000383773 30 0.000577261 30 0.000674408 30 0.000359682 30 0.000720992 30 0.00108393 30 0.00126602 30 0.000484403 30 0.000970226 30 0.00145746 30 0.001701640 0.000255375 40 0.000512646 40 0.000771825 40 0.000902134 40 0.000479938 40 0.000962772 40 0.00144851 40 0.00169247 40 0.000646187 40 0.00129489 40 0.0019461 40 0.0022726350 0.000319515 50 0.000641996 50 0.00096747 50 0.00113134 50 0.000600375 50 0.00120528 50 0.00181472 50 0.00212115 50 0.000808128 50 0.00162018 50 0.00243613 50 0.0028455560 0.000383773 60 0.000771825 60 0.0011642 60 0.00136203 60 0.000720992 60 0.00144851 60 0.00218258 60 0.00255207 60 0.000970226 60 0.0019461 60 0.00292755 60 0.0034203570 0.00044815 70 0.000902134 70 0.00136203 70 0.00159421 70 0.000841792 70 0.00169247 70 0.00255207 70 0.00298523 70 0.00113248 70 0.00227263 70 0.00342035 70 0.0039970280 0.000512646 80 0.00103293 80 0.00156095 80 0.00182791 80 0.000962772 80 0.00193716 80 0.00292321 80 0.00342063 80 0.00129489 80 0.00259979 80 0.00391452 80 0.0045755390 0.000577261 90 0.0011642 90 0.00176098 90 0.00206311 90 0.00108393 90 0.00218258 90 0.003296 90 0.00385829 90 0.00145746 90 0.00292755 90 0.00441005 90 0.00515588100 0.000641996 100 0.00129596 100 0.00196212 100 0.00229984 100 0.00120528 100 0.00242872 100 0.00367045 100 0.00429819 100 0.00162018 100 0.00325593 100 0.00490694 100 0.00573805
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000183203 10 0.000366466 10 0.000549788 10 0.000641471 10 0.000182497 10 0.000364925 10 0.000547285 10 0.000638439 10 0.000159417 10 0.000318673 10 0.000477769 10 0.00055725620 0.000366466 20 0.000733168 20 0.0011001 20 0.00128365 20 0.000364925 20 0.000729576 20 0.00109395 20 0.00127603 20 0.000318673 20 0.000636703 20 0.000954087 20 0.0011125430 0.000549788 30 0.0011001 30 0.00165093 30 0.00192652 30 0.000547285 30 0.00109395 30 0.00163997 30 0.00191274 30 0.000477769 30 0.000954087 30 0.00142895 30 0.0016658340 0.000733168 40 0.00146726 40 0.00220224 40 0.00257005 40 0.000729576 40 0.00145804 40 0.00218534 40 0.00254854 40 0.000636703 40 0.00127082 40 0.00190235 40 0.0022171350 0.000916606 50 0.00183464 50 0.00275403 50 0.0032142 50 0.000911797 50 0.00182183 50 0.00273003 50 0.00318342 50 0.000795476 50 0.00158691 50 0.00237428 50 0.0027664360 0.0011001 60 0.00220224 60 0.00330627 60 0.00385895 60 0.00109395 60 0.00218534 60 0.00327404 60 0.00381733 60 0.000954087 60 0.00190235 60 0.00284473 60 0.0033137170 0.00128365 70 0.00257005 70 0.00385895 70 0.00450427 70 0.00127603 70 0.00254854 70 0.00381733 70 0.00445027 70 0.00111254 70 0.00221713 70 0.00331371 70 0.0038589780 0.00146726 80 0.00293806 80 0.00441205 80 0.00515013 80 0.00145804 80 0.00291144 80 0.00435991 80 0.00508219 80 0.00127082 80 0.00253126 80 0.0037812 80 0.004402290 0.00165093 90 0.00330627 90 0.00496555 90 0.00579651 90 0.00163997 90 0.00327404 90 0.00490174 90 0.00571308 90 0.00142895 90 0.00284473 90 0.0042472 90 0.0049434100 0.00183464 100 0.00367468 100 0.00551943 100 0.00644337 100 0.00182183 100 0.00363631 100 0.00544282 100 0.0063429 100 0.00158691 100 0.00315755 100 0.00471171 100 0.00548255
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5
A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE A RMSE10 0.000118077 10 0.000235982 10 0.000353716 10 0.000412519 10 0.0000672356 10 0.000134367 10 0.000201393 10 0.000234868 10 0.0000210739 10 0.0000421238 10 0.0000631498 10 0.000073653820 0.000235982 20 0.000471279 20 0.000705891 20 0.000822941 20 0.000134367 20 0.000268316 20 0.00040185 20 0.000468461 20 0.0000421238 20 0.0000841519 20 0.000126085 20 0.00014701530 0.000353716 30 0.000705891 30 0.00105653 30 0.00123127 30 0.000201393 30 0.00040185 30 0.000601374 30 0.000700788 30 0.0000631498 30 0.000126085 30 0.000188805 30 0.00022008640 0.000471279 40 0.00093982 40 0.00140563 40 0.00163752 40 0.000268316 40 0.000534969 40 0.000799971 40 0.000931857 40 0.0000841519 40 0.000167922 40 0.000251313 40 0.00029286750 0.000588671 50 0.00117307 50 0.00175321 50 0.00204169 50 0.000335135 50 0.000667676 50 0.000997647 50 0.00116168 50 0.00010513 50 0.000209665 50 0.000313609 50 0.00036536160 0.000705891 60 0.00140563 60 0.00209926 60 0.0024438 60 0.00040185 60 0.00079997 60 0.00119441 60 0.00139026 60 0.000126085 60 0.000251313 60 0.000375694 60 0.0004375770 0.000822941 70 0.00163752 70 0.0024438 70 0.00284384 70 0.000468461 70 0.000931857 70 0.00139026 70 0.0016176 70 0.000147015 70 0.000292867 70 0.00043757 70 0.00050949480 0.00093982 80 0.00186873 80 0.00278682 80 0.00324183 80 0.000534969 80 0.00106334 80 0.0015852 80 0.00184372 80 0.000167922 80 0.000334328 80 0.000499236 80 0.00058113690 0.00105653 90 0.00209926 90 0.00312833 90 0.00363777 90 0.000601374 90 0.00119441 90 0.00177924 90 0.00206863 90 0.000188805 90 0.000375694 90 0.000560695 90 0.000652497100 0.00117307 100 0.00232912 100 0.00346833 100 0.00403167 100 0.000667676 100 0.00132507 100 0.00197239 100 0.00229232 100 0.000209665 100 0.000416968 100 0.000621948 100 0.000723578
165
Appendix
B.Tran
sientmem
ristormodel
Tab
leB.6:
Ran
geof
theparam
etersx0 ,
Aan
da.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-60 10-40 10-30 10-70 10-30 10-20 10-20 10-90 10-40 10-30 10-20x0 = 0.4 x0 = 0.5 x0 = 0.6
a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-60 10-30 10-20 10 10-100 10-50 10-30 10-30 10-70 10-30 10-20 10-20x0 = 0.7 x0 = 0.8 x0 = 0.9
a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-70 10-40 10-40 10-100 10-50 10-30 10-20 10-100 10-90 10-60 10-50
166
Tab
leB.7:
Ran
geof
mem
ristance
usin
gdifferen
tparam
etervalu
es.ω=1.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-60 10-40 10-30 10-70 10-30 10-20 10-20 10-90 10-40 10-30 10-20M M M M M M M M M M M M
14.41-14.067 14.41-13.656 14.41-13.167 14.41-12.89 12.82-12.188 12.82-11.466 12.82-10.662 12.82-10.235 11.23-10.403 11.23-9.52 11.23-8.613 11.23-8.16214.41-7.451 14.41-5.806 14.41-5.806 14.41-6.988 12.82-7.324 12.82-8.054 12.82-8.054 12.82-7.324 11.23-4.072 11.23-4.776 11.23-4.072 11.23-5.422
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-60 10-30 10-20 10 10-100 10-50 10-30 10-30 10-70 10-30 10-20 10-20M M M M M M M M M M M M
9.64-8.733 9.64-7.832 9.64-6.972 9.64-6.563 8.05-7.177 8.05-6.367 8.05-5.637 8.05-5.304 6.46-5.72 6.46-5.068 6.46-4.504 6.46-4.2549.64-4.668 9.64-4.668 9.64-4.668 8.05-2.635 8.05-2.635 8.05-2.782 8.05-2.66 6.46-3.27 6.46-3.365 6.46-3.365 6.46-3.27
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-70 10-40 10-40 10-100 10-50 10-30 10-20 10-100 10-90 10-60 10-50M M M M M M M M M M M M
4.87-4.332 4.87-3.872 4.87-3.48 4.87-3.307 3.28-2.974 3.28-2.713 3.28-2.488 3.28-2.387 1.69-1.592 1.69-1.505 1.69-1.426 1.69-1.394.87-2.002 4.87-1.078 4.87-1.723 4.87-1.078 3.28-1.244 3.28-1.2424 3.28-1.482 3.28-1.819 1.69-1.050 1.69-1.101 1.69-1.101 1.69-1.061
167
Appendix
B.Tran
sientmem
ristormodel
Tab
leB.8:
Ran
geof
mem
ristance
usin
gdifferen
tparam
etervalu
es.ω=10.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
14.41-14.378 14.41-14.346 14.41-14.313 14.41-14.297 12.82-12.76 12.82-12.701 12.82-12.64 12.82-12.609 11.23-11.15 11.23-11.07 11.23-10.989 11.23-10.94814.41-14.067 14.41-13.656 14.41-13.167 14.41-12.89 12.82-12.188 12.82-11.466 12.82-10.662 12.82-10.235 11.23-10.403 11.23-9.52 11.23-8.613 11.23-8.162
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
9.64-9.55 9.64-9.459 9.64-9.369 9.64-9.324 8.05-7.96 8.05-7.871 8.05-7.783 8.05-7.739 6.46-6.382 6.46-6.305 6.46-6.229 6.46-6.1919.64-8.733 9.64-7.832 9.64-6.972 9.64-6.563 8.05-7.177 8.05-6.367 8.05-5.637 8.05-5.304 6.46-5.72 6.46-5.068 6.46-4.504 6.46-4.254
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
4.87-4.812 4.87-4.755 4.87-4.7 4.87-4.672 3.28-3.247 3.28-3.215 3.28-3.183 3.28-3.167 1.69-1.679 1.69-1.669 1.69-1.659 1.69-1.6544.87-4.332 4.87-3.872 4.87-3.48 4.87-3.307 3.28-2.974 3.28-2.713 3.28-2.488 3.28-2.387 1.69-1.592 1.69-1.505 1.69-1.426 1.69-1.39
168
Tab
leB.9:
Ran
geof
mem
ristance
usin
gdifferen
tparam
etervalu
es.ω=100.
x0 = 0.1 x0 = 0.2 x0 = 0.3a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
14.41-14.4 14.41-14.4 14.41-14.4 14.41-14.39 12.82-12.814 12.82-12.808 12.82-12.802 12.82-12.799 11.23-11.222 11.23-11.214 11.23-11.206 11.23-11.20214.41-14.378 14.41-14.346 14.41-14.313 14.41-14.297 12.82-12.76 12.82-12.701 12.82-12.64 12.82-12.609 11.23-11.15 11.23-11.07 11.23-10.989 11.23-10.948
x0 = 0.4 x0 = 0.5 x0 = 0.6a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
9.64-9.631 9.64-9.622 9.64-9.613 9.64-9.608 8.05-8.041 8.05-8.032 8.05-8.023 8.05-8.018 6.46-6.452 6.46-6.444 6.46-6.436 6.46-6.4329.64-9.56 9.64-9.459 9.64-9.369 9.64-9.324 8.05-7.96 8.05-7.871 8.05-7.783 8.05-7.739 6.46-6.382 6.46-6.305 6.46-6.229 6.46-6.191
x0 = 0.7 x0 = 0.8 x0 = 0.9a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5 a = 1 a = 2 a = 3 a = 3.5A A A A A A A A A A A A
10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100 10-100M M M M M M M M M M M M
4.87-4.864 4.87-4.858 4.87-4.852 4.87-4.849 3.28-3.276 3.28-3.273 3.28-3.27 3.28-3.268 1.69-1.688 1.69-1.687 1.69-1.686 1.69-1.6864.87-4.812 4.87-4.755 4.87-4.7 4.87-4.672 3.28-3.247 3.28-3.215 3.28-3.183 3.28-3.167 1.69-1.679 1.69-1.669 1.69-1.699 1.69-1.654
169