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Modeling of Memristive Systems and Its Applications Modeling of memristive systems and its applications By Jose´ Balaam Alarcon´ Angulo Thesis submitted as a partial requirement for the degree of Master in Science with specialty in Electronics at the Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ San Andres´ Cholula, Puebla Adviser: Dr. Librado Arturo Sarmiento Reyes, INAOE c INAOE 2017 The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole or in part Modeling of memristive systems and its applications Master's Thesis By: Jos´eBalaam Alarc´onAngulo Adviser: Dr. Librado Arturo Sarmiento Reyes Instituto Nacional de Astrof´ısica Optica´ y Electr´onica Electronics Department San Andres´ Cholula, Puebla. January 16, 2018 i Agradezco a mis padres Guadalupe Alarc´ony Ma. Teresa y a mi hermana Dayanira Alarc´onpor su incondicional apoyo a lo largo de toda mi vida. Al doctor Arturo Sarmiento por su orientaci´ondurante el desarrollo de esta tesis.Y a todos mis amigos que siempre est´anah´ıe hicieron invaluables aportes para el desarrollo de este trabajo. Modeling of memristive systems and its applications iii To my future readers. I hope you will enjoy reading this work as much as I did when writing it and that it will serves as a guide for future work. Modeling of memristive systems and its applications Resumen Desde el advenimiento del memristor como elemento b´asicode circuito real, ha habido un significativo impulso en la investigaci´onorientada al desarrollo de aplicaciones del memristor en dise~node circuitos y procesamiento de se~nales.Amplificadores con retroalimentac´onnegativa basados en nullores se encuentran entre las aplicaciones m´as factibles debido a que la propiedad de memoria del memristor puede ser incorporada a la transferencia de amplificaci´onal colocar el memristor directamente en el lazo de retroalimentaci´on. Esta tesis introduce un modelo del memristor que resulta ad hoc para el an´alisis de amplificadores de retroalimentaci´onnegativa basados en nullores. El modelo ha sido desarrollado utilizando el m´etodo de homotop´ıamodificado que resulta en una expresi´ontotalmente simb´olicade la memristancia con una estructura de funciones arm´onicas.El modelo desarrollado puede ser codificado en Verilog-A para fines de simulaci´onel´ectricade los amplificadores. Se ha realizado el an´alisisde distorsi´on arm´onicay el an´alisisde ruido para los cuatro tipos de amplificadores basados en nullores, de voltaje, de transmemristancia, de transmemductancia y de corriente. En un paso posterior, el nullor se implementa con un memistor, el cual consiste en la conexi´onanti-serie de dos memristores. Obteni´endosede esta manera, un amplificador puramente memristivo. Finalmente un amplificador de transmemristencia se analiza como caso de estudio. [v] Abstract Since the advent of the memristor as an actual basic element, a significant thrust of the research has been oriented to develop applications of the memristor in circuit design and signal processing. Nullor-based negative-feedback amplifiers are among the most directly feasible applications due to the fact that the memory property of the memristor can be incorporated to the overall transfer gain by placing a memristive feedback. This thesis introduces a tailored memristor model for the analysis of nullor-based negative-feedback memristive amplifiers. The model has been developed by using a modified homotopy method that yields a fully symbolic harmonic expression of the memristance. This model can be recast in a piece of Verilog-A code for electric simu- lation of the amplifiers. Harmonic distortion and noise analyses are carried out for the four types of nullor-based memristive amplifiers, namely, voltage, transmemristance, transmemductance and current amplifiers. In a further step, the nullor is implemented by a memistor which consists in the back-to-back connection of two memristors. As a result, it yields a full memristive amplifier. Finally, a transmemristance memistor- based configuration is tackled as a case study. [vii] Contents Resumenv Abstract vii List of Figures xiii List of Tables xv 1 Introduction1 1.1 Objective.................................2 1.2 Hypothesis.................................2 1.3 Methodology of the research.......................2 2 Fundamentals5 2.1 Memristor and memristive systems...................5 2.2 The Nullor.................................9 2.3 Nullor-based amplifiers.......................... 10 3 Memristor model generation 13 3.1 Modelling methodology.......................... 13 3.2 Model characterization.......................... 16 4 The memistor 23 4.1 Introduction................................ 24 4.2 Memistor with HPM memristor model................. 24 4.2.1 Characterization......................... 25 [ix] x CONTENTS 5 Memristive nullor-based amplifiers 31 5.1 Basic topologies.............................. 32 5.1.1 Memristive voltage amplifier................... 32 5.1.2 Transmemristance amplifier................... 33 5.1.3 Transmemductance amplifier................... 35 5.1.4 Memristive current amplifier................... 36 5.1.5 Summary............................. 37 6 Harmonic analysis 39 6.1 Introduction................................ 39 6.2 Basic amplifiers & symbolic harmonic analysis............. 40 6.2.1 Memristive voltage amplifier................... 40 6.2.2 Transmemristance amplifier................... 41 6.2.3 Transmemductance amplifier................... 42 6.2.4 Memristive current amplifier................... 43 6.3 Simulation................................. 44 6.3.1 Memristive voltage amplifier................... 44 6.3.2 Transmemristance amplifier................... 44 6.3.3 Transmemductance amplifier................... 45 7 Noise Analysis 47 7.1 Introduction................................ 47 7.1.1 Thermal noise........................... 47 7.2 Basic Amplifiers.............................. 49 8 Case of study 53 8.1 Noise in memistor............................. 53 8.2 Transmemristance amplifier....................... 55 8.2.1 Noise simulations......................... 55 8.2.2 Harmonic analysis........................ 56 9 Conclusions and future work 59 Appendices 62 Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ CONTENTS xi A Appendix A 65 A.1 Code for Order-1k=7 ........................... 65 A.2 Code for Order-3k=1 ........................... 66 B Appendix B 67 B.1 Positive-gain transmemristance amplifier................ 67 B.2 Positive-gain transmemductance amplifier............... 69 Bibliography 71 Modeling of memristive systems and its applications xii CONTENTS Electronics Department Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ List of Figures 2.1 Basic circuit elements...........................6 2.2 Struture of HP memristor.........................7 2.3 Basic fingerprint for memristors......................7 2.4 Structure of a nullor............................9 2.5 Single-loop configuration of nullor-based negative-feedback amplifiers. 10 3.1 Joglekar window with some discrete values to k............. 14 3.2 Equivalent circuit of the coupled ohmic-tunnelin variable-resistor cir- cuit model, where Ron is de ohmic resistance and Roff is the tunnelling resistance.................................. 15 3.3 Pinched Hysteresis Loop for both models and differents values of !.. 17 3.4 M-I characteristics for several values of !................ 18 3.5 Memristance vs frequency......................... 19 3.6 Current & voltage of the memristor for ! = 1.............. 20 3.7 Characterization of memristance vs Xo.................. 20 3.8 Passivity vs Xo .............................. 21 3.9 Monotonicity vs Xo ............................ 21 4.1 The Widrow memistor........................... 23 4.2 Memristor-based memistor........................ 24 4.3 Circuit for the characterization of the memistor............. 25 4.4 Ix for several cases............................ 26 4.5 Ix for several cases............................ 27 π 4.6 Ix-VD characteristic for t = 0; 2 ; π..................... 28 π 4.7 Ix-VG characteristic for t = 0; 2 ; π..................... 29 [xiii] xiv LIST OF FIGURES 5.1 Basic topologies for the nullor as ideal amplifier............ 31 5.2 Pinched hysteresis transfer loop for the voltage amplifier at ! = 1.. 32 5.3 Voltage gain for ! = 1.......................... 33 5.4 Pinched hysteresis transfer loop for the transmemristance amplifier at ! = 1.................................... 34 5.5 Gain for ! = 1.............................. 34 5.6 Pinched hysteresis transfer loop for the transmemductance amplifier at ! = 1.................................. 35 5.7 Gain for ! = 1.............................. 36 5.8 Pinched hysteresis transfer loop for the voltage amplifier at ! = 1.. 36 5.9 Current gain for ! = 1.......................... 37 6.1 Voltage amplifier with input output voltage for ! = 1.......... 40 6.2 Transmemristance amplifier with input current and output voltage for ! = 1.................................... 42 6.3 Transmemductance amplifier with input voltage and output current for ! = 1.................................. 43 6.4 Memristive current amplifier with input and output current for ! = 1. 43 7.1 Noise model of a resistor......................... 48 7.2 Noise model of a memristor....................... 49 7.3 Basice memristive amplifiers: noisy configurations............ 50 8.1 Nullor implementation with memistor.................. 53 8.2 Memistor with noise contributions.................... 54 8.3 Transmemristance amplifier with memistor..............
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