비선형 멤리스터 모델을 사용한 부분활성화 이론 기반 발진기 설계 방법

국문 초록 (Abstract)

전기 기본 소자에는 전자회로 설계와 신경회로망 응용에 사용되는 저항, 인덕터 그리고 커패시터 등 3가지 소자 외에도 4 번 째의 전기소자라고 할 수 있는 멤리스터가 있다. 멤리스터는 자기 장과 전하 사이의 비선형 관계를 형성하는 소자로서 Leon Chua교수에 의해 1971년에 처음 그 존재가 알려졌다. 멤리스터의 발견 후 여러 종류의 멤리스터들이 발견되었는데, 상태 종속 오옴의 법칙과 상태 방정식에 따라 이상형(Ideal) 멤리스터, 이상형 일반 (Ideal Generic) 멤리스터, 일반 (Generic) 멤리스터 그리고, 확장 (Extended) 멤리스터 등 4 종류의 멤리리스터 등이 소개되었고, 이 4 개의 종류의 멤리스터로부터 상태방정식 만으로 결정되는 휘발성 및 비휘발성 멤리스터도 소개되었다.
본 논문에서는 여러 종류의 멤리스터와 그 들의 특징들을 연구하고 해석하였다. 또, POP (Power-Off Plot)와 멤리스터가 휘발성인지 비 휘발성인지 를 결정하는데 도움이 되는 Dynamic Route Map (DRM) 에 대해서도 예제와 함께 자세히 설명하였다.
또, 이 논문에서 두 개의 멤리스터 모델에 기반한 전기 진동기를 지역적 활동 원리 (Local Activity Principle)와 홉 발진 현상 (Hopf Bifurcation Phenomenon)에 의거하여 설계하였다. 이 논문의 첫 번 째 부분은 저주파수 멤리스터 모델에 기반한 전자 진동기를 지역활성형 PTC 멤리스터 모델과 밧데리, 그리고 인덕터를 직렬연결하여 설계할 수 있음을 보였다. PTC 멤리스터 모델은DC V-I 곡선의 부저항 영역 상에 있는 지역 활성화 영역에 존재한다. DC 동작점 Q는 PTC 멤리스터 모델의 지역활성화 영역 내에서 선택하고, Q점에서의 소신호 등가회로는 테일러 시리즈 를 전개하여 유도한다. 또, Q 점에서의 상기 소신호 등가회로와 외부 인덕터를 직렬 연결할 때, 이에 대한 총 임피던스 값 에서의 저항 값 성분이 0되게하는 조건을 만족하는 인덕터 값을 외부 인덕터의 값으로 정한다. 이 회로에서의 진동 주파수는 임피던스의 허수 부분도 0가 되는 조건에서 결정된다.
본 논문의 두 번 째 부분에서는 따개비의 근육섬유에 대한 3 차원 Morris-Lecar 모델을 단순화하여 간단한 3 차 멤리스터 Morris-Lecar 모델을 설계하였다. 또, 3차 Moris-Lecar 모델을 대형 따개비의 근육섬유의 3 차원 멤리스티브 Morris-Lecar 를 사용함으로써 칼슘과 칼륨이온에 관련된 다양한 진동동작을 분석하였다. Original third-order Morris-Lecar 모델에서의 칼슘과 칼륨 이온들은 대형 따개비의 근육섬유에서의 신호전달 활동전위 (Action Potential)에 관련이 있고, 이 칼슘과 칼륨 이온들은 사실 전기회로적 관점에서 볼 때 칼슘 멤리스터 와 칼륨 멤리스터들 이다. 즉, 3 차 Morris-Lecar 모델의 칼슘이온과 칼륨이온들은 멤리스터의 특징을 모두 포함하고 있기 때문에 시변 칼슘이온과 칼륨이온들은 시불변 칼슘 멤리스터 과 칼륨 멤리스터 라고 할 수 있다. 따개비의 근육에서 생성되는 활동전위 (Action Potential)는 생체시스템에서의 신호전달이 주 응용분야이다. 이 연구에서 2 차 와 3차 멤리스터의 Morris-Lecar 모델에 있어서의 작은 고정 크기의 오실레이션의 존재는 두 개의 부임계 Hopf bifurcation 점들과 관련된 점진 안정화된 균형점(asymptotically stable equilibrium point) 의 흡인영역(basin of attraction) 을 계산함으로서 수치 해석적으로 확인하였다. 또, 3 차 멤리스터 Morris-Lecar 모델의 오실레이션 발생에 대한 완전한 분석을 소신호 분석과 부임계 Hopf bifurcation 현상을 통하여 수행하였다.

다국어 초록 (Multilingual Abstract)

Memristor is the fourth basic electrical circuit element in addition to the typical three fundamental electrical circuit elements namely, resistor, inductor, and capacitor which is essential for electronic circuit design and neuromorphic applications. Memristor has the nonlinear relationship between magnetic flux and electric charge and it was first postulated by Prof. Leon Chua in 1971.
After the invention of memristors, four different classes of memristors namely, Ideal memristors, Ideal Generic memristors, Generic memristors and Extended memristors were introduced which are defined by the state-dependent Ohm’s law and the state equation and from the above four classes of memristors, another different class of memristors especially, Volatile and Non-volatile memristors were introduced which are determined by the state equation only. Hence, it is necessary to study and interpret the various classes of memristors and its characteristics. Several new concepts, including POP (Power-Off plot) and Dynamic Route Map (DRM) which helps to determine whether a memristor is volatile or non-volatile is explained in detail with examples.
In this thesis, we designed the two memristive-model based electronic oscillators via local activity principle and Hopf bifurcation phenomenon.
The first part of the thesis demonstrates the design of low-frequency memristive-model based electronic oscillator circuit by connecting an inductor in series with a locally-active PTC Memristor model and a battery. The PTC Memristor model used in this thesis represents the model of a physical device called Positive Temperature Coefficient (PTC) thermistor.
The PTC Memristor model is locally active on the negative resistance region of its DC VM -IM curve. A DC operating point Q is chosen on the locally-active region of the PTC Memristor model and a small-signal equivalent circuit at Q is derived via Taylor series. The small-signal admittance YCM(s; V) of the composite one-port of locally-active first-order PTC memristor oscillator is derived using the small-signal equivalent circuit at Q, in series with an inductor whose value is chosen such that Z_CM(iw_0 ; V) = 0 at some s = iw_0 . The sinusoidal oscillation computed numerically from this circuit is shown to emerge from a super-critical Hopf bifurcation.
The second part of the thesis describes the design of a simple third-order memristive Morris-Lecar model of the original third-order Morris-Lecar model of a barnacle muscle fiber. We presented the detailed analysis of various oscillatory behaviors (Action Potential) observed in relation to the calcium and potassium ions in the original third-order Morris-Lecar model via third-order memristive Morris-Lecar model of giant barnacle muscle fiber.
It reveals that the calcium and potassium ions in the original third-order Morris-Lecar model, which are responsible for generating action potential (AP) in giant barnacle muscle fiber are in fact calcium and potassium memristors in the perspective of electrical circuit theory. Since, both the calcium and potassium ions in the original third-order Morris-Lecar model exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the original third-order Morris-Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris-Lecar model. The main application of action potential generated in barnacle muscle fiber is used for signal transmission in biological systems.
We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order memristive Morris-Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two sub-critical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris-Lecar model via small-signal circuit analysis and a sub-critical Hopf bifurcation phenomenon.

목차 (Table of Contents)

• CHAPTER 1 1
• Introduction 1
• 1.1 Dissertation Objectives 2
• 1.2 Dissertation Organization 3
• CHAPTER 2 4
• Basics of Memristor and its types 4
• 2.1 Memristor 4
• 2.1.1 Memristor: The Fourth Basic Electrical Circuit Element 4
• 2.1.2 Memristor: Definition and Symbol 7
• 2.1.3 Fingerprints of Memristor 8
• 2.1.3.1 Pinched Hysteresis Loop 8
• 2.1.3.2 Frequency-dependent Pinched Hysteresis Loop 9
• 2.2 Types of Memristors 10
• 2.2.1 Types of Memristors : Based on its State-dependent Ohms law and State Equations 10
• 2.2.1.1 Ideal Memristor 10
• 2.2.1.2 Ideal Generic Memristor 13
• 2.2.1.3 Generic Memristor 13
• 2.2.1.4 Extended Memristor 16
• 2.2.2 Types of Memristors : Based on its Volatility 20
• 2.2.2.1 Power-Off Plot (POP) and Dynamic Route Map (DRM) 20
• 2.2.2.2 Volatile Memristors 22
• 2.2.2.3 Non-Volatile Memristors 23
• CHAPTER 3 26
• Conventional Electronic Oscillators and Memristor-Based Oscillators 26
• 3.1 Conventional Electronic Oscillators 26
• 3.1.1 Tunnel Diode Oscillator 27
• 3.1.2 Wein Bridge Oscillator 28
• 3.2 Memristor-Based Oscillators 30
• 3.2.1 First-Order and Second-Order Generic Memristors 30
• 3.2.1.1 Examples of First-Order Generic Memristors 30
• 3.2.1.2 Examples of Second-Order Generic Memristors 35
• 3.2.2 Generic PTC Memristor Oscillator 39
• 3.2.3 Generic Second-Order Memristor Oscillator 41
• CHAPTER 4 42
• Design of Low-Frequency PTC Memristor Oscillator 42
• 4.1 DC VM -IM Curve of the PTC Memristor 45
• 4.2 Small-Signal Equivalent Circuit of the PTC Memristor 46
• 4.3 Pole-Zero diagram of the admittance function YM(s ; Q) of the PTC Memristor 49
• 4.4. Frequency Response of the admittance function YM(s , Q) of the PTC Memristor 51
• 4.5 Computation of Eigen values of Jacobian Matrix of the PTC Memristor Oscillator 53
• 4.6 Computation of Composite Admittance Function YCM(s;VQ) of the Combined Inductor and the PTC Memristor One-Port 57
• 4.7 Pole-zero diagram of the Composite Admittance Function YCM(s;VQ) of the PTC Memristor Oscillator 59
• 4.8 Frequency Response of the Composite Admittance Function YCM(s;VQ) of the PTC Memristor Oscillator 65
• 4.9 Local Activity, Edge of Chaos and Hopf Bifurcation of the PTC Memristor Oscillator 68
• 4.9.1 Local Activity 68
• 4.9.2 Edge of Chaos 70
• 4.9.3 Hopf Bifurcation 70
• CHAPTER 5 79
• Design of Second-Order and Third-Order Memristive Morris-Lecar (M-L) Model of the Barnacle Muscle Fiber 79
• 5.1 Third-Order and Second-Order Memristive M-L Model 79
• 5.2 First-Order Potassium Ion-Channel Memristor 83
• 5.2.1 Pinched Hysteresis Loop of the First-order Potassium Ion-channel Memristor 85
• 5.3 First-Order Calcium Ion-Channel Memristor 87
• 5.3.1 Pinched Hysteresis Loop of the First-order Calcium Ion-channel Memristor 88
• 5.4 Third-Order Memristive and DC Memristive M-L Circuit Model 90
• 5.5 DC V-I Curve of Third-Order and Second-Order Memristive M-L Model 91
• 5.6 Small-Signal Equivalent Circuit of the Third-Order Memristive M-L Model 93
• 5.6.1 Small-signal Equivalent Circuit of the Calcium Ion-channel Memristor 93
• 5.6.2 Small-signal Equivalent Circuit of the Potassium Ion-channel Memristor 97
• 5.6.3 Small-signal Equivalent Circuit of the Third-order Memristive M-L Model 100
• 5.7 Frequency Response of the admittance function YM(s , Vm(Q)) of the Third-Order Memristive M-L Model 101
• 5.8 Pole-Zero diagram of the Small-Signal admittance function YM(s , Vm(Q)) of the Third-Order Memristive M-L Model 107
• 5.8.1 Comparison of Pole-Zero diagram of the Small-Signal admittance function YM(s , Vm(Q)) of the Third-Order and Second-Order Memristive M-L Model 110
• 5.9 Local Activity, Edge of Chaos and Hopf Bifurcation in the Third-Order Memristive M-L Model 110
• 5.9.1 Local Activity and Edge of Chaos in the Third-Order Memristive M-L Model 110
• 5.9.2 Hopf Bifurcation in the Third-Order Memristive M-L Model 118
• 5.9.2.1 Comparison of Hopf Bifurcation in the Third-Order and the Second-Order Memristive M-L Model 122
• 5.10 Oscillations Confirming the Existence of Globally Stable Equilibrium Point, and the Large Stable Limit Cycle 124
• 5.11 Subcritical Hopf Bifurcation and Small Unstable Limit Cycle Oscillation 127
• 5.11.1 Subcritical Hopf Bifurcation in the Second-order and the Third-order Memristive M-L Model 127
• 5.11.2 Small Unstable Limit Cycle Oscillation in the Second-order and the Third-order Memristive M-L Model 132
• CHAPTER 6 144
• Conclusions and Future Research Directions 144
• 6.1 Conclusions 144
• 6.2 Future Research Directions 146
• References 148