An Overview of Sinusoidal Oscillators Based on Memristive Devices

Abstract

Memristive devices include memristor, memcapacitor, and meminductor. Due to the adjustable resistance of the memristor, adjustable capacity of memcapacitor and adjustable inductance of meminductor, these devices can be used in the design of many analog circuits, including sinusoidal oscillators. Designing and implementation of a low-frequency voltage-controlled oscillator to achieve a wide tuning range, while meeting practical constraints such as small area and low power consumption, is a challenge. This challenge is overcome by replacing the resistors that occupy a large Silicon area in the conventional design with memristors, and hence smaller values of capacitances are used. Therefore, this chapter proposes and characterizes an overview of the implementation of memristive-based oscillators that are used in Electrical Neural Stimulation. In this chapter, an overview of the use of memristive devices in the design of sinusoidal oscillators and voltage-controlled oscillators is presented.

1. Introduction

An oscillator or waveform generator is essential in any electronic component. For example, sinusoidal oscillators are part of the frequency converter system in superheterodyne receivers. Oscillators are used in erasing and generating magnetism in magnetic recording and for timing clock pulses in digital circuits. Many electronic measuring devices, such as capacitance meters, have an oscillator. There are many types of sine wave oscillators, but they all consist of two basic parts: a frequency-determining part and a holding part. The frequency-determining part can be a resonant circuit or a capacitive-resistive network. The resonant circuit can be a combination of an inductor and a capacitor, a length of transmission line, or a cavity resonator, depending on the required frequency. Capacitive-resistive networks do not have a natural frequency, but their phase shift can be used to determine the oscillation frequency. The holding part provides the energy to the resonant circuit to keep it in the oscillating state. In many oscillators, this part can be an active component such as a FET or BJT transistor, a combination of these two, or a gain block, such as a high-bandwidth amplifier. Broadband amplifiers can, for example, cover frequencies from several hundreds of KHz to several GHz. Oscillators convert DC energy into RF energy and do this at a required frequency and with acceptable efficiency. The efficiency of a low-noise electronic oscillator can vary between 10 and 70%, depending on the frequency and circuit combination used. In many cases, when the goal is to have an output signal with stable frequency, clean, with low phase noise, and sufficient amplitude, efficiency can become a secondary issue [1, 2, 3]. The term stability refers to both short-term and long-term stability, and the purity of the oscillator means that unwanted and spurious responses should not occur in the circuit. Various noise sources such as noise generated by the transistors, modulated noise on the power supply, and noise caused by the frequency-regulating capacitors can generate noise in the oscillator and, as a result, cause phase noise destruction.

In different applications, oscillators should generally have various characteristics such as operating frequency or frequency range, phase noise, power consumption, frequency regulation range, output power, output power as a function of temperature, unwanted response, sensitivity to load changes, harmonic suppression, post-tuning drift, adjustment characteristic, adjustment linearity, adjustment sensitivity, and adjustment speed to improve system performance. On the other hand, with the issue of frequency modulation and demodulation in telecommunication transmitter and receiver circuits from years ago, the main need tended toward oscillators that can produce alternating oscillations with a pure sinusoidal waveform at a specific frequency. In a certain frequency, the three main requirements that must be met are phase noise, power consumption, and frequency adjustment range. Of course, meeting the first two requirements is more preferable to the third one [2, 4]. On the other hand, the history of phase noise, both in general and especially in oscillators, indicates that optimizing this factor has always been a primary goal. Because, phase noise precisely determines the purity of the resulting signal in oscillators and voltage-controlled oscillators, and hence, the accuracy of the circuits containing them [5].

In classical circuit theory, there are three circuit elements including: resistor, capacitor, and inductor, each of which has two terminals. The first element expresses the relationship between current and voltage, the second element expresses the relationship between the load and voltage, and the third element expresses the relationship between current and magnetic flux. These two-ended passive elements are the foundation of modern electronics and, therefore, are present in all circuits. On the other hand, these elements are not able to store information, even if only the state of one of the mentioned elements changes, if the circuit is turned off (disconnecting the power supply of the circuit) and a little time passes, the information of the new state will be lost [6]. In addition, if the information contains a range of interconnected values, it is possible to replace analog calculations with digital ones. On the other hand, in electronic oscillator design, it is emphasized that, in addition to establishing Moore’s law, they should go toward parts that are not only very small in nano size, but also have many capabilities [7]. Thus, it is necessary to expand all types of nanoscale memory cells in non-volatile memory [6, 7]. One of these circuit elements is the memristor, which was proposed by Chua in 1971 by analyzing the mathematical relationships between the basic variables of the circuit [6]. A few years later, solid-state memristors were developed in HP laboratories using thin films of titanium. In these elements, the resistance changes according to the voltage applied to the system, through the displacement of the atomic deficiency, which consists of the lack of oxygen atoms (oxygen vacancies) in certain areas of the film [8, 9]. Due to this effect, when the power supply is off, the oxygen vacancy cannot easily return to its original position and the system remains in the new resistance state. Recently, extensive research has been conducted in the field of modeling and implementation of nanoscale memory resistors (memristors) [6, 7, 8, 9, 10], modeling and analysis of memristor-based oscillators, and analysis and design of memristor-based functional circuits [11, 12, 13]. Recently, many oscillator circuits have been designed using memristive devices [5, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In [5, 14, 15, 16, 17, 18, 19, 20], the authors propose the well-known Wien and phase-shift oscillators using memristors, in which the resistors are replaced by memristors. They also show that the poles of the system oscillate, however, sustained oscillation is achieved due to the ring oscillator properties of the memristor [22]. The most recent memristor-based oscillator is based on replacing a reactive element such as a capacitor with a memristor [29, 30, 31, 32, 33, 34]. As the increasing and decreasing of the memristor resistance can represent the charging and discharging of the capacitor. Therefore, the memristor is applied as a resistance storing element, and it can replace the energy-storing elements. In this chapter, the performance of memristive elements is described, and finally, the application of these memristor elements in the design of non-linear oscillators is discussed.

2. Memristive device

2.1 Memristor

As we know, the basic elements in electrical circuits are resistance, inductor, and capacitor. In 1971, Dr. Leon Chua mathematically proposed a theory based on which the fourth fundamental element of the circuit must exist, based on the symmetry of the governing equations of the passive circuit theory [6, 8]. Chua called this device a memristor (a compound word from memory resistance). In 2008, the HP laboratory created the memristor as a physical device [30]. A physical memristor is a nanoscale device with unique properties. These properties can be used to improve the behavior of electronic systems and computer architecture. A memristor can be thought of as a time-varying resistor, where the resistance changes due to changes in the current passing through the component. The resistance value of a memristor remains unchanged when the current passing through it is zero. This unique property makes the memristor a promising candidate for a non-volatile memory element.

Based on the equations governing resistance, capacitor, and inductor, Chua assumed that there must be a fourth element that establishes the relationship between magnetic flux and charge. Resistance is the relationship between electric current and voltage, inductor is the relationship between electric current and magnetic flux, and capacitor is the relationship between electric charge and voltage. Chua discovered that the capability of the memristor could not be replicated by any of the other three passive elements and that an active circuit that mimics the memristor’s performance would require approximately 25 transistors. As mentioned, the memristor is the fourth circuit element alongside other circuit elements including capacitor, resistor, and inductor. These four elements form the four basic elements of the circuit. In 2008, the memristor was implemented and produced [8, 9, 10], and as a result, it was recognized as the fourth basic circuit element. The memristor is a two-end element in which the magnetic flux($\phi$) between its terminals is a function of the electric charge $\left(q\right)$ that passes through it. The relationship between voltage $v\left(t\right)$ and current $i\left(t\right)$ in a memristor is described by the following Eq. (1):

$$i\left(t\right)=\frac{\mathit{dq}}{\mathit{dt}}=\frac{\mathit{dq}}{\mathit{d\phi }}\frac{\mathit{d\phi }}{\mathit{dt}}=W\left(\phi \left(t\right)\right)v\left(t\right)$$E1

which in this relation, $W\left(\phi \left(t\right)\right)$ is called the incremental memductance function because it describes the conductivity and is defined as follows:

$$W\left(\phi \left(t\right)\right)=\frac{\mathit{dq}}{\mathit{d\phi }}$$E2

According to these, it can be concluded that the behavior of a memristor cannot be replicated by any combination of three other types of inactive circuit components, such as resistance, inductor, and capacitor, even if those components exhibit non-linear behavior. Figure 1 shows the geometric schematic of a memristor created by the HP company. As you can see, the simple structure of this element consists of two layers of titanium dioxide that are placed between two platinum electronic oscillator designs. The basis of this piece’s work is to shift the boundary between two different parts of titanium dioxide: the part that is essentially pure and the adjacent part that contains impurities. Unlike most examples of impurity in semiconductor elements, which typically involve foreign substances, the impurity in this part of the titanium dioxide is simply a lack of a number of oxygen atoms. It is a very simple form of impurity. In the crystal structure of titanium dioxide, these vacancies are displaced to a certain extent from their normal position. The part with this impurity is more conductive than the pure part because electronic oscillators can move between these vacancies. But at the same time, the possibility of moving these empty positions is limited. This means that, in the absence of an external voltage, they are almost stationary and do not move. This property makes this piece act like a variable resistor, where the change in resistance is caused by the voltage applied to the piece. While it is possible to build such a system using a number of active and passive components of the circuit, it is a new discovery that a single piece has such a property. When a positive voltage is applied to this part, the oxygen vacancies expand, which causes the layer with a lack of oxygen to be thicker, so the resistance value of the part decreases. On the contrary, when a negative voltage is applied to the part, the oxygen vacancies shrink and the resistance value of the part increases.

Figure 1.

Physical model of Memristor [35].

The relationship of i-v for the memristor can be considered as follows. This equation shows that the memristor can be modeled by using two series resistors, the resistance of each of them depends on the thickness of the oxygen-deficient layer. In the model shown in Figure 1, a thin semiconductor film can be seen that has two regions: one with a high impurity concentration that behaves like low resistance, $R_{\mathit{ON}}$, and the other region with a low impurity concentration and high resistance, which is called $R_{\mathit{OFF}}$ [13]. The $R_{\mathit{ON}}$ is the resistance of the piece when $\frac{w\left(t\right)}{D}=1$ and the $R_{\mathit{OFF}}$ is the resistance of the piece when $\frac{w\left(t\right)}{D}=0$; D is the total thickness of these two oxide layers.

$$v\left(t\right)=\left(R_{\mathit{ON}}\frac{w\left(t\right)}{D}+R_{\mathit{OFF}}\left(1-\frac{w\left(t\right)}{D}\right)\right)i\left(t\right)$$E3

By the following equation, the drift speed ($v_d$) is given [13]:

$$\frac{\mathit{dw}\left(t\right)}{\mathit{dt}}=v_d=\frac{{\mathit{\eta \mu }}_d{R}_{\mathit{ON}}}{D}i\left(t\right)$$E4

where $\eta =\pm 1$, which depends on the position of the memristor. If $\eta =1$, then the width of the oxygen-deficient layer will expand under positive bias, otherwise it will shrink. The ${\mu }_d$ provides the mobility of charges in the layer with low resistance. The above Equation describes the linear drift speed for the memristor, where charges move at the same rate regardless of the position of the barrier between the layers. A very detailed model is presented in [13], where the nonlinear drift velocity predicts the velocity of the current flow based on the position of the barrier between the low and high resistivity layers. The function used to achieve the desired speed based on the position of the barrier is as follows [13]:

$$F\left(\frac{w\left(t\right)}{D}\right)=1-{\left(2\left(\frac{w\left(t\right)}{D}\right)-1\right)}^{2\rho }$$E5

where $\rho$, is a positive integer that controls the intensity in the function that drives the velocity toward zero at the end of the device. The following equation shows that the drift speed is defined by adding a window function as follows [36]:

$$\frac{\mathit{dw}\left(t\right)}{\mathit{dt}}=v_d=\frac{{\mathit{\eta \mu }}_d}{D}i\left(t\right)F\left(\frac{w\left(t\right)}{D}\right)$$E6

where $i\left(t\right)$, is the current through the memristor.

2.2 Meminductor

Meminductor is a special case of meminductive systems [35, 37]. The dynamic equation of a charge-controlled meminductive system is defined as follows [38]:

$$\phi \left(t\right)=L_M\left(q\right)i\left(t\right)$$E7

where

$$L_M\left(q\right)=\frac{\mathit{d\rho }}{\mathit{dq}}$$E8

For

$$\rho \left(t\right)=\int \phi \left(t\right)\mathit{dt},\text{and}q\left(t\right)=\int i\left(t\right)\mathit{dt}$$

$$\frac{\mathit{dq}}{\mathit{dt}}=\frac{\mathit{dq}}{\mathit{d\rho }}\frac{\mathit{d\rho }}{\mathit{dt}}$$E9

Using Eq. (7), so,

$$i\left(t\right)=\frac{\mathit{dq}}{\mathit{d\rho }}\phi \left(t\right)$$E10

Thus,

$$\phi \left(t\right)=\frac{\mathit{d\rho }}{\mathit{dq}}i\left(t\right)$$E11

The charge of memristor ($q_{\mathit{MR}}$) and the charge of meminductor ($q_{\mathit{ML}}$) have a linear relationship. Moreover, the time-domain integration of the electric flux of meminductor $\left({\rho }_{\mathit{ML}}\right)$ and the flux of memristor (${\phi }_{\mathit{MR}})$ also have a linear relationship:

$$\left\{\begin{array}{c}{q}_{\mathit{ML}}=k_1{q}_{\mathit{MR}}\\ {\rho }_{\mathit{ML}}=k_2{\phi }_{\mathit{MR}}\end{array}\right.$$E12

where $k_1$ and $k_2$ are appropriate real constants. Therefore,

$$i_{\mathit{ML}}=k_1{i}_{\mathit{MR}}$$E13

$${\phi }_{\mathit{ML}}=k_2{v}_{\mathit{MR}}$$E14

where $i_{\mathit{MR}}$ and $v_{\mathit{MR}}$ are the current and voltage across the memristor, respectively; $i_{\mathit{ML}}$ and ${\phi }_{\mathit{ML}}$ represent the current and flux through the meminductor. From Eq. (14), the voltage across the meminductor ($v_{\mathit{ML}})$ is calculated by

$$v_{\mathit{ML}}=k_2\frac{d{v}_{\mathit{MR}}}{\mathit{dt}}$$E15

Therefore,

$$L_M\left(q\right)=\frac{d{\rho }_{\mathit{ML}}}{d{q}_{\mathit{ML}}}=\frac{k_{2}d{\phi }_{\mathit{MR}}}{k_{1}d{q}_{\mathit{MR}}}=\frac{k_2}{k_1}M\left(q\right)$$E16

According to Eq. (16), the charge-controlled meminductance is calculated as:

$$L_M\left(q\right)=\frac{k_2}{k_1}\left(R_{\mathit{OFF}}+\left(R_{\mathit{ON}}-R_{\mathit{OFF}}\right)\frac{{\mu }_v{R}_{\mathit{ON}}}{D^2}q\left(t\right)\right)$$E17

Let $\alpha =\frac{k_2}{k_1}{R}_{\mathit{OFF}}$ and $\beta =\frac{k_2}{k_1}\left(R_{\mathit{ON}}-R_{\mathit{OFF}}\right)\frac{{\mu }_v{R}_{\mathit{ON}}}{D^2}$, so,

$$L_M=\alpha +\mathit{\beta q}\left(t\right)$$E18

3. Application of memristor in oscillators

Since the memristor is a device that can maintain its previous state even after the power is cut off, it can store information indefinitely and consume energy only when we want to read the information. This property makes it a suitable candidate for use in oscillators, such as the Ring oscillator, where the state of the memristor can be used to control the frequency of oscillation. The memristor can replace multiple transistors in certain circuits and occupy less space. This element is manufactured on a nanoscale and its resistance depends on the amplitude, polarity, and duration of the voltage applied to it. According to these, it can be concluded that no combination of the three types of passive circuit elements (resistance, inductor, and capacitor) can fully replicate the behavior of a memristor, even if those elements exhibit nonlinear behavior. In recent years, researchers have done a lot of work in the field of oscillator design using memristor devices [5, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 39, 40, 41]. We will review some of them.

3.1 The use of memristor in the design of Chua oscillator

As an example, the memristor has been used in the design of the Chua oscillator [10]. Figure 2 shows the circuit schematic and phase space curve of this oscillator. The dynamic equations of this oscillator are as follows:

Figure 2.

Memristor-based oscillator (a) oscillator circuit schematic (b) phase curve in oscillatory mode.

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=\mathrm{k\alpha }\left(\mathrm{y}-\mathrm{W}\left(\mathrm{w}\right)\mathrm{x}\right)\\ \dot{\mathrm{y}}=\mathrm{k}\left(\mathrm{z}-\mathrm{x}\right)\\ \dot{\mathrm{z}}=\mathrm{k}\left(-\mathrm{\beta y}+\mathrm{\gamma z}\right)\end{array}\right.$$E19

Where, $\mathrm{x}=v_1,y=i_3,z=v_2,w=\phi ,\alpha =\frac{1}{C_1},\beta =\frac{1}{C_2},\gamma =\frac{G}{C_2},L=1$ and k is the time scale factor.

3.2 The use of memristor in Wien bridge and phase shift oscillators

The memristor, which is a type of variable resistance that changes with voltage, can be utilized in constructing oscillators like the Wien bridge oscillator. As we know, the Wien bridge oscillator is one of the types of electronic oscillators, that can be used to generate sinusoidal waveforms in many frequency ranges [14, 15, 16, 17, 18, 22]. Figure 3 shows the circuit schematic of the Wien bridge oscillator [15, 16]. The characteristic equation of this circuit is as follows:

Figure 3.

Wien bridge oscillator circuit (a) schematic of Wien bridge oscillator (b) Wien bridge oscillator based on memristor [15].

$$s^2+\mathit{bs}+d=0$$E20

Where

$$b=\frac{1}{R_2{C}_2}+\frac{1}{R_1{C}_1}-\frac{R_3}{R_1{R}_4{C}_2},d=\frac{1}{R_1{R}_2{C}_1{C}_2}$$E21

According to the above equation, the fluctuation condition is given by:

$$\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1}+\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{{\mathrm{R}}_3}{{\mathrm{R}}_4}$$E22

The oscillation frequency is as follows:

$$\mathrm{f}=\frac{1}{2\mathrm{\pi }\sqrt{{\mathrm{R}}_1{\mathrm{R}}_2{\mathrm{C}}_1{\mathrm{C}}_2}}$$E23

Figure 4 shows the simulation results of the memristor-based Wien bridge oscillator and its frequency spectrum.

Figure 4.

Simulation results (a) output waveform (b) output harmonic distribution.

Figure 5 shows another oscillator that utilizes a memristor, which is the phase shift oscillator [21, 42]. In this oscillator, resistors are replaced with memristors.

Figure 5.

Phase shift oscillator based on memristor [38].

3.3 Using memristor in oscillator using Deboo integrator and oscillators based on digital gates

The oscillator based on the Deboo integrator is shown in Figure 6 [43].

Figure 6.

Oscillator using Deboo integrator and two op-amp [43].

The first stage is a non-linear amplifier with two resistors connected to the negative input and the second stage uses its integral output. Resistance R1 affects the conditions of the oscillator, while R2 affects the frequency of the oscillator. The R3 resistance affects both the frequency and condition of the oscillator. The oscillator output for a certain frequency is shown in Figure 7.

Figure 7.

Oscillator output at the frequency of 15.91 Hz [43].

The zeros and poles of the system determine the stability of the oscillator. In this regard, another example of these oscillators based on 6 memristors in Figure 8 [44].

Figure 8.

Schematic of the Op-amps oscillator based on memristor [44].

3.4 Reactance-less oscillator based on memristor

Despite the implementation of a memristor, the oscillator still requires the use of reactive elements. This section provides a comprehensive mathematical derivation that explains a diverse range of memristor-based reactance-less oscillators (MRLOs), while also presenting a physical implementation of an MRLO for the first time [27, 30, 32, 34]. The initial reactance-less oscillator is presented in this letter. The utilization of a memristor in the oscillator design enables complete on-chip implementation, eliminating the requirement for capacitors or inductors. This leads to a compact and fully integrated solution.

The various types of memristor-based clinometer structures without reactance are depicted in Figure 9, alongside their schematic representations [27].

Figure 9.

(a) General schematic of the MRLO. (b-d) Schematic of different sub-types of MRLO family connected in the positive configuration [27].

Figure 9(a) displays the overall structure of the suggested oscillator architecture. The oscillator comprises of a voltage divider consisting of two fundamental elements ($E_1$ and $E_2$), and a transfer function.

$${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathrm{V}}_{\mathrm{o}}\left(\mathrm{t}\right)\frac{\mathrm{R}\left({\mathrm{E}}_2\right)}{\mathrm{R}\left({\mathrm{E}}_1\right)+\mathrm{R}\left({\mathrm{E}}_2\right)}$$E24

Here, $R\left(E_X\right)$ denotes the resistance of device $E_X$. The type of oscillator is identified based on the type of the two fundamental elements ($E_1$ and $E_2$). The two basic devices, $E_1$ and $E_2$, can take the form of either two memristors, a floating memristor and a grounded resistor, or a floating resistor and a grounded memristor [29, 30, 31]. The oscillator based on digital gates is a type of oscillator that is designed without using active elements such as inductors and capacitors. One of the benefits of this oscillator is its ability to operate at higher frequencies compared to other oscillators designed without inductors. It also provides a wide range of resistance [45]. Figure 10 shows the oscillator circuit with three gates. The duty cycle of the circuit is set to 50% to achieve the desired frequency [45].

Figure 10.

Oscillator circuit with three gates [45].

In the next step, the circuit is designed to use the memristor. Figure 11 shows the desired circuit.

Figure 11.

Oscillator circuit with memristor [45].

3.5 Ring oscillator

Ring oscillator circuits consist of an odd number of delay elements, which can be implemented as logical NOT gates, that are connected in cascade to create the oscillator circuit. The configuration of the ring oscillator circuit mentioned earlier is fundamentally a closed-loop system, as illustrated in Figure 12a. The Barkhausen criterion represents a vital prerequisite for the continuation of sustained oscillations. According to the Barkhausen criterion, at the frequency ω0, the loop gain must be equal to or greater than unity, and the frequency-dependent phase shift should be equivalent to 2π. To satisfy these two requirements, a minimum of three inverter stages is necessary. Each stage of the Ring oscillator provides a phase shift of ${120}^{\circ }$ degrees, resulting in an overall phase shift of the system of ${360}^{\circ }$ degrees [46, 47]. As a result, the circuit becomes self-generating, and the element operating at frequency ω0 produces continuous oscillations, making the Ring oscillator circuit an essential building block in many electronic applications. The overall output of the system is stable since the only DC operating point is inherently unstable, and even the slightest noise disturbance can initiate free-running oscillations.

Figure 12.

(a) N-stage voltage controlled ring oscillator. (b) Memristive load inverter delay cell. (c) Two-Memristor based inverter delay cell [46].

The frequency at which the circuit oscillates is commonly referred to as the:

$$f=\frac{1}{\left(2N{\tau }_P\right)}$$E25

Where $N$ represents the number of delay elements or stages in the circuit, and ${\tau }_P$ refers to the average propagation delay of one stage, calculated by the Eq. (26), with ${\tau }_{{}_{P}HL}$ and ${\tau }_{{}_{P}LH}$ representing the high-to-low and low-to-high propagation delays of a single-stage inverter [48].

$${\tau }_P=\frac{{\tau }_{{}_{P}HL}+{\tau }_{{}_{P}LH}}{2}$$E26

Due to the noisy behavior of the MOSFETs, CMOS ring oscillators exhibit inherent jitter in the overall period. In [46] a ring oscillator is constructed using two memristors and a MOSFET-based inverter as the delay element. Two other types of ring oscillator circuits were considered for comparison: one based on a simple CMOS inverter delay unit, and another based on a memristive load inverter delay unit [13]. The frequency of operation of the oscillator is directly influenced by the design of the delay cell used in the circuit. Two different structures for the inverter delay cell are shown in Figure 12. Figure 12c demonstrates that during a low input pulse, the memristor $X1$ ‘s resistance $R_{\mathit{off}}$ is very high, which limits the current flow. Similarly, when the pulse is applied to the gate of PMOS M1, which is the same pulse as before, the transistor M1 turns ON during a low pulse, causing $V_{\mathit{\text{out}}}$ to become high. In contrast, during a high input pulse, the PMOS turns OFF, and the output is determined by the memristor’s divider circuit. With a much higher resistance $R_{\mathit{off}}$ compared to $R_{\mathit{on}}$, $R_{\mathit{off}}\gg R_{\mathit{on}}$, $V_{\mathit{\text{out}}}$ is pulled toward the potential of the Ring oscillator. Figure 12b demonstrates that when the input pulse is in a high state, the M4 turns ON, and $V_{\mathit{\text{out}}}$ is connected to the ground potential. Conversely, during a low input pulse, M4 is cut-off, and $V_{\mathit{\text{out}}}$ assumes the potential created by the current flowing through the memristor $X2$ [41, 49]. The design employs a memristor as the load component. The fundamental schematic of a CMOS ring oscillator is depicted in Figure 13a. To address issues such as leakage current and leakage power in a three-stage loop oscillator based on CMOS technology, Figure 13b demonstrates the incorporation of the memristor technique as a means of improvement [40, 46].

Figure 13.

CMOS-based three stage ring oscillator; (a) without Memristor. (b) with Memristor.

Ring oscillators based on memristors are employed as a means of generating true random numbers, known as true random number generators [41].

3.6 PTC Memristor based low-frequency oscillator

Strukhov et al. reported the discovery of a memristor based on Resistive Random Access Memory (ReRAM) on May 1, 2008. Their discovery sparked renewed interest in memristors and their diverse applications across various fields, leading to a continuous growth of research and development in the area. Over the past 5 years, Rajamani et al. examined an electronic oscillator circuit that involved connecting an inductor in series with a “locally-active” Positive Temperature Coefficient (PTC) memristor and a battery [28]. Sah et al. subsequently investigated a “second order” memristor that represents the model of a physical device consisting of a Positive Temperature Coefficient (PTC) and Negative Temperature Coefficient (NTC) thermistor connected in series [50]. The objective of this study is to examine a circuit that includes a linear passive capacitor, a linear passive inductor, a nonlinear resistor, and a Negative Temperature Coefficient thermistor, which is a non-linear and locally-active volatile memristor [51].

The definition of a first-order locally active Positive Temperature Coefficient (PTC) memristor is given by [28, 52]:

$$i_M=G\left(x_m\right)v_M$$E27

Where $i_M$ shows current through PTC memristor, $v_M$ represents the voltage across PTC memristor, and so,

$$G\left(x_m\right)=\frac{1}{R_{\mathit{OM}}{e}^{{\beta }_M\left(x_M-{\gamma }_m\right)}}$$E28

Where

$$\frac{{\mathit{dx}}_M}{\mathit{dt}}=\frac{1}{{\sigma }_M}\left[{\delta }_M\left({\gamma }_M-x_M\right)+G\left(x_M\right){v_M}^2\right]=f\left(x_M,v_M\right)$$E29

where ${\sigma }_M$, ${\delta }_M$, ${\gamma }_M$, ${\beta }_M$, and $R_{\mathit{OM}}$ are device parameters.

Conventional electronic oscillator circuits typically require a combination of energy storage elements, such as inductors and/or capacitors, and an active nonlinear resistor with two terminals, or a three-terminal resistor, in addition to a power source like a battery. The tunnel diode is an example of a locally-active, two-terminal, nonlinear resistor. The transistor is an example of a locally-active 3-terminal resistor. Sah et al. proposed a locally-active second-order memristor oscillator without using any inductors or capacitors in which the memristor is connected directly across a battery [26, 28]. In Figure 14, an oscillator circuit is shown that consists of a linear inductor (L*) connected in series with a first-order, locally-active Positive Temperature Coefficient (PTC) memristor and a battery. The PTC memristor used in Figure 14 is a first-order, locally-active generic memristor that has a simpler state equation and a less complicated small-signal equivalent circuit than the second-order memristor presented in [26]. The oscillator circuit presented in the text is similar to the circuit introduced in [52]. In both circuits, a memristor is used as the nonlinear element to create feedback and generate oscillations. The use of the PTC memristor offers a simpler and more practical approach to creating oscillator circuits without the need for complex memristor models. The Chua’s Corsage memristor used in [52], is a hypothetical memristor that is used as a mathematical model to study memristive systems. On the other hand, the first-order, locally-active PTC memristor used in the text represents the model of a physical device called a Positive-Temperature Coefficient (PTC) thermistor. The PTC thermistor and inductor can be used to create an oscillator circuit, where the PTC thermistor acts as the temperature-dependent resistor and the inductor provides the necessary feedback for oscillation. When the circuit is connected to a battery, it can generate sustained oscillations at a specific frequency determined by the values of the components used [28].

Figure 14.

Locally-active first-order PTC Memristor oscillator.

In order to have sustained oscillations in a circuit, there must be at least two nonlinear differential equations governing the behavior of the system. However, the addition of a linear inductor alone may not be sufficient to create the necessary nonlinear behavior for oscillation. It is more likely that the combination of the nonlinear behavior of the PTC memristor and the inductance of the inductor together creates the necessary nonlinear dynamics for sustained oscillations, as shown in Figure 14 [28].

4. Application of memristor in voltage-controlled oscillator

Voltage-controlled oscillators (VCO) are oscillators that work in wide frequency ranges and change their frequency value by means of one of the frequency-determining circuits. A voltage-controlled oscillator is an electronic oscillator in which the frequency of oscillation changes proportionally to the input voltage. Voltage-controlled oscillators are widely used in high-frequency and electronic circuits. In recent years, various studies have been done in the field of using memristors in the design of voltage-controlled oscillators [53, 54, 55, 56, 57, 58].

The basic hardware circuit schematic diagrams for both resistor-based and memristor-based VCOs are shown in Figure 15 [55].

Figure 15.

Circuit schematic for both resistors-based and memristors-based VCO [55].

5. Meminductor based sinusoidal oscillator

In ref. [25], meminductor is used to design a sinusoidal Colpitts oscillator circuit. it can be concluded that using meminductor in a Colpitts oscillator has two major advantages over previous designs:

1. There has been no previous work conducted on the design of meminductor-based sinusoidal oscillators.

2. Nanoscale meminductors can be manufactured to significantly reduce the physical dimensions of the oscillator circuit.

In [25, 38, 59, 60], the meminductor emulator used in the experiments can be easily implemented. Demonstrating the performance of sinusoidal oscillators to undergraduate students can be easily achieved by reproducing the results in a laboratory setting.

Figure 16a shows a simple Colpitts-based oscillator. Meminductors are considered to be on the nanoscale and their inductance varies according to the current passing through them or the magnetic flux they experience. Therefore, to improve the behavior of the circuit shown in Figure 16a, a meminductor is used instead of a conventional inductor. Figure 16b shows an improved meminductor-based Colpitts oscillator.

Figure 16.

(a) Conventional Colpitts oscillator, and (b) meminductor-based Colpitts oscillator.

6. Conclusions

Undoubtedly, adequate knowledge of the behavior of oscillators and their analysis is of great importance in the electronics industry. The general structure of oscillators and their important characteristics were investigated in this chapter, followed by an exploration of the application of memristors and meminductors in these circuits.

Conflict of interest

There is no conflict of interest between the authors.

Declarations

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. This research did not receive any specific grant from any funding agency in the public, commercial, or not-for-profit sectors.