Charged Controlled Mem-Element Emulator and Its Application in a Chaotic System

This article proposes a charged controlled emulator model for memristor and memcapacitor using second-generation Current Conveyor (CCIIs) and Analog Multiplier (AM). The grounded and floating mem-element circuits have been designed using two CCIIs and one multiplier in addition to some passive components. The grounded type of design requires three resistors and three capacitors while floating design necessitates only two resistors and two capacitors. With the help of a switch, the proposed design can be easily switched into a memristor or memcapacitor. The proposed emulator model has been theoretically analyzed and simulated in PSpice to substantiate the effectiveness and accuracy. The practicability of the circuit has been established using commercially available ICs AD844AN and AD633JN. Non-linear characteristics of the proposed memristor emulator have been used to design a chaotic system.


I. INTRODUCTION
With the advancement in VLSI technology, demand for high speed and low power technology has increased manifold. One of the major challenges in designing nanoscale transistors is the design of gates. As the gate size decreases, it becomes difficult to control the current flow through a thin channel. In 2008, Strukov and his team at HP Lab [1] developed a working memristor using Titanium Dioxide (TiO 2 ), a fourth fundamental passive element of the circuit after the resistor, capacitor and inductor which were earlier predicted by Leon Chua in 1971 [2] in his paper 'The Missing Memristor Found.' He found the missing relationship between charge and magnetic flux. After the discovery of memristor, the notion of passive memory device is not limited to memristor and has further extended to the memcapacitor and meminductor by Di Ventra et al. in 2009 [3]. The memcapacitor can possibly be the most valuable and novel segment on account of its efficacy and dynamic semblance to the memristor.

Memcapacitor gives a pinched hysteresis loop in charge and
The associate editor coordinating the review of this manuscript and approving it for publication was Di He . voltage plain as current and voltage in the case of a memristor. Similarly, meminductor depicts pinched hysteresis behaviour in the current-flux plan, where it acts as a memory device. Since these elements store the past value of their input signals, pinched hysteresis loops are generated when bipolar inputs are applied to these elements.
The attributes of the pinched hysteresis loop are unique in these mem-elements, which enable them to be used as nonvolatile memories. Although these mem-elements have the potential for memory applications, they are not expected to appear soon due to difficulties of fabrication with current technologies. Therefore, the utilization of emulators would be an alternative solution to develop application circuits. In fact, some research groups have proposed many emulator circuits for memristor and memcapacitor. Kim et al. proposed a memristor emulator using off-the-shelf solid-state devices in 2012 [4], who, later in 2015, extended its experimental results for the replacement of real memristor [5]. Some researchers have recommended a memristor emulator based on operational amplifiers [4], [6]. However, an operational amplifier based emulator has many shortcomings like low linearity, large supply voltage, small voltage fluctuation, and VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ limited operating frequency. These drawbacks can be overcome using current mode circuits. These circuits have a better performance than voltage-mode circuits in terms of speed, dynamic range, bandwidth, slew rate, high linearity, and better accuracy. They require less operating voltage than voltagemode circuits, thus enabling them to produce higher current swing at less supply voltage, and resulting in reduced distortion. Due to better performance than voltage-mode circuits in terms of operating speed, CMOS current-mode circuits have grabbed the attention of the industry [7]. The results illustrate considerable progress in current-mode analog signal processing, multiprocessors, telecommunication systems, instrumentation, computer interfaces with high speed. In complex electronic systems, various current mode building blocks have been reported. The second-generation Current Conveyor (CCII), which is the basic current mode building block, is used to design emulator model for the memcapacitor. In 2014, Lopez et al., have proposed CCII based floating memristor emulator [8]. However, the proposed memristor emulator is a flux-controlled model that is different from real charged controlled HP-memristor. In [9], [10], Ranjan et al., have advocated a memristor emulator that uses only one kind of an element. In 2010, Krems et al., had built a memcapacitor using solid-state materials i.e. nano-pores [11]. Under this, when ions are allowed to enter nano-pores and an electric field is subjected across the material, memcapacitive behaviour is observed. In 2010, Biolek et al., proposed SPICE modeling of memcapacitor [12] followed by behavioral modeling in [13]. In this modeling, width of the dielectric is controlled by the charge flowing through it. However, this model cannot be used for practical circuits. Mutators were developed for transforming memristor to memcapacitor or meminductor in [14]- [19]. However, the properties of these memcapacitor depend upon the used memristor which uses excessive components. Memristor-less Memcapacitor circuits were introduced in [20]. Vista and Ranjan had proposed a simple charge controlled floating memcapacitor emulator using Dual X current conveyor differential input transconductance amplifier (DXCCDITA) [21]. In 2018, Babacan et al., proposed OTA based memcapacitor and meminductor [22]. [23], [24] includes applications based on memcapacitor. Yu et al., recommended a memcapacitor based relaxation oscillator in 2016 [23]. In 2018, Feali et al., implemented adaptive neuron based on memristor and memcapacitor [24]. Vista and Ranjan proposed MOSFET based floating memristor [25]. Rajagopal et al., proposed a hyper-chaotic memcapacitor oscillator with infinite equilibrium and co-existing attractors [26]. In 2020, Raj et al., proposed memristor based hyper chaotic system [27]. Voltage-tunable fully balanced voltage differencing buffered amplifier (FB-VDBA) based memristor emulator was presented by Yadav et al.. in 2019 [28]. Yesil et al., had proposed an electronically tunable memristor using only one voltage differencing current conveyor (VDCC) [29]. In 2019, a universal emulator was proposed by Zhao et al., for memristor, memcapacitor, and meminductor and its chaotic circuit [30].
In this article, a charged controlled switch-based memelement emulator has been proposed. With the help of switches, a mode for memristor and memcapacitor can be selected. Proposed circuit, as shown in Fig. 1, first acts as a grounded mem-element emulator. It requires two CCIIs, one Multiplier, three resistors, three capacitors, and three switches. The second proposed circuit is depicted in Fig. 2 acting as floating mem-element emulator. It requires two CCIIs, one Multiplier, two resistors, two capacitors, and two switches. Proposed emulators are charged controlled and can be easily shifted to incremental and decremental modes using a simple switch. Effectiveness and precision of the proposed emulator model are theoretically analyzed and simulated in PSpice. The practicability of the circuit is experimentally verified. Chaotic system has been designed using floating type memristor emulator as part of an application of proposed emulator. Chua's diode is replaced with floating memristor for realization of chaotic circuit. Chua's diode and memristor both exhibit non-linear properties. The negative resistance in Chua's circuit is realized using IC3080.

II. MATHEMATICAL MODEL OF CHARGED CONTROLLED EMULATOR
The mathematical relations of mem-element are realized by analyzing the relationships among current (i), charge (q), voltage (v), and flux (ϕ). Charge (q) and flux (ϕ) are the time integral of the current I (q = t t 0 i(τ )dτ ) and the voltage v The memristance M (q), can be defined as: Recently, the concept of memristor was extended to the memcapacitor (C M ) and is defined as: where σ = t t 0 q(τ )dτ . Likewise, memcapacitor also shows the pinched hysteresis loop in charge-voltage plane. It acts as a memory device. Two simple dedicated mem-element emulator circuits are presented using two AD844AN blocks [31] and one AD633JN block [32]. IC AD844AN represents port relationship V X = V Y , I Z = I X , V W = V Z , I Y = 0 of the second-generation Current Conveyor (CCII) whereas IC AD633JN acts as an analog multiplier. The proposed grounded mem-element is described in Section 2.A while Section 2.B takes care of a floating mem-element.

A. GROUNDED CHARGED CONTROLLED MEM-ELEMENT EMULATOR
The proposed grounded charged controlled mem-element emulator circuits are shown in Fig. 1, which consist of two CCIIs (AD844AN), one analog multiplier (AD633JN), three resistors and three capacitors. The switches S 0 and S 1 are used for memristor and memcapacitor switching. If S 0 and S 1 switches are connected to point A and V R1 respectively, the circuit will work as a memristor and if they are connected to point B & V C1 respectively, it will work like a memcapacitor. Switch S is used for incremental and decremental behaviours. When switch S is connected to Y 2 terminal and Y 1 is grounded of AD633JN, it emulates incremental behaviour, and if switch S is connected to the Y 1 terminal and Y 2 is grounded, it follows decremental behaviour.

1) GROUNDED MEMRISTOR EMULATOR AND ITS FREQUENCY ANALYSIS
For memristor operation, S 0 switch is connected to point A and S 1 switch is connected to V R1 . Fig. 1 displays a proposed grounded type memristor.
Input voltage can be written as: The voltage at x terminal of CCII-2 is given as: The voltage at the Z terminal of CCII-2 is shown as: With the properties of AD633JN V w can be written as: As V w = V X , substitute the value of V X in Eqn. 1 we get, From Eqn. 5, the memristance value can be written as: It is essential to note that the input signal V in (t), and the values of passive components manage the memristance value. The frequency characteristic of the presented grounded memristor circuit can be analyzed by assuming an input voltage V in (t) = V m sinωt, where V m is the amplitude, and ω is the frequency of the signal. The average input current can be calculated by substituting the time-varying part to zero in Eqn. 6. So, it is stated as: As a result, q(t) can be expressed as: By substituting equation (8) in equation (6), memristance of the circuit can be established as: Eqn. 9 illustrates that R M (q(t)) has a linear time-invariant part and linear time-variant part. Here, the linearity is described in terms of voltage and current. The linear time-variant part is dominated by the linear time-invariant part as the frequency increases. Also, the linear curve is attained between current and voltage for higher frequencies.
2) GROUNDED MEMCAPACITOR EMULATOR AND ITS FREQUENCY ANALYSIS S 0 switch is now changed into point B and S 1 switch is connected to V C1 in Fig.1 for grounded memcapacitor emulator operation. Input voltage can be written as: The voltage at x terminal of CCII-2 is given as: The voltage at the Z terminal of CCII-2 is written as: With the properties of AD633JN (used as an analog multiplier) V w expression can be presented as: VOLUME 8, 2020 As V w = V X , substitute the value of V X in Eqn. 10 we get, From Eqn. 14, the mem-capacitance value can be put up as: It is important to note that the memcapacitance value can be directed by an input signal V in (t), and the values of passive components. Frequency response can be analyzed by providing a sinusoidal input V in (t) = V m sinωt, where V m is the amplitude, and ω is frequency of the signal. The average stored charge can be calculated by replacing the time-varying part to zero in Eqn. 15. Therefore, it is expressed as: As a result, σ (t) can be given as: Substituting Eqn. 17 in Eqn. 6, memcapacitance of the circuit can be written as: Eqn. 18 depicts that C −1 M (q(t)) is having linear time-variant and linear time-invariant parts. When the frequency increases, the linear time-invariant part dominates the linear timevariant part. During this, a linear curve is achieved between the charge and the voltage for higher frequency values.

B. FLOATING CHARGED CONTROLLED MEM-ELEMENT EMULATOR
The proposed floating charged controlled mem-element emulator circuit is shown in Fig. 2, containing two CCIIs (AD844AN), one analog multiplier (AD633JN), two resistors, and two capacitors. S 0 switch is used for memristor and memcapacitor switching. If switch S 0 is connected to point A, the circuit will work as a memristor and if it is connected to point B, it will work as a memcapacitor. Switch S is used for switching incremental and decremental behaviours. When switch S is connected to the Y 2 terminal and Y 1 is grounded of AD633JN, it emulates incremental behaviour and if switch S is connected to the Y 1 terminal and Y 2 is grounded, it illustrates decremental behaviour.

1) FLOATING MEMRISTOR EMULATOR AND ITS FREQUENCY ANALYSIS
For the memristor operation, S 0 switch is connected to point A in proposed floating mem-element as shown in Fig. 2.
Input voltage can be written as: In the case of memristor, V Y can be put up as: The voltage at the Z terminal of CCII-2 is given as: With the properties of AD633JN, V w expression can be explained as: Substituting the value of V Y and V W in Eqn. 19 we get, From Eqn. 23, the memristance value can be written as: Memristance value can be controlled by the input signal V in (t), and the values of passive components. For frequency analysis, let V in (t) = V m sinωt, where V m is the amplitude, and ω is frequency of the input signal. The average input current can be estimated by substituting the time-varying part to zero in Eqn. 24. Therefore, it is expressed as: As an effect, q(t) can be expressed as: By substituting Eqn. 26 in Eqn. 24, the circuit's memristance can be determined as: Eqn. 27 shows that R M (q(t)) contains a linear timeinvariant part and a linear time-variant part. With the increase in frequency, the linear time-invariant part starts dominating the linear time-variant part. In contrast, the linear curve is obtained between current and voltage for higher frequency.

2) FLOATING MEMCAPACITOR EMULATOR AND ITS FREQUENCY ANALYSIS
For floating memcapacitor emulator operation, S 0 switch is now connected to point B as shown in Fig. 2. Input voltage can be put up as: In the case of memcapacitor, V Y can be written as: The voltage at the Z terminal of CCII-2 is given as: With the properties of AD633JN, V w expression can be shown as: After substituting the value of V Y and V W in Eqn. 19, we get: From Eqn. 32, the memcapacitance value can be written as: The memcapacitance value can be managed by an input signal V in (t), and the values of passive components. Frequency analysis of proposed floating memcapacitor can be studied by providing an input signal V in (t) = V m sinωt, where V m is the amplitude, and ω is frequency of the input signal. The average stored charge can be assessed by substituting the time-varying part to zero in Eqn. 33. Therefore, it is expressed as: As a result, σ (t) can be expressed as: By substituting Eqn. 35 in Eqn. 33, memcapacitance of the circuit can be written as: Eqn. 36 conveys that C −1 M (q(t)) is having linear time-invariant and linear time-variant parts. When the frequency increases, linear time-invariant part governs the linear time-variant part. This leads to a linear curve between charge and voltage for higher values of frequencies.

A. SIMULATION RESULTS OF GROUNDED TYPE MEMRISTOR
Pspice simulation has been performed to verify the functioning of the grounded type memristor circuit as illustrated in Fig. 1. For the grounded type memristor emulator, switch S 0 is connected to point A, and S 1 is connected to V R1 .
Grounded memristor behaviour is tested by providing a sinusoidal input signal with V m = 1.5 V. The parameter taken for implementing grounded type memristor is as follows: R 0 = 10 k , R 1 = 10 k , and R 2 =75 k . The capacitor value C 2 is a variable that varies with the input frequency. Fig. 3 shows the pinched hysteresis loop obtained from proposed grounded memristor. The supply voltage given to CCII and analog multiplier is ± 10 V.
From Fig.3, we can deduce the following: 1) With increase and decrease in capacitor value C 2, the area of the pinched hysteresis curve decreases and increases respectively as shown in Fig. 3(a). This happens because the linear time-invariant part of the memristor is dominated by the linear time-variant part, as mentioned in Eqn. 9. 2) With an increase in frequency, the pinched hysteresis curve area decreases, and the memristor behaves like an ordinary resistor, i.e., a linear curve is observed in a current-voltage plane at higher frequencies. Thus, simulated results are well agreed upon with a theoretical analysis as discussed in Section 2.A.1.

B. SIMULATION RESULTS OF GROUNDED TYPE MEMCAPACITOR EMULATOR
To implement a grounded type capacitor, S 0 switch is connected to point B, and S 1 switch is connected to V C1 as shown in Fig. 1. Pspice simulation has been performed to verify practicality of the proposed grounded capacitor. The values taken for simulation are C 0 = 500 nF, C 1 = 1 nF, and R 1 = 75 k . The capacitor value C 2 is a variable that varies with the input frequency. VOLUME 8, 2020 Hysteresis loop for memcapacitor is obtained by providing a sinusoidal input signal with V m = 1.5 V. The supply voltage given to CCII, and an analog multiplier is ± 10 V. Fig. 4 displays the pinched hysteresis loop obtained at different frequencies. The capacitor voltage is taken at C 1 .
From Fig.4, we receive two observations: 1) With increase and decrease in capacitor value C 2 , the pinched hysteresis curve area decreases and increases respectively. This happens because the linear time-invariant part of the memcapacitor is dominated by the linear time-variant part, as mentioned in Eqn.18. 2) With an increase in frequency, the pinched hysteresis curve area decreases, and the memcapacitor behaves like an ordinary capacitor, i.e., a linear charge-voltage curve is observed at higher frequencies.
Thus, simulated results are well agreed upon with a theoretical analysis, as discussed in Section 2.A.2.

C. SIMULATION RESULTS OF FLOATING TYPE MEMRISTOR EMULATOR
Floating type memristor emulator is acquired by attaching S 0 switch to point A as shown in Fig. 2. Pspice simulation has been executed to verify functioning of the proposed floating type memristor circuit. The parameter values taken for simulation are R 0 = 10 k , and R 2 =50 k . The capacitor value C 1 is a variable that varies with the input frequency. Floating memristor emulator hysteresis behaviour is tested by providing a sinusoidal input signal with V m = 1.5 V. The supply voltage given to CCII and an analog multiplier is ± 10 V. Fig. 5 demonstrates the pinched hysteresis loop obtained at different frequencies and capacitor values.
From Fig. 5, we can easily ascertain that with an increase in frequency, the hysteresis loop becomes narrower. Besides, we can study that with reduction of the capacitor C 1 , the hysteresis loop becomes wider. The above two observations are well agreed upon with Eqn. 27, as discussed in Section 2.B.1. It can be observed that some offset are present in the simulation results. However, these offsets are because of the nonlinearity and tracking error of the building blocks CCII and Analog Multiplier. However, these can be reduced by using the offset reduction method.

D. SIMULATION RESULTS OF FLOATING TYPE MEMCAPACITOR
Pspice simulation has been conducted to validate the proposed floating memcapacitor emulator circuit obtained by connecting S 0 switch to point B as given in Fig. 2. C 0 = 15 nF, and R 1 = 50 k are taken for simulation. Capacitor value C 1 is a variable that varies with the input frequency. The input sinusoidal with V m = 1.5 V has been provided to obtain hysteresis behaviours. Supply voltage is given to CCII and analog multiplier is ± 10 V. Fig. 6 shows the pinched hysteresis loop obtained at different frequencies.
The capacitor voltage is taken at C 0 . Fig.6 displays that when capacitor value C 1 increase or decreases, the pinched hysteresis curve area decreases and increases respectively, and as the frequency increases, the pinched hysteresis curve area decreases, i.e., a linear charge-voltage curve is observed at a higher frequency. The above two observations are well agreed upon with Eqn. 36 as discussed in Section 2.B.2.

IV. NON-VOLATILITY TEST
A critical feature of the mem-element emulator circuit is its non-volatility, i.e., the emulator circuit should retain its value when no input signal is applied. To test the non-volatility of the proposed memristor shown in Fig. 2, an input pulse train has been applied. The input pulse has 1.5 V amplitude with 25 µ sec duration and 175 µ sec pulse period. During the simulation, other parameters of the memristor emulator were considered with R 0 = 10 k , R 1 = 50 k , and C 1 = 5 nF. Fig. 7 shows the memristance value and applied input pulse voltage with respect to time. One can observe that there is no alteration in memristance during the off period of an applied input pulse. During the pulse period, there is an abrupt change in voltage, so the proposed emulator shows a non-volatile nature.

V. COMPARISON TABLE
A comparative study of the proposed mem-element emulator with an existing mem-element emulator has been undertaken in Table 1.
From the table, it can be observed that, • Unlike circuits proposed in [14], [16] and [18] whose memcapacitor properties depend upon the memristor, the proposed grounded and the floating circuit is designed without using a memristor.
• Memristor, Memcapacitor, and Meminductor proposed in [19] was independent. But it uses more active and passive component along with op-amp and varactor diode (V D ).
• Universal emulator for memristor, memcapacitor, and meminductor proposed in [30] required 5 CCIIs,  one analog multiplier, one voltage buffer along with 7 passive components for floating configuration. This proposed design involved only 2CCIIs, one analog multiplier, and 4 passive components. Also, in [30] the hysteresis loop is limited up to 5 kHz, but in the proposed emulator, it is obtained up to 20 kHz.

VI. EXPERIMENTAL VERIFICATION FOR MEMRISTOR AND MEMCAPACITOR EMULATOR
It is important to verify the performance and simulation results of the proposed memristor and memcapacitor emulator experimentally. Therefore, a prototype circuit of mem-element is assembled using commercially available ICs i.e. AD844AN and AD633JN. An input voltage V in = 0.8sin (20000π t) is used to verify the mem-element emulator. The other parameters for experimental testing of emulator are configured same as kept during simulation. The supply voltage of ICs is ±10 V. Fig. 8 and Fig. 9 depict a pinched hysteresis loop obtained at 10 kHz for grounded memristor and grounded memcapacitor, respectively. In Fig. 8, the input voltage vs. resistive voltage (V R1 ) is plotted as V R1 reflecting input current. In Fig. 9, the input voltage is plotted against the capacitor voltage (V C1 ), which is reflecting the charge.  In Fig. 10, the experimental and simulation results for floating memcapacitor have been compared at 10 kHz operating frequency. The experiment has been carried out by supplying input voltage V in = 1.5sin (20000π t). The other passive parameters value are kept the same as in simulation. The experimental and simulation.csv data are obtained from digital storage oscilloscope and Pspice simulation, respectively. The.csv data are transferred and plotted in Origin 2019b.

VII. APPLICATION
In this section, we have realized a non-linear Chua's oscillator system using a charged controlled memristor as presented in Fig. 11. The non-linear characteristics of a memristor make it suitable for chaotic circuits. Chua's oscillator is achieved by replacing Chua's diode with a charged controlled memristor. The negative resistance is designed using commercially existing IC CA3080. Chua's circuit consists of one negative resistor, one capacitor, two inductors and one charged controlled memristor. The Chua's circuit displays an ample variety of chaos outputs for different values of circuit components. The dynamic state equation of canonical Chua's oscillator for charged controlled memristor emulator is expressed as: The value of passive components L 1 , L 2, and C is taken as 5.5 mH, 55 mH, and 3.5nF, respectively. The negative resistance is created by using IC CA3080. The biasing values selected for IC CA3080 to make it negative resistance are V b = 2.5 V and R b = 52 . Fig. 12 displays the chaotic output plotted for capacitor voltage vs. current through L 1 inductor. As chaotic outputs are very sensitive to components value, even a small change in the value will create a significant change in the output. Fig. 13 shows the output of chaotic circuit with a small change in capacitor voltage. All other component values are the same, and the capacitor value is now changed to 5.5 nF. Chua's circuit can be used in many applications such as: secure communication, analog noise generator, or to improve performance of ultrasonic device in the presence of crosstalk and noise.

VIII. CONCLUSION
This article presents two types of mem-element emulators, memristor and memcapacitor, using CCIIs, multiplier, and some passive components. Both grounded and floating mem-elements are presented. The memristor and memcapacitor emulators can be easily changed by simple switching. Also, the incremental and decremental configurations of the mem-element emulator can be simply changed by merely switching the connection. The different values of a parameter VOLUME 8, 2020 are required for the circuit to work in both the configurations. The theoretical analysis and frequency analysis have been carried out. Simulation and experiment of the proposed memelement emulator have been undertaken. Simulation results are matched with the experimental results, thus validating the theoretical proposition. A chaotic system has been designed using floating type memristor emulator as part of an application of the proposed emulator. His research interests include analog and digital integrated circuits, discrete time analog filters, non-linear circuits, data converters, and ultralow-voltage building blocks for biomedical applications. VOLUME 8, 2020