Electronically Charge-Controlled Tunable Meminductor Emulator Circuit With OTAs and Its Applications

Studies on new passive elements with memory properties such as memristor, meminductor, and memcapacitor have increased recently. In this article, an operational transconductance amplifier (OTA) based meminductor simulator circuit without using a memristor, which is suitable for the physically produced memristor structure, has charge control, floating structure, and characteristics can be controlled electronically, is proposed. The OTA IC used is based on $0.18\mu \text{m}$ CMOS technology. The meminductor emulator circuit works in high-frequency regions and has low power consumption. Furthermore, the meminductor emulator circuit can be operated both as a decreasing model and an increasing model. Temperature, Monte-Carlo and worst-case analyzes are performed to verify the robustness of the proposed circuit. In this study, a chaotic circuit is designed by taking advantage of the nonlinearity of the proposed meminductor circuit. In addition, a neuromorphic circuit application has been carried out to demonstrate the memory ability of the proposed meminductor circuit. The electronic tunability of the proposed meminductor circuit in both chaotic and neuromorphic applications are demonstrated. The electronic simulations of the applications realized with the proposed meminductor emulator circuit, which are obtained by using the LTspice program.


I. INTRODUCTION
Memory circuit elements have become an interesting research subject in recent years, thanks to their dynamic properties. Examining the relations of current, voltage, load and flux equations with each other, Prof. L. Chua concluded that besides the existing passive circuit elements, there should be another two-prong basic element. Thus, in 1971 for the first time, working on the relationship between flux (ϕ) and charge (q), memristor was presented as a new circuit element by Prof. L. Chua [1]. In 2008, Hewlett Packard Labs researchers introduced the first physical-state device, which satisfies the theoretical definition of the memristor [2]. Since the current flowing through the memristor element at any The associate editor coordinating the review of this manuscript and approving it for publication was Ludovico Minati .
instant depends not only on the current voltage, but also on the voltages applied in the past, where it exhibits non-linear characteristics.
When no voltage is applied to the memristor, no current can flow through it, so the memristor element cannot store energy as in the resistor element. Memcapacitor and meminductor, which are modeled based on the concept of memristor and can store energy unlike memristor, have been proposed as two new generalized memory elements (memelements) [3]. In recent years, memristive circuit elements, have been widely used in areas such as non-volatile memory [4], adaptive learning [5], [6], [7], [8], [9], and artificial neural networks [10], [11], [12] etc. Memristors, changing their conductance as a function of the history of voltage differences across the device and retaining the memory of the last resistance, replicate several aspects of synaptic plasticity, like weight evolution, and combine it with weight storage and weight effect [13]. In 2010, Jo et al. demonstrated the usefulness of memristors in neuromorphic execution circuits by experimentally realizing synaptic functions using nanoscale silicon-based memristors at a two-terminal synapse with a crossbar configuration [14].
Due to the cost and technical difficulties of microelectronic production, memristive elements will not be commercially available soon. Therefore, studies on these three memristive elements have focused on their simulation models and emulator circuits. This study focuses on meminductor emulator circuits. Some meminductor studies are given in Table 1. A grounded meminductor emulator circuit operating in a narrow frequency range using a memristor and an opamp active element is presented in [15]. It is quite easy to design meminductor or memcapacitor emulator circuits by using memristors. However, in this case, a design depending on the electrical characteristics of the memristor is required and the complexity of the circuit will increase. In [16], a meminductor emulator circuit is presented using only 1 MO-OTA active element, but it is not suitable for integrating the capacitor and the inductor in series for the emulator circuit. A general circuit structure is proposed in [17] that implements memristor, meminductor and memcapacitor emulator circuits individually. However, it is a quite disadvantage that increasing number of different types of active elements are given in the equivalent emulator circuit in the floating structure. Due to the OTA in the structure, the meminductor emulator circuit, which has electronic controllability and is designed only as an incremental model, is proposed in [18]. The floating meminductor element, which can operate up to 1 MHz and is designed with only 2 VDTA elements, is presented in [19]. The grounded meminductor circuit in which OTAs and an analog multiplier are used as active elements is given in [20]. The emulators in the studies [21], [22], [23] are not electronically controllable.
Thanks to the dynamic properties of meminductor elements, it is expected to be used as a key element in chaotic circuits. Various chaotic attractors have been designed by using the meminductor emulator circuits presented in the VOLUME 11, 2023 53291 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
literature [22], [24], [25]. Simulations in the studies have revealed that not only the memristor but also the meminductor can be used to create a chaotic oscillator.
In this study, a floating charge-controlled meminductor emulator circuit is presented for both decreasing and increasing models by using active and passive elements. The results obtained from the LTspice program are given for the different input currents, frequencies and bias currents considering the OTAs of the meminductor emulator circuit. This paper also presents a new 3D chaotic oscillator using the meminductor emulator circuit. Phase portrait and time domain analyses of the chaotic circuit application designed with the proposed meminductor are given. Moreover, an adaptive learning application is carried out with the meminductor proposed in this study.
This paper is organized as follows. In Sec. II, the charge-controlled floating meminductor emulator circuit is proposed. The circuit simulation results and analyses are shown in Sec. III. In Sec. IV, the meminductor chaotic circuit is designed and the simulation is presented. In Sec. V, adaptive learning application and electronic simulations of the application are given. Finally, the conclusion part of this paper has given in Sec. VI.

II. CHARGE-CONTROLLED FLOATING MEMINDUCTOR EMULATOR CIRCUIT
In the study proposed by Di Ventra et al. in 2009, the definition relations of the n-th order charge-controlled meminductor are given in (1) and (2) [3]. x Here ϕ L (t) and i(t) on the meminductor at time t; represent the accumulated flux value and the flowing current respectively. By using (1) and (2), the following arrangements can be made.
Here q is the charge on the meminductor and is defined as the integral ofi (t) with respect to time t.
Here a and b are real numbers.
In the literature, a grounded meminductor emulator circuit is proposed using one MO-OTA and one OTA [20]. In this study, a floating structure has been obtained by directly designing the input terminals of the OTA as the input terminals of the meminductor element. In addition, both the increasing model and the decreasing model are applied by utilizing the outputs of MO-OTA. The proposed charge-controlled meminductor emulator circuit is shown   in Fig. 1, which consist of OTA, MO-OTA, analog multiplier and 3 passive elements. In addition, the 0.18µm CMOS technology-based circuit structure of MO-OTA is given in Fig. 2. The analog multiplier circuit used in this study is as in [26] and is given in fig. 3. The relation between input and output parameters of MO-OTA is given in (8).
The analog multiplier circuit in fig. 3 is connected to the proposed circuit in fig. 1 as follows: port V 2 is connected to the V C2 node and port V 1 is connected to the V R node. V 1 ' and V 2 ' ports are connected to GND.
Here, the input currents of the OTA element I P and I N are zero and it can be verified from (8). The meminductor input current is transmitted to the output. g m1 and g m2 are the transconductance gain of OTA and MO-OTA, respectively, where OTA's output current is I 1 ; The input voltage of the meminductor is V in ; The (9) and (10) can be arranged as; The flux of the meminductor emulator circuit can be calculated as follows; The input voltage of MO-OTA at P 2 terminal; According to (12) and (13) V P2 is rewritten as follows; Hereq is the charge of the C 2 . V M is the input voltage of the N 2 terminal of the MO-OTA, according to (16) and (17); The flux value of the meminductor is obtained as follows; The inductance value, which is obtained by using (2) and (20) equations of the meminductor depending on the charge is computed as follows; As it can be seen from (21), a decreasing type of meminductor has been obtained. If the R 1 is connected to the I ZP3 terminal of the MO-OTA, an incremental type of meminductor is obtained as given in (22) and (23).  Table 2.  A pinched hysteresis curve should be formed on the current flux graph of the meminductor. The flux obtained against the input current for both increasing and decreasing models of the presented meminductor emulator circuit is given in fig. 4. Where the amplitude of the input current is 100nA and the frequency is 1KHz. The parameters for the meminductor are selected as follows; I bias = 75µA, C 1 = 5µF and C 2 = 1.5nF.
The hysteresis curves of the proposed meminductor emulator circuit obtained against input currents of different amplitudes at 500kHz frequency are shown in Fig. 5. Here I bias = 300µA, C 1 = 250pF and C 2 = 200pF are selected. VOLUME 11, 2023 53293 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  The hysteresis curves of the proposed meminductor emulator circuit obtained by the constant amplitude input current as 5µA at different frequencies are given in Fig. 6. Here I bias = 350µA, C 1 = 120pF and C 2 = 90pF are selected.
The hysteresis curves of the proposed meminductor emulator circuit obtained against input currents of different amplitudes at 1MHz frequency are shown in Fig. 7. Here I bias = 350µA, C 1 = 120pF and C 2 = 90pF are selected.
One of the important features of the presented meminductor circuit is that it can be controlled electronically. In Fig. 8, a pinched hysteresis curves are given for different bias currents of OTAs versus input current at 500KHz frequency and 5µA amplitude. Here I bias = 300µA, C 1 = 250pF and C 2 = 200pF are selected.
Finally, the memory feature of the proposed meminductor circuit has been tested. A square wave current with an amplitude of −1µA and a period of 1µs is applied to the input of the proposed decreasing type meminductor emulator circuit, and the response of the meminductor flux to the input current is shown in Fig. 9.   Similarly, the proposed increasing type meminductor is tested with a square wave current of 1µA amplitude and 53294 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. 1µs period and the response of the meminductor flux to the input current is shown in Fig. 10. In fig. 9 and fig. 10, I bias = 350µA, C 1 = 120pF and C 2 = 90pF are selected.
The fluxes specified in Fig. 4-10 are calculated automatically over a period through the LTspice program as given in (12). For the meminductor circuit presented in fig. 11, the current-flux relationship at different temperatures is given.
In Fig. 12, the current-flux relationship obtained from the Monte-Carlo simulation is given for the proposed meminductor circuit. Here, resistors are selected with 10% tolerance and capacitors are selected with 20% tolerance, and a simulation is conducted for 100 steps.
A worst-case analysis is given for the meminductor circuit presented in fig. 13. Here, as in the Monte-Carlo analysis; resistors are selected with a tolerance of 10% and capacitorsare selected with a 20% tolerance, and a simulation of 100 steps is made.   For the figs. 11-14, the input current is a sinusoidal with 100nA amplitude, the operating frequency is 1 kHz and the bias current of OTA is 75µA. All simulations are obtained by using the LTspice program.

IV. MEMINDUCTOR BASED CHAOTIC CIRCUIT APPLICATION
The memristive elements are frequently used in chaotic applications, thanks to their nonlinearity. A flux controlled memristor-based chaotic circuit is presented for the Lorenz system in [24]. In [25], the mathematical models of the meminductor and memcapacitor were analyzed through the Chua circuit. A chaotic circuit using a flux controlled memristor, a flux controlled meminductor and a capacitor is proposed in [22]. The meminductor was designed by using two memristors and one op-amp and its chaotic implementation is presented in [15]. The chaotic application of flux controlled meminductor designed by using CDTA and OTA is presented in [9].
The main challenge of designing chaotic circuits based on meminductors lies in the control method. As it is known, the meminductor is designed in two different types chargecontrolled and flux-controlled. Since flux controlled meminductor circuits are easily applicable to circuit node equations, their chaotic applications are easier than charge controlled meminductor circuits. The physically produced memristor is produced in a charge-controlled structure. We estimate that the physical state of the meminductor will also be produced in charge-controlled type. For this reason, charge-controlled meminductor circuit is designed in this study. In this study, a chaotic circuit is designed based on the nonlinear relationship between current and flux of the proposed charge-controlled meminductor emulator circuit. The chaotic oscillator circuit consisting of a negative resistor (−R 2 ), a capacitor (C), an inductor (L), the proposed meminductor element (L M ) and a resistor (R 3 ) is shown in Fig. 15. The negative resistor (−R 2 ) circuit is made up of an op-amp (AD711), positive feedback resistor, negative feedback resistor, and a series resistor.
According to Kirchhoff circuit laws, the following differential equations can be obtained: where the current of the inductor (L) and the meminductor (L M ) is represented by I L and I LM , respectively. The values of the chaotic circuit elements simulated using the LTspice program are as follows: L=55mH, C=0.75nF,  in Fig. 17. In both fig. 16 and fig. 17, the bias current of OTA (I Bias ) is 75µA.
For secure communication, changing parameters such as amplitude and phase against any threat in chaotic communication circuits can be a solution. By controlling the bias currents of the OTAs in the structure of the proposed meminductor-based chaotic circuit, the characteristic of the phase portrait of the chaotic circuit can be adjusted. To show the electronic controllability of the chaotic circuit designed with the proposed meminductor, the phase portraits obtained in the LTspice program for the different bias currents of the OTAs are given in Fig. 18-20.

V. MEMINDUCTOR BASED ADAPTIVE LEARNING CIRCUIT APPLICATION
When the behavior of living things is to be modeled, this task will be quite complex due to the brain structure of humans and animals. For this reason, existing since the formation of the world; Modeling the behavior of amoeba single-celled organism, which has intelligence abilities such as memorization, timing and prediction, would be a suitable start. The anatomy and behavior of an amoeba cell is studied in [27].
An amoeba slows down when the ambient temperature drops. When the temperature is no longer falling, the amoeba can predict the time of the next temperature drop by slowing down again at times when the temperature drop would occur. Also, the amoeba can trigger oscillations when the temperature drops again. Thus, the amoeba cell remembers the past and can predict the timing of future events that are like past events. In order to model this feature of the amoeba, scientists have tried to produce the electronic circuit equivalent of the behavioral model of the amoeba by using memory circuit elements. An adaptive learning application with a memristor is designed by using DVCC and OTA is presented in [5]. An adaptive learning application with a memcapacitor VOLUME 11, 2023  implemented with DXCCDITA is presented in [8]. An adaptive learning application with a meminductor developed using a memristor and a VDCC is demonstrated in [8].
The proposed meminductor-based adaptive learning circuit is presented in fig. 21. The temperature changes that occur in the environment around the amoeba are represented by the applied input voltage (V in ). The resistor (R), capacitor (C) and meminductor (L M ) form the electronic model of the decision-making mechanism of the amoeba cell. The change in locomotive speed in response to the change in the ambient temperature of the amoeba cell is also represented by the output voltage (V out ).
In Fig. 21, R=1 and C=0.5pF are selected for adaptive learning circuit. The pulses applied to the input of the circuit are given in fig. 22 a). The voltages obtained at the output of the adaptive learning circuit are given in fig. 22 b) and fig. 22 c) for fixed bias current and different bias currents, respectively. In Fig. 22 b), the bias current of the OTAs in  the meminductor circuit is taken as 75µA. In Fig. 22 c), the bias current of OTAs is applied for 3 different values, 25µA, 50µA and 75µA.
In Fig. 22b), the response of the first pulse signal decreased to -120mV. Learning starts with the second pulse signal and the response signal drops down to -126.25mV. It outputs as -127.02mV in response to the 3rd pulse signal. The 4th and 5th Pulse responses are at -120mV. A similar learning situation is also valid for fig. 22 c). For the first three consecutive pulses, the second and third pulse responses increased in the negative direction. In addition, it is understood from fig. 22 c) that the pulse response intensity and speed can be controlled by the bias currents of the OTA active element.

VI. CONCLUSION
In this paper, a new floating both increasing and decreasing model meminductor emulator circuit is proposed. The presented meminductor emulator circuit consists of MO-OTA, OTA, an analog multiplier, one resistor and two capacitors. Thanks to the bias currents of the OTAs used, electronic controllability is provided on the meminductor. All passive elements are grounded. However, due to the use of the multiplier circuit in the proposed circuit, the complexity of the circuit has increased, and it has created a disadvantage for the proposed circuit. During tests of the memory feature of the proposed meminductor emulator circuit, it has been shown that the flux value is stored at the output by applying input currents with positive and negative square waveforms.
In addition, the operation of the presented meminductor emulator circuit for input currents from 100nA to 7µA is shown. The operation of the proposed meminductor emulator circuit from 1KHz to 1MHz is provided by simulations. The minimum power consumed in simulation analysis is 3.87mW.
Thanks to the nonlinearity of the meminductor element, a meminductor-based chaotic circuit is proposed in this study. The time domain and phase portrait simulation results of the chaotic circuit application are given. In addition, thanks to the bias current of the OTAs in the structure of the proposed meminductor circuit; It has been shown that the chaotic circuit characteristic can be controlled for both the time domain and the phase portrait. In this study, an adaptive learning application is also carried out by using the proposed meminductor circuit. An equivalent electronic circuit is designed based on the behavior of an amoeba cell; It has been observed that it gives responses similar to the behavior of amoeba cells to the applied pulses.
The potential application areas of this study are: It can be used as a source of randomness in random number generator circuits in encrypted communication systems. Since the electronic characteristic is adjustable, the characteristic of the random generator instantly becomes changeable. In addition, it can be used as a simplified brain structure for adaptive learning processes. The effects of the learning process in terms of time and electrical intensity can be analyzed thanks to its electronically adjustable feature.