High-Frequency Tunable Grounded&Floating Incremental-Decremental Meminductor Emulator and Application

This paper proposes a new design for realizing grounded and floating meminductor emulators built with two operational transconductance amplifiers (OTAs) and two second-generation current conveyors. The proposed grounded and floating emulators claim that the circuits are much simpler in design and can be utilized in incremental and decremental topologies. The proposed circuits' performance has been verified with Cadence Virtuoso Spectre using standard CMOS 180nm technology. Furthermore, the layout of the proposed circuits has been designed, and post-layout simulations have been performed. The non-ideal and Monte Carlo analyses have been carried out in detail. This paper also proposes the application of a meminductor as an Amplitude Modulator (AM). Moreover, the experimental results are presented to verify the theoretical and simulation analyses of proposed meminductor emulator circuits.


Introduction
Resistor, inductor, and capacitor were three traditional fundamental basic electrical elements; now, the memristor represents the fourth fundamental element.Chua postulated the memristor in 1971 [1] as the fourth basic electrical element.In 1980 this postulation was then generalized to an infinite variety of basic circuit elements [2] and can be generalized into elements quadrangle.It was highlighted only after 2008 when HP fabricated a memristor based on thinfilm TiO2.From then onward, there has been a boom of research in this field.Chua's circuit element quadrangle was then extended to propose higher-order elements (which require two or more than two Cs ), such as memcapacitors (MCs) and meminductors (MLs).The meminductor provides a relationship between the charge q and the time integral of flux ρ.Unlike capacitors and inductors, meminductors can store information for a long without power because of their non-volatile property.Although the device is still a theoretical concept, some device-level memelements (only memristor) have been fabricated [3], and emulators are essential to analyze the characteristics and study their applications.The research on solid-state memelements is yet to mature completely, especially for MCs and MLs.Solid-state MCs have not been commercialized, and there has been no information on solid-state MLs.Some models, though directly labeled as "memristors" or "memcapacitors," are essentially practical MR emulators [3].
Therefore, a substantial number of circuit implementations have been proposed through the use of emulators.In [4], a relationship on the doubly periodic table of circuit elements, also called the four elements torus, is given in correlation with the basic circuit element quadrangle of all four basic electrical elements.Furthermore, an extension of the memristive system to capacitive and inductive elements whose properties depend on the state and history of the system is presented in [5].Physical characteristics analysis of these memory-based elements and mathematical examples for memristor, meminductor, and memcapacitors are presented in [6,7].
Several circuits for emulating memristor-less meminductors are proposed in [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], while meminductors formed using mutators are proposed in [25][26][27][28].In [8], a memristor-less current and voltage-controlled meminductor emulator are reported using a second-generation current conveyor (CCII), adder, multiplier, and several passive components in the count.A chargecontrolled meminductor emulator using an inductor, op-amps, multiplier, transistors, and several other passive components is reported in [9].In 2014, a practical implementation of the meminductor using many active and passive components was reported [10].It consists of four current feedback operational amplifiers (CFOAs), one buffer, two op-amps, one multiplier, and some passive components making the circuit quite complex and bulky.A flux-controlled meminductor is reported [11] but consists of many active blocks and passive components.The meminductor reported in [12] is based on six op-amps and one multiplier, whereas the design in [13] employs three CFOAs, one op-amp, one operational transconductance amplifier (OTA), and one multiplier.The design reported in [14] is based on two voltage differencing transconductance amplifiers (VDTAs) and one multiplier.In 2017, a much simpler circuit for emulating a meminductor was reported using multioutput OTA [15], but it uses an inductor and has a low frequency of operation.All the reported circuits [8][9][10][11][12][13][14] are complex as they employ multiplier along with an excessive number of active blocks, and [15] have a shallow frequency of operation, which very much limits the practical use and, further, it realizes only grounded meminductor.
The meminductor design in [16] is based on three OTAs and two capacitors.The design reported in [17] is based on two OTAs and one differential voltage current conveyor (DVCC), and the design in [18] is based on one OTA and one VDTA.The topology in [19] reports a memeinductor employing two OTAs and one current differencing buffered amplifier (CDBA), whereas [20] is based on two OTAs and one current differencing transconductance amplifier (CDTA).
Meminductor design in [21] employs two CCIIs and one OTA, whereas design [22] is based on two VDTAs.Moreover, [23] reports a design based on one modified voltage differencing current conveyor (MVDCC) and one OTA.However, all the designs reported in [8-12, 15-18, 20-23] realize only one type of meminductance emulator, i.e., the grounded or floating meminductor; only [19] realizes both grounded and floating meminductance emulator.Furthermore, the designs reported in [21][22][23] realize only one type of meminductance emulator and possess a low frequency of operation.Moreover, [22,23] realize only incremental type of configuration.
Another method of emulating meminductors is with mutators proposed in [24][25][26][27][28]. Mutators simulating second-order elements using their inherent relationship are reported in [24,25] and represent the simplified meminductor emulator using the multiplier approach, but the memristor used here is bulky, complex, and has a low operating frequency.A mutator based on one CCII and three op-amps is reported in [26].A universal mutator using many active and passive components is also available [27].Moreover, a mutator circuit based on two current buffered transconductance amplifiers (CBTAs) and one multiplier is reported in [28].Apart from these mutators, the PSpice model of meminductor and its nonlinear model with its study on device parameter variations are available in [29,30].A detailed composite behavior in series and parallel meminductor topologies is reported [31].The applications of meminductors in chaotic oscillators and their dynamic studies are reported in [32][33][34], whereas application as a low-power filter design is available in [35].This paper proposes a meminductor emulator built with active blocks consisting of two OTAs and two-second generation current conveyors.The proposed memristor emulators possess the following important features: (i) simple circuitry with no requirement of multiplier, (ii) grounded and floating configuration from the same topology, (iii) option for both incremental and decremental configurations to increase the range of values of meminductance ( the value of meminductance can be increased and decreased from its base value in incremental and decremental types of topology respectively), and also application flexibility, (iv) high-frequency range of operation, (v) electronic control of meminductance value in addition to the control by frequency and amplitude of the applied voltage signal across emulator.The port relationships of OTA are expressed as:

Employed Analog Building Blocks and General Meminductor Model
Where Gm is the transconductance of OTA.The routine analysis results in the following expression for Gm: Here, k is a parameter of the MOS device given by: The W, L, µn, Cox, and Vth are, respectively, channel width, length, the mobility of the carrier, capacitance per unit area, and the threshold voltage of MOS.
The port relationship CCII± is given by: Similarly, port relationship CCCII± is given by: where,   = ρ(t) is the integral of ϕ(t), i.e.
The relation between input current I(t) and ϕ(t) of meminductor is defined as; −1 is inverse meminductance and the general representation of flux controlled meminductor having an initial value of inverse meminductance given by 'm' and decremental or incremental product term given by n is expressed as [14,18]; or, () = ( ± ρ(t))ϕ(t) It can be inferred from (3b) that a meminductor model contains 'm' as a fixed term and nρ(t) as a time varying term.

Proposed Grounded and Floating Meminductor Emulator Circuit
Schematic diagrams of the proposed grounded and floating meminductor emulators are shown in Fig. 4 and Fig.
Similarly,  2 =  2 −  2  2 (by the port relationship of CCCII) Substituting (4) in (3) Dividing ( 5) by ( 2) Bias voltage VB3 is given by Where   is the total flux obtained by the meminductor, and it is expressed as Substituting ( 7) into (1), the transconductance Gm3 is obtained as Also, Using ( 10) and ( 9) results in an expression On substituting (11) in (6) and by incorporating the relation of ( 8), meminductance (LM) of the proposed grounded incremental meminductor emulator is obtained as Similarly, the change of switch connections to w-z and y-x changes the polarity of the timevariant part of the meminductance of ( 12), resulting in a decremental type meminductance as Equations ( 12) and ( 13) can be combined and rewritten as If the operating frequency is much less than where RX2 is the resistance at input port X of CCCII and is usually RX2 kept low [39], then we can write (1 +  2  2 ) ≈ 1.
Hence, equation ( 14) can then be simplified in terms of the inverse of LM as: In equation ( 15), Gm4 can be controllable by external bias voltage VB4, which makes the proposed circuit electronically tunable.The equations ( 15) represent incremental and decremental meminductance, where for a fixed operating frequency 2 ) is the time-varying term as in  is the function of the time-varying input signal.
For 0 = in  meminductance attains a constant value in both the topologies (incremental and decremental), where for the operator±, the + is for decremental andis for incremental configuration.

Comparison of meminductor emulators
A comparison of available meminductor emulators is given in The proposed work presents both grounded and floating types of meminductor realizations using simple basic blocks, two CCII/CCCII, and two OTAs with only two capacitors and one resistor.All the passive elements in both of the proposed meminductor circuits are grounded.
Moreover, both the incremental and decremental properties are present in the proposed emulators.Further, the proposed grounded and floating meminductors are valid for frequencies of 1 MHz and 10 MHz, respectively.An important feature of the proposed meminductor emulator is its ability to control the meminductance value by controlling the transconductance, Gm4 with the bias voltage, VB4; hence both the emulator circuits are electronically tunable.Power consumption of the proposed circuits is greater than that of [14,15], however smaller than that of [21] among the available literature.

Simulations results and discussion
This section deals with verifying the hysteresis loop between flux and the current, one of the fingerprints of the meminductor.Various simulations with 180 nm CMOS technology have been performed to verify the meminductive nature of proposed emulator circuits.Supply voltages of +1.2V and -1.2 V are used for VDD and VSS, respectively, for grounded and floating meminductors.The aspect ratios of MOS transistors are given in Table 3, and they operate in the saturation region.

Grounded meminductor simulation result
The grounded meminductor emulator in Figure .4 is simulated for different frequencies.The results for the pinched hysteresis loop obtained for a sinusoidal signal of amplitude (Am=140 mV) with frequencies of 100 kHz, 200 kHz, 300 kHz, 400 kHz, and 500 kHz are shown in Figure .6. Here, the product of the capacitor (C2) value and frequency (f) is kept constant (75 x 10 -6 FaradHz) and VB4=450 mV.On increasing the frequency, the pinched hysteresis loop area of Φ-I curves decreases, which satisfies (14), suggesting that the time-varying nature of the loop decreases and ultimately vanishes at a specific frequency.Figure .7 shows the relationship between charge q(t) and ρ(t), where ρ() = ∫ () and q() = ∫ ().Additionally, the current, () flowing in a meminductor can be expressed as per [6].The single valued function q(ρ) has been illustrated in detail in [6].This can also be verified graphically from Figure .7 as a single valued curve is obtained, implicitly implying that the corresponding device is a meminductor.FaradHz.

Floating meminductor simulation result
The floating emulator's simulation results for the pinched hysteresis loop obtained for frequencies of 1 MHz, 2 MHz, 4 MHz, 6 MHz, and 8 MHz are shown in Figure .8. Here, the product of capacitor(C2) value and frequency (f) is kept constant (75 x 10 -6 FaradHz) with Am=200 mV, VB4=500 mV.On increasing the frequency, the pinched hysteresis loop area of Φ-I curves decreases; this validates the meminductive behavior of emulators as obtained in (34).
Fig. 9 shows the relationship between charge q(t) and integral of flux, ρ(t).It shows that q(t) is a single-valued function of ρ(t).Therefore, the device current goes through a meminductor.On increasing the frequency, the pinched hysteresis loop area of Φ-I curves decreases, which satisfies (14), suggesting that the time-varying nature of the loop decreases and ultimately vanishes at a specific frequency.Figure 9 shows the relationship between charge q(t) and ρ(t).
As discussed earlier for Figure 7, the curve in Figure 9 is also a single valued.Therefore, the device current flows through a meminductor.FaradHz.

Effect of variation of bias voltage (VB4) of OTA on the pinched hysteresis loop
It is seen in ( 15) and ( 34) that the meminductance of emulators depends on transconductance Gm4, which is electronically tunable by the external bias voltage, VB4,.

Effect of varying frequency and capacitances on the pinched hysteresis loop
The effect on Φ-I characteristics for variation of applied signal frequency for a fixed capacitance shows that as the frequency increases from 800k Hz -4 MHz for a fixed capacitance value, the area under the hysteresis loop gradually decreases as predicted by (34).Similar to the effect of frequency variation, area of hysteresis loop of meminductance changes for the variation in capacitances (C1 and C2) and this can also be verified by observing these parameters in ( 15) and ( 34).On varying C1 and C2 with a fixed frequency of 500 kHz for grounded topology and 800 kHz for floating topology, the results are obtained as shown in Figure .11(c-f).meminductor for variable C2 at 800 kHz frequency, (e) grounded meminductor for variable C1 at 500 kHz frequency, (f) floating meminductor for variable C1 at 800 kHz frequency.

Layout, post-layout simulations, and Monte Carlo analysis
Layouts are obtained, and post-layout simulations are carried out for both grounded and floating topologies to check the effect of parasitics on the hysteresis.Further, the simulation results of the Monte Carlo analysis of proposed emulator circuits are also presented.

Nonideality Analysis
Non-ideal transfer gains and parasitics of active building blocks will have an impact practically.
Sections 6.1 discusses the effect due to non-ideal transfer gains, and sections 6.2 and 6.3 discuss the effect due to OTA and CCII/CCCII parasitics.

Nonideality effect of OTA transconductance gain and CCII/CCCII current transfer gains
Due to the OTA's non-ideal transfer gain, the port relationship is modified as follows: is the non-ideal transconductance gain coefficient from the input terminal to the output terminal of OTA, which is ideally considered unity.
Similarly, the port relationships of CCII and CCCII due to non-ideal transfer gains modify as follows: [ Where j=1 stands for the first CCCII and j=2 for the second CCCII. and  are current and voltage transfer gains from X to Z and Y to X terminals for CCII/CCCII, respectively.Ideally, and  are unity.
The routine analysis of Fig. 4 considering non-ideal port relationships from (35), (36), and (37) for grounded meminductor configuration results in the following: Similar analysis for the floating topology of Fig. 5 results in : [where 1   2 are the transconductance gains of grounded and floating meminductors' first and second OTAs.It is observed from (38) and ( 39) that non-ideal transfer gains affect meminductance.Fig. 15 shows the non-ideal model of OTA where (Ri, Ci) and (R0, C0) are input and output parasitic capacitances and resistances, respectively.The capacitances across the input ports are assumed to be equal.16.Non-Ideal model of the CCII/CCCII [38].
Also, on applying KCL at node C and by port relationship Using (43) in (42) results in On applying KCL at node B Using ( 44) and (45) results in ] ≈ 0 as ( 3 ≈ ∞).The substitution of these values in (48) results in the following: (49)This is similar to (14).Hence it can be concluded that the effect of parasitic on the proposed grounded circuit in the frequency range of a few MHz is negligible.
By applying the port relationship at the X2 terminal of CCCII and using (52), we get: On substituting (62) in (58) results for both the incremental and decremental topologies as; It is inferred from (63) that meminductance will be affected by parasites, however on applying the typical practical values of CMOS OTA parasitic obtained from routine analysis and [37]  (64) This is similar to (32) and (33).Hence it can be concluded that the effect of parasitic on the proposed floating meminductor in the frequency range of low to 10 MHz is negligible.

Application of meminductor as amplitude modulator (AM)
An AM modulation scheme with a meminductor is carried out as an application of the proposed meminductive device.Schematic of various circuits along with a floating meminductor emulator is shown in Figure .19.In Figure .19(a), a multifunction filter [39] using OTA is given, which can implement both bandpass filter (BPF) and low pass filter (LPF) responses using the components, as given in Table 4, for Y1, Y2, Y3, Y4, and Y5.
Thus, the output current of the meminductor, IMI(t), of Figure 19

Experimental results
As monolithic implementations of the proposed meminductor are not available, the possible experimental realization of meminductors using commercially available ICs, AD844 and CA3080, is shown in Fig. 21 (a, b), followed by the assembled grounded meminductor emulator on a breadboard is given in Fig. 21 (c).This circuit can broadly verify the functionality; however, detail parameters cannot be comparable with monolithic implementation.To verify experimentally, the proposed meminductor emulator is implemented with capacitance (C1) as 40nF (in a parallel combination of four 10nF), resistance (R1) of 100 Ω, and commercially available BJT-based OTA ICs (CA3080), and CFOA ICs (AD844).The pinched hysteresis loops of the emulator are obtained for the operating frequency of 820 kHz for a 4V peak-to-peak input signal, as shown in Fig. 22(a).Moreover, an AM modulator has been realized using the proposed meminductor emulator circuit as an application.Finally, a prototype of the proposed circuit is assembled, and experimental results are given and discussed.

Figure. 10 (Figure. 10 .
Figure.10.Φ-I characteristic curves obtained with sinusoidal current signal with 500 kHz, for both grounded and floating meminductor emulators are shown in Figures.11(a) and 11(b), respectively.Figure.11(a) shows that as frequency increases from 400 kHz -800 kHz for a fixed capacitance value, the hysteresis loop becomes more and more linear, which satisfies (15), suggesting the time-varying nature of the loop decreases.Figure.11(b)

Figure. 11 .
Figure.11.Φ-I characteristic curves for a sinusoidal current signal of Am=140 mV and

Fig. 16
Fig.16shows the non-ideal model of CCII/CCCII.RX represents a low-value parasitic resistance

Fig. 17
Fig.17shows the non-ideal model with device parasitics of proposed grounded meminductor
Figure.19(c) is used to demodulate the AM signal to recover the message signal.

Fig. 20 .
Fig. 20.The plot of (a) carrier signal, message signal, and modulated signal, (b) spectrum of the

Fig. 22 (
b) shows the pinched hysteresis loops for a floating meminductor (hardware implementation not shown) emulator at an operating frequency of 910 kHz for a 4V peak-to-peak input signal.Fig.22(c) shows the time domain waveform for the grounded meminductor's current and charge (integration of current).Fig.22(d)shows the time domain waveform for input phi and rho (integration of phi) for the grounded meminductor.It can further be noted that both charge and rho curves tend to increase as time increases which justifies that the proposed circuit is a meminductor.It can further be noted that the integration of the input current and phi across the capacitor is performed through the inbuilt integration function in the oscilloscope to prevent any loading effect and loss of signal.Fig. 22(e) shows the time domain waveform for flux and current for the grounded meminductor.Analysis of Fig. 22(e) reveals that the proposed circuit possesses a nonlinear relationship between flux and current and is simultaneously zero, thus verifying passivity.

Table 1 .
These mechanisms apply to both grounded and floating meminductor emulators.

Table 1 .
Connection topology for pins M, N, O, and P for two modes of operations.

Table 3 .
Design Parameters for analog blocks in grounded meminductor 3 =  4 = ∞,   = 1,   = 50,  0 = 100.Similarly, values of CCII/CCCII obtained from routine analysis and[40]can be assumed to be a few ohms for Rx, a few hundreds of MΩ for RY, and few MΩ for RZ, and the Cy, Cz are in the range of a few femtofarads.If the operating frequency is within the range of a few MHz, then we may use can be assumed approximately as;   = ∞,   = 1,   = 50,  0 = 100, while that of CCII/CCCII as discusses in above.If the operating frequency is within the range of a few MHz (say 10 MHz), then we may approximate the following as follows:  2 =  2 , Z 4 =  1 ,  1 = , = 1 ,  3 ≈ very high, and hence, [