- 1School of Information and Communications Engineering, Xi’an Jiaotong University, Xi’an, China
- 2Department of Communications Engineering, Graduate School of Engineering, Tohoku University, Sendai, Japan
- 3Department of Electronic Science, Xiamen University, Xiamen, China
- 4Key Laboratory of Cognitive Radio and Information Processing (Ministry of Education), Guilin University of Electronic Technology, Guilin, China
In this paper, a Hewlett-Packard (HP) memristor model with a new window function and its versatile characteristics are presented. SPICE behaviors of the linear and nonlinear memristor model are studied through PSpice simulation. High flexibility is demonstrated for emulating the behaviors of the practical HP memristors. Furthermore, the characteristics of the composite SPICE behaviors are both investigated when two memristors are connected in series and in parallel. The polarity of each memristor is also taken into consideration. The relationships among flux, charge, voltage, current, and memristance of the double memristor circuits are simulated and analyzed.
1 Introduction
There are three fundamental two-terminal passive circuit elements, i.e., resistors R, capacitors C and inductors L, which can be found and introduced from any basic circuit books. Chua first theoretically postulated the existence of the fourth passive electrical circuit element in 1971 [1], named as memristor, whose memristance (i.e., an acronym for memory resistance) M can be defined by the relation of magnetic flux
No great progress was developed in the research field of memristor until the group of Williams at Hewlett-Packard (HP) lab successfully fabricated the first nanoscale memristive device based on titanium dioxide thin film in 2008 [2]. Since then, many researchers or scholars have focused on the memristor research [3–7] and its numerous potential applications in nonvolatile random access memories [8, 9], neuromorphic systems [10, 11], chaotic circuits [12], reconfigurable logics [13, 14] and RF/microwave devices [15, 16].
In order to emulate these memristor applications, various memristor nonlinear drift models using SPICE have been presented before fabricated realization [17–24]. Window functions need to be used to tackle the boundary conditions of the memristors [17–22]. In ref. 17, a switching window function was proposed, where the current i is involved, but it has limited scalability. Two different window functions were presented to mimic nonlinear effects in refs. 18 and 20. However, as the boundary between the doped and undoped layers approaches either end of the memristive device, it cannot be excited by external stimulus. In refs. 20 and 22, the piecewise functions were utilized to construct the window functions, which were relatively complicated when applying in the memristive devices.
In this paper, a memristor model using a new window function, called cosine window function, is proposed along with its versatile characteristics. To illustrate the significance of the proposed window function, main characteristics of state-of-the-art ones in recent years are listed in Table 1 for comparisons. Then, the SPICE behaviors of the linear and nonlinear memristor model are investigated through PSpice simulation. Furthermore, the characteristics of the composite SPICE behaviors are both studied when two memristors are connected in series and in parallel. The relationships among flux, charge, voltage, current, and memristance are simulated and analyzed using PSpice.
2 Memristor Model Using Cosine Window Function
The concept of memristive devices or systems was generalized by Chua and Kang using the following equations [25],
where v is the voltage, i is the current, and
At time t, the width of the doped region w depends on the amount of charge that passed through the memristor. Thus, the time derivative of w is proportional to the current i, which can be expressed by
where
The Eqs. 3, 4 is actually an expression of a linear drift model of memristor, but it does not take the boundary effects of the memristor into account. Nonlinear dopant drift phenomena exist at the boundaries of the physically realized nanoscale memristor, therefore, an appropriate window function f(x) need to be introduced in ref. 4, i.e.,
where f(x) ensures zero drift at the boundaries, i.e., f (0) = f (1) = 0.
On the other hand, the memristance can be expressed by
where
where p is the control parameter. This new and simple cosine window function is demonstrated in Figure 2 for various values of p. As the parameter p rises, a unity window function f(x) will occur, and thus the model tends to a linear model. Here, the parameter p is set with a large range of variation, because the cosine window function is not changed significantly near the region of boundary.
Using this new cosine window function, the behaviors of the memristor can be observed in Figures 3–6 with versatile characteristics. Here, we set
FIGURE 3. Time dependence of (A) the state variable and (B) memristance with voltage sinusoidal stimulus of
Actually, the minimum state variable
FIGURE 4. Influence of the frequency variation of the biased voltage on the time-varying (A) state variable and (B) memristance.
Figure 5 illustrates a nonlinear calculated current-voltage (i-v) curve with p set as 8, where a periodic pinched hysteresis loop with hard switching emerges. This memristive model has zero-crossing property in a form of i-v Lissajous figure, which represents no current through the system when the voltage value is zero. Figure 6 shows the versatility of our model, where the i-v characteristic curves are plotted for eight different values of p. This model demonstrates high flexibility for emulating the behaviors of the practical HP memristors, which can be beneficial in characterizing memristive devices before fabrication.
FIGURE 5. Calculated i-v characteristic of our proposed memristor with p = 8. The inset is the nonlinear single-valued function between charge and magnetic flux.
3 SPICE Behaviors of Double Memristor Circuits
The HP memristor model using cosine window function and its characteristics have been addressed in the above section. To further investigate the relationships among charge, flux, and memristance of composite memristors connected in different topologies, simulated SPICE behaviors of single and double memristor circuits will be presented in details in this section.
3.1 SPICE Behaviors of Single Memristor
The single memristor circuit using the above-mentioned cosine window function has been proposed, and the corresponding SPICE code was written for use in PSpice [26], as seen in Figure 7. The single memristor’s SPICE behaviors can be seen in Figures 8, 9, 10, where the initial state of the linear and nonlinear memristor models are both M(0) = 15.9 kΩ with a sinusoidal input signal of
FIGURE 8. (A) Memristance versus charge curves and (B) flux versus charge curves for the linear and nonlinear model of memristor.
FIGURE 10. (A) Current versus voltage curves, (B) memristance versus charge curves, and (C) flux versus charge curves of the single memristor and two memristors in series with opposite polarities.
As shown in Figure 8A, when the value of p increases, the curves of the nonlinear models will be toward that of linear model, which means that the nonlinearity will be weakened. At the middle area of the whole charging process, the slopes of all the curves with various p are identical including that of the linear model. In Figure 8B, as the value of p increases for a fixed electrical charge, the magnetic flux will decrease, i.e., the nonlinearity will be weakened.
3.2 SPICE Behaviors of Two Serial Memristor Circuits
Then, we construct a circuit with two nonlinear memristors (p = 8), in series with same polarities in PSpice as shown in Figure 11A. The initial states of these two memristors are both M(0) = 11.25 kΩ. Due to the movement of the doping ions caused by the bias voltage, the memristance associated with the doping boundary in the HP model definition will be changed accordingly. Therefore, when the magnitude Vm of the sinusoidal input signal
FIGURE 11. (A) Current versus voltage curves, (B) memristance versus charge curves, and (C) flux versus charge curves of the single memristor and twomemristors in series with same polarities.
Figure 9A shows the relationships between voltage and current of the single memristor (M1 or M2) and two serial memristors with same polarities in Figure 11A, where the magnitude of the sinusoidal input signal Vm = 2.3V. Due to the two memristors connected in series, equal current flows through both memristors. Therefore, the behaviors of each single memristor are identical. Moreover, the applied voltage is distributed across each memristor identically at any instant of time due to same polarities of these two memristors M1 and M2. Consequently, the slope of pinched hysteresis loop of the composite memristor circuit is half of that of the single memristor.
Figures 9B,C illustrate the curves of memristance versus charge (M-q) and flux versus charge (
For a two-serial-memristor circuit with opposite polarities in PSpice shown in Figure 11B, the current versus voltage curves of the individual memristors and two serial memristors are plotted in Figure 10A. The initial states of the two memristances are both set as M(0) = 11.25 kΩ in PSpice software. In contrast to the case of the same polarities in Figures 9A, 10A has three separate hysteresis loops, denoting the individual memristor M1, M2 and composite M of the two memristors. Due to the two memristors connected with opposite polarities, they have the opposite variations. The specific expression is that when the memristance M1 decreases, the other memristance M2 will increase, and vice versa.
Figure 10B shows the relationship between memristance and charge. The initial memristances of M1 and M2 are both M(0) = 11.25 kΩ. Therefore, the memristance of composite M is 22.5 kΩ at first. In the linear region, the amplitude changes of the two memristors are identical, but the trends are opposite. Therefore, the memristance of composite M remains unchanged. When the shift is close to the boundary, memristors are in the nonlinear region. Thus, the memristance of composite M will decrease accordingly. This phenomenon is caused from the boundary effects in physical memristor model. At this time, the two memristors reach the minimum and maximum memristances, respectively, resulting in the composite memristance of approximately 16 kΩ. For the relationship between flux and charge in Figure 10C, it is also different from that of Figure 9C. The magnetic fluxes of M1 and M2 are not the same any more due to the two memristors connected with opposite polarities. In Figure 10C, the slope of composite flux in black is always equal to the arithmetic sum of the two slopes of individual memristor fluxes. Moreover, the composite flux is varied more linearly as a function of charge, compared with those of the individual memristor fluxes.
3.3 SPICE Behaviors of Two Parallel Memristor Circuits
Similarly, the parallel circuits of two nonlinear memristors (p = 8) are constructed in PSpice, where the initial states of the memristiors are M1(0) = M2 (0) = 11.25 kΩ. Since the two memristors are connected in parallel with same and opposite polarities as shown in Figures 12A,B, respectively, equal voltage passes through the two memristors, where the magnitude Vm of the sinusoidal input signal is set as 1.15V.
The relationships between voltage and current of the single memristor (M1 or M2) and two memristors in parallel with same polarities are shown in Figure 13A. Since the initial states of the two memristors are identical, the applied voltage is distributed across each memristor equally at any instant of time. Hence, the behaviors of each single memristor are identical. In addition, the slope of pinched hysteresis loop of the composite memristor circuit is twice as big as that of the single memristor.
FIGURE 13. (A) Current versus voltage curves, (B) memristance versus flux curves, and (C) charge versus flux curves of the single memristor and two memristors in parallel with same polarities.
Figure 13B exhibits the variations of memristance versus flux (M-φ). In contrast to composite memristor circuit in series, the memristance of composite memristor circuit in parallel is half of that of the individual memristor at any instant of flux. As the memristance decreases, the magnetic flux will rise accordingly and its variations (i.e., slopes) for single memristor and composite memristor circuit are both increasingly large until the memristance reaches the minimum value. Figure 13C plots the charge versus flux (q-φ) curves of the single memristor and two memristors in parallel with same polarities. Seen from Figure 13C, the slope of the q-φ curve of the single memristor is also half of that of the composite memristor circuit.
For a double memristor circuit connected in parallel with opposite polarities shown in Figure 12B, the curves of current versus voltage are plotted in Figure 14A. It has three different hysteresis loops in contrast to the case of the same polarities in Figure 13A. The inset of Figure 14A shows that the curve of M2 has a relatively linear hysteresis loop compared with those of the other two. However, the composite current is always equal to the sum of the individual branch currents of the two memristors at any time whatever the applied voltage is.
FIGURE 14. (A) Current versus voltage curves, (B) memristance versus flux curves, and (C) charge versus flux curves of the single memristor and two memristors in parallel with opposite polarities.
The curves of memristance versus flux (M-φ) and charge versus flux (q-φ) are plotted in Figures 14B,C, respectively. As seen in Figure 14B, the variation trends of the two individual memristors M1 and M2 are opposite, due to opposite polarities of these two memristors. Additionally, the composite memristance is always lower than the memristance of each single memristor at any time. Seen from Figure 14C, it can be found that the q-φ curves of the two individual memristors with opposite polarities are symmetrical about the initial origin point. Moreover, the slope of composite charge is always equal to the arithmetic sum of the two slopes of individual memristor charges.
4 Conclusion
A new window function has been created and applied into the HP memristor model in this paper. To validate its advantages, versatile characteristics and SPICE behaviors of the memristor have been demonstrated. Moreover, we also investigate the composite SPICE behaviors when two memristors are connected in series and in parallel with same or opposite polarities, and make comparisons with those of the individual memristors. Their relationships among flux, charge, voltage, current, and memristance are analyzed using PSpice simulation.
Data Availability Statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author Contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This work was supported in part by the FY2019 JSPS Postdoctoral Fellowship for Research in Japan under Grant P19350, Grant-in-Aid for JSPS Research Fellow under Grant JP19F19350, National Natural Science Foundation of China under Grant 61601390, Opening project of Key Laboratory of Cognitive Radio and Information Processing under Grant CRKL180202, Natural Science Foundation of Guangxi under Grant 2019GXNSFFA245002, and Dean Project of Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing under Grant GXKL06190118.
Conflict of Interest
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.
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Keywords: memristive devices, memristor circuits, simulation program with integrated circuit emphasis model, window function, memristor
Citation: Xu K-D, Li D, Jiang Y and Chen Q (2021) SPICE Behaviors of Double Memristor Circuits Using Cosine Window Function. Front. Phys. 9:648737. doi: 10.3389/fphy.2021.648737
Received: 01 January 2021; Accepted: 21 January 2021;
Published: 03 March 2021.
Edited by:
Gang Zhang, Nanjing Normal University, ChinaCopyright © 2021 Xu, Li, Jiang and Chen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yannan Jiang, eW5qaWFuZ0BndWV0LmVkdS5jbg==