Hindawi Active and Passive Electronic Components Volume 2021, Article ID 5582774, 8 pages https://doi.org/10.1155/2021/5582774 Research Article A Novel Passive Circuit Emulator for a Current-Controlled Memristor Leonardo Barboni Department of Electronica, Instituto de Ingenieria Electrica-FING, Julio Herrera y Reissig 565 11300, Montevideo, Uruguay Correspondence should be addressed to Leonardo Barboni; lbarboni@fing.edu.uy Received 2 February 2021; Revised 23 March 2021; Accepted 16 April 2021; Published 22 April 2021 Academic Editor: Luigi Di Benedetto Copyright © 2021 Leonardo Barboni. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A memristor is an electrical element, which has been conjectured in 1971 to complete the lumped circuit theory. Currently, researchers use memristor emulators through diodes, inductors, and other passive (or active) elements to study circuits with possible attractors, chaos, and ways of implementing nonlinear transformations for low-voltage novel computing paradigms. However, to date, such passive memristor emulators have been voltage-controlled. In this study, a novel circuit realization of a passive current-controlled passive inductorless emulator is established. It overcomes the lack of passive current-controlled memristor commercial devices, and it can be used as part of more sophisticated circuits. Moreover, it covers a gap in the state of the art because, currently, only passive circuit voltage-controlled memristor emulators and active current-controlled emulators have been developed and used. (e emulator only uses two diodes, two resistors, and one capacitance and is passive. (e formal theory and simulations validate the proposed circuit, and experimental measurements were performed. (e parameter conditions of numerical simulations and experiments are consistent. Simulations were performed with an input current amplitude of 15mA and frequencies of up to 3kHz and measurements were carried out with an input current amplitude of 0:74mA and frequency of 1:5kHz in order to compare with the state of the art. 1. Introduction potential for building novel integrated circuits and com- puting systems, as has been proposed in [1–7]. Memristors A memristor is an electrical two-terminal passive nonlinear allow the memory and time-varying processing of infor- resistance element that exhibits a well-known pinched mation through nonlinear transformations in a unique hysteresis loop at the origin of the voltage-current plane passive element. when any bipolar periodic zero-mean excitatory voltage or (ere is neither implementation nor a proof-of-concept current of any value is applied across it. However, there are for a passive current-controlled memristor system-on-chip disagreements regarding whether a memristor can be but yes for an active current-controlled memristor [8]. With considered a fundamental element and whether its dynamic an increasing need to better understand a pinched hysteresis is purely electromagnetic, as originally conjectured, or if attractor, researchers implement memristor emulators there are other mechanisms involved such as ionic transport. through diodes and other passive (or active) elements. Such Despite the controversy surrounding the technological re- emulators allow studying (theoretically and numerically) alization, the amount of research into the properties of the possible attractors and ways of implementing nonlinear pinched hysteresis loop continues to increase. Irrespective of transformations for novel computing paradigms. (e first whether the memristor is implemented by emulating its voltage-controlled memristor emulator has been proposed behavior through circuitry composed of other active or in [9] and was based on a diode bridge with a parallel R − C passive components, new studies continue to generate and filter as a load. Other studies [10–16] used voltage-controlled sustain optimistic expectations in the scientific community memristor emulators as vital building blocks of other cir- about the use and advantages of the memristor. (e research cuits for the in-depth study of their bifurcations and chaotic interest in the memristor is motivated by its promising behaviors. 2 Active and Passive Electronic Components To summarize, (a) background: to the best of our 2. Proposed Current-Controlled Memristor knowledge and after searching in database providers, we have concluded that a passive and current-controlled Novel current-controlled memristor emulators have been memristor emulator has not been published previously (only introduced recently in [8, 17, 18]; however, all of them are active current-controlled memristor [8]) (although there are active (it means that a voltage power supply is needed). In a lot of voltage-controlled memristor emulators) and (b) this work, the first passive circuit realization of a current- motivation: design a passive and current-controlled mem- controlled emulator is established in Figure 1. It uses two ristor emulator for neuromorphic computing and as a diodes, two resistors, and one capacitor. candidate for mimicking synaptic functions and inductorless We construct the equations by beginning with the emulator that could be easily integrated into CMOS tech- Shockley diode equation (both diodes are equal and without nology. It is similar in spirit to some circuits in the literature, considering high-frequency effects that produce unwanted which combine a rectifier with a LC low pass filter and not an dynamic effects): RC as it is here. (is memristor enables its utilization in 2αVjD iD � Is�e 1 − 1 � ; programmable circuits and systems controlled by digital 1 (3) 2αVjD pulses. iD � Is�e 2 − 1 �; Memristors exhibit three characteristics for any bipolar 2 periodic signal excitation: (i) there is a pinched hysteresis where 2α � 1/nVT and Is denote the reverse saturation loop in the voltage-current plane, (ii) the area of the hys- current, n is the emission coefficient, VT is the thermal voltage, and V and V are the junction diode voltages. If teresis loop decreases and shrinks to a single-valued V-I jD1 jD2 function when the signal excitation frequency tends to in- we consider their series parasitic resistance Rp, the diode finity, and (iii) for voltage-controlled generalized memristive voltages become time-invariant systems, the following equations apply [1–7]: 1 iD 1 iD V � R i + ln 1 + 1! and V � R i + ln 1 + 2!: D1 p D1 D2 p D2 8> im � Gx;Vm �Vm and G(x; 0) ≠ ∞∀x; 2α Is 2α Is > < (4) > > dx : � fx;V �x represents the inner state variables; (erefore, dt m 2α�V − R I � 2α�V − R I � (1) i � I e D1 p D1 − 1! and i � I e D2 p D2 − 1!: D1 s D2 s where i is the current across the memristor, V is the m m (5) voltage across the terminals, G(x;Vm) is bounded, and f(x; im) is the equation of state, which must be also bounded According to the voltage drop, to guarantee the existence of a solution x(t). (e area of the Vm � Ri + VD and Vm � Ri − VD ; (6) lobes, shapes, and orientation of the hysteresis loop evolves 1 1 2 2 with frequency. All of the abovementioned references de- −V � V + V : ( ) veloped voltage-controlled memristor emulators (i.e., c D1 D2 7 equations such as equation (1)). However, in this study, we (e current i corresponds to i � i − i . Using implement the first-ever built passive current-controlled m m D1 D2 memristor emulator for which the following equations equation (5), apply: V � R ; i i R( ; ) ; >8 m x m � m and x 0 ≠ ∞∀x <> > > dx : � fx; i �x represents the inner state variables: dt m (2) α�V +V − R �i +i �� α�V − V − R �i − i �� − α�V − V − R �i − i �� D1 D2 p D1 D2 D1 D2 p D1 D2 D1 D2 p D1 D2 im � Ise e − e !: (8) Using equation (7), it becomes convenient to express However, from equation (6), − αV − αRp�iD +iD � 2Vm � Rim + VD − VD : (10) i � 2I e c e 1 2 sinh�α�V − V − R i ��: 1 2 m s D1 D2 p m (9) (en, the �Vm − im � relation is provided, Active and Passive Electronic Components 3 i1 i2 + + im RRvR vR V 1 2 + m + – C – i C def. V im m + + – – v – D v C v D 1 D1 D2 2 – + i i D1 D2 (a) (b) Figure 1: (a) Proposed current-controlled memristor circuit emulator and (b) generalized symbol of the memristor device. Now, to complete the state equation, we have to calculate 1 1 − 1 im αVc+αRp�iD +iD � 1 V � Ri + e 1 2 ! + R i : m m sinh p m iD , which is straightforward (note that according to 2 2α 2Is 2 1 equation (2), x � Vc): (11) V − V m D1 Because i � i − i + i and i � i + i , we obtain � i1; c D1 m 2 c D2 2 R iD + iD � 2ic + im − 2i2 and then, 1 2 V i i � c + m⇒V ; (17) 1 1 − 1 m αVc+αRp()2ic+im− 2i2 1 D1 Vm � Rim + sinh e ! + Rpim: 2R 2 2 2α 2Is 2 Ri V (12) � V − m − c: m 2 2 Next, we focus on the i expression. By taking V � 2 m (erefore, from equation (16) and VD + VC + Ri2 and Vm � −VD − VC + Ri1, we obtain 2α(V − R i ) 1 2 i � I (e D1 p D1 − 1), we obtain D1 s VD + VD + 2VC + R i2 − i1 � � 0: (13) 1 2 dVc 2αV − αRi − αV − 2αR i im Vc C � I �e m e m e c e p D1 − 1� − − : t s R Because i2 − i1 � i2 − 2i1 + i1 � im − 2i1, and d 2 2 −V � V + V , from equation (13), we obtain (18) c D1 D2 V i V i i � c + m and i � − c + m: (14) By introducing equation (15), we obtain 1 R 2 R 2 2 2 2 V −1 αVc+αRp()2ic+Vc/R d c sinh im/2Ise �+αRpim− 2αRpiD − αVc C � Ise 1 Finally, equation (12) becomes dt (19) i im Vc 1 1 − 1 m αVc+αRp()2ic+Vc/R − Is − − : Vm � �R + Rp �im + sinh e !: 2 2R 2 2α 2Is Because i � i + (V /2R) + (i /2), we obtain (15) D1 c c m Next, we focus on the state equation: