Memristor, Memcapacitor, Meminductor: Models and Experimental Circuit Emulators

Before 1971, the number of passive electrical components was limited to three: resistor, capacitor, and inductor. In 1971, Pr. Chua predicted the existence of a fourth element, called memristor, since it corresponds to a resistor with memory behavior. Several years later, the concept of memory circuit was extended to capacitors and inductors. This paper proposes mathematical models for mem-elements, validated by Matlab and experimental circuit emulators for memcapacitor and meminductor. The experimental results show a good fit between theory, Ltspice simulations, and emulation circuits. Keywords-memristor; memcapacitor; meminductor; models and simulations; experimental emulator


INTRODUCTION
The memristor, or memory resistor, is a two-terminal passive electronic component. Memristor, as a concept, was introduced in 1971 by Leon Chua [1]. Chua defined the behavior of the memristor by a constitutive relationship between flux and charge linkage. However, the concept of memristor did not catch the attention of the scientific community until 2008. The electronic community continued to develop analog and digital circuits without considering the tremendous potential of memristors [2][3][4]. In fact, Hewlett Packard (HP) Labs manufactured the first memristor component with layers of TiO 2 between two layers of Pt in 2008 [5]. HP also introduced a mathematical model of component operation. The scientific community criticized this model in several points but subsequently the model was widely used [6]. Since then, thanks to the memory effect and nonlinear behavior, the memristor found use in various domains, such as neural networks [7][8][9][10][11], programmable analog circuits [12][13][14][15], logic gates [16], adaptive filters [17], chaotic circuits [18,19], and non-volatile memories [20][21][22].

II. MATHEMATICAL MODELS
Chua, Pershin, and Di Ventra established the electrical relations for memristor, mecapacitor, meminductor as can be seen in [23] and in Figure 1 of [24], where the charge (q), the time-integral of the voltage (ϕ), the time-integral of the charge (σ), and the time-integral of the flux (ρ) are visualized.

A. Memristor Models
Two pairs of equations define the memristor: electrical load control and the magnetic flux control [1,25].

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According to the mathematical relations, the electrical resistance of the memristor is not constant but depends on the current that was previously circulated through the device. In 1976, L. Chua introduced dynamic equations to define a memristive system with the property of "pinched hysteresis loop" between current and voltage [25,26]. According to Biolek, this footprint of the memristif is "self-crossing" which corresponds to a cross with coordinates (0:0) [27]. The pinched hysteris loop/self-crossing was extended to memcapacitor and meminductor.
The HP model's is based on the "modulation" of the size of the doping zone w of the memristor between the value R on and R off as a function of the electrical loads (current) that runs through the component. where M(q) is the instantaneous value of the memristor, w the size of the doped zone, D the total size of the doped zone, and µ v the mobility of the electric charge [5,6].

B. Memcapacitor Models
The memcapacitance is defined with the electrical relation between the time-integral of the charge (σ) and the timeintegral of the voltage (ϕ). It can be either voltage-controlled or charge-controlled depending on its constitutive input variable [25,28]. In the case of voltage controle, The memcapacitance Cm is:

C. Meminductor Models
The meminductor represents the link between charge q(t) and the time integral of the flux ρ(t). The mathematical representations of meminductors have two forms: currentcontrolled meminductor or flux-controlled meminductor [22,25]. Thus, the meminductance Lm is: (5) Note that, the pinched hysteris loop/self-crossing which corresponds to a cross with coordinates (0 : 0) is footprint of the mem-elements [26,27].

A. Memristor Implementation
The model was implemented on Matlab. Figure 1 shows the pinched hysteresis loop between current and voltage and the variation of memristor versus voltage. For this simulation, the electrical load control was chosen. Note that if the frequency increases beyond 100Hz, the characteristic becomes a linear line corresponding to Ohm's law.

B. Memcapacitor Implementation
The model was implemented in Matlab and the footprint (pinched hysteresis loop) of the memcapacitor appears under sinusoidal input between voltage and electrical load. Figure 2 shows the results.

C. Meminductor Implementation
The model was implemented in Matlab and the footprint (pinched hysteresis loop) of meminductor appears under sinusoidal input between flux and current. Figure 3 shows the numerical results.

IV. EXPERIMENTAL CIRCUIT EMULATORS
To the best of our knowledge, there are no physical implementations of memcapacitor or meminductor circuits. Due to this, there has been an emerging research devoted to the development of emulator circuits of these devices during the recent years. In the literature, different approaches have been proposed for the emulation of memcapacitors and  [24,[29][30][31][32]. In our case, we chose to develop memcapacitor and meminductor circuit emulators that do not require the use of a memristor or its emulator. In fact, our emulators are based on the same design and are able to emulate either a memcapacitor or a meminductor with minimal changes. The circuit emulators are based on voltage/current converter with operational amplifier. The operating principle of the circuits is based on a second output feedback which is connected to the non-inverting input of the amplifier through the capacitor C2. Thus, in both cases, the C2 capacitor provides the memory effect. Frequency simulations of the emulator circuits show that they act as filters. The memcapacitor emulator has a high-pass filter behavior and the meminductor emulator acts as a lowpass filter. In the Figure 6, the frequency simulation (Bode diagram) results can be seen.  In fact, we can show that the circuit emulators have equivalent circuit filters. These circuits can easily be built from passive elements. Figure 7 presents the equivalent circuit filters and Bode diagram results. In order to validate the memcapacitor and meminductor emulators, circuit implementations were made. Figures 8-9 show the implementation and the experimental responses of the circuits. The obtained results show a good fit between the simulations and the experimental tests. However, during the tests, the memcapacitor and meminductor effects were obtained at lower frequencies than those of the Ltspice simulations. We assume that the implementation of emulators on PCB will achieve better results in terms of operating frequency. Indeed, some parasitic effects will be reduced.
V. CONCLUSION In this paper, the concepts of memory-elements memristor, memcapacitor, and meminductor were presented. After the presentation of the mathematical models describing the behavior of these components, the circuit emulators for memcapacitor and meminductor, based-on operational amplifier voltage/current converter, were shown. The results show a good fit between the experimental tests and theory simulations.