## 1. Introduction

Prof. L. Chua first presented the notion of a memristor (memory-resistor) in 1971, defining a new circuit element which established the relationship between flux (

ϕ, time-integral of the voltage) and charge (

q) [

1]. The memristor produces a non-linear relation between the current (

i) and voltage (

v), where the memory effect is manifested through a characteristic pinched hysteresis loop in its

i–

v curve. In recent times, with the advent of actual physical memristors [

2,

3,

4,

5], the topic of passive memory devices has raised a high interest for researches and it has generated transversal studies involving different science and technologies areas such as chaotic and logic circuits [

6,

7], or neuromorphic computing [

8,

9].

Nearly 40 years later, the concept of passive memory-devices (memdevices) was generalized, again by Chua, for capacitive and inductive non-linear systems [

10], expanding the relation between the different physical magnitudes schematized in

Figure 1. Thus, the memcapacitance (

C_{M}, memory-capacitance) established the electrical relation between the time-integral of the charge (

σ) and the flux (

ϕ), whereas the relation between the charge (

q) and the time-integral of the flux (

ρ) was defined by the meminductance (

L_{M}, memory-inductance).

Because of their inherent memory and dynamic storage ability, these devices are expected to play an important role in diverse fields of science and technology, such as non-volatile memories and simulations of learning, adaptive, and spontaneous behavior circuits [

11]. In the particular case of meminductors, several works have reported promising results for future electronic circuits, e.g., adaptative filters [

12], chaotic circuits for encryption [

13] or neuromorphic computation [

14]. However, the solid-state implementation of these latter devices (memcapacitor and meminductor), in contrast to memristors, presents a challenge due to the yet elusive nature of their physical existence. For this reason, the emulators of these devices are leading the way to envisage real-practical implementations until single-device solid-state demonstrators become available. Thus, different examples of memcapacitor and meminductor emulators implemented with off-the-shelf devices or based on mutator approaches have been presented in the literature [

15,

16,

17,

18,

19,

20,

21], all of them following the same principle, the designing of a circuit which satisfies the constitutive equations of the emulated device.

Following this principle, in this work we addressed a simple approach to emulate a meminductor based on one of the most famous inductor electrical simulators: the gyrator circuit proposed by Antoniou almost 50 years ago [

22] which, using two op-amps and five passive elements, has been widely used in the literature to emulate different systems, such as Chua’s circuit: a simple electronic circuit that exhibits classic chaotic behavior [

23]. Thus, the emulator proposed in this work, in contrast to the aforementioned approaches collected in the literature, presents the advantage of not requiring any four-quadrant analog multipliers, current conveyors or memristive devices, which make it one of the simplest and cheapest meminductor emulators to date.

The manuscript is structured as follows: after this introduction,

Section 2 presents the analytical relations between the modified Antoniou’s circuit and meminductance.

Section 3 proposes a design valid for emulating a two-state meminductor as well as a continuous transition of inductances together with their electrical characterization.

Section 4 illustrates the feasibility of the meminductor emulator to study neuromorphic applications with a long-term potentiation example, and finally, the main conclusions are drawn in

Section 5.

## 2. Meminductance and Antoniou’s Circuit

The canonical meminductor definition from

Figure 1 can be generalized to a more general meminductance (

L_{M}) concept which establishes a non-linear

n^{th}-order relation between the current (

i) and the flux (

ϕ) that can be expressed in terms of a current-controlled meminductive system [

10]:

or a flux-controlled meminductive system:

where

t is the time and

$\overrightarrow{{x}_{n}}$ the history (

nth-order state-vector) of the system.

The first-order meminductive system whose inductance only depends on the time-integral of the input flux (TIF,

ρ) is also known as flux-controlled meminductor and its definition is reduced to Equation (5):

with

$\rho ={\int}_{{t}_{0}}^{t}\varphi \left(\tau \right)d\tau $, proved that

${\int}_{-\infty}^{{t}_{0}}\varphi \left(\tau \right)d\tau =0$.

Therefore, as Equation (5) indicates, the i-ϕ characteristic of meminductors is given by a pinched hysteresis loop, the current being zero whenever the input flux is zero.

In the case of flux-controlled meminductors, Equation (5) can be expressed as

On this basis, hereinafter we demonstrate that the circuit proposed in this work, shown in

Figure 2, can be modelled by Equation (6) when its different passive elements are properly selected, and therefore it can be used to emulate a grounded meminductor.

Antoniou’s simulator is widely described in the literature and it is considered one of the best options to simulate grounded inductors since it is very tolerant to the nonideal properties of the op-amps [

24]. In addition, this configuration requires neither actual inductors (with the potential advantage of reduced form-factor, lower losses and cost) nor memristor for its implementation. In this case, to implement the meminductor emulator we drew on the equivalent input impedance (

Z_{IN}) of the circuit, considering the original

Z_{1} as a TIF-dependent impedance,

Z_{1}(

ρ), then:

From this basic circuit, a flux-controlled inductor could be modelled replacing

Z_{1} by a TIF-dependent resistor (controlled by the time-integral of the input flux),

Z_{4} by a capacitor

C_{4} and finally,

Z_{2},

Z_{3} and

Z_{5} by the resistors

R_{2},

R_{3} and

R_{5}, respectively. Therefore, Equation (7) can be expressed as

which corresponds to the impedance of a flux-controlled meminductor given by the following expression:

Moreover, this meminductor can be connected directly with the meminductance by means of the constitutive equation of the inductor:

which can also be expressed as:

The first term of Equation (10) can be further developed introducing the input charge (i.e., the charge circulating through the equivalent inductor):

Combining the definition of flux-controlled meminductance (Equation (5)) with Equations (11) and (12), the resulting meminductance seen at the input of the circuit is derived as follows:

## 3. Meminductor Circuit Demonstrator

Aiming to illustrate the feasibility of this circuit, we developed a simple two-state meminductor demonstrator to study its electrical behavior based on the general schematic of

Figure 2. In this circuit,

R(ρ) takes two different values depending on the time-integral of the input flux (

ρ) and the triggering value of the switch (triggering value of the TIF,

ρ_{TH}). The double integrator circuit was simulated by means of two Miller integrators in cascade, considering that in a practical implementation a resistor should be used in the feedback loop to provide a DC feedback path, reducing the low frequency gain of the op-amp and hence avoiding the saturation of the output signal [

24].

Therefore, as indicated in Equation (14), this particularized circuit yields a two-state meminductor. If the TIF is lower or equal to the triggering value (

ρ_{TH}), then

R_{6} is not connected in the feedback loop, otherwise

R_{6} is connected in parallel with

R_{1}_{.} Thus, according to Equation (13), the equivalent meminductance seen from the input terminal takes the following values:

The circuit of

Figure 3 has been simulated with SPICE to confirm its meminductive behavior. For this, we considered the following values for the discrete components:

R_{1} = R_{2} = R_{3} = R_{5} = R_{6} = 1 kΩ,

C = 47 nF and

ρ_{TH} = 0 V·s

^{2}, while the input was a sinusoidal signal,

V_{m}sin(2πft), with

V_{m} = 1 V considering two different frequencies,

f = 1 kHz and

f = 10 kHz. Thus, the resulting values of meminductance, according to Equation (14), would be

L_{M} = 47 mH for

ρ ≤ ρ_{TH} and

L_{M} = 23.5 mH for

ρ > ρ_{TH}.

The results shown in

Figure 4 corroborate the two-state meminductive behavior of the proposed circuit. On the one hand,

Figure 4a depicts the relevant signals of the circuit implemented. As seen, and in agreement with Equation (5), the input current takes the same waveform that the input flux (

ϕ) but modulated according to the two values of meminductance as a function of its time time-integral (

ρ). On the other hand, the

i-

ϕ plot, represented in

Figure 4b, depicts the closed pinched hysteresis loop passing through the origin, which is the signature of meminductors [

10]. It can also be noted that, as expected and as occurs for memristor and memcapacitors, the hysteresis collapses with increasing frequency [

10,

25,

26].

The time-domain impedance of a time-dependent inductor (

Z_{L}(

t)

= v_{L}(

t)/

i_{L}(

t)) can be determined from the fundamental relationship between flux and inductance:

which, considering a sinusoidal input voltage (

V_{L} = V_{m} sin(ωt)) and a TIF-dependent inductance, produces the following time-domain impedance:

This equation matches with the results obtained for the time-domain input impedance extracted from the simulations presented above, as shown in

Figure 5. It can be noted that the input impedance corresponds to two different values of inductance according to the time-integral of the input-flux and both Equations (14) and (16).

Once the operation of the circuit has been exposed, hereafter we demonstrate that this circuit could also be further extended to a continuous transition of inductances (instead of discrete states) using the configuration shown in

Figure 6. For that, we considered

Z_{1} as a TIF-controlled resistor whose values are given by Equation (17), while the rest of the components remain unchanged with respect to the previous configuration.

For the simulations we considered

R_{0} = 2 kΩ and the constant factor

K = 1 kΩ to satisfy that

R_{1,max} = 3 kΩ and

R_{1,min} = 1 kΩ, given that

ρ was normalized as

|ρ| ≤ 1. Under this configuration, and as represented in

Figure 7a,b, the circuit shown in

Figure 6 also fulfils the constitutive equations of meminductors, but in this case, the behavior of the meminductor emulator does not present discrete transitions between states but rather continuous ones, since

R_{1}(

ρ) is a continuous function of

ρ, as indicated in Equation (17).

Moreover, to round up the flexibility of the proposed emulator, we present in

Figure 8 the operation of the circuit shown in

Figure 6 with a more complex excitation signal. In this particular example, we used the same continuous meminductance transition given by Equation (17), but considering a square input signal with a peak-to-peak amplitude of

V_{pp} = 1 V and a frequency of 1 kHz.

Finally, we present a design of the continuous-states meminductor using commercial off-the-shelf devices. In this case, the TIF-controlled resistor is implemented by means of a photoresistive opto-isolator (which consists of an LED input optically coupled to a photocell, such as the model NSL-32 by Advanced Photonix [

27]), as shown in

Figure 9. With this particular optocoupler, we can achieve resistances from 40 Ω to 500 kΩ. A constant DC voltage (

V_{offset}) is added to the TIF to prevent the LED turning off (since it would reduce the current thorough

R_{1} down to values close to zero) and to work in the light-dependent resistor’s linear region [

28].

The proper working of this approach using the same configuration as the previous examples with a

V_{offset} = 3 V is demonstrated in

Figure 10, where it can be appreciated that the circuit behaves as a continuous-state meminductor as a consequence of the change in

R_{1}(

ρ) according to the TIF (taking values from ~60 Ω to ~150 Ω).

## 4. Long-Term Potentiation Example

In this last section, we demonstrated that this circuit is also feasible for the emulation of neuromorphic circuits. In neural activity, neurons act as signal spike generators to pass an electrical signal to another neuron, and synapses are the means by which they do so [

29]. Then, in particular, we show in

Figure 11 a long-term potentiation (LTP) and long-term depression (LTD) example, which emulates the long-term memory in biological systems. For that, we firstly applied pre-synaptic stimuli, modelled by successive current spikes with positive sign, progressively increasing the TIF of the device (potentiation). After that, a series of current spikes of negative value, acting as post-synaptic stimuli, induced the depression of the device, hence progressively reducing the TIF.

The confirmation of the learning/forgetting processes of the device can be recognized in the flux that the device stores and removes during each spike, reflecting an increase and decrease in the meminductance during the potentiation and depression, respectively. Thus, the repeated application of these pulses produces LTP or LTD behavior, in which the flux does not recover its pre-pulses value over a long period of time, and therefore neither does the meminductance.