# PSPICE modeling of meminductor

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## Abstract

PSPICE models of current- and flux-controlled meminductor are described. The models consist of two parts, one of which represents the state-space description of the memory effect of the device, and the other part is an inductor whose inductance depends on the system state. The basic fingerprints of the meminductor, i.e. the flux–current pinched hysteresis loops, the unambiguous constitutive relation between the time integral of flux and electric charge, and identical zero-crossing points of flux and current waveforms are demonstrated on the example of current-controlled meminductor.

### Keywords

Meminductor Model SPICE## 1 Introduction

The discovery of solid-state memristor in Hewlett-Packard (HP) labs, reported in Nature in May 2008 [1], initiated a growing interest in computer modeling and simulation of memristor [2, 3, 4, 5, 6, 7, 8, 9], the fourth fundamental passive element, theoretically predicted by Chua in 1971 [10]. The reason consists in the fact that the above device is not currently available as off-the-shelf circuit. Then its model, particularly when implemented in current programs for circuit simulation such as SPICE, can serve as an important tool in computer experiments with such devices.

In addition to the memristor, other hypothetical “mem-devices” have come into consideration within the last year, namely the memcapacitor and the meminductor [11, 12, 13, 14]. As noted in [15], “…combined with the already known memristor, such elements open up new and unexplored functionalities in electronics…”.

In order to enable simulation experiments with the above mem-elements, the existing PSPICE models of memristor should be complemented with similar models of memcapacitor and meminductor. The first step was performed in [16], where a general methodology for PSPICE modeling of mem-devices was introduced. The very first PSPICE model of the memcapacitor was published in [17]. To the best of our knowledge, no similar PSPICE model of the meminductor has been hitherto reported in the literature. Thereby, this Letter introduces foolproof models of current- and flux-controlled meminductors.

## 2 General models of current- and flux-controlled meminductors for PSPICE

*ρ*is the time-domain integral of magnetic flux

*φ*(TIF) of the meminductor, and

*q*is the electric charge, i.e. the time integral of electric current

*i*(TIC):

*q*as a single-valued function \( \hat{q} \) of the TIF,

It is shown on the example of HP memristor [1] that there is a significant difference between the behavior of the ideal hypothetical device (memristor) and its physical implementation. Since the key parameter of the device, resistance in the case of memristor, can be changed only within bounds that are given by the physical limitations of a concrete implementation, the HP memristor must be modeled as a more general memristive system [19]. Analogously, it is shown in [12, 16] that the memcapacitor and the meminductor are special cases of more general memcapacitive and meminductive systems. In order to model properly the meminductor also in its limit states, it should be regarded as a current- or flux-controlled meminductive system (CCMLS or FCMLS), respectively.

*L*

_{M}( ) and

*f*

_{i}( ) are nonlinear functions, which depend on the concrete physical implementation of the current-controlled meminductive system or meminductor. The suffix

*i*denotes the current-controlled system.

_{M}( ) and

*f*

_{φ}( ) are nonlinear functions. The suffix

*φ*denotes the flux-controlled system.

The functions *f*_{i}( ) and *f*_{φ}( ) should also model the so-called boundary effects, which cause the decrease of the speed of state variable to zero if the device state approaches its physical limits [20]. Details will be given in the following section.

Port equations (9) and (11) are modeled by inductors with inductances controlled through the blocks *L*_{M}( ) and Λ_{M}( ). Input signals of these blocks are the system state *x* and the meminductor current (Fig. 1a) or flux (Fig. 1b). The flux is computed as a time-domain integral of meminductor voltage (see the block *Int*_{φ} in Fig. 1b). Differential state equations (10) and (12) are modeled by the integrator *Int*_{x} and the nonlinear blocks *f*_{i}( ) and *f*_{φ}( ), respectively.

*L*

_{M}or with varying inverse inductance Λ

_{M}= 1/

*L*

_{M}

*φ*(0) and

*i*(0) are the initial flux and current at time 0, can be rewritten in the form

## 3 Example of SPICE modeling of meminductor

*l*

_{min}and

*l*

_{max}from the fixed terminal. The slider positions

*l*

_{min}and

*l*

_{max}determine the limiting values of the coil inductances

*L*

_{min}and

*L*

_{max}

*.*Let us define the dimensionless state variable

*x*

*L*is, roughly speaking, proportional to the square of the number of turns

*N*:

*R*

_{M}is the magnetic resistance of the magnetic part of the coil.

*L*

_{M}of the meminductor in Fig. 2 depends on the state variable

*x*approximately as follows:

*x*of the coil slider from the fixed terminal in Fig. 2, must depend on the inductor current. Analogously to the charge-controlled HP memristor in [3] and memcapacitor in [17], consider state equation (10) of the current-controlled meminductor in the form

*k*. The purpose of the nonlinear function

*window*(

*x*) is to model the rate decrease to zero when the slider approaches the limit positions

*l*

_{min}and

*l*

_{max}. As known from papers on HP memristor, the modeling of such boundary effects can be accomplished by several types of window functions, which provide a transition to zero values such as the rectangular [12, 21], Joglekar [20], or Biolek [3] window, denoted by

*R*,

*J*, and

*B*subscripts in the formulae below:

*stp*(

*x*) is a step function, i.e.

*stp*(

*x*) = 1 when

*x*≥ 0,

*stp*(

*x*) = 0 when

*x*< 0,

*p*is a positive integer, and

*xd*is a quantity which evaluates the direction of the change of the state variable, i.e.

*xd*> 0 when

*x*increases, and

*xd*≤ 0 when

*x*decreases or is constant.

*p*increases, the Joglekar window converges to the rectangular window. The basic problem of this window consists in the fact that when the state variable of the device is in its terminal value 0 or 1, no external stimulus can change this state because the time derivative of the state variable in (17) is zero, independently of the terminal signals. Such a discrepancy between the behavior of the model and the requirements for the operation of a real circuit element is resolved by the Biolek window, which reflects the fact that the speeds of receding from and approaching the limit positions of the slider are different. An illustration of how this window works is given in Fig. 3(b) for the control parameter

*p*= 2. Details can be found in [3, 9]. As a conclusion, the more sophisticated Biolek window models well the device behavior also near its boundary states, but numerical problems in transient analysis due to the discontinuities at boundary points can appear more frequently than in the case of the Joglekar window. The Joglekar window can be advantageously used for cases when the mem-element operates far enough from its boundary states. Note that when applying the Biolek window, constitutive relation (1) will now depend on the way the meminductor interacts with the surrounding networks. However, it is only a consequence of the fact that the device in Fig. 2 is not an ideal meminductor but a more general meminductive system.

PSPICE subcircuit of the meminductor from Fig. 2

No. | PSPICE code |
---|---|

1 | .SUBCKT memL Plus Minus PARAMS: |

2 | + Lmin=1mH Lmax=20mH Linit=5mH k=10 |

3 | * Input port * |

4 | Vsense Plus + 0 V; sensing of the meminductor current |

5 | Gml + Minus value={(V(flux)+IC*Linit)/LM(V(x))}; see Eq. 13 |

6 | *Flux computation via time-domain integration of meminductor voltage* |

7 | Gflux 0 flux value={V(plus,minus)} |

8 | Cflux flux 0 1 |

9 | Rflux flux 0 1G |

10 | *State-space equation (17). Intx from Fig. 1 is formed by Cx and Gx* |

11 | .param xinit {(sqrt(Linit)−sqrt(Lmin))/(sqrt(Lmax)−sqrt(Lmin))}; results from Eq. 16 |

12 | Gx 0 x value={I(Vsense)*k*windowJ(V(x),p)}; version for Joglekar window, see Eq. 17 |

13 | ;Gx 0 x value={I(Vsense)*k*windowB(V(x),I(Vsense),p)}; version for Biolek window, see Eq. 17 |

14 | Cx x 0 1 IC={xinit} |

15 | Rx x 0 1G |

16 | *Functions for defining meminductance and boundary effects |

17 | .func LM(x)={(sqrt(Lmin)+x*(sqrt(Lmax)−sqrt(Lmin)))^2}; see Eq. 16 |

18 | .func windowJ(x,p)={1−(2*x−1)^(2*p)}; Joglekar window, see Eq. 18 |

19 | .func windowB(x,xd,p)={1−(x−stp(−xd))^(2*p)}; Biolek window, see Eq. 18 |

20 | *Computing charge and time-domain integral of flux (TIF) |

21 | Gcharge 0 0 value={SDT(I(Vsense))} |

22 | Gintflux 0 0 value={SDT(V(flux))} |

23 | .ENDS memL |

The Vsense voltage source, defined on line 4, serves to sense the meminductor current. The value of this current is needed for modeling the state equation on line 12 or 13 as well as for computing the charge on line 21. The code on line 12 can be replaced by the text on line 13 when the Biolek window is used instead of the Joglekar window. The integrator *Int*_{x} in Fig. 1(a) is accomplished by a capacitor *C*_{x} with 1F capacitance, charged from a current source *G*_{x}, which provides current according to Eq. 17. A shunting resistor *R*_{x} is needed in order to provide the DC path to the ground. The initial state *x*_{init} of the device is derived from the initial value of inductance *L*_{init}, see line 11. The flux computation as a time-domain integral of meminductor voltage is done similarly on lines 7–9.

In order to easily visualize constitutive relation (1), the charge and TIF computations are performed on lines 21 and 22. For simplicity, the time-domain integrations are provided via a PSPICE function SDT, not by means of the current source and capacitor. To save model nodes, the computation is arranged via current sources with each terminal grounded. The values of these currents are equal to the charge and TIF and thus they can be used for the visualization of the CR. Note that the SDT function was not utilized in the above-mentioned models of integrators. The main reason for not using the SDT function is that the precision of these integrators can then be adjusted via a proper selection of the component parameters.

*x*due to changing the terminal current and voltage.

*x*. Note that whereas the initial transients are almost identical (see Fig. 6a, b), in the steady-state the meminductor modeled by the Joglekar window operates in the regime of

*x*= 1, due to the effect of locking in on the boundary state in which the time derivative of the state variable is zero, and thus it behaves as a conventional linear inductor. The corresponding flux–current characteristic in Fig. 7(a) is a line without any hysteretic behavior. By contrast, the operation of the meminductor with the Biolek window exhibits a steady-state variation of the state variable

*x*(see Fig. 6d), and the hysteretic loop in the flux–current characteristic is evident in Fig. 7(b). Note that the constitutive TIF-charge relation in Fig. 7(d) is not ambiguous because the meminductor now behaves as a more complicated meminductive system.

## 4 Conclusions

The methodology described here facilitates effective modeling of meminductive systems of various physical natures in the SPICE-family programs for circuit simulation. An example of the modeling of a current-controlled meminductor whose inductance is changed depending on a state variable, controlled by the meminductor current, is analyzed in detail. The SPICE modeling of flux-controlled meminductive systems can be figured out analogously, following the general model in Fig. 1(b).

## Notes

### Acknowledgments

Research described in the paper was supported by the Czech Science Foundation under grant No. P102/10/1614, and by the research programmes of BUT Nos MSM0021630503/13 and UD Brno No. MO FVT0000403, Czech Republic.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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