PSM of ideal memristors, memcapacitors, and meminductors

The classical ﬁ ngerprint of the ideal memristor with differentiable constitutive relation has recently been extended with a piece of knowledge that the area of the pinched hysteresis loop of a memristor driven by sinusoidal signal with ﬁ xed amplitude and increasing frequency disap-pears with an integer power of the frequency. This ﬁ ngerprint is extended to the memcapacitor and meminductor, and it is also shown that the way the hysteresis disappears can be determined directly from the behaviour of the parameter versus state map (PSM) of the element in the neighbourhood of the initial operating point.


Introduction:
In [1], the memristor (MR), memcapacitor (MC), and meminductor (ML) are defined uniformly as two-terminal elements, which can be described by the equations where y and u are the terminal quantities, x is the internal state variable, and ẋ is its derivative with respect to time.The function g(x), also called the parameter versus state map (PSM) [2], represents the interrelationship between the dominant parameter of the element and its state.For the MR, MC, and ML, the PSM and the variables y, u, x are specified in Table 1.
Table 1: Meaning of variables in (1) for ideal MRs, MCs, and MLs controlled by voltage (Vc), current (Cc), charge (Qc), and flux (Fc) The unified form of the defining equation (1) of the MR, MC, and ML enables extending the results published hitherto only for the MR also to the remaining two memory elements.
Typical y-u odd-symmetric [1] pinched hysteresis loops are generated when driving the element of type (1) by a sinusoidal signal u as illustrated in Fig. 1.The non-zero area of the loop is a measure of the memory effect, which originates due to the dependence of the parameter of the element on the time-domain integral of the driving signal.The gradual disappearance of the hysteresis if the frequency of excitation increases beyond all bounds, while its magnitude is maintained within predefined limits, belongs to the well-known fingerprints of MR, MC, and ML [3].A new piece of knowledge was published in [4] about the way the hysteresis disappears at high frequencies.Consider the initial state x = x 0 of the element before applying sinusoidal excitation.This state represents the starting operating point on the PSM characteristic g(x) (see Fig. 2).During the excitation, the state variable will vary within the interval x ∈ X.Also assume that the function g(x) can be approximated on the interval x by its Taylor expansion around the point x 0 , thus where g i are the respective coefficients of the Taylor series, and g 0 = g (x 0 ).The area S of one lobe of the y-u hysteresis loop is given by a formula which results from [4] where U MAX and ω are the amplitude and the angular frequency of sinusoidal excitation, respectively.It follows from (3) that if g m is the first non-zero term of the sequence g 1 , g 2 , g 3 , .., then the loop area must disappear at high frequencies according to the ω −m law.Inset is the loop area versus frequency Fig. 2 Impact of the location of the initial operating point x 0 on the disappearance of the hysteresis Black (ω −1 )strictly monotonic PSM, m = 1 Red (ω −2 , ω −4 )local minimum and maximum, m ≥ 2, even Orange (ω −3 , ω −5 )inflection points with zero-valued first derivative (saddle) [6], m ≥ 3, odd The objective of this Letter is to show that the information about the order m of the disappearance of the hysteresis can be obtained directly from the PSM characteristic g(x) of the element.
Connection between the way the hysteresis disappears and the PSM: It follows from the definition of the Taylor series (2) that Substituting ( 4) into (3) and a simple rearrangement yield the loop area The formula (5) provides the following connection between the course of the function g(x) in the neighbourhood of the initial operating point and the high-frequency disappearance of the hysteresis.
Theorem of the disappearance of hysteresis: Assume that the element of type (1) with the initial state x = x 0 is driven by a sinusoidal signal u(t).Consider the sequence of cases when the excitation frequency ω gradually increases beyond all bounds while the amplitude U MAX remains unchanged.Then the high-frequency disappearance of the area of the pinched hysteresis loop is governed by the law S ≈ ω −m if and only if m > 0 is the order of the first non-zero term of the sequence In other words, the order m of the hyperbolic drop must be equal to the order of the first non-zero derivative of the PSM at the initial operating point.For m = 1, the point is located in the area of monotonicity of the PSM curve.If m > 1, then m determines whether the initial point on the PSM curve is a stationary (m even) or an inflection (m odd) point [5].
The way the hysteresis disappears at high frequencies can therefore be deduced from what area of the PSM the initial operating point is located in.For illustration, examples of the operating points in Fig. 2 are intentionally sequenced from left to right such that they correspond to the gradually increasing number m: the higher the number m (or the order of the first non-zero derivative), the flatter the characteristic is in the neighbourhood of the corresponding operating point.

Simulation:
The set of pinched hysteresis loops in Fig. 1 was generated under sinusoidal excitation of the element (1) with the PSM characteristic (2) according to Fig. 3.The initial operating point, which corresponds to the state x = x 1 , is denoted in the inset of Fig. 3 by the number ①. Seeing that the point lies on the descending part of the PSM, the high-frequency disappearance of the hysteresis is governed by the ω −1 law, as shown in Fig. 1.According to [7], the area of the lobe of the loop is defined as the integral of y with respect to u within one half-period.This means that the area can be positive or negative (it is negative for the case ① from Fig. 1) depending on the loop orientation.It follows from the theorem of the disappearance of hysteresis that the log-log diagram of area versus frequency changes at high frequencies to a line with a drop of m units per decade.Fig. 3 demonstrates this regularity for three initial operating states x 1 , x 2 , and x 3 , which coincide with the cases m = 1, 2, and 3.

Conclusion:
As is obvious from the theorem of the disappearance of hysteresis, the loop area S of the ideal MR, MC, and ML disappears at high frequencies along the classical hyperbola, i.e. by the law S ≈ ω −1 if and only if the initial operating point is located in a strictly monotonic part of the PSM characteristic.It is the case, for example, of the classical HP MR whose PSM is a linear function [8].The location of the initial operating point at some of the stationary points of the PSM characteristic automatically involves a higher order of the disappearance of the hysteresis, thus m > 1.If the order is even, the initial operating point must lie at the local minimum or maximum of the PSM characteristic.On the contrary, for the odd-order disappearance (with the exception of m = 1), the initial operating point must be placed into the saddle, i.e. into the inflection point with zero-valued first derivative of the PSM.

Fig. 1
Fig. 1 Pinched hysteresis loops for sinusoidal excitation Example of a possible evolution of the loop area for the frequency of sinusoidal excitation increasing from 0.01 Hz (①) to 10 Hz (⑥) with the fixed amplitude U MAX = 0.1 PU.Inset is the loop area versus frequency