Memristive Fingerprints of Electric Arcs
Abstract
We discuss the memristive fingerprints of the hybrid CassieMayr model of electric arcs. In particular, it is shown that (i) the voltagecurrent characteristic of the model has the pinched hysteresis nature, (ii) the voltage and current zero crossings occur at the same instants, and (iii) when the frequency of the power supply increases, the voltagecurrent pinched hysteresis characteristic tends closer to a singlevalued one, meaning that the voltagecurrent graph becomes that of a resistor (with an increased linearity for ). The conductance of the CassieMayr model decreases when the frequency increases. The hybrid CassieMayr model describes therefore an interesting case of a memristive phenomenon.
1 Introduction
Consider the CassieMayr hybrid model of electric arcs [1][12]
(1) 
driven by a power circuit with the voltage source , resistor and inductor , described by
(2) 
where and are the arc voltage and current, respectively, is the conductance of the arc with and , , , , , , , , are real positive constants, , with , and being constants such that . When the current is small, one can consider , while for large current [3]. Another frequent simplification (however not assumed in this paper) is to have , which means that no energy dissipation occurs due to plasma radiation.
Also, the positive constant plays the role of a minimum value of , as many authors assume that when the current is small. The is a very small conductance between two electrodes when the arc is absent. In general, the value of depends on the distance between the electrodes, their geometry, type of gas used and temperature. Detailed physical assumptions about the above model can be found, for example, in [3][7].
The literature on electric arcs in welding, foundry, gas discharge lamps, lighting as well as voltaic, iron, cobalt, nickel, titunium and mercury arcs is particularly immense over the last 150 years. For example, many papers on electric arcs were published in the Journal of the Franklin Institute over a period of more than hundred years  since 1850s to 1950s  see [8][12] for a few examples of such papers. A list of papers on the topic of electric arcs available in the literature can really be made impressive and long.
Impressive are also the very recent discoveries in the area of nanotechnology related to memristors and memristive circuits and their properties. The announcement by a group of Hewlett Packard researchers [13] about a succesful construction of ’the missing memristor’ renewed interest in the earlier theoretical work of L. O. Chua and others on memristors [14],[15]. That research has been significantly expanded in the last few years, see [16][23] and references therein.
The two seemingly distant areas of electric arcs and memristors are, in fact, close to each other and this paper addresses that issue through the analysis of the properties of the models of electric arcs and the models of memristors.
In particular, it is shown in this paper that the model (1),(2) has the three fingerprints of memristors (see [16],[18],[19]), as follows:

The and characteristic is of the pinched hysteresis type.

The and zero crossings occur at the same instants.

As , then the  pinched hysteresis characteristic becomes that of a resistor, meaning that the  graph is a singlevalued one with no memory effect.
2 Memristors
Memristors, as passive elements, complement the widely used other passive elements: resistors, capacitors and inductors. Each of the four passive twoport elements uses a pair of the current, voltage, charge or flux variables as their inputs and outputs. Memristors are nonlinear elements whose present state at any instant depends on the past (i.e. memory). For example, the currentcontrolled voltage memristor is described by a relationship between the flux and charge , as follows: with some function . This gives the Ohm’s law for such a memristor in the form: , with , the memristance, while and denote the voltage and current, respectively. The memory effect is due to the dependence of memristance on . Other types of memelements are also possible (see [17] and references therein). The recent papers [16],[18][20] show interesting electrical, mechanical and biological devices and phenomena, all having the features (the socalled fingerprints) of memristors. This paper goes in the same direction and proves mathematically that the hybrid CassieMayr model of an electric arc [3] has the three fingerprints of memristors (in the time and frequency domains).
First, it is worth pointing out that the fundamental idea behind the model (1),(2) is to have the conductance as a combination of the conductances and , as follows [4],[24]
(3) 
where and are the conductances obtained from the Cassie and Mayr models, respectively, and the weighting function is monotonically increasing when increases. Also, typically . The most common choice is to have in (1), with being the transition current. When is much smaller than , the Mayr model is dominant in (1), while for large, the Cassie model dominates in (1). This feature of the hybrid CassieMayr model is similar to that of the Hewlett Packard (HP) memristor’s model in which the total memristance is obtained as a weighted sum of and resistances [13]
(4) 
where and denote the resistances of the region with a high concentration of dopants (having low resistance ), and the region with a low dopant concentration (having much higher resistance ), respectively. Thus, and . As a consequence, we have a onetoone correspondence between (3) and (4). Namely, gives resulting in a small memristor’s current. A small , that is in an electric arc, gives . On the other hand, close to yields and a large memristor’s current. A large , that is in an arc, results in .
The used in (3) is not the only possible function used in hybrid arc models. Other monotonical functions used in modeling of electric arcs are , or for constants , and [24].
The above onetoone correspondence between an electric arc and a memristor is further obvious by analyzing the three fingerprints of memristive phenomena, mentioned above and analyzed in detail in the next section.
3 The three memristive fingerprints of electric arcs
Remark 1: The pinched property occurs at the origin, so and . By using and the fact that the pinched property occurs at , it is possible to show that in (1),(2) has one positive value as and , but has two different values (positive and negative) at the origin. The two opposite values of occur half a period apart (see Fig.2(a)). This indicates that there are two different trajectories of the  characteristic at the origin, one that is concave up (with ) and another that is concave down (with ). See Fig.2(b) showing clearly two tangential trajectories with different concavity around the origin.
Proof of fingerprint 1: Notice that the pinched hysteresis occurs as . Thus, the Mayr model is in effect. Let and (see the second fingerprint below). We have . Also, since , therefore we have when . This yields . At we have , and since , therefore . Thus, the pinched hysteresis has slope at .
Now, we shall show that is a twovalued quantity at , that is and are of opposite signs. Using the fact that yields , we obtain . Note that the derivative is positive at and also from (2) we have , since and . Thus, the Mayr model predicts that when the periodic, zeroaverage current crosses the zero value at , it is of a cosine type, with opposite signs of slope at and . If , then, half a period later we have . This yields and . On the other hand, if , then, half a period later we have . This yields and . This proves that the concavity of the trajectory (pinched hysteresis) is opposite at than at . The trajectory moving in time along the  characteristic is of different type of concavity when passing through , every half of the period , as illustrated in Figs.2(a) and 2(b). This completes the proof of the first memristive fingerprint of the hybrid CassieMayr model.
The facts that the slope has the same positive value at and at and opposite values of at and yield the pinched hysteresis of type II, as discussed in [25][27].
Fingerprint 2: The and zero crossings occur at the same instants.
Remark 2: Since , then, if for some , then when . Therefore, to avoid the symbol , it suffices to show that the model (1),(2) yields the conductance . See Fig.1(d) for an illustration.
Proof of fingerprint 2: If for some , then, obviously, and the hybrid arc model follows that of Mayr. Thus, from (3) we have and the results from the Mayr model . When , then we have . Since the must be greater or equal for small (see a remark in section 1), therefore we must have that . This yields .
Fingerprint 3: As , then the  pinched hysteresis characteristic becomes that of a memoryless resistor, meaning that the  graph is a singlevalued one.
Remark 3: The third fingerprint can be proved, by analyzing the area enclosed by the hysteresis  as . Notice that is the frequency of in (2). We shall show that the enclosed area shrinks to zero as . This gives a singlevalued relationship between and .
Proof of fingerprint 3: Since in (2) and the fact that and are periodic, we can assume that
(5) 
for some real numbers , , and , .
The area, say , enclosed by the pinched hysteresis loop  over half period , , equals
(6) 
Using (5) and from (2) we obtain
(7) 
Notice that the righthand side of (7) contains integrals of various products of the cosine and sine terms. The integrals are computed over half of the period , that is for . Such integrals can be computed according to the wellknown formulas
(8) 
(9) 
In addition, the same righthand side holds true if we replace both sine terms by cosine terms in (8).
Notice that by using the above integrals and the fact that is present in the nonzero righthand sides in (8) and (9), we obtain the righthand side of (7) in the form of an infinite series with each term proportional to , reciprocal of frequency. Thus, if , then the area of the pinched hysteresis decays to zero. This means that the  characteristic becomes a singlevalued one and the proof of the third fingerprint is complete.
Fig.1(c) indicates that when the frequency increases, then the  characteristic not only becomes closer to a singlevalued one, but it becomes a linear with decreasing values of and (assuming fixed value of ). The is practically constant in Fig.1(c) for a particular large value of . Note that since for large values, therefore the Mayr model dominates in (1). One can estimate the constant value of for large , by using the wellknown frequency formula for the Mayr model [30]. Namely, the sinusoidal current in the Mayr model yields with . Thus . Fig.3 and and the associated Table 1 illustrate the use of the above limit in a simple numerical example.
[kHz]  [A]  [S] 

3  3.821  0.3650 
5  2.264  0.1281 
7  1.568  0.0615 
9  1.152  0.0332 
11  0.790  0.0156 
4 Variation of hystereses with parameters
Figs. 1(b) and 2(a) show various shapes of the hysteresis loops of (1),(2) when the frequency and current change, respectively. Other parameters in the CassieMayr model impact the hystereses, too. Fig. 4(a), 4(b) and 4(c) illustrate such an impact when the parameters , and vary, respectively. The constant parameters in all three figures were: , , , , , and . In addition, , and in Fig.4(a). Also, , and in Fig.4(b). Finally, , and in Fig.4(c). The hysteresis loops are shown in the first quadrant only (positive voltage and current values). Symmetric graphs exist in the third quadrant with negative voltage and current values.
5 Conclussion
The seemingly distant and disjoint areas of electric arcs from welding, electric furnaces and circuit breakers on one side and memristors from nanometer electronics on the other side have been linked together through their identical mathematical properties (fingerprints). It was shown through mathematical analysis that the hybrid CassieMayr model of electric arcs has all the fingerprints of memristors, passive nonlinear nanoelements with memory. By linking the electric arcs with memristors one can now apply various techniques and methods from the nanoscale electronics (i.e. to analyze energy and power [17],[28],[29]) to the nonlinear plasma phenomena in electric arc furnaces, circuit breakers and welding processes [4][7].
6 Acknowledgement
The author would like to thank Prof. Z. Trzaska from Warsaw (Poland) for his discussion on the topic of electric arcs.
References
 [1] A. M. Cassie, Arc rupture and circuit serverity: a new theory, Paris, CIGRE Rep. 102 (1939)
 [2] O. Mayr, Beitrage zur theorie des statischen und des dynamischen lichbogens, Archiv für Elektrotechnik 37 (12) (1943) 588–608.
 [3] K.J. Tseng, Y. Wang, D. M. Vilathgamuwa , An experimentally verified hybrid CassieMayr electric arc model for power electronics simulations, IEEE Trans. Power Electronics 12 (3) (1997) 429–436.
 [4] M. Moghadasian, E. AlNasser, Modelling and control of electrode system for an electric arc furnace, 2nd Int. Conf. Research in Science, Engineering and Technology (ICRSET), pp. 129–133, March 2122, 2014 Dubai (UAE) doi: 10.15242/IIE.E0314558.
 [5] S. Nitu, C. Nitu, C. Mihalache, P. Anghelita, D. Pavelescu, Comparison between model and experiment in studying the electric arc, J. Optoelectronics and Advanced Materials 10 (5) (2008) 1192–1196.
 [6] D. C. Bhonsle, R. B. Kelkar, New time domain electric arc furnace model for power quality study, J. Electr. Eng. 14 (3) (2014) 240246.
 [7] A. Sawicki, L. Switon, R. Sosinski, Process simulation in the AC welding arc circuit using a CassieMayr hybrid model, Supplement to the Welding Journal 90 (2011) 41s–44s.
 [8] M.L. Foucault, Experiments with the light of the voltaic arc, Journal of the Franklin Institute, 48 (1) 1849, 5052.
 [9] E. Thomson, E, J. Houston, The electric arc – its resistance and illuminating power, Journal of the Franklin Institute, 108 (1) (1879) 48–51.
 [10] E. Karrer, The luminous efficiency of the radiation from the electric arc, Journal of the Franklin Institute, 183 (1) (1917) 61–72.
 [11] J. Slepian, The electric arc in circuit interrupters, Journal of the Franklin Institute, 214 (4) (1932) 413–442.
 [12] K.B. McEachron, Lightning protection since Franklin’s day, Journal of the Franklin Institute, 253 (5) (1952) 441470.
 [13] D. S. Strukov, G. S. Snider, D. R. Stewart, R. S. Williams, The missing memristor found, Nature 453 (2008) 80–83.
 [14] L. O. Chua, Memristor  the missing circuit element, IEEE Trans. Circuit Theory 18 (1971) 507–519.
 [15] L. O. Chua, K. Sung Mo, memristive devices and systems, Proc. IEEE 64 (1976) 209–223.
 [16] S. P. Adhikari, M. P. Sah, H. Kim, L. O. Chua, Three fingerprints of memristor, IEEE Trans. Ciruits and SystemsIL Regular Papers 80 (11) (2013) 3008–021.
 [17] W. Marszalek, T. Amdeberhan, Least action principle for memelements, J. Circ., Syst. Comp. 24 (10) (2015) 1550148.
 [18] M. P. Sah, H. Kim, L. O. Chua, Brains are made of memristors, IEEE Circuits and Systems Magazine 14 (1) (2014) 12–36.
 [19] D. Lin, S. Y. Ron Hui, L. O. Chua, Gas discharge lamps are volatile memristors, IEEE Trans. Circuits and SystemsI: Regular Papers 61 (7) (2014) 2066–2073.
 [20] S. Asapu, Y. V. Pershin, Electromechanical emulator of memristive systems and devices, IEEE Trans. Electron Devices 62 (11) (2015) 3678–3684.
 [21] W. Marszalek and Z. W. Trzaska, Memristive circuits with steadystate mixedmode oscillations, Electr. Lett. 50 (2014) 1275–1277.
 [22] W. Marszalek and Z. W. Trzaska, Properties of memristive circuits with mixedmode oscillations, Electr. Lett. 51 (2015) 140–141.
 [23] W. Marszalek, Bifurcations and Newtonian properties of Chua’s circuits with memristors, DeVry Univ. J. of Scholarly Research 2 (2) (2015) 13–21.
 [24] A. Sawicki, Assessement of power parameters of asymmetric arcs by means of the Cassie and Mayr models, Electrical Review 87 (2) (2011) 131–134.
 [25] D. Biolek, Z. Biolek, V. Biolkova, Pinched hysteretic loops of ideal memristors, memcapacitors and meminductors must be ‘selfcrossing,’ Electr. Lett. 47 (25) (2011) 1385–1387.
 [26] Z. Biolek, D. Biolek, V. Biolkova, Analytical computation of the area of pinched hysteresis loops of ideal memelements, Radioengineering 22 (1) (2013) 32–35.
 [27] W. Marszalek, T. Amdeberhan, Memristive jounce (Newtonian) circuits, Appl. Math. Modelling (2015), doi: 10.1016/j.apm.2015.10.012.
 [28] W. Marszalek, On the action parameter and oneperiod loops of oscillatory memristive circuits, Nonlinear Dynamics 82 (1) (2015) 619–628.
 [29] H. Podhaisky and W. Marszalek, Bifurcations and synchronization of singularly perturbed oscillators: an application case study, Nonlinear Dynamics 69 (3) (2012) 949–959.
 [30] V. M. Myakishev, M. S. Zhevaev, E. M. Shishkov, A method for determining the time constant of a welding arc, Russian Electrical Engineering, 80 (2) (2009) 78–80.