Memristive Fingerprints of Electric Arcs
We discuss the memristive fingerprints of the hybrid Cassie-Mayr model of electric arcs. In particular, it is shown that (i) the voltage-current 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 voltage-current pinched hysteresis characteristic tends closer to a single-valued one, meaning that the voltage-current graph becomes that of a resistor (with an increased linearity for ). The conductance of the Cassie-Mayr model decreases when the frequency increases. The hybrid Cassie-Mayr model describes therefore an interesting case of a memristive phenomenon.
driven by a power circuit with the voltage source , resistor and inductor , described by
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 . 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 -.
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 - 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  about a succesful construction of ’the missing memristor’ renewed interest in the earlier theoretical work of L. O. Chua and others on memristors ,. That research has been significantly expanded in the last few years, see - 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.
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 single-valued one with no memory effect.
Memristors, as passive elements, complement the widely used other passive elements: resistors, capacitors and inductors. Each of the four passive two-port 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 current-controlled 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 mem-elements are also possible (see  and references therein). The recent papers ,- show interesting electrical, mechanical and biological devices and phenomena, all having the features (the so-called fingerprints) of memristors. This paper goes in the same direction and proves mathematically that the hybrid Cassie-Mayr model of an electric arc  has the three fingerprints of memristors (in the time- and frequency domains).
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 Cassie-Mayr 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 
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 one-to-one 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 above one-to-one 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 two-valued 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, zero-average 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 Cassie-Mayr model.
Fingerprint 2: The and zero crossings occur at the same instants.
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 single-valued 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 single-valued relationship between and .
Proof of fingerprint 3: Since in (2) and the fact that and are periodic, we can assume that
for some real numbers , , and , .
The area, say , enclosed by the pinched hysteresis loop - over half period , , equals
Notice that the right-hand 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 well-known formulas
In addition, the same right-hand 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 non-zero right-hand sides in (8) and (9), we obtain the right-hand 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 single-valued 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 single-valued 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 well-known frequency formula for the Mayr model . 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.
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 Cassie-Mayr 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.
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 Cassie-Mayr 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 ,,) to the nonlinear plasma phenomena in electric arc furnaces, circuit breakers and welding processes -.
The author would like to thank Prof. Z. Trzaska from Warsaw (Poland) for his discussion on the topic of electric arcs.
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