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Fractional order inductive phenomena based on the skin effect

J.A. Tenreiro Machado · Alexandra M.S.F. Galhano

Abstract The Maxwell equations play a fundamental This phenomenon shows characteristics well mod- role in the electromagnetic theory and lead to models elled by the Fractional Calculus (FC) tools [1, 15, 17, useful in physics and engineering. This formalism in- 18, 20] revealing a dynamics of half order. Moreover, volves integer-order differential calculus, but the elec- the model development based on Maxwell’s equations tromagnetic diffusion points towards the adoption of a suggests the possibility of implementing devices with fractional calculus approach. This study addresses the inductive characteristics of variable fractional order. skin effect and develops a new method for implement- This characteristic is of utmost importance since re- ing fractional-order inductive elements. Two genetic searchers have been devoting attention mainly to the algorithms are adopted, one for the system numeri- electrical elements with fractional-order capacitances cal evaluation and another for the parameter identifi- [5, 6, 11, 22], while the inductive case has been ad- cation, both with good results. dressed only as an effect occurring in some electrical machines [2, 4, 14, 21]. Keywords Skin effect · Electromagnetism · Having these ideas in mind this paper is organized Inductance · Fractional calculus · Genetic algorithms as follows. Section 2 summarizes the classical math- ematical description of the SE based on the Maxwell equations. Section 3 re-evaluates the results demon- 1 Introduction strating its fractional dynamics of half order. After clarifying the fundamental concepts Sect. 4 addresses The tendency of a high-frequency electric current to the case of implementing inductive electrical elements distribute itself in a conductor so that the current den- of variable fractional order. For that purpose two ge- sity near the surface is greater than that at its core is netic algorithms are adopted. A first genetic algorithm called the skin effect (SE). The SE can be reduced by is adopted for yielding the initial conditions required using stranded instead of solid wire, increasing the ef- in the calculation of the model differential equations. fective surface area of the wire for a given wire gauge. The second genetic algorithm is used for identifying the parameters of a FC approximate model. Finally, Sect. 5 draws the main conclusions.

2 The skin effect

In the differential form the Maxwell equations are [19] √ ∂B = ˜ ∇×E =− (1) For a sinusoidal field E√ 2E sin(ωt) we√ adopt ∂t the complex notation E = 2Ee˜ jωt, where j = −1, ∂D yielding ∇×H = δ + (2) ∂t ∇· = d2E˜ 1 dE˜ D ρ (3) + + q2E˜ = 0 (11) dr2 r dr ∇·B = 0(4) with q2 =−jωγμ. where E, D, H, B, δ represent the vectors of elec- Equation (11) is a particular case of the Bessel tric field intensity, electric flux density (or electric equation that has a solution: displacement), magnetic field intensity, magnetic flux density and the current density, respectively, ρ and t q J0(qr0) are the charge density and time, and ∇ is the nabla E = I, 0 ≤ r ≤ r (12) 2πr γ J (qr ) 0 operator. 0 1 0 For a homogeneous, linear and isotropic media, we where J0 and J1 are complex valued Bessel functions have of the first kind of orders 0 and 1, respectively. Equation (12) establishes the SE phenomenon that D = εE (5) consists on having a non-uniform current density, B = μH (6) namely a low density near the conductor axis and an δ = γ E (7) high density on surface, the higher the frequency ω. Therefore, for a conductor of length l0 the total voltage ˜ ˜ ˜ where ε, μ and γ are the electrical permittivity, the drop is ZI = El0 and the equivalent electrical com- magnetic permeability and the conductivity, respec- plex impedance Z is given by tively. In order to study the SE we consider a cylindrical ql0 J0(qr0) Z = (13) conductor with radius r0 conducting a current I along 2πr0γ J1(qr0) its longitudinal axis. In a conductor, even for high fre- ∂D For small values of x the Taylor series [16] leads to quencies, the term ∂t is negligible in comparison with the conduction term δ, that is, the displacement cur- 2 rent is much lower than the conduction current, and (2) x J0(x) = 1 − +··· (14) simplifies to ∇×H = δ. Therefore, for a radial dis- 22 tance r

−6 −1 3 3 The fractional calculus perspective μ0 = 1.257 × 10 Hm and μr = 10 . For model (13) the numerical values were obtained through a The standard approach in electrical engineering in or- symbolic mathematical package, while for (19)–(20) der to avoid handling the transcendental equation (13) it is straightforward to attain the results. We verify is to assign a resistance R and an inductance L given that the explicit (19) is inferior to the implicit (20) ˜ = + by Z R jωL. Nevertheless, this method is inade- fractional-order approximation, and that this one leads { } quate because the model parameters L,R must vary to a very good curve fitting. with the frequency (see Fig. 1). Expression (18) reveals the half-order nature of the dynamic phenomenon, at high frequencies (i.e., ˜ 1 Z ∼ ω 2 ) which is not captured by the classical integer- 4 Fractional-order inductive elements order approach. The FC eliminates those problems [9, 13]. A simple method is to join the two asymp- totic expressions (17)–(18) through the so-called ex- In the previous section we observed that the SE leads plicit and implicit fractional-order approximations: to a fractional model of the electrical field of order 1 α = . A possible question is if, and how, other values α 2 ˜ jω of α can be designed, either by varying the geometry Zapp = Z0 1 + (19) a of the conductor, or by modifying its electromagnetic jω α properties. Z˜ = Z 1 + (20) app 0 a Equation (11) has now to be integrated numerically since it does not follow any more the Bessel equation. = l0 = 4 = 1 Adopting the Euler forward approximation for the first where Z0 2 , a 2 and α 2 .Itmustbe πr0 γ r0 γμ noted that, while other approximations are possible, and second order derivatives and making explicit the expressions (19)–(20) have simple analytical struc- real and imaginary components of the sinusoidal elec- ˜ l0 ˜ l0 ωμ tric field, E = Er + jEi , we get the approximation tures yielding Zapp = and Zapp = (1 + πr2γ 2πr0 2γ → →∞0 j) for ω 0 and ω , respectively. r For example, Fig. 2 compares the Bode diagrams Er (k + 2) + −2 + Er (k + 1) r of amplitude and phase of E(k0) based on expres- sions (13), (19) and (20) in the case of a conductor + − r + 2 = 7 −1 −3 1 Er (k) ωμ r γ(r)Ei(k) 0 (21) with γ = 10 m, l0 = 1m,r0 = 3.02 × 10 m, r Fig. 2 Amplitude and phase Bode diagrams of E(k0) for the theoretical and the approximate expressions with 7 −1 γ = 10 m, l0 = 1m, −3 r0 = 3.02 × 10 m, −6 −1 μ0 = 1.257 × 10 Hm 3 and μr = 10

r The calculation of (21)–(22) requires initial condi- Ei(k + 2) + −2 + Ei(k + 1) r tions compatible with (23)–(24). Therefore, for esti- mating the (unknown) initial conditions it was imple- r 2 + 1 − Ei(k) − ωμ r γ(r)Er (k) = 0 (22) mented a Genetic Algorithm (GA) [7, 8, 10], whose r population are the values {Er (1), Er (0), Ei(1), Ei(0)}, where k and k + 1 represent two consecutive sampling with fitness function: points in space and r is the integration step along the 2 1 Ei(k + 1]−Ei(k) conductor radius. J = init ωμ r The numerical initialization must be obtained from 2 the boundary conditions + Er (k + 1) − Er (k) (25)

1 dE 1 Ei(k0) − Ei(k0 − 1) H = (23) H0 = (26) jωμ dr ωμ r

I where k0 r = r0. It was adopted a GA popula- H0 = H(r = r0) = (24) 2πr0 tion of nGA = 2000 elements, crossover and muta- Fig. 3 Amplitude and phase Bode diagrams of E(k0) for a annular conductor with inner and outer radius r1 and r0,such that r1 ={0, 0.3, 0.5, 0.7}×r0, with γ = 107 −1 m, l0 = 1m, −3 r0 = 3.02 × 10 m, −6 −1 μ0 = 1.257 × 10 Hm 3 and μr = 10

⎧ = = ⎪ r0 ≤ 1 tion probabilities of pc 0.5 and pm 0.1, respec- ⎨ 66 i 3 nGA tively, and elitism. Furthermore, the GA was executed r(i) = r0 1 n

The first approach for modifying the properties of sion was considered: the SE consists of adopting a conductor with a differ- ent geometry. One simple possibility is, for example, β = − r −∞ +∞ to have a annular conductor with inner and outer ra- γ(r) γ0 1 , <β< (28) r0 dius r1 and r0, respectively. Figure 3 shows the am- plitude and phase Bode diagrams of E(k ) for r = 0 1 Obviously, β = 0 yields the case of constant elec- {0, 0.3, 0.5, 0.7}×r . We verify that by eliminating 0 trical conductivity which was analyzed analytically in the flow of current in the inner part of the conductor we can shift the frequency response. Sects. 2 and 3. = The second approach for a different SE consists of During the experiments the numerical values γ 7 −1 −3 varying the electrical conductivity with the conductor 10 m, r0 = 3.02 × 10 m, μ0 = 1.257 × −6 −1 = 3 = −2 −1 = radial distance, that is, to have γ = γ(r),0≤ r ≤ r0. 10 Hm , μr 10 , ωmin 10 s and ωmax For the electrical conductivity the following expres- 103 s−1 were adopted. Fig. 5 Variation of the parameters {E0,a,α} versus β

jω α Figure 4 depicts the Bode diagrams of amplitude E = E 1 + , ={− − 1 1 } app 0 a and phase of E(k0) for β 1, 2 , 0, 2 , 1 . It was observed that when moving far away from E0 > 0,a>0,α>0 (29) the central case of β = 0 the results became more and { } more ‘unstable’, that is, with considerable variations in For the estimation of the three parameters E0,a,α an identification GA with fitness function: the plots. Therefore, in the study were considered only 2 those cases that depicted a sound numerical response. Jident = Re(Eapp) − Er (k0) The Bode plots reveal that at low frequencies we + − 2 get the usual resistive behaviour, but at high frequen- Im(Eapp) Ei(k0) (30) cies we have inductive effects of different fractional was implemented, where Ω represents the set of nΩ order. 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