Equivalent Analytical Functions of Sums of Sigmoid like Transcendental Functions

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Abstract

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The Brillouin and Langevin as well as the error function, the tanh, sigmoid and the tan-1 functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.

Abstract

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The Brillouin and Langevin as well as the error function, the tanh, sigmoid and the tan-1 functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.

1 Introduction

One of the unsolved problems of mathematics is to add up transcendent functions analytically to be used in calculations. Functions used for calculating magnetisation and for its mathematical modelling of hysteresis, almost exclusively are falling into this category. The different models usually provide close approximations to the Brillouin B j(x) [1] and Langevin L(x) [2] functions, at the same time, giving approximate solutions to Everett integral [3,4] as a replacement of the Preisach distribution [5]. The most well known and most frequently used analytical functions for this kind of modelling, leading to the most likely distributions, are the Erf (error function) [6], Tanh (hyperbolic tangent) [7, 8], Si (sigmoid) [9] and the tan-1 (inverse tangent) [10] functions. Although other functions are also used and the method described here may apply to those functions as well, we concentrate on these four functions to demonstrate the useful nature of the process for mathematicians and physics researchers alike. These can be a useful approximating replacement to other numerical methods in mathematical calculations. These may be applicable in other applied mathematical calculations as well. The sigmoid like character of the magnetisation process obviated the choice of the subject area [11].

To remind the reader, the functions investigated in this paper are as follows:

The Brillouin function:

BJx=2J+12Jcoth2J+12Jx12Jcothx2J

The Langevin function:

Lx=cothx1x

The error function:

akfE(kx)=akπkx+kxet2dt

The tangent hyperbolic function:

akfT(kx)=aktanh(kx)

The sigmoid function:

akfSkx=ak1+ekxak2

The inverse tangent function:

akfAkx=aktan1(kx)

2 Equivalent function

In Magnetics, in most occasions, composite materials are used, which unavoidably calls in modelling, for the summation of the functions above [11]. Single phase materials are, almost solely, used only in research and calibration purposes. When ak fz(akx) is representing any of the investigated functions and a0f0z(a0x) describes the equivalent function of their sum respectively, then the numerical values of a0 and a0 parameters need to be determined to complete the characteristic formulation of the equivalent function in a normalized analytical form. For the calculation of these two parameters two criteria will be used. The first criterion calls for the equivalence of a0 and the sum of the amplitudes of the n number of component functions. The second criterion prescribes that a0, the inclination of the curve with the horizontal axis should be the same as that of composite curve at x = 0.

The mathematical formulations of the two criteria are as follows:

The first criteria in mathematical form:

a0f0z(0x)=knakfzkx

At x = xm

a0=knak

Here n is the maximum number of k functions to be added up and xm is the maximum numerical value of x where the functions satisfy the second criterion. The user has the freedom to select xm within the saturation region. This choice will slightly change numerical values of a0 and a0without effecting the mathematical formulations described above.

To satisfy the second criteria the first derivatives of the functions have to be taken:

ddx[a0f0z0x]=ddxknakfzkx

At x = 0

a00=knakk

Therefore

0=1a0knakk

With the knowledge of the two parameters the equivalent function a0 f0z0x) can be constructed.

3 Error of the approximations

Let us assume that the composite functions have four components with the following identical normalized parameters for all four component functions:

a1= 5 α1 = 0.75

a2 = 3 α2 = 0.9

a3 =2 α3 = 1.5

a4= 0.5 α4 = 3

By using (8) and (11) the numerical (computer aided) values of the two parameters can be calculated as: a0= 10.5 and a0 = 1.043.

The numerical and the modeling functions are shown in Fig. 1.

Fig. 1

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Fig. 1

The numerical and analytical sums of sigmoid functions Solid lines numerical broken lines analitical

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00030

Due to the inherent different shapes of the sampled functions all functions need to be normalized to approximate the shape of the Bj0/Lr0 functions. The following normalization to models (3), (4), (5) and (6) make the closest fit to the Brillouin/Langevin functions:

BJ0=knak2J+12Jcoth2J+12J2.5kx12Jcoth2.5kx2J
Lr0=knakcoth2.5kx12.5kx
Erf0=kn0.88akπ0.75kx+0.75kxet2dt
Tanh0=kn0.936aktanh(0.955kx)
Si0=kn1.79ak1+e1.75kx1.79ak2
At0=kn0.7tan1(1.2kx)

By forming the difference Δ between numerical sum of the Br/Lr and ck a0 f0z ( γk0x) functions we can assess the accuracy of the each of the modeling functions. Fig. 2 shows that Δ deviation is maximum from the numerical (computer aided) curve of Br/Lr in case of the Erf function (14) followed by the Si (16) and the Tanh (15) functions. The smallest difference created by the tan-1(17) function.

Fig. 2

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Fig. 2

The difference Δ between the normalized numerical Br/Lr and the curves of the equivalent functions shown in Fig. 1.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00030

The maximum linear and the integrated error IΔ, (see equ. (18)) between the Br0/Lr0 numerical (computer aided) and equivalent analytical functions, in the first quadrant is a good indicator for the most accurate and usable function for mathematical operations.

I=0[a0c0f00γ00xknakckfzγkkx]dx

Here ck and γk represent the factors needed to normalize the amplitudes and inclination for the various functions respectively and f00 is the sum of the four Brillouin functions of the same parameters as the modeling functions.

The IΔ integral function in (18) shown in Fig. 3 for each modeling function, depicts a slightly different picture, that the tan-1 function has the smallest error, followed by tanh and Si functions. The smallest error makes the tan-1 the most accurate, however both the Si function and the tan-1 function are hard functions, not easy to apply them in mathematical operations. Considering accuracy and convenience the Tanh function looks the best selection to use for modeling composite hysteretic materials.

Fig. 3

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Fig. 3

The difference IΔ between the numerical Br/Lr and the equivalent curves, integrated for the first quadrant.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00030

4 Conclusion

In modeling practice often necessary to use sums of transcendental functions. One of mathematics unsolved problems however is adding up analytically transcendental functions. There is a need in applied mathematics, used in physics and engineering for an analytical approximation of the process which can be incorporated into mathematical operations. Presently only numerical calculation by computer is available for this. In this paper an analytical process is proposed for sigmoid like functions, mostly applied for calculations in Magnetics. The error between the computer generated numerical result and the proposed analytical equivalents are illustrated graphically, indicating a good accuracy of the proposed analytical approximations. Considering the accuracy and the mathematical usefulness, the Tanh function has the most practical advantage over the other three functions, the Erf, Si and tan-1 functions, for inclusion in mathematical calculations.

Communicated by Juan L.G. Guirao

References

  • [1]

    Brillouin M.L. (1927) Les moments de rotation et le magnetisme dans la mechanique ondolatoire. J.de Phys. Radium Vol.VIII. p.74

  • [2]

    Langevin M.P. (1905) Magnetisme et theorie des electrons. Ann. Chim. Et Phys. Vol. 94. p. 277.

  • [3]

    Takacs J. The Everett integral and its analytical approximation in: L. Malinski (Ed.) pp: 203- 230 in Advanced Magnetic Materials, Intech Publication, (2012).

  • [4]

    Finocchio G., Carpentieri M., Cardelli E., Azzerboni B. Analytical solution of Everett integral using Lorentzian Preisach function approximation. JMMM Vol. 300 No.2 (2006) pp. 451-470.

  • [5]

    Preisach F., Uber die magnetische Nachwirkung. (1935) Zeitschrift fur Physik, pp. 227-302

  • [6]

    Della Torre E., (1999) Magnetic Hysteresis IEEE Press N.Y.

  • [7]

    J. Takács. Mathematics of Hysteretic Phenomena (2003) Wiley, Weinheim.

  • [8]

    J. Takács and A. Nemcsics (2014) Hysteretic behaviour in some natural processes pp: 89-106, In Hysteresis: Types, Applications and Behaviour Patterns in complex systems, Edited by Jose Carlos Dias Nova S. P. N.Y.

  • [9]

    Petrescu L., Cazacu E., Petrescu K. Sigmoid functions used in hysteresis phenomenon modeling. IEEE Advanced Topics in Electrical Engineering (ATEE), 9th International Symposium, (2015) Location: Bucharest, Romania INSPEC Accession Number: 15240830.

  • [10]

    Perez-Lojas C. Fitting saturation and hysteresis via arctangent function. IEEE Power Engineering Review, Vol. 20. No. 11 (2000) pp. 55-57.

  • [11]

    J. Takács, Analytical way of transforming the hyperbolic hysteresis model into the T(x) model. COMPEL, Vol. 37 No. 3 (2018) pp: 1131-1138

[1]

Brillouin M.L. (1927) Les moments de rotation et le magnetisme dans la mechanique ondolatoire. J.de Phys. Radium Vol.VIII. p.74

[2]

Langevin M.P. (1905) Magnetisme et theorie des electrons. Ann. Chim. Et Phys. Vol. 94. p. 277.

[3]

Takacs J. The Everett integral and its analytical approximation in: L. Malinski (Ed.) pp: 203- 230 in Advanced Magnetic Materials, Intech Publication, (2012).

[4]

Finocchio G., Carpentieri M., Cardelli E., Azzerboni B. Analytical solution of Everett integral using Lorentzian Preisach function approximation. JMMM Vol. 300 No.2 (2006) pp. 451-470.

[5]

Preisach F., Uber die magnetische Nachwirkung. (1935) Zeitschrift fur Physik, pp. 227-302

[6]

Della Torre E., (1999) Magnetic Hysteresis IEEE Press N.Y.

[7]

J. Takács. Mathematics of Hysteretic Phenomena (2003) Wiley, Weinheim.

[8]

J. Takács and A. Nemcsics (2014) Hysteretic behaviour in some natural processes pp: 89-106, In Hysteresis: Types, Applications and Behaviour Patterns in complex systems, Edited by Jose Carlos Dias Nova S. P. N.Y.

[9]

Petrescu L., Cazacu E., Petrescu K. Sigmoid functions used in hysteresis phenomenon modeling. IEEE Advanced Topics in Electrical Engineering (ATEE), 9th International Symposium, (2015) Location: Bucharest, Romania INSPEC Accession Number: 15240830.

[10]

Perez-Lojas C. Fitting saturation and hysteresis via arctangent function. IEEE Power Engineering Review, Vol. 20. No. 11 (2000) pp. 55-57.

[11]

J. Takács, Analytical way of transforming the hyperbolic hysteresis model into the T(x) model. COMPEL, Vol. 37 No. 3 (2018) pp: 1131-1138

Journal Information

Figures

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    The numerical and analytical sums of sigmoid functions Solid lines numerical broken lines analitical

  • View in gallery

    The difference Δ between the normalized numerical Br/Lr and the curves of the equivalent functions shown in Fig. 1.

  • View in gallery

    The difference IΔ between the numerical Br/Lr and the equivalent curves, integrated for the first quadrant.

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