Equivalent Analytical Functions of Sums of Sigmoid like Transcendental Functions

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:

(1)

\begin{matrix} B_{J} (x) = \frac{2 J + 1}{2 J} \coth (\frac{2 J + 1}{2 J} x) - \frac{1}{2 J} \coth (\frac{x}{2 J}) \end{matrix}

$$\begin{align} B_J\left(x\right)=\frac{2J+1}{2J}\coth{\left(\frac{2J+1}{2J}x\right)}-\frac{1}{2J} \coth\left(\frac{x}{2J}\right) \end{align}$$

The Langevin function:

(2)

\begin{matrix} L (x) = \coth (x) - \frac{1}{x} \end{matrix}

$$\begin{align} L\left(x\right)=\coth{\left(x\right)}-\frac{1}{x}\end{align}$$

The error function:

(3)

\begin{matrix} a_{k} f_{E} (\propto_{k} x) = \frac{a_{k}}{\sqrt{π}} \int_{- \propto_{k} x}^{+ \propto_{k} x} e^{t^{2}} d t \end{matrix}

$$\begin{align} a_kf_E({\propto{}}_kx)=\frac{a_k}{\sqrt{\pi{}}}\int_{-{\propto{}}_kx}^{+{\propto{}}_kx}e^{t^2}dt \end{align}$$

The tangent hyperbolic function:

(4)

\begin{matrix} a_{k} f_{T} (\propto_{k} x) = a_{k} \tanh (\propto_{k} x) \end{matrix}

$$\begin{align} a_kf_T({\propto{}}_kx)=a_k\tanh{({\propto{}}_kx}) \end{align}$$

The sigmoid function:

(5)

\begin{matrix} a_{k} f_{S} (\propto_{k} x) = \frac{a_{k}}{1 + e^{- \propto_{k} x}} - \frac{a_{k}}{2} \end{matrix}

$$\begin{align} a_kf_S\left({\propto{}}_kx\right)=\frac{a_k}{1+e^{-{\propto{}}_kx}}-\frac{a_k}{2} \end{align}$$

The inverse tangent function:

(6)

\begin{matrix} a_{k} f_{A} (\propto_{k} x) = {a_{k} t a n}^{- 1} (\propto_{k} x) \end{matrix}

$$\begin{align} a_kf_A\left({\propto{}}_kx\right)={a_ktan}^{-1}{({\propto{}}_kx}) \end{align}$$

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 a_k f_z(a_kx) is representing any of the investigated functions and a₀f_0z(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:

(7)

\begin{matrix} a_{0} f_{0 z} (\propto_{0} x) = \sum_{k}^{n} a_{k} f_{z} (\propto_{k} x) \end{matrix}

$$\begin{align} a_0f_{0z}({\propto{}}_0x)=\ \sum_k^na_kf_z\left({\propto{}}_kx\right)\end{align}$$

At x = x_m

(8)

\begin{matrix} a_{0} = \sum_{k}^{n} a_{k} \end{matrix}

$$\begin{align} a_0=\ \sum_k^na_k\end{align}$$

Here n is the maximum number of k functions to be added up and x_m is the maximum numerical value of x where the functions satisfy the second criterion. The user has the freedom to select x_m 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:

(9)

\frac{d}{d x} [a_{0} f_{0 z} (\propto_{0} x)] = \frac{d}{d x} \sum_{k}^{n} a_{k} f_{z} (\propto_{k} x)

$$\begin{equation} \frac{d}{dx}[a_0f_{0z}\left({\propto{}}_0x\right)]=\ \frac{d}{dx}\sum_k^na_kf_z\left({\propto{}}_kx\right) \end{equation}$$

At x = 0

(10)

\begin{matrix} a_{0} \propto_{0} = \sum_{k}^{n} a_{k} \propto_{k} \end{matrix}

$$\begin{align} a_0{\propto{}}_0=\ \sum_k^na_k{\propto{}}_k\end{align}$$

Therefore

(11)

\begin{matrix} \propto_{0} = \frac{1}{a_{0}} \sum_{k}^{n} a_{k} \propto_{k} \end{matrix}

$$\begin{align} {\propto{}}_0=\ \frac{1}{a_0}\sum_k^na_k{\propto{}}_k \end{align}$$

With the knowledge of the two parameters the equivalent function a₀f_0z (µ ₀x) can be constructed.

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.

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

Due to the inherent different shapes of the sampled functions all functions need to be normalized to approximate the shape of the B_j0/L_r0 functions. The following normalization to models (3), (4), (5) and (6) make the closest fit to the Brillouin/Langevin functions:

(12)

B_{J 0} = \sum_{k}^{n} a_{k} (\frac{2 J + 1}{2 J} \coth (\frac{2 J + 1}{2 J} 2.5 \propto_{k} x) - \frac{1}{2 J} \coth (\frac{2.5 \propto_{k} x}{2 J}))

$$\begin{equation} B_{J0}=\sum_k^na_k\left(\frac{2J+1}{2J}\coth{\left(\frac{2J+1}{2J}2.5{\propto{}}_kx\right)}-\frac{1}{2J}\coth{\left(\frac{2.5{\propto{}}_kx}{2J}\right)}\right) \end{equation}$$

(13)

\begin{matrix} L_{r 0} = \sum_{k}^{n} a_{k} \coth (2.5 \propto_{k} x) - \frac{1}{2.5 \propto_{k} x} \end{matrix}

$$\begin{align} L_{r0}=\sum_k^na_k\coth{\left(2.5{\propto{}}_kx\right)}-\frac{1}{2.5{\propto{}}_kx}\end{align}$$

(14)

{E r f}_{0} = \sum_{k}^{n} \frac{{0.88 a}_{k}}{\sqrt{π}} \int_{- {0.75 \propto}_{k} x}^{+ {0.75 \propto}_{k} x} e^{t^{2}} d t

$$\begin{equation} {Erf}_0=\sum_k^n\frac{{0.88a}_k}{\sqrt{\pi{}}}\int_{-{0.75\propto{}}_kx}^{+{0.75\propto{}}_kx}e^{t^2}dt\ \end{equation}$$

(15)

{T a n h}_{0} = \sum_{k}^{n} {0.936 a}_{k} \tanh ({0.955 \propto}_{k} x)

$$\begin{equation} {Tanh}_0=\sum_k^n{0.936a}_k\tanh{({0.955\propto{}}_kx}) \end{equation}$$

(16)

\begin{matrix} {S i}_{0} = \sum_{k}^{n} \frac{1.79 a_{k}}{1 + e^{- {1.75 \propto}_{k} x}} - \frac{{1.79 a}_{k}}{2} \end{matrix}

$$\begin{align} {Si}_0=\sum_k^n\frac{1.79a_k}{1+e^{-{1.75\propto{}}_kx}}-\frac{{1.79a}_k}{2} \end{align}$$

(17)

{A t}_{0} = \sum_{k}^{n} 0.7 {t a n}^{- 1} ({1.2 \propto}_{k} x)

$$\begin{equation} {At}_0=\sum_k^n 0.7{tan}^{-1}{({1.2\propto{}}_kx}) \end{equation}$$

By forming the difference Δ between numerical sum of the Br/Lr and c_k a₀f_0z ( γ_k∝ ₀x) 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.

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

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.

(18)

I △ = \int_{0}^{\infty} [{{a_{0} c}_{0} f}_{00} ({γ_{0} \propto}_{0} x) - \sum_{k}^{n} {{a_{k} c}_{k} f}_{z} ({γ_{k} \propto}_{k} x)] d x

$$\begin{equation} I\bigtriangleup{}=\int_0^{\infty{}}[{{a_0c}_0f}_{00}\left({{\gamma{}}_0\propto{}}_0x\right)-\sum_k^n{{a_kc}_kf}_z\left({{\gamma{}}_k\propto{}}_kx\right)]dx \end{equation}$$

Here c_k 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.

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

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.

eISSN:: 2444-8656
Lingua:: Inglese

Frequenza di pubblicazione:: Volume Open
Argomenti della rivista:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Equivalent Analytical Functions of Sums of Sigmoid like Transcendental Functions

Pubblicato online: 19 lug 2018

Pagine: 403 - 408

Ricevuto: 03 mar 2018

Accettato: 19 lug 2018

DOI: https://doi.org/10.21042/AMNS.2018.2.00030

Parole chiavetranscendental functions, sigmoid functions, modelling

© 2018 J. Takács, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

Parole chiave
transcendental functions, sigmoid functions, modelling