Approximation of the Nonlinear B-H Curve by Complex Exponential Series

The paper presents an accurate and simple method for the approximation of the nonlinear B-H curves using expansion into complex exponential series. The least-squares fit of the model is obtained by the application of the Moore-Penrose pseudoinverse. Due to the completeness of the orthogonal basis of the approximation, any desired accuracy can be achieved with increasing the number of terms. Using the measurements acquired by an automated system based on the PXI digitizer boards, the feasibility of the method is proved.


I. INTRODUCTION
The numerical methods like finite element method (FEM) need accurate expressions representing the B-H curve. Although it is possible to introduce hysteresis models like Jiles-Atherton [1] or Preisach model of ferromagnetic materials into the FEM analysis [1]- [14], it is still relevant to use the single-valued B-H curve in such an analysis. For the magnetically soft materials, the hysteresis loops are relatively narrow. Therefore, the single-valued B-H curve represented by the initial magnetization curve adequately characterizes such materials in many applications like the calculation of the magnetic field in motors and generators [4]. Consequently, the computational effort is reduced, and the speed of the calculations is increased. In such a case, the losses are usually estimated using analytical formulae, while the nonlinear problem is solved using the Newton-Raphson or successive substitution method [15]- [17]. The accurate modeling of single-valued B-H curve is also important in equivalent magnetic circuit modeling [18]. Moreover, the anhysteretic, single-valued B-H models [19], [20] are important in the analysis of magnetic recording systems [21], while the anhysteretic B-H curve is the basis of the Jiles-Atherton model of the magnetic hysteresis [22].
The analytical expressions used for the singlevalued representation of a B-H curve are traditionally based on empirical formulas [23]- [25] or power series The associate editor coordinating the review of this manuscript and approving it for publication was Wuliang Yin .
representation [26], [27]. The other approaches include representation by hyperbolas [26], transcendental functions [26], exponential series or sums of exponentials [28], [29], and rational functions [30]- [32]. The possibility of very accurate modeling of the whole B-H curve (including Ralyeigh region and saturation knee) launched the approximation using Fourier series [26], [33]. The Fourier series approximation converges well, and any desired accuracy can be achieved [26]. However, the calculation of Fourier coefficients [34] needs the integration of the B-H curve, which is available only at discrete (measured) points. Thus, the numerical integration and the interpolation of the curve between the points must be applied [33]. The Fourier analysis is widely applied in the time to frequency transformations and representation of signals and functions [35]. In the wider context, the representation of the discrete-time systems and signals using the z-transformation and finite impulse response (FIR) models is a well-established discipline [35], [36]. Applying the Fourier transformation to the Lth-order FIR system approximation which is a sum of complex exponentials. Here, f denotes frequency, z is complex variable in z-transformation, a r is set of coefficients χ is sampling frequency, and A is the approximation of the system function. The Fourier transform of the FIR model is a periodic function of frequency with period χ [35]. In the sequel (section 3) it will be demonstrated that the proposed complex exponential series form the complete orthogonal system. The methods for the determination of the coefficients a r are well established [37], [38] and thus can be applied in the proposed modeling of B-H curve. While the system modeling usually imposes additional requirements on the causality [39] or the real character of the coefficients a r , these requirements are weakened in the B-H curve modeling, which allows direct application of the unconstrained least-squares methods in the evaluation of the coefficients.
The main purpose of this paper is to present a method for the approximation of the nonlinear B-H curves using expansion into complex exponential series, where the least-squares fit to the data is obtained by the application of the Moore-Penrose pseudoinverse. The proposed approximation easily allows use of the non-uniformly sampled points in the measured B-H curve. Moreover, it is analytically derivable, which can be applied in the calculation of the magnetic permeability and the energy stored in the magnetic material [27], [28].
The paper is organized as follows: in Section II the measurement system based on NI 4461 DAQ cards is described. Section III presents the proposed method for the B-H curve approximation. Section IV gives the results. Finally, Section V are conclusions.

II. MEASUREMENT SYSTEM
The measurement system is based on commercial National Instruments PXI modular measurement system equipped with digitizer board, Toellner TOE 7621 four-quadrant power supply working in voltage-controlled mode, R-C integrator, precise shunt and a ferromagnetic coil test sample.
The NI-PXI4461 (named hereafter as NI4461) module is used as a digitizer unit placed in PXI chasis. It is a dynamic signal acquisition device equipped with two simultaneously sampled input and output channels, integrating sigma-delta A/D and D/A converters, both with 24-bit resolution and sampling rates up to 204.8 kS/s. The PXI system generates the excitation signal, which is fed to the input of the TOE 7621 power supply. The amplitude of the excitation current is measured as voltage drop present on the shunt R s . The resistance of the shunt is considered constant, independent of the ambient temperature and applied voltage.
The control of measurement process and measured data processing is performed in LabVIEW environment. Aim was to use fully automated measurement of the first magnetization curve to prepare measured data for export in a format ready for the import to a computer program for the B-H curve approximation.   The measurement set-up is based on the IEEE standard procedures for magnetic cores, where the RC integrator is connected on the search (pick-up) coil. The single-valued initial B-H curve is measured using the locus points of the symmetric hysteresis cycles [40], [41]. Instead of using a ring core specimen, measurements were performed using a SU30 ferromagnetic C-core (Iskra). C-cores also have reasonably uniform flux density and therefore are useful for the B-H curve measurement [42]. Fig. 2 and Table 1 give the dimension of the core.
The number of turns was N = 169 in both coils, while the wire diameter was 0.3 mm. Using the Ampère's law, the magnetic field in the core is where l m is the average length of the flux lines, and u S (t) is the voltage drop across the shunt resistor R s . VOLUME 8, 2020 The magnetic flux density is Here, S denotes the cross-section of the coil, R i is the resistor value of the RC integrator, C i is the capacitance of the capacitor in the RC integrator, and u C (t) is the voltage across the capacitor C. Using the data from the Table 1, the crosssection and the average length of the flux lines is S = c · f = 9.388 · 10 −5 m 2 l m = 2(e + g) + cπ = 0.116 m In characterizing an analog-to-digital converter (ADC), one of the important parameters is the integral nonlinearity (INL). As this experiment was done at frequency of 20 Hz, work provided in [43] declares INL that is below 1 µV/V.
For simultaneous sampling of 1 V signal amplitude, input range ±1V and sampling frequency of 51.2 kS/s, the NI4461 metrological characteristics are provided [44]: -temperature coefficient of the magnitude is 7.4 µV/V·K, -temperature coefficient of the phase is 0.000008 • /K, -standard deviation of the magnitude of the voltage ratio is 2.3 µV/V, -deviation of the magnitude of the voltage ratio from the nominal value is within ±3 µV/V, -phase deviation is under 16 µ • . Following components: -current shunt, -integrator resistor, -integrator capacitor were measured in slow mode using the LCR Bridge Rohde & Schwarz (Hameg) HM8118, applying the average and 4-wire connection and the open-short calibration.
The resistance of the current shunt is R s = (1.0084 ± 0, 0041) .

III. COMPLEX FIT
Let have the set of M measured pairs (H k , B k ), k = 0, . . . M − 1, which satisfy Using the model representation given by the equation (3), we seek the representation of the sampled nonlinear curve using the approximation where b denotes the modeled magnetic flux density and h is the magnetic field strength. Moreover, a k is a set of complex-valued coefficients, L is the order of the approximation, and χ is a parameter that defines the periodicity of the model in terms of the variable h.
The set of orthogonal functions form the complete orthogonal system. It can be seen using the substitution where the expansion (8) (8) is periodic in h with the period χ , which is easily obtained using the substitution which yields The impact of the periodicity of the model on the upper bound of the parameter χ can be illustrated with Figs. 3 and 4. In Fig. 3, H max is lower than χ, and consecutive replicas of the model does not overlap. In Fig. 4, H max exceeds χ and the consecutive replicas of the model overlap in the intervals H ϑ . Thus, accurate modeling of the B-H curve cannot be achieved in the pink regions, since the model function cannot be uniquely defined for these regions. To ensure that the model does not overlap in h the parameter χ has to satisfy the condition: (13) 49612 VOLUME 8, 2020 Nevertheless, the experiments with the varying the parameter χ from 1.1H max to 2H max has shown that the quality of the approximation gradually degrades for χ lower than 1.5H max . Therefore, in all further presented modeling the parameter χ was chosen to be χ = 2H max (14) It should be noted that the interval we can express the error function as [45], [46] Here, T denotes vector transpose. Equation (13) can be expressed for M (B k , H k ) pairs as The design problem is to minimize the sum of error squares: which can be expressed in matrix notation as: In the sequel we use the mathematical equivalence of the least-squares problem (18) and the system of normal equations [47] (F H F)A = (F H D) (25) Here, H denotes the Hermitian conjugate transpose operator.
Although it is tempting to solve (14) directly as it is impossible for L M , since in this case F becomes ill-conditioned [37], and a direct solution will probably have large errors. Therefore we propose to solve the system using the matrix pseudoinverse.
For a m × n matrix F, its pseudoinverse F + gives the minimum-norm [48] solution to the least-squares problem as The explicit list of the properties of the Moore-Penrose (MP) pseudoinverse can be found in [47]. The calculation of the MP pseudoinverse is generally based on the singular value decomposition, which is efficiently implemented in LAPACK and also in LAPACK based MATLAB routine pinv (15).
Using the proposed approximation, the derivative of the b with respect to h is from which the permeability can be expressed as Remark: If the set of coefficients a k would be restricted to real numbers, the magnitude of the model function would be even function of parameter h [26]. In line with the Whittaker-Shannon theorem this would impose a stronger condition on the parameter χ [49]- [51]:

IV. RESULTS
Using the proposed method, the approximation of a measured B-H curve was calculated. The measurements were performed using the setup from Section 2. In the origin of the b-h plane, the point (0,0) was added to measured pairs. The overall number of data point was M = 1274, while the order of the approximation was L = 100. Since the method allows application of the non-equidistant sampling of the data points (in h-or b-axis), the measurements were done with denser sampling for lower values of the magnetic flux densities. The algorithm for the approximation was programmed in MATLAB, and the pseudoinverse was calculated using the VOLUME 8, 2020  routine pinv. Fig. 5 presents the comparison of the measured data and the approximation results. To avoid an imaginary component in the model, produced by the numerical and round-off errors, the absolute value of the model is used in comparison. Since the two curves are indistinguishable, the Fig. 6 presents the difference between measurement and modeling.

V. COMPARISON WITH OTHER MODELS
The comparison of the proposed method and several established methods for the approximation of the initial B-H curve was based on the root-mean-square error (RMSE) criterion. RMSE is defined as where b i denotes the approximated value given by the model, B i denotes the measured value, and M denotes the number of measured data points. The models used for the comparison were [26]: The nonlinear regression for models (a-c) was performed in MATLAB, applying the Levenberg-Marquardt algorithm with the MATLAB function nlinfit. The same set of the measured points was applied in the regression and the evaluation of all analyzed models. While Figs. 7-9 are giving a graphical comparison of the measurement data and models, Table 2 summarizes the quality of the modeling results based on the RMSE criterion. It is shown that the proposed model achieves the best fit (lowest RMSE), comparing to the analyzed established models. It should be noted that this is paid by the higher complexity of the proposed model.

VI. CONCLUSION
An accurate and simple least-squares solution is presented for the analytical approximation of single-valued B-H curves.
Applying the Moore-Penrose pseudoinverse, a larger number of points in measurements can be used to obtain a lower order approximation. Due to the completeness of the orthogonal basis of the approximation, any desired accuracy can be achieved with increasing the number of terms. In line with the method of B-H curve modeling, an automated system for its measurement is presented, allowing the remote measurement through the arbitrarily number of points. In this way, a single PXI platform can be shared between several users with their own PC platforms. Thus, the measurements are performed remotely, while the approximation is obtained locally. Using the model, the derivation of B with respect to H is easily obtained analytically, which is important in calculations of permeability and energy stored in the ferromagnetic material.