**2. Actual modelling of the nonlinear inductances under MATLAB-SIMULINK code**

In the following of this chapter, we take as example the equivalent circuit of the HV power supply of microwaves generators used for the industrial applications. The three nonlinear inductances (primary, secondary and shunt) of the special HV power transformer of this alimentation (model in Fig.1) are function of the reluctances of the magnetic circuit portion which it represents. Each one of them is represented by its characteristic Ф (i) outcome of the relation, ( ) ( ( )/ ) <sup>2</sup> *Li n i i* which can be determined from the magnetization curve B (H).

**Figure 1.** Equivalent circuit of HV power supply for microwaves generators

Not having analytical expression representing this B (H) curve for high values of the magnetic field H, we have introduced point by point the values of this curve using a linear interpolation between two consecutive points in the iterations of the code under SIMULINK. However, this method of interpolation can lead in case of temporal simulations including the traditional models of transformers to a numerical instability as quoted in [1]-[9]. A specific routine was elaborated in MATLAB to deduce the values couple (i, Φ) from those (H, B) and geometrical data of three inductances. The implementation of each nonlinear inductance of this circuit under SIMULINK software was realized by using the following blocks (Fig. 2):

**Figure 2.** Implementation of each nonlinear inductance under SIMULINK, a: Voltage measurement, b: Integrator, c: Lookup Table, d: current source


**2. Actual modelling of the nonlinear inductances under MATLAB-**

**Figure 1.** Equivalent circuit of HV power supply for microwaves generators

conclusion.

**SIMULINK code** 

blocks (Fig. 2):

Integrator, c: Lookup Table, d: current source

order to derive analytic expressions of the nonlinear inductances. In the last one a

In the following of this chapter, we take as example the equivalent circuit of the HV power supply of microwaves generators used for the industrial applications. The three nonlinear inductances (primary, secondary and shunt) of the special HV power transformer of this alimentation (model in Fig.1) are function of the reluctances of the magnetic circuit portion which it represents. Each one of them is represented by its characteristic Ф (i) outcome of the relation, ( ) ( ( )/ ) <sup>2</sup> *Li n i i* which can be determined from the magnetization curve B (H).

Not having analytical expression representing this B (H) curve for high values of the magnetic field H, we have introduced point by point the values of this curve using a linear interpolation between two consecutive points in the iterations of the code under SIMULINK. However, this method of interpolation can lead in case of temporal simulations including the traditional models of transformers to a numerical instability as quoted in [1]-[9]. A specific routine was elaborated in MATLAB to deduce the values couple (i, Φ) from those (H, B) and geometrical data of three inductances. The implementation of each nonlinear inductance of this circuit under SIMULINK software was realized by using the following

**Figure 2.** Implementation of each nonlinear inductance under SIMULINK, a: Voltage measurement, b:

To ensure the convergence of nonlinear model simulation under SIMULINK, the simulation steps which is a crucial problem in the numerical simulations has been appropriately chosen Te= 0.01 ms after many trials.

We superimpose in Fig.3A, Fig.3B the simulation results obtained by SIMULINK code with those simulated by EMTP under the same conditions. These shapes resulting from the two codes under nominal operation (U1=220 V et f=50 Hz) are consistent with those obtained in

**Figure 3.** A. Simulation with EMTP and SIMULINK code: Forms of voltage waves (at the nominal operation); B. Simulation with EMTP and SIMULINK code: Forms of current waves (at the nominal operating)

practice (Fig.7), especially the magnetron current curves which respect the maximum current magnetron constraint (Ipeak<1.2A) recommended by the manufacturer. Precisely, the current patterns resulting from SIMULINK are closer to practice than EMTP current patterns. Indeed, The current magnetron peak value reaches approximately -0.96 A (Fig.3.B.3) using SIMULINK code which is near -1 A from experimental results (Fig.7) while the peak value obtained from EMTP code equals -0.92 A. In general, between peak to peak values (Fig.3.A, Fig.3.B, Fig.7), the relative variations never exceed 8 % for EMTP code while those resulting from SIMULINK do not exceed 4%. The accuracy of the outcome resulting from SIMULINK can be justified by the large number of points N=100=2x50 (including the negative values) used to feature B (H) in the table (see lookup table in Fig.2) while in our EMTP code version [18], we were limited to use a restricted number N=17. Taking into account the precision of the various data and acceptable tolerances on operation of the magnetron, modelling was considered to be satisfactory with the two codes.

#### **3. New modelling of the nonlinear inductances**

308 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

B.1

B.2

B.3

**Figure 3.** A. Simulation with EMTP and SIMULINK code: Forms of voltage waves (at the nominal operation); B. Simulation with EMTP and SIMULINK code: Forms of current waves (at the nominal

practice (Fig.7), especially the magnetron current curves which respect the maximum current magnetron constraint (Ipeak<1.2A) recommended by the manufacturer. Precisely, the current patterns resulting from SIMULINK are closer to practice than EMTP current patterns. Indeed, The current magnetron peak value reaches approximately -0.96 A (Fig.3.B.3) using SIMULINK code which is near -1 A from experimental results (Fig.7) while the peak value obtained from EMTP code equals -0.92 A. In general, between peak to peak values (Fig.3.A, Fig.3.B, Fig.7), the relative variations never exceed 8 % for EMTP code while those resulting from SIMULINK do not exceed 4%. The accuracy of the outcome resulting from SIMULINK can be justified by the large number of points

operating)

The field problem solution involving ferromagnetic materials is complicated by the nonlinear relationship between B and H. One of the problems encountered is the absence of single mathematical expression, to represent the magnetization curve characteristic over a wide range of magnetic fields, having a smooth variation of the incremental permeability. As well, in our previous works [2]-[3], the B-H curve introduced in the old model of this special transformer of the HV power supply for magnetrons and based on a set of measurement data, is approximated under software tools (EMTP and SIMULINK) by several straight line segments connecting the points of measurements. However, the B-H curve obtained is not smooth at the joints of the segments and the slopes of the straight lines representing the permeability are discontinuous at these joints [4]-[5]. Hence, the accuracy of permeability computed using such an approximation of the B-H curve is limited by the number of the straight-line segments. In order to improve this model of this special transformer, we present a more accurate modelling of the nonlinear B-H curve of ferromagnetic material used for fabrication (SF19). In fact, the nonlinear B-H curve is represented by analytic expressions considering two parts of this curve: the first one is the linear region which is fitted by a non integer power series and the second one representing the saturation region is approximated by polynomial representation.

#### **3.1. Fitting H-B curve by a hybrid method**

A set of N discrete measurements data In and Фn or Hn-Bn of this special transformer (n=1,2,3,…N=100) is given as depicted in Fig.4. Two parts are considered for this analytic representation:

 The linear region is fitted by a non-integer power series [1]. It is based on selected powers of B (not generally integer) with positive coefficients giving a power series. This linear part of this curve can be expressed by the power curve :

$$H = \sum\_{\vec{l}} k\_{\vec{l}} B^{\frac{\eta\_{\vec{l}}}{\vec{l}}} \text{ with } \mathbf{k} \succeq 0 \text{, } \mathbf{n} \succeq 0 \text{, for all } \mathbf{i}. \tag{1}$$

An adequate procedure of determining initial ki and ni values from logarithmic plots is adopted ensuring that these initial estimates of the parameters will be positive. Once an initial estimate of parameters (ki, ni) has been obtained, the parameters are optimized using regression analysis to get the best fit which is defined as corresponding to the minimum sum of squares of absolute errors. The resulting analytic expression of H is given by:

$$\text{for}H \le H\_{18}\text{ H(B)} = 220.65\text{B}^{0.96} + 19.5\text{B}^{11} \tag{2}$$

$$\text{for } H > H\_{18}$$

$$\text{d} = \text{a}\_1.\text{B} + \text{a}\_3.\text{B}^3 + \text{a}\_5.\text{B}^5 \dots + \text{a}\_{2^{\text{p}+1}}\text{B}^{2^{\text{p}+1}} \tag{3}$$

$$\mathbf{H(B)} = 62967\mathbf{B} - 59157\mathbf{B}^3 + 17475\mathbf{B}^5 - 1409\mathbf{B}^7 \tag{4}$$

$$L\_p' = L\_S = \frac{n\_2 \Phi\_P}{l\_P} = \frac{n\_2^2 B S\_1}{H l\_P} = \frac{n\_2^2 S\_1}{l\_P} \cdot \frac{B}{H(B)}$$

$$L\_p' = L\_S = 106338,46 \frac{B}{220.65 B^{0.96} + 19.5 B^{11}}\tag{5}$$

$$L\_p' = 106338,46 \frac{B}{62967B - 59157B^3 + 17475B^5 - 1409B^7}$$

$$L'\_{Sh} = \frac{n\_2 \Phi\_3}{l'\_{Sh}} = \frac{n\_2.2.\Phi\_{Sh}}{H l\_{Sh}} = \frac{n\_2.2(B.S\_3)}{H l\_{Sh}} = \frac{2n\_2 B.S\_3}{H l\_{Sh}}$$

$$L'\_{Sh} = 139693 \frac{B}{H(B)}$$

$$L'\_{Sh} = 138693 \frac{B}{220.65 B^{9.96} + 19.5 B^{11}}\tag{6}$$

$$L\_{Sh}^{\prime} = 138693 \frac{B}{62967 \text{B} - 59157 \text{B}^3 + 17475 \text{B}^3 - 1409 \text{B}^7} \tag{7}$$

this model. It leads to continuous slopes representing permeability while with the old model under EMTP introducing the B-H curve point by point and approximating it by several straight line segments, the slopes representing the permeability are discontinuous at the joints. Taking into account the precision of the various data (software) and acceptable tolerances in the operation of the magnetron, the validation of this improved modelling based on analytic inductances is considered to be more satisfactory than the old one.

**Figure 6.** A. Simulation with EMTP (- - -) and SIMULINK ( ) code: Forms of voltage waves (nominal operation); B. Simulation with EMTP(- - -) and SIMULINK( ) code: Forms of current waves (nominal operating)

#### **4. Experimental results**

312 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

inductances is considered to be more satisfactory than the old one.

A.1

A.2

A.3

this model. It leads to continuous slopes representing permeability while with the old model under EMTP introducing the B-H curve point by point and approximating it by several straight line segments, the slopes representing the permeability are discontinuous at the joints. Taking into account the precision of the various data (software) and acceptable tolerances in the operation of the magnetron, the validation of this improved modelling based on analytic

> We sought to integrate the transformer new model (with analytic expressions of inductances) in the circuit of the HV supply from the source to the magnetron (Fig. 1),

where we represented the tube microwave by the equivalent diagram deduced from its electric characteristic [2]-[3]-[6] which is formally similar to that of a diode of dynamic resistance ܴ ൌ ઢ ઢ neighbor of 350 Ohms and threshold voltage E of about 3800 Volts. We validate this new model by carrying out tests that have been set up previously [2] on generator microwaves composed of the following elements (Fig.1):


**Figure 7.** Experimental forms curves of the voltages and currents waves (nominal operating)
