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Comparative analysis of (HF) non-linear circuits modelled by different environments.

I. INTRODUCTION

Many electric circuits feature some type of non-linearity of their used devices. Non-linear resistors or inductors could be typical examples. Also, all semiconductor devices are in their nature non-linear ones. From the point of view of the circuit solution they are presented by non-linear static and dynamical system described by differential equations. Particularly, in steady state, it could be system of non-linear algebraic equations.

II. NON-LINEAR STATIC SYSTEMS

Let's assume electric circuit in Fig. 1 considering steady state when capacitor current [i.sub.CD] is equal zero.

General solution of the non-linear static system described in compact form as

x = F(x). (1)

It is possible using iterative methods. Then one-step stationary iterative method yields

[x.sup.k + 1] = F([x.sup.k]). (2)

Numerical solution of non-linear algebraic equation system can be done by Newton-Raphson method using two members of Taylor expansion and Jacobi matrix

f(x) = f([x.sup.(0))] + [[??].sup.(x(0))] x (x - x(0)) + ..., (3)

where f(x) is Jacobian matrix of n-dimensional function at point [x.sup.(0)] [1], [2], [5]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The simplest method of circuit solution is graphical one. According to electrical scheme in Fig. 1 and assuming diode can be modelled as a voltage controlled current source in the forward characteristics one can write

[i.sub.D] = [I.sub.s][exp([theta] x [u.sub.D]) - 1], (5)

where [theta] = [Q.sub.e]/MK[upsilon], [Q.sub.e]--charge of electron 1.59 x [10.sup.-19] [C]; M --idealist or emission factor (typically varies from 1.02 to 1.6); K - Boltzmann constant 1.38 x [10.sup.-23] [J/K]; [theta] - PN junction temperature [K]; [I.sub.s]__saturation current in reverse direction and [u.sub.D]--voltage of diode.

Applying Kirchhoff law we obtain

[RI.sub.s][exp([theta][u.sub.D]) - 1] + [u.sub.D] - 4 = 0, (6)

where f([u.sub.D]) = U - [RI.sub.s][exp([theta][u.sub.D]) - 1] and for one-step stationary iterative method

[u.sup.k + 1.sub.D] = U - [RI.sub.s][exp([theta][u.sup.k.sub.D]) - 1], (7)

where R - resistance of load resistor and U - input voltage of direct source (= [U.sub.IN]).

The resulting solution for [u.sub.D] quantity under steady-state condition is depicted in Fig. 2.

The Schottky diode model can be described [3]

[i.sub.SD] = [I.sub.S]{exp[[theta]([V.sub.bi] - [IR.sub.s]) - 1}, (8)

where [I.sub.s] - the saturation current; [R.sub.s] - the series resistance; [V.sub.bi] - the built in potential.

Note: By similar way circuit with SiC PIN diode, based on [1] can be modeled:

[i.sub.SiCD] = [I.sub.S] x exp(-[theta][V.sub.bi])[exp([theta][u.sub.D]) - 1] (9)

or

[i.sub.SiCD] = [I.sub.S][[([upsilon] - [V.sub.bi])/k].sup.1/(a + 1)], k = [k.sub.0]/(a + 1). (10)

III. NON-LINEAR DYNAMIC SYSTEMS

Many applications in technical practice which models are presented in following text should be described as nonlinear differential equation (DE) systems. In compact form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Such a system of ODE can be solved analytically and/or also numerically (e.g. by Euler explicit method) [1], [2]. If the matrix elements are non-stationary (e.g. time dependent ones) then system of equations cannot be solved by the methods using the matrix operation. As a solver for their solution the following numerical methods can serve, completed by fictitious exciting functions method making possible numerical solution of this DE system with nonstationary matrices [1], [2], [5]. It deals with Euler's--and Taylor expansion methods for consequent numerical solution in Matlab environment.

Simple electrical circuit given in Fig. 3 comprises a nonlinear resistor (e.g. varistor--resistor depending on voltage, thermistor--resistor depending on temperature). In this case it is [R.sub.non] with non-linear dependency on its current [i.sub.R]

[R.sup.non]([i.sub.R]) = R.exp(-[i.sub.R]/K) (12)

and consequently its voltage will be [u.sub.R] = [R.sup.non]([i.sub.R])[i.sub.R].

The following differential equations can written for quantities of this simple electrical circuit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Using fictitious exciting functions method and adapting system equations into matrix form yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

and it in discrete form using Euler explicit formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

where h - integration step.

Another example of R-L-C circuit, with non-linear inductor [L.sup.non], is shown in Fig. 4.

The following differential equations can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The system equations into matrix discrete form using Euler explicit formula will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The non-linear dynamical system with serial rectifier diode is presented in Fig. 5.

Dynamical model of diode is presented by Q-V characteristic and non-linear capacitor C2on model as follow:

[q.sub.D] = [q.sub.0][exp([theta][u.sub.D]) - 1], (18)

[C.sup.non.sub.D] = [C.sub.D0]exp([theta][u.sub.D], (19)

where [C.sub.D0] = [q.sub.0][theta]0 and q - charge of capacitor (C)

Applying Kirchhoff law for resistor current [i.sub.R], diode current [i.sub.D] and capacitor current [i.sub.CD] we obtain

[i.sub.CD] = [C.sub.D0] d[u.sub.D]/dt = [i.sub.R] - [i.sub.D], (20)

thus the resulting differential equation will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Rather complex electrical circuit with serial rectifier diode is shown in Fig. 6.

Based on above given approach the non-linear model of the circuit can be created and in discrete form using Euler explicit formula with [R.sub.T] = (R + [r.sub.T]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

IV. SIMULATION AND EXPERIMENTAL VERIFICATION

Simulation experiments have been done with non-linear inductor which inductance depends on its current. It has been measured using static biased method with bifilar

of desired magnetic field strength and the other for precise inductance measuring, Fig. 7.

Parameters of the measured inductor [10]: L = 32.5 [micro]H; [S.sub.Fe] = 31 cm2; N1 = N2 = 6.8; magnetic material 3F3; air gap [delta] = 28.5 [micro]m; [L.sub.sat] = 0.

Values of the inductance [L.sup.non] have been calculated from measured data as follow

[L.sup.non.sub.k] = [u.sub.L(k)] - [u.sub.L(k - 1)]/[i.sub.L(k)] - [i.sub.L(k - 1)][DELTA]t. (23)

Calculated L - [i.sub.L] characteristic from measured data is given in graphic form in Fig. 8(a), the pre-calculated and simulated by different non-linear models [5] from data sheet in Fig. 8(b).

Simulation results for the electric circuit given by Fig. 4 are shown using both Matlab and OrCAD-PSpice programming environment in Fig. 9.

Parameters of simulations: L = 32.5 [micro]H; C = 45 nF; f = 132 kHz; [u.sub.IN] = 6 V sin([omega]t), rL = 0.001[omega]L; rC = 1000R; R = 33 [ohm]

V. CONCLUSIONS

The non-linear static and dynamical systems described by differential equations were presented with non-linear components. Particularly, in steady state, it could be system of non-linear algebraic equations using fictitious exciting functions method. The analysis of electrical circuit with real non-linear inductor was presented provided by different modelling environments--Matlab and PSpice. The inductance of inductor has been measured and next used for simulation. Simulation results of both modelling environments have proved very good agreement. The comparison of measured proved simulation results will be given in future work.

http://dx.doi.org/ 10.5755/j01.eee.19.4.1754

Manuscript received May 26, 2012; accepted November 13, 2012. The authors wish to thank for the financial support to the APVV Agency for the project No. APVV-0138-10 and CEX 2 R&D operational program Centre of excellence of power electronics systems and materials for their components II. No. OPVaV-2009/2.1/02-SORO, ITMS 26220120046 funded by European regional development fund (ERDF).

REFERENCES

[1] H. Mann, Using Computer for Design of Electrical Circuits and Systems. SNTL/ALFA Publisher, Prague /Bratislava, 1984, p. 480. (in Czech).

[2] D. Mayer, B. Ulrych, "Numerical Analysis of Non-Linear Electrical Circuits by State Variables Method", Electrical Review, vol. 65, no. 1, pp. 30-38, 1976. (in Czech).

[3] The Metal-Semiconductor Junction/Schottky Diode Ohmic Contacts manual, 2011. [Online]. Available: http://www.tmi.vu.lt/legacy/pfk/.../diode/ schottky.htm

[4] M. Marcokova, V. Guldan, "On Jacobi Polynomials and Related Functions, in Proc. of MMA'10 Int'l Symposium on Mathematical Methods in Engineering, Coimbra, 2010, pp. 339-346.

[5] B. Dobrucky, M. Benova, D. Gombarska, "Comparison of Euler's-and Taylor's Expansion Methods for Numerical Solution of Non-Linear system of Differential Equation", in Proc. of 19th TCP--Int'l Technical Computing Conference, Prague, 2011, pp. 1-34.

[6] M. Luft, R. Cioc, D. Pietruszczak, "Fractional Calculus of Measuring Transducers", Elektronika ir Elektrotechnika (Electronics and Electrical Engineering), no. 4, pp. 97-100, 2011.

[7] R. Radvan, B. Dobrucky, M. Frivaldsky, P. Rafajdus, "Modelling and Design of HF 200 kHz Transformers for Hard- and Soft- Switching Application", Elektronika ir Elektrotechnika (Electronics and Electrical Engineering), no. 4, pp. 7-12, 2011.

[8] B. Dobrucky, M. Benova, S. Kascak, "Transient Analysis and Modelling of 2nd- and 4th-Order LCLC Filter under Non-Symmetrical Control", Elektronika ir Elektrotechnika (Electronics and Electrical Engineering), no. 5, pp. 89-94, 2011.

[9] M. Luft, E. Szychta, L. Szychta, "Method of designing ZVS boost converter", in Proc. of the 13th Power Electronics and Motion Control Conference (EPE-PEMC), Poznan, 2008, pp. 463-467.

[10] FerroxCube--3F3, 0.2-0.5 MHz. Material Specification, FerroxCube --A Yageo Company, 2008.

B. Dobrucky (1), M. Benova (1), M. Frivaldsky (1), R. Radvan (1), D. Gombarska (1)

(1) Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia dobrucky@fel.uniza.sk
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Author:Dobrucky, B.; Benova, M.; Frivaldsky, M.; Radvan, R.; Gombarska, D.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:4EXSV
Date:Apr 1, 2013
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