Comparative Analysis of ( HF ) Non-linear Circuits Modelled by Different Environments

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 non-linear static and dynamical system described by differential equations. Particularly, in steady state, it could be system of non-linear algebraic equations. DOI: http://dx.doi.org/10.5755/j01.eee.19.4.1754


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 CD is equal zero.General solution of the non-linear static system described in compact form as It is possible using iterative methods.Then one-step stationary iterative method yields = . ( Numerical solution of non-linear algebraic equation system can be done by Newton-Raphson method using two members of Taylor expansion and Jacobi matrix where is Jacobian matrix of n-dimensional function at point [1], [2], [5] 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 where # = Applying Kirchhoff law we obtain where 7 $ = 4 − 3 exp #$ − 1& and for one-step stationary iterative method where R -resistance of load resistor and U -input voltage of direct source (= U IN ).The resulting solution for $ 2 quantity under steady-state condition is depicted in Fig. 2.
The Schottky diode model can be described [3] where 1 -the saturation current; 3 1 -the series resistance; ; <= -the built in potential.Note: By similar way circuit with SiC PIN diode, based on [1] can be modelled: or

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: 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 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 3 LML with non-linear dependency on its current N The following differential equations can written for quantities of this simple electrical circuit Using fictitious exciting functions method and adapting system equations into matrix form yields and it in discrete form using Euler explicit formula where h -integration step.
Another example of R-L-C circuit, with non-linear inductor L non , is shown in Fig. 4. The following differential equations can be written The system equations into matrix discrete form using Euler explicit formula will be The non-linear dynamical system with serial rectifier diode is presented in Fig. 5.
where c 2 = w # and q -charge of capacitor (C) Applying Kirchhoff law for resistor current N , diode current 2 and capacitor current A2 we obtain thus the resulting differential equation will be 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 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 windings of the inductor.One of them serves for providing of desired magnetic field strength and the other for precise inductance measuring, Fig. 7. ( Calculated Li 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  Simulation results for the electric circuit given by Fig. 4 are shown using both Matlab and OrCAD-PSpice programming environment in Fig. 9.

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.

Fig. 1 .
Fig. 1.Electric circuit with serial rectifier diode with non-linear static I-V characteristic.

Fig. 3 .
Fig. 3. Dynamical system with non-linear resistor 3 LML and its parasitic inductance lR and serial capacitor C.

Fig. 4 .
Fig. 4. Dynamical system with non-linear inductor k LML and its parasitic resistance rL.

Fig. 5 .
Fig. 5. Dynamical system with serial rectifier diode D with respect of its non-linear dynamical model.Dynamical model of diode is presented by Q-V characteristic and non-linear capacitor c 2 LML model as follow:

Fig. 6 .
Fig. 6.Output part of SMPS (DC/DC) converter with isolating HF transformer and non-linear rectifier diode and inductor.

Fig. 8 .
Non-linear dependency of the inductor inductance -measured (a) and pre-calculated by different models (b).