Modelling and identification of a non-linear saturated magnetic circuit: Theoretical study and experimental results
Introduction
In conventional models used in electrical engineering to represent electrical machines, it is quite a common thing to neglect the magnetic saturation phenomenon, which allows to make the simplifying assumption that inductances depend on the value of the flux imposed by currents, thanks to the following non-linear relation ϕ(i) = L(i)·i, where ϕ(i) is a picture of the non-linear magnetization characteristic B(H) (Fig. 1) of the employed ferromagnetic material.
Moreover taking into account the saturation phenomenon leads to differential equations which coefficients are time-varying and consequently to fundamentally non-linear in respect to parameters models. The main difficulty is also to find a non-linear model structure suited to the studied system and of a complexity level straightforward enough so as to develop an identification procedure.
As for the parameter estimation, a Non-linear Programming algorithm is imperative.
Firstly, before extending this study to swivel alternating current machines, we contemplate analysing the saturation phenomenon in the primary winding of a single-phase transformer.
The last mentioned is also represented firstly by a non-linear inductance connected in series with a resistance.
However, it will be seen later that the identification results obtained in the experimental case can be improved if iron losses are not neglected and taken into account by a resistance in parallel with the inductance. First of all, the emphasis is put on the mathematical model chosen to represent the magnetic saturation of the inductance.
After developing an iterative identification procedure, both simulation and experimental results are given to validate the minimization algorithm as well as the model of the saturated inductance.
Section snippets
Saturated inductances modelling
In this study, the magnetizing flux will be described by a simple magnetization curve like Fig. 1 and not a hysteresis cycle [6]. Thus, iron losses through hysteresis as well as eddy current will be at first neglected. On the same way the leakage flux is neglected.
A grey-box approach has been carried out to determine a set of analytical functions which approximate at best the magnetization characteristic B(H) (or the curve ϕ(i)).
The example of an iron core coil fed by a sinusoidal voltage u(t)
Parameters estimation algorithm
As the model is non-linear in respect to its parameters, the technique employed to estimate the vector relies on the minimization of a quadratic criterion thanks to a Non-linear Programming algorithm.
This Non-linear Programming algorithm is the Marquardt one [5] and realizes automatically the compromise between a Gradient type algorithm far from the optimum so as to get closer to it without instability and a Newton type algorithm in order to increase the convergence speed in the optimum
Simulation results
The digital simulation of the system constituted by iron core coils fed by a sinusoidal voltage 250 V/50 Hz is carried out by a fourth-order Runge–Kutta algorithm.
Fig. 6 shows the shape of the simulated current i(t) for a working in saturation mode.
As for the identification protocol, the excitation of the system is achieved by an additional random disturbance, such as a Pseudo Random Binary Sequence (P.R.B.S.), applied to the amplitude of the feeding voltage.
Moreover, the main property of this
Experimental results
This section puts the emphasis on some practical aspects with regard to the identification of a dynamical continuous model and presents the experimental set-up which has been carried out on an iron core coil so as to validate the dynamical inductance model given in Section 2 and the identification algorithm described previously in Section 3.
The experiment is made up of a measurement and formatting system as well as a signals reading–recording system (Fig. 11).
The first one is composed of
Conclusion
In this paper, a solution to model and identify dynamical inductances which vary widely with the magnetic saturation level is proposed.
A sensible choice of the dynamical inductance model along with a well-adapted estimation technique allow to know with a better accuracy the behaviour of winding inductances.
This study will be extended to more complicated electrical systems such as the induction motor even if a new difficulty appears in this case since the number of parameters to estimate becomes
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