A Pulse Compression procedure for power inductors modeling up to moderate non-linearity
Introduction
The achievement of increasing power density levels is a main concern in Switch-Mode Power Supply (SMPS) design [1], especially in applications where weight and size are major constraints. Integrated hardware solutions like power modules, systems-in-package and systems-on-chip have recently allowed the impact of power devices and magnetic components on high-frequency integrated DC-DC converters to be reduced [2,3]. Power inductors and transformers however remain the largest devices in high-frequency discrete power supplies, and their minimization is indispensable to obtain higher-power-density SMPSs.
Ferrite Core (FC) inductors are widely used in high-efficiency SMPSs designs, due to their low core losses. They exhibit a strong non-linear behavior. In particular, their L-I inductance versus current characteristic is almost constant for low current values (weak-saturation region), then it sharply drops as the current increases (roll-off region) [4]. The weak-saturation region is typically wider for bigger (and thus heavier) inductors. Standard SMPSs design relies on inductors operating in the weak-saturation region. This approach is quite conservative and involves the use of relatively big and heavy inductors. In recent years, some experimental tests have proved that smaller-volume FC inductors working in moderate saturation can help to achieve more compact power supplies, still satisfying the overall design specifications [[5], [6], [7], [8]]. Moreover, inductor peak-to-peak ripple current and power losses stabilization are achievable over wide load ranges by means of viable control solutions [9,10]. In particular, hysteretic-derived control techniques have been developed and implemented into commercial control chips, ensuring fixed-frequency operation. Dynamic performances of DC-DC regulators exploiting moderately saturated inductors have been recently verified, with simulations and experiments [9,10], proving good efficiency and transient response with reduced inductor size, compared to solutions based on conventional non-saturated inductors. New methods to analyze the effects of moderate saturation of FC power inductors in SMPS applications have also been proposed [4]. However, enhanced behavioral models of non-linear FC inductors are still needed, to reliably predict the magnitude of the ripple current for whatever saturation level and voltage applied to the inductor in the charge and discharge intervals of a FC inductor and assess its impact on the operation of high-power-density SMPSs.
The analytical models presented in Refs. [[11], [12], [13]] allow to analyze FC power inductors behavior in saturation and are suitable for SMPSs steady-state analysis and power stage design exploiting moderately saturated inductors. Models [11,12] consider exponential and trigonometric functions, respectively, and provide global accurate modeling of inductors saturation. Low-order polynomial approximations of saturation, like those discussed in Ref. [13], mainly rely on simple local approximants, whose coefficients change with the operating conditions. For circuit simulations in dynamic conditions, computationally effective non-linear models of FC inductors are still required, to prevent slow or not convergent simulations.
The Hammerstein modeling has been successfully adopted to provide dynamic behavioral modeling of non-linear circuits and systems, such as electronic devices [14,15] and power converters [16,17]. This modeling approach has a convenient blocks representation, relying on the decomposition of the input-output relationship of a system into two or more interconnected elements. In particular, the system dynamics can be represented by a linear transfer function and the inherent non-linearity introduced by using non-linear functions of inputs and outputs of the linear system.
The Hammerstein model of non-linear FC power inductors is proposed in this paper. The aim of this study is to discuss the accuracy and reliability of the Hammerstein model compared to analytical non-elementary behavioral models based on transcendent functions. The technique, whose preliminary application experiments are reported in Ref. [18], has been thoroughly investigated, thus verifying its effectiveness on FC inductors with different L-I characteristics. We have also verified the model capability to operate with approximant orders lower than those ones previously considered, choosing orders characterized by a number of parameters comparable with relevant Taylor and Padé approximants.
The paper is organized as follows. In Section 2, the Hammerstein modeling of non-linear systems based on a Pulse Compression (PuC) identification technique is described. In Section 3, the PuC modeling is applied to define the Hammerstein model of non-linear FC inductors. In Section 4, the results of the Hammerstein modeling approach for FC power inductors are compared to the Taylor and Padé approximations of the saturation behavioral model discussed in Ref. [12], with different order levels.
Section snippets
A PuC identification technique for the Hammerstein modeling of non-linear systems
In specific operating conditions, the non-linear behavior of the real system becomes not negligible, and models considering the non-linearity must be adopted [19]. Volterra models of non-linear systems have been frequently considered, since they can serve as universal approximators to a large variety of non-linear systems [20]. They have the advantage to be linear in the parameters, but the price for this is, however, the high dimensionality of the universal non-linear space and thus, the
PuC modeling of FC power inductors
In the present work, the PuC method has been adopted to define the Hammerstein model of non-linear FC power inductors. Once the model has been defined, i.e. once identified the functions, it can be used to calculate the non-linear system response to any input signal. In particular, by sending sinusoids of different amplitudes and frequencies, we can study the evolution of the frequency response according to the amplitude, or plot x-y diagrams that highlight the link between input and
Case studies, results and discussion
The effectiveness of the proposed modeling technique has been verified in numerous tests by referring to measured data directly provided by the manufacturer, and comparing the obtained numerical results to the Taylor and Padé numerical approximations of the saturation behavioral model discussed in Ref. [12].
The discussion is herein referred to power inductors with different ferrite materials and core types. In fact, shielded, partially-shielded and unshielded magnetic cores have been
Conclusions
A reliable modeling of the saturation curves of Ferrite Core (FC) inductors is needed for an accurate prediction of their operation in Switch-Mode Power Supplies (SMPSs). In this paper, the Pulse Compression (PuC) technique is adopted for the Hammerstein modeling identification of the FC inductors saturation profile. The results show that the proposed modeling method allows obtaining a much more reliable model of the non-linear inductance versus current curve, compared to Taylor and Padé
Acknowledgements
Work supported by the University of Salerno through the project funds “Fondo di Finanziamento per le Attività Base di Ricerca” (300638FFABR18DICAPUA) and “Sviluppo di sistemi innovativi di alimentazione a elevata densità di potenza per applicazioni aeronautiche, militari e spaziali” (300638GRIDICAPUA).
The Authors also thank Coilcraft Inc., for providing power inductors relevant data.
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