Calculation of eddy current and hysteresis losses during transient states in laminated magnetic circuits

Abstract

Purpose The purpose of this paper is to develop the method of taking the eddy current losses in the laminated magnetic circuits into account during implicit transient calculations. The nonlinear magnetization characteristic of iron and the hysteresis losses can also be considered in the simulations done with the developed method. Design/methodology/approach The paper presents complex equivalent magnetic permeability derived from the presumed angular frequency in a laminated magnetic circuit. On this basis, the synthesis of a magnetic permeability as a function of the Laplace variable “s” is presented. After transformation of the variable “s” to a variable “z” of the Z transformation, it is possible to conduct discrete time calculation of transient states of magnetic circuits including the eddy current losses. An iterative process is developed to take the saturation of the magnetic circuit in these calculations into account. As regards hysteresis losses, the scalar model of magnetic hysteresis by Juhani Tellinen was implemented. The new method is validated by calculations of a two-coil transformer. Findings It is important to take into account the losses in sheet metal directly in the implicit transient calculations. This possibility is provided by the presented method based on the synthesis of the equivalent magnetic permeability μ^(s). The presented method was proved to be correct and efficient. The calculated sheet metal losses were compared with the results presented in literature. Good conformance of results was attained. Practical implications The method enables calculation of eddy current and hysteresis losses in laminated magnetic circuits during calculations of transient states. It does not need, unlike the previous methods, previously provided information (“a priori”) about the content of higher harmonics in waveforms. The method takes into account mutual dependence of transient waveforms of currents in the analysed system and losses of laminated magnetic circuit, expressed by eddy currents and hysteresis losses. Its implementation comes down to using in calculations a filter of the IIR type and corresponds to its calculation complexity. The author plans to use the presented method in the finite elements method transient calculations. Originality/value A new approach is a synthesis of the equivalent magnetic permeability in Laplace domain, which creates an equivalent RC circuit for permeability. Analytic equations for parameters of this equivalent circuit are original. A method for considering nonlinear magnetization characteristic and hysteresis losses was presented. In calculations of transient states of systems with magnetic circuits, one can use the developed equivalent circuit of magnetic permeability in a form of the IIR filter. Operator magnetic permeability includes fractional derivative of Laplace’s variable “vs”. Therefore, the equivalent IIR filter includes “history” of the processes that take place in the laminated magnetic circuit to the current, calculated time moment. This “history” in terms of its content is limited only by the degree of the applied IIR filter. It enables to calculate “step by step”, without previous (“a priori”) knowledge about harmonic components of the whole waveforms. It was necessary in the previously used methods, when determining parameters of magnetic permeability. The method proposed in the paper allows for calculations with taking into account direct dependence of an electric part of the system on its magnetic part.

Direct Consideration of Eddy Current Losses in Laminated Magnetic Cores in Finite Element Method (FEM) Calculations Using the Laplace Transform

“…These circuits should be treated as IIR filters, the transmittance of which comprises the representation of the equivalent magnetic permeability of a laminated core in the domain of the complex variable s of the Laplace transform. The equivalent magnetic permeability can be derived from Figure 3 and Formula (8) [29,30]. In order to include the saturation of the magnetic circuit in these calculations, an iterative process was developed.…”

“…where µ is the magnetic permeability of iron plate and γ is the electric conductivity. According to the considerations in [29], the formula was obtained for the mean value of the magnetic flux density in the plate B avr and for the complex magnetic permeabilityμ:…”

“…The magnetic permeability in Equation 10assumes an operational form, in which there occurs √ s. In order to transform this permeability to the form of a rational function of Laplace's variable s, it is possible to make use of the methods connected with fractional derivatives. When deriving Equation 11, an inverse Laplace transform [29] was used.…”

“…According to [29], the equivalent magnetic permeability of sheets from the Formula (10) can be represented in the form of the relationship:…”

“…In Figure 4 the difference between the curves marked as 1 and 2 for the assumed value of the magnetic flux density B determines the hysteresis losses. It presents the implementation of Tellinen's hysteresis model in the time calculations [29,35].…”