Domains of validity of quasistatic and quasistationary field approximations

Abstract

Purpose -The paper aims at proposing a uniform and demonstrative description of two well-known and widely used approximations of slowly time-varying electromagnetic fields, i.e. the electro-quasistatic and the magneto-quasistationary approximation to Maxwell's equations. Design/methodology/approach -Under both approximations, the orders of magnitude of the relative errors of the dominant fields are analyzed by using three characteristic time constants. These time constants are determined by considering the material properties, the characteristic length scale and the characteristic time scale. Findings -Limiting curves which show the domains of applicability of the two approximations are retrieved from the estimation of their relative errors. The relation between the domains of validity of the electro-quasistatic and magneto-quasistationary approximations was found and depicted in a combined diagram.Research limitations/implications -The study is restricted to slowly time-varying electromagnetic fields. Heuristic and local estimates based on local material properties were used for the analysis. Rigorous estimations of the errors (e.g. also considering the field problem's topology) of the magneto-quasistationary approximation are already known in the literature. A rigorous estimation of the error of the electro-quasistatic approximation is, therefore, suggested for future research. Originality/value -The combined diagram showing the domains of validity of both approximations considered here in a uniform way is novel. It gives rise to an intuitive and easily accessible understanding of their applicability.

Implications of Galilean electromagnetism in numerical modeling

“…[t can be easily shown that the two scalings (1) and (2) obtained here by dimensional analysis are in agreement with those expressed in terms of fields and time scalings given in [5]. Note howe ver that the existence of two Galilean limits is not so obvious if we naively take the limit c ----+ +00 in equations (6) and (7) (see [1] for the details).…”

Time domain methods for slowly varying fields

“…Therefore, unlike in the magnetoquasistatic case, charge conservation is maintained [11]. Limits for the validity of the electroquasistatic approximation in terms of values of the electric permittivity as well as the electric conductivity based on characteristic time constants are also derived in [7], [9], [8]. Again, these limits are more strict than the quasi-stationary condition (1), as, starting from the Darwin model, a further reduction in the level of mutual coupling between the separate equations is carried out.…”

“…As it constitutes a further simplification based on the Darwin model, the range of validity of this approximation is restricted to a smaller spectrum of practical problems regarding the allowed rate of change of the electromagnetic fields. Estimations based on characteristic time constants are provided in [7], [9], [8]. Note that resonance cannot be observed within this model as no electric but rather magnetic energy is stored in the fields.…”

“…The prerequisite mentioned above can be expressed based on the diameter d of the sphere containing the geometrical structure of interest and the minimum expected cycle time τ of the electromagnetic waves [7], [8]. When…”

Different types of quasistationary formulations for time domain simulations

“…[7] If radiation as well as propagation effects are a-priori known to be negligible for a specific problem under consideration, alternative mathematical descriptions of electromagnetic phenomena can be found. The prerequisite mentioned above can be expressed based on the diameter d of the sphere containing the geometrical structure of interest and the minimum expected cycle time t of the electromagnetic waves as shown by Haus and Melcher [1989] and Steinmetz et al [2009]. When…”