The fundamental importance of dynamic modeling of electromagnetic fields in space and time

The fundamental importance of dynamic modeling of electromagnetic fields in space and time,10.1109/MWSYM.2011.5972739,Wolfgang J. R. Hoefer

The fundamental importance of dynamic modeling of electromagnetic fields in space and time  
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Frequency domain or time-harmonic models are ubiquitous in microwave engineering because they are elegant, familiar to microwave engineers, and appropriate for many practical applications. However, they break down in highly nonlinear scenarios or in situations where the steady state can never be reached. In this paper, three examples illustrate the importance of using time domain approaches in such cases to preserve causality and to avoid drawing conclusions that conflict with the laws of Physics. Since Heaviside introduced the time-harmonic formalism into Maxwell's theory of electromagnetic fields, several generations of physicists, electrical and microwave engineers have learned and practiced their profession in the frequency domain. Indeed, the time-harmonic (frequency domain) model of both fields and networks has several advantages. The most obvious is the reduction by one of the number of independent variables, since the time dimension can be omitted in all expressions, a time dependence of the form t j e ω being implied throughout. Another advantage is the complex notation that reduces differentiation and integration with respect to time to multiplication and division by ω j , and allows both the active/passive and the reactive properties of materials, structures and circuits to be described by a single complex number or function. Even though the complex domain is purely mathematical and fictitious, it has become a familiar, almost tangible mindscape in which most microwave theories and techniques have been conceived, developed and realized. Finally, the time-harmonic form of Maxwell's equations can also yield time-dependent solutions of Maxwell's equations through the Fourier or Laplace transform, as long as all properties are linear, and spectral data are known for a sufficiently large frequency range to yield a causal time response. Indeed, if these conditions are satisfied, a general transient problem can be reduced to a number of time- harmonic problems; the time-harmonic solutions are then simply recombined through the inverse transform to yield the time domain solution. One might thus conclude that solutions in time domain are not really needed. Indeed, the two requirements of linearity and wideband spectral information are reasonably well satisfied in many practical situations. Even nonlinear scenarios can be treated in frequency domain using harmonic balance and Volterra series, provided that the nonlinearity is not too stiff. Nevertheless, the two requirements of linearity and information over a wide bandwidth hint at the fundamental limitations of the time- harmonic model. In this paper, three representative examples will be discussed, in which the time harmonic approach breaks down and, at worst, may lead to conclusions that are at odds with the laws of Physics.
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