![]() ![]() In this case, the Fourier transform will be applied to products of the fields. In nonlinear media, the dielectric function and the magnetic permeability will be functions of E and B. Finally, the time dependence is recovered by taking an inverse Fourier transform of the solution. Only spatial derivatives need to be considered when determining the eigenfunctions of the system, so there is only one step’s worth of separation of variables in the above equations. For homogeneous media, these problems can often be solved by hand. The benefit of applying a Fourier transform to Maxwell’s equations is that the resulting differential equations could be easier to solve in the frequency domain. We also lose any information on the transient response this is one of the reasons FDTD might be used with a digital signal as the transient behavior can be calculated. Random noise is very difficult to simulate in the time domain without a specialized solver, but it is quite easy to simulate in the frequency domain as it will have a continuous power spectrum.īy working in the frequency domain, we lose any nonlinear temporal information such as we might have with a piecewise source. When broadband noise is present in the system. Systems driven with modulated sources can be more easily solved in the frequency domain as their Fourier spectra are often sums of delta functions. Continuous signals that may have broad bandwidth are easy to work with as they will be represented by continuous functions in the frequency domain. When the sources are continuous and broadband. This is related to the previous point, but the sources will be sums of delta functions. When the sources are sums of harmonic functions. In this case, the frequency-domain representation of J and ⍴ will be a delta function. The above equations are much easier to work with in several cases: In the above equation, the sources (J and ⍴), electric field E, and magnetic flux density B are functions of frequency rather than time. When to Use Maxwell’s Equations in the Frequency Domain Maxwell’s equations in the frequency domain for macroscopic media. Using the derivative identity, we have Maxwell’s equations in the frequency domain: This is the form of Maxwell’s equations normally used to simulate the electromagnetic field in PCBs or ICs with an FDTD field solver.īy applying the Fourier transform operator to the above equations, the time-dependent terms are immediately converted into the frequency-domain. This is the most general description of the electromagnetic field in an LTI system. However, because we are in an LTI system, the dielectric properties do not vary in time, but they might vary in space. In the above equations, we have time-dependent sources (J and ⍴) that may also vary in space. Maxwell’s equations in the time domain for macroscopic media. Maxwell’s equations in the time domain are: This is easiest with the differential form of Maxwell’s equations, although the Fourier transform operator can also be applied to the integral form by switching the order of integration once the F operator is applied. In this case, the time-dependent Maxwell’s equations are written in such a way that the Fourier transform operator F can be applied to them directly. Understanding how to apply a Fourier transform to Maxwell’s equations is easiest when we consider an LTI system driven with arbitrary sources (such as an external antenna or current source). Derivation: Maxwell’s Equations Fourier Transform The interpretation of Maxwell’s equations in the frequency domain may be unintuitive, but the analysis benefits are clear when working with linear time-invariant (LTI) systems.īelow you’ll find the definition of Maxwell’s equations in the frequency domain and their interpretation in terms of signal behavior. However, many problems are often easier to solve in the frequency domain, and you can use Maxwell’s equations with a Fourier transform to move to the frequency domain. Maxwell’s equations are fundamental for describing electromagnetism, and they are normally presented in the time domain with sources. James Clerk Maxwell on a San Marino stamp. Some problems are easier to solve in the frequency domain, such as when we have sources that are superpositions of harmonic waves. Maxwell’s equations can be used in the time domain or the frequency domain.ĭescribing electromagnetism in the frequency domain requires using a Fourier transform with Maxwell’s equations.
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