Different types of quasistationary formulations for time domain simulations



[1] For many practical examples in electromagnetics, quasistationary approximations to the full set of the Maxwell equations are valid. Applying such approximations reduces the level of mutual coupling between the separate equations and leads to different sets of differential equations for the specific case. Based on the resulting subsets of the Maxwell equations, numerous formulations in terms of fields and potentials can be obtained. As the type of differential equation is changed due to the approximation in general, appropriate time-integration schemes need to be selected for the numerical solution of the specific problem. An overview of existing quasistationary formulations is given while a simple test model is used to compare the different formulations.

1. Introduction

[2] The full set of the Maxwell equations provides a general description of the electromagnetic fields. In the following, the view is restricted to the non-relativistic case. Different levels of approximations based on these equations can be applied in many cases in order to reduce the complexity of the mathematical description. Each simplification leads to a modeling error which, however, can be shown to be sufficiently small for practically relevant scenarios.

[3] Even though for many cases, in particular regarding devices at power frequency, such as electrical machines and power generators, frequency domain (FD) approaches are well suited, the main emphasis of this paper is on time-domain (TD) methods in the quasistationary regime. Even though TD methods are in general computationally more expensive than the FD counterparts, their application can be beneficial, e.g., in the following situations:

[4] 1. In the context of eddy-current simulations often nonlinear material laws, e.g., for ferromagnetic materials, need to be considered. Here, the magnetic permeability μ(|B|) depends locally on the actual magnetic flux density. Furthermore, also electric diffusion problems where material might show a nonlinear dependence of the electric conductivity σ(|E|) on the electric field strength E receive increasing interest [see, e.g., van Rienen et al., 2003; Weida et al., 2009]. As an alternative in FD, harmonic balance methods can be applied in the nonlinear case as reported by Chang and Steer [1990] for nonlinear microwave circuits.

[5] 2. In presence of non-sinusoidal, non-smooth excitations a Fourier transform needs to be applied in order to make use of FD methods for each spectral component of the signal. However, the benefit of FD methods decreases if more than a few frequency points need to be considered. This is given in particular in presence of non-smooth forcing terms. In this case, the magnitude of the spectral coefficients of a Fourier transform applied to the respective excitation signal might decay very slowly. As a consequence, a large number of FD simulations is required in order to achieve a sufficient accuracy. Therefore, the use of TD methods is more appropriate regarding the complexity of the related numerical solution process.

[6] 3. When electromagnetic field models are coupled to network models within a single simulation, the solution can be carried out conveniently in FD in case of passive components in the circuit section [see, e.g., Tsukerman et al., 1993; De Gersem et al., 2004]. In presence of switching elements as, e.g., reported by De Gersem et al. [2007], TD methods provide a competitive alternative to the harmonic balance approach mentioned above.

2. Quasistationary Approximation

[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 τ of the electromagnetic waves as shown by Haus and Melcher [1989] and Steinmetz et al. [2009]. When

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with cmin = c0/equation image the minimum wave propagation speed and c0 the speed of light in vacuum, holds, the quasistationary assumption is valid [cf. Dirks, 1996]. Here, μr,max and ɛr,max denote the maximum values of the relative magnetic permeability and the relative electric permittivity inside the volume of interest, respectively. Under the assumption (1), appropriate sets of partial differential equations can be found, which approximate the full set of the Maxwell equations up to a certain extent.

2.1. The Darwin Model

[8] The idea behind the application of approximations to the full set of the Maxwell equations consists in reducing the complexity of the underlying system of partial differential equations. This is done by means of relaxing the level of mutual coupling between the separate equations. Condition (1) implies that the process under consideration is much slower than the propagation time of an electromagnetic wave through the respective domain of interest. While introducing a small modeling error, it is possible to exclude the effects of wave propagation at the level of the underlying partial differential equations. Namely, the hyperbolic character of the set of partial differential equations is removed by reducing the mutual coupling between electric and magnetic field quantities. This procedure can be illustrated using the electrodynamic potentials A and ϕ, introduced as

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in the Coulomb gauge [see, e.g., Rothwell and Cloud, 2001]. For a homogeneous distribution of the linear electric permittivity, the Maxwell equations can then be written as

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in combination with appropriate initial and boundary conditions. Here, σ denotes the electric conductivity and Js is the source current density. Moreover, the first term on the right-hand side in (4) describes the displacement current density, whereas the second term corresponds to the conduction current density. Under the chosen gauge condition it is now obvious that (3) provides a Helmholtz decomposition

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of the electric field strength into an irrotational part Ei and a solenoidal one Es. Taking the divergence on both sides of (3) shows that the irrotational fraction Ei is solely determined by the electric scalar potential ϕ as ∇ · A = 0. On the other hand applying the curl operator to (3) demonstrates that the solenoidal fraction of E can be attributed directly to the magnetic vector potential A [cf. Dirks, 1996; Larsson, 2007]. Moreover, Ei can be identified as the static part of E while Es provides the induced electric field strength. Using these results, the right-hand side R of (4) can be rewritten as

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[9] If the term equation imageequation imageEs corresponding to the induced displacement current density is small compared to the remaining terms of R in (7), it can be neglected [cf. Dirks, 1996]. This condition is met as long as the quasistationary assumption (1) holds. Changing (4) accordingly reveals that the system (4)–(5) is no longer hyperbolic as the second order time derivative of the magnetic vector potential vanishes. This is, however, consistent to the initial assumption, i.e. avoiding the modeling of wave propagation and radiation effects in the quasistationary case. A more detailed derivation based on the integral solutions for the electric scalar potential as well as the magnetic vector potential in time and frequency domain can be found in the work of Dirks [1996].

[10] According to the approximation described above, the mutual coupling between Ampère's law and Faraday's law of induction is reduced. The two equations in the Darwin model read

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[11] Namely, the solenoidal part Es of E in (8) does not contribute to the displacement current density in (9).

[12] The Darwin model has early been used in the context of plasma and particle simulations [see, e.g., Kaufman and Rostler, 1971; Busnardo-Neto et al., 1977]. Furthermore, additional theoretical aspects of its application can be found in the work of Degond and Raviart [1992]. As the displacement current density is not neglected entirely, effects of resonance can still be observed as indicated by Larsson [2007]. Therefore, in the context of applications in the quasistationary limit (1) where capacitive effects become relevant [see, e.g., De Gersem et al., 2010], the Darwin model seems to be well suited as an alternative to the full hyperbolic set of equations.

2.2. The Modified Darwin Model

[13] The original Darwin model described in the preceding subsection was developed for plasma and particle simulations in free space. In particular the chosen Coulomb gauge naturally leads to a decomposition of the electric field strength into an irrotational and a solenoidal fraction whereas each part can be assigned to either the electric scalar potential or the magnetic vector potential. This feature enables to interpret the neglected term in (4) physically as the induced displacement current density. In order to transfer the concept to a larger class of quasistationary problems involving, e.g., massive conductors coated by a dielectric insulation, slight modifications to the original model are necessary. Specifically, the Coulomb gauge ∇ · A = 0 cannot directly be applied to the formulation in case of a space-dependent distribution of the electric permittivity as it is not allowed to bring ɛ in front of the divergence operator. However, applying the same procedure as described in the previous section to the non-gauged counterpart of (4)–(5) leads to the system

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with ϱi = ∇ · (ɛA). Using the continuity equation

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which describes the conservation of electric charge, a slightly different formulation can be derived. Substituting Gauss's law

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into (12) and rearranging the terms yields

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which can be used alternatively to (11) itself in the formulation. Here, the source current density Js is assumed to be divergence-free.

[14] In the original Darwin model, the solenoidal part Es of the electric field strength related to the magnetic vector potential A is identified to cause the hyperbolic character of the partial differential equation. As no gauge condition is applied on the formulation level here, the decomposition of E into an irrotational and a solenoidal part does not carry over from the original model. Hence the direct physical interpretation of the neglected term is also lost. Nevertheless, following the same argumentation as in the previous section, the contribution of the terms involving the second order time derivative of A is again neglected with respect to the remaining terms in (10) and (14). The resulting system of partial differential equations, consisting of (10) and (14) under the mentioned approximation is again parabolic.

[15] Note that irrespective of the missing gauge condition the magnetic flux density as well as the electric field strength are uniquely defined according to (2) and (3). Merely the interpretation of the electric scalar potential as the electrostatic potential needs to be abandoned. Note that within this formulation the latter would be questionable anyway as the electric field strength is not conservative. Moreover, a direct physical interpretation of the two potentials used in the formulation is not mandatory for the solution of the electromagnetic problem. Of course, drawbacks exist in terms of the lack of a straightforward physical interpretation of the neglected terms in contrast to the original Darwin model.

[16] The modified Darwin model is suited for the simulation of electromagnetic fields for more general cases than the original model as the derivation above is not restricted to a homogeneous distribution of the electric permittivity. A similar strategy based on a FD field formulation is used by Doliwa et al. [2007] to calculate the electromagnetic fields in particle accelerator structures following a perturbation approach with respect to the contribution of the second order time derivative.

2.3. The Magnetoquasistatic Model

[17] A further reduction in the level of mutual coupling between electric and magnetic quantities based on the modified Darwin model is obtained, if conditions exist which allow for neglecting additional terms in (10) and (14). In particular, the displacement current density on the right-hand side in (10) is assumed to be small compared to the conduction current density within the magnetoquasistatic model. In turn, capacitive effects are also neglected in (14) with respect to conductive phenomena. Considering these approximations the system of partial differential equations reads

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[18] As this procedure comprises a further simplification based on the modified 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 by Haus and Melcher [1989], Dirks [1996], and Steinmetz et al. [2009].

[19] Due to the additional approximation, charge conservation in terms of the time-dependent continuity equation implied by taking the divergence on both sides of (15), leading to (16), is not guaranteed anymore [cf. Larsson, 2007]. This is due to the fact that a temporal change of the charge density, e.g., related to the movement of charged particles, is no longer reflected in the partial differential equation for the vector potential A. Thus, free charges as well as charge accumulation during time evolution cannot be modeled in terms of the magnetoquasistatic approximation. Moreover, resonance cannot be observed within the magnetoquasistatic model as no electric but rather magnetic energy is stored in the fields.

[20] Based on the approximation of entirely neglecting capacitive effects described above, numerous alternative formulations in terms of fields or different potentials can be derived. An overview is given [e.g., Carpenter, 1977; Bíró and Preis, 1989; Albanese and Rubinacci, 1990]. The formulation given by (15) and (16) is recognized as the non-gauged A − ϕ-formulation [Biddlecombe et al., 1982].

2.4. The Electroquasistatic Model

[21] In order to address a different class of electromagnetic problems regarding the predominant effects, an alternative simplification can be applied to the modified Darwin model. For the latter the terms related to the time derivative of the magnetic vector potential A contributing to the displacement current density on the right-hand side of (10) as well as in (14) are neglected. However, the conduction current density in both equations is still related to the magnetic vector potential. If the remaining contribution of the magnetic vector potential for a certain expected characteristic cycle is small compared to the related terms, it can be neglected in addition [cf. Dirks, 1996]. In this case the modified Darwin model can be simplified to

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[22] Due to the applied approximation, the system is decoupled with respect to the potentials. Therefore, (18) can be solved independently from (17) and constitutes the electroquasistatic model. The electric field strength is now determined only by the electric scalar potential. In a subsequent step the magnetic field strength arising from the calculated electroquasistatic currents can be determined.

[23] 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 by Haus and Melcher [1989], Dirks [1996], and Steinmetz et al. [2009]. Again, these limits are more strict than the quasistationary condition (1), as, starting from the modified Darwin model, a further reduction in the level of mutual coupling between the separate equations is carried out.

[24] Similar to the magnetoquasistatic model, resonance in terms of oscillation between magnetic and electric energy is absent as only the latter is present in this model [cf. Larsson, 2007].

3. Numerical Simulation

[25] In order to solve the parabolic initial values problems originating from the different quasistationary models, the method of lines is applied. Spatial discretization is carried out by means of volume-based schemes such as the finite integration technique (FIT) or the finite element method (FEM).

3.1. Discretization

[26] The continuous formulations in terms of the magnetic vector potential and the electric scalar potential arising from the different quasistationary models described in sections 2.2–2.4 are discretized using the finite element method. On a tetrahedral grid, tangentially continuous vectorial shape functions wke related to the edge k are used to represent the magnetic vector potential. For the electric scalar potential standard nodal shape functions win are applied. Here, the edge k connects the nodes with the indices i and j.

[27] For the modified Darwin model, the Galerkin procedure leads to the semi-discrete block system

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whereas a and u collect the degrees of freedom for the magnetic vector potential and the electric scalar potential, respectively and js represents the source current density. The entries of the different stiffness matrices Kξn/e and mass matrices Mξn/e are determined by

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with ξ either the magnetic reluctivity ν = 1/μ, the electric permittivity ɛ or the electric conductivity σ. Moreover, the matrix G denotes the discrete gradient operator. Based on the discretization of the modified Darwin model the different further approximations in the formulation described in the preceding sections are obtained by modifying the discrete system (19) accordingly.

[28] This procedure in general leads to stiff systems of differential algebraic equations (DAE) [see, e.g., Clemens and Weiland, 1999]. Typically, the system (19) is solved by means of implicit time-integration schemes [cf. Hairer and Wanner, 1996]. The basic class of single-step implicit schemes is given by the Θ-method, with 1/2 ≤ Θ ≤ 1. Choosing Θ = 1 yields the backward Euler method also known as BDF1 scheme which is L-stable and suited for the integration of DAE of maximum index 1 [cf. Hairer and Wanner, 1996]. It is first order accurate with respect to the size of the time step. For Θ = 1/2 the second order accurate Crank-Nicholson scheme is obtained which, however, tends to produce non-physical oscillations due to the absence of numerical damping as reported, e.g., by Wood and Lewis [1975] and Tsukerman [1995].

3.2. Example

[29] In order to illustrate the effects of the different quasistationary approximations reported in sections 2.2–2.4, a simple test model as shown in Figure 1 is considered for TD simulations.

Figure 1.

Test model comprising conductive, dielectric as well as magnetic parts. A massive conductor is connected to two plates which form a capacitor in combination with the dielectric material in between. Two ferrite rings surround the conductor symmetrically.

[30] It consists of a massive cylindrical conductor (σc = 50 kS/m, μr = ɛr = 1) which is separated by a gap of 0.5 cm in the center while the radius is 1 cm. The gap is filled by dielectric material (σiso = 10 mS/m, μr = 1, ɛr = 100). In the thin region, two ferrite rings (σfer = 100 μS/m, μr = 1000, ɛr = 1) are placed concentrically around the conductor. Moreover, a small conductivity of σreg is introduced the remaining domain. Excitation is carried out by means of a voltage Vs = ϕ1 − ϕ0 applied between the two cartesian coordinate planes at x = xmin and x = xmax. The excitation signal is defined as a linear step function with variable rise time trise and duration T. Evaluating the constraint for the quasistationary assumption (1) for this case yields the limit τem = 0.2μs ≪ τ for the allowed rise time triseτ.

[31] For the spatial discretization, FE shape functions of lowest order are applied. The coefficients of the respective approximations of the magnetic vector potential and the electric scalar potential are the primary unknowns in the numerical simulation. Time integration is carried out by means of the backward Euler method using a constant time step Δt.

[32] Thanks to the regularization of the system by means of a small conductivity σreg a weighted Coulomb gauge ∇ · (σA) can be applied on the discrete level. The resulting system is non-symmetric but regular. Therefore, a direct solution technique based on LU-decomposition is used to solve the system of linear equations (19) within each time step.

[33] For the following simulations, the model in Figure 1 is discretized by 7972 tetrahedral elements resulting in a total number of 12101 unknowns for the Darwin model as well as for the magnetoquasistatic model. In the electroquasistatic case only 1717 nodal degrees of freedom are required. A constant time step of Δt = 1 μs is used to discretize the linear step (trise = 20 μs, T = 30 μs). In order to compare the results under the different quasistationary approximations, the x-component of the electric field strength is recorded at two positions. The first point is located in the center of the dielectric model part on the axis of the cylindrical conductor. At the border of the conductive ring forming the capacitor, the second point is positioned on the same axis. Figure 2 shows the x-component of the electric field strength with respect to time for the first point inside the capacitor.

Figure 2.

Comparison of the x-component of the electric field strength in the center of the test structure for the Darwin model, the magnetoquasistatic (MQS) formulation and the electroquasistatic (EQS) formulation. A full-wave FDTD solution (FW-FIT) is included as a reference.

[34] As a reference value, the result obtained using a full-wave finite difference scheme (FIT) based on 8450 hexahedral cells for the identical model are included. Due to the adverse setup regarding the length of the simulation interval and the geometrical dimensions, 6 · 106 (explicit) time steps are required in this case. While the computational time for the quasistationary simulations is in the range of minutes, the full-wave solution takes about two hours on a state-of-the-art workstation (4 cores at 3 GHz, 16 GB RAM). The results for the different methods agree very well at the ascending slope as well as on the flat top. However, significantly larger errors are observed at the discontinuities of the excitation signal. At the beginning of the flat top, the difference between the quasistationary and the full-wave solution is still below 1%. Though not visually distinguishable, the relative error is very large near t = 0 s due to the small absolute values of the quantities to compare. The according results for the second point located inside the conductive domain are given in Figure 3.

Figure 3.

Comparison of the x-component of the electric field strength inside the conductor at the end of the plate: For the Darwin model, the magnetoquasistatic (MQS) formulation and the electroquasistatic (EQS) formulation. A full-wave FDTD solution (FW-FIT) is included as a reference.

[35] Despite the spatial proximity to the first point, here the electroquasistatic solution differs obviously from the remaining ones. However, the curves related to the Darwin model as well as to the magnetoquasistatic formulation agree well with the full-wave solution.

[36] In order to point out the limits of the modified Darwin formulation, the parameters of the test example are slightly changed. A artificial relative permittivity of ɛr is assigned to the dielectric material while the other parameters remain unchanged. In turn, the characteristic cycle time τ gets closer to the quasistationary limit. The respective results are shown in Figure 4 for the second point.

Figure 4.

Test model equipped with artificial relative permittivity of 10000: Comparison of the x-component of the electric field strength inside the conductor at the end of the plate: Darwin model, magnetoquasistatic (MQS) formulation, electroquasistatic (EQS) formulation and full-wave FDTD (FW-FIT) solution.

[37] Clearly, neither of the quasistationary approximations is suitable for this parameter setup. The magnetoquasistatic model and the electroquasistatic model seem to bound the two other formulations from below and above, respectively. All quasistationary models fail to consider the oscillations emanating from the broadband excitation. However, a common stationary state at the flat top, namely the stationary current flow in this case, is reached for the three approximative models.

3.3. Discussion

[38] The methodology outlined in this paper is only one way among others to deal with the well-known disadvantageous properties of finite-difference time-domain (FDTD) methods. Of course, unconditionally stable implicit time integration methods are also available for the full set of the Maxwell equations, e.g., the Newmark-Beta scheme. Moreover, modifications tailored to specific applications, such as the calculation of electromagnetic fields in the human body, have been proposed [see, e.g., De Moerloose et al., 1997]. In this case, the skin depth is usually large compared to the dimension of the objects of interest. On the other hand, if the displacement current density is negligible in relation to the conduction current, i.e. in the magnetoquasistatic case, scaling down the speed of light also results in parabolic systems of partial differential equations [cf. Holland, 1995].

4. Conclusion

[39] Applying quasistationary approximations to the full set of the Maxwell equations leads to parabolic partial differential equations as opposed to the hyperbolic ones provided by the general description. Different strategies exist in order to approximate the original set of equations accordingly. In this paper, an approach based on the Darwin model is outlined. It is tested and compared to other quasistationary formulations as well as to a full-wave time-domain scheme. Using the Darwin formulation, capacitive as well as inductive effects can be considered within a single simulation. However, the range of application is limited. When approaching the quasistationary limit depending on the respective set of parameters, the results become inaccurate as the physical modeling of wave propagation is excluded. On the other hand, for parameter setups resulting in a large separation from the quasistationary limit, the additional computational effort for the complete Darwin formulation would be unnecessary. However, if applicable, it leads to a significant reduction of the number of time steps in transient simulations of slowly varying electromagnetic fields compared to explicit schemes while still accounting for capacitive and inductive effects.