reference.md

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Reference

Mathematical background

The solver computes a finite element approximation to the three-dimensional, time-harmonic Maxwell's equations in second-order form. The nondimensionalized, source-free, boundary value problem for \bm{E}(\bm{x})\in\mathbb{C}^3, \bm{x}\in\Omega, \partial\Omega = \Gamma, where \bm{\mathscr{E}}(\bm{x},t) = \text{Re}\{\bm{E}(\bm{x})e^{i\omega t}\} denotes the electric field, is written as

$$\begin{aligned} \nabla\times\mu_r^{-1}\nabla\times\bm{E} + i\omega\sigma\bm{E} - \omega^2\varepsilon_r\bm{E} &= 0 \,,\; \bm{x}\in\Omega \\\ \bm{n}\times\bm{E} &= 0 \,,\; \bm{x}\in\Gamma_{PEC} \\\ \bm{n}\times(\mu_r^{-1}\nabla\times\bm{E}) &= 0 \,,\; \bm{x}\in\Gamma_{PMC} \\\ \bm{n}\times(\mu_r^{-1}\nabla\times\bm{E}) + \gamma\bm{n}\times(\bm{n}\times\bm{E}) &= \bm{U}^{inc} \,,\; \bm{x}\in\Gamma_{Z} \end{aligned}$$

where the nondimensionalization has been performed with respect to a characteristic length L_0, time L_0/c_0, magnetic field strength H_0, and electric field strength Z_0 H_0. Here, c_0 and Z_0 are the speed of light and impedance of free space, respectively. This nondimensionalization will be used throughout this entire reference. For more details, see [1] and [2].

Given the electric field solution, the time-harmonic magnetic flux density can be calculated as

$$\bm{B} = -\frac{1}{i\omega}\nabla\times\bm{E} \,.$$

The flux density is related to the magnetic field, \bm{H}, by the standard linear constitutive relationship \bm{H} = \mu_r^{-1}\bm{B}.

In general, the material property coefficients may be scalar- or matrix-valued. In the matrix-valued (anisotropic) case, the material property coefficients should still always be symmetric.

For a general isotropic lossy dielectric, the relative permittivity \varepsilon_r is a complex-valued quantity:

$$\varepsilon_r = \varepsilon_r' (1-i\tan{\delta})$$

where \varepsilon_r' is the real relative permittivity and \tan{\delta} is the loss tangent. Alternatively, conductor loss is modeled by Ohm's law \bm{J} = \sigma\bm{E} with electrical conductivity \sigma > 0.0. For a superconducting domain, the constitive current-field relationship given by Ohm's law is replaced by that given by the London equations:

$$\frac{\partial \bm{J}}{\partial t} = \frac{1}{\mu_r\lambda_L^2}\bm{E}$$

where \lambda_L = \sqrt{m/\mu n_s e^2}/L_0 is the nondimensionalized London penetration depth. In this case, the term +i\omega\sigma \bm{E} arising for a normal conductor in the time-harmonic Maxwell's equations becomes +(\mu_r \lambda_L^2)^{-1}\bm{E}.

The domain boundary \Gamma = \Gamma_{PEC}\cup\Gamma_{PMC}\cup\Gamma_{Z}, is separated into perfect electric conductor (PEC), perfect magnetic conductor (PMC), and impedance boundaries, respectively. The PEC boundary condition is a homogeneous Dirichlet condition, while the PMC boundary condition is the natural boundary condition for the problem and is satisfied at all exterior boundaries by the finite element formulation. Impedance boundaries are modeled using a Robin boundary condition with \gamma = i\omega/Z_s, in which Z_s the surface impedance of the boundary, with units of impedance per square.

Time domain formulation

A time-dependent formulation is also available to compute the electric field response \bm{E}(\bm{x},t) for a given time-dependent source excitation \bm{U}^{inc}(\bm{x},t). The governing equations in this case are

$$\nabla\times\mu_r^{-1}\nabla\times\bm{E} + \sigma\frac{\partial\bm{E}}{\partial t} + \varepsilon_r\frac{\partial^2\bm{E}}{\partial t^2} = 0 \,,\; \bm{x}\in\Omega$$

subject to the same boundary conditions as the frequency-dependent case except for the Robin boundary condition which is written for a lumped resistive port boundary, for example, as

$$\bm{n}\times(\mu_r^{-1}\nabla\times\bm{E}) + Z_s^{-1}\bm{n}\times\left(\bm{n}\times\frac{\partial\bm{E}}{\partial t}\right) = \bm{U}^{inc} \,,\; \bm{x}\in\Gamma_{Z} \,.$$

The second-order electric field formulation is chosen to take advantage of unconditionally stable implicit time-integration schemes without the expense of a coupled block system solve for \bm{E}(\bm{x},t) and \bm{B}(\bm{x},t). It offers the additional benefit of sharing many similarities in the spatial discretization as the frequency domain formulation outlined above.

Eigenmode calculations

For eigenmode problems, the source term is zero and a quadratic eigenvalue problem for the eigenvalues \omega is solved:

$$(\bm{K}+i\omega\bm{C}-\omega^2\bm{M})\bm{x} = 0$$

where the matrix \bm{K} represents the discretized curl-curl operator, \bm{M} the mass term, and \bm{C} the port impedance boundary conditions. The damped frequency \omega_d and quality factor Q are postprocessed from each of the resulting eigenvalues as

$$\omega_d = \text{Re}\{\omega\} \,, \qquad Q = \frac{|\omega|}{2|\text{Im}\{\omega\}|} \,.$$

Lumped ports and wave ports

For lumped port boundaries, the surface impedance can be related to an equivalent circuit impedance, Z. There are two common cases:

1. Rectangular ports: Z = Z_s l / w, where l and w are the length and width of the port, respectively (length here is defined as the distance between the two conductors).

2. Coaxial ports: Z = Z_s \ln(b/a) / 2\pi, where a and b denote the inner and outer radii of the port, respectively.

A lumped parallel RLC circuit boundary has a circuit impedance

$$\frac{1}{Z} = \frac{1}{R}+\frac{1}{i\omega L}+i\omega C \,.$$

Thus, the relationships between the circuit and surface element parameters for the user to specify are given by R_s = \alpha R, L_s = \alpha L, and C_s = C/\alpha, where \alpha = w/l for a rectangular port or \alpha = 2\pi / \ln(b/a) for a coaxial port.

For multielement lumped ports, the effective circuit impedance is given by

$$\frac{1}{Z} = \sum_k \frac{1}{Z_k} \,.$$

That is, the circuit impedances of each port contributing to the multielement port add in parallel. For the specific case of a two element multielement port with two identical lumped elements, we have Z = (1/Z_1 + 1/Z_2)^{-1} = Z_k / 2, where Z_k is the circuit impedance of a single port element.

The source term \bm{U}^{inc} in a driven frequency-response problem is related to the incident field at an excited port boundary by

$$\bm{U}^{inc} = -2\gamma(\bm{n}\times\bm{E}^{inc})\times\bm{n}$$

where (\bm{n}\times\bm{E}^{inc})\times\bm{n} is just the projection of the excitation field onto the port surface. The incident fields for lumped ports depend on the port shape:

1. Rectangular ports: \bm{E}^{inc} = E_0 \, \hat{\bm{l}}, where E_0 is a uniform constant field strength and \hat{\bm{l}} a unit vector defining the direction of polarization on the port (typically should be the direction between the two conductors).

2. Coaxial ports: \bm{E}^{inc} = \frac{E_0 r_0}{r} \, \hat{\bm{r}}, where E_0 is again a uniform constant field strength, r_0 is a characteristic length for the port, r is the distance from the port center, and \hat{\bm{r}} a unit vector specifying the port radial direction.

In the time domain formulation, the source term \bm{U}^{inc} appears as

$$\bm{U}^{inc} = -2 Z_s^{-1}\left(\bm{n}\times\frac{\partial\bm{E}^{inc}}{\partial t}\right) \times\bm{n} \,.$$

The incident field \bm{E}^{inc}(\bm{x},t) is

$$\bm{E}^{inc}(\bm{x},t) = p(t)\bm{E}^{inc}(\bm{x})$$

where \bm{E}^{inc}(\bm{x}) is identical to the spatial excitation in the frequency domain formulation, and p(t) describes the temporal shape of the excitation. Possible options include a sinusoidal, Gaussian, modulated Gaussian, or step excitation.

In the frequency domain, the scattering parameters can be postprocessed from the computed electric field for each lumped port with boundary \Gamma_i as

$$S_{ij} = \frac{\displaystyle\int_{\Gamma_i}\bm{E}\cdot\bm{E}^{inc}_i\,dS} {\displaystyle\int_{\Gamma_i}\bm{E}^{inc}_i\cdot\bm{E}^{inc}_i\,dS} - \delta_{ij} \,.$$

In the time domain, the time histories of the port voltages can be Fourier-transformed to get their frequency domain representation for scattering parameter calculation.

Numeric wave ports assume a field with known normal-direction dependence \bm{E}(\bm{x}) = \bm{e}(\bm{x}_t)e^{ik_n x_n} where k_n is the propagation constant. For each operating frequency \omega, a two-dimensional eigenvalue problem is solved on the port yielding the mode shapes \bm{e}_m and associated propagation constants k_{n,m}. These are used in the full 3D model where the Robin port boundary condition has coefficient \gamma = i\text{Re}\{k_{n,m}\}/\mu_r and the computed mode is used to compute the incident field in the source term \bm{U}^{inc} at excited ports. Scattering parameter postprocessing takes the same form as the lumped port counterpart using the computed modal solutions. Since the propagation constants are known for each wave port, scattering parameter de-embedding can be performed by specifying an offset distance d for each port:

$$\tilde{S}_{ij} = S_{ij}e^{ik_{n,i}d_i}e^{ik_{n,j}d_j} \,.$$

For more information on the implementation of numeric wave ports, see [3].

Other boundary conditions

The first-order absorbing boundary condition, also referred to as a scattering boundary condition, is a special case of the general impedance boundary condition described above:

$$\bm{n}\times(\mu_r^{-1}\nabla\times\bm{E}) + i\omega\sqrt{\mu_r^{-1}\varepsilon_r}\bm{n}\times(\bm{n}\times\bm{E}) = 0 \,.$$

This is also known as the Sommerfeld radiation condition, and one can recognize the dependence on the impedance of free space Z_0^{-1} = \sqrt{\mu_r^{-1}\varepsilon_r}. The second-order absorbing boundary condition is

$$\bm{n}\times(\mu_r^{-1}\nabla\times\bm{E}) + i\omega\sqrt{\mu_r^{-1}\varepsilon_r}\bm{n}\times(\bm{n}\times\bm{E}) - \beta\nabla\times[(\nabla\times\bm{E})_n\bm{n}] = 0$$

where assuming an infinite radius of curvature \beta = \mu_r^{-1}c_0/(2i\omega), and the contribution depending on (\nabla\cdot\bm{E}_t) has been neglected.

Additionally, while metals with finite conductivity can be modeled using an impedance boundary condition with constant impedance Z_s, a more accurate model taking into account the frequency dependence of the skin depth is

$$Z_s = \frac{1+i}{\delta\sigma}$$

where \delta = \sqrt{2/\mu_r\sigma\omega} is the skin depth and \sigma is the conductivity of the metal. Another model, which takes into account finite thickness effects, is given by

$$Z_s = \frac{1}{\delta\sigma}\left(\frac{\sinh{\nu}+\sin{\nu}}{\cosh{\nu}+\cos{\nu}} + i\frac{\sinh{\nu}-\sin{\nu}}{\cosh{\nu}+\cos{\nu}}\right)$$

where \nu = h/\delta and h is the layer thickness. This model correctly produces the DC limit when h\ll\delta.

Energy-participation ratios

The energy-participation ratio (EPR) for lumped inductive elements is computed from the electric and magnetic fields corresponding to eigenmode m, \bm{E}_m and \bm{H}_m, using the formula

$$p_{mj} = \frac{1}{\mathcal{E}^{elec}_m} \, \frac{1}{2} \, L_j I_{mj}^2$$

where p_{mj}\in[-1,1] denotes the signed participation ratio for junction j in mode m, L_j is the provided junction circuit inductance, I_ {mj} is the peak junction current for mode m, and \mathcal{E}^{elec}_m is the electric energy in mode m. The junction current is computed using the mean voltage across the port, \overline{V}_{mj}, as I_{mj} = \overline{V}_{mj}/Z_{mj}, where Z_{mj} = 1/(i\omega_m L_j) is the impedance of the inductive branch of the lumped circuit. The mean port voltage depends on the computed electric field mode and the shape of the port:

1. Rectangular ports: \overline{V}_{mj} = \frac{1}{w_j}\int_{\Gamma_j}\bm{E}_m\cdot\hat{\bm{l}}_j\,dS.

2. Coaxial ports: \overline{V}_{mj} = \frac{1}{2\pi}\int_{\Gamma_j}\frac{\bm{E}_m}{r}\cdot\hat{\bm{r}}_j\,dS.

Finally, the total electric energy in mode m is

$$\mathcal{E}^{elec}_m = \frac{1}{2} \, \text{Re}\left\{\int_\Omega\bm{D}_m^*\cdot\bm{E}_m\,dV\right\} + \sum_j \frac{1}{2} \, C_jV_{mj}^2$$

where \bm{D}_m = \varepsilon_r\bm{E}_m is the electric flux density for mode m and the second term on the right-hand side accounts for any lumped capacitive boundaries with nonzero circuit capacitance C_j.

The EPR can also be used to estimate mode quality factors due to input-output (I-O) line coupling. The mode coupling quality factor due to the j-th I-O port is given by

$$Q_{mj} = \frac{\omega_m}{\kappa_{mj}}$$

where the port coupling rate \kappa_{mj} is calculated as

$$\kappa_{mj} = \frac{1}{\mathcal{E}^{elec}_m} \, \frac{1}{2}\,R_j I_{mj}^2 \,.$$

Bulk and interface dielectric loss

The quality factor due to bulk dielectric loss resulting from an electric field \bm{E} present in domain j with associated loss tangent \tan{\delta}_j is given by

$$\frac{1}{Q_j} = p_j \tan{\delta}_j = \frac{1}{\mathcal{E}^{elec}} \, \frac{1}{2} \, \tan{\delta}_j \, \text{Re}\left\{\int_{\Omega_j}\bm{D}^*\cdot\bm{E}\,dV\right\}$$

where, as above, \mathcal{E}^{elec} is the total electric field energy in the domain, including the contributions due to capacitive lumped elements.

Likewise, the quality factor due to surface interface dielectric loss for interface j is given by

$$\frac{1}{Q_j} = p_j \tan{\delta}_j = \frac{1}{\mathcal{E}^{elec}} \, \frac{1}{2} \, t_j\tan{\delta}_j \, \text{Re}\left\{\int_{\Gamma_j}\bm{D}^*\cdot\bm{E}\,dS\right\}$$

where t_j is the thickness of the layer and \bm{D} = \varepsilon_{r,j}\bm{E} is the electric displacement field in the layer evaluated using the relative permittivity of the interface \varepsilon_{r,j}. For an internal boundary, this integral is evaluated on a single side to resolve abiguity due to the discontinuity of \bm{E} across the boundary.

The above formula for interface dielectric loss can be specialized for the case of a metal-air, metal-substrate, or substrate-air interface [4]. In each case, the quality factor for interface j is given by

• Metal-air:

$$\frac{1}{Q^{MA}_j} = \frac{1}{\mathcal{E}^{elec}} \, \frac{1}{2} \, \frac{t_j\tan{\delta}_j}{\varepsilon_{r,j}^{MA}} \, \text{Re}\left\{\int_{\Gamma_j}\bm{E}_n^*\cdot\bm{E}_n\,dS\right\}$$

• Metal-substrate:

$$\frac{1}{Q^{MS}_j} = \frac{1}{\mathcal{E}^{elec}} \, \frac{1}{2} \, \frac{t_j\tan{\delta}_j(\varepsilon_{r,j}^{S})^2}{\varepsilon_{r,j}^{MS}} \, \text{Re}\left\{\int_{\Gamma_j}\bm{E}_n^*\cdot\bm{E}_n\,dS\right\}$$

• Substrate-air:

$$\frac{1}{Q^{SA}_j} = \frac{1}{\mathcal{E}^{elec}} \, \frac{1}{2} \, t_j\tan{\delta}_j\left(\varepsilon_{r,j}^{SA} \, \text{Re}\left\{\int_{\Gamma_j}\bm{E}_t^*\cdot\bm{E}_t\,dS\right\} + \frac{1}{\varepsilon_{r,j}^{SA}} \, \text{Re}\left\{\int_{\Gamma_j}\bm{E}_n^*\cdot\bm{E}_n\,dS\right\}\right)$$

where \bm{E}_n denotes the normal field to the interface and \bm{E}_t = \bm{E}-\bm{E}_n denotes the tangential field.

Lumped parameter extraction

For electrostatic simulations, the Maxwell capacitance matrix is computed in the following manner. First, the Laplace equation subject to Dirichlet boundary conditions is solved for each terminal with boundary \Gamma_i in the model, yielding an associated voltage field V_i(\bm{x}):

$$\begin{aligned} \nabla\cdot(\varepsilon_r\nabla V_i) &= 0 \,,\; \bm{x}\in\Omega \\\ V_i &= 1 \,,\; \bm{x}\in\Gamma_i \\\ V_i &= 0 \,,\; \bm{x}\in\Gamma_j \,,\; j\neq i \,. \end{aligned}$$

The energy of the electric field associated with any solution is

$$\mathcal{E}(V_i) = \frac{1}{2}\int_\Omega\varepsilon_r\bm{E}_i\cdot\bm{E}_i\,dV$$

where \bm{E}_i=-\nabla V_i is the electric field. Then, the entries of the Maxwell capacitance matrix, \bm{C}, are given by

$$\bm{C}_{ij} = \mathcal{E}(V_i+V_j)-\frac{1}{2}(\bm{C}_{ii}+\bm{C}_{jj}) \,.$$

Magnetostatic problems for inductance matrix extraction are based on the magnetic vector potential formulation:

$$\begin{aligned} \nabla\times(\mu_r^{-1}\nabla\times\bm{A}_i) &= 0 \,,\; \bm{x}\in\Omega \\\ \bm{n}\times(\mu_r^{-1}\nabla\times\bm{A}_i) = \bm{n}\times\bm{H}_i &= \bm{J}_s^{inc} \,,\; \bm{x}\in\Gamma_i \\\ \bm{n}\times(\mu_r^{-1}\nabla\times\bm{A}_i) &= 0 \,,\; \bm{x}\in\Gamma_j \,,\; j\neq i \,. \end{aligned}$$

For each port with boundary \Gamma_i, a unit source surface current \bm{J}_s^{inc} is applied, yielding an associated vector potential solution \bm{A}_i(\bm{x}). Homogeneous Dirichlet boundary conditions \bm{n}\times\bm{A}_i=0 are also imposed on specified surfaces of the model. The magnetic field energy associated with any solution is

$$\mathcal{E}(\bm{A}_i) = \frac{1}{2}\int_\Omega\mu_r^{-1}\bm{B}_i\cdot\bm{B}_i\,dV$$

where \bm{B}_i = \nabla\times\bm{A}_i is the magnetic flux density. Then, the entries of the inductance matrix, \bm{M}, are given by

$$\bm{M}_{ij} = \frac{1}{I_i I_j}\mathcal{E}(\bm{A}_i+\bm{A}_j) - \frac{1}{2}\left(\frac{I_i}{I_j}\bm{M}_{ii}+\frac{I_j}{I_i}\bm{M}_{jj}\right)$$

where I_i is the excitation current for port i, computed by integrating the source surface current \bm{J}_s^{inc} over the surface of the port.

Error estimation and adaptive mesh refinement (AMR)

Error estimation is used to provide element-wise error estimates for AMR, as well as a global error indicator used to terminate AMR iterations or provide an estimate for solution accuracy. A Zienkiewicz–Zhu (ZZ) error estimator based on [5] is implemented, which measures the error in the recovered magnetic field and electric flux density. On element K, we have

$$\eta^2_K = \eta_{m,2}^2+\eta_{e,K}^2 = \|\mu_r^{1/2}\bm{R}_{ND}(\mu^{-1}\bm{B}) - (\mu_r^{-1/2}\bm{B})\|_{L^2(\Omega_K)}^2 + \|\varepsilon_r^{-1/2}\bm{R}_{RT}(\varepsilon_r\bm{E}) - (\varepsilon_r^{1/2}\bm{E})\|_{L^2(\Omega_K)}^2$$

where \bm{R}_{ND} and \bm{R}_{RT} are the smooth-space recovery operators which orthogonally project their argument onto H(\text{curl}) and H(\text{div}), discretized by Nédélec and Raviart-Thomas elements, respectively.

References

[1] J.-M. Jin, The Finite Element Method in Electromagnetics, Wiley-IEEE Press, Hoboken, NJ, Third edition, 2014.
[2] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, Oxford, 2003.
[3] L. Vardapetyan and L. Demkowicz, Full-wave analysis of dielectric waveguides at a given frequency, Mathematics of Computation 72 (2003) 105-129.
[4] J. Wenner, R. Barends, R. C. Bialczak, et al., Surface loss of superconducting coplanar waveguide resonators, Applied Physics Letters 99, 113513 (2011).
[5] S. Nicaise, On Zienkiewicz-Zhu error estimators for Maxwell’s equations, Comptes Rendus Mathematique 340 (2005) 697-702.