Treatment of Various Boundary Conditions

Up to now, the procedure has been outlined to calculate an inhomogeneous substrate consisting of an arbitrary number of homogeneous regions. Many practical problems and modeling approaches further deal with artificial, effective, or equivalent boundary conditions.

These boundary conditions model idealized conditions or account for physical effects outside the computational domain. It is common to model ground vias connecting the top and bottom metalizations as PEC barrels, enforcing the voltage to be zero on the circumference.

Edges of PCBs are commonly modeled as Perfect Magnetic Conductor (PMC), accounting for the fact that no conduction current can flow between the upper and the lower plate at the edges. Sometimes, the radiation from edges is taken into account and modeled with an equivalent impedance taking the capacitive effect of the edge, as well as, the radiation resistance into account [103, Chapter 5]. This practice motivates to consider the boundary conditions of PEC, PMC, and arbitrary impedances.

Consider a general problem as the one illustrated in Figure 2.6. It incorporates dielectric interfaces, PMC, and PEC boundaries representing a PCB. For the sake of consistency with [5, 62–64, 103, 119], we will consider PEC boundaries to model ground vias, PMC boundaries to model the edges at finite distance of PCBs, and impedances to include the effect of decoupling capacitors,e.g. for a Power Delivery Network (PDN), by placing impedance conditions on boundaries. The notation and the examples are motivated by the application of the CIM for the modeling of SI/PI on PCB interconnects. Nevertheless, the techniques are suitable in all fields of application.

Impedance Boundary Conditions

Impedance boundary conditions are used to model lumped elements, such as decoupling capacitors in PCBs [5, 62] or radiation at the edges of PCBs [104, 105] and can also be used to model a radiating boundary condition. With this approach, a fixed ratio between voltage and current, and therefore also electric and magnetic field is enforced at a port. The procedure is similar to the one of the connection of regions of homogeneous dielectric. The impedances are written into a, usually diagonal⁸, impedance matrix Z^(L). This impedance matrix is treated as if it is modeling a homogeneous region in the segmentation procedure.

The matrix is added to the initial set of impedance matrices and treated in the same way as the other impedance matrices. The result is a global impedance matrix Z^{(1,···,Nr,L)} taking into account the inhomogeneous substrate and the impedance boundaries.

8The diagonal property reflects the property that the ports are only coupled inside the regime computed with the CIM. Sometimes, e.g., when the radiation shall be modeled with high accuracy, additionally backscattering from exterior objects needs to be taken into account. In this case, the matrix is not diagonal [105]. For the proposed procedure, there is no difference in considering diagonal or full matrices.

2.3 Extension to Modeling of Inhomogeneous Substrates

Figure 2.6: The inhomogeneous substrate (a) is decomposed into regions, which are bounded by contours. The regions are represented by impedance matrices (b) with respect to external ports pI, pII, internal ports c, and boundary ports q1, q2, q3 defined on the corresponding contours. Figure and caption taken from [5].

In order to model a radiating boundary condition⁹ it is possible to use the characteristic port impedance ((2.9) for linear segments, (2.12) for circular cutouts, and (2.50) for outer circular boundaries) as a termination impedance. This is helpful for validation and test of numerical code, but is of limited importance for practical simulations because the CIM intrinsically models unbounded problems.

PEC and PMC Boundary Conditions

Commonly, ports that are used to enforce a boundary condition are denoted with the subscript q [5, 62–64, 103, 119]. In Figure 2.6, the ports denoted with q₁ refer to the PMC boundary condition at the edges and q₂ and q₃ refer to PEC boundary conditions at ground vias. During the segmentation procedure to connect the homogeneous materials, these ports are considered as external ports. Generally, we refer to ports with a PEC boundary condition with the subscript p_e and to ports with a PMC boundary condition with the subscript p_m. The global impedance matrix after the segmentation of the regions of homogeneous dielectric and impedance boundary conditions can be sorted as



9In context of FDTD this is conventionally called Perfectly Matched Layer (PML)

with a corresponding retention matrix

M(1,···,N,L)=^hM^(1,···_p ^,N,L) M(1,···,N,L)

pe M^(1,···_pm ^,N,L)i. (2.37)

The PEC boundary condition is represented by the voltages V_pe being zero. The dual boundary condition of PMC is represented by the currents I_pm being zero. The whole matrix can be rewritten as an impedance matrix connecting the voltages and currents on the external ports as V_p =ZI_p and a corresponding retention matrix. After some algebra, these matrices are found to be

Z=Z^(1,···_pp ^,N,L)−Z(1,···,N,L)

ppe F, (2.38)

M=M(1,···,N,L)

p −M^(1,···_pe ^,N,L)F, (2.39)

where

F=Z(1,···,N,L) pepe

−1

Z(1,···,N,L)

pep . (2.40)

The retention matrix M provides the voltages and currents on the internal ports as a function of the currents on the external ports. The currents on ports with PEC boundary condition and the voltages on ports with PMC boundary condition are found to be

I_pe =−FI_p, (2.41)

V_pm =

"

Z^(1,···_pmp^,N,L)−Z^(1,···_pmpe^,N,L)F

#

·I_p. (2.42)

With this procedure, it is possible to model inhomogeneous substrates and various boundary conditions in the CIM. The final result is an impedance matrix for the external ports and a retention matrix to retain the voltage and currents on internal ports. This procedure is used throughout this thesis to model all kinds of dielectric inclusions. Hence, the application examples for dielectric inclusions in Section 2.4.1 and concentric circular contours in Section 2.5 should be seen as validation examples for the proposed approach.