As already discussed, the far field model considers a 2D problem of wave propagation in the cavity. Either through the concatenation of the near field models or directly, it also describes the scattering, i.e. the overall field, due to incident fields from every signal via and reflected fields from all vias (signal vias and ground vias) and other terminated ports such as the plane contours. The “direct” description of scattering refers to the property that the reflection at ground vias and plane contours can typically be simplified to simple boundary conditions of the planar problem and therefore mostly have simpler near field models. Some important modeling approaches that are mostly applied for homogeneously filled cavities are discussed in the following. Adaptation to the inhomogeneously filled cavity are outlined in the following and presented in detail in Chapter 4.
3.6.1 Radial Waveguide Method
In the radial waveguide method, the radial wave propagation is considered without reflec-tions from scatterers such as ground vias and plane edges, see [88, 89]. The formulation considers a finite size of the radial port for which a field is evaluated but neglects the size of the source port [61, Sec. 4.3.2]. Using image theory the case of finite planes with open boundary condition can also be computed [88].
3.6.2 Cavity Resonator Method
The cavity resonator method is applicable to planar circuits with simple shapes and in-cludes the reflection from either open-circuited boundaries [90], [91, Sec. 2.4.1]. The origi-nal formulation defines rectangular ports but adaptations to circular ports are also avail-able [92, Sec. III], [56, Appendix B]. The accuracy of the method can be controlled by the number of modes that are used. The method can be implemented relatively easily and provides a good accuracy if a large enough number of modes is ensured by observing the
3.6 Far-Field Modeling convergence behavior. The cavity resonator method has proven itself to be a good reference for the validation of techniques for arbitrarily shaped contours such as the contour integral method discussed in the following.
3.6.3 Contour Integral Method for Planar Circuits
The method of choice for this thesis is the contour integral method (CIM) for planar cir-cuits because it provides a good trade-off between the generality of its application and the achievable numerical efficiency. This numerical technique includes both the scattering from arbitrarily shaped contours with arbitrary terminations and finite port sizes of both source and observation ports. The CIM is a frequency-domain method for microwave and optical frequencies for the computation of the N-port network parameters and field information of planar circuits [91,93]. Circuits are considered planar if only two spatial dimensions are larger than the guided wavelength in these structures while the third dimension, which is referred to as the height, is significantly shorter. This leads to the assumption for homoge-neous fillings that the transverse field components are constant across the height. Thus, the electromagnetic fields propagating in this structure can be described by a 2D Helmholtz equation.
The method has numerous possible applications and extensions. The typical general field is the application for signal and power integrity analysis. Coupling of the method with descriptions for connected devices (e.g. decoupling capacitors) is possible and can be used for the optimal placement of decoupling capacitors for reduction of simultaneous switching noise [93]. The method can further be used to investigate the resonant frequencies of planar circuits [91]. Also, as an advanced application, the numerical efficiency allows application in optimal circuit pattern synthesis [91, Ch. 7].
Because the CIM is a frequency domain method, the properties of the circuit at discrete frequencies are determined. Continuous wave problems where the structure is excited with a time-harmonic signal can directly be investigated. Therefore, time domain analysis can in general only be performed after Fourier transformation of the frequency domain infor-mation. Nonlinearities in form of the dependence of the electromagnetic properties on the amplitudes of the fields are not included. This would need to be considered in the most general case of modeling of the silicon properties but is not further considered here.
A single computational domain of the CIM consists of a planar structure of which the stackup is invariant with respect to the directions which span the plane. In the most simple case, a planar stackup consists of only a dielectric of a certain thickness and real permittivity which is bounded by perfect electric conductor (PEC) layers. Thus no
dielec-tric losses and conductivity are considered for the dielecdielec-tric and no losses through limited conductivity of the conducting layers. The fundamental mode which can propagate in this parallel plate structure is TEM.
For the application with realistic structures, such as silicon interposers, PCBs, and package substrates, a finite conductivity of the bounding metal layers and dielectric and conductive losses in the dielectrics need to be considered. As is discussed further in the following chapter, the fundamental mode which can propagate in these structures is not a true TEM mode because the electric field does not vanish at the interfaces of bounding layers and dielectric layers and also has a component in propagation direction. If the these effects are relatively small, this component in propagation direction is still small and the mode is qualified as “quasi-TEM”. In contrast to the parallel plate structure with homogeneous filling, even the fundamental mode of the layered lossless structure has longidudinal field components. In the presented work it is aimed at extending the CIM to the treatment of structures with a larger variety of stackups with a focus on those found in TSV structures.
The “classical” CIM considers piecewise linear contours which are segmented into linear ports. This is illustrated in Fig. D.1a of Appendix D. The planar circuit is treated as an N-port where N is the number of segments: the ports at the outer contour are typically considered to be open circuited which corresponds to a perfect magnetic conductor (PMC) boundary condition of the corresponding electromagnetic problem and models the typically small radiation resistance. The CIM interrelates the electric and magnetic fields, i.e., their tangential components, of all boundary ports. By assuming the fields to be constant on each port, the fields can be replaced with modal currents and voltages which are defined as integral quantities of the fields. Self impedance is defined as the ratio of voltage and current at one port and transfer impedances for voltages and currents at different ports.
The extended CIM is based on both line ports from which waves emanate that are decom-posed and approximated by one or more isotropic radial waves and circular ports on which both isotropic modes with azimuthal symmetry and anisotropic modes with azimuthally varying fields are defined:
• The fundamental isotropic mode features constant fields along the port perimeter.
This mode is dominant in many practical applications and its sole consideration is justified in many cases where ports are sufficiently distant from each other. The sufficient distance depends on the geometrical and material properties.
• The higher-order anisotropic modes feature an azimuthal variation, i.e. a functional dependence on angle, of the fields on the circular port. Their application is significant for cases of closely spaced vias or vias close the plane edges. They can also be relevant due to specific excitations of the fields at the ports, e.g., the excitation of a via by
3.7 Embedding of the Physics-Based Via Model