Efficient modeling of power and signal integrity for semiconductors and advanced electronic package systems 2

Chapter Electrical Performance Modeling of Power-Ground Layers with Multiple Vias The outline of the efficient approach for system-level modeling of advanced electronic packages is presented in Chapter 1, in which power distribution network (PDN) and signal distribution network (SDN) are separately analyzed by using mode decomposition for the entire problem The analytical method for analysis of the power-ground plane pair is also presented in the previous chapter Although, the method is efficient to calculate the impedance of the package, it is only applicable to the rectangular structure of power-ground planes In this chapter, the semi-analytical scattering matrix method (SMM) based on the N-body scattering theory is proposed for multiple scattering of vias Using the modal expansion of fields in a parallel-plate waveguide, the formula derivation of the SMM is presented in details In the conventional SMM, the power-ground planes are assumed to be infinitely large so it cannot capture the resonant behavior of the realworld packages In this research study, an important extension to the SMM is made to simulate the finite domain of power-ground planes A novel boundary modeling 41 Chapter Modeling for Power-Ground Planes with Multiple Vias 42 method is proposed based on factitious layer of PMC cylinders with frequencydependent radii at the periphery of an electronic package Hence, the extended SMM is capable to handle the real-world package structures In the latter part of the chapter, numerical examples are presented for validation of the implemented SMM algorithm with the proposed frequency-dependent cylinder layer (FDCL) The extended method is not only capable to simulate the finitesized power-ground planes and it can also simulate the irregular-shaped planes and cutout structure in the planes This is one prominent feature of the FDCL modeling method 3.1 Problem Statement for Modeling of Multiple Vias An advanced electronic package consisting of signal traces, power-ground planes and plenty of vias, as shown in Fig 3.1, can be subdivided into two problem/design sets: the signal distribution network (SDN) and the power distribution network (PDN) For such a complex package, it is essential to consider the coupling impact of the power-ground vias in the PDN on the electrical performance of the signal in order to characterize the SDN more accurately Due to complexity of each network, it is very difficult and time consuming to model both networks simultaneously As the methodology outline for analysis of the entire problem has been discussed earlier; the inner domain of the package, which consists of parallel power-ground planes and vias, is analyzed by using the semi-analytical scattering matrix method (SMM) The SMM based on the N-body scattering theory is developed to extract its multi-port admittance matrix parameters Vias are usually employed in the electronic packages with the shape of circular cylinders Thus, the theory of multiple scattering among many parallel conducting cylinders [88] can be used to model them efficiently The theory of scattering by con- Chapter Modeling for Power-Ground Planes with Multiple Vias 43 Figure 3.1: Schematic diagram of a multilayered advanced electronic package ducting cylinders (vias) in the presence of two PECs (perfect electric conductors) [55] has been applied to study the problem of vias in multilayered structures [56, 57] In this research, instead of using the Green’s function approach in [56, 57] to obtain the corresponding formulae, we will directly apply the parallel-plate waveguide theory, which is a relatively simple and straightforward way to tackle the problem of scattering by cylinders in the presence of two or more PEC planes Without loss of generality, we assume that the power-ground planes in an electronic package are made of PECs, which may be of finite thickness; and the vias are circular PEC cylinders 3.2 Modal Expansion of Fields in a Parallel-Plate Waveguide The source-free Maxwell equations are given by ì E = jàH (3.1) × H = jωεE (3.2) ∇·E = (3.3) ∇ · H = (3.4) Two adjacent conductor planes either power or ground can be considered as a parallel-plate waveguide Assume that the z-axis is normal to the surface of the P-G Chapter Modeling for Power-Ground Planes with Multiple Vias 44 planes and the electromagnetic fields have e−jβz dependence where β is the propagation wavenumber along the guiding direction z For the parallel-plate waveguide structure, two independent solutions of the above Maxwell equations in cylindrical coordinate are expressed as ∞ ∞ Ez (ρ, φ, z) = (2) aE Jn (kρ ρ) + bE Hn (kρ ρ) Cm ejnφ for T M waves , mn mn n=−∞ m=0 ∞ ∞ Hz (ρ, φ, z) = (3.5) (2) aH Jn (kρ ρ) + bH Hn (kρ ρ) Sm ejnφ for T E waves , mn mn n=−∞ m=1 (3.6) where aE and bE are the expansion coefficients of the incoming and outgoing TM mn mn waves, aH and bH are the expansion coefficients of the incoming and outgoing TE mn mn mπ 2 , where d is the spacing waves, respectively k = ω µε = kρ + βm, βm = kz = d of the adjacent power-ground planes, and µ and ε represent the permeability and permittivity of the dielectric sandwiched between the P-G planes The terms Cm and Sm stand for Cm = cos (βm z) and Sm = sin (βm z), respectively An ejωt time dependence is assumed throughout the formulation herein and subsequently Other components of E and H related to Ez and Hz are calculated by jàì s Ez ⎥ z Es ⎥ ⎢ ⎢ ⎥⎣ ∂z ⎣ ⎦= 2⎢ ⎦ ⎦ ∂ kρ ⎣ −jωεˆ× Hs ∇s Hz z ∂z ⎡ ⎤ ⎡ (3.7) The operator ∇s represents the gradient in the transverse direction and in cylindrical coordinates, and it can be written as ∇s = ρ ˆ ∂ ∂ +ϕ ˆ ∂ρ ρ ∂φ (3.8) Then, by using the modal expansion approach, the Ez and Hz components of an incident wave are expressed as: aE cos (βmz) Jn (kρ ρ) ejnφ mn m for TEz mode n inc Hz = m for TMz mode , aH sin (βm z) Jn (kρ ρ) ejnφ mn inc Ez = (3.9) n scat scat The modal expansion of the scattered fields Ez and Hz can be expressed, similar to those in (3.9), by using bE and bH as the unknown expansion coefficients Submn mn Chapter Modeling for Power-Ground Planes with Multiple Vias 45 stituting (3.9) into (3.7), we can obtain all other components of the electromagnetic fields corresponding to TMz and TEz modes Since the total field is a summation of the incident and scattered fields, we can finally obtain the following expressions for the total tangential electromagnetic fields in cylindrical coordinates, normal to ρ in the ith parallel-plate waveguide formed by ˆ pair of power-ground planes ∞ ∞ n=−∞ ∞ m=0 (i) E(i) (i) (i) aE(i)Jmn + bE(i)Hmn etmn mn mn Et = + (3.10) (i) aH(i)Jmn mn + (i) bH(i)Hmn mn H(i) etmn e jnφ m=1 ∞ ∞ n=−∞ ∞ (i) Ht m=0 E(i) (i) (i) aE(i)Jmn + bE(i)Hmn htmn mn mn = + (3.11) (i) aH(i)Jmn mn + (i) bH(i)Hmn mn H(i) htmn e jnφ m=1 where the eigen-vectors are defined as E(i) (i) ˆ etmn = Cm z − E(i) htmn =− (i) jnβm 2(i) kρ ρ (i) ˆ Sm ϕ (3.12) jωε (i) Cm ϕ ˆ (i) kρ for the mnth TM mode, and H(i) etmn = H(i) htmn = jωµ (i) kρ (i) Sm (i) Sm ϕ ˆ z+ ˆ (i) jnβm 2(i) kρ ρ (3.13) (i) ˆ Cm ϕ (i) (i) (i) (i) for the mnth TE mode The terms Cm and Sm are defined as Cm = cos βm (z − zi) (i) (i) and Sm = sin βm (z − zi ) , respectively, where z ∈ [zi, zi + hi ]; and hi is the height (i) (i) (i) (i) of the waveguide Symbols Jmn , Jmn , Hmn and Hmn represent the following Bessel and Hankel functions: (i) (i) Jmn = Jn km ρ , (i) Hmn = (2) Hn 2(i) 2(i) 2(i) where km = kρ = k − βm (i) km ρ (i) (i) Jmn = Jn km ρ , , (i) Hmn = (2) Hn (i) km ρ (3.14) , Chapter Modeling for Power-Ground Planes with Multiple Vias 3.3 46 Multiple Scattering Coefficients among Cylindrical PEC and PMC Vias The boundary condition for the perfect magnetic conductor (PMC) is given as n × ˆ H = The total magnetic field on the surface of q th PMC cylinder with radius rq in the ith parallel-plate layer is given by ∞ ∞ n=−∞ (i) m=0 (2) aE(i)Jn (kρ rq ) + bE(i)Hn (kρ rq ) mn mn Ht (rq , φ, z) = ∞ aH(i)Jn mn + (kρ rq ) + (2) bH(i)Hn mn (kρ rq ) m=1 −jωε Cm ϕ ˆ kρ jnβm C m ϕ + Sm z ˆ ˆ kρ rq e jnφ (3.15) =0 for any value of z ∈ [ 0, d ] Then, H(i) E(i) H(i) (3.16) E(i) H(i) E(i) (3.17) bmn(q) = Tmn(q) amn(q) bmn(q) = Tmn(q) amn(q) where Jn (kρ rq ) E(i) Tmn(q) = − Jn (kρ rq ) H(i) Tmn(q) = − (2) Hn (kρ rq ) (3.18) (2) Hn (kρ rq ) =− Jn+1 (kρ rq ) − Jn−1 (kρ rq ) (2) (2) Hn+1 (kρ rq ) − Hn−1 (kρ rq ) (3.19) k − βm The equations can be written in matrix form as with kρ = km = ⎡ ⎤ E(i) ⎢ bmn(q) ⎥ ⎣ H(i) bmn(q) ⎦ ⎡ ⎤⎡ H(i) 0 Tmn(q) ⎢ Tmn(q) =⎣ E(i) E(i) ⎤ ⎥ ⎢ amn(q) ⎥ ⎦⎣ H(i) amn(q) ⎦ (3.20) The boundary condition for the perfect electric conductor (PEC) is given as n × E = The total electric field on the surface of the q th PEC cylinder with a ˆ radius rq in the ith parallel-plate layer is given by ∞ ∞ n=−∞ m=0 (i) Et (rq , φ, z) = ∞ + aH(i)Jn mn (2) aE(i)Jn (kρ rq ) + bE(i)Hn (kρ rq ) mn mn (kρ rq ) + (2) bH(i)Hn mn m=1 −jnβm Sm ϕ + C m z ˆ ˆ kρ rq jωµ (kρ rq ) Sm ϕ ejnφ = ˆ kρ (3.21) for any value of z ∈ [ 0, d ] Then, E(i) E(i) E(i) bmn(q) = Tmn(q) amn(q) (3.22) Chapter Modeling for Power-Ground Planes with Multiple Vias H(i) H(i) 47 H(i) bmn(q) = Tmn(q) amn(q) (3.23) The equations can be also written in matrix form as ⎡ ⎢ ⎣ E(i) bmn(q) H(i) bmn(q) ⎤ ⎡ ⎥ ⎦ =⎣ ⎢ ⎤⎡ E(i) Tmn(q) 0 H(i) Tmn(q) ⎥⎢ ⎦⎣ E(i) amn(q) H(i) amn(q) ⎤ ⎥ ⎦ (3.24) For the scattering analysis from the PMC and PEC cylinders, the different zdirection modes (related to different index m) are decoupled, and different φ-direction modes (related to different index n) are decoupled, and then, the TM (E-) and TE (H-) modes for the cases of PMC and PEC cylinders are considered as decoupled The following short discussion proves that TE and TM modes generated by PEC cylinder are decoupled in the parallel-plate waveguide The boundary condition at the surface of the PEC cylinder is : Et|ρ=a = 0, i.e., ∞ E(i) H(i) (i) (i) (i) (i) aE(i)Jmn + bE(i)Hmn et,mn + aH(i)Jmn + bH(i)Hmn et,mn = mn mn mn mn (3.25) m=0 Substituting the corresponding equations in (3.12) and (3.13) into (3.25), we have ∞ (i) cos βm (z − zi ) z − ˆ (i) (i) aE(i)Jmn + bE(i)Hmn mn mn m=0 (i) (i) + aH(i)Jmn + bH(i)Hmn mn mn jωµ (i) km (i) jnβm (i) (km )2 ρ (i) sin βm (z − zi ) ϕ ˆ (i) sin βm (z − zi) ϕ = ˆ (3.26) Grouping all the terms in (3.26) w.r.t z and ϕ components, we get ˆ ˆ ∞ m=0 ∞ (i) (i) (i) aE(i)Jmn + bE(i)Hmn cos βm (z − zi ) z + ˆ mn mn (i) (i) aE(i)Jmn + bE(i)Hmn mn mn m=0 (i) (i) aH(i)Jmn + bH(i)Hmn mn mn jωµ (i) km (i) −jnβm (i) (km )2 ρ + (3.27) (i) sin βm (z − zi ) ϕ = ˆ (i) (i) The sine and cosine functions, sin βm (z − zi ) and cos βm (z − zi ) in (3.27), are not always zero, so we have (i) (i) aE(i)Jmn + bE(i)Hmn = mn mn (3.28) and (i) (i) aE(i)Jmn + bE(i)Hmn mn mn (i) −jnβm (i) km ρ (i) (i) + aH(i)Jmn + bH(i)Hmn mn mn jωµ (i) km = (3.29) Chapter Modeling for Power-Ground Planes with Multiple Vias 48 Because of (3.28), the expansion coefficients in (3.29) for TE and TM modes become independent, i.e., TE and TM modes for the PEC cylinders are totally decoupled; and different modes n are also decoupled Finally, we have (i) (i) aE(i)Jmn + bE(i)Hmn = mn mn (3.30) (i) (i) aH(i)Jmn + bH(i)Hmn = , mn mn (3.31) or bE(i) = − mn (i) Jmn (i) Hmn aE(i), mn bH(i) = − mn (i) Jmn (i) Hmn aH(i) mn (3.32) Hence, the wave scattering in TE and TM modes can be considered separately Consider a set of randomly distributed cylindrical vias as shown in Fig 3.2, where the vias can have different radii and may be present in different layers of an electronic package Figure 3.2: A set of random cylindrical vias (2D view) By taking into account the multiple scattering among Nc cylindrical vias, the scattered field at an observation point p can be expressed as E scat (ρ) = Nc M q Nq (2) bqmn Hn (kρ ρq )ejnφq cos(βmz) (3.33) q=1 m=0 n=−Nq where (ρq , φq ) are the local coordinates with ρq = |ρ − ρq | and φq = arg(ρ − ρq ) Mq + represents the truncation number of modes in the parallel-plate waveguide structure, and 2Nq + is that of the Hankel functions used to express the scattered waves of the q th via bqmn denotes the unknown expansion coefficients for the scattered field Chapter Modeling for Power-Ground Planes with Multiple Vias 49 Figure 3.3: A schematic of cylindrical coordinates for translational addition theorem The addition theorem of the Bessel functions for the translation of cylindrical coordinates from cylinder p to cylinder q is given as (2) Hm (kρ ρp ) ejmφp = ∞ Hn−m (kρ dqp ) e−j(n−m)θqp Jn (kρ ρq ) ejnφq (2) (3.34) n=−∞ where ρq < dqp ; [ρp , ρq , φp, φq , dqp , θqp ] ∈ Real; kρ ∈ Complex; kρ = 0, and the terms here are expressed in the global coordinate system The detailed expressions are given in Appendix A According to (3.5), we define the following incoming and outgoing modes for TM case as follows (a)E Ezmn = Jn (kρ ρ) Cm ejnφ (3.35) (b)E (2) Ezmn = Hn (kρ ρ) Cm ejnφ (3.36) Substituting (3.35) and (3.36) into (3.7), we get the tangential modes for TM case ∂ (a)E ∇sEzmn kρ ∂z (3.37) −jωε (a)E z × ∇s Ezmn ˆ kρ (3.38) ∂ (b)E ∇sEzmn kρ ∂z (3.39) −jωε (b)E z × ∇s Ezmn ˆ kρ (3.40) E(a)E = smn H(a)E = smn for incoming wave; and E(b)E = smn H(b)E = smn for outgoing wave Chapter Modeling for Power-Ground Planes with Multiple Vias 50 According to (3.6), we define the following incoming and outgoing modes for TE case (a)H Hzmn = Jn (kρ ρ) Sm ejnφ (3.41) (b)H (2) Hzmn = Hn (kρ ρ) Sm ejnφ (3.42) Substituting (3.41) and (3.42) into (3.7), we get the tangential modes for TE case E(a)H = smn jà (a)H z ì s Hzmn kρ (3.43) ∂ (a)H ∇s Hzmn kρ ∂z (3.44) jà (b)H z ì s Hzmn k (3.45) ∂ (b)H ∇s Hzmn kρ ∂z (3.46) H(a)H = smn for incoming wave; and E(b)H = smn H(b)H = smn for outgoing wave The outgoing T M wave from the pth cylinder can be written as (b)E Ezm(p) Np (b)E bE p (p)Ezmnp (p) mn = np =−Np Nq ⎡ ⎤ Hnq −np (kρ dqp ) e−j(nq −np )θqp bE p (p)⎦ Jnq (kρ ρq ) Cm ejnq φq mn np =−Np (3.47) ⎣ = Np nq =−Nq Nq (2) (a)E aE q (q) Ezmnq (q) mn = nq =−Nq (a)E = Ezm(q) Since ρq ∈ q th cylinder’s boundary, so ρq < dqp The incoming wave coefficient for the q th cylinder is then given as Np Hnq −np (kρ dqp ) e−j(nq −np )θqp bE p (p) , mn (2) aE q (q) = mn (3.48) np =−Np and aE q (q) is independent of the terms ρp , ρq , φp , and φq mn In different coordinates, the value of ∇s should not change, which means for any function f(ρ), ∇(p)f (ρ) = ∇(q)f (ρ) s s (3.49) Chapter Modeling for Power-Ground Planes with Multiple Vias 77 Figure 3.17: Implementation of the PMC cylinders placed at periphery of an electronic package in the boundary modeling method (2D view) Figure 3.18: Comparison of the extended SMM results with fixed and dynamic radii of the PMC cylinders in the FDCL and the reference solution Chapter Modeling for Power-Ground Planes with Multiple Vias 78 Figure 3.19: Effects of the different values of ζ on the accuracy of the simulation results by the FDCL boundary modeling method Chapter Modeling for Power-Ground Planes with Multiple Vias 3.8.2 79 Experimental Validations of the Extended SMM Algorithm For validation of the extended SMM with proposed FDCL, we fabricate several test boards as shown in Figs 3.20 and 3.24 The first test board has dimension of 156 × 106 mm, thickness of 1.2 mm, relative dielectric constant of 4.1, and loss tangent of 0.015 The locations of the ports are given as Port (46, 26), Port (122, 53), and Port (46, 76); all dimensions are in mm, as shown in Fig 3.20 The S-parameters between the ports are simulated using the SMM algorithm with the FDCL approach and the results are compared with measurement data in Figs 3.21, 3.22 and 3.23 The simulated results of the proposed method are agreed well with the measurement data for the S-parameters The resonant frequency points are matched very well but the magnitudes of the simulation are slightly higher than ones of the measurement The reason is that the dielectric constant of the substrate shows some frequency dependence and the loss tangent becomes larger than what it is assumed in the simulation Since the dielectric loss in the dielectric substrate layer is related to the loss tangent, the magnitude variation in the S-parameters was expected Assuming a fixed dielectric constant εr of 4.1 with loss tangent tan δ = 0.015 in our simulation, the slight magnitude variation in the high frequency band should be found in comparison of the results Other examples of test printed circuit boards (PCBs) are considered for the analysis of power-ground vias between the planes to reduce the coupling effect between the signal vias or consider the low transfer impedance in the package for stable power distribution over frequency band The test PCBs for the analysis are given in Fig 3.24 The test PCB1 has two signal vias (P1 and P2) only, and no power-ground (shorting) vias as shown in Fig 3.24(a) In the test PCB2 and PCB3, the shorting vias are placed with different configurations in order to reduce the coupling effect between the two signal vias as shown in Figs 3.24(b) and 3.24(c) The features of the boards are dimension of 75 × 50 mm, thickness of 1.1 mm, relative dielectric constant of 4.1, and loss tangent of 0.015 The port locations P1 and P2 are shown Chapter Modeling for Power-Ground Planes with Multiple Vias 80 Figure 3.20: Test printed circuit board (PCB) for measurement Figure 3.21: The scattering (S11) parameter for Port of the test board in Fig 3.20 The simulated result by the SMM with FDCL method is compared against the measurement data Chapter Modeling for Power-Ground Planes with Multiple Vias 81 Figure 3.22: The scattering (S21) parameter for Ports and of the test board in Fig 3.20 The simulated result by the SMM with FDCL method is compared against the measurement data Figure 3.23: The scattering (S22) parameter for Port of the test board in Fig 3.20 The simulated result by the SMM with FDCL method is compared against the measurement data Chapter Modeling for Power-Ground Planes with Multiple Vias 82 as in the figures The simulated results for the S-parameters are presented with the measurement data The S11 at Port and the S21 between Ports and for both boards are compared with measured data in Figs 3.25 and 3.26, respectively The results are shown good agreement The top and bottom conductor planes and the PMC boundary at the periphery of the test board form a cavity, whose resonant frequencies are accurately captured by the FDCL modeling method The slight difference in the magnitude of the S parameters between the simulation and the measurement is probably due to the variation of the loss tangent over frequencies Experimental Setup for S-Parameter Measurements Figure 3.27 shows the experimental setup for S-parameter measurement of the test PCBs in Figs 3.20 and 3.24 We use a vector network analyzer HP 8510C and S-parameter test set HP from Agilent to measure the S-parameters of the test vehicles in a frequency range from 45 MHz to 50 GHz with 2.4 mm coaxial cable Chapter Modeling for Power-Ground Planes with Multiple Vias 83 (a) PCB1 (b) PCB2 (c) PCB3 Figure 3.24: Test printed circuit boards (PCBs) for analysis of the coupling effect between the signal vias Chapter Modeling for Power-Ground Planes with Multiple Vias 84 (a) PCB1 and PCB2 (b) PCB1 and PCB3 Figure 3.25: Comparison of the S11 parameters between the simulated results and measurement data for the test boards in Fig 3.24 Chapter Modeling for Power-Ground Planes with Multiple Vias 85 (a) PCB1 and PCB2 (b) PCB1 and PCB3 Figure 3.26: Comparison of the S21 parameters between the simulated results and measurement data for the test boards in Fig 3.24 Chapter Modeling for Power-Ground Planes with Multiple Vias 86 Figure 3.27: Experimental setup using Agilent HP 8510C Vector Network Analyzer and HP 8517B S-parameter Test Set to measure the test vehicles in Figs 3.20 and 3.24 Chapter Modeling for Power-Ground Planes with Multiple Vias 3.8.3 87 Irregular-shaped Power-Ground Planes and Cut-out Structure One prominent feature of the FDCL modeling method is that it is versatile in handling arbitrarily shaped boundary of power-ground planes and cut-out structures in the planes We develop a special formation procedure of the PMC cylinders for the arbitrary-shaped periphery of the packages and plane cut-out structures According to the dimensions of arbitrarily shaped boundaries, the radii of the PMC cylinders located at the boundary’s corners of the packages may be slightly larger or smaller than the radius (calculated by (3.108)) for other PMC cylinders located between them in the special formation procedure The value of ζ is chosen within the optimal range of 0.4 to 0.5 to calculate the radius of the PMC cylinders except the ones at the boundary’s corners The radii of the corner PMC cylinders are slightly larger or smaller according to the lengths of the boundary Then, the number of modes for the cylindrical waves is defined based on the largest radius among the cylinders Figure 3.28: An irregular-shaped power-ground planes and cut-out structure (unit: mm) An L-shaped test board with cutout is used as an example to demonstrate the flexibility of the FDCL for modeling irregular shaped boundary The example, Chapter Modeling for Power-Ground Planes with Multiple Vias 88 take from [93], is shown in Fig 3.28 On the power plane there is a rectangular cutout whose diagonal points are (35, 25) and (50, 10); all dimensions are in mm, respectively The dielectric material between the planes has a thickness of mm, a relative permittivity of 2.65, and a loss tangent of 0.003 The source and observation SMA ports are located as shown in Fig 3.28 Some diagrams for the formation of the PMC cylinders in the FDCL, which will change in accordance with operating frequencies, are presented in Figs 3.29 and 3.30 Figure 3.29: PMC cylinders formation in the FDCL at operation frequency of GHz for the finite power-ground planes Figure 3.31 shows well agreement between the result of the SMM algorithm with FDCL and the measurement data [93] in a wide frequency band except first resonant frequency point have a slight shift Again, the difference may be caused by the frequency-dependent dielectric material characteristics However, the difference between the simulation result and the measurement data are within well acceptable margin of applications Chapter Modeling for Power-Ground Planes with Multiple Vias 89 Figure 3.30: PMC cylinders formation in the FDCL at operation frequency of GHz for the finite power-ground planes Figure 3.31: Comparison of the transfer impedance by the SMM simulation with the FDCL method against the measurement for the test board in Fig 3.28 Chapter Modeling for Power-Ground Planes with Multiple Vias 90 FDCL method for modeling of finite-domain power-ground planes The conventional scattering matrix method (SMM) using the model expansions of waves in parallel-plate waveguide is normally used to analyze the multiple scattering among the cylinders in the electronic package However, the power-ground planes of the package are assumed to be infinitely large, which cannot capture the resonant behavior of real-world packages In this research work, we have contributed the important extension of the scattering matrix method (SMM) using a novel factitious cylinders layer (FDCL) to simulate the finite-sized power-ground planes in advanced electronic packages In the FDCL approach, the radius of the PMC cylinders must be changed in accordance with operating frequencies Numerical examples showed that the FDCL approach used in the SMM provides accurate prediction of the resonant frequencies of the electronic packages By introducing the novel factitious layer of cylinders with dynamic radius at the periphery of an electronic package, the SMM is able to handle the real-world package structures The simulated results of the SMM with FDCL show well agreement with the measured data Finally, the proposed algorithm, SMM with FCDL, is also able to address the problem of multiple via coupling effects on the electrical performance for the entire electronic package in efficient and correct way 3.9 Summary The scattering matrix method (SMM) for the parallel-plate waveguide modes are developed to analyze electrical performance of the power-ground (P-G) planes due to the multiple scattering among the vias in electronic packages The scattering T -matrices for coupling among the power-ground vias and the outgoing wave coefficients for each via are computed to model the equivalent admittance (Y) matrix of the power-ground planes including the multiple scattering effects of the P-G vias with signal vias The integral equation method is used to extract the equivalent circuit model for the coupling between the external signal traces and the vertical vias The SPICE-like simulation is made using the extracted circuit model and the Chapter Modeling for Power-Ground Planes with Multiple Vias 91 Y matrix to analyze the signal response of the system The novel method in this chapter is also developed to transform the modal expansion with the SMM into a viable and efficient method for the analysis of multiple via coupling in finite-sized parallel-plate structures The frequency-dependent cylinder layer (FDCL) method is a simple yet powerful method for boundary modeling The radii of the cylinders in the FDCL vary according to the simulation frequency, which ensure the accuracy of the simulation results All the cylinders in the FDCL can have the same radius for each frequency, which makes the FDCL a uniform layer; they can also have slightly different radii at different regions along the boundary, which facilitate modeling sharp corners at the boundary and enhance the accuracy of the simulation results Numerical examples show that the proposed FDCL method used in the SMM provides accurate prediction of the resonant frequencies of electronic packages and then the extended SMM algorithm is able to analyze the real-world package structures The simulated results of the extended algorithm with the FDCL show well agreement with the measured data Unlike the absorbing boundary or radiation boundary used in other computational electromagnetic methods, the FDCL utilizes physical structures, i.e., cylinders with finite radii to enforce the boundary Although all the examples in this research employ the FDCL for the PMC boundary, the FDCL is not limited to that and it is straightforward to implement it for PEC boundary and other types of boundaries ... ⎢ H (2) (·) ⎢ ⎢ ⎢ ⎢ H (2) (·) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ (2) ⎢H ⎢ 2N −1 (·) ⎣ (2) H2N (·) (2) −H1 (·) (2) H2 (·) ··· (2) H0 (·) (2) −H1 (·) (2) H2 (·) (2) (2) ··· H1 (·) H2 (·) (2) (2) H2N (·)... H2N (·) (2) H2 (·) (2) −H1 (·) (2) H2 (·) ⎤ ⎥ ⎥ (2) −H2N −1(·) ⎥ ⎥ ··· (2) ··· H2N −1(·) (2) −H2N −1(·) 61 (2) H1 (·) ⎥ ⎥ ⎥ ⎥ ⎥(3.99) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ H0 (·) with the argument (·) for the Hankel... results and measurement data for the test boards in Fig 3 .24 Chapter Modeling for Power- Ground Planes with Multiple Vias 85 (a) PCB1 and PCB2 (b) PCB1 and PCB3 Figure 3 .26 : Comparison of the S21