Efficient modeling of power and signal integrity for semiconductors and advanced electronic package systems 4

Chapter Hybrid Modeling of Signal Traces in Power Distribution Network by Using Modal Decomposition Via structures in multilayered electronic packages are used to connect the signal or power supply traces residing in the different layers. Since parallel-plate waveguide modes are introduced by layered structures, the signals on active vias can excite the waveguide modes within the layers, and also can affect other vias including powerground (P-G) vias. The affected vias can, in turn, interfere with the original signals. Such coupling may even cause unreliable behavior or complete signal failure, along with signal integrity loss, inappropriate switching, and longer signal delay. Another potential problem in multilayered packages is that when harmonics of the transient signal coincide with the resonant frequency of the power-ground planes, it will cause electromagnetic compatibility (EMC) problems in the microprocessor packages. To minimize such noise behavior, pre-layout and post-layout verifications of the power distribution network (PDN) are necessary. Thus, accurate analysis of the signal traces in the PDN has become of vital important for optimizing the performance of high-speed digital circuits. 130 Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 131 In this chapter, an efficient modeling technique based on the modal decomposition of electromagnetic fields is proposed to analyze the power distribution network of an electronic package, which includes the signal traces and the multilayered power-ground planes with multiple through-hole vias. The total electromagnetic fields inside the package are decomposed into two modes: the parallel-plate mode and the transmission line mode. The propagation of the fields between the P-G planes is considered as parallel-plate mode and is efficiently analyzed by using the scattering matrix method, as presented in Chapters and 4. The latter, which comprises the stripline mode for the traces between the P-G planes and the microstrip line mode for the traces on top/bottom of the package, is modeled as admittance (Y) networks by using the multiconductor transmission line theory. The discontinuities of the signal traces at the through-hole vias are analyzed by using analytical modeling of equivalent circuits. Finally, by cascading the equivalent networks, the overall network parameter for the system is obtained to analyze the coupling effects of the P-G vias to the signal traces. 5.1 Methodology for Hybridization of SMM and Modal Decomposition A typical structure of the signal trace routed in a power distribution network of an electronic package is shown in Fig. 5.1. In multilayered structures, the metal power-ground planes and vias are to provide a low-impedance path for the power distribution system between the printed circuit board and the die. The signal traces reside between the different layers and their return currents flow in the reference planes just below them. When the traces pass through different layers, the vertical through-hole vias are placed to ensure the continuity of the return current. Consider that substrates sandwiched between the metal planes are very thin compared to operating wavelengths, and are usually uniform and isotropic; hence, the electromagnetic fields not change in the z-direction. Each pair of the metal Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 132 Figure 5.1: Signal trace route in power distribution network of an electronic package. planes serves as parallel-plate waveguide providing the transverse electromagnetic (TEM) mode. The signal traces routed in the PDN are modeled as multiconductor transmission line (MTL) and divided into two parts: the traces between the P-G planes as strip line mode and the traces on top/bottom of the package as microstrip line mode. It is assumed that the MTL has a uniform cross section, allowing the propagation of quasi-TEM waves along the traces [95]. Hence, the total electromagnetic fields propagating inside the package can be decomposed into two independent modes: the parallel-plate mode and the transmission line mode. Mode conversions usually occur at the transition between the signal trace and the through-hole via. At the via hole, the parallel-plate mode gets excited due to the switching signal currents, and conversely, the noise voltages between the P-G planes gets coupled to the stripline mode. A novel modal decomposition approach is applied at the discontinuities of all through-hole signal vias as mode transition ports. In Fig. 5.2, the entire domain of the problem is decomposed into three sub-domains: the parallel-plate planes with P-G vias; the microstrip lines and striplines; and the through-hole signal vias. The parallel-plate mode of the P-G planes with a large number of vias is analyzed by using the scattering matrix method (SMM), which is based on the N-body scattering theory, to model the equivalent Y network. During the research study, we develop the SMM to facilitate the modeling of coupling effects among densely populated vias in the multilayered package with finite power-ground planes. The transmission line mode of the microstrip lines and striplines can be Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 133 (a) (b) (c) Figure 5.2: Three sub-domains applied in the modal decoupling; (a) multilayered P-G planes, (b) signal traces, and (c) through signal vias. Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 134 analyzed by using the MTL theory [96], and extracted the equivalent circuit models. A simple and accurate analytical formula for the discontinuity of through-hole signal via is derived based on [97] to calculate the signal via’s parasitic capacitances and its equivalent LC Π-circuit including the via pad’s inductance and capacitance is modeled. Then, the equivalent circuit models of all sub-domains are recombined as cascading of the multi-ports networks at the transitions of signal traces to the through-hole signal vias. 5.2 Modeling of Power-Ground Planes with Multiple Vias Vias are widely 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 conducting cylinders (vias) in the presence of PEC (perfect electric conductor) [55] planes has been applied to model the coupling effects of the power-ground vias in a multilayered package [56, 57]. In this research study, instead of using the Green’s function approach in [56, 57] to obtain the corresponding formulas, we have directly applied the parallel-plate waveguide theory to resolve the problem and developed the semi-analytical scattering matrix method (SMM) for modeling of multiple scattering of the vias in the electronic package as we discussed in Chapter 3. The proposed method is reported in [98–101]. Since the P-G planes of the package are assumed infinitely large in the conventional SMM, an important extension to the SMM has been made to handle the finite-sized power-ground planes in advanced packages. We assume a PMC (perfect magnetic conductor) boundary on the periphery of a package. This assumption is made considering one of the major geometric features of the advanced package structures, i.e., the separation of the metal plates in the package is far less than its operating wavelength. Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 135 By adding the PMC boundary, we confine a problem domain to a finite region. A layer of the PMC cylinders is used at the periphery of the package to simulate the finite domain of the P-G planes. Hence, we extend the SMM algorithm with the boundary of the PMC layer cylinders and the algorithm is now able to handle real-world package structures. The detailed formulation and validation of the SMM for modeling of multiple scattering among the P-G vias in multilayered structure is presented in Chapters and 4. 5.3 Modeling of Multiconductor Signal Traces Consider a multiconductor transmission line (MTL) consisting of N conductors and reference conductor immersed in homogeneous medium [102]. The per-unit-length equivalent circuit model for derivation of multiconductor transmission line equations for the N + conductors is shown in Fig. 5.3. Figure 5.3: The per-unit-length equivalent circuit model for derivation of the transmission line equations. Writing Kirchhoff’s voltage law around the ith circuit consisting of the ith con- Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 136 ductor and the reference conductor yields [102] N Vi (z, t) = ri ∆zIi (z, t)+r0∆z N Ik (z, t)+∆z k=1 lik k=1 ∂Ik (z, t) +Vi (z + ∆z, t) . (5.1) ∂t Dividing both sides by ∆z and taking the limit as ∆z → 0, the first transmission line (MTL) equation for the ith conductor is given as N N ∂Ik (z, t) ∂Vi (z, t) = −ri ∆zIi (z, t) − r0∆z . Ik (z, t) − ∆z lik ∂z ∂t k=1 k=1 (5.2) With the collection for all conductors, it can be written in a compact form using matrix notations [R] and [L] as ∂ ∂ V (z, t) = −[R]I (z, t) − [L] I (z, t) . ∂z ∂t (5.3) Similarly, the second MTL equation can be obtained by applying Kirchhoff’s current law to the ith conductor in the per-unit-length equivalent circuit yields Ii (z + ∆z, t) − Ii (z, t) N = −∆z gik [Vi (z + ∆z, t) − Vk (z + ∆z, t)] − ∆z gii Vi (z + ∆z, t) k=1,k=i N −∆z cik k=1,k=i N ∂ ∂Vi (z + ∆z, t) [Vi (z + ∆z, t) − Vk (z + ∆z, t)] − ∆z cii (5.4) ∂t ∂t gik Vk (z + ∆z, t) − ∆z = ∆z k=1,k=i cik k=1,k=i gik Vi (z + ∆z, t) k=1 N +∆z n n ∂Vk (z + ∆z, t) ∂Vi (z + ∆z, t) cik − ∆z . ∂t ∂t k=1 Dividing both sides by ∆z and taking the limit as ∆z → 0, the first transmission line (MTL) equation for the ith conductor is given as N N ∂Ii (z, t) gik Vk (z + ∆z, t) − gik Vi (z + ∆z, t) = ∂z k=1,k=i k=1 N N ∂Vk (z + ∆z, t) ∂Vi (z + ∆z, t) − . + cik cik ∂t ∂t k=1,k=i k=1 (5.5) With the collection for all conductors, it can be written in a compact form using matrix notations [G] and [C] as ∂ ∂ I (z, t) = −[G]V (z, t) − [C] V (z, t) . ∂z ∂t (5.6) Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 137 Then, the per-unit-length parameter matrices of resistance [R], inductance [L], conductance [G], and capacitance [C] are given as follows: ⎛ ⎞ r0 ··· ⎜ r1 + r0 ⎜ . ⎜ r0 r2 + r0 . . ⎜ ⎜ [R] = ⎜ ⎜ . . . ⎜ ⎝ ··· r0 ⎛ [L] = ⎛ [G] = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ r0 r0 . ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ r0 (5.7) rN + r0 ⎞ l12 · · · l1N ⎟ ⎟ l22 · · · l2N ⎟ ⎟ ⎟ . . ⎟ . . . ⎟ l11 l12 . l1N l2N · · · lN N (5.8) ⎟ ⎠ ⎞ N ⎜ g1k ⎜ ⎜ k=1 ⎜ ⎜ ⎜ −g 12 ⎜ ⎜ ⎜ ⎜ ⎜ . ⎜ ⎜ ⎜ ⎝ −g 1N −g12 ··· −g1N ⎟ ⎟ N . −g2N . . g2k k=1 . −g2N ··· N gN k ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (5.9) k=1 ⎛ [C] = ⎞ N ⎜ c1k ⎜ ⎜ k=1 ⎜ ⎜ ⎜ −c ⎜ 12 ⎜ ⎜ ⎜ ⎜ . ⎜ ⎜ ⎜ ⎝ −c 1N −c12 ··· −c1N ⎟ ⎟ N . −c2N . . c2k k=1 . −c2N ··· N cN k k=1 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (5.10) Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 5.3.1 138 Properties of the Per-Unit-Length Parameters For the case of multiconductor transmission line (MTL) consisting of N + lossless conductors immersed in a homogeneous medium characterized by permeability µ, permittivity ε and conductivity σ, the per-unit-length parameter matrices are related by [L][C] = [C][L] = µε[U ] and (5.11) [L][G] = [G][L] = µσ[U ] , (5.12) where [U ] is the N × N identity matrix. For the case of MTL consisting N + low-loss conductors ([R][C] is the larger of ρ and ρ (see Fig. A.1). Figure A.1: Translation in the cylindrical coordinate system. By using the raising operator for the Bessel functions, the translational addition 185 Appendix A. Translational Addition Theorem in Cylindrical Coordinates 186 theorem can be generally established so that (2) Hm (kρ |ρ − ρ |) ejmφ = ⎧ ∞ ⎪ ⎪ ⎪ Jn−m (kρ ρ ) e−j(n−m)φ Hn(2) (kρ ρ) ejnφ , ⎪ ⎪ ⎨ n=−∞ ∞ ⎪ ⎪ (2) ⎪ ⎪ Hn−m (kρ ρ ) e−j(n−m)φ Jn (kρ ρ) ejnφ , ⎪ ⎩ ρ>ρ (A.2) ρ[...]... impedances, and modal voltages and currents representing the multiconductor signal traces Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 156 Figure 5.12: Overall equivalent network for the signal trace routed in the power distribution network Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 5.5 157 Numerical Simulations of Hybrid Modeling Algorithm for Signal. .. dimensions of a signal trace routed between the power- ground planes (top view and side view) (unit: mm) Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 159 (a) S11 (b) S21 Figure 5. 14: Reflection and transmission characteristics of the signal trace shown in Fig 5.13 Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 160 Figure 5.15: The dimensions of a signal. .. illustrated Finally, the combination of all equivalent networks for the entire signal trace is demonstrated The proposed combination method can be straightforwardly extended for the case Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 148 of the multiple striplines (signal traces) 5 .4. 1 Modeling of Striplines between Power- Ground Planes For the stripline commonly used in microwave... top/bottom microstrip lines and the power- ground planes combined with the stripline model, and the closed-form equivalent circuit of through-hole signal via Figure 5.5: Signal trace route in the power- ground planes of power distribution network In the following sections, the modeling of striplines combined with power- ground planes and the equivalent circuit of through-hole signal vias are illustrated... (a−b) e b (5.71) Combination of Equivalent Networks for Modeling of Entire Signal Trace As we have demonstrated the re-coupling procedure of stripline mode and parallelplate mode for the stripline and the power- ground planes and the equivalent circuit model of through-hole signal via in the previous sections, the entire equivalent network for a typical structure of the signal trace routed in the PDN... between Port 1 and 4) of the coupled signal traces are simulated and compared with the results of Ansoft HFSS simulations as shown in Figs 5.18, 5.19 and 5.20 Good agreements for all the results S11 , S21 and S41 are observed This demonstrates the accuracy of the proposed hybrid algorithm for simulation of the closely placed signal traces In such cases, both the transmission line mode and parallel-plate... example of two coupled signal traces (Case1) routed between the power- ground planes Four through-hole vias are used to connect the signal traces Several P-G vias are also placed between the planes to analyze the coupling effects between them and the traces The dielectric material for all substrates is FR4 with a dielectric constant of 4. 4 and loss Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution... (5.51) ⎤ 0 T −1 V 0 146 ⎥ ⎦ (5.52) For lossless and homogeneous transmission lines, [T V ] = [T I ] = [T ] and [T ] is orthogonal matrix, then ⎡ ⎢ Z net = ⎣ ⎤ T 0 ⎥ ⎢ ⎢ T ⎦ Z net ⎣ m ⎤ 0 T ⎡ Y net = ⎣ ⎡ T ⎡ 0 ⎥ 0 T ⎤ ⎦ ⎢ Y net ⎣ m t 0 T t 0 0 ⎥ Tt ⎦ (5.53) ⎤ 0 ⎥ Tt ⎦ (5. 54) Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 5 .4 147 Modeling of Entire Signal Traces in Power Distribution... and currents IL/R, IL/R defined in Figs 5.8 and 5.9 These four ports are considered for the connections between the stripline and the possible external signal traces Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 151 Figure 5.8: Port voltages and currents defined for three equivalent networks Figure 5.9: Combination for the equivalent Y-networks of the power- ground planes and. .. characteristic of the two coupled signal traces routed between the power- ground planes shown in Fig 5.17 Figure 5.20: Crosstalk characteristic of the two coupled signal traces routed between the power- ground planes shown in Fig 5.17 Chapter 5 Hybrid Modeling of Signal Traces in Power Distribution Network 165 analytical SMM simulation needs only few mode numbers for each of the PEC/PMC vias: PEC for the signal and . detailed formulation and validation of the SMM for modeling of multiple scattering among the P-G vias in multilayered structure is presented in Chapters 3 and 4. 5.3 Modeling of Multiconductor Signal. T ⎤ ⎥ ⎦  Y net m   ⎡ ⎢ ⎣ T t 0 0 T t ⎤ ⎥ ⎦ . (5. 54) Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 147 5 .4 Modeling of Entire Signal Traces in Power Distribution Network As we discussed. effects of the P-G vias to the signal traces. 5.1 Methodology for Hybridization of SMM and Modal Decomposition A typical structure of the signal trace routed in a power distribution network of an electronic