Efficient modeling of power and signal integrity for semiconductors and advanced electronic package systems 3

Chapter Modeling for Multilayered Power-Ground Planes in Power Distribution Network The scattering matrix method (SMM) with FDCL for analysis of multiple vias in the single layer package (a pair of power-ground planes) has been presented in the previous chapter. Using the several numerical examples, the developed algorithm is validated by comparing the simulated results with analytical solutions and measurement data. However, there are multiple layers (pairs of power-ground planes) in practical structure of power distribution network for an advanced electronic package. In this chapter, the formula derivation for multilayered structure of powerground planes in an advanced electronic package is presented. The procedure is illustrated using the modal expansions of parallel-plate waveguide (PPWG) and the mode matching in the anti-pad region of the via. Firstly, a case of two-layer structure of the power-ground planes is considered for formula derivation as shown in Fig. 4.1. It has a case of three PPWGs - PPWG I, II, and III. Later, the formulation of the multilayered power-ground planes is given for general case. Numerical simulations for the multilayered power-ground planes with vias are presented and validated with full-wave numerical method. 92 Chapter 4. Modeling for Multilayered Power-Ground Planes 93 Figure 4.1: A though-hole via in two-layer structure and forming three PPWGs. 4.1 Modal Expansions and Boundary Conditions As discussed in Chapter 3, the tangential fields w.r.t ρ inside the two-layer structure (Fig. 4.1) can be expressed by modal expansions as ⎧ ⎪ ⎨ (i) ae(i) mn Jmn ⎪ n=−∞ m=0 ⎩ (i) ah(i) mn Jmn ⎧ ⎪ ⎨ (i) ae(i) mn Jmn ⎪ n=−∞ m=0 ⎩ (i) ah(i) mn Jmn ∞ ∞ (i) Et = ∞ ∞ (i) Ht = h(i) e(i) et,mn + ⎫ ⎪ ⎬ h(i) et,mn ⎪ ⎭ + (i) be(i) mn Hmn + (i) bh(i) mn Hmn + (i) be(i) mn Hmn e(i) ht,mn + ⎫ ⎪ ⎬ + (i) bh(i) mn Hmn h(i) ht,mn ⎪ ⎭ ejnφ (4.1) ejnφ (4.2) h(i) where we have a0n = b0n = for TE mode, and e(i) (i) (z − zi ) zˆ − et,mn = cos βm e(i) ht,mn = − jωε (i) km h(i) h(i) jωµ (i) km (i) (km )2 ρ (i) sin βm (z − zi ) ϕ ˆ (i) cos βm (z − zi) ϕ ˆ (i) (z − zi ) zˆ + ht,mn = sin βm et,mn = (i) jnβm (i) jnβm (i) (km )2 ρ (4.3) (i) cos βm (z − zi ) ϕˆ (i) sin βm (z − zi) ϕˆ 2 + βm , k = ω µε = km (4.4) and βm = mπ . hi (4.5) Chapter 4. Modeling for Multilayered Power-Ground Planes 94 For the structure in Fig. 4.1, the following boundary conditions are applied EIII t (ρ, φ, z) ρ=b HIII t (ρ, φ, z) ρ=b = = ⎧ ⎪ ⎨ EIt (ρ, φ, z) ⎪ ⎩ EII t (ρ, φ, z) , z ∈ [h1 , h] , , z ∈ [0, h1 ] ρ=b ⎧ ⎪ ⎨ HIt (ρ, φ, z) ⎪ ⎩ HII t (ρ, φ, z) z ∈ [0, h1 ] , ρ=b ρ=b z ∈ [h1 , h] . , ρ=b (4.6) (4.7) Because of the decoupling of different modes n, we will only consider mode m in the following derivation. Substituting (4.1) and (4.2) into (4.6) and (4.7), respectively, we have ∞ ⎛ ⎝ m=0 = ⎞ e,III e,III III e,III (2) III ⎜ amn Jn (km b) + bmn Hn (km b) et,mn + ⎟ h,III III h,III (2) III ah,III mn Jn (km b) + bmn Hn (km b) et,mn ⎧ ⎛ ⎪ ⎪ ∞ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎨ m=0 ⎛ ⎪ ⎪ ∞ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎩ m=0 ⎠ ⎞ e,I I e,I (2) I ae,I mn Jn (km b) + bmn Hn (km b) et,mn + ⎟ II ae,II mn Jn (km b) + (2) II be,II mn Hn (km b) z ∈ [0, h1 ] ⎠, h,I I h,I (2) I ah,I mn Jn (km b) + bmn Hn (km b) et,mn ee,II t,mn + h,II II h,II (2) II ah,II mn Jn (km b) + bmn Hn (km b) et,mn (4.8) ⎞ ⎟ ⎠, z ∈ [h1 , h] . For convenience, we drop all the subscripts n in the following derivation and use the following notations: I I I = Jmn = Jn km b , Jm I I I and Hm = Hmn = Hn(2) km b . (4.9) So Eq. (4.8) can be written as ∞ m=0 = ⎛ ⎜ ⎝ III ae,III m Jm ah,III JmIII m ⎧ ⎛ ⎪ ∞ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎨ m=0 ⎛ ⎪ ⎪ ∞ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎩ m=0 + III be,III m Hm + ee,III t,m + bh,III HmIII m eh,III t,m ⎞ ⎟ ⎠ ⎞ e,I I e,I I ae,I m Jm + bm Hm et,m + ⎟ h,I I h,I I ah,I m Jm + bm Hm et,m II ae,II m Jm + II ah,II m Jm II be,II m Hm + II bh,II m Hm z ∈ [0, h1 ] ⎠, ee,II t,m + eh,II t,m ⎞ ⎟ ⎠, z ∈ [h1 , h] . (4.10) Chapter 4. Modeling for Multilayered Power-Ground Planes 95 Similarly, the following equation is obtained from (4.7) as ∞ ⎛ ⎝ m=0 = ⎞ e,III III e,III III he,III t,m + ⎟ ⎜ am Jm + bm Hm III III ah,III Jm + bh,III Hm hh,III t,m m m ⎧ ⎛ ⎪ ⎪ ∞ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎨ m=0 ⎛ ⎪ ⎪ ∞ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎩ m=0 ⎠ ⎞ e,I I e,I I ae,I m Jm + bm Hm ht,m + ⎟ h,I I h,I I ah,I m Jm + bm Hm ht,m II ae,II m Jm + he,II t,m + II be,II m Hm z ∈ [0, h1 ] ⎠, h,II II h,II II ah,II m Jm + bm Hm ht,m (4.11) ⎞ ⎟ ⎠, z ∈ [h1 , h] . Replacing the tangential unit vectors in (4.10) by those in (4.3) and (4.4), we obtain the L.H.S of (4.10) as ∞ LHS|Et = III III ae,III Jm + be,III Hm . m m m=0 III jnβm III ˆ + sin βm z ϕ III (km ) b jωµ III + bh,III HmIII III sin βm z ϕ ˆ m km III cos βm z zˆ − ah,III JmIII m ∞ III III III ae,III Jm + be,III Hm cos βm z zˆ + m m = m=0 ∞ III III ae,III Jm + be,III Hm m m m=0 ah,III JmIII + bh,III HmIII m m (4.12) III −jnβm + III )2 b (km jωµ III sin βm z ϕˆ . III km Similarly, we can obtain the L.H.S of (4.11) ∞ LHS|Ht = III ae,III + be,III HmIII m Jm m m=0 −jωε III cos βm z ϕ+ ˆ III km III III ah,III Jm + bh,III Hm . m m III sin βm z zˆ + ∞ = m=0 ∞ III jnβm III z ϕ ˆ cos βm III )2 b (km III III III ah,III Jm + bh,III Hm sin βm z zˆ + m m III III ah,III Jm + bh,III Hm m m m=0 III ae,III + be,III HmIII m Jm m III jnβm + III )2 b (km −jωε III cos βm z ϕ ˆ. III km (4.13) Chapter 4. Modeling for Multilayered Power-Ground Planes 96 The zˆ and ϕˆ components in (4.10) and (4.11) are separated as shown in the following equations ∞ III e,III III III ae,III Hm cos βm z m Jm + b m m=0 = ⎧ ∞ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ m=0 ∞ I e,I I I ae,I m Jm + bm Hm cos βm z , z ∈ [0, h1 ] II e,II II II ae,II m Jm + bm Hm cos βm z , z ∈ [h1 , h] , m=0 ⎛ ⎞ III III III −jnβm ae,III Jm + be,III Hm + ⎟ ∞ ⎜ m m III )2 b ⎜ ⎟ (km ⎜ ⎟ sin β III z ⎜ ⎟ m jωµ ⎠ III h,III III m=0 ⎝ ah,III J + b H m m m m III km ⎛ ⎞ ⎧ I −jnβm ⎪ ⎪ e,I I e,I I ⎪ + ⎟ ⎪ ∞ ⎜ am Jm + bm Hm ⎪ I )2 b ⎜ ⎟ ⎪ (k ⎪ ⎜ ⎟ sin β I z , m ⎪ ⎪ m ⎜ ⎟ ⎪ ⎪ jωµ ⎠ ⎪ h,I I h,I I m=0 ⎝ ⎪ a J + b H ⎪ m m m m ⎨ I km ⎛ ⎞ = II ⎪ −jnβm ⎪ e,II II e,II II ⎪ ⎪ + ⎟ ⎪ ∞ ⎜ am Jm + bm Hm ⎪ II )2 b ⎜ ⎟ ⎪ (k ⎪ m ⎜ ⎟ sin β II z , ⎪ ⎪ m ⎜ ⎟ ⎪ ⎪ jωµ ⎝ ⎠ ⎪ II h,II II ⎪ ah,II J + b H ⎩ m=0 m m m m II km ∞ (4.14) z ∈ [0, h1 ] (4.15) z ∈ [h1 , h] , III III III ah,III Jm + bh,III Hm sin βm z m m m=0 ⎧ ∞ ⎪ ⎪ ⎪ ⎪ ⎨ =⎪ ⎪ ⎪ ⎪ ⎩ m=0 ∞ I h,I I I ah,I m Jm + bm Hm sin βm z , z ∈ [0, h1 ] II h,II II II ah,II m Jm + bm Hm sin βm z , z ∈ [h1 , h], m=0 ⎛ ⎞ (4.16) III jnβm + + ⎟ ⎜ ∞ III )2 b ⎜ ⎟ (km ⎜ ⎟ cos β III z ⎜ ⎟ m −jωε ⎝ ⎠ m=0 III e,III III ae,III J + b H m m m m III km ⎛ ⎞ ⎧ I jnβm ⎪ h,I I h,I I ⎪ ⎪ am Jm + bm Hm + ⎟ ⎪ ∞ ⎜ ⎪ I )2 b ⎜ ⎟ ⎪ (km ⎪ ⎜ ⎟ cos β I z , ⎪ (4.17) z ∈ [0, h1 ] ⎪ m ⎜ ⎟ ⎪ ⎪ ⎠ ⎪ e,I I e,I I −jωε m=0 ⎝ ⎪ am Jm + bm Hm ⎪ ⎨ I km ⎛ ⎞ = II ⎪ jnβ ⎪ m h,II II h,II II ⎪ ⎪ + ⎟ ⎪ ∞ ⎜ am Jm + bm Hm ⎪ II )2 b ⎜ ⎟ ⎪ (km ⎪ ⎜ ⎟ cos β II z , z ∈ [h1 , h] . ⎪ ⎪ m ⎜ ⎟ ⎪ ⎪ −jωε ⎠ ⎪ ⎪ m=0 ⎝ ae,II J II + be,II H II ⎩ m m m m II km Equations (4.14), (4.15), (4.16), and (4.17) are correspondent to Ez , Eφ , Hz , and III ah,III Jm m Hφ , respectively. III bh,III Hm m Chapter 4. Modeling for Multilayered Power-Ground Planes 97 As referred to Section 3.3, for the PEC cylinder (ρ = a) in PPWG-III (Fig. 4.1), we have the relationship between the incoming and outgoing wave coefficients as be,III m =− III a Jn km (2) III a) Hn (km , ae,III m bh,III m =− III a Jn km ah,III m (2) III Hn (km a) . (4.18) Then, we designate the following notations ⎛ III e,III III ⎝J III − ae,III m Jm + bm Hm = m ⎛ III + be,III HmIII = ⎝JmIII − ae,III m Jm m ⎛ III III III Jm + bh,III Hm = ⎝Jm − ah,III m m ⎛ ah,III JmIII + bh,III HmIII = ⎝JmIII − m m III III a Hm Jn km (2) Hn III a) (km ⎞ ∆ ⎠ ae,III = m III a HmIII Jn km (2) Hn III a) (km III III a Hm Jn km (2) Hn III a) (km III a) (km (4.19) ⎞ ∆ ⎠ ae,III = III e,III Je,m am , m (4.20) ⎞ ∆ ⎠ ah,III = III a HmIII Jn km (2) Hn III e,III Je,m am , m III h,III Jh,m am , (4.21) ⎞ ∆ ⎠ ah,III = m III h,III Jh,m am , (4.22) III III where Jm and Hm are defined as those in (4.9). We can now rewrite (4.14) to (4.17) as follows: ∞ III ae,III Je,m m cos ⎧ ∞ ⎪ ⎪ ⎪ ⎪ ⎨ III βm z = m=0 m=0 ∞ ⎪ ⎪ ⎪ ⎪ ⎩ I e,I I I ae,I m Jm + bm Hm cos βm z , z ∈ [0, h1 ] II e,II II II ae,II m Jm + bm Hm cos βm z , z ∈ [h1 , h] m=0 ∞ m=0 = ∞ m=0 III III −jnβm ae,III Je,m m III )2 b (km ⎛ ⎧ ⎪ ⎪ ⎪ ⎪ ∞ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ m=0 ⎝ ⎪ ⎪ ⎨ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ m=0 ⎪ ⎩ III ah,III Jh,m m (4.23) III Jh,m + ah,III m jωµ III sin βm z III km ⎞ I −jnβm + + ⎟ I )2 b ⎟ (km ⎟ sin β I z , m ⎟ jωµ ⎠ I h,I I ah,I J + b H m m m m I km ⎞ II −jnβ m II e,II II ae,II + ⎟ m Jm + bm Hm II )2 b ⎟ (km ⎟ sin β II z , m ⎟ ⎠ h,II II h,II II jωµ am Jm + bm Hm II km I ae,I m Jm sin III βm z I be,I m Hm ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ = ⎪ ⎪ ⎪ ⎪ ⎩ ∞ m=0 ∞ z ∈ [0, h1 ] (4.24) z ∈ [h1 , h] I h,I I I ah,I m Jm + bm Hm sin βm z , z ∈ [0, h1 ] II h,II II II ah,II m Jm + bm Hm sin βm z , z ∈ [h1 , h] m=0 (4.25) Chapter 4. Modeling for Multilayered Power-Ground Planes ∞ m=0 = 4.2 III jnβm e,III III −jωε III cos βm z + a m Je,m III III k (km ) b m ⎞ I jnβ m I h,I I ah,I + ⎟ m Jm + bm Hm I )2 b ⎟ (km ⎟ cos β I z , ⎟ m −jωε ⎠ I e,I I ae,I J + b H m m m m I km ⎞ II jnβm h,II II h,II II + ⎟ am Jm + bm Hm II )2 b ⎟ (km ⎟ cos β II z , ⎟ m −jωε ⎠ II e,II II ae,II J + b H m m m m II km 98 III ah,III Jh,m m ⎛ ⎧ ⎪ ⎪ ⎪ ⎪ ∞ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ m=0 ⎪ ⎪ ⎨ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩ m=0 z ∈ [0, h1 ] (4.26) z ∈ [h1 , h] . Mode Matching in Parallel-plate Waveguides (PPWGs) In this section, we focus on derivation of the following generalized T matrix: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎡ ⎥ ⎥ e,II ⎥ b ⎥ ⎥ ⎥ bh,I ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ b e,I bh,II = ee TI,I ee TI,II eh TI,I ee TII,I ee TII,II eh TII,I he TI,I he TI,II hh TI,I eh TI,II ⎤⎡ ⎥⎢ ⎥⎢ ⎢ eh TII,II ⎥ ⎥⎢ ⎥⎢ ⎥⎢ hh ⎥ ⎢ TI,II ⎥⎢ ⎦⎣ he he hh hh TII,I TII,II TII,I TII,II a e,I ⎤ ⎥ ⎥ ⎥ a ⎥ ⎥ ⎥ ah,I ⎥ ⎥ ⎦ e,II (4.27) ah,II where the size of the matrices depends on the number of modes used for each PPWGs. The T matrix in (4.27) can be derived using the mode matching technique [94]. The orthogonality relations for the Fourier series used in the mode matching technique are given as ⎧ a a mπx mπx a nπx nπx ⎪ ⎪ ⎪ cos dx = sin dx = δnm , for m, n = cos sin ⎨ a a a a 0 (4.28) a ⎪ mπx nπx ⎪ ⎪ sin dx = . cos ⎩ a a For numerical calculation, we also truncate the infinite summation to a finite one and the numbers of modes are M1, M2, and M3 for PPWG- I, II, and III, respectively. For performing the mode matching, we can either test it over [0, h] (or [0, h1 ] and [h1, h]). Here we choose the testing functions as being those of PPWG-III to Chapter 4. Modeling for Multilayered Power-Ground Planes 99 enforcing Ez and Hz , and those of PPWG-I and II to enforcing Eφ and Hφ . By performing the testing by cos(βpIII z) on (4.23): h h (4.23) × cos βpIII z dz → pπ z dz → h (4.23) × cos M1 M2 h I e,I I I,III II e,II II II,III ae,I J + b H I + ae,II (4.29) = m m mp m Jm + bm Hm Imp m=0 m m m=0 III Je,p ae,III p where I,III = Imp II,III = Imp h1 I cos βm z cos βpIII z dz = h2 II cos βm z cos βpIII z dz = h1 cos mπ pπ z cos z dz h1 h (4.30) cos mπ pπ z cos z dz . h2 h (4.31) h2 By performing the testing on (4.24) over [0, h1 ] and [h1, h] by sin(βpI z) and sin(βpII z), respectively: M3 III ae,III m Je,m m=0 ⎛ ⎜ ⎝ M3 III −jnβm III jωµ III,I Jh,m Imq = + ah,III m III III km (km ) b −jnβqI kqI b I e,I I ae,I q Jq + bq Hq I h,I I + ah,I q Jq + bq Hq ⎞ jωµ ⎟ h1 ⎠ kqI III −jnβm h,III III jωµ III,II Jh,m III Imr = + a m III k (km ) b m II jωµ II −jnβr + be,II H JrII + bh,II HrII II + ah,II r r r r II kr (kr ) b (4.32) III ae,III m Je,m m=0 II ae,II r Jr h2 (4.33) where III,I Imq = III,II = Imr h1 h2 III sin βm z sin βqI z dz = III sin βm z sin βrII z dz = h1 sin mπ qπ z sin z dz h h1 (4.34) sin mπ rπ z sin z dz . h h2 (4.35) h2 By performing the testing by sin(βpIII z) on (4.25): h (25) × sin βpIII z dz → III Jh,p ah,III p h pπ z dz → h (25) × sin M1 M2 h I h,I I I,III II h,II II II,III = ah,I J + b H I + ah,II (4.36) m m mp m Jm + bm Hm Imp m=0 m m m=0 where I,III = Imp h1 I sin βm z sin βpIII z dz = h1 sin mπ pπ z sin z dz h1 h (4.37) Chapter 4. Modeling for Multilayered Power-Ground Planes h2 II,III Imp = II sin βm z sin βpIII z dz = h2 sin 100 mπ pπ z sin z dz . h2 h (4.38) By performing the testing on (4.26) over [0, h1 ] and [h1 , h] by cos(βpI z) and cos(βpII z), respectively. M3 III ah,III Jh,m m m=0 ⎛ ⎜ ⎝ III jnβm e,III III −jωε III,I Je,m III Imq = + am III km (km ) b I h,I I ah,I q Jq + bq Hq jnβqI kqI b I e,I I + ae,I q Jq + bq Hq ⎞ −jωε ⎟ h1 ⎠ kqI III jnβm e,III III −jωε III,II Imr = + a m Je,m III III k (km ) b m jnβrII II h,II II e,II II e,II II −jωε ah,II J + b H r r r r + ar Jr + br Hr II krII (kr ) b M3 (4.39) III ah,III Jh,m m m=0 h2 (4.40) where III,I = Imq III,II = Imr h1 h2 III cos βm z cos βqI z dz = III cos βm z cos βrII z dz = h1 cos mπ qπ z dz z cos h h1 (4.41) cos mπ rπ z cos z dz . h h2 (4.42) h2 We introduce the following notations to make the subsequent derivation concisely: III jωµ ∆ µ,III jnβm ∆ β,III = τm , = τm , III III km (km ) b Thus, we have (4.32), (4.33), (4.39) and (4.40): M3 jωε ∆ ε,III = τm . III km (4.43) β,III III µ,III III III,I ae,III (−τm )Je,m + ah,III τm Jh,m Imq = m m m=0 h1 (4.44) II e,II II ae,II (−τrβ,II ) + ah,II JrII + bh,II HrII τrµ,II r Jr + br Hr r r h2 (4.45) I e,I I β,I h,I I h,I I µ,I ae,I q Jq + bq Hq (−τq ) + aq Jq + bq Hq τq M3 β,III III µ,III III III,II ae,III )Je,m + ah,III τm Jh,m Imr = m (−τm m m=0 M3 β,III III ε,III III III,I ah,III τm Jh,m + ae,III (−τm )Je,m Imq = m m m=0 I h,I I β,I I e,I I ε,I ah,I + ae,I q Jq + bq Hq τq q Jq + bq Hq (−τq ) h1 (4.46) Chapter 4. Modeling for Multilayered Power-Ground Planes M3 101 β,III III ε,III III III,II ah,III τm Jh,m + ae,III (−τm )Je,m Imr = m m m=0 II e,II II ah,II JrII + bh,II HrII τrβ,II + ae,II (−τrε,II ) r r r Jr + br Hr h2 . (4.47) The unknown coefficients are derived by manipulating (4.29), (4.32), (4.33), (4.36), (4.39) and (4.40). Equations (4.29) and (4.36) can be rewritten as ⎛ ⎞ M1 ⎜ =⎜ ⎜ ae,III p m=0 M2 ⎝ I ae,I m Jm ⎜ =⎜ ⎜ ah,III p ⎝ M1 m=0 M2 m=0 I,III Imp + ⎟ ⎟ ⎟ ⎠ e,II II e,II II II,III a J +b H I m m=0 ⎛ + I be,I m Hm m m m mp III hJe,p (4.48) . III hJh,p (4.49) ⎞ I ah,I m Jm + II ah,II m Jm I bh,I m Hm + I,III Imp + II bh,II m Hm ⎟ ⎟ ⎟ ⎠ II,III Imp First changing the subscript m in (4.44) to p, then M3 III III ae,III (−τpβ,III )Je,p + ah,III τpµ,III Jh,p IpqIII,I = p p p=0 h1 . I e,I I β,I h,I I h,I I µ,I ae,I q Jq + bq Hq (−τq ) + aq Jq + bq Hq τq (4.50) Substituting (4.48) and (4.49) into (4.50), we obtain ⎧⎛ M1 ⎪ ⎪ ⎪⎜ M3 ⎨ ⎜ m=0 ⎜ ⎪ ⎝ M2 p=0 ⎪ ⎪ ⎩ ⎧⎛ ⎪ ⎪ M3 ⎪ ⎨⎜ ⎜ ⎜ ⎪ ⎝ p=0 ⎪ ⎪ ⎩ m=0 M1 m=0 M2 m=0 ⎫ ⎪ ⎪ ⎪ + ⎟ −2τ β,III I III,I ⎬ ⎟ p pq + ⎟ ⎪ ⎠ h ⎪ e,II II e,II II II,III ⎪ a J +b H I ⎭ ⎞ I ae,I m Jm m I be,I m Hm m I ah,I m Jm m + II ah,II m Jm I bh,I m Hm + I,III Imp + m mp I,III Imp + II bh,II m Hm ⎞ ⎟ ⎟ ⎟ ⎠ II,III Imp ⎫ ⎪ ⎪ III III,I ⎪ ⎬ I 2τpµ,III Jh,p III hJh,p I e,I I β,I h,I I h,I I µ,I ae,I q Jq + bq Hq (−τq ) + aq Jq + bq Hq τq pq h1 . ⎪ ⎪ ⎪ ⎭ (4.51) = Chapter 4. Modeling for Multilayered Power-Ground Planes I III I,III III,II −2τpµ,III Jm Jh,p Imp Ipr = , III Jh,p h p=0 M3 hI,2 Prm hI,3 Pqm r(m) = 1, · · · M2(M1) I I,III III,I −2τpβ,III Jm Imp Ipq τqβ,I JqI h1 + δmq , h M3 = p=0 hI,4 = Prm I I,III III,II −2τpβ,III Jm Imp Ipr , h p=0 M3 114 (4.136) q(m) = 1, · · · M1 (4.137) r(m) = 1, · · · M2(M1) (4.138) Column of T Matrix: T C4 = A −1 P hII (4.139) where T P hII hII,1 Pqm = hII,2 = Prm N ×M = hII,1 PM 1×M hII,2 PM 2×M hII,3 PM 1×M II III II,III III,I −2τpµ,III Jm Jh,p Imp Ipq , III Jh,p h p=0 M3 hII,4 PM 2×M (4.140) q(m) = 1, · · · M1(M2) (4.141) II III II,III III,II −2τpµ,III Jm Jh,p Imp Ipr τrµ,II JrII h2 , + δ mr III Jh,p h M3 p=0 r(m) = 1, · · · M2 (4.142) hII,3 Pqm = M3 II II,III III,I −2τpβ,III Jm Imp Ipq h p=0 M3 hII,4 Prm = p=0 q(m) = 1, · · · M1(M2) , II II,III III,II −2τpβ,III Jm Imp Ipr τ β,II JrII h2 , + δmr r h (4.143) r(m) = 1, · · · M2 (4.144) Summary of all the variables in the above equations: I(II)(III) = βm mπ h1(2)(3) β,I(II)(III) τm = I(II)(III) ; km I(II)(III) jnβm I(II)(III) km b µ,I(II)(III) ; τm = jωµ I(II)(III) km mπ I(II)(III) = k − βm = ω µε − h1(2)(3) ε,I(II)(III) ; τm = (4.145) jωε I(II)(III) km (4.146) I(II) I(II) I(II) I(II) I(II) I(II) = Jmn = Jn km b ; Hm = Hmn = Hn(2) km b Jm JmI(II) = I(II) Jmn ⎛ III III = ⎝Jm − Je,m = I(II) b ∂Jn km I(II) ∂ km b III III a Hm Jn km (2) III a) Hn (km ; ⎞ ⎠; HmI(II) = I(II) Hmn = ⎛ III Je,m = ⎝JmIII − (4.147) I(II) b ∂Hn(2) km I(II) ∂ km b III a HmIII Jn km (2) III a) Hn (km ⎞ ⎠ (4.148) Chapter 4. Modeling for Multilayered Power-Ground Planes ⎛ III III Jh,m = ⎝Jm − III III a Hm Jn km (2) Hn III a) (km ⎞ ⎠; ⎛ III Jh,m = ⎝JmIII − III a HmIII Jn km (2) 115 ⎞ ⎠ III a) Hn (km (4.149) I,III II,III Formulas to be used for computing the integrals such as Imp , Imp etc.: ⎧ ⎪ ⎪ ⎪ ⎨ I1 = sin [(a − b)z] sin [(a + b)z] + , for |a| = |b| 2(a − b) 2(a + b) cos(az) cos(bz)dz = ⎪ ⎪ z sin(2az) ⎪ ⎩ + , for |a| = |b| 4a ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ I2 = 4.5 sin(az) sin(bz)dz = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ sin [(a − b)z] sin [(a + b)z] − , for |a| = |b| 2(a − b) 2(a + b) z sin(2az) − , for |a| = |b| & ab ≥ 4a z sin(2az) , for |a| = |b| & ab ≤ − + 4a (4.150) (4.151) Formulas Summary for Multi-layer Problem In this section, we have summarized the formulae of generalized T matrix for multilayered structure in Fig. 4.2. The following formulas are consolidated for source-free via and source via comprised in multiple layer. Figure 4.2: A though-hole via in multi-layer structure and forming PPWGs. Chapter 4. Modeling for Multilayered Power-Ground Planes 116 For Source-free Via, Generalized T matrix for a multilayered structure: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ be,R ⎥ ⎥ ⎥ ⎥ bh,I ⎥ ⎥ ⎥ . ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ be,I . = bh,R = ee TI,I ··· . . ee eh TI,I TI,R . . eh · · · TI,R . . ee ··· TR,I ee eh TR,I TR,R eh · · · TR,R he TI,R . hh TI,I . hh · · · TI,R . . he hh TR,R TR,I hh · · · TR,R he TI,I . ··· . he TR,I ··· T e,C1 · · · T e,CR T h,C1 · · · T h,CR ⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ae,I . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ae,R ⎥ ⎥ ⎥ ⎥ ah,I ⎥ ⎥ ⎥ . ⎥ ⎥ ⎦ ah,R ae,I . ⎤ (4.152) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ e,R ⎥ a ⎥ ⎥ h,I ⎥ ⎥ a ⎥ ⎥ . ⎥ ⎥ ⎦ ah,R Columns of T Matrix: T e(h),Cκ = A −1 P e(h),κ (4.153) where Matrix [A]: ⎡ [A] ⎢ ⎢ ⎢ ⎢ ⎡ ⎤ ⎢ ⎢ ⎢ ee eh ⎢ A A ⎥ ⎢ =⎣ ⎦=⎢ ⎢ he hh ⎢ A A ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ee,I:I eh,I:I eh,I:R AM · · · Aee,I:R M1×MR AM1×M1 · · · AM1 ×MR ×M1 . . . . . . ee,R:R eh,R:I eh,R:R Aee,R:I MR ×M1 · · · AMR ×MR AMR ×M1 · · · AMR ×MR he,I:R hh,I:I hh,I:R Ahe,I:I M1 ×M1 · · · AM1×MR AM1×M1 · · · AM1 ×MR . . . . . . he,R:I he,R:R hh,R:R · · · AM Ahh,R:I AM MR ×M1 · · · AMR ×MR R ×M1 R ×MR and the size of Matrix [A] is R k=1 = Aee,Υ:Ψ qm MΠ p=0 = Aeh,Υ:Ψ qm MΠ p=0 Mk by R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.154) Mk . The elements are given as k=1 Ψ Ψ,Π Π,Υ −2τpβ,Π Hm Imp Ipq τqβ,ΥHqΥ hΥ + δΥΨ δmq , q(m) = 1, · · · MΥ (MΨ ) h (4.155) µ,Π Ψ Π Ψ,Π Π,Υ µ,Υ Υ 2τp Hm Jh,p Imp Ipq −τ Hq hΥ , q(m) = 1, · · · MΥ (MΨ ) + δΥΨ δmq q Π Jh,p h (4.156) Chapter 4. Modeling for Multilayered Power-Ground Planes Ahe,Υ:Ψ qm MΠ = p=0 Ahh,Υ:Ψ qm MΠ = p=0 117 Ψ Π Ψ,Π Π,Υ −2τpε,Π Hm Je,p Imp Ipq τqε,Υ HqΥ hΥ , q(m) = 1, · · · MΥ (MΨ ) + δΥΨ δmq Π h Je,p (4.157) β,Π Ψ Ψ,Π Π,Υ β,Υ Υ 2τp Hm Imp Ipq −τq Hq hΥ , q(m) = 1, · · · MΥ (MΨ ) . + δΥΨ δmq h (4.158) Matrix [P ]: P e,κ = P ee,κ P he,κ T = P ee,κ:1 · · · P ee,κ:R P he,κ:1 · · · P he,κ:R P h,κ = P eh,κ P hh,κ T = P eh,κ:1 · · · P eh,κ:R P hh,κ:1 · · · P hh,κ:R T T (4.159) (4.160) where κ κ,Π Π,Ω 2τpβ,Π Jm Imp Ipq −τqβ,κJqκ hκ + δκΩ δmq , q(m) = 1, · · · Mκ (MΩ ) h p=0 (4.161) MΠ ε,Π κ Π κ,Π Π,Ω ε,κ κ 2τ J J I I −τ J h κ p m e,p mp pq q q he,κ:Ω = , q(m) = 1, · · · Mκ (MΩ ) Pqm + δκΩ δmq Π Je,p h p=0 (4.162) MΠ µ,Π κ Π κ,Π Π,Ω µ,κ κ −2τ J J I I τ J h κ p m h,p mp pq q q eh,κ:Ω = , q(m) = 1, · · · Mκ (MΩ ) Pqm + δκΩ δmq Π Jh,p h p=0 (4.163) MΠ β,Π κ κ,Π Π,Ω β,κ κ −2τ J I I τ J h p m mp pq q q κ hh,κ:Ω + δκΩ δmq = , q(m) = 1, · · · Mκ (MΩ ) Pqm h p=0 (4.164) ee,κ:Ω = Pqm MΠ For Source Via, If the via is a source via, then we have = 0, (i = 1, · · · , R). Two-layer Problem: For the two-layer problem, we can rewrite (4.44)-(4.47) as two set of linear equations ⎡ ⎢ ⎢ ⎢ ⎣ HqI (−τqβ,I )h1 HqI (−τqε,I )h1 ⎡ M3 = ⎢ ⎢ ⎢ p=0 ⎢ M3 ⎢ ⎣ p=0 HqI τqµ,I h1 HqI τqβ,I h1 III ae,III (−τpβ,III )Je,p p ⎤ ⎤ ⎡ ⎥ be,I ⎥⎢ q ⎥ ⎥⎣ ⎦ ⎦ bh,I q ⎤ + III ah,III τpµ,III Jh,p p IpqIII,I ⎥ ⎥ ⎥ ⎥, ⎥ III,I ⎦ III III ae,III (−τpε,III )Je,p + ah,III τpβ,III Jh,p Ipq p p (4.165) Chapter 4. Modeling for Multilayered Power-Ground Planes ⎡ ⎢ ⎢ ⎢ ⎣ HrII (−τrβ,II )h2 II Hr (−τrε,II )h2 ⎡ M3 = ⎢ ⎢ p=0 ⎢ ⎢ ⎢ M3 ⎣ ⎤ ⎡ ⎥ ⎥⎢ ⎥⎣ ⎦ HrII τrµ,II h2 II β,II Hr τr h2 III ae,III (−τpβ,III )Je,p p + 118 ⎤ be,II r ⎥ bh,II r ⎦ ⎤ III ah,III τpµ,III Jh,p p IprIII,II ⎥ ⎥ ⎥ ⎥. ⎥ III,II ⎦ (4.166) III III ae,III (−τpε,III )Je,p + ah,III τpβ,III Jh,p Ipr p p p=0 We can solve for b in the above two matrix equations. Multilayer-layer Problem: We can find the reflection coefficients of the κth parallel-plate waveguide by solving the following linear system of equations ⎡ ⎢ ⎢ ⎢ ⎣ Hrκ (−τrβ,κ)hκ Hrκ (−τrε,κ )hκ ⎡ MΠ = ⎢ ⎢ ⎢ p=0 ⎢ ⎢ MΠ ⎢ ⎣ ⎤ ⎡ ⎤ ⎥ be,κ ⎥⎢ r ⎥ ⎥⎣ ⎦ ⎦ h,κ Hrκ τrµ,κ hκ Hrκ τrβ,κ hκ br ⎤ (4.167) ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ Π h,Π β,Π Π Π,κ ⎦ +a τ J I β,Π Π ae,Π p (−τp )Je,p + µ,Π Π ah,Π p τp Jh,p ε,Π ae,Π p (−τp )Je,p p p h,p IprΠ,κ r = 1, · · · Mκ . pr p=0 Recall that: we have the following formula to solve the linear system of equations ⎡ ⎢ ⎣ where ⎡ ⎤ ⎤⎡ a b1 ⎥ ⎢ x ⎥ ⎡ c2 b a b2 ⎣ y ⎤ ⎢ a b1 ⎥ ⎦ ⎦⎣ a b2 ⎢ c1 b ⎥ x=⎣ ⎤ ⎦ ⎡ ⎤ ⎢ c1 ⎥ =⎣ c2 ⎡ ⎦ ⎤ ⎢ a c1 ⎥ ⎦, y=⎣ a c2 ⎦ (4.168) ⎡ ⎤ ⎢ a b1 ⎥ ⎣ ⎦. a b2 (4.169) Reorganizing (4.167) into a generalized T matrix: ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Hrκ (−τrβ,κ)hκ κ Hr (−τrε,κ )hκ ⎡ Hrκ τrµ,κ hκ κ β,κ Hr τr hκ ⎤ ⎡ ⎥ ⎥⎢ ⎥⎣ ⎥ ⎦ ⎤ be,κ r ⎥ bh,κ r ⎦ ⎤ β,Π Π,κ β,Π Π,κ µ,Π Π Π,κ µ,Π Π Π,κ Π Π ⎢ (−τ0 )Je,0 I0r · · · (−τMΠ )Je,MΠ IMΠ r τ1 Jh,1 I1r · · · τMΠ Jh,MΠ IMΠ r ⎥ =⎣ ε,Π β,Π Π Π Π,κ Π Π Π,κ (−τ0ε,Π )Je,0 I0r · · · (−τM )Je,M I Π,κ τ1β,Π Jh,1 I1r · · · τM J I Π,κ Π Π MΠ r Π h,MΠ MΠ r e,Π h,Π h,Π . ae,Π · · · a MΠ a · · · a MΠ T , ⎦ where r = 1, · · · Mκ (4.170) Chapter 4. Modeling for Multilayered Power-Ground Planes ⎡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ H0κ (−τ0β,κ)hκ [H1 ] . [HMκ ] where ⎡ ⎢ ⎢ [Hr ] = ⎢ ⎢ ⎣ ⎡ ⎤ ⎤⎢ ⎢⎡ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎣ ⎥⎢ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢⎡ ⎦⎢ ⎢⎢ ⎢⎣ ⎣ be,κ be,κ bh,κ . be,κ Mκ bh,κ Mκ Hrκ (−τrβ,κ )hκ κ Hr (−τrε,κ )hκ 119 ⎤ ⎡ ⎤⎥ ⎥ ⎥ ⎥⎥ ⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤⎥ ⎥ ⎥ ⎥⎥ ⎦⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎡ ⎤⎢ ⎢ ⎢ e C ⎢ ⎥⎢ ⎣ ⎦⎢ ⎢ h ⎢ C ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ = Hrκ τrµ,κ hκ Hrκ τrβ,κhκ ae,Π ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ e,Π ⎥ a MΠ ⎥ ⎥ h,Π ⎥ a0 ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ h,Π ⎦ . (4.171) a MΠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (4.172) ⎡ ⎤ β,Π µ,Π Π Π Π Π Π,κ e (−τ0β,Π)Je,0 I0rΠ,κ · · · (−τM )Je,M I Π,κ τ1µ,Π Jh,1 I1r · · · τM J I Π,κ Π Π h,MΠ MΠ r ⎥ Π MΠ r ⎢C ⎥ ⎢ ⎣ ⎦=⎣ ⎦. ε,Π β,Π Π Π Π,κ Π Π Π,κ (−τ0ε,Π )Je,0 I0r · · · (−τM )Je,M I Π,κ τ1β,Π Jh,1 I1r · · · τM J I Π,κ Ch Π Π MΠ r Π h,MΠ MΠ r (4.173) As discussed in Chapter 3, we recall that the magnetic frill current source for the packaging problem will not excite H-mode around the source via. Hence, Eq. (4.170) is reduced to the following equation ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Hrκ (−τrβ,κ)hκ Hrκ (−τrε,κ )hκ ⎡ = ⎢ ⎢ ⎣ Hrκ τrµ,κ hκ Hrκ τrβ,κ hκ ⎤ ⎡ ⎥ ⎥⎢ ⎥⎣ ⎥ ⎦ ⎤ be,κ r bh,κ r ⎥ ⎦ ⎤ β,Π Π Π (−τ0β,Π)Je,0 I0rΠ,κ · · · (−τM )Je,M I Π,κ ⎢ ⎢ Π Π MΠ r ⎥ ⎢ ⎥ Π Π,κ (−τ0ε,Π )Je,0 I0r ε,Π Π · · · (−τM )Je,M I Π,κ Π Π MΠ r ⎤ ⎡ ⎦⎢ ⎢ ⎣ ae,Π . ae,Π MΠ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦ r = 1, · · · Mκ . (4.174) Then, we have ⎡ ⎢ ⎢ [Hr ] = ⎢ ⎢ ⎣ ⎡ ⎢ hκ ⎢ ⎢ = ⎢ ⎣ ⎤ ⎡ ⎤ Hrκ τrµ,κ hκ ⎥ κ β,κ κ µ,κ Hr (−τr ) Hr τr ⎥ ⎥ h ⎥= κ⎢ ⎣ ⎦ ⎥ κ β,κ Hr τr hκ ⎦ H κ (−τ ε,κ ) H κ τ β,κ r r r r ⎤ jnβrκ jnµ (2) κ (2) κ Hn (kr b) κ ⎥ −Hn (kr b) κ (kr ) b kr ⎥ κ ⎥ ⎥ jnε jnβ r ⎦ (2) κ (2) κ −Hn (kr b) κ Hn (kr b) κ kr (kr ) b Hrκ (−τrβ,κ)hκ Hrκ (−τrε,κ )hκ (4.175) Chapter 4. Modeling for Multilayered Power-Ground Planes 120 β,Π Π Π Π I0rΠ,κ · · · (−τpβ,Π)Je,p IprΠ,κ · · · (−τM )Je,M I Π,κ C e = (−τ0β,Π)Je,0 Π Π MΠ r (4.176) ε,Π Π Π,κ Π Π,κ Π I0r · · · (−τpε,Π )Je,p Ipr · · · (−τM )Je,M I Π,κ C h = (−τ0ε,Π )Je,0 Π Π MΠ r (4.177) ⎛ Cpe = Π (−τpβ,Π)Je,p IprΠ,κ ⎛ ⎞ Jn kpΠ a Hn(2) kpΠ b jnβpΠ Π ⎝ ⎠ I Π,κ = − Π Jn kp b − pr (2) Π (kp ) b Hn kp a ⎞ Π (2) Π Π (2) Π jnβpΠ Jn kp b Hn kp a − Jn kp a Hn kp b ⎠ I Π,κ ⎝ =− Π pr (2) Π (kp ) b Hn kp a ⎛ Π Π,κ Ipr = − Cph = (−τpε,Π )Je,p ⎛ =− jnε ⎝ kpΠ Jn jnε ⎝ Jn kpΠ b − Π kp kpΠ a Hn(2) Hn(2) kpΠ a Jn kpΠ b Hn(2) kpΠ a − Jn kpΠ a Hn(2) kpΠ b (2) Hn (4.178) ⎞ ⎠ I Π,κ pr ⎞ (4.179) ⎠ I Π,κ pπ rπ z sin z dz hΠ hκ hκ pπ rπ = cos z cos z dz . hΠ hκ IprΠ,κ = Π,κ Ipr hκ kpΠ a kpΠ b sin The integration can be computed by using (4.150) and (4.151). pr (4.180) (4.181) Chapter 4. Modeling for Multilayered Power-Ground Planes 4.6 121 Numerical Simulations for Multilayered Powerground Planes with Multiple Vias An example is given to demonstrate the combination of the modal expansion scattering matrix method (SMM) with the FDCL boundary modeling method and the generalized T matrix approach for the analysis of multiple via coupling in multilayered parallel-plate structures. The geometry of a multilayered parallel-plate structure for the example is shown in Fig. 4.3. It has three conductor power-ground planes. The relative permittivity of the substrate is 4.2 with a loss tangent of 0.02. The total of 100 vias are distributed as 64 vias in center block and 36 vias in four corner blocks as shown in Fig. 4.3. The simulated result of the input impedance seen from top end of the active via by our method agree quite well with the result by the Ansoft HFSS software, as shown in Fig. 4.4. In Table 4.1, the comparison of the memory usage and computing time is presented for the extended SMM algorithm with the FDCL and the Ansoft HFSS simulation. The simulation time of our SMM algorithm is much faster than one of the HFSS and the memory usage is also much lesser. Hence, the developed algorithm is very much efficient compared to the full-wave simulation tools and still provides the correct solution. Another example considered is to discuss a bottleneck of the conventional fullwave simulation methods. Figure 4.5 shows the geometry of three conductor powerground planes which has more vias, compared to Example 1. The signal via is also at the same location as in the previous example, and the total of 221 vias are distributed as 64 vias in center block and 156 vias in outer ring block as shown in Fig. 4.5. For this example, the HFSS simulation cannot be performed due to the memory insufficient while the SMM algorithm with FDCL can simulate with no difficulty. The simulated result for the input impedance is shown in Fig. 4.6. The comparison of the memory usage and computing time between the algorithm of the SMM algorithm with FDCL and the Ansoft HFSS simulation is presented in Table 4.3. Chapter 4. Modeling for Multilayered Power-Ground Planes 122 Figure 4.3: Example - a multilayered parallel-plate structure with three conductor power-ground planes and 101 vias (unit: mm). Chapter 4. Modeling for Multilayered Power-Ground Planes 123 k k Figure 4.4: Comparison of the input impedance seen from the top end of the active via in Example 1: SMM algorithm with FDCL vs. HFSS simulation. k k Table 4.1: Comparison of memory usage and computing time for Example Ansoft HFSS SMM with FDCL No. of vias 101 101 No. of unknowns 41467 tetrahedrons 404 modes Memory usage 420 MB 70 MB CPU time hr 21 20 14 sec *simulated on the machine of Intel Centrino 1.3 GHz, 512 MB. k Chapter 4. Modeling for Multilayered Power-Ground Planes 124 Figure 4.5: Example - a multilayered parallel-plate structure with three conductor power-ground planes and 221 vias (unit: mm). Chapter 4. Modeling for Multilayered Power-Ground Planes 125 Figure 4.6: Input impedance seen from the top end of the active via in Example 2. Table 4.2: Comparison of memory usage and computing time for Example Ansoft HFSS SMM with FDCL No. of vias 221 221 No. of unknowns 74111 tetrahedrons 884 modes Memory usage Insufficient 180 MB CPU time - hr 16 sec *simulated on the machine of Intel Centrino 1.3 GHz, 512 MB. Chapter 4. Modeling for Multilayered Power-Ground Planes 126 The example of two active vias in multilayered parallel-plate structure is also considered as shown in Fig. 4.7. It has six conductor power-ground planes. The relative permittivity of the substrate is 4.2 with a loss tangent of 0.02. The two active vias’ locations are given in the figure and the rest of 16 power-ground vias are located at (7.5, 11), (8.5, 10.5), (9, 10), (9, 11), (8, 12), (8, 10), (8.25, 11), (7, 11.5), (9.5, 10.5), (8.75, 11.75), (7.5, 9.5), (10, 11), (8.5, 12.5), (6, 11), (8.5, 9.5), and (11, 11.5); all dimensions are in mm. In Figs. 4.8, 4.9 and 4.10, the S-parameters simulation results by the SMM algorithm implemented for analysis of multilayered power-ground planes are plotted and compared with those from the HFSS simulation. 4.7 Summary A generalized T matrix model for the scattering matrix method is derived by the mode matching technique to analyze the vias penetrating more than two conductor planes. The generalized T matrix model obviates the use of multiple equivalent magnetic sources to model the plated-through vias. It facilitates modeling the coupling of multilayered vias. The modal expansion of the scattering matrix method (SMM) incorporating with the FDCL boundary modeling and the generalized T matrix approach is a powerful numerical method. The simulation time and memory usage are greatly reduced as compared to the full-wave methods and it still yields accurate simulation results. Chapter 4. Modeling for Multilayered Power-Ground Planes 127 Figure 4.7: Example - a multilayered parallel-plate structure with six conductor power-ground planes (unit: mm). Chapter 4. Modeling for Multilayered Power-Ground Planes 128 Figure 4.8: Comparison of the Z11 parameter simulated results for multilayered structure of Example 3: SMM algorithm with FDCL vs. HFSS simulation. Figure 4.9: Comparison of the Z21 parameter simulated results for multilayered structure of Example 3: SMM algorithm with FDCL vs. HFSS simulation. Chapter 4. Modeling for Multilayered Power-Ground Planes 129 Figure 4.10: Comparison of the Z22 parameter simulated results for multilayered structure of Example 3: SMM algorithm with FDCL vs. HFSS simulation. Table 4.3: Comparison of memory usage and computing time for Example Ansoft HFSS SMM with FDCL No. of vias 18 18 No. of planes No. of unknowns 22487 tetrahedrons 180 modes Memory usage 294 MB 52 MB CPU time hr 54 12 17 sec *simulated on the machine of Intel Centrino 1.3 GHz, 512 MB. [...]... M2(M2) (4.1 13) Row 3 of Matrix [A]: A31 = qm ε,III I III I,III III,I −2τp Hm Je,p Imp Ipq τqε,I HqI h1 , + δmq III Je,p h 2 M3 p=0 q(m) = 1, · · · M1 (4.114) A32 = qm A 33 = qm ε,III II III II,III III,I −2τp Hm Je,p Imp Ipq , III Je,p h p=0 M3 q(m) = 1, · · · M1(M2) β,III I I,III III,I I 2τp Hm Imp Ipq −τqβ,I Hq h1 , + δmq h 2 M3 p=0 A34 = qm β,III II II,III III,I 2τp Hm Imp Ipq , h p=0 M3 (4.115) q(m)... p=0 M3 eII,4 Prm = p=0 M3 q(m) = 1, · · · M1(M2) ε,III II III II,III III,II 2τp Jm Je,p Imp Ipr −τ ε,II JrII h2 , + δmr r III Je,p h 2 (4. 131 ) r(m) = 1, · · · M2 (4. 132 ) Column 3 of T Matrix: T C3 = A −1 P hI (4. 133 ) where P hI hI,1 Pqm = M3 p=0 T N ×M 1 = hI,1 hI,2 hI ,3 hI,4 PM 1×M 1 PM 2×M 1 PM 1×M 1 PM 2×M 1 µ,III I III I,III III,I −2τp Jm Jh,p Imp Ipq τ µ,I J I h1 , + δmq q q III Jh,p h 2 (4. 134 )... · · M1 (4. 135 ) Chapter 4 Modeling for Multilayered Power- Ground Planes µ,III I III I,III III,II −2τp Jm Jh,p Imp Ipr = , III Jh,p h p=0 M3 hI,2 Prm hI ,3 Pqm r(m) = 1, · · · M2(M1) β,III I I,III III,I I −2τp Jm Imp Ipq τqβ,I Jq h1 + δmq , h 2 M3 = p=0 hI,4 Prm = β,III I I,III III,II −2τp Jm Imp Ipr , h p=0 M3 114 (4. 136 ) q(m) = 1, · · · M1 (4. 137 ) r(m) = 1, · · · M2(M1) (4. 138 ) Column 4 of T Matrix:... The geometry of a multilayered parallel-plate structure for the example is shown in Fig 4 .3 It has three conductor power- ground planes The relative permittivity of the substrate is 4.2 with a loss tangent of 0.02 The total of 100 vias are distributed as 64 vias in center block and 36 vias in four corner blocks as shown in Fig 4 .3 The simulated result of the input impedance seen from top end of the active... machine of Intel Centrino 1 .3 GHz, 512 MB Chapter 4 Modeling for Multilayered Power- Ground Planes 126 The example of two active vias in multilayered parallel-plate structure is also considered as shown in Fig 4.7 It has six conductor power- ground planes The relative permittivity of the substrate is 4.2 with a loss tangent of 0.02 The two active vias’ locations are given in the figure and the rest of 16 power- ground... elements of the generalized T matrix in (4.60) one column by another For such case, we first let ae,I = 0 and ae,II = ah,I = ah,II = 0 in (4.56)(4.59) Then, we can project (4.56)-(4.59) into a linear system of equation: C1 [A] {b} = P eI ae,I , (4.61) or ⎡ ⎤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 31 32 33 34 AM 1×M 1 AM 1×M 2 AM 1×M 1 AM 1×M 2 ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎣ 11 12 13 14 ⎢ AM 1×M 1 AM 1×M 2 AM 1×M 1 AM 1×M 2 ⎢ ⎢ 21 22 23. .. entries in Row 3 of the matrices [A] and [P ] are given as follows: ε,III I III I,III III,I −2τp Hm Je,p Imp Ipq τqε,I HqI h1 , + δmq III Je,p h 2 M3 A31 = qm p=0 q(m) = 1, · · · M1 (4.74) A32 = qm A 33 qm ε,III II III II,III III,I −2τp Hm Je,p Imp Ipq , III Je,p h p=0 M3 = p=0 M3 β,III I I,III III,I I 2τp Hm Imp Ipq −τqβ,I Hq h1 , + δmq h 2 A34 = qm eI ,3 Pqm = M3 p=0 q(m) = 1, · · · M1(M2) β,III II... − b)z] sin [(a + b)z] − , for |a| = |b| 2(a − b) 2(a + b) z sin(2az) − , for |a| = |b| & ab ≥ 0 2 4a z sin(2az) , for |a| = |b| & ab ≤ 0 − + 2 4a (4.150) (4.151) Formulas Summary for Multi-layer Problem In this section, we have summarized the formulae of generalized T matrix for multilayered structure in Fig 4.2 The following formulas are consolidated for source-free via and source via comprised in... by using (4.150) and (4.151) pr (4.180) (4.181) Chapter 4 Modeling for Multilayered Power- Ground Planes 4.6 121 Numerical Simulations for Multilayered Powerground Planes with Multiple Vias An example is given to demonstrate the combination of the modal expansion scattering matrix method (SMM) with the FDCL boundary modeling method and the generalized T matrix approach for the analysis of multiple via... ⎡ = T C1 T C2 T C3 T C4 ⎤ a h,I ah,II ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ e,I ⎢ a ⎥ ⎢ ⎥ ⎢ e,II ⎥ ⎢ a ⎥ ⎢ ⎢ ⎢ ⎢ ⎣ a h,I ah,II (4.1 03) ⎥ ⎥ ⎥ ⎥ ⎦ Column 1 of T Matrix: T C1 = A where −1 P eI (4.104) ⎡ ⎤ [A] = ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 34 AM 1×M 2 ⎥ ⎥ ⎦ 11 12 13 14 ⎢ AM 1×M 1 AM 1×M 2 AM 1×M 1 AM 1×M 2 ⎢ ⎢ 21 ⎢ AM 2×M 1 A22 2×M 2 A 23 2×M 1 A24 2×M 2 M M M ⎢ ⎢ ⎣ A31 1×M 1 A32 1×M 2 A 33 1×M 1 M M M A41 2×M 1 A42 2×M 2 A 43 2×M 1 A44 2×M 2 . to Chapter 4. Modeling for Multilayered Power- Ground Planes 99 enforcing E z and H z , and those of PPWG-I and II to enforcing E φ and H φ .By performing the testing by cos(β III p z) on (4. 23) :  h 0 (4. 23) . solutions and mea- surement data. However, there are multiple layers (pairs of power- ground planes) in practical structure of power distribution network for an advanced electronic package. In. 4 Modeling for Multilayered Power- Ground Planes in Power Distribution Network The scattering matrix method (SMM) with FDCL for analysis of multiple vias in the single layer package (a pair of power- ground