Development and Verification of Multi-Level Sub- Meshing Techniques of PEEC to Model High- Speed Power and Ground Plane-Pairs of PFBS

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Rose-Hulman Institute of Technology Rose-Hulman Scholar Graduate Theses - Electrical and Computer Engineering Graduate Theses Spring 5-2015 Development and Verification of Multi-Level Sub- Meshing

1 Rose-Hulman Institute of Technology Rose-Hulman Scholar Graduate Theses - Electrical and Computer Engineering Graduate Theses Spring Development and Verification of Multi-Level Sub- Meshing Techniques of PEEC to Model High- Speed Power and Ground Plane-Pairs of PFBS Leihao Wei Rose-Hulman Institute of Technology, weil@rose-hulman.edu Follow this and additional works at: Part of the Electrical and Electronics Commons Recommended Citation Wei, Leihao, "Development and Verification of Multi-Level Sub-Meshing Techniques of PEEC to Model High-Speed Power and Ground Plane-Pairs of PFBS" (2015). Graduate Theses - Electrical and Computer Engineering. Paper 4. This Thesis is brought to you for free and open access by the Graduate Theses at Rose-Hulman Scholar. It has been accepted for inclusion in Graduate Theses - Electrical and Computer Engineering by an authorized administrator of Rose-Hulman Scholar. For more information, please contact bernier@rose-hulman.edu.

ii ROSE-HULMAN INSTITUTE OF TECHNOLOGY Final Examination Report Leihao Wei Electrical Engineering Name Graduate Major Thesis Title Development and Verification of Multi-Level Sub-Meshing Techniques

3 ii ROSE-HULMAN INSTITUTE OF TECHNOLOGY Final Examination Report Leihao Wei Electrical Engineering Name Graduate Major Thesis Title Development and Verification of Multi-Level Sub-Meshing Techniques of PEEC to Model High-Speed Power and Ground Plane-Pairs of PCBs DATE OF EXAM: April 24, 2015 EXAMINATION COMMITTEE: Thesis Advisor: Thesis Advisory Committee Edward Wheeler Jianjian Song Maarij Syed Department ECE ECE PHOE x PASSED FAILED

9 u Ω ϵ r =4.3 Ω

11 dc

16 1 1

17 x y R L

19 s I B = A A L loop = Ψ I = s B d s I = C A d l I = i C i A d l I C i A A L pi = C i I i A d l M pij = C j I i A d l

20 I i I j A A A A C j M pij I j I i C i A A A A Ψ C I i = = = C A d l I i C i A d l + A d l + A d l + A d l I i C i A d l I i = L pi

21 A A Aij C M pij = j d l I i L pi IL pi IM pij I(L pi M pij ) L loop = L pi M pij + L pj M pji L loop =2(L pi M pij )

22 Lp

23 N1 I y 1 N3 I x 1 I y 1 I x 2 y x N 2 I x1 I I y2 y 2 N4 I x 2 I s 4 1 L 44 = V 4 /si s

24 sl px11 sl px sl px21 sl px sl py11 sl py sl py21 sl py V N1 V N2 V N3 V N4 I x1 I x2 I y1 I y2 I sh = I s > Ω µ

25 z x k k k x x d r km z y x m m x d z x m x + Lp kk + + I m Lp mm + v V a b c d V I m V Lp k k Lp m m

26 V a V b = si m (Lp km Lp km + Lp k m Lp k m) Ls km = V a V b I m = Vs k si m =2(Lp km Lp km ) 1/Distance 2 Ls kk =2(Lp kk Lp kk ). r>>h k m h r k m q2 Ls km 0.1 r

27 w =0.211 Ls km = 1(Lp 4 k1m1 + Lp k1m2 + Lp k2m1 + Lp k2m2 ) Ls km = 1 n 2 n n i=1 j=1 Lp ki mj Ls km =2(Ls km Ls km ) =2 1 4 [(Lp k1m1 + Lp k1m2 + Lp k2m1 + Lp k2m2 ) (Lp k1m 1 + Lp k1m 2 + Lp k2m 1 + Lp k2m 2)] = 1 4 [2(Lp k1m1 Lp k1m 1)+2(Lp k1m2 Lp k1m 2) +2(Lp k2m1 Lp k2m 1)+2(Lp k2m2 Lp k2m 2)]

28 cell to cell coupling (ph) LpMutZeroT 1 filament 2 filaments section distance (mm)

29 a l l d =2a k = d/l l<d

30 5 contacts 3 contacts 2 contacts Lp 11 = µl 4 [( k k ) ( ) 1 π k2 π 24 +( 2 (l)+6 (2) (a) 4 (kπ)) 1 π + 8a ] lπ 3 [ ( Lp 11 = µl ) l 2π a + ( l ] a ) ( a l )2 + a l \ V g V p I g I p g p s L gg sl gp s L pg sl pp

31 z z 1 x e1 x x s1 xs 2 1 y 1 y 2 2 x e2 z 2 y Lp 12 = µ 4 4π k=1 ] a 2k + ρ2 ( 1) k+1 [a k ( ) a k + a 2k + ρ2 a 1 = x e2 x s1 a 2 = x s2 x s1 a 3 = x s2 x e1 a 4 = x e2 x e1 ρ = (y 2 y 1 ) 2 +(z 2 z 1 ) 2

32 Ltot 44 =(V4 Vp)/sI s V4 V1 A/4 Is A A/2 Vp Ip Lpg Ig I p = I s V 1 sl gg I g sl gp I p =0 V p sl pp I p sl pg I g =0

33 Cp = ϵa/d A d A A/4 A/2 A

34 Corner cap Full cap N2 Edge cap N3 Via Y X δ <T T σ δ = 1 πfµσ

35 R c 1D R c =2 x σ yδ. δ <T

36 sc + G sc + G sc + G sc + G slpx11 R slpx slpx21 slpx22 R slpy11 R slpy slpy21 slpy22 R 0 0 V1 V2 V3 V4 Vp Ix1 Ix2 Iy1 Iy2 Ig Ip = Is Is slgg slgp slpg slpp

37 h p i C = C d(v k V l ) dt v L = L di L dt + Ri L α dx p dt = k px p + k p 1 x p 1 + k p 2 x p 2 β dx p 1 dt k p k p 1 k p 2 α β 1/h p 1/h p 1/h p 1/h p 1.5/h p 2/h p 1/2h p

38 C n dv n dt + C n k p V (p) n + i n i n i b = Is i (p) b = C n k p 1 V (p 1) n + C n k p 2 V (p 2) n + Is (p) di m V l V k = L pm dt + R mi m + di k L pkm dt k m V (p) l V (p) k k p L pm i (p) m R m i (p) m k p L pkm i (p) k = L pm k p 1 i (p 1) m k p 2 k m L pkm i (p 2) k k m k p C A A T k p L R V (p) I (p) = Ck p 1 V (p 1) + Ck p 2 V (p 2) Lk p 1 I (p 1) Lk p 2 I (p 2) + I (p) s 0 h p

41 V G6 sc G I G6,G2 + I G6,S6 + I G6,S I G6,G5 =0 V S3 sc S I G6,G I G7,G3 + I S3,S7 + I S3,G7 + I S3,G6 =0 V G7 sc G7 +0.5I G7,G3 + I G7,S8 + I G7,S4 + I G7,S3 =0 V G8 sc G I G8,S1 + I G8,S10 + I G8,S I G8,S4 =0 V S1 sc S1 + I S1,G4 + I S1,G8 + I S1,S I G4,G I G8,S4 =0 2/3 I S1,G8 (G8,S10)

42 = Ls km Ls km(apprx) Ls km 100

43 10 0 error % section distance (mm)

44 normalized required distance n=0 n=1 n=2 n=3 n=4 n= section size (mm) 10 3 L n = Ls km Ls kk k m q2 =0.1 rls kk

45 normalized inductance section distance (mm) normalized required distance n=0 n=1 n=2 n=3 n=4 n= section size (mm)

46 > d

48 \ \

49 Source L Short Lp via = Lp via11 Lp via12

50 Inductance (ph) Lplane h=0.2mm Lplane h=1mm Lvia h=0.2mm Lvia h=1mm Spacing between vias (L)

52 r

53 Z 11

54 log(leq (ph)) 10 4 r=0.5'' PPP r=1'' PPP r=2'' PPP r=3'' PPP r=0.5'' PDN r=1'' PDN r=2'' PDN r=3'' PDN r=0.5'' CST r=1'' CST r=2'' CST r=3'' CST number of caps

55 ϵ r =4.3 δ =0.025 σ = S/m

56 Impedance (Ω) CST PPP frequency (MHz) s = jω d/dt t p h p hx p =1.5x p 2x p x p 2

57 v(t)/i(t) 10 8 PPP CST PPP CST v(t)/i(t) [Ω] v(t) [V] t [ns] t [ns] u Ω ϵ r =4.3 Ω

59 > uniform one level two level Time [sec] side length [mm]

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