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

Modeling for MultilayeredPower-Ground Planes in Power Distribution Network The scattering matrix method SMM with FDCL for analysis of multiple vias inthe single layer package a pair of p

Modeling for Multilayered

Power-Ground Planes in Power

Distribution Network

The scattering matrix method (SMM) with FDCL for analysis of multiple vias inthe single layer package (a pair of power-ground planes) has been presented in theprevious chapter Using the several numerical examples, the developed algorithm

is validated by comparing the simulated results with analytical solutions and surement data However, there are multiple layers (pairs of power-ground planes) inpractical structure of power distribution network for an advanced electronic package

mea-In this chapter, the formula derivation for multilayered structure of ground planes in an advanced electronic package is presented The procedure isillustrated using the modal expansions of parallel-plate waveguide (PPWG) and themode matching in the anti-pad region of the via Firstly, a case of two-layer struc-ture of the power-ground planes is considered for formula derivation as shown inFig 4.1 It has a case of three PPWGs - PPWG I, II, and III Later, the formula-tion of the multilayered power-ground planes is given for general case Numericalsimulations for the multilayered power-ground planes with vias are presented andvalidated with full-wave numerical method

power-92

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

For the structure in Fig 4.1, the following boundary conditions are applied

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,

For convenience, we drop all the subscripts n in the following derivation and use

the following notations:

Similarly, the following equation is obtained from (4.7) as

The ˆz and ˆ ϕ components in (4.10) and (4.11) are separated as shown in the following

jωµ

k III m

jωµ

k II m

 jnβ III m

−jωε

k III m

−jωε

k I m

 jnβ II m

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

Then, we designate the following notations

a e,III m J m III + b e,III m H m III =

∆

= J e,m III a e,III m , (4.19)

a e,III m J m
III + b e,III m H m
III =

H n(2)(k III

m a)

⎞

⎠a e,III m

∆

= J e,m
III a e,III m , (4.20)

a h,III m J m III + b h,III m H m III =

H n
(2)(k III

m a)

⎞

⎠a h,III m

H n
(2)(k III

m a)

⎞

⎠a h,III m

∆

= J h,m
III a h,III m , (4.22)

where J III

m and H III

m are defined as those in (4.9)

We can now rewrite (4.14) to (4.17) as follows:

jωµ

k I m

 jnβ I m

−jωε

k I m

−jωε

k II m

II,I T eh II,II

T he I,I T he I,II T hh

I,I T hh I,II

T he II,I T he II,II T hh

II,I T hh II,II

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

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

h2

2 (4.33)where

2 (4.40)where

∆

= τ m µ,III , jωε

k III m

∆

= τ m ε,III (4.43)Thus, we have (4.32), (4.33), (4.39) and (4.40):

Equations (4.29) and (4.36) can be rewritten as



I II,III mp

hJ III h,p

Similarly, we can eliminate a e,III



I
II,III mp

hJ III h,p

hJ III e,p



I II,III mp

⎞

⎟

⎟−2τ ε,III

p J
III e,p I III,II pr

hJ III e,p



I
II,III mp

Reorganizing all the terms in (4.51):

−2τ β,III

p J m I I mp I,III I pq
III,I h

e,II m

−2τ β,III

p J II

m I II,III

mp I
III,I pq

mp I
III,I pq

J III h,p h + a

h,I m

2τ µ,III

p J I

m J
III h,p I
I,III

mp I
III,I pq

J III h,p h

mp I
III,I pq

J III h,p h + a

h,II m

2τ µ,III

p J II

m J
III h,p I
II,III

mp I
III,I pq

J III h,p h

e,I q

mp I
III,I pq

J III h,p h + δ mq b

h,I q

mp I
III,I pq

J III h,p h + δ mq a

h,I q

mp I
III,I pq

J III h,p h .

(4.56)

Similarly, we have for (4.52)-(4.54):

e,II r

mp I
III,II pr

J III h,p h + δ mr b

h,II r

e,II r

mp I
III,II pr

J III h,p h +

h,II r

e,I q

mp I III,I pq

J III e,p h +

mp I III,I pq

J III e,p h + δ mq a

e,I q

mp I III,I pq

J III e,p h +

h,I q

mp I III,II pr

J III e,p h +

e,II r

h,II r

mp I III,II pr

J III e,p h +

mp I III,II pr

J III e,p h + δ mr a

e,II r

4.3 Generalized T Matrix for Two-layer Problem

As we have discussed the modal expansion and boundary conditions, and the modematching in the previous two sections, we are ready to formulate the following

I,I T eh I,II

T II,I ee T II,II ee T II,I eh T II,II eh

T he I,I T he I,II T hh

I,I T hh I,II

T he II,I T he II,II T hh

II,I T hh II,II

We will derive the elements of the generalized T matrix in (4.60) one column by

another For such case, we first let a e,I = 0 and a e,II = a h,I = a h,II = 0 in (4.59) Then, we can project (4.56)-(4.59) into a linear system of equation:

where N = 2(M 1 + M 2), The superscript C1 in (4.61) indicates that the entries in

the vector b are corresponding to the entries in the first column of the T matrix in

h , q(m) = 1, · · · M1(M2) (4.65)

mp I
III,I pq

J III h,p h + δ mq

mp I
III,I pq

J III h,p h , q(m) = 1, · · · M1(M2) (4.67)

mp I
III,II pr

J III h,p h + δ mr

mp I III,I pq

J III e,p h + δ mq

mp I III,I pq

J III e,p h , q(m) = 1, · · · M1(M2) (4.75)

mp I III,I pq

J III e,p h + δ mq

Those entries in Row 4 of the matrices [A] and [P ] are given as follows:

mp I III,II pr

J III e,p h + δ mr

mp I III,II pr

J III e,p h , r(m) = 1, · · · M2(M1) (4.83)

We can now obtain from (4.61):

where [A] −1 [P eI ] correspond to the first column of the T matrix in (4.60).

The derivation for the other three columns of the T matrix in (4.60) follows exactly the same procedure as that for Column 1 of the T matrix We can easily notice that the matrix [A] for deriving all the other three columns of the T matrix in (4.60) is identical to the one in (4.84) for deriving the first column of the T matrix

in (4.60) The only difference is the matrix [P ], so we only present here the entries

of the matrix [P ].

The Column 2 of the T matrix in (4.60) can be derived by setting a e,II = 0 and

a e,I = a h,I = a h,II = 0 in (4.56)-(4.59) The expression is as follows:

mp I III,II pr

J III e,p h + δ mr

Similarly, the Column 3 of the T matrix in (4.60) is derived by setting a h,I = 0 and

a e,I = a e,II = a h,II = 0 in (4.56)-(4.59) The formula relevant to the third column

mp I
III,I pq

J III h,p h + δ mq

h , r(m) = 1, · · · M2(M1) (4.96)

The Column 4 of the T matrix in (4.60) is derived by setting a h,II = 0 and a e,I =

a e,II = a h,I = 0 in (4.56)-(4.59) The formula relevant to the fourth column of the

where [A] −1 [P hII ] correspond to the fourth column of the T matrix in (4.60), and

mp I
III,I pq

J III h,p h , q(m) = 1, · · · M1(M2) (4.99)

4.4 Formulas Summary for Two-layer Problem

I,I T eh I,II

T II,I ee T II,II ee T II,I eh T II,II eh

T he I,I T he I,II T hh

I,I T hh I,II

T II,I he T II,II he T II,I hh T II,II hh

mp I
III,I pq

J III h,p h + δ mq

Row 2 of Matrix [A]:

h , r(m) = 1, · · · M2(M1) (4.110)

mp I
III,II pr

J III h,p h , r(m) = 1, · · · M2(M1) (4.112)

mp I
III,II pr

J III h,p h + δ mr

mp I III,I pq

J III e,p h + δ mq

mp I III,II pr

J III e,p h + δ mr

mp I III,I pq

J III e,p h + δ mq

mp I III,I pq

J III e,p h , q(m) = 1, · · · M1(M2) (4.131)

mp I III,II pr

J III e,p h + δ mr

mp I
III,I pq

J III h,p h + δ mq

mp I
III,II pr

J III h,p h , r(m) = 1, · · · M2(M1) (4.136)

mp I
III,II pr

J III h,p h + δ mr

H n(2)(k III

m a)

⎞

⎠ (4.148)

4.5 Formulas Summary for Multi-layer Problem

In this section, we have summarized the formulae of generalized T matrix for

multi-layered structure in Fig 4.2 The following formulas are consolidated for source-freevia and source via comprised in multiple layer

Figure 4.2: A though-hole via in multi-layer structure and forming PPWGs

For Source-free Via,

Generalized T matrix for a multilayered structure:

T ee I,R T eh I,I

.

· · · T eh I,R

.

T ee R,I · · ·

T I,I he · · ·

T ee R,R T eh R,I

T I,R he T I,I hh

· · · T eh R,R

· · · T hh I,R

For Source Via,

If the via is a source via, then we have a i = 0, (i = 1, · · · , R).

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

−H
(2)

n (k κ r b) jnε

k κ r

H n(2)(k κ r b) jnβ

κ r

4.6 Numerical Simulations for Multilayered

Power-ground Planes with Multiple Vias

An example is given to demonstrate the combination of the modal expansion tering matrix method (SMM) with the FDCL boundary modeling method and the

scat-generalized T matrix approach for the analysis of multiple via coupling in

mul-tilayered parallel-plate structures The geometry of a mulmul-tilayered parallel-platestructure 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 fourcorner blocks as shown in Fig 4.3 The simulated result of the input impedanceseen 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 rithm with the FDCL and the Ansoft HFSS simulation The simulation time of ourSMM algorithm is much faster than one of the HFSS and the memory usage is alsomuch lesser Hence, the developed algorithm is very much efficient compared to thefull-wave simulation tools and still provides the correct solution

algo-Another example considered is to discuss a bottleneck of the conventional wave simulation methods Figure 4.5 shows the geometry of three conductor power-ground planes which has more vias, compared to Example 1 The signal via isalso at the same location as in the previous example, and the total of 221 vias aredistributed as 64 vias in center block and 156 vias in outer ring block as shown

full-in Fig 4.5 For this example, the HFSS simulation cannot be performed due tothe 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 inTable 4.3

Figure 4.3: Example 1 - a multilayered parallel-plate structure with three conductorpower-ground planes and 101 vias (unit: mm).

*simulated on the machine of Intel Centrino 1.3 GHz, 512 MB.

k

Figure 4.5: Example 2 - a multilayered parallel-plate structure with three conductorpower-ground planes and 221 vias (unit: mm).

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 2

No of unknowns 74111 tetrahedrons 884 modes

*simulated on the machine of Intel Centrino 1.3 GHz, 512 MB

The example of two active vias in multilayered parallel-plate structure is alsoconsidered as shown in Fig 4.7 It has six conductor power-ground planes The rel-

ative 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 di-

mensions are in mm In Figs 4.8, 4.9 and 4.10, the S-parameters simulation sults by the SMM algorithm implemented for analysis of multilayered power-groundplanes are plotted and compared with those from the HFSS simulation

re-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 pling of multilayered vias The modal expansion of the scattering matrix method

cou-(SMM) incorporating with the FDCL boundary modeling and the generalized T

matrix approach is a powerful numerical method The simulation time and memoryusage are greatly reduced as compared to the full-wave methods and it still yieldsaccurate simulation results

Figure 4.7: Example 3 - a multilayered parallel-plate structure with six conductorpower-ground planes (unit: mm).

Figure 4.8: Comparison of the Z11 parameter simulated results for multilayeredstructure of Example 3: SMM algorithm with FDCL vs HFSS simulation.

Figure 4.9: Comparison of the Z21 parameter simulated results for multilayeredstructure of Example 3: SMM algorithm with FDCL vs HFSS simulation

Figure 4.10: Comparison of the Z22 parameter simulated results for multilayeredstructure of Example 3: SMM algorithm with FDCL vs HFSS simulation.

Table 4.3: Comparison of memory usage and computing time for Example 3

*simulated on the machine of Intel Centrino 1.3 GHz, 512 MB