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

Electrical Performance Modelingof Power-Ground Layers with Multiple Vias The outline of the efficient approach for system-level modeling of advanced electronicpackages is presented in Chap

Electrical Performance Modeling

of Power-Ground Layers with

Multiple Vias

The outline of the efficient approach for system-level modeling of advanced electronicpackages is presented in Chapter 1, in which power distribution network (PDN) andsignal distribution network (SDN) are separately analyzed by using mode decompo-sition for the entire problem The analytical method for analysis of the power-groundplane 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 rectangularstructure 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 theSMM is presented in details In the conventional SMM, the power-ground planes areassumed to be infinitely large so it cannot capture the resonant behavior of the real-world 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

method is proposed based on factitious layer of PMC cylinders with dependent radii at the periphery of an electronic package Hence, the extendedSMM is capable to handle the real-world package structures.

frequency-In the latter part of the chapter, numerical examples are presented for validation

of the implemented SMM algorithm with the proposed frequency-dependent cylinderlayer (FDCL) The extended method is not only capable to simulate the finite-sized power-ground planes and it can also simulate the irregular-shaped planes andcutout structure in the planes This is one prominent feature of the FDCL modelingmethod

Vias

An advanced electronic package consisting of signal traces, power-ground planes andplenty 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 thepower-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 isvery difficult and time consuming to model both networks simultaneously As themethodology outline for analysis of the entire problem has been discussed earlier;the inner domain of the package, which consists of parallel power-ground planes andvias, 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 circularcylinders Thus, the theory of multiple scattering among many parallel conductingcylinders [88] can be used to model them efficiently The theory of scattering by con-

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] Inthis research, instead of using the Green’s function approach in [56, 57] to obtainthe corresponding formulae, we will directly apply the parallel-plate waveguide the-ory, which is a relatively simple and straightforward way to tackle the problem ofscattering 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 aremade of PECs, which may be of finite thickness; and the vias are circular PECcylinders

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

planes and the electromagnetic fields have e −jβz dependence where β is the gation wavenumber along the guiding direction z For the parallel-plate waveguide

propa-structure, two independent solutions of the above Maxwell equations in cylindricalcoordinate are expressed as

mn are the expansion coefficients of the incoming and outgoing TM

waves, a H mn and b H mn are the expansion coefficients of the incoming and outgoing TE

waves, respectively k2 = ω2µε = k2

ρ + β2

m , β m = k z = mπ

d , where d is the spacing

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 C m and S m stand for C m = cos (β m z) and S m = sin (β m z), respectively An e jωt timedependence is assumed throughout the formulation herein and subsequently

Other components of E and H related to E z and H z are calculated by

Then, by using the modal expansion approach, the E z and H z components of

an incident wave are expressed as:

Sub-stituting (3.9) into (3.7), we can obtain all other components of the electromagneticfields corresponding to TMz and TEz modes.

Since the total field is a summation of the incident and scattered fields, we canfinally obtain the following expressions for the total tangential electromagnetic fields

in cylindrical coordinates, normal to ˆρ in the ith parallel-plate waveguide formed bypair of power-ground planes

where the eigen-vectors are defined as

eE(i) tmn = C m (i) zˆ− jnβ m (i)

, respectively, where z ∈ [z i , z i + h i ]; and h i is the height

of the waveguide Symbols J (i)

3.3 Multiple Scattering Coefficients among

Cylin-drical PEC and PMC Vias

The boundary condition for the perfect magnetic conductor (PMC) is given as ˆn ×

H = 0 The total magnetic field on the surface of qth PMC cylinder with radius r q

in the ith parallel-plate layer is given by

for any value of z ∈ [ 0, d ] Then,

b H (i) mn(q) = T mn(q) E(i) a H (i) mn(q) (3.16)

b E(i) mn(q) = T mn(q) H (i) a E(i) mn(q) (3.17)where

n × E = 0 The total electric field on the surface of the qth PEC cylinder with a

radius r q in the ith parallel-plate layer is given by

for any value of z ∈ [ 0, d ] Then,

b E(i) mn(q) = T mn(q) E(i) a E(i) mn(q) (3.22)

b H (i) mn(q) = T mn(q) H (i) a H (i) mn(q) (3.23)The equations can be also written in matrix form as

z-The following short discussion proves that TE and TM modes generated by PECcylinder are decoupled in the parallel-plate waveguide

The boundary condition at the surface of the PEC cylinder is : Et | ρ=a = 0, i.e.,

a E(i) mn J mn (i) + b E(i) mn H mn (i) = 0 (3.28)

Because of (3.28), the expansion coefficients in (3.29) for TE and TM modes becomeindependent, i.e., TE and TM modes for the PEC cylinders are totally decoupled;

and different modes n are also decoupled Finally, we have

a E(i) mn J mn (i) + b E(i) mn H mn (i) = 0 (3.30)

a H (i) mn J mn
(i) + b H (i) mn H mn
(i) = 0 , (3.31)or

Figure 3.2: A set of random cylindrical vias (2D view)

By taking into account the multiple scattering among N c cylindrical vias, the

scat-tered field at an observation point p can be expressed as

where (ρ q , φ q ) are the local coordinates with ρ q = |ρ − ρ q | and φ q = arg( ρ − ρ q)

M q+ 1 represents the truncation number of modes in the parallel-plate waveguide

structure, and 2N q + 1 is that of the Hankel functions used to express the

scat-tered waves of the qth via b qmn denotes the unknown expansion coefficients for thescattered field

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

where ρ q < d qp ; [ρ p , ρ q , φ p , φ q , d qp , θ qp]∈ Real; k ρ ∈ Complex; k ρ = 0, and the terms

here are expressed in the global coordinate system The detailed expressions aregiven in Appendix A

According to (3.5), we define the following incoming and outgoing modes for

TM case as follows

E zmn (a)E = J n (k ρ ρ) C m e jnφ (3.35)

E zmn (b)E = H n(2)(k ρ ρ) C m e jnφ (3.36)Substituting (3.35) and (3.36) into (3.7), we get the tangential modes for TM case

According to (3.6), we define the following incoming and outgoing modes for

TE case

H zmn (a)H = J n (k ρ ρ) S m e jnφ (3.41)

H zmn (b)H = H n(2)(k ρ ρ) S m e jnφ (3.42)Substituting (3.41) and (3.42) into (3.7), we get the tangential modes for TE case

for outgoing wave

The outgoing T M wave from the pth cylinder can be written as

Since ρ q ∈ qth cylinder’s boundary, so ρ q < d qp The incoming wave coefficient for

the qth cylinder is then given as

and a E mn q (q) is independent of the terms ρ p , ρ q , φ p , and φ q

In different coordinates, the value of∇ sshould not change, which means for any

function f (ρ),

∇ (p)

Substituting (3.47) into (3.39), we get

The outgoing waves away from the pth cylinder are translated into the incoming

waves of the qth cylinder The relationship between these coefficients is derived in(3.48)

Similarly, we can get the exact same relationship between the translational efficients for TE case,

whereT E/H m(q) is a diagonal matrix with its elements T mn(q) E/H given in (3.18) and (3.19).

Finally, the unknown coefficient vector b q is summarized in the following tion:

whereT q stands for the transition T -matrix of the qth via; a q denotes the expansion

coefficients of the wave incident on the qth via Matrix α qpis the translation matrix

representing the wave scattered by the pth via incident onto the qth via The matrix

elements of α qp can be obtained as

α qp (n q , n p ) = H n(2)q −n p (k ρ d qp ) e −j(n q −n p )θ qp (3.58)Consolidating (3.57) for all the vias yields the following equation for multiple scat-tering of cylinders:

where I is the unit matrix and T is the block diagonal matrix consisting of the

T q matrices for all cylinders (q = 1, · · · , N c ) b = [b1, b2, · · · , b N c]T stands for the

unknown expansion vector of scattered waves and a = [a1, a2, · · · , a N c]T is the

expansion vector of incident waves on all the vias The matrix S is the combined

translation matrix written as

We can obtain the unknown coefficient vector b by solving (3.59) The boundary

of the package is modeled with a proposed novel boundary modeling technique Thedetailed discussion for the modeling technique will be presented in Section 3.7

3.4 Excitation Source and Network Parameter

For using the formulation in the previous section, the equivalence principle is plied to replace the annular via anti-pad with PECs in the P-G planes and an equiv-alent magnetic source is added at the original via anti-pad region (see Fig 3.4(b)).The structure shown in Fig 3.4 can be considered as a two-port network and the topand bottom via anti-pad regions are designated as Port 1 and Port 2, respectively

ap-In order to facilitate the subsequent signal and power integrity analysis, we need toevaluate the admittance parameters of the two-port network shown in Fig 3.4(a).The magnetic current source considered here is an angular magnetic current ringsource, which is also called a magnetic frill current The magnetic frill current isplaced on a perfectly conducting ground plane (PEC) (see Fig 3.5) For modeling

Figure 3.5: A magnetic frill current on the bottom PEC plane of an infinite parallelplate waveguide.

of the packaging problem in this research, the magnetic frill current is an equivalentsource which is due to the electric field at the aperture on the bottom PEC plane.The field at the aperture is assumed to be of the TEM mode:

E(ρ, z = 0) = V0

ρ ln (b/a) ˆ for a ≤ ρ ≤ b (3.61)

where V0 is the modal voltage at the aperture

The equivalent current, which is a magnetic frill current, is given by

M(ρ, z) = E × ˆn = M φ ϕˆ for a ≤ ρ ≤ b (3.62)and

we have only three electromagnetic components - H φ , E ρ and E z; and the other

three components are zero - E φ = H ρ = H z = 0 (see [89], page 266)

H φ can be found from the following differential equation:

∂

∂ρ

1

The associated electric field components are

Considering the coefficients in the T-matrix method for the incident wave due to the magnetic frill current

Recall that for the TMz mode, we have the following expression for the E z

mn= 0 for all the TEz modes

Assume that a magnetic frill current is at via ‘s’, then the H φ incident wave on

the via ‘s’ due to the current using (3.71) (for the case of ρ ≤ a) is

Comparing (3.77) with (3.73), we obtain the coefficient of the incident wave for the

Similarly, we can derive the coefficient of the incident wave on via ‘q’ (q = s) due to

the magnetic frill current at via ‘s’ as follows

cos(k z z)

−H (2)

0 (k ρ |ρ − ρ s |).

(3.79)

Referring to (3.34) together with Fig 3.3, the addition theorem for translation

of the coordinates is given as

of the incident wave for the via ‘q’

In order to derive I1, a testing ring of unity amplitude magnetic current Mt isapplied around the signal via at the anti-pad region The magnetic frill current is

equivalent to a delta-gap source and the electric field Esrc of the delta-gap source

and sinc(x) = sin(x)/x.

By applying the Esrc expression for the signal (source) via ‘q’, the incident field

at any P-G via ‘p’ can be expressed as

Einc from the delta-gap source is expressed in TM mode only

From the above equation, we obtain the coefficient for the incident wave, which

is used to calculate the scattered waves as outlined in the previous section TheY-matrix elements are calculated as

The other two entries of the admittance parameters of the two-port network can beobtained by repeating the same procedures.

The calculation for mutual admittance Y between the pth and qth P-G vias can

be performed in the following procedure

For pth PEC via,

3.5 Implementation of Effective Matrix-Vector

Mul-tiplication in Linear Equations

To solve the matrix equation for multiple scattering of the vias given in (3.59), theeffective matrix-vector multiplication for the linear equation system is implemented

in this section For the number of vias N c, the unknown wave coefficient vector for

the qth via can be rewritten in the following form:

represents the excitation coefficient vector for the source via ‘s’ Equation (3.94)

can be solved for each mode m The diagonal matrix T (q) m is expressed, if the qth via

0 0 .

with the argument (·) for the Hankel function of second kind being given as (k m ρ qp).

The matrix-vector multiplication (MVM) of the matrix H (qp) m is

.0

.0

.0

The whole matrix can be obtained as

b E m = T m · α m ·b E m + b E inc

m



(3.104)where α m can be written as the sum of a lower triangular matrix and an uppertriangular matrix as α m = α L m + α U m , and the wave coefficient vector b E m(s) inc of the

source via ‘s’ is given by

3.6 Numerical Examples for Single-layer

Power-Ground Planes

3.6.1 Validation of the SMM Algorithm

Figure 3.6 shows an example to verify the SMM algorithm for modeling ground planes with multiple vias where a signal trace passes through a pair ofconductor planes with 12 shorting vias The conductor planes are assumed to beinfinitely large

power-Figure 3.6: Vias of a signal trace passing through two conductor planes The via isenclosed by 12 shorting vias connecting the two planes

Figure 3.7: Comparison of the E z field distribution at 1 GHz: SMM simulationresult (left) vs HFSS simulation result (right) The vias are drawn as white dots

The simulation results are checked against those from the Ansoft HFSS The E z field distribution is plotted in Fig 3.7 The normalized value of the E z componentalong a horizontal line from the edge of the central via is given in Fig 3.8 In bothcases the results agree quite well The admittance parameter (Y11) of the structure

is also calculated up to 5 GHz The results again match well with the HFSS solutions

as shown in Fig 3.9

To simulate the coupling effect from the multiple vias, one example with a largenumber of shorting vias is presented in Fig 3.10 The vias are formed in the centerblock of 256 vias and the outer ring of 348 vias and the signal via is located between

them The SMM algorithm is used to calculate the E-field distribution and the

computing is done within few minutes In Fig 3.10, the field distribution is plotted

to show the effect of multiple scattering from the P-G vias

Figure 3.8: Validation of the simulated results by SMM algorithm for E z with thosefrom the HFSS simulation.

Figure 3.9: Y11 for the two-port network formed by the plate-through via