Unified Formulation for Multiple Vias With or Without Circular Pads in High Speed Interconnects

1226 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 8, AUGUST 2011

Unified Formulation for Multiple Vias with orwithout Circular Pads in High Speed InterconnectsZhonghai Guo, Guangwen “George” Pan, Senior Member, IEEE, and Helen K. Pan, Senior Member, IEEE

Abstract— A unified full-wave characterization of massivenumber of vias with or without circular pads is formulatedanalytically by means of equivalent magnetic frill array modeland Galerkin’s procedure. The proposed method takes advantageof the parallel-plate structure and employs image theory andFourier transform to simplify the problem from 3-D configurationinto 2-D frame. Based on the cylindrical symmetry with Bessel’sfunctions and addition theorem, the final matrix equation isformulated analytically which can be used immediately for sen-sitivity analysis in both via dimensions and pad size. As a result,the new method is simple, efficient, and accurate. Numericalexamples demonstrate good agreement between our analyticalsolution and the results obtained by commercial software (highfrequency structure simulator) over a frequency range up to20 GHz.

Index Terms— Galerkin’s procedure, image theory, integratedcircuits, magnetic frill.

I. INTRODUCTION

VERTICAL vias have been widely used in electronicpackaging. They provide DC power delivery and signal

transmission through dielectric layers. Over the past halfcentury, electronic devices and systems have been persistingin the trend of higher density and faster operating frequency,in which the multilayered via-in-pad interconnects play animportant role. Modern silicon complementary metal–oxide–semiconductor transistors have gate lengths on the order of30 nm and a gate oxide thickness of 1–2 nm, yielding unprece-dented integrated circuits (ICs) integration densities. The fastgrowth in the bandwidth of wireless data links, in recentyears, has led the industry toward utilizing the 57–64 GHzunlicensed frequency band, for high-speed internet, data,and voice channels. Nonetheless, the high oxygen absorptionat 60 GHz makes this frequency range particularly attrac-tive for short range and secure communication links, and

Manuscript received August 2, 2010; revised January 28, 2011; acceptedMarch 15, 2011. Date of publication July 22, 2011; date of current versionAugust 12, 2011. This work was supported in part by the Intel ResearchLaboratory, Hillsboro, under Grant DW9-1040. Recommended for publicationby Associate Editor S. L. Dvorak upon evaluation of reviewers’ comments.

Z. Guo was with the Department of Electrical Engineering, Arizona StateUniversity, Tempe, AZ 85281 USA. He is now with the Apache DesignSolution Inc., San Jose, CA 95134 USA. (e-mail: [email protected]).

G. Pan is with the Department of Electrical Engineering, Arizona StateUniversity, Tempe, AZ 85281 USA (e-mail: [email protected]).

H. K. Pan is with the Intel Research Laboratory, Hillsboro, OR 97124 USA(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCPMT.2011.2149520

the millimeter-wave band has much higher channel capac-ity, avoiding the overcrowded channels in the low giga-hertz frequency range of WWAN (GSM 850/1 800, WWAN,WCD-MA, GPS), WLAN (802.11.b/g, Bluetooth), WiMAX(802.16e), WLAN (802.11a), etc. Recently, vias have foundapplications in microstrip patched antennas. The traditionalquasi-static methods [1], [2] cannot address radiation anddispersion. In particular, at high frequencies, impedance dis-continuities may generate undesired local modes, which in turnmay radiate power to the environment. Therefore, full-wavesolution is required [3]– [14].

Due to massive number of vias in high density ICs, theanalytical method is preferable to the numerical method. Atypical vertical via structure can be decomposed into interiorand exterior regions based on the equivalence principle. Forthe interior problem, a semi-analytic approach was conductedto analyze cross talk between coupled vias [4]. It employedthe model of a cylinder antenna excited by an infinite frillarray of equivalent magnetic current based on the imageprinciple and even-odd mode decomposition. While simpleand accurate, this method only handles two vias without pads,neglecting the influence of other vias. Recent years, a general-purpose full-wave analytical method appeared in the scene[6], employing the Foldy–Lax multiple scattering equationin vector spherical/cylindrical wave functions. The methodis general, robust but requires the Foldy–Lax theorem as aprerequisite. The Foldy–Lax was originally used for multiplescattering by densely packed media, where particles are notlocated in each other’s far-field zone [9].

Recently, a simplified full-wave algorithm was developedin [7], applying the mode-matching method [8] to augmentthe algorithm in [4] into a full-wave solution for multiplevias in a much simpler and more comprehensive manner.One of the significant differences between [6] and [7] is theGreen’s functions related to the equivalent magnetic currentsource. In [7] only the 1-D scalar radial Green’s functionis used, which easily leads to the analytical expression ofthe admittance matrix. However, in [6] the dyadic Green’sfunction, which is expressed in terms of vector cylindricalwaves, and waveguide modes are applied. Nonetheless, [7]only addressed the couping of vias without pads whereasthe influence of via pads on the signal transmission can beremarkable.

This paper presents a unified formulation for vias with orwithout circular pads, which is derived in a simplified andself-consistent manner (only one set of formulas). Every entryof the final admittance matrix is analytically expressed in

2156–3950/$26.00 © 2011 IEEE

GUO et al.: UNIFIED FORMULATION FOR MULTIPLE VIAS WITH OR WITHOUT CIRCULAR PADS 1227

z

h

2c2b

2a

2a

2h

padvia

magnetic frill

x

antipad

Fig. 1. Equivalent magnetic frill array model of a via with pad in an layerof multiple-layer vertical through hole via structure.

closed forms, allowing immediate sensitivity analysis of signalintegrity in terms of via dimension and pad size variations.This paper is an augment over the former work [7], suchthat geometry and boundary conditions are more general.The derived formulas are different from those in [7] and cannotbe found in any other via related papers. The testing results ofthis paper are in good agreement with that of commercial finiteelement method (FEM) software, high frequency structuresimulator (HFSS).

II. FORMULATIONS

A. Single Via

Fig. 1 depicts the equivalent magnetic frill array model[15] of a via with pads at its bottom and top ends based onthe equivalence principle and image theory. The equivalenceprinciple is invoked with equivalent magnetic current sourcesin the antipads regions located at z = 0 and z = h wheretwo infinite conductor planes are assumed. Approximately, themagnetic current at z = 0 is

�Mb(ρ, z) = − 2Vb

ρ ln cb

δ(z)φ (1)

and the magnetic current at z = h is

�Mu(ρ, z) = − 2Vu

ρ ln cb

δ(z − h)φ (2)

where Vb and Vu are the exciting voltages at the bottom andtop of via, respectively, b ≤ ρ ≤ c for both (1) and (2). Byapplying either one of these two equivalent magnetic sourcesto excite the via, we can obtain the full admittance matrix dueto the symmetry. So in the following derivation, we supposeonly one voltage V is applied to the via bottom.

With the source at the bottom and the usual condition h <<λ/2, only transverse magnetic modes are excited. Therefore,the unknown electric current on the via surface has onlythe z-component. Using the image principle, the via can beconsidered as an infinitely long cylinder in a homogeneousspace filled with the same material. The infinitely long cylinderis excited by periodically located magnetic current frills atz = ±2nh shown in Fig. 1. Based on this model, the currentintensity on each cylinder is periodic along the z direction andcan be expressed as

Jz(φ, z) =+∞∑

n=−∞J ∗

z (φ, z + 2nh) (3)

where J ∗z is the unknown electric current intensity on the infi-

nitely long cylinder with the excitation source only at z = 0.To obtain the summation above, we take the Fourier seriesexpansion of the periodically distributed current intensity onthe cylinder as

Jz(φ, z) = 1

2hJz(φ; 0) + 1

h

+∞∑

m=1

Jz

(φ; mπ

h

)cos

mπz

h(4)

where Jz(φ; α) is the Fourier transform of J ∗z in terms of z.

Further, as the periodic natural in the φ direction on each via,the coefficient Jz(φ; α) is expanded in harmonics as

Jz(φ; α) =+∞∑

n=−∞Cn(α)e jnφ. (5)

To get the unknown current intensity Jz(φ; α) on the cylinder,we need to solve the electric field integral equation (EFIE) in2-D. Namely

Escz ( �ρ; α) = −E inc

z ( �ρ; α), �ρ ∈ � (6)

where � is the profile of the cross-section of the infinitely longcylinder. Esc

z ( �ρ; α) and E incz ( �ρ; α) are the Fourier transform

of the scattered electric field and incident electric field at �ρ,respectively, namely

Escz ( �ρ; α) = k2

ρa

jωε

∫ 2π

0Jz(φ; α)Ge( �ρ, �ρ′)dφ′ (7)

and

E incz (ρ; α) = − 1

ρ

∂

∂ρρ

∫ c

bMφ(ρ′)Gm(ρ, ρ′)dρ′ (8)

where k2ρ = k2 − α2, Mφ(ρ′) is the Fourier transform of (1)

with respect to z

Mφ(ρ′) = − 2V

ρ′ ln cb

. (9)


y

ρ� − ρ�'

ρ�'

ρ�l

φρ'ρl

φρρl

ρ�

x

Fig. 2. Addition theory in cylindrical coordinate system for single cylinder.

The 2-D Green’s function, Ge( �ρ, �ρ′), satisfies

(∇2t + k2

ρ)Ge( �ρ, �ρ′) = −δ( �ρ − �ρ′) (10)

and is given in [16] as

Ge( �ρ, �ρ′) = 1

4 jH (2)

0

(kρ

∣∣ �ρ − �ρ′∣∣) . (11)

The 1-D radial Green’s function Gm(ρ, ρ′) satisfies theequation

(∇2t + k2

ρ)Gm(ρ, ρ′) = −δ(ρ − ρ′) (12)

and can be shown [7] to have the form

Gm(ρ, ρ′) = −π

2jρ′

{H (2)

1 (kρρ′)J1(kρρ), if ρ ≤ ρ′,H (2)

1 (kρρ)J1(kρρ′), if ρ ≥ ρ′.(13)

In order to perform the integration in (7) analytically, weapply the addition theorem [17] in cylindrical coordinates. Byreferring to Fig. 2, for ρ ≥ ρ′, we have

H (2)0 (kρ

∣∣ �ρ − �ρ′∣∣) =+∞∑

m=−∞Jm(kρρ′)H (2)

m (kρρ)e jm(φ−φ′).

(14)Substituting (5) and (14) into (7) and (11) and making theintegration, we arrive at

Escz ( �ρ; α) = −k2

ρπaη

2k

+∞∑

n=−∞Cn(α)Jn(kρa)H (2)

n (kρρ)e jnφ.

(15)Substituting (9) and (13) into (8) and making the integration,

we arrive at

E incz (ρ; α) = π j V

ln cb

[H (2)

0 (kρc) − H (2)0 (kρb)

]J0(kρρ) (16)

for a ≤ ρ ≤ b

Eincz (ρ; α) = π j V

ln cb

[J0(kρc) − J0(kρb)

]H (2)

0 (kρρ) (17)

for ρ ≥ c. Substituting (15) and (16) into (6) and testing bothside with e− j n′φ, n′ = −∞ · · · + ∞ and integrate in [0, 2π]yields

Cn(α) = j2kV

aηk2ρ ln c

b

H (2)0 (kρc) − H (2)

0 (kρb)

H (2)0 (kρa)

δ0,n. (18)

(a) (b) (c) (d)

Fig. 3. Four types of vias can be characterized by the model.

Substituting (5) and (18) into (4) yields

Jz(φ, z) = 1

2hC0(0) + 1

h

+∞∑

m=1

C0

(mπ

h

)cos

mπz

h

= j V

aηhk ln cb

H (2)0 (kc) − H (2)

0 (kb)

H (2)0 (ka)

− j2kV

aηh ln cb

+∞∑

m=1

1

q2m

K0(qmc) − K0(qmb)

K0(qma)cos

mπz

h

(19)

where q2m = (mπ/h)2 − k2. From (19), we may conclude

that due to symmetry, the current intensity on a single via isindependent of φ, or equivalently, only the fundamental modealong φ exists. Finally, we obtain the admittance as

Ybb = Yuu =∫ 2π

0 Jz(φ, 0)adφ

V

= j2π

ηhk ln cb

H (2)0 (kc) − H (2)

0 (kb)

H (2)0 (ka)

− j4πk

ηh ln cb

+∞∑

m=1

1

q2m

K0(qmc) − K0(qmb)

K0(qma)(20)

Yub = Ybu =∫ 2π

0 Jz(φ, h)adφ

V

= j2π

ηhk ln cb

H (2)0 (kc) − H (2)

0 (kb)

H (2)0 (ka)

− j4πk

ηh ln cb

+∞∑

m=1

(−1)m

q2m

K0(qmc) − K0(qmb)

K0(qma). (21)

The model reported here is applicable to four types of viasshown in Fig. 3. The via in Fig. 3(a) is ended with two pads,the via in (b) and (c) is ended with only one pad at eitherend and the via in (d) has no pads. To model these fourtypes of vias, let us set a = b in (20) and (21) and denotethe corresponding admittance as Y ′

bb = Y ′uu and Y ′

ub = Y ′bu .

Then, the admittance matrices of the four types of vias are,respectively

Y a =[

Ybb Ybu

Yub Yuu

], Y b =

[Y ′

bb Ybu

Y ′ub Yuu

],

Y c =[

Ybb Y ′bu

Yub Y ′uu

], Y d =

[Y ′

bb Y ′bu

Y ′ub Y ′

uu

]. (22)

Note that the last equation of Y d is the case of vias withoutpads reported in [7].


B. Multiple Vias

To address the coupling of massive number of vias, wetake the assembly of all via surfaces as just one 2-Dscattering surface. On this surface, there are induced electricalcurrents that are to be solved in terms of the equivalentmagnetic current excitation source located on the anti-padsaround all vias. Namely, we modify the 2-D EFIE (6) as

Escz ( �ρ; α) = −E inc

z ( �ρ; α), �ρ ∈P⋃

l=1

�l (23)

where Escz ( �ρ; α) and E inc

z ( �ρ; α) are, respectively, the net(total) scattered and incident electric field at a field point, �ρ onany via surface, P is the total number of vias with or withoutpads, �l is the cross-section profile of via barrel l. Applyingthe linear superposition principle, we may express (23) as

P∑

l=1

Esc,lz ( �ρ; α) = −

P∑

l=1

E inc,lz ( �ρ; α), �ρ ∈

P⋃

l=1

�l (24)

where Esc,lz ( �ρ; α) is the partial scattered electric field at

�ρ contributed by the electric current on the via barrel l,E inc,l

z ( �ρ; α) is the partial incident electric field at �ρ due tothe magnetic current in the antipad of via l. It is suggested todistinguish the net (total) field in (23) and partial field in (24),and the superposition nature will be revealed clearly in (29).

Expanding the unknown electric current intensityJz(φρρl ; α) on each cylinder in the same way of thesingle via case (5), we obtain

J lz (φρρl ; α) =

+∞∑

n=−∞Cl

n(α)e jnφρρl . (25)

By the same token, (15) yields

Esc,lz ( �ρ; α) = −k2

ρπaη

2k

+∞∑

n=−∞Cl

n(α)Jn(kρa)

×H (2)n (kρ | �ρ − �ρl |)e jnφρρl (26)

where �ρl is the center position of via l. Modifying (16) and(17) accordingly, we arrive at

E inc,lz ( �ρ; α) = π j V l

ln cb

[H (2)

0 (kρc) − H (2)0 (kρb)

]

×J0(kρ | �ρ − �ρl |) (27)

for a ≤ | �ρ − �ρl | ≤ b, and

E inc,lz ( �ρ; α) = π j V l

ln cb


]

×H (2)0 (kρ | �ρ − �ρl |) (28)

for | �ρ − �ρl | ≥ c. For convenience, we rewrite (24)explicitly as

P∑

i=1i =l

E sc,iz ( �ρ; α) + Esc,l

z ( �ρ; α) = −P∑

i=1i =l

E inc,iz ( �ρ; α)

−E inc,lz ( �ρ; α), �ρ ∈ �l , l = 1, . . . , P. (29)

y

ρ�i − ρ�

l

ρ� − ρ�i

ρ�l

ρ�i

φρiρl

φρρl

ρ�

x

Fig. 4. Addition theory in cylindrical coordinate system for multiplecylinders.

Substituting (15), (27), and (28) into (29) we get

−k2ρπaη

2k

+∞∑

n=−∞Cl

n(α)Jn(kρa)H (2)n (kρ | �ρ − �ρl |)e jnφρρl

−k2ρπaη

2k

P∑

i=1i =l

+∞∑

n=−∞Ci

n(α)Jn(kρa)H (2)n (kρ | �ρ − �ρi |)e jnφρρi

= π j V l

ln cb

[H (2)

0 (kρb) − H (2)0 (kρc)

]J0(kρ | �ρ − �ρl |)

−P∑

i=1i =l

π j V i

ln cb


]H (2)

0 (kρ | �ρ − �ρi |),

�ρ ∈ �l , l = 1, . . . , P. (30)

For �ρ ∈ �l or equivalently, | �ρ− �ρl | = a, we apply the additiontheorem again [17] in the cylindrical coordinate system to (30).By referring to Fig. 4, for | �ρ − �ρl | ≤ | �ρi − �ρl | we have

H (2)n (kρ | �ρ − �ρi |)e jnφρρi = (−1)n

+∞∑

m=−∞Jm(kρ | �ρ − �ρl |)

×H (2)m−n(kρ | �ρi − �ρl |)e jm(φρρl −φρi ρl )+ j nφρi ρl . (31)

After substituting (31) into (30), we arrive at

−k2ρπaη

2k

+∞∑

n=−∞Cl

n(α)Jn(kρa)H (2)n (kρa)e jnφρρl

−k2ρπaη

2k

P∑

i=1i =l

+∞∑

n=−∞C(i)

n (α)Jn(kρa)(−1)n+∞∑

m=−∞

Jm(kρa)H (2)m−n(kρ | �ρi − �ρl |)e jm(φρρl −φρi ρl )+ j nφρi ρl

= π j V l

ln cb

[H (2)

0 (kρb) − H (2)0 (kρc)

]J0(kρa)

−P∑

i=1i =l

π j V i

ln cb


] +∞∑

m=−∞Jm(kρa)

×H (2)m (kρ | �ρi − �ρl |)e jm(φρρl −φρi ρl ). (32)

Suppose we choose the highest order mode number of interestin φρρl direction as N . we test both sides of (32) with


1

2

3

4

Fig. 5. 3 × 3 via-in-pad array in a layer of (PCBs).

2 4 6 8 10 12 14 16 18 20−20

−10

0

Frequency (GHz)

Ret

urn

Los

s (d

B)

−10

−5

0

Inse

rtio

n L

oss

(dB

)

Analytic SB1B1 & SU1B1Analytic SB2B2 & SU2B2Analytic SB3B3 & SU3B3Analytic SB4B4 & SU4B4HFSS SB1B1 & SU1B1HFSS SB2B2 & SU2B2HFSS SB3B3 & SU3B3HFSS SB4B4 & SU4B4

Fig. 6. Return loss and insertion loss of 3 × 3 via-in-pad array.

e− j n′φρρl , n′ = −N, . . . , N and integrate over [0, 2π], yielding

−k2ρπaη

2kCl

n(α)Jn(kρa)H (2)n (kρa)δn,n′

−k2ρπaη

2k

P∑

i=1i =l

N∑

n=−N

C(i)n (α)Jn(kρa)(−1)n Jn′(kρa)

×H (2)n′−n(kρ | �ρi − �ρl |)e− j (n′−n)φρi ρl

= π j V l

ln cb

[H (2)

0 (kρb) − H (2)0 (kρc)

]J0(kρa)δn′,0

−P∑

i=1i =l

π j V i

ln cb


]Jn′ (kρa)

×H (2)n′ (kρ | �ρi − �ρl |)e− j n′φρi ρl . (33)

Conducting such testing procedures for all vias, we finallyobtain a matrix equation

¯Z I = V (34)

where matrix ¯Z is of dimension (2N + 1)P × (2N + 1)P andvectors I and V are of dimension (2N +1)P ×1. By referring

2 4 6 8 10 12 14 16 18 20−34

−32

−30

−28

−26

−24

−22

−20

−18

−16

Frequency (GHz)

Cro

ssta

lks

(dB

)

Analytic SU2B1Analytic SU3B1Analytic SU3B2Analytic SU4B2HFSS SU2B1HFSS SU3B1HFSS SU3B2HFSS SU4B2

Fig. 7. Cross talks between vias of 3 × 3 via-in-pad array.

2 4 6 8 10 12 14 16 18 20−40

−20

0

Frequency (GHz)

Ret

urn

Los

s (d

B)

−4

−2

0

Inse

rtio

n L

oss

(dB

)

Analytic SB1B1 & SU1B1Analytic SB2B2 & SU2B2Analytic SB3B3 & SU3B3Analytic SB4B4 & SU4B4HFSS SB1B1 & SU1B1HFSS SB2B2 & SU2B2HFSS SB3B3 & SU3B3HFSS SB4B4 & SU4B4

Fig. 8. Return loss and insertion loss of 3 × 3 via array without pads.

2 4 6 8 10 12 14 16 18 20

−34

−32

−30

−28

−26

−24

−22

−20

−18

−16

Frequency (GHz)

Cro

ssta

lks

(dB

)

Analytic SU2B1Analytic SU3B1Analytic SU3B2Analytic SU4B2HFSS SU2B1HFSS SU3B1HFSS SU3B2HFSS SU4B2

Fig. 9. Cross talks between vias of 3 × 3 array without pads.

to (4), we solve (34) for every mode α = (mπ/h), m =1, 2, . . . , M separately, which keeps the size of matrix ¯Zrelatively small.


TABLE I

POSITIONS OF NINE VIAS (UNIT : Mil)

via1 via2 via3 via4 via5

(0, 0) (50, −30) (80, 90) (30, 0) (15, 25.981)

via6 via7 via8 via9

(−15, 25.981) (−30, 0) (−15, −25.981) (15, −25.981)

1

2

3

4

56

7

89

Fig. 10. Coaxial grounded vias in a layer of PCBs.

To get the current on cylinder l, we integrate (4), rendering

I (l)(z) =∫ 2π

0J (l)

z (φ′, z)adφ′

= πa

hC(l)

0 (0) + 2πa

h

M∑

m=1

C(l)0

(mπ

h

)cos

mπz

h. (35)

Eventually, the normalized admittance matrix ¯Y is acquired as

¯Y =⎡

⎣¯Y bb ¯Y bu

¯Y ub ¯Y uu

⎤

⎦ (36)

where

Y uul,i = I (l,u)

V (i,u)

√Y (l,u)

0 Y (i,u)0

∣∣∣∣ V (i,b)=0V (m)=0,m =i

(37)

Y ubl,i = I (l,u)

V (i,b)

√Y (l,u)

0 Y (i,b)0

∣∣∣∣ V (i,u)=0V (m)=0,m =i

(38)

Y bul,i = I (l,b)

V (i,u)

√Y (l,b)

0 Y (i,u)0

∣∣∣∣ V (i,b)=0V (m)=0,m =i

(39)

Y bbl,i = I (l,b)

V (i,b)

√Y (l,b)

0 Y (i,b)0

∣∣∣∣ V (i,u)=0V (m)=0,m =i

(40)

l, i, m = 1, 2, . . . , P . The scattering matrix can be obtainedfrom the admittance matrix by

¯S ={ ¯I2P + ¯Y

}−1 { ¯I2P − ¯Y}

(41)

where Y (l,u)0 , Y (l,b)

0 , Y (i,u)0 , and Y (i,b)

0 are characteristic admit-tances of the corresponding ports at the bottom and top planes.

For the modal expansion and image method used in thispaper, convergence is an important issue. Fortunately, they

TABLE II

COMPARISON OF CPU TIME AND MEMORY

Case1: 3 × 3 via-in-pad array in a layer of PCBs

Item HFSS simulation Method in this paper

Memory 440M byte 70M byte

Total time 65 minutes 168 seconds

Case2: Coaxial grounded vias in a layer of PCB

Item HFSS simulation Method in this paper

Memory 500M byte 70M byte

Total time 82 minutes 168 seconds

2 4 6 8 10 12 14 16 18 20−40

−30

−20

−10

0R

etur

n L

oss

(dB

)

−4

−3

−2

−1

0

Frequency (GHz)

Inse

rtio

n L

oss

(dB

)

Analytic SB1B1 & SU1B1Analytic SB2B2 & SU2B2Analytic SB3B3 & SU3B3HFSS SB1B1 & SU1B1HFSS SB2B2 & SU2B2HFSS SB3B3 & SU3B3

Fig. 11. Return loss and insertion loss of vias with coaxial grounded barrels.

converge relatively fast. Our experience reveals that the trun-cation error is negligible for the geometry and dimensions ofall examples in this paper if we set M = 20 (image term) andN = 6 (modal number).

III. NUMERICAL RESULTS AND DISCUSSION

To validate our method, we conduct simulations of twocases in an interior layer of a PCB shown in Figs. 5 and 10,respectively. The layer thickness is h = 30 mil with relativepermittivity εr = 4.4 and loss tangent tan δ = 0.02. The radiusof all vias a = 5 mil, radius of all circular pads b = 10 mil,and radius of all holes c = 15 mil. The pitch in Fig. 5is 60 mil. The positions of vias for Fig. 10 are shown inTable I. The comparisons are made between our analyticalresults and the results by using commercial FEM software,HFSS. For our analytical method, the highest order moderelated φ is N = 6. The Fourier series in (35) is truncated


2 4 6 8 10 12 14 16 18 20

−65

−60

−55

−50

−45

−40

−35

−30

−25

Frequency (GHz)

Cro

ssta

lks

(dB

)

Analytic SU2B1Analytic SU3B1Analytic SU3B2HFSS SU2B1HFSS SU3B1HFSS SU3B2

Fig. 12. Cross talks between vias with coaxial grounded barrels.

at M = 20. All results show good agreement between ourmethods and HFSS results over a frequency range of up to20 GHz. Table II shows the comparison of the CPU timeand memory consumption of the HFSS and method in thispaper. Our method consumes approximately 15% memoryand uses 4% CPU time as the HFSS does. For comparisonpurpose, the results of the same via array as Fig. 5 but withoutpads are presented in Figs. 8 and 9. By comparing Figs. 6and 7 with Figs. 8 and 9, it demonstrates that pads in via-in-pad array degrade signal integrity quite significantly. Forinstance, pads raised the insertion loss from 3 to 8.5 dB, i.e.,an increase about 5.5 dB at 20 GHz. Fig. 10 illustrates asystem consisting of nine vias, where via 1 to 3 are signalvias with pads, while vias 4 through 9 are grounded. Groundvias are popular in patched antennas, electric band gap andcavities, acting as isolation walls. Figs. 11 and 12 represent,respectively, the return/insetion loss and crosstalk of the threesignal vias in Fig. 10. By comparing Figs. 7 and 12, onecan see the general trend that ground vias reduce cross talkvalues.

Although all simulations were performed only for a singlelayer, this method is readily to be extended to multilayer casesby using the cascade rule of the transmission (generalizedABCD) matrices.

IV. CONCLUSION

In this paper, we proposed a unified full-wave formulationfor the frequency-dependent propagation characteristics ofmultiple through hole vias, with or without pads, in themultilayered packaging environment. The current distributionson the vias are acquired analytically, from which the insertionloss, return loss, and crosstalk among different types of viasare investigated. The proposed algorithm is simple, accurate,and fast and is applicable to high-density and high-speedmultilayered ICs, system-on-chip, and wireless systems. Theresults are obtained for vias in one layer and are comparedagainst that of HFSS with excellent agreement, while thetypical memory consumption and CPU are one to two ordersof magnitude less.

REFERENCES

[1] T. Wang, R. F. Harrington, and J. R. Mautz, “Quasi-static analysis of amicrostrip via through a hole in a ground plane,” IEEE Trans. Microw.Theory Tech., vol. 36, no. 6, pp. 1008–1013, Jun. 1988.

[2] G.-W. Pan, X. Zhu, and B. K. Gilbert, “Analysis of transient behavior ofvertical interconnects in stacked circuit board layers using quasi-statictechniques,” IEEE Trans. Comp., Packag., Manuf., Technol., Part B: Adv.Packag., vol. 18, no. 3, pp. 521–531, Aug. 1995.

[3] S. W. Ho, S. W. Yoon, Q. Zhou, K. Pasad, V. Kripesh, and J. H.Lau, “High RF performance TSV silicon carrier for high frequencyapplication,” in Proc. Electron. Comp. Technol. Conf., Lake Buena Vista,FL, May 2008, pp. 1946–1952.

[4] Q. Gu, Y. E. Yang, and M. A. Tassoudji, “Modeling and analysis of viasin multilayered integrated circuits,” IEEE Trans. Microw. Theory Tech.,vol. 41, no. 2, pp. 206–214, Feb. 1993.

[5] X. Gu, A. E. Ruehli, and M. B. Ritter, “Impedance design for multilay-ered vias,” in Proc. Electr. Perform. Electron. Packag. Syst., San Jose,CA, Oct. 2008, pp. 317–320.

[6] B. Wu and L. Tsang, “Modeling multiple vias with arbitrary shape ofantipads and pads in high speed interconnect circuits,” IEEE Microw.Wireless Comp. Lett., vol. 19, no. 1, pp. 12–14, Jan. 2009.

[7] Z. Guo and G. Pan, “On simplified fast modal analysis for throughsilicon vias in layered media based upon full-wave solutions,” IEEETrans. Adv. Packag., vol. 33, no. 2, pp. 517–523, May 2010.

[8] Z. Guo, G. Pan, S. Hall, and C. Pan, “Broadband characterization ofcomplex permittivity for low-loss dielectrics: Circular PC board diskapproach,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3126–3135, Oct. 2009.

[9] M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scatteringof Light by Particles. Cambridge, U.K.: Cambridge Univ. Press, 2006,p. 18.

[10] H. Gan, Q. He, and D. Jiao, “Hierarchical and adaptive finite-elementreduction-recovery method for large-scale power and signal integrityanalysis of high-speed IC and packaging structures,” in Proc. Electr.Perform. Electron. Packag. Syst., San Jose, CA, Oct. 2008, pp. 127–130.

[11] R. Chen, H. Wang, and R. Abhari, “Suppression of power/ground noiseusing differential vias,” in Proc. Electr. Perform. Electron. Packag. Syst.,San Jose, CA, Oct. 2008, pp. 325–328.

[12] Z.-Z. Oo, E.-X. Liu, and E.-P. Li, “A semi-analytical approach forsystem-level electrical modeling of electronic packages with large num-ber of vias,” IEEE Trans. Adv. Packag., vol. 31, no. 2, pp. 267–274,May 2008.

[13] E.-X. Liu, E.-P. Li, Z. Z. Oo, and X. Wei, “Novel methods for modelingof multiple vias in multilayered parallel-plate structures,” IEEE Trans.Microw. Theory Tech., vol. 57, no. 7, pp. 1724–1733, Jul. 2009.

[14] S.-G. Hsu and R.-B. Wu, “Full wave characterization of a through holevia using the matrix-penciled moment method,” IEEE Trans. Microw.Theory Tech., vol. 42, no. 8, pp. 1540–1547, Aug. 1994.

[15] D. V. Otto, “The admittance of cylindrical antennas driven from a coaxialline,” Rad. Sci., vol. 2, no. 9, pp. 1031–1042, 1967.

[16] R. F. Harrington, Field Computation by Moment Methods. Piscataway,NJ: IEEE Press, 1992.

[17] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed.Cambridge, U.K.: Cambridge Univ. Press, 1948.

Zhonghai Guo received the M.S. degree in radiophysics from Sichuan University, Chengdu, China,in 2002, and the Ph.D. degree in electrical engi-neering from Arizona State University, Tempe, in2010.

He was an Electrical Engineer in the South-west Research Institute of Electronics Technology,Chengdu, China (10th Research Institute of CertifiedTechnical Education Centre), from 2002 to 2005. Hewas a Graduate Research Associate at Arizona StateUniversity, from 2005 to 2010. Currently, he is a

Principal Engineer in Apache Design Solutions Inc., San Jose, CA. His currentresearch interests include computational electromagnetics, microwave circuitdesign, antenna design, and signal integrity analysis.


Guangwen “George” Pan (S’80–M’84–SM’93)received the M.S. and Ph.D. degrees in electri-cal engineering from the University of Kansas,Lawrence, in 1982 and 1984, respectively.

He was a Post-Doctoral Fellow at the Universityof Texas, Arlington, from 1984 to 1985, an Engineerat the Mayo Foundation, Scottsdale, AZ, from 1985to 1986, and an Associate Professor at South DakotaState University, Brookings, from 1986 to 1988.From 1988 to 1995, he was with the Universityof Wisconsin-Milwaukee, Milwaukee, where he was

promoted to Professor in 1993. He joined Arizona State University, Tempe,in 1995, as a Professor and the Director of the Electrical EngineeringDepartment, Electric Packaging of the Laboratory. His current research inter-ests include computational electromagnetics, collimated and array antennas,electronic packaging, and device modeling.

Dr. Pan is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION, and an Associate Editor of the International Journal ofComputational Electronics.

Helen K. Pan (S’99–M’01–SM’07) received theB.S. degree in computer science from Beijing Poly-technic University, Beijing, China, and the M.S.degree in electrical engineering from the Universityof Illinois at Urbana-Champaign, Urbana, in 1996and 2000, respectively.

She was with the Agilent Technologies, SantaRosa, CA, from 2001 to 2002, developing high-performance spectrum analyzers. From 2003 to2004, she was with Maxim Integrated Products,Hillsboro, OR, developing and testing high-speed

digital-to-analog converter chips. Since 2004, she has been with Intel Labora-tory of Intel Corporation, Hillsboro, investigating various multi-radio wirelessplatforms. She has six filed and pending patents, and she published 37 papersin IEEE journals and international conferences. Her current research interestsinclude radio frequency/microwave circuits, systems, 60 GHz technology, andreconfigurable antenna designs.

Dr. Pan has served as a technical program committee member or sectionchair for various IEEE conferences, and as a reviewer for IEEE TRANSAC-TIONS ON ANTENNA PROPAGATION. She received the IEEE Antennas andPropagation Society H. A. Wheeler Prize Paper Award.

Documents

Unified Formulation for Multiple Vias With or Without Circular Pads in High Speed Interconnects