Efficient modeling of power and signal integrity for semiconductors and advanced electronic package systems 4

In this chapter, an efficient modeling technique based on the modal sition of electromagnetic fields is proposed to analyze the power distribution net-work of an electronic package, which i

Hybrid Modeling of Signal Traces

in Power Distribution Network by Using Modal Decomposition

Via structures in multilayered electronic packages are used to connect the signal orpower supply traces residing in the different layers Since parallel-plate waveguidemodes are introduced by layered structures, the signals on active vias can excite thewaveguide modes within the layers, and also can affect other vias including power-ground (P-G) vias The affected vias can, in turn, interfere with the original signals.Such coupling may even cause unreliable behavior or complete signal failure, alongwith signal integrity loss, inappropriate switching, and longer signal delay Anotherpotential problem in multilayered packages is that when harmonics of the transientsignal coincide with the resonant frequency of the power-ground planes, it will causeelectromagnetic compatibility (EMC) problems in the microprocessor packages Tominimize such noise behavior, pre-layout and post-layout verifications of the powerdistribution network (PDN) are necessary Thus, accurate analysis of the signaltraces in the PDN has become of vital important for optimizing the performance ofhigh-speed digital circuits

130

In this chapter, an efficient modeling technique based on the modal sition of electromagnetic fields is proposed to analyze the power distribution net-work of an electronic package, which includes the signal traces and the multilayeredpower-ground planes with multiple through-hole vias The total electromagneticfields inside the package are decomposed into two modes: the parallel-plate modeand the transmission line mode The propagation of the fields between the P-Gplanes is considered as parallel-plate mode and is efficiently analyzed by using thescattering matrix method, as presented in Chapters 3 and 4 The latter, which com-prises the stripline mode for the traces between the P-G planes and the microstripline mode for the traces on top/bottom of the package, is modeled as admittance(Y) networks by using the multiconductor transmission line theory The disconti-nuities of the signal traces at the through-hole vias are analyzed by using analyticalmodeling of equivalent circuits Finally, by cascading the equivalent networks, theoverall network parameter for the system is obtained to analyze the coupling effects

decompo-of the P-G vias to the signal traces

Modal Decomposition

A typical structure of the signal trace routed in a power distribution network of

an electronic package is shown in Fig 5.1 In multilayered structures, the metalpower-ground planes and vias are to provide a low-impedance path for the powerdistribution system between the printed circuit board and the die The signal tracesreside between the different layers and their return currents flow in the referenceplanes just below them When the traces pass through different layers, the verticalthrough-hole vias are placed to ensure the continuity of the return current

Consider that substrates sandwiched between the metal planes are very thincompared to operating wavelengths, and are usually uniform and isotropic; hence,

the electromagnetic fields do not change in the z-direction Each pair of the metal

Figure 5.1: Signal trace route in power distribution network of an electronic package.planes serves as parallel-plate waveguide providing the transverse electromagnetic(TEM) mode The signal traces routed in the PDN are modeled as multiconductortransmission line (MTL) and divided into two parts: the traces between the P-Gplanes as strip line mode and the traces on top/bottom of the package as microstripline mode It is assumed that the MTL has a uniform cross section, allowing thepropagation of quasi-TEM waves along the traces [95] Hence, the total electromag-netic fields propagating inside the package can be decomposed into two independentmodes: the parallel-plate mode and the transmission line mode.

Mode conversions usually occur at the transition between the signal trace andthe through-hole via At the via hole, the parallel-plate mode gets excited due tothe switching signal currents, and conversely, the noise voltages between the P-Gplanes gets coupled to the stripline mode A novel modal decomposition approach isapplied at the discontinuities of all through-hole signal vias as mode transition ports

In Fig 5.2, the entire domain of the problem is decomposed into three sub-domains:the parallel-plate planes with P-G vias; the microstrip lines and striplines; and thethrough-hole signal vias The parallel-plate mode of the P-G planes with a largenumber of vias is analyzed by using the scattering matrix method (SMM), which is

based on the N -body scattering theory, to model the equivalent Y network During

the research study, we develop the SMM to facilitate the modeling of coupling effectsamong densely populated vias in the multilayered package with finite power-groundplanes The transmission line mode of the microstrip lines and striplines can be

(b)

(c)Figure 5.2: Three sub-domains applied in the modal decoupling; (a) multilayeredP-G planes, (b) signal traces, and (c) through signal vias

analyzed by using the MTL theory [96], and extracted the equivalent circuit models.

A simple and accurate analytical formula for the discontinuity of through-hole signalvia is derived based on [97] to calculate the signal via’s parasitic capacitances and

its equivalent LC Π-circuit including the via pad’s inductance and capacitance is

modeled Then, the equivalent circuit models of all sub-domains are recombined

as cascading of the multi-ports networks at the transitions of signal traces to thethrough-hole signal vias

Mul-tiple Vias

Vias are widely employed in the electronic packages with the shape of circularcylinders Thus, the theory of multiple scattering among many parallel conduct-ing cylinders [88] can be used to model them efficiently The theory of scattering byconducting cylinders (vias) in the presence of PEC (perfect electric conductor) [55]planes has been applied to model the coupling effects of the power-ground vias in

a multilayered package [56, 57] In this research study, instead of using the Green’sfunction approach in [56, 57] to obtain the corresponding formulas, we have directlyapplied the parallel-plate waveguide theory to resolve the problem and developed thesemi-analytical scattering matrix method (SMM) for modeling of multiple scattering

of the vias in the electronic package as we discussed in Chapter 3 The proposedmethod is reported in [98–101]

Since the P-G planes of the package are assumed infinitely large in the ventional SMM, an important extension to the SMM has been made to handle thefinite-sized power-ground planes in advanced packages We assume a PMC (perfectmagnetic conductor) boundary on the periphery of a package This assumption

con-is made considering one of the major geometric features of the advanced packagestructures, i.e., the separation of the metal plates in the package is far less than itsoperating wavelength

By adding the PMC boundary, we confine a problem domain to a finite region.

A layer of the PMC cylinders is used at the periphery of the package to simulatethe finite domain of the P-G planes Hence, we extend the SMM algorithm withthe boundary of the PMC layer cylinders and the algorithm is now able to handlereal-world package structures The detailed formulation and validation of the SMMfor modeling of multiple scattering among the P-G vias in multilayered structure ispresented in Chapters 3 and 4

Consider a multiconductor transmission line (MTL) consisting of N conductors and

reference conductor immersed in homogeneous medium [102] The per-unit-lengthequivalent circuit model for derivation of multiconductor transmission line equations

for the N + 1 conductors is shown in Fig 5.3.

Figure 5.3: The per-unit-length equivalent circuit model for derivation of the mission line equations

trans-Writing Kirchhoff’s voltage law around the ith circuit consisting of the ith

con-ductor and the reference concon-ductor yields [102]

With the collection for all conductors, it can be written in a compact form using

matrix notations [R] and [L] as

∂

∂z V (z, t) = −[R]I (z, t) − [L] ∂

∂t I (z, t) (5.3)Similarly, the second MTL equation can be obtained by applying Kirchhoff’s

current law to the ith conductor in the per-unit-length equivalent circuit yields

With the collection for all conductors, it can be written in a compact form using

matrix notations [G] and [C] as

∂

∂z I (z, t) = −[G]V (z, t) − [C] ∂

∂t V (z, t) (5.6)

Then, the per-unit-length parameter matrices of resistance [R], inductance [L], conductance [G], and capacitance [C] are given as follows:

5.3.1 Properties of the Per-Unit-Length Parameters

For the case of multiconductor transmission line (MTL) consisting of N + 1 lossless conductors immersed in a homogeneous medium characterized by permeability µ, permittivity ε and conductivity σ, the per-unit-length parameter matrices are related

by

[L][C] = [C][L] = µε[U] and (5.11)

[L][G] = [G][L] = µσ[U ] , (5.12)

where [U ] is the N × N identity matrix.

For the case of MTL consisting N + 1 low-loss conductors ([R][C] << [L][G],

[C][R] << [G][L], and [R][G] → 0, [G][R] → 0) immersed in a homogeneous

medium characterized by permeability µ, permittivity ε, and conductivity σ, the

per-unit-length parameter matrices are related by

[L][C] = [C][L] ≈ µε[U ] and (5.13)

[L][G] = [G][L] ≈ µσ[U] (5.14)

The parameter matrices [L], [G], and [C] are symmetric and positive definite The inductance matrix [L] for the MTL with surrounding medium replaced by

another medium with the same permeability µ is the same as the original inductance

value [L] Thus, for the surrounding medium with permeability µ0, the inductance

matrix [L] is calculated by

[L] = µ0ε0[C0]−1 (5.15)

with [C0] being the per-unit-length parameter for the same MTL with the

surround-ing medium replaced by free-space

Therefore, for an inhomogeneous medium, these per-unit-length parameter trices are determined by a procedure of (1) computing the capacitance matrix with

ma-the inhomogeneous medium present, [C], (2) computing ma-the per-unit-length

capaci-tance matrix with the inhomogeneous medium removed and replaced with free space,

[C0], and then (3) computing the inductance matrix [L] from (5.15).

Also, the computing of conductance matrix [G] for an inhomogeneous medium

is developed from a modified capacitance calculation In order to obtain the unit-length capacitance and conductance matrices, we could use the capacitancesolver with each dielectric replaced by its complex permittivity:

per-ˆ

where tan δi is the loss tangent (at the particular frequency of interest) of the ithdielectric layer and tan δ i = (σ ef f,i /ε i) This will give a complex capacitance matrixas

[ ˆC] = [CR] + j[CI] (5.17)Then, we obtain the per-unit-length capacitance and conductance matrices as

and

where ω = 2πf is the radian frequency of interest.

5.3.2 Mode Decoupling of the Parameters in Frequency

notation t denotes for transpose matrix The resulting equations in (5.20) are a set

of coupled first-order ordinary differential equations with complex coefficients.Alternatively, the coupled first-order phasor MTL equations in (5.20) can beplaced in the form of uncoupled second-order ordinary differential equations by

differentiating one with respect to line position z and substituting the other, and

vice versa, to yield

observed In differentiating (5.20) with respect to line position z, we assumed that

the per-unit-length parameter matrices [R], [L], [G] and [C] are independent of z.

Thus, we have assumed the cross-sectional line dimensions and surrounding media

properties are invariant along the line (independent of z) or, in other words, the line

is a uniform line

Consider on solving the second-order differential equations given in (5.21)

No-tice that the equations in (5.21) are coupled together because [Z][Y ] and [Y ][Z]

are full matrices; i.e., each set of voltages and currents, V i (z) and I i (z), affects all the other sets of voltages and currents, V j (z) and I j (z) A change of variables is

used to decouple the second-order differential equations in (5.21) by putting them

into the form of N separate equations describing N isolated two-conductor lines.

Apply the solution techniques in [102] to these individual two-conductor lines andthen use the change of variables to return to the original voltages and currents Thismethod of using a change of variables is perhaps the most frequently used techniquefor generating the general solution to the MTL equation [103]

In implementing the method, we transform to mode quantities as

V (z) = [T V ] V m (z) (5.22a)

I (z) = [T I ] I m (z) (5.22b)

The N ×N complex matrices [T V ] and [T I] define a change of variables between the

actual phasor line voltages currents, V and I, and the mode voltages and currents,

V m and I m In order for this to be valid, these N ×N matrices must be non-singular;

i.e., [T V]−1 and [T I]−1 must exist, where we denote the inverse of an N × N matrix

[M] as [M ] −1 in order to go between both sets of variables Substituting these intothe second-order MTL equations in (5.21) gives

The objective is to decouple these second-order equations by finding a [T V ] and [T I]

that simultaneously diagonalize [Z][Y ] and [Y ][Z] via similarity transformations

where the mode impedance [Zm ] and mode admittance [Y m] are diagonal matrices.

From (5.26) and (5.27), we can get

[T V]−1 [Z][Y ][T V ] = [T I]−1 [Y ][Z][T I ] = [Z m ][Y m ] (5.28)Let

with γ2

m,i = Z m,i Y m,i The entries in γ2

m , γ2

m,i for i = 1, · · · , N, are the eigenvalues of

[Z][Y ] and [Y ][Z]1 The column vectors of [T V ] are eigenvectors of [Z][Y ], and

the column vectors of [T I ] are eigenvectors of [Y ][Z] The mode equations in (5.23)

and (5.24) will therefore be decoupled This will yield the N propagation constants

γ2

m,i of the N modes.

This is the classic eigenvalue/eigenvector problem of matrices Suppose we need

to find an N × N non-singular matrix [T ] that diagonalizes an N × N matrix [M]

as

[T ] −1 [M ][T ] = Λ , (5.31)

where Λ is an N × N diagonal matrix with Λi on the main diagonal and zeros

elsewhere Multiplying both sides of (5.31) by [T ] yields

[M][T ] − [T ]Λ = 0 , (5.32)

where 0 is the N × N zero matrix The N columns of [T ], [T i], are the eigenvectors

of [M ] and the N values Λi are the eigenvalues of [M] Equation (5.32) gives N

equations for the N eigenvectors as

([M] − Λ i1n ) [T i ] = 0, i = 1, · · · , N (5.33)

where 0 is the N ×1 zero vector that contains all zeros and the N ×1 column vectors

of the eigenvectors [T i] contain the unknowns to be solved for Equation (5.33) of a

set of homogeneous, algebraic equations are finally solved for the mode decoupling

matrices [T V ] and [T I ] using [Z][Y ] and [Y ][Z], respectively.

Solving the following first-order decoupled equations derived from (5.20),

d

d

dz I m (z) = −[Y m ] V m (z) (5.34b)

yields the ith mode voltages and currents, Vm,i and Im,i,

Vm,i (z) = A+m,i e −γ m,i z

+ A − m,i e γ m,i z

(5.35)

1If [P ] −1 [A][P ] = [D], [D] is a diagonal matrix, then [A][P ] = [P ][D] so that [A][P i] =

[P i ][D], where [P i ] is the column vector of [P ].

Z m,i Y m,i is the propagation constant, and Z m0,i =

Z m,i /Y m,i is the

mode characteristic impedance Let γm,i = αm,i + jβm,i, the velocity of the ithmode

is v m,i = ω/β m,i If the velocity of each mode is different, the signal distortion will

be introduced

For lossless case ([R] = [G] = 0), [Z] = jω[L] and [Y ] = jω[C], and then [T V]

and [T I]∈ R n×n.

For lossless ([R] = [G] = 0) and homogeneous surrounding medium (ε, µ) case,

[L][C] = [C][L] = µε[U ] Therefore, let [T V ] = [T I ] = [T ], the column vectors of [T ] are the eigenvectors of [L].

[T ] −1 [L][T ] = [L m] (5.37)

[T ] −1 [C][T ] = [C m ] = µε[L m]−1 (5.38)

Since [L] is symmetric matrix, [T ] is orthogonal matrix, i.e., [T ] −1 = [T ] t, where

the notation t denotes for transpose matrix In summary,

[C m ][L m ] = µε[U ] [Z m ] = jω[L m]

5.3.3 Impedance Matrix of the MTLs with Same Length l

Figure 5.4: The equivalent network for multiconductor transmission lines.The equivalent network for multiconductor transmission lines (MTLs) is given

in Fig 5.4 For the decoupled system, it can be written as

For lossless transmission lines,

Suppose [A]
is the permuted matrix of [A] with the size of N by N, then

⎧

⎪

⎪

2j
− 1, j
≤ N 2(j
− N), j
> N .Provide a N × N matrix [C] = [A]
[B], then

For lossless and homogeneous transmission lines, [T V ] = [T I ] = [T ] and [T ] is

orthogonal matrix, then

5.4 Modeling of Entire Signal Traces in Power

Distribution Network

As we discussed in the previous sections, the equivalent network parameters for thepower-ground planes with a large number of vias and the multiconductor signaltraces are efficiently modeled using the extended scattering matrix method (SMM)and the multiconductor transmission line (MTL) theory In this section, we willpresent efficient modeling technique to model an entire signal trace in power distri-bution network The equivalent network of the power-ground planes calculated inSection 5.2 is combined with the equivalent network of the signal traces, especiallywith the network of the striplines For the sake of simplicity, as shown in Fig 5.5, anexample of a signal trace passing through inside the P-G planes is used to present

a detailed procedure of the proposed method As demonstrated in Fig 5.2, anoverall equivalent network of the entire signal trace is considered as composition ofthe equivalent networks of the top/bottom microstrip lines and the power-groundplanes combined with the stripline model, and the closed-form equivalent circuit ofthrough-hole signal via

Figure 5.5: Signal trace route in the power-ground planes of power distributionnetwork

In the following sections, the modeling of striplines combined with power-groundplanes and the equivalent circuit of through-hole signal vias are illustrated Finally,the combination of all equivalent networks for the entire signal trace is demonstrated.The proposed combination method can be straightforwardly extended for the case

of the multiple striplines (signal traces).

5.4.1 Modeling of Striplines between Power-Ground Planes

For the stripline commonly used in microwave engineering, its two parallel referenceplanes are considered shorted and hence equal potential The cross section of thestripline (signal trace route) sandwiched between a pair of power-ground planes can

be seen in Fig 5.6 The thicknesses of the lower and upper substrates are hl and hu, whereas the widths of the planes and the signal conductor are w p and w s Supposethe conductor losses are negligible, the substrate between the planes is homogeneousand the electromagnetic fields are confined between the planes When the striplinesare routed between the P-G planes, the parallel-plate modes are excited and thevoltage drops are accumulated between the power and ground planes Based on theskin effect approximation, we split each stripline into upper and lower transmissionlines, as shown in Fig 5.6, by considering the potential difference between the planes

The subscripts L and R represent the left and right ports of the striplines and the superscripts Su and Sl denote the split upper and lower striplines, respectively.

Figure 5.6: Cross-section view of the stripline route

For the original stripline in Fig 5.7 (a), the admittance matrix is defined as

(a) Original stripline

(b) Split modelFigure 5.7: Transmission line representations of the stripline and its split model.For the split model of the stripline in Fig 5.7 (b), the admittance matrices aredefined as follows: ⎡

⎢ I L Su

I Su R

⎤

⎥=

Y ustrip

⎡⎢ V Su L

V Su R

L/R , I Sl L/R stand for port voltages and currents de-fined in Fig 5.7

The split stripline is equivalent to the unsplit (original) stripline when the powerand ground planes are shorted By shorting the power and ground planes of the splitstripline, we get ⎡

⎢

⎣ V

Su L

V Su R

V Sl R

When the stripline approaches the power plane (h u → 0 in Fig 5.6), I Su

L/R = I L/R and I Sl

L/R = 0 Conversely, when the stripline approaches the ground plane (h u → h

in Fig 5.6), I L/R Sl = I L/R and I L/R Su = 0 Based on this observation of the skin effect

approximation and the dimension h is much small, the relations of the port voltages

and currents are concluded that

⎡

⎢ I L Su

I Su R

⎤

⎥= h l h

⎤

⎥= h u h

The combined network is a four-port network with the admittance matrix

I u R

V u R

L/R , I l L/R defined in Figs 5.8 and5.9 These four ports are considered for the connections between the stripline andthe possible external signal traces

Figure 5.8: Port voltages and currents defined for three equivalent networks.

Figure 5.9: Combination for the equivalent Y-networks of the power-ground planesand the split stripline

Analyzing with the extended SMM, the admittance matrix

Yplane



of the P-Gplanes with multiple vias, as shown in Fig 5.9, can be expressed as

⎡

⎢

⎣ I

a L

Figure 5.9 demonstrates the combination for the

equiv-alent Y networks of the power-ground planes and the split stripline, in other words,

the re-coupling of the stripline mode and parallel-plate mode for the stripline routed

between the P-G planes in terms of the Y networks: Y ustrip, Y lstrip and Yplane.From Figs 5.8 and 5.9, we have

V Sl L

V Sl R

V L a

V a R

V u R

V l L

I u R

I l L

I Sl L

I Sl R

I L a

I a R

and [U] is a 2 × 2 unit matrix The superscript t in (5.64) means the transpose to

the matrix Substituting (5.56), (5.57), (5.63) and (5.64) into (5.62), we get

strip+ Yplane −Yplane

−Yplane Y Slstrip+ Yplane

⎤

⎥

The proposed re-coupling method for the stripline mode and the parallel-plate mode

is conveniently extended to include multiple striplines In such cases, the scalar

values of the voltages V L/R u , V L/R l , V L/R Su , V L/R Sl and the currents I L/R u , I L/R l , I L/R Su , I L/R Sl

will be extended to vectors

5.4.2 Equivalent Circuit Model of Through-Hole Signal Vias

One of the important discontinuity structures in an electronic package is a hole via connecting the signal traces in the different layers [104, 105] In Fig 5.10,the through-hole via and its equivalent Π-circuit are presented

through-Figure 5.10: Through-hole signal via and its equivalent circuit

The discontinuity structure of the though-hole signal via is modeled as the LC

Π-network including the via pad’s inductance and capacitance as shown in Fig 5.10.The via pad is the connection point of the microstrip line and the through-hole

signal via The inductance L pad and capacitance C pad values in the via pad’s model

are given from an optimization technique as L pad = (802 × c/2) pH and Cpad =(132× c/2 + 54) fF [39], where c denotes the pad radius in mm.

The closed-form value of the inductance L in Fig 5.10 can be obtained from [69].

In [97], the boundary of the via region is assumed as a perfect magnetic cylinder

with radius R, as shown in Fig 5.11 Cylindrical wave expansion is then used to calculate the capacitors C1 and C2 In this research, we assume R → ∞, and apply

the asymptotic expansions of Bessel functions to derive the closed-form formulas of

capacitors C1 and C2 The inductance L is calculated as follows:

L = µ 2π

where the PEC boundary is used for ρ = a and the PMC boundary is used for ρ = R

as shown in Fig 5.11 I1 and I0 are the modified Bessel functions of first kind; and

K1 and K0 are the modified Bessel functions of second kind

where h = h1+ h2 and K0 is the modified Bessel function of second kind with zeroorder.

For x > 0, since K0
(x) = −K1(x) < 0, then K0(x) monotonously decreases for

x > 0 If knb > kna > 30, due to avoiding the computing error for large arguments

in Bessel functions and using the asymptotic expansions, then the capacitances in(5.70) can be rewritten as

5.4.3 Combination of Equivalent Networks for Modeling of

Entire Signal Trace

As we have demonstrated the re-coupling procedure of stripline mode and plate mode for the stripline and the power-ground planes and the equivalent circuitmodel of through-hole signal via in the previous sections, the entire equivalent net-work for a typical structure of the signal trace routed in the PDN can be finallyillustrated as shown in Fig 5.12 A novel procedure for combination of the equiv-alent networks of the parallel-plate mode, stripline mode and microstrip line mode

parallel-is developed to model the entire signal traces routed in the power dparallel-istribution work For combination procedure, we use the equivalent ABCD parameters for eachnetworks and the generalized cascading of ABCD matrices for the entire system isdiscussed details in Appendix B

net-Finally, the proposed hybrid algorithm is developed based on the semi-analyticalapproach of scattering matrix method with the FDCL and the transmission linetheory for extraction of equivalent circuit parameters of the signal traces Hence, thehybrid algorithm simulation performs very much faster than the conventional full-wave simulation methods For the case of multiple signal traces coupling in the PDN,the procedure here can simply be extended by using modal propagation constants,modal characteristic impedances, and modal voltages and currents representing themulticonductor signal traces

Figure 5.12: Overall equivalent network for the signal trace routed in the powerdistribution network.

5.5 Numerical Simulations of Hybrid Modeling

Algorithm for Signal Traces in PDN

For validation of the developed hybrid algorithm, an example of a signal trace ing through the power-ground planes is considered as shown in Fig 5.13 The

pass-dielectric material of all layers is considered FR4 with a pass-dielectric constant of 4.1.

All the dimensions are shown in Fig 5.13 By using the modal decomposition andcombination procedure of the algorithm, the entire signal trace can be described asequivalent networks as shown in Fig 5.12 The reflection and transmission coeffi-cients (S11 at Port 1 and S21 between Ports 1 and 2) of the trace are simulated andcompared the simulated results with the ones of the Ansoft HFSS Good agreement

of the results is observed in Fig 5.14 The sharp drops of S21observed in the resultsare due to the strong resonances of the power-ground planes At these resonantfrequencies, the impedance of the return current path for the signal is greatly in-creased To eliminate those resonances, the decoupling capacitors are commonlyused in practical electronic package design

In next example, we consider the six decoupling capacitors mounted on thetop plane in the previous example of Fig 5.13 The locations of the decouplingcapacitors are shown in Fig 5.15 Each decoupling capacitor has the value of 10 nF,

an equivalent series inductance of 0.1 nH, an equivalent series resistance of 0.5 Ω.

The simulated results of the S-parameters for the signal trace with the presence of thedecoupling capacitors are plotted in Fig 5.16 From the results, it is observed thatthe resonances in the package are greatly eliminated by the decoupling capacitors

We also consider a series of examples for analysis of coupled signal traces routed

in the PDN including the P-G vias Figure 5.17 shows an example of two coupledsignal traces (Case1) routed between the power-ground planes Four through-holevias are used to connect the signal traces Several P-G vias are also placed betweenthe planes to analyze the coupling effects between them and the traces The di-

electric material for all substrates is FR4 with a dielectric constant of 4.4 and loss

Figure 5.13: The dimensions of a signal trace routed between the power-groundplanes (top view and side view) (unit: mm).

(a) S11

(b) S21

Figure 5.14: Reflection and transmission characteristics of the signal trace shown inFig 5.13

Figure 5.15: The dimensions of a signal trace routed between the power-groundplanes (top view and side view) The black dots represent the decoupling capacitors(unit: mm).

(a) S11

(b) S21

Figure 5.16: Reflection and transmission characteristics of the signal trace with thedecoupling capacitors shown in Fig 5.15

Figure 5.17: The dimensions of two coupled signal traces routed between the ground planes (top view and side view) - Case 1 All black dots represent the P-Gvias (unit: mm).