Chapter 5
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 or
power supply traces residing in the different layers. Since parallel-plate waveguide
modes are introduced by layered structures, the signals on active vias can excite the
waveguide 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, along
with signal integrity loss, inappropriate switching, and longer signal delay. Another
potential problem in multilayered packages is that when harmonics of the transient
signal coincide with the resonant frequency of the power-ground planes, it will cause
electromagnetic compatibility (EMC) problems in the microprocessor packages. To
minimize such noise behavior, pre-layout and post-layout verifications of the power
distribution network (PDN) are necessary. Thus, accurate analysis of the signal
traces in the PDN has become of vital important for optimizing the performance of
high-speed digital circuits.
130
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 131
In this chapter, an efficient modeling technique based on the modal decompo-
sition of electromagnetic fields is proposed to analyze the power distribution net-
work of an electronic package, which includes the signal traces and the multilayered
power-ground planes with multiple through-hole vias. The total electromagnetic
fields inside the package are decomposed into two modes: the parallel-plate mode
and the transmission line mode. The propagation of the fields between the P-G
planes is considered as parallel-plate mode and is efficiently analyzed by using the
scattering 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 microstrip
line 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 analytical
modeling of equivalent circuits. Finally, by cascading the equivalent networks, the
overall network parameter for the system is obtained to analyze the coupling effects
of the P-G vias to the signal traces.
5.1 Methodology for Hybridization of SMM and
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 metal
power-ground planes and vias are to provide a low-impedance path for the power
distribution system between the printed circuit board and the die. The signal traces
reside between the different layers and their return currents flow in the reference
planes just below them. When the traces pass through different layers, the vertical
through-hole vias are placed to ensure the continuity of the return current.
Consider that substrates sandwiched between the metal planes are very thin
compared 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
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 132
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 multiconductor
transmission line (MTL) and divided into two parts: the traces between the P-G
planes as strip line mode and the traces on top/bottom of the package as microstrip
line mode. It is assumed that the MTL has a uniform cross section, allowing the
propagation of quasi-TEM waves along the traces [95]. Hence, the total electromag-
netic fields propagating inside the package can be decomposed into two independent
modes: the parallel-plate mode and the transmission line mode.
Mode conversions usually occur at the transition between the signal trace and
the through-hole via. At the via hole, the parallel-plate mode gets excited due to
the switching signal currents, and conversely, the noise voltages between the P-G
planes gets coupled to the stripline mode. A novel modal decomposition approach is

applied 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 the
through-hole signal vias. The parallel-plate mode of the P-G planes with a large
number of vias is analyzed by using the scattering matrix method (SMM), which is
basedontheN-body scattering theory, to model the equivalent Y network. During
the research study, we develop the SMM to facilitate the modeling of coupling effects
among densely populated vias in the multilayered package with finite power-ground
planes. The transmission line mode of the microstrip lines and striplines can be
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 133
(a)
(b)
(c)
Figure 5.2: Three sub-domains applied in the modal decoupling; (a) multilayered
P-G planes, (b) signal traces, and (c) through signal vias.
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 134
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 signal
via 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 the
through-hole signal vias.
5.2 Modeling of Power-Ground Planes with Mul-
tiple Vias
Vias are widely employed in the electronic packages with the shape of circular
cylinders. Thus, the theory of multiple scattering among many parallel conduct-
ing cylinders [88] can be used to model them efficiently. The theory of scattering by
conducting 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’s
function approach in [56,57] to obtain the corresponding formulas, we have directly
applied the parallel-plate waveguide theory to resolve the problem and developed the
semi-analytical scattering matrix method (SMM) for modeling of multiple scattering
of the vias in the electronic package as we discussed in Chapter 3. The proposed
method is reported in [98–101].
Since the P-G planes of the package are assumed infinitely large in the con-
ventional SMM, an important extension to the SMM has been made to handle the
finite-sized power-ground planes in advanced packages. We assume a PMC (perfect
magnetic conductor) boundary on the periphery of a package. This assumption
is made considering one of the major geometric features of the advanced package
structures, i.e., the separation of the metal plates in the package is far less than its
operating wavelength.
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 135
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 simulate
the finite domain of the P-G planes. Hence, we extend the SMM algorithm with
the boundary of the PMC layer cylinders and the algorithm is now able to handle
real-world package structures. The detailed formulation and validation of the SMM
for modeling of multiple scattering among the P-G vias in multilayered structure is
presented in Chapters 3 and 4.
5.3 Modeling of Multiconductor Signal Traces
Consider a multiconductor transmission line (MTL) consisting of N conductors and
reference conductor immersed in homogeneous medium [102]. The per-unit-length
equivalent 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 trans-
mission line equations.
Writing Kirchhoff’s voltage law around the i
th

circuit consisting of the i
th
con-
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 136
ductor and the reference conductor yields [102]
V
i
(z,t)=r
i
∆zI
i
(z,t)+r
0
∆z
N

k=1
I
k
(z,t)+∆z
N

k=1
l
ik
∂I
k
(z,t)
∂t
+V

i
(z +∆z,t) . (5.1)
Dividing both sides by ∆z and taking the limit as ∆z → 0, the first transmission
line (MTL) equation for the i
th
conductor is given as
∂V
i
(z,t)
∂z
= −r
i
∆zI
i
(z,t) − r
0
∆z
N

k=1
I
k
(z,t) − ∆z
N

k=1
l
ik
∂I
k

(z,t)
∂t
. (5.2)
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 i
th
conductor in the per-unit-length equivalent circuit yields
I
i
(z +∆z,t) − I
i
(z,t)
= −∆z
N

k=1,k=i
g
ik
[V
i
(z +∆z,t) − V
k

(z +∆z,t)] − ∆zg
ii
V
i
(z +∆z,t)
−∆z
N

k=1,k=i
c
ik
∂
∂t
[V
i
(z +∆z,t) − V
k
(z +∆z,t)] − ∆zc
ii
∂V
i
(z +∆z,t)
∂t
=∆z
N

k=1,k=i
g
ik
V

k
(z +∆z,t) − ∆z
n

k=1
g
ik
V
i
(z +∆z,t)
+∆z
N

k=1,k=i
c
ik
∂V
k
(z +∆z,t)
∂t
− ∆z
n

k=1
c
ik
∂V
i
(z +∆z,t)
∂t

.
(5.4)
Dividing both sides by ∆z and taking the limit as ∆z → 0, the first transmission
line (MTL) equation for the i
th
conductor is given as
∂I
i
(z,t)
∂z
=
N

k=1,k=i
g
ik
V
k
(z +∆z,t) −
N

k=1
g
ik
V
i
(z +∆z,t)
+
N


k=1,k=i
c
ik
∂V
k
(z +∆z,t)
∂t
−
N

k=1
c
ik
∂V
i
(z +∆z,t)
∂t
.
(5.5)
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)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 137
Then, the per-unit-length parameter matrices of resistance [R], inductance [L],
conductance [G], and capacitance [C] are given as follows:

[R] =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
r
1
+ r
0
r
0
··· r
0
r
0
r
2
+ r
0
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
r
0
r
0
··· r
0
r
N
+ r
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(5.7)
[L] =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
l
11
l
12
··· l
1N
l
12
l
22
··· l
2N
.
.
.
.
.
.

.
.
.
.
.
.
l
1N
l
2N
··· l
NN
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(5.8)
[G] =
⎛
⎜
⎜
⎜
⎜
⎜

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
N

k=1
g
1k
−g
12
··· −g
1N
−g
12
N

k=1
g
2k
.
.
.
−g

2N
.
.
.
.
.
.
.
.
.
.
.
.
−g
1N
−g
2N
···
N

k=1
g
Nk
⎞
⎟
⎟
⎟
⎟
⎟
⎟

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(5.9)
[C] =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
N

k=1

c
1k
−c
12
··· −c
1N
−c
12
N

k=1
c
2k
.
.
.
−c
2N
.
.
.
.
.
.
.
.
.
.
.
.

−c
1N
−c
2N
···
N

k=1
c
Nk
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (5.10)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 138
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 µ
0
, the inductance
matrix [L] is calculated by
[L] = µ
0
ε
0
[C
0
]
−1
(5.15)
with [C

0
] 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 ma-
trices are determined by a procedure of (1) computing the capacitance matrix with
the inhomogeneous medium present, [C], (2) computing the per-unit-length capaci-
tance matrix with the inhomogeneous medium removed and replaced with free space,
[C
0
], and then (3) computing the inductance matrix [L] from (5.15).
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 139
Also, the computing of conductance matrix [G] for an inhomogeneous medium
is developed from a modified capacitance calculation. In order to obtain the per-
unit-length capacitance and conductance matrices, we could use the capacitance
solver with each dielectric replaced by its complex permittivity:
ˆε
i
= ε
i
(1 − j tan δ
i
) (5.16)
where tan δ
i
is the loss tangent (at the particular frequency of interest) of the i
th
dielectric layer and tan δ
i
=(σ
ef f,i

/ε
i
). This will give a complex capacitance matrix
as
[
ˆ
C] = [C
R
] + j[C
I
] . (5.17)
Then, we obtain the per-unit-length capacitance and conductance matrices as
[C] = [C
R
] (5.18)
and
[G] = −ω [C
I
] (5.19)
where ω =2πf is the radian frequency of interest.
5.3.2 Mode Decoupling of the Parameters in Frequency Do-
main
By using (5.3) and (5.6), the frequency-domain MTL equations can be written as
d
dz
V (z)=−[Z] I (z) (5.20a)
d
dz
I (z)=−[Y ] V (z) (5.20b)
with [Z] = [R] + jω[L] and [Y ] = [G] + jω[C].Since[R], [L], [G] and [C] are

symmetric, [Z] and [Y ] are symmetric and [Z] = [Z]
t
and [Y ] = [Y ]
t
, where the
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 be
placed in the form of uncoupled second-order ordinary differential equations by
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 140
differentiating one with respect to line position z and substituting the other, and
vice versa, to yield.
d
2
dz
2
V (z)=[Z][Y ] V (z) (5.21a)
d
2
dz
2
I (z)=[Y ][Z] I (z) (5.21b)
Ordinarily, the per-unit-length parameter matrices [Z] and [Y ] do not commute,
i.e., [Z][Y ] = [Y ][Z], so that the proper order of multiplication in (5.21) must be
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)andI
i
(z), affects all
the other sets of voltages and currents, V
j
(z)andI
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 and
then use the change of variables to return to the original voltages and currents. This
method of using a change of variables is perhaps the most frequently used technique
for 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
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 141
[M] as [M]
−1
in order to go between both sets of variables. Substituting these into
the second-order MTL equations in (5.21) gives
d
2
dz
2
V
m
(z)=[T

V
]
−1
[Z][Y ][T
V
] V
m
(z)
= γ
2
m
V
m
(z)
(5.23)
d
2
dz
2
I
m
(z)=[T
I
]
−1
[Y ][Z][T
I
] I
m
(z)

= γ
2
m
I
m
(z) .
(5.24)
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
as
[T
V
]
−1
[Z][Y ][T
V
] = γ
2
m
(5.25a)
[T
I
]
−1
[Y ][Z][T
I

] = γ
2
m
. (5.25b)
Also, considering non-singular matrices [T
V
] and [T
I
] to diagonalize [Z] and
[Y ] is given as
[Z
m
] = [T
V
]
−1
[Z][T
I
] (5.26)
[Y
m
] = [T
I
]
−1
[Y ][T
V
] , (5.27)
where the mode impedance [Z
m

] 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
[D] = [Z
m
][Y
m
] = γ
2
m
(5.29)

where γ
2
m
is an N × N diagonal matrix:
γ
2
m
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
γ
2
m,1
0 ··· 0
0 γ
2
m,2
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 γ
2
m,N
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(5.30)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 142
with γ
2
m,i
= Z

m,i
Y
m,i
. The entries in γ
2
m
, γ
2
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
γ
2
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 Λ

i
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 Λ
i
are the eigenvalues of [M ]. Equation (5.32) gives N
equations for the N eigenvectors as
([M] − Λ
i
1
n
) [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

dz
V
m
(z)=−[Z
m
] I
m
(z) (5.34a)
d
dz
I
m
(z)=−[Y
m
] V
m
(z) (5.34b)
yields the i
th
mode voltages and currents, V
m,i
and I
m,i
,
V
m,i
(z)=A
+
m,i
e

−γ
m,i
z
+ A
−
m,i
e
γ
m,i
z
(5.35)
1
If [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 ].
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 143
I
m,i
(z)=
1
Z
m0,i


A
+
m,i
e
−γ
m,i
z
− A
−
m,i
e
γ
m,i
z

, (5.36)
where γ
m,i
=

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 γ
m,i
= α
m,i
+ jβ
m,i
,thevelocityofthei
th
mode
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],andthen[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
]
[Y
m
] = jω[C
m
]
γ
m,i
= jβ
m,i
= jω
√
µε
Z
m0,i
=

L
m,i
C
m,i
.
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 144
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


Z
net
m



I
net
m

=

V
net
m

, (5.39)
where

V
net
m

=
⎡
⎢
⎣
V
net
mL

V
net
mR
⎤
⎥
⎦
,

V
net
mL

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
V
net
mL,1
.
.
.
V
net
mL,N
⎤

⎥
⎥
⎥
⎥
⎥
⎦
,

V
net
mR

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
V
net
mR,1
.
.
.
V
net
mR,N
⎤

⎥
⎥
⎥
⎥
⎥
⎦
(5.40)

I
net
m

=
⎡
⎢
⎣
I
net
mL
I
net
mR
⎤
⎥
⎦
,

I
net
mL


=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I
net
mL,1
.
.
.
I
net
mL,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, and

I
net
mR


=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I
net
mR,1
.
.
.
I
net
mR,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (5.41)

Z
net
m


is the modal impedance matrix for all decoupled transmission lines as dis-
cussed in the previous section.

Z
net
m

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Z
net
m,1
0
.
.
.
0 Z
net
m,N
⎤
⎥
⎥
⎥

⎥
⎥
⎦
(5.42)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 145
For lossless transmission lines,

Z
net
m,i

=
Z
m0,i
j sin (β
m,i
l)
⎡
⎢
⎣
cos (β
m,i
l)1
1cos(β
m,i
l)
⎤
⎥
⎦
(5.43)


Y
net
m,i

=
1
jZ
m0,i
sin (β
m,i
l)
⎡
⎢
⎣
−cos (β
m,i
l)1
1 −cos (β
m,i
l)
⎤
⎥
⎦
. (5.44)
The notation prime “

” in (5.39) means the permutation defined as follows.
Suppose [A]


is the permuted matrix of [A] with the size of N by N,then
a

i

,j

= a
i,j
,wherei =
⎧
⎪
⎨
⎪
⎩
2i

−1,i

≤ N
2(i

−N),i

>N
and j =
⎧
⎪
⎨
⎪

⎩
2j

−1,j

≤ N
2(j

−N),j

>N
.
Provide a N × N matrix [C] = [A]

[B],then
c
i

,j

=
N

k

=1
a

i


,k

b
k

,j

=
N

k

=1
a
i,k
b
k

,j

=
N

k

=1
a

2i


−1
2(i

−N)

,

2k

−1
2(k

−N)

b
k

,j

. (5.45)
If [C] = [B][A]

,then
c
i

,j

=
N


k

=1
b
i

,k

a

k

,j

=
N

k

=1
b
i

,k

a
k,j
=
N


k

=1
b
i

,k

a

2k

−1
2(k

−N)

,

2j

−1
2(j

−N)

. (5.46)
For original coupled transmission lines system, it has


Z
net

I
net

=

V
net

, (5.47)
with

V
net

=
⎡
⎢
⎣
V
net
L
V
net
R
⎤
⎥
⎦

,

V
net
L

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
V
net
L,1
.
.
.
V
net
L,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦

,

V
net
R

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
V
net
R,1
.
.
.
V
net
R,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦

(5.48)

I
net

=
⎡
⎢
⎣
I
net
L
I
net
R
⎤
⎥
⎦
,

I
net
L

=
⎡
⎢
⎢
⎢
⎢

⎢
⎣
I
net
L,1
.
.
.
I
net
L,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, and

I
net
R

=
⎡
⎢
⎢
⎢
⎢

⎢
⎣
I
net
R,1
.
.
.
I
net
R,N
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (5.49)
Since we have

V
net

=
⎡
⎢
⎣
T
V

0
0 T
V
⎤
⎥
⎦

V
net
m

and

I
net

=
⎡
⎢
⎣
T
I
0
0 T
I
⎤
⎥
⎦

I

net
m

, (5.50)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 146
then

Z
net

=
⎡
⎢
⎣
T
V
0
0 T
V
⎤
⎥
⎦

Z
net
m


⎡
⎢

⎣
T
−1
I
0
0 T
−1
I
⎤
⎥
⎦
(5.51)

Y
net

=
⎡
⎢
⎣
T
I
0
0 T
I
⎤
⎥
⎦

Y

net
m


⎡
⎢
⎣
T
−1
V
0
0 T
−1
V
⎤
⎥
⎦
. (5.52)
For lossless and homogeneous transmission lines, [T
V
] = [T
I
] = [T ] and [T ] is
orthogonal matrix, then

Z
net

=
⎡

⎢
⎣
T 0
0 T
⎤
⎥
⎦

Z
net
m


⎡
⎢
⎣
T
t
0
0 T
t
⎤
⎥
⎦
(5.53)

Y
net

=

⎡
⎢
⎣
T 0
0 T
⎤
⎥
⎦

Y
net
m


⎡
⎢
⎣
T
t
0
0 T
t
⎤
⎥
⎦
. (5.54)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 147
5.4 Modeling of Entire Signal Traces in Power
Distribution Network
As we discussed in the previous sections, the equivalent network parameters for the

power-ground planes with a large number of vias and the multiconductor signal
traces are efficiently modeled using the extended scattering matrix method (SMM)
and the multiconductor transmission line (MTL) theory. In this section, we will
present efficient modeling technique to model an entire signal trace in power distri-
bution network. The equivalent network of the power-ground planes calculated in
Section 5.2 is combined with the equivalent network of the signal traces, especially
with the network of the striplines. For the sake of simplicity, as shown in Fig. 5.5, an
example 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, an
overall equivalent network of the entire signal trace is considered as composition of
the equivalent networks of the top/bottom microstrip lines and the power-ground
planes combined with the stripline model, and the closed-form equivalent circuit of
through-hole signal via.
Figure 5.5: Signal trace route in the power-ground planes of power distribution
network .
In the following sections, the modeling of striplines combined with power-ground
planes 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
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 148
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 reference
planes are considered shorted and hence equal potential. The cross section of the
stripline (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 h
l
and h
u
,

whereas the widths of the planes and the signal conductor are w
p
and w
s
. Suppose
the conductor losses are negligible, the substrate between the planes is homogeneous
and the electromagnetic fields are confined between the planes. When the striplines
are routed between the P-G planes, the parallel-plate modes are excited and the
voltage drops are accumulated between the power and ground planes. Based on the
skin effect approximation, we split each stripline into upper and lower transmission
lines, 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
⎡
⎢
⎣
I
L
I
R
⎤
⎥
⎦
=

Y
strip


⎡
⎢
⎣
V
L
V
R
⎤
⎥
⎦
. (5.55)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 149
(a) Original stripline
(b) Split model
Figure 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 are
defined as follows:
⎡
⎢
⎣
I
Su
L
I
Su
R
⎤
⎥
⎦
=


Y
u
strip

⎡
⎢
⎣
V
Su
L
V
Su
R
⎤
⎥
⎦
, (5.56)
⎡
⎢
⎣
I
Sl
L
I
Sl
R
⎤
⎥
⎦

=

Y
l
strip

⎡
⎢
⎣
V
Sl
L
V
Sl
R
⎤
⎥
⎦
, (5.57)
where V
L/R
,V
Su
L/R
,V
Sl
L/R
and I
L/R
,I

Su
L/R
,I
Sl
L/R
stand for port voltages and currents de-
finedinFig.5.7.
The split stripline is equivalent to the unsplit (original) stripline when the power
and ground planes are shorted. By shorting the power and ground planes of the split
stripline, we get
⎡
⎢
⎣
V
Su
L
V
Su
R
⎤
⎥
⎦
=
⎡
⎢
⎣
V
Sl
L
V

Sl
R
⎤
⎥
⎦
=
⎡
⎢
⎣
V
L
V
R
⎤
⎥
⎦
. (5.58)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 150
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
Sl
L/R
= I
L/R
and I
Su
L/R
= 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
Su
L
I
Su
R
⎤
⎥
⎦
=
h
l
h
⎡
⎢

⎣
I
L
I
R
⎤
⎥
⎦
and
⎡
⎢
⎣
I
Sl
L
I
Sl
R
⎤
⎥
⎦
=
h
u
h
⎡
⎢
⎣
I
L

I
R
⎤
⎥
⎦
. (5.59)
Substituting (5.58) and (5.59) into (5.55)-(5.57), we get

Y
u
strip

=
h
l
h

Y
strip

and (5.60)

Y
l
strip

=
h
u
h


Y
strip

, (5.61)
where

Y
strip

is the characteristic admittance of the original stripline. The model
used in Fig. 5.7(b) ensures that the magnetic field produced by the signal trace is
confined between the transmission line and the corresponding plane, and is valid for
high frequencies. We also observed that

Y
strip

=

Y
u
strip

+

Y
l
strip


.
For the stripline inside the P-G planes shown in Fig. 5.6, it can be modeled as
the combination of three networks: the upper stripline, bottom stripline and the
equivalent network of the P-G planes with vias. Figure 5.8 shows the port voltages
and currents of these three networks and the connections of their equivalent networks
are shown in Fig. 5.9.
The combined network is a four-port network with the admittance matrix de-
fined as
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I
u
L
I
u
R
I
l
L
I
l
R

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=

Y

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
V
u
L
V
u
R

V
l
L
V
l
R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(5.62)
with the port voltages V
u
L/R
,V
l
L/R
and currents I
u
L/R
,I
l
L/R
defined in Figs. 5.8 and

5.9. These four ports are considered for the connections between the stripline and
the possible external signal traces.
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 151
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 planes
and the split stripline.
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 152
Analyzing with the extended SMM, the admittance matrix

Y
plane

of the P-G
planes with multiple vias, as shown in Fig. 5.9, can be expressed as
⎡
⎢
⎣
I
a
L
I
a
R
⎤
⎥
⎦
=

Y
plane


⎡
⎢
⎣
V
a
L
V
a
R
⎤
⎥
⎦
(5.63)
with

Y
plane

=

Z
plane

−1
. 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
u

strip
, Y
l
strip
and Y
plane
.
From Figs. 5.8 and 5.9, we have
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
V
Su
L
V
Su

R
V
Sl
L
V
Sl
R
V
a
L
V
a
R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

=

P

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
V
u
L
V
u
R
V
l
L
V
l
R
⎤
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎦
and
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I
u
L
I
u
R
I
l
L
I
l
R
⎤

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=

P

t
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣
I
Su
L
I
Su
R
I
Sl
L
I
Sl
R
I
a
L
I
a
R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎦
, (5.64)
where

P

=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
U 0
0 U
−UU
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, (5.65)

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

Y

=

P

t
·
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Y
Su
strip
00
0 Y
Sl
strip
0
00Y
plane
⎤
⎥

⎥
⎥
⎥
⎥
⎦
·

P

. (5.66)
Finally, the total admittance for the combination of the stripline model and the P-G
planes model is given as

Y

=
⎡
⎢
⎣
Y
Su
strip
+ Y
plane
−Y
plane
−Y
plane
Y
Sl

strip
+ Y
plane
⎤
⎥
⎦
. (5.67)
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 153
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
u
L/R
,V
l
L/R
,V
Su
L/R
,V
Sl
L/R
and the currents I
u
L/R
,I
l
L/R
,I
Su

L/R
,I
Sl
L/R
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 through-
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.
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 C
pad
=
(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 C
1
and C

2
. In this research, we assume R →∞, and apply
the asymptotic expansions of Bessel functions to derive the closed-form formulas of
Chapter 5. Hybrid Modeling of Signal Traces in Power Distribution Network 154
capacitors C
1
and C
2
. The inductance L is calculated as follows:
L =
µ
2π

h · log

h +
√
a
2
+ h
2

/a

−
√
a
2
+ h
2

+ a

. (5.68)
Figure 5.11: PEC/PMC boundaries defined for analysis of the via region as a
bounded coaxial cavity.
The values of the capacitances C
1
and C
2
are given as
C
1(2)
=
4πε
h
1(2)
ln (b/a)
2N −1

n=1,3,5,
1
k
2
n

1 −
I
1
(k
n

R) K
0
(k
n
b)+K
1
(k
n
R) I
0
(k
n
b)
I
1
(k
n
R) K
0
(k
n
a)+K
1
(k
n
R) I
0
(k
n
a)


,
(5.69)
where the PEC boundary is used for ρ = a and the PMC boundary is used for ρ = R
as shown in Fig. 5.11. I
1
and I
0
are the modified Bessel functions of first kind; and
K
1
and K
0
are the modified Bessel functions of second kind.
k
n
=





nπ
2h

2
−
1
λ
2

g
with λ
g
=
c
f
√
ε
r
being the wavelength in the dielectric sub-
strate.
Let R →∞, the calculation of the capacitances in (5.69) can be reduced to
C
1(2)
=
4πε
h
1(2)
ln (b/a)
2N −1

n=1,3,5,
1
k
2
n

1 −
K
0

(k
n
b)
K
0
(k
n
a)

, (5.70)