Efficient modeling of power and signal integrity for semiconductors and advanced electronic package systems 3
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Chapter 4
Modeling for Multilayered
Power-Ground Planes in Power
Distribution Network
The scattering matrix method (SMM) with FDCL for analysis of multiple vias in
the single layer package (a pair of power-ground planes) has been presented in the
previous chapter. Using the several numerical examples, the developed algorithm
is validated by comparing the simulated results with analytical solutions and mea-
surement data. However, there are multiple layers (pairs of power-ground planes) in
practical structure of power distribution network for an advanced electronic package.
In this chapter, the formula derivation for multilayered structure of power-
ground planes in an advanced electronic package is presented. The procedure is
illustrated using the modal expansions of parallel-plate waveguide (PPWG) and the
mode matching in the anti-pad region of the via. Firstly, a case of two-layer struc-
ture of the power-ground planes is considered for formula derivation as shown in
Fig. 4.1. It has a case of three PPWGs - PPWG I, II, and III. Later, the formula-
tion of the multilayered power-ground planes is given for general case. Numerical
simulations for the multilayered power-ground planes with vias are presented and
validated with full-wave numerical method.
92
Chapter 4. Modeling for Multilayered Power-Ground Planes 93
Figure 4.1: A though-hole via in two-layer structure and forming three PPWGs.
4.1 Modal Expansions and Boundary Conditions
As discussed in Chapter 3, the tangential fields w.r.t ρ inside the two-layer structure
(Fig. 4.1) can be expressed by modal expansions as
E
(i)
t
=
∞
n=−∞
∞
m=0
⎧
⎪
⎨
⎪
⎩
a
e(i)
mn
J
(i)
mn
+ b
e(i)
mn
H
(i)
mn
e
e(i)
t,mn
+
a
h(i)
mn
J
(i)
mn
+ b
h(i)
mn
H
(i)
mn
e
h(i)
t,mn
⎫
⎪
⎬
⎪
⎭
e
jnφ
(4.1)
H
(i)
t
=
∞
n=−∞
∞
m=0
⎧
⎪
⎨
⎪
⎩
a
e(i)
mn
J
(i)
mn
+ b
e(i)
mn
H
(i)
mn
h
e(i)
t,mn
+
a
h(i)
mn
J
(i)
mn
+ b
h(i)
mn
H
(i)
mn
h
h(i)
t,mn
⎫
⎪
⎬
⎪
⎭
e
jnφ
(4.2)
wherewehavea
h(i)
0n
= b
h(i)
0n
=0forTEmode,and
e
e(i)
t,mn
=cos
β
(i)
m
(z −z
i
)
ˆz −
jnβ
(i)
m
(k
(i)
m
)
2
ρ
sin
β
(i)
m
(z −z
i
)
ˆϕ
h
e(i)
t,mn
= −
jωε
k
(i)
m
cos
β
(i)
m
(z −z
i
)
ˆϕ (4.3)
h
h(i)
t,mn
=sin
β
(i)
m
(z −z
i
)
ˆz +
jnβ
(i)
m
(k
(i)
m
)
2
ρ
cos
β
(i)
m
(z −z
i
)
ˆϕ
e
h(i)
t,mn
=
jωµ
k
(i)
m
sin
β
(i)
m
(z −z
i
)
ˆϕ (4.4)
k
2
= ω
2
µε = k
2
m
+ β
2
m
, and β
m
=
mπ
h
i
. (4.5)
Chapter 4. Modeling for Multilayered Power-Ground Planes 94
For the structure in Fig. 4.1, the following boundary conditions are applied
E
III
t
(ρ, φ, z)
ρ=b
=
⎧
⎪
⎨
⎪
⎩
E
I
t
(ρ, φ, z)
ρ=b
,z∈ [0,h
1
]
E
II
t
(ρ, φ, z)
ρ=b
,z∈ [h
1
,h] ,
(4.6)
H
III
t
(ρ, φ, z)
ρ=b
=
⎧
⎪
⎨
⎪
⎩
H
I
t
(ρ, φ, z)
ρ=b
,z∈ [0,h
1
]
H
II
t
(ρ, φ, z)
ρ=b
,z∈ [h
1
,h] .
(4.7)
Because of the decoupling of different modes n, we will only consider mode m in the
following derivation. Substituting (4.1) and (4.2) into (4.6) and (4.7), respectively,
we have
∞
m=0
⎛
⎜
⎝
a
e,III
mn
J
n
(k
III
m
b)+b
e,III
mn
H
(2)
n
(k
III
m
b)
e
e,III
t,mn
+
a
h,III
mn
J
n
(k
III
m
b)+b
h,III
mn
H
(2)
n
(k
III
m
b)
e
h,III
t,mn
⎞
⎟
⎠
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎝
a
e,I
mn
J
n
(k
I
m
b)+b
e,I
mn
H
(2)
n
(k
I
m
b)
e
e,I
t,mn
+
a
h,I
mn
J
n
(k
I
m
b)+b
h,I
mn
H
(2)
n
(k
I
m
b)
e
h,I
t,mn
⎞
⎟
⎠
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎝
a
e,II
mn
J
n
(k
II
m
b)+b
e,II
mn
H
(2)
n
(k
II
m
b)
e
e,II
t,mn
+
a
h,II
mn
J
n
(k
II
m
b)+b
h,II
mn
H
(2)
n
(k
II
m
b)
e
h,II
t,mn
⎞
⎟
⎠
,z∈ [h
1
,h] .
(4.8)
For convenience, we drop all the subscripts n in the following derivation and use
the following notations:
J
I
m
= J
I
mn
= J
n
k
I
m
b
, and H
I
m
= H
I
mn
= H
(2)
n
k
I
m
b
. (4.9)
So Eq. (4.8) can be written as
∞
m=0
⎛
⎜
⎝
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
e
e,III
t,m
+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
e
h,III
t,m
⎞
⎟
⎠
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎝
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
e
e,I
t,m
+
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
e
h,I
t,m
⎞
⎟
⎠
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎝
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
e
e,II
t,m
+
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
e
h,II
t,m
⎞
⎟
⎠
,z∈ [h
1
,h] .
(4.10)
Chapter 4. Modeling for Multilayered Power-Ground Planes 95
Similarly, the following equation is obtained from (4.7) as
∞
m=0
⎛
⎜
⎝
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
h
e,III
t,m
+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
h
h,III
t,m
⎞
⎟
⎠
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎝
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
h
e,I
t,m
+
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
h
h,I
t,m
⎞
⎟
⎠
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎝
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
h
e,II
t,m
+
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
h
h,II
t,m
⎞
⎟
⎠
,z∈ [h
1
,h] .
(4.11)
Replacing the tangential unit vectors in (4.10) by those in (4.3) and (4.4), we obtain
the L.H.S of (4.10) as
LHS|
E
t
=
∞
m=0
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
.
cos
β
III
m
z
ˆz −
jnβ
III
m
(k
III
m
)
2
b
sin
β
III
m
z
ˆϕ
+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
jωµ
k
III
m
sin
β
III
m
z
ˆϕ
=
∞
m=0
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
cos
β
III
m
z
ˆz + (4.12)
∞
m=0
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
−jnβ
III
m
(k
III
m
)
2
b
+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
jωµ
k
III
m
sin
β
III
m
z
ˆϕ.
Similarly, we can obtain the L.H.S of (4.11)
LHS|
H
t
=
∞
m=0
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
−jωε
k
III
m
cos
β
III
m
z
ˆϕ+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
.
sin
β
III
m
z
ˆz +
jnβ
III
m
(k
III
m
)
2
b
cos
β
III
m
z
ˆϕ
=
∞
m=0
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
sin
β
III
m
z
ˆz + (4.13)
∞
m=0
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
jnβ
III
m
(k
III
m
)
2
b
+
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
−jωε
k
III
m
cos
β
III
m
z
ˆϕ.
Chapter 4. Modeling for Multilayered Power-Ground Planes 96
The ˆz and ˆϕ components in (4.10) and (4.11) are separated as shown in the following
equations
∞
m=0
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
cos
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∞
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
cos
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
cos
β
II
m
z
,z∈ [h
1
,h] ,
(4.14)
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
−jnβ
III
m
(k
III
m
)
2
b
+
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
jωµ
k
III
m
⎞
⎟
⎟
⎟
⎟
⎠
sin
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
−jnβ
I
m
(k
I
m
)
2
b
+
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
jωµ
k
I
m
⎞
⎟
⎟
⎟
⎟
⎠
sin
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
−jnβ
II
m
(k
II
m
)
2
b
+
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
jωµ
k
II
m
⎞
⎟
⎟
⎟
⎟
⎠
sin
β
II
m
z
,z∈ [h
1
,h] ,
(4.15)
∞
m=0
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
sin
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∞
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
sin
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
sin
β
II
m
z
,z∈ [h
1
,h],
(4.16)
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
jnβ
III
m
(k
III
m
)
2
b
+
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
−jωε
k
III
m
⎞
⎟
⎟
⎟
⎟
⎠
cos
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
jnβ
I
m
(k
I
m
)
2
b
+
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
−jωε
k
I
m
⎞
⎟
⎟
⎟
⎟
⎠
cos
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
jnβ
II
m
(k
II
m
)
2
b
+
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
−jωε
k
II
m
⎞
⎟
⎟
⎟
⎟
⎠
cos
β
II
m
z
,z∈ [h
1
,h] .
(4.17)
Equations (4.14), (4.15), (4.16), and (4.17) are correspondent to E
z
, E
φ
, H
z
,and
H
φ
, respectively.
Chapter 4. Modeling for Multilayered Power-Ground Planes 97
As referred to Section 3.3, for the PEC cylinder (ρ = a) in PPWG-III (Fig. 4.1),
we have the relationship between the incoming and outgoing wave coefficients as
b
e,III
m
= −
J
n
k
III
m
a
H
(2)
n
(k
III
m
a)
a
e,III
m
,b
h,III
m
= −
J
n
k
III
m
a
H
(2)
n
(k
III
m
a)
a
h,III
m
. (4.18)
Then, we designate the following notations
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
a
e,III
m
∆
= J
III
e,m
a
e,III
m
, (4.19)
a
e,III
m
J
III
m
+ b
e,III
m
H
III
m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
a
e,III
m
∆
= J
III
e,m
a
e,III
m
, (4.20)
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
a
h,III
m
∆
= J
III
h,m
a
h,III
m
, (4.21)
a
h,III
m
J
III
m
+ b
h,III
m
H
III
m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
a
h,III
m
∆
= J
III
h,m
a
h,III
m
, (4.22)
where J
III
m
and H
III
m
are defined as those in (4.9).
We can now rewrite (4.14) to (4.17) as follows:
∞
m=0
a
e,III
m
J
III
e,m
cos
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∞
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
cos
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
cos
β
II
m
z
,z∈ [h
1
,h]
(4.23)
∞
m=0
a
e,III
m
J
III
e,m
−jnβ
III
m
(k
III
m
)
2
b
+ a
h,III
m
J
III
h,m
jωµ
k
III
m
sin
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
−jnβ
I
m
(k
I
m
)
2
b
+
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
jωµ
k
I
m
⎞
⎟
⎟
⎟
⎟
⎠
sin
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
−jnβ
II
m
(k
II
m
)
2
b
+
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
jωµ
k
II
m
⎞
⎟
⎟
⎟
⎟
⎠
sin
β
II
m
z
,z∈ [h
1
,h]
(4.24)
∞
m=0
a
h,III
m
J
III
h,m
sin
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∞
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
sin
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
sin
β
II
m
z
,z∈ [h
1
,h]
(4.25)
Chapter 4. Modeling for Multilayered Power-Ground Planes 98
∞
m=0
a
h,III
m
J
III
h,m
jnβ
III
m
(k
III
m
)
2
b
+ a
e,III
m
J
III
e,m
−jωε
k
III
m
cos
β
III
m
z
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
jnβ
I
m
(k
I
m
)
2
b
+
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
−jωε
k
I
m
⎞
⎟
⎟
⎟
⎟
⎠
cos
β
I
m
z
,z∈ [0,h
1
]
∞
m=0
⎛
⎜
⎜
⎜
⎜
⎝
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
jnβ
II
m
(k
II
m
)
2
b
+
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
−jωε
k
II
m
⎞
⎟
⎟
⎟
⎟
⎠
cos
β
II
m
z
,z∈ [h
1
,h] .
(4.26)
4.2 Mode Matching in Parallel-plate Waveguides
(PPWGs)
In this section, we focus on derivation of the following generalized T matrix:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
b
e,I
b
e,II
b
h,I
b
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
T
ee
I,I
T
ee
I,II
T
eh
I,I
T
eh
I,II
T
ee
II,I
T
ee
II,II
T
eh
II,I
T
eh
II,II
T
he
I,I
T
he
I,II
T
hh
I,I
T
hh
I,II
T
he
II,I
T
he
II,II
T
hh
II,I
T
hh
II,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
a
e,II
a
h,I
a
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.27)
where the size of the matrices depends on the number of modes used for each PP-
WGs.
The T matrix in (4.27) can be derived using the mode matching technique
[94]. The orthogonality relations for the Fourier series used in the mode matching
technique are given as
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
a
0
cos
nπx
a
cos
mπx
a
dx =
a
0
sin
nπx
a
sin
mπx
a
dx =
a
2
δ
nm
, for m, n =0
a
0
cos
nπx
a
sin
mπx
a
dx =0.
(4.28)
For numerical calculation, we also truncate the infinite summation to a finite one and
the numbers of modes are M1, M2, and M3 for PPWG- I, II, and III, respectively.
For performing the mode matching, we can either test it over [0,h](or[0,h
1
]
and [h
1
,h]). Here we choose the testing functions as being those of PPWG-III to
Chapter 4. Modeling for Multilayered Power-Ground Planes 99
enforcing E
z
and H
z
, and those of PPWG-I and II to enforcing E
φ
and H
φ
.By
performing the testing by cos(β
III
p
z) on (4.23):
h
0
(4.23) × cos
β
III
p
z
dz →
h
0
(4.23) ×cos
pπ
h
z
dz →
a
e,III
p
J
III
e,p
h
2
=
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
(4.29)
where
I
I,III
mp
=
h
1
0
cos
β
I
m
z
cos
β
III
p
z
dz =
h
1
0
cos
mπ
h
1
z
cos
pπ
h
z
dz (4.30)
I
II,III
mp
=
h
2
0
cos
β
II
m
z
cos
β
III
p
z
dz =
h
2
0
cos
mπ
h
2
z
cos
pπ
h
z
dz . (4.31)
By performing the testing on (4.24) over [0,h
1
]and[h
1
,h]bysin(β
I
p
z) and sin(β
II
p
z),
respectively:
M3
m=0
a
e,III
m
J
III
e,m
−jnβ
III
m
(k
III
m
)
2
b
+ a
h,III
m
J
III
h,m
jωµ
k
III
m
I
III,I
mq
=
⎛
⎜
⎝
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
−jnβ
I
q
k
I
q
2
b
+
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
jωµ
k
I
q
⎞
⎟
⎠
h
1
2
(4.32)
M3
m=0
a
e,III
m
J
III
e,m
−jnβ
III
m
(k
III
m
)
2
b
+ a
h,III
m
J
III
h,m
jωµ
k
III
m
I
III,II
mr
=
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
−jnβ
II
r
(k
II
r
)
2
b
+
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
jωµ
k
II
r
h
2
2
(4.33)
where
I
III,I
mq
=
h
1
0
sin
β
III
m
z
sin
β
I
q
z
dz =
h
1
0
sin
mπ
h
z
sin
qπ
h
1
z
dz (4.34)
I
III,II
mr
=
h
2
0
sin
β
III
m
z
sin
β
II
r
z
dz =
h
2
0
sin
mπ
h
z
sin
rπ
h
2
z
dz . (4.35)
By performing the testing by sin(β
III
p
z) on (4.25):
h
0
(25) × sin
β
III
p
z
dz →
h
0
(25) × sin
pπ
h
z
dz →
a
h,III
p
J
III
h,p
h
2
=
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
(4.36)
where
I
I,III
mp
=
h
1
0
sin
β
I
m
z
sin
β
III
p
z
dz =
h
1
0
sin
mπ
h
1
z
sin
pπ
h
z
dz (4.37)
Chapter 4. Modeling for Multilayered Power-Ground Planes 100
I
II,III
mp
=
h
2
0
sin
β
II
m
z
sin
β
III
p
z
dz =
h
2
0
sin
mπ
h
2
z
sin
pπ
h
z
dz . (4.38)
By performing the testing on (4.26) over [0,h
1
]and[h
1
,h]bycos(β
I
p
z)andcos(β
II
p
z),
respectively.
M3
m=0
a
h,III
m
J
III
h,m
jnβ
III
m
(k
III
m
)
2
b
+ a
e,III
m
J
III
e,m
−jωε
k
III
m
I
III,I
mq
=
⎛
⎜
⎝
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
jnβ
I
q
k
I
q
2
b
+
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
−jωε
k
I
q
⎞
⎟
⎠
h
1
2
(4.39)
M3
m=0
a
h,III
m
J
III
h,m
jnβ
III
m
(k
III
m
)
2
b
+ a
e,III
m
J
III
e,m
−jωε
k
III
m
I
III,II
mr
=
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
jnβ
II
r
(k
II
r
)
2
b
+
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
−jωε
k
II
r
h
2
2
(4.40)
where
I
III,I
mq
=
h
1
0
cos
β
III
m
z
cos
β
I
q
z
dz =
h
1
0
cos
mπ
h
z
cos
qπ
h
1
z
dz (4.41)
I
III,II
mr
=
h
2
0
cos
β
III
m
z
cos
β
II
r
z
dz =
h
2
0
cos
mπ
h
z
cos
rπ
h
2
z
dz . (4.42)
We introduce the following notations to make the subsequent derivation con-
cisely:
jnβ
III
m
(k
III
m
)
2
b
∆
= τ
β,III
m
,
jωµ
k
III
m
∆
= τ
µ,III
m
,
jωε
k
III
m
∆
= τ
ε,III
m
. (4.43)
Thus, we have (4.32), (4.33), (4.39) and (4.40):
M3
m=0
a
e,III
m
(−τ
β,III
m
)J
III
e,m
+ a
h,III
m
τ
µ,III
m
J
III
h,m
I
III,I
mq
=
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
(−τ
β,I
q
)+
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
τ
µ,I
q
h
1
2
(4.44)
M3
m=0
a
e,III
m
(−τ
β,III
m
)J
III
e,m
+ a
h,III
m
τ
µ,III
m
J
III
h,m
I
III,II
mr
=
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
(−τ
β,II
r
)+
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
τ
µ,II
r
h
2
2
(4.45)
M3
m=0
a
h,III
m
τ
β,III
m
J
III
h,m
+ a
e,III
m
(−τ
ε,III
m
)J
III
e,m
I
III,I
mq
=
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
τ
β,I
q
+
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
(−τ
ε,I
q
)
h
1
2
(4.46)
Chapter 4. Modeling for Multilayered Power-Ground Planes 101
M3
m=0
a
h,III
m
τ
β,III
m
J
III
h,m
+ a
e,III
m
(−τ
ε,III
m
)J
III
e,m
I
III,II
mr
=
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
τ
β,II
r
+
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
(−τ
ε,II
r
)
h
2
2
. (4.47)
The unknown coefficients are derived by manipulating (4.29), (4.32), (4.33), (4.36),
(4.39) and (4.40).
Equations (4.29) and (4.36) can be rewritten as
a
e,III
p
=
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2
hJ
III
e,p
(4.48)
a
h,III
p
=
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2
hJ
III
h,p
. (4.49)
First changing the subscript m in (4.44) to p,then
M3
p=0
a
e,III
p
(−τ
β,III
p
)J
III
e,p
+ a
h,III
p
τ
µ,III
p
J
III
h,p
I
III,I
pq
=
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
(−τ
β,I
q
)+
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
τ
µ,I
q
h
1
2
. (4.50)
Substituting (4.48) and (4.49) into (4.50), we obtain
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
−2τ
β,III
p
I
III,I
pq
h
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
+
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2τ
µ,III
p
J
III
h,p
I
III,I
pq
hJ
III
h,p
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
=
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
(−τ
β,I
q
)+
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
τ
µ,I
q
h
1
2
.
(4.51)
Chapter 4. Modeling for Multilayered Power-Ground Planes 102
Similarly, we can eliminate a
e,III
m
and a
h,III
m
in (4.45), (4.46) and (4.47):
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
−2τ
β,III
p
I
III,II
pr
h
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
+
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2τ
µ,III
p
J
III
h,p
I
III,II
pr
hJ
III
h,p
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
=
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
(−τ
β,II
r
)+
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
τ
µ,II
r
h
2
2
(4.52)
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
−2τ
ε,III
p
J
III
e,p
I
III,I
pq
hJ
III
e,p
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
+
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2τ
β,III
p
I
III,I
pq
h
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
=
a
e,I
q
J
I
q
+ b
e,I
q
H
I
q
(−τ
ε,I
q
)+
a
h,I
q
J
I
q
+ b
h,I
q
H
I
q
τ
β,I
q
h
1
2
(4.53)
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
e,I
m
J
I
m
+ b
e,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
e,II
m
J
II
m
+ b
e,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
−2τ
ε,III
p
J
III
e,p
I
III,II
pr
hJ
III
e,p
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
+
M3
p=0
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎝
M1
m=0
a
h,I
m
J
I
m
+ b
h,I
m
H
I
m
I
I,III
mp
+
M2
m=0
a
h,II
m
J
II
m
+ b
h,II
m
H
II
m
I
II,III
mp
⎞
⎟
⎟
⎟
⎠
2τ
β,III
p
I
III,II
pr
h
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
=
a
e,II
r
J
II
r
+ b
e,II
r
H
II
r
(−τ
ε,II
r
)+
a
h,II
r
J
II
r
+ b
h,II
r
H
II
r
τ
β,II
r
h
2
2
.
(4.54)
Chapter 4. Modeling for Multilayered Power-Ground Planes 103
Reorganizing all the terms in (4.51):
M3
p=0
M1
m=0
b
e,I
m
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ a
e,I
m
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+
M3
p=0
M2
m=0
b
e,II
m
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
+ a
e,II
m
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
+
M3
p=0
M1
m=0
b
h,I
m
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ a
h,I
m
2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+
M3
p=0
M2
m=0
b
h,II
m
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
+ a
h,II
m
2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
=
a
e,I
q
−τ
β,I
q
J
I
q
h
1
2
+ b
e,I
q
−τ
β,I
q
H
I
q
h
1
2
+ a
h,I
q
τ
µ,I
q
J
I
q
h
1
2
+ b
h,I
q
τ
µ,I
q
H
I
q
h
1
2
.
(4.55)
Then, all the terms with unknown coefficients for scattered fields are put to the
L.H.S of (4.55) and all the remaining terms to the R.H.S:
M3
p=0
M1
m=0
b
e,I
m
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
b
e,I
q
τ
β,I
q
H
I
q
h
1
2
+
M3
p=0
M2
m=0
b
e,II
m
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
+
M3
p=0
M1
m=0
b
h,I
m
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
b
h,I
q
−τ
µ,I
q
H
I
q
h
1
2
+
M3
p=0
M2
m=0
b
h,II
m
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
=
M3
p=0
M1
m=0
a
e,I
m
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
a
e,I
q
−τ
β,I
q
J
I
q
h
1
2
+
M3
p=0
M2
m=0
a
e,II
m
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
+
M3
p=0
M1
m=0
a
h,I
m
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
a
h,I
q
τ
µ,I
q
J
I
q
h
1
2
+
M3
p=0
M2
m=0
a
h,II
m
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
.
(4.56)
Chapter 4. Modeling for Multilayered Power-Ground Planes 104
Similarly, we have for (4.52)-(4.54):
M3
p=0
M1
m=0
b
e,I
m
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
+
M3
p=0
M2
m=0
b
e,II
m
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
b
e,II
r
τ
β,II
r
H
II
r
h
2
2
+
M3
p=0
M1
m=0
b
h,I
m
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
+
M3
p=0
M2
m=0
b
h,II
m
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
b
h,II
r
−τ
µ,II
r
H
II
r
h
2
2
=
M3
p=0
M1
m=0
a
e,I
m
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
+
M3
p=0
M2
m=0
a
e,II
m
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
a
e,II
r
−τ
β,II
r
J
II
r
h
2
2
+
M3
p=0
M1
m=0
a
h,I
m
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
+
M3
p=0
M2
m=0
a
h,II
m
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
a
h,II
r
τ
µ,II
r
J
II
r
h
2
2
(4.57)
M3
p=0
M1
m=0
b
e,I
m
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
b
e,I
q
τ
ε,I
q
H
I
q
h
1
2
+
M3
p=0
M2
m=0
b
e,II
m
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
+
M3
p=0
M1
m=0
b
h,I
m
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
b
h,I
q
−τ
β,I
q
H
I
q
h
1
2
+
M3
p=0
M2
m=0
b
h,II
m
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
=
M3
p=0
M1
m=0
a
e,I
m
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
a
e,I
q
−τ
ε,I
q
J
I
q
h
1
2
+
M3
p=0
M2
m=0
a
e,II
m
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
+
M3
p=0
M1
m=0
a
h,I
m
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
a
h,I
q
τ
β,I
q
J
I
q
h
1
2
+
M3
p=0
M2
m=0
a
h,II
m
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
(4.58)
Chapter 4. Modeling for Multilayered Power-Ground Planes 105
M3
p=0
M1
m=0
b
e,I
m
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
+
M3
p=0
M2
m=0
b
e,II
m
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
b
e,II
r
τ
ε,II
r
H
II
r
h
2
2
+
M3
p=0
M1
m=0
b
h,I
m
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
+
M3
p=0
M2
m=0
b
h,II
m
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
b
h,II
r
−τ
β,II
r
H
II
r
h
2
2
=
M3
p=0
M1
m=0
a
e,I
m
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
+
M3
p=0
M2
m=0
a
e,II
m
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
a
e,II
r
−τ
ε,II
r
J
II
r
h
2
2
+
M3
p=0
M1
m=0
a
h,I
m
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
+
M3
p=0
M2
m=0
a
h,II
m
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
a
h,II
r
τ
β,II
r
J
II
r
h
2
2
.
(4.59)
Chapter 4. Modeling for Multilayered Power-Ground Planes 106
4.3 Generalized T Matrix for Two-layer Problem
As we have discussed the modal expansion and boundary conditions, and the mode
matching in the previous two sections, we are ready to formulate the following
generalized T matrix (cf. (4.27)):
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
b
e,I
b
e,II
b
h,I
b
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
T
ee
I,I
T
ee
I,II
T
eh
I,I
T
eh
I,II
T
ee
II,I
T
ee
II,II
T
eh
II,I
T
eh
II,II
T
he
I,I
T
he
I,II
T
hh
I,I
T
hh
I,II
T
he
II,I
T
he
II,II
T
hh
II,I
T
hh
II,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
a
e,II
a
h,I
a
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
. (4.60)
We will derive the elements of the generalized T matrix in (4.60) one column by
another. For such case, we first let a
e,I
=0anda
e,II
= a
h,I
= a
h,II
= 0 in (4.56)-
(4.59). Then, we can project (4.56)-(4.59) into a linear system of equation:
[A] {b} =
P
eI
C1
a
e,I
, (4.61)
or
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A
11
M1×M1
A
12
M1×M2
A
13
M1×M1
A
14
M1×M2
A
21
M2×M1
A
22
M2×M2
A
23
M2×M1
A
24
M2×M2
A
31
M1×M1
A
32
M1×M2
A
33
M1×M1
A
34
M1×M2
A
41
M2×M1
A
42
M2×M2
A
43
M2×M1
A
44
M2×M2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×N
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
b
e,I
b
e,II
b
h,I
b
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
eI,1
M1×M1
P
eI,2
M2×M1
P
eI,3
M1×M1
P
eI,4
M2×M1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×M 1
a
e,I
M1
,
(4.62)
where N =2(M1+M2), The superscript C1 in (4.61) indicates that the entries in
the vector b are corresponding to the entries in the first column of the T matrix in
(4.60), and
b
e(h),I
=
b
e(h),I
T
,m=1, ···M1
b
e(h),II
=
b
e(h),II
T
,m=1, ···M2 .
(4.63)
The entries in Row 1 of the matrices [A]and[P ] in (4.62) are given as
A
11
qm
=
M3
p=0
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
τ
β,I
q
H
I
q
h
1
2
,q(m)=1, ···M1 (4.64)
A
12
qm
=
M3
p=0
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.65)
Chapter 4. Modeling for Multilayered Power-Ground Planes 107
A
13
qm
=
M3
p=0
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
−τ
µ,I
q
H
I
q
h
1
2
,q(m)=1, ···M1
(4.66)
A
14
qm
=
M3
p=0
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
,q(m)=1, ···M1(M2) (4.67)
P
eI,1
qm
=
M3
p=0
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
−τ
β,I
q
J
I
q
h
1
2
,q(m)=1, ···M1 . (4.68)
Those entries in Row 2 of the matrices [A]and[P ] are given as follows:
A
21
rm
=
M3
p=0
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.69)
A
22
rm
=
M3
p=0
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
τ
β,II
r
H
II
r
h
2
2
,r(m)=1, ···M2(M2)
(4.70)
A
23
rm
=
M3
p=0
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
,r(m)=1, ···M2(M1) (4.71)
A
24
rm
=
M3
p=0
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
−τ
µ,II
r
H
II
r
h
2
2
,
r(m)=1, ···M2(M2) (4.72)
P
eI,2
rm
=
M3
p=0
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) . (4.73)
Those entries in Row 3 of the matrices [A]and[P ] are given as follows:
A
31
qm
=
M3
p=0
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
τ
ε,I
q
H
I
q
h
1
2
,q(m)=1, ···M1
(4.74)
A
32
qm
=
M3
p=0
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
,q(m)=1, ···M1(M2) (4.75)
A
33
qm
=
M3
p=0
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
−τ
β,I
q
H
I
q
h
1
2
,q(m)=1, ···M1 (4.76)
A
34
qm
=
M3
p=0
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.77)
P
eI,3
qm
=
M3
p=0
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
−τ
ε,I
q
J
I
q
h
1
2
,q(m)=1, ···M1 .
(4.78)
Chapter 4. Modeling for Multilayered Power-Ground Planes 108
Those entries in Row 4 of the matrices [A]and[P ] are given as follows:
A
41
rm
=
M3
p=0
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
,r(m)=1, ···M2(M1) (4.79)
A
42
rm
=
M3
p=0
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
τ
ε,II
r
H
II
r
h
2
2
,
r(m)=1, ···M2(M2) (4.80)
A
43
rm
=
M3
p=0
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.81)
A
44
rm
=
M3
p=0
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
−τ
β,II
r
H
II
r
h
2
2
,r(m)=1, ···M2(M2)
(4.82)
P
eI,4
rm
=
M3
p=0
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
,r(m)=1, ···M2(M1) . (4.83)
We can now obtain from (4.61):
b
=
A
−1
P
eI
a
e,I
, (4.84)
where [A]
−1
[P
eI
] correspond to the first column of the T matrix in (4.60).
The derivation for the other three columns of the T matrix in (4.60) follows
exactly the same procedure as that for Column 1 of the T matrix. We can easily
notice that the matrix [A] for deriving all the other three columns of the T matrix in
(4.60) is identical to the one in (4.84) for deriving the first column of the T matrix
in (4.60). The only difference is the matrix [P ], so we only present here the entries
of the matrix [P ].
The Column 2 of the T matrix in (4.60) can be derived by setting a
e,II
=0and
a
e,I
= a
h,I
= a
h,II
= 0 in (4.56)-(4.59). The expression is as follows:
b
=
A
−1
P
eII
a
e,II
, (4.85)
where [A]
−1
[P
eII
] correspond to the second column of the T matrix in (4.60), and
P
eII
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
eII,1
M1×M2
P
eII,2
M2×M2
P
eII,3
M1×M2
P
eII,4
M2×M2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×M 2
a
e,II
M2
, (4.86)
Chapter 4. Modeling for Multilayered Power-Ground Planes 109
P
eII,1
qm
=
M3
p=0
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.87)
P
eII,2
rm
=
M3
p=0
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
−τ
β,II
r
J
II
r
h
2
2
,r(m)=1, ···M2
(4.88)
P
eII,3
qm
=
M3
p=0
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
,q(m)=1, ···M1(M2) (4.89)
P
eII,4
rm
=
M3
p=0
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
−τ
ε,II
r
J
II
r
h
2
2
,r(m)=1, ···M2 .
(4.90)
Similarly, the Column 3 of the T matrix in (4.60) is derived by setting a
h,I
=0and
a
e,I
= a
e,II
= a
h,II
= 0 in (4.56)-(4.59). The formula relevant to the third column
of the T matrix in (4.60) is
b
=
A
−1
P
hI
a
h,I
, (4.91)
where [A]
−1
[P
hI
] correspond to the third column of the T matrix in (4.60), and
P
hI
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
hI,1
M1×M1
P
hI,2
M2×M1
P
hI,3
M1×M1
P
hI,4
M2×M1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×M 1
a
h,I
M1
, (4.92)
P
hI,1
qm
=
M3
p=0
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
τ
µ,I
q
J
I
q
h
1
2
,q(m)=1, ···M1
(4.93)
P
hI,2
rm
=
M3
p=0
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
,r(m)=1, ···M2(M1) (4.94)
P
hI,3
qm
=
M3
p=0
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
τ
β,I
q
J
I
q
h
1
2
,q(m)=1, ···M1 (4.95)
P
hI,4
rm
=
M3
p=0
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) . (4.96)
The Column 4 of the T matrix in (4.60) is derived by setting a
h,II
=0anda
e,I
=
a
e,II
= a
h,I
= 0 in (4.56)-(4.59). The formula relevant to the fourth column of the
T matrix in (4.60) is
b
=
A
−1
P
hII
a
h,II
, (4.97)
Chapter 4. Modeling for Multilayered Power-Ground Planes 110
where [A]
−1
[P
hII
] correspond to the fourth column of the T matrix in (4.60), and
P
hII
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
hII,1
M1×M2
P
hII,2
M2×M2
P
hII,3
M1×M2
P
hII,4
M2×M2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×M 2
a
h,II
M2
, (4.98)
P
hII,1
qm
=
M3
p=0
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
,q(m)=1, ···M1(M2) (4.99)
P
hII,2
rm
=
M3
p=0
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
τ
µ,II
r
J
II
r
h
2
2
,r(m)=1, ···M2
(4.100)
P
hII,3
qm
=
M3
p=0
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.101)
P
hII,4
rm
=
M3
p=0
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
τ
β,II
r
J
II
r
h
2
2
,r(m)=1, ···M2 .
(4.102)
Chapter 4. Modeling for Multilayered Power-Ground Planes 111
4.4 Formulas Summary for Two-layer Problem
Generalized T matrix:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
b
e,I
b
e,II
b
h,I
b
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
T
ee
I,I
T
ee
I,II
T
eh
I,I
T
eh
I,II
T
ee
II,I
T
ee
II,II
T
eh
II,I
T
eh
II,II
T
he
I,I
T
he
I,II
T
hh
I,I
T
hh
I,II
T
he
II,I
T
he
II,II
T
hh
II,I
T
hh
II,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
a
e,II
a
h,I
a
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
T
C1
T
C2
T
C3
T
C4
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
a
e,II
a
h,I
a
h,II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.103)
Column 1 of T Matrix:
T
C1
=
A
−1
P
eI
(4.104)
where
[A]=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A
11
M1×M1
A
12
M1×M2
A
13
M1×M1
A
14
M1×M2
A
21
M2×M1
A
22
M2×M2
A
23
M2×M1
A
24
M2×M2
A
31
M1×M1
A
32
M1×M2
A
33
M1×M1
A
34
M1×M2
A
41
M2×M1
A
42
M2×M2
A
43
M2×M1
A
44
M2×M2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N×N
(4.105)
Row 1 of Matrix [A]:
A
11
qm
=
M3
p=0
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
τ
β,I
q
H
I
q
h
1
2
,q(m)=1, ···M1 (4.106)
A
12
qm
=
M3
p=0
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.107)
A
13
qm
=
M3
p=0
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
−τ
µ,I
q
H
I
q
h
1
2
,q(m)=1, ···M1
(4.108)
A
14
qm
=
M3
p=0
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
,q(m)=1, ···M1(M2) (4.109)
Row 2 of Matrix [A]:
A
21
rm
=
M3
p=0
−2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.110)
Chapter 4. Modeling for Multilayered Power-Ground Planes 112
A
22
rm
=
M3
p=0
−2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
τ
β,II
r
H
II
r
h
2
2
,r(m)=1, ···M2(M2)
(4.111)
A
23
rm
=
M3
p=0
2τ
µ,III
p
H
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
,r(m)=1, ···M2(M1) (4.112)
A
24
rm
=
M3
p=0
2τ
µ,III
p
H
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
−τ
µ,II
r
H
II
r
h
2
2
,
r(m)=1, ···M2(M2) (4.113)
Row 3 of Matrix [A]:
A
31
qm
=
M3
p=0
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
τ
ε,I
q
H
I
q
h
1
2
,q(m)=1, ···M1
(4.114)
A
32
qm
=
M3
p=0
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
,q(m)=1, ···M1(M2) (4.115)
A
33
qm
=
M3
p=0
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
−τ
β,I
q
H
I
q
h
1
2
,q(m)=1, ···M1 (4.116)
A
34
qm
=
M3
p=0
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.117)
Row 4 of Matrix [A]:
A
41
rm
=
M3
p=0
−2τ
ε,III
p
H
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
,r(m)=1, ···M2(M1) (4.118)
A
42
rm
=
M3
p=0
−2τ
ε,III
p
H
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
τ
ε,II
r
H
II
r
h
2
2
,
r(m)=1, ···M2(M2) (4.119)
A
43
rm
=
M3
p=0
2τ
β,III
p
H
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.120)
A
44
rm
=
M3
p=0
2τ
β,III
p
H
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
−τ
β,II
r
H
II
r
h
2
2
,r(m)=1, ···M2(M2)
(4.121)
Matrix [P ]:
P
eI
N×M 1
=
P
eI,1
M1×M1
P
eI,2
M2×M1
P
eI,3
M1×M1
P
eI,4
M2×M1
T
(4.122)
Chapter 4. Modeling for Multilayered Power-Ground Planes 113
P
eI,1
qm
=
M3
p=0
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
−τ
β,I
q
J
I
q
h
1
2
,q(m)=1, ···M1 (4.123)
P
eI,2
rm
=
M3
p=0
2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.124)
P
eI,3
qm
=
M3
p=0
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,I
pq
J
III
e,p
h
+ δ
mq
−τ
ε,I
q
J
I
q
h
1
2
,q(m)=1, ···M1
(4.125)
P
eI,4
rm
=
M3
p=0
2τ
ε,III
p
J
I
m
J
III
e,p
I
I,III
mp
I
III,II
pr
J
III
e,p
h
,r(m)=1, ···M2(M1) (4.126)
Column 2 of T Matrix:
T
C2
=
A
−1
P
eII
(4.127)
where
P
eII
N×M 2
=
P
eII,1
M1×M2
P
eII,2
M2×M2
P
eII,3
M1×M2
P
eII,4
M2×M2
T
(4.128)
P
eII,1
qm
=
M3
p=0
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.129)
P
eII,2
rm
=
M3
p=0
2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
−τ
β,II
q
J
II
q
h
2
2
,r(m)=1, ···M2
(4.130)
P
eII,3
qm
=
M3
p=0
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,I
pq
J
III
e,p
h
,q(m)=1, ···M1(M2) (4.131)
P
eII,4
rm
=
M3
p=0
2τ
ε,III
p
J
II
m
J
III
e,p
I
II,III
mp
I
III,II
pr
J
III
e,p
h
+ δ
mr
−τ
ε,II
r
J
II
r
h
2
2
,r(m)=1, ···M2
(4.132)
Column 3 of T Matrix:
T
C3
=
A
−1
P
hI
(4.133)
where
P
hI
N×M 1
=
P
hI,1
M1×M1
P
hI,2
M2×M1
P
hI,3
M1×M1
P
hI,4
M2×M1
T
(4.134)
P
hI,1
qm
=
M3
p=0
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,I
pq
J
III
h,p
h
+ δ
mq
τ
µ,I
q
J
I
q
h
1
2
,q(m)=1, ···M1
(4.135)
Chapter 4. Modeling for Multilayered Power-Ground Planes 114
P
hI,2
rm
=
M3
p=0
−2τ
µ,III
p
J
I
m
J
III
h,p
I
I,III
mp
I
III,II
pr
J
III
h,p
h
,r(m)=1, ···M2(M1) (4.136)
P
hI,3
qm
=
M3
p=0
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,I
pq
h
+ δ
mq
τ
β,I
q
J
I
q
h
1
2
,q(m)=1, ···M1 (4.137)
P
hI,4
rm
=
M3
p=0
−2τ
β,III
p
J
I
m
I
I,III
mp
I
III,II
pr
h
,r(m)=1, ···M2(M1) (4.138)
Column 4 of T Matrix:
T
C4
=
A
−1
P
hII
(4.139)
where
P
hII
N×M 2
=
P
hII,1
M1×M2
P
hII,2
M2×M2
P
hII,3
M1×M2
P
hII,4
M2×M2
T
(4.140)
P
hII,1
qm
=
M3
p=0
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,I
pq
J
III
h,p
h
,q(m)=1, ···M1(M2) (4.141)
P
hII,2
rm
=
M3
p=0
−2τ
µ,III
p
J
II
m
J
III
h,p
I
II,III
mp
I
III,II
pr
J
III
h,p
h
+ δ
mr
τ
µ,II
r
J
II
r
h
2
2
,r(m)=1, ···M2
(4.142)
P
hII,3
qm
=
M3
p=0
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,I
pq
h
,q(m)=1, ···M1(M2) (4.143)
P
hII,4
rm
=
M3
p=0
−2τ
β,III
p
J
II
m
I
II,III
mp
I
III,II
pr
h
+ δ
mr
τ
β,II
r
J
II
r
h
2
2
,r(m)=1, ···M2
(4.144)
Summary of all the variables in the above equations:
β
I(II)(III)
m
=
mπ
h
1(2)(3)
;
k
I(II)(III)
m
2
= k
2
−
β
I(II)(III)
m
2
= ω
2
µε −
mπ
h
1(2)(3)
2
(4.145)
τ
β,I(II)(III)
m
=
jnβ
I(II)(III)
m
k
I(II)(III)
m
2
b
; τ
µ,I(II)(III)
m
=
jωµ
k
I(II)(III)
m
; τ
ε,I(II)(III)
m
=
jωε
k
I(II)(III)
m
(4.146)
J
I(II)
m
= J
I(II)
mn
= J
n
k
I(II)
m
b
; H
I(II)
m
= H
I(II)
mn
= H
(2)
n
k
I(II)
m
b
J
I(II)
m
= J
I(II)
mn
=
∂J
n
k
I(II)
m
b
∂
k
I(II)
m
b
; H
I(II)
m
= H
I(II)
mn
=
∂H
(2)
n
k
I(II)
m
b
∂
k
I(II)
m
b
(4.147)
J
III
e,m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
; J
III
e,m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
(4.148)
Chapter 4. Modeling for Multilayered Power-Ground Planes 115
J
III
h,m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
; J
III
h,m
=
⎛
⎝
J
III
m
−
J
n
k
III
m
a
H
III
m
H
(2)
n
(k
III
m
a)
⎞
⎠
(4.149)
Formulas to be used for computing the integrals such as I
I,III
mp
, I
II,III
mp
etc.:
I
1
=
cos(az)cos(bz)dz =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
sin [(a − b)z]
2(a − b)
+
sin [(a + b)z]
2(a + b)
, for |a| = | b|
z
2
+
sin(2az)
4a
, for |a| = | b|
(4.150)
I
2
=
sin(az)sin(bz)dz =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
sin [(a −b)z]
2(a −b)
−
sin [(a + b)z]
2(a + b)
, for |a| = |b|
z
2
−
sin(2az)
4a
, for |a| = |b| & ab ≥ 0
−
z
2
+
sin(2az)
4a
, for |a| = |b| & ab ≤ 0
(4.151)
4.5 Formulas Summary for Multi-layer Problem
In this section, we have summarized the formulae of generalized T matrix for multi-
layered structure in Fig. 4.2. The following formulas are consolidated for source-free
via and source via comprised in multiple layer.
Figure 4.2: A though-hole via in multi-layer structure and forming PPWGs.
Chapter 4. Modeling for Multilayered Power-Ground Planes 116
For Source-free Via,
Generalized T matrix for a multilayered structure:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
b
e,I
.
.
.
b
e,R
b
h,I
.
.
.
b
h,R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
T
ee
I,I
···
.
.
.
.
.
.
T
ee
I,R
T
eh
I,I
.
.
.
.
.
.
··· T
eh
I,R
.
.
.
.
.
.
T
ee
R,I
···
T
he
I,I
···
T
ee
R,R
T
eh
R,I
T
he
I,R
T
hh
I,I
··· T
eh
R,R
··· T
hh
I,R
.
.
.
.
.
.
T
he
R,I
···
.
.
.
.
.
.
T
he
R,R
T
hh
R,I
.
.
.
.
.
.
··· T
hh
R,R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
.
.
.
a
e,R
a
h,I
.
.
.
a
h,R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
T
e,C
1
··· T
e,C
R
T
h,C
1
··· T
h,C
R
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
e,I
.
.
.
a
e,R
a
h,I
.
.
.
a
h,R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.152)
Columns of T Matrix:
T
e(h),C
κ
=
A
−1
P
e(h),κ
(4.153)
where Matrix [A]:
[A]=
⎡
⎢
⎣
A
ee
A
eh
A
he
A
hh
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A
ee,I:I
M
1
×M
1
··· A
ee,I:R
M
1
×M
R
A
eh,I:I
M
1
×M
1
··· A
eh,I:R
M
1
×M
R
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
ee,R:I
M
R
×M
1
··· A
ee,R:R
M
R
×M
R
A
eh,R:I
M
R
×M
1
··· A
eh,R:R
M
R
×M
R
A
he,I:I
M
1
×M
1
··· A
he,I:R
M
1
×M
R
A
hh,I:I
M
1
×M
1
··· A
hh,I:R
M
1
×M
R
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
he,R:I
M
R
×M
1
··· A
he,R:R
M
R
×M
R
A
hh,R:I
M
R
×M
1
··· A
hh,R:R
M
R
×M
R
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.154)
and the size of Matrix [A]is2
R
k=1
M
k
by 2
R
k=1
M
k
. The elements are given as
A
ee,Υ:Ψ
qm
=
M
Π
p=0
−2τ
β,Π
p
H
Ψ
m
I
Ψ,Π
mp
I
Π,Υ
pq
h
+ δ
ΥΨ
δ
mq
τ
β,Υ
q
H
Υ
q
h
Υ
2
,q(m)=1, ···M
Υ
(M
Ψ
)
(4.155)
A
eh,Υ:Ψ
qm
=
M
Π
p=0
2τ
µ,Π
p
H
Ψ
m
J
Π
h,p
I
Ψ,Π
mp
I
Π,Υ
pq
J
Π
h,p
h
+ δ
ΥΨ
δ
mq
−τ
µ,Υ
q
H
Υ
q
h
Υ
2
,q(m)=1, ···M
Υ
(M
Ψ
)
(4.156)
