Page 227 - System on Package_ Miniaturization of the Entire System

Mixed-Signal (SOP) Design 201

parameters can be extracted analytically from the physical dimensions. The nodal
admittance matrix for a single plane pair in Figure 4.48 can be written as

A ⎡ B ⎤
⎢ ⎥ Y ⎡ + 2 / Z −1 / Z ⎤
B ⎢ A + 1 / Z B ⎥ ⎢ ⎥
⎢ ⎥ ⎢ −1 / Z Y + 3 / Z −1 / Z ⎥
⎢ ⎥ ⎥ ⎢ ⎥
Y = ⎢ B O O ⎥ A = ⎢ −1 / Z Y + 3 / Z O ⎥
⎢ O O B ⎥ ⎢ O O O −1 / Z ⎥ ⎥
⎢ ⎥ ⎢ Z ⎥
⎢ B A + 1 / Z B⎥ ⎢ −1 / Z Y + 3 / Z −1 / ⎥
⎢ ⎥ ⎣ −1 / Z Y + 2 / Z⎦
⎣ ⎢ B A⎦ ⎥
B = 1/ Z
(4.18)

where Y and Z are the per-unit cell admittance and impedance, which can be obtained
from the complex permittivity (e), permeability (m), distance between the planes (d),
thickness of each plane (t), mesh length (h), and conductivity (s) as [55c]:

h 2
Y = jωε
d
2 jωμ (4.19)
Z = jωμ d + + 2
σ t σ

In case of multiple plane pairs as in Figure 4.49, Y and Z themselves are also matrices
that reflect the multilayered structure.
This formulation is based on the multilayered finite-difference method (MFDM)
[55c] and is very useful when analyzing planes with complex shapes and apertures.
When a stripline is introduced into the three planes, as shown in Figure 4.49, modal
decomposition can still be applied based on superposition. Now, the following
matrix has to be added to the admittance matrix of the planes based on modal
decomposition:

I ⎛ ⎞ ⎛ ⎛ V ⎞
1
⎜ ⎟ ⎛ ⎞ ⎜ 1 ⎟
2
⎜ I 2 ⎟ ⎜ kY str − ( k − k)Y str ) kY str ⎟ ⎜ ⎜ V 2 ⎟
2
)
I ⎜ ⎟
⎜
⎟
−
⎜ ⎟ = ( k − ) k Y str (k + 2 k + ) 1 ( k − 1 )Y str ⎜ V ⎟ ⎟ (4.20)
−
3
2
2
3
⎜ I 4⎟ ⎜ ⎜ ⎟ ⎟ ⎜ V 4⎟
)
1
Y
I ⎜ ⎟ ⎝ kY str (−− k Y str Y str ⎠ ⎜ V ⎟
⎜ ⎟ ⎜ 5 ⎟
5
I ⎝ ⎠ ⎝ V ⎠
6 6
where the currents and voltages are defined in Figure 4.50.
As an example, consider the structure in Figure 4.51, which consists of a stripline
between a power-ground plane. The bottom ground plane contains an aperture. Ports
are defined at the input (port 1) and output (port 2) of the stripline with an additional