A simplified structure is shown in Fig. 2.2 for analyzing the problems of power-ground planes with practical source ports [22]. A pair of close-spaced power-ground planes with length l and width w are separated by a dielectric substrate with a thickness h and a relative dielectric constant
ε
r. Two source ports, port A and port B, are placed in the plane for the modeling of via structures through the planes.Current injecting into port A may induce a power-ground bounce noise between the planes. The noise wave travels throughout the entire planes and eventually the noise will couple to port B, causing a slightly variation to the port voltage.
Fig. 2.3 shows the modeling of possible paths of current flow between the two practical source ports at high frequencies compared with the electrical dimension by adding virtual ports on the planes [22]. As the wavelength of interests approaches the
x y
z
P ra ctica l s ource ports
w
l
h
P ort
A
P ort
B
ε
rFig. 2.2 A simplified structure for analyzing the problems of power-ground planes with
practical source ports [22].
electrical dimensions of the interconnection structures, the current does not flow straightly from port A to port B but also includes the paths spreading radially outward from port A and then reflecting from the edges of power-ground planes to port B.
This phenomenon can be effectively modeled by the novel method proposed in [22]
using an equivalent lumped circuit model with virtual ports that play the role of the distributed current transition. The values of lumped circuit elements are associated
P ra ctica l s ource ports Virtua l ports P os s ible pa ths of curre nt flow
Fig. 2.3 Modeling possible paths of current flow between the two practical source ports
at high frequencies compared with the electrical dimension by adding virtual
ports on the planes [22].
with the geometry of the triangular mesh and can be obtained by employing the Delaunay triangularization for the mesh generation and applying Voronoi tesserlation at each node.
P ra ctica l s ource ports
Fig. 2.4 Typical setting for the Delaunay-Vononoi modeling of power-ground planes. (a)
The arrangement of triangular meshes and the placement virtual ports. (b) The
equivalent lumped circuit associated with the practical source port
i
and thevirtual ports connected.
2.2.1 Delaunay-Vononoi Modeling of Power-Ground Planes
For the typical setting of Delaunay-Vononoi modeling of power-ground planes shown in Fig. 2.4, if the arrangement of triangular meshes and the placement virtual
ports is given in Fig. 2.4a, the equivalent lumped circuit associated with the practical source port i and its connecting neighbor virtual ports, which are embraced in the dashed circle, can be determined as Fig. 2.4b, assume a lossless condition. The values of the elements in the equivalent lumped circuit model can be determined by matching the terms derived from the perspective of circuit and electromagnetic theory.
Applying the Kirchhoff’s Current Law (KCL), the relation between nodal voltages and the lumped circuit elements at the i-th node in Fig. 2.4 can be obtained as
0
6 1
− = +
∑
=
j ij
j i i
i
j L
V V V
C
j ω ω
(2.7)with the notations shown in Fig. 2.4b, where the nodal voltages at the i- and j-th node are denotes by
V and
iV , respectively. The shunt capacitance at the i-th node is
j denoted byC and
iL represents the series inductance connecting the i- and j-th
ij node.On the other hand, if the thickness h in Fig. 2.2 is small enough compare to the wavelength of interest, the voltage distribution can be approximate as a function of
coordinates (x, y), the electric and magnetic field can be written as Fig. 2.4, with integration along the path connecting the circumcenters of the triangular mesh surrounding the node, the integral on both sides of the equation can be derived as
between the circumcenters of the two triangles with a common edge connecting the j- and i-th node, and
A is the area of the integration loop. Detailed derivation can be
i found in [22].By matching the terms in (2.7) and (2.9), the values of the lumped circuit elements is then determined as
h
Ci =ε Ai (2.10a)
and
ij ij
ij l
L μhd
= , (2.10b)
which shows a relation only with the geometric parameters. Therefore once the meshes for the analysis are set, these values can then be uniquely defined.
2.2.2 Time Marching Scheme for Delaunay-Vononoi Modeling of Power-Ground Planes
For the i-th node of the equivalent lumped circuit model of power-ground planes shown in Fig. 2.4b, the node voltage and the branch current can be related with the lumped circuit elements in time domain as
dt C dV
I
i = i i (2.11a)and
dt L dI V
V
i − j = ij ij , (2.11b)where
V and
iV is the node voltage at the i- and j-th node, respectively,
jI is the
i current through the shunt capacitanceC , and
iI is the current through the series
ij inductanceL connecting the i- and j-th node. These two current can be related by
ijKCL as
Since the current flows from the i- to j-th node in the opposite direction as from j- to
i-th node, it is obvious that I
ij =−I
ji when (2.11) is applied to the j-th node.Let the time in (2.11) be dicretized as n⋅Δt. If the node voltages is located at the integer multiples as
V
in =V
i( n
⋅Δt )
and the branch current is located at the center of every two contiguous integer multiples as ⎟⎠
, applying central difference to the differentiation in time in
(2.11)yields
(
jn)
(2.11c) can also be rewritten as
∑
+with branch currents 2
1
I . For the time marching scheme of Delaunay- Vononoi modeling of power-ground planes, the iteration at time step n begins with (2.12b), where the branch currents through the series inductances connecting the nodes are updates to the next half time step with those at previous half time step and the voltages at all nodes at current time step. The current through the shunt capacitance at each node can then be update by (2.12c). After that all node voltages can be update to the next time step with their value at current time step and the currents at next half time step. Time step is then moved from n to n+1 for the beginning of a new iteration.