Page 222 - System on Package_ Miniaturization of the Entire System

196 Cha pte r F o u r

B. Reverse Mapping Reverse mapping is the process of mapping the electrical parameters
to the geometrical parameters. Trained and validated neuromodels can be employed to
extract the geometrical parameters that meet the area constraints and maximize the
inductor parameters such as quality factor. During reverse mapping, the geometrical
parameters are extracted from electrical specifications while satisfying the design
constraints. This solution is nonunique as multiple combinations of layout parameters
of the inductor can correspond to the same inductance value (with a different Q factor,
and SRF). A knowledge base of equations or constraints for the inductor parameters can
be included in the model to allow for fast convergence and generate unique solutions in
such a multivariable optimization problem. The geometries synthesized by reverse
mapping may not be feasible for fabrication. In such a case, the design values are
rounded off to the closest processable physical dimensions in accordance with process
ground rules. This can be enabled by enforcing constraints on the physical dimensions
during synthesis.
As an example consider the inductor in Figure 4.6. The physical dimensions that
dictate the electrical response of the inductor are line width, line spacing, and number
of turns. The distance to the ground plane is fixed since it depends on the stackup that
in general has a fixed thickness. The equivalent circuit for the inductor with parasitic
elements is shown in Figure 4.7. The circuit parameters can be extracted using an
optimizer that is available in most RF circuit simulators using the frequency response
as the input. The frequency response can be generated using an electromagnetic solver.
Since, the equivalent circuit parameters in Figure 4.7 are frequency independent, a
mapping can be created both in the forward and reverse directions between the variable
physical dimensions in Figure 4.6 and circuit parameters in Figure 4.7. The forward
mapping generates a functional relationship between the physical parameters
(independent variable) as input and the circuit parameters (dependent variable) as
output. This is reversed during the reverse mapping. If the data are monotonic, artificial
neural networks are a good way to establish this relationship. A functional relationship
can now be established between the circuit parameters in Figure 4.7 and the inductor
specifications such as L and Q (SRF is rarely used as a specification as long as it is two
times the frequency at which the inductor is used). This approach can also be used for
capacitors. The synthesis method described can be used to generate inductor and
capacitor libraries. This captures the three-level scheme described in Figure 4.44, where
Level 3 has the inductor layout. The parasitics of this layout get mapped to the circuit
components in Level 2 from which the Level 1 parameters inductance L and Q can be
derived.
Scaling of Filter Layouts

A. Forward Mapping Based on Lumped-Circuit Models and Polynomial Fitting For the layout
scaling of RF circuits with reduced design cycle time, it is important to extract accurate
lumped-element models that capture the physical effects of layout. For example, in the
scaling of an RF bandpass filter, an extensive lumped-circuit model can be generated
based on segmentation of a physical layout, as shown in Figure 4.45. The layout is
decomposed into circuit sections that are isolated from each other, under the assumption
that the coupling is weak. The dotted lines represent the segmented sections. For
example, the L resonators are segmented into the coupled inductor section L_cp and
uncoupled inductor sections L1 and L2. This technique allows separate scaling and
mapping of geometrical sections that have little electromagnetic interaction between