Page 230 - System on Package_ Miniaturization of the Entire System

204 Cha pte r F o u r

Finally, using a full-wave simulator for analysis of practical geometries is computationally
very expensive, so full-wave simulators are generally used for simulating small
segments of the geometry.

4.5.3 Rational Functions, Network Synthesis, and Transient Simulation
For digital signal lines, transient waveforms need to be generated from the frequency
response that can be generated using the methods in the previous section. This is
possible by directly simulating the matrix equations developed in the previous section.
A more useful approach is to embed the extracted frequency responses in SPICE, as
shown in Figure 4.47. This requires the conversion of the frequency response into a
SPICE circuit, which is possible using rational functions. For interconnections, the
transfer function H(s) (S, Y, or Z parameters) can be represented in the form:
P
∑ as k
k
Hs() = k=0 (4.21)
Q
∑ bs l
l
l=0
where s = jw, w is the angular frequency, and a and b are coefficients. Given the frequency
l
k
response H(s) available at the discrete frequency points, the goal is to compute the coefficients
such that the response of the rational function [right side in Equation (4.21)] matches the
frequency response at the available discrete points. The coefficients can be computed by
solving a matrix equation, the details of which are available in [57–58]. To ensure that the
resulting rational function can result in an equivalent circuit that retains all of the properties
of the original data, the stability and passivity criteria need to be satisfied [58]. Since
interconnections are stable and passive, the poles resulting from Equation (4.21) should
reside on the left half-plane and the resulting frequency response from the rational functions
should be positive real. These conditions can be satisfied by following a number of methods
that have been developed in the EDA community [58]. The resulting rational function, also
called as a macromodel or black-box model, can be very useful since these functions can be
used to synthesize networks that can be simulated in SPICE or any other circuit simulator.

Network Synthesis
By rewriting Equation (4.21) as a continued fraction, networks can be synthesized. The
resulting networks are typically nonunique and nonphysical, meaning that the circuit
models do not necessarily match the physical structure. However, the response of the
equivalent circuit will match exactly the frequency response modeled.
As an example consider a one-port embedded inductor, whose magnitude and
phase of impedance is shown in Figure 4.53. This response can be used to generate a
rational function that can be synthesized into an equivalent circuit. Since, the frequency
response in Figure 4.52 has only one resonance, the number of poles required is small
(in this example, a real pole and a complex conjugate pole pair are required). In general,
for embedded inductors and capacitors, the number of poles required is between three
to five. Hence, the number of components in the synthesized circuit is small as well. The
synthesized circuits for five different one-port embedded inductor designs are shown
in Figure 4.54 [57]. In all cases, the inductor is first simulated using an electromagnetic
simulator to extract the frequency response, a rational function is then generated from
the frequency response and the equivalent circuit is synthesized from the frequency
response. Constraints have been imposed to ensure that a relationship could be