Interconnect Modeling and Simulation:Interconnect Simulation Issues

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Interconnect Simulation Issues

As pointed out earlier, simulation of interconnects is associated with two major bottlenecks: mixed frequency/time problem and the CPU expense.

Mixed Frequency/Time Problem

The major difficulty in simulating these high-frequency models lies in the fact that distributed/full-wave elements, while formulated in terms of partial differential equations, are best described in the frequency domain. Non-linear terminations, on the other hand, can only be given in the time domain. These simultaneous formulations cannot be handled by a traditional ordinary differential equation solver such as SPICE [8,38].

CPU Expense

In general, interconnect networks consist of hundreds or thousands of components, such as resistors, capacitors, inductors, transmission lines, and other levels of interconnect models. At the terminations, there generally exist some nonlinear elements such as drivers and receivers. If only lumped RLC models are considered, ordinary differential equation solvers such as SPICE may be used for simulation purposes. However, the CPU cost may be large, owing to the fact that SPICE is mainly a non-linear simulator and it does not handle large RLC networks efficiently.

Background on Circuit Simulation

Prior to introducing interconnect simulation algorithms, it would be useful to have a glimpse at the basic circuit simulation techniques. Conventional circuit simulators are based on the simultaneous solution of linear equations which are obtained by applying Kirchoff ’s current law (KCL) to each node in the network. Either for frequency- or time-domain analysis, the first step is to set up the modified nodal analysis matrix (MNA) [68]. For example, consider the small circuit in Figure 14.18. Its MNA equations are

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The above equation, representing a simple two-node circuit, has the same form as any other MNA matrix representing a large linear lumped network. Hence, MNA equations in general for lumped linear networks can be written as

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Discussion of CPU Cost in Conventional Simulation Techniques Frequency-domain simulation is conventionally done by solving Eq. 14.39 at each frequency point through LU decomposition and forward-backward substitution. For time-domain simulation, linear multi-step techniques [69] are used. The most common of these integration formulas is the trapezoidal rule, which gives the following difference formula

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Note that the trapezoidal rule is an integration formula of order of 2, which is the highest possible order that could ensure absolute stability [69]. Due to such relatively low order, simulators are forced to use small step sizes to ensure the accuracy during transient simulation. Transient response computation requires the LU decomposition of Eq. 14.42 at every time step. It gets further complicated in the presence of nonlinear elements, in which case the Eq. 14.40 gets modified as

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In the case of nonlinear elements, Newton iterations are used to solve Eq. 14.44, which requires two to three LU decompositions at each time step. This causes (note that W and G matrices for interconnect networks are usually very large) the CPU cost of a time-domain analysis to be very expensive. This led to the development of model-reduction algorithms, which effects a reduction in the order of the linear subnetwork before performing a non-linear analysis so as to yield fast simulation results.

Circuit Equations in the Presence of Distributed Elements

Now consider the general case in which the network f also contains arbitrary linear subnetworks along with lumped components. The arbitrary linear subnetworks may contain lossy coupled transmission lines and also measured subnetworks. Let there be Nt lossy coupled transmission line sets, with na coupled conductors in the linear subnetwork a. Without loss of generality, the modified nodal admit- tance (MNA) [68,69] matrix for the network f with an impulse input excitation can be formulated as

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To illustrate the above formulation, consider the network shown in Figure 14.19, which is a modified version of the previous example with introduction of a coupled transmission line. The network can be described by Eq. 14.46 as

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Conventional simulation methods obtain the frequency response of a circuit by solving Eq. 14.46 using LU decomposition and forward-backward substitution at various frequency points (usually thousands of points are required to get an accurate response over a desired frequency range). However, moment-matching techniques such as AWE extract the poles and residues of Eq. 14.46 using one LU decomposition only. The transfer function of the network and its frequent response can then be deduced from poles and residues. In addition to speeding up the simulation, MMTs provide a convenient way to handle mixed frequency/time simulation problem through macromodeling.

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