Interconnect Modeling and Simulation:Interconnect Simulation Techniques
Interconnect Simulation Techniques
The main objective behind the interconnect simulation algorithms is to address the mixed frequency/ time problem as well as ability to handle large linear circuits without too much CPU expense. There have been several algorithms proposed for this purpose, and they are discussed below.
Method of Characteristics
The method of characteristics transforms partial differential equations of a transmission line into ordinary differential equations [28,29]. Extensions to this method to allow it to handle lossy transmission lines can also be found in the literature; for example, the iterative waveform relaxation techniques (IWR) [28,29]. The method of characteristics is still used as one of the most practical techniques for simulation of lossless lines.
An analytical solution for Eqs. 14.21 and 14.22 was derived in Reference 28 for two conductor lines which provides the U parameters of the two-port transmission line network
where g is the propagation constant, and Z0 is the characteristic impedance. V1 and I1 are the terminal voltage and current at the near end of the line, V2 and I2 are the terminal voltage and current at the far end of the line. The U parameters of the transmission line are complex functions of s, and in most cases cannot be directly transformed into an ordinary differential equation in the time domain. The method of characteristics succeeded in doing such a transformation, but only for lossless transmission lines. Although this method was originally developed in the time domain using what was referred to as characteristic curves (hence, the name), a short alternative derivation in the frequency domain will be presented here. The frequency-domain approach gives more insight as to the limitations of this technique and possible solutions to those limitations.
By rearranging the terms in Eq. 14.54, we can write
A lumped model of the transmission line can then be deduced from Eqs. 14.55 and 14.56, as shown in Figure 14.20. If the lines were lossless (in which case the propagation constant is purely imaginary; g = jb), the frequency-domain expression (Eq. 14.57) can be analytically converted into time domain using inverse Laplace transform as
Recently, there have been several publications based on approximating the frequency characteristics of transmission-line equations using rational polynomials [27]. Analytical techniques to directly convert partial differential equations into time-domain macromodels based on Padé rational approximations of exponential matrices have also been reported [26].
Moment-Matching Techniques
Interconnect networks generally tend to have large number of poles, spread over a wide frequency range. Although the majority of these poles would normally have very little effect on simulation results, they make the simulation CPU extensive by forcing the simulator to take smaller step sizes.
Dominant Poles
Dominant poles are those that are close to the imaginary axis and significantly influence the time as well as the frequency characteristics of the system. The moment-matching techniques (MMTs) [34–67] capitalize on the fact that, irrespective of the presence of large number of poles in a system, only the dominant poles are sufficient to accurately characterize a given system. This effect is demonstrated in Figure 14.21, where it is clear that pole P2 will have little effect on the final transient result.
A brief mathematical description of the underlying concepts of moment-matching techniques is given below. Consider a single input/single output system and let H(s) be the transfer function. H(s) can be represented in a rational form as
Equations 14.67 and 14.68 yield an approximate transfer function in terms of rational polynomials. Alternatively, an equivalent pole-residue model can be found as follows. Poles pi are obtained by applying a root-solving algorithm on denominator polynomial Qˆ (s). In order to obtain ki, expand the approximate transfer function given by Eq. 14.62 using Maclaurin series as
Comparing Hˆ (s) from Eqs. 14.65 and 14.69, we note that
In the above equations, cˆ represents the direct coupling between input and output. There are more exact ways to compute cˆ which are not detailed here; interested readers can refer to Reference 35.
Computation of Moments
Having outlined the concept of MMTs, to proceed further, we need to evaluate the moments of the system. Consider the circuit equation represented by Eq. 14.46 and expand the response vector X(s) using Taylor series as
The above equations give a closed form relationship for the computation of moments. The moments of a particular output node of interest (which are represented by mi in Eqs. 14.65 to 14.71), are picked from moment-vectors Mi. As seen, Eq. 14.75 requires only one LU decomposition and few forward- backward substitutions during the recursive computation of higher-order moments. Since the major cost involved in circuit simulation is due to LU decomposition, MMTs yield very high speed advantage (100 to 1000 times) compared to conventional simulators.
Generalized Computation of Moments
In the case of networks containing transmission lines and measured subnetworks, moment computation is not straightforward. A generalized relation for recursive computation of higher-order moments at an expansion point s = a can be derived [44] using Eq. 14.46 as:
where the superscript r denotes the rth derivative at s = a. It can be seen that the coefficient on the left- hand side does not change during higher-order moment computation and, hence, only one LU decom- position would suffice. It is also noted that the lumped networks are a special case of Eq. 18.76 (where Y(r) = 0 for r ³ 2, in which case Eq. 14.76 reduces to the form given by Eq. 14.75).
Having obtained a recursive relationship Eq. 14.76 for higher-order moments, in order to proceed further, we need the derivatives of (Y). The derivatives A(r) lines can be computed using the matrix exponential-based method [34,38]. Efficient techniques for computation of moments of transmission lines with frequency-dependent parameters [56], full-wave [12], and measured subnetworks [63,65] can also be found in the literature.
Computation of Time-Domain Macromodel
Once a pole-residue model describing the interconnect network is obtained, a time-domain realization in the form of state-space equations can be obtained as [39–41,75,76]:
where ip and vp are the vectors of terminal currents and voltages of the linear subnetwork p. Using standard non-linear solvers or any of the general-purpose circuit simulators, the unified set of differential equations represented by Eq. 14.77 can be solved to yield unified transient solutions for the entire non-linear circuit consisting of high-frequency interconnect subnetworks. For those sim- ulators (such as HSPICE) that do not directly accept the differential equations as input, the macro- model represented by Eq. 14.77 can be converted to an equivalent subcircuit, and is described in the next section. Figure 14.22 summarizes the computational steps involved in the MMT algorithm.
Limitations of Single-Expansion MMT Algorithms
Obtaining a lower-order approximation of the network transfer function using a single Padé expansion is commonly referred as asymptotic waveform evaluation (AWE) in the literature. However, due to the
inherent limitations of Padé approximants, MMTs based on single expansion often give inaccurate results. The following is a list of those properties that have the most impact on MMTs.
• The matrix in Eq. 14.67 (which is known as Toeplitz matrix) is ill-conditioned if its size is large. This implies that we can only expect to detect six to eight accurate poles from a single expansion.
• Padé often produces unstable poles on the right-hand side of the complex plane.
• Padé accuracy deteriorates as we move away from the expansion point.
• Padé provides no estimates for error bounds.
Generally, AWE gives accurate results for RC networks, but often fails for non-RC networks. This is due to the fact that the poles of an RC network are all on the real axis and therefore an expansion at the origin could clearly determine which ones are dominant and which ones are not. However, in the case of general RLC networks, it is possible to have some of the dominant poles outside the radius of convergence of the Padé expansion. In systems containing distributed elements, the number of dominant poles will be significantly higher, and it is very difficult to capture all with a single Padé expansion.
In addition, there is no guarantee that the reduced-model obtained as above is passive. Passivity implies that a network cannot generate more energy than it absorbs, and no passive termination of the network will cause the system to go unstable [70–73]. The loss of passivity can be a serious problem because transient simulations of reduced networks may encounter artificial oscillations.
Recent Advances in Moment-Matching Techniques
In order to address the above difficulties, recent research in the circuit simulation area has focused on arriving at compact, accurate, as well as passive macromodels for high-speed interconnects. The problem of accuracy is addressed using multi-point expansion techniques such as complex frequency hopping (CFH) [44–46]. Also, in order to enhance the accuracy range of an approximation at a given expansion and to reduce the number of hops, several techniques based on Krylov-space formulations are developed [47–55]. Also, efficient schemes based on congruent transformation for preservation of passivity during the reduction of interconnect networks is available in the literature [47–55]. For further readings, interested readers can look at the recent proceedings of ICCAD, DAC, or IEEE transactions on computer-aided design of ICs (T-CAD), circuits and systems (T-CAS), and microwave theory and techniques (T-MTT).
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