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1 AWEsymbolic: Compiled Analysis of Linear(ized) Circuits using Asymptotic Waveform Evaluation John Y. Lee and Ronald A. Rohrer SRC-CMU Center for Computer Aided Design Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, Pennsylvania Abstract Asymptotic Waveform Evaluation (AWE) is shown to be effective in the symbolic analysis of lineariired) circuits. AWEsymbolic efficiently and accurately produces reduced order models of both frequency and time domain behavior of linear(ized) circuits. By use of moment level partitioning, the numeric and symbolic computations are substantially decoupled, hence increasing computation speed. Results for AWEsymbolic show a marked decrease in execution time as compared to a full AWE analysis, while producing identical results. This portends the use of AWEsymbolic as an efficient modeling methodology, where AWE approximations are compiled into a reduced set of operations. 1. Introduction HE goal of symbolic analysis tools, as opposed to tradi- T tional circuit simulators [5], is to obtain symbolic forms for network functions in the frequency domain. Some or all of the elements in a circuit are treated as symbols that have no assigned numerical values. The resulting network functions are pure symbolic, or mixed symbolic-numeric expressions. Such symbolic forms have served many roles [lo], including modeling, sensitivity calculation and pedagogy. Most efforts, including [121, 191, [lo], [21, 181 compute an exact form of the network functions in the frequency domain. For high order systems, this can lead to complex symbolic forms, even when the number of symbols is low. In order to simplify these expressions, heuristic pruning can be employed [81. These pruning mechanisms can be unreliable, because suppressed terms may play an important role in the response of a circuit. Determining which terms will be significant and which not, over a range of element values, is difficult. In particular, pruning at the level of the coefficients of the numerator and denominator polynomials of the impulse response is dangerous as the effects on pole-zero locations are difficult to discern. This paper proposes the use of Asymptotic Waveform Evaluation (AWE) [7] for the symbolic analysis of linear and linear(ized) circuits. AWE affords the ability to reliably and efficiently compute reduced order symbolic expressions, without the use of unreliable pruning methods. Efficient pole-zero sensitivity analysis in AWE provides a mechanism to identify elements that have a significant effect on the dominant poles and zeros or residues of the circuit. Such sensitivities may serve as an automatic means for identifying suitable symbolic elements. Through the use of moment level circuit partitioning [ll, AWEsymbolic is able to significantly decouple the numeric and symbolic computations. The advantage of this approach is decreased computation time. In essence, the symbolic form provides a compiled set of operations which can quickly produce a final AWE approximation, where the operands are the values of the symbols. Examples later in the paper will show that the cost of evaluating such a model is much lower than that of a full AWE analysis, and hence can be useful in highly iterative applications. Since the order of a reasonably accurate AWE approximation is typically low, often less than five, it is possible to factor the symbolic forms obtained. Not only can the symbolic forms for the dominant poles and zeros of a circuit be computed, but the transient response of a circuit can be expressed symbolically as well. Given the low iterative computational cost, and the ability to model in the time domain, AWEsymbolic should serve as a useful mechanism for modeling interconnect delay in physical CAD design tools. 2. Methodology This section covers the methodology of AWEsymbolic. Notation is introduced below, and AWE is discussed briefly (a reader unfamiliar with Asymptotic Waveform Evaluation is encouraged to consult [7]). AWEsensitivity [41 and AWE partitioning [I], as they apply to AWEsymbolic are then discussed Definitions Consider a single input, single output, linear time invari- M38-100X/ Q 1992 IEEE 29th ACM/IEEE Design Automation Conference@ 213

2 ant system, represented in state variable [6] form: x = Ax+ bu (1) y = clx+du (2) The ftequency domain system function (the Laplace transform of the impulse response) is expressed as follows: H(s) = ct(sl-a) b+d (3) Given a set of symbolic elements e : e = {el, e2,..., e,) (4) the goal of symbolic analysis is to compute the transfer function as H(s, e). Given that not all variables of H(s) are treated as symbols, the final result will be in mixed numericsymbolic form. It is easily shown that the coefficients of the numerator and denominator polynomials of H(s) are multi-linear in the symbolic elements [lo]. For example, consider the simple circuit shown in Figure 1. The full symbolic system function (where all elements are treated as symbols) is: GIG2 H(s) = (5) C,C2d+ (G2C,+G2C,+G,C2)s+G,G2 If the value of G1 is set to 5, the mixed numeric-symbolic expression is: H(s) = 5% ClC2s2 + (G2C1 + G2C, + 5C2) s + 5G, (6) The coefficients of the numerators and denominators of (5) and (6) are linear in each of the elements. Often times, it is not necessary or desirable to compute an exact solution, whether the answer be numeric or symbolic. AWE addresses this issue. Figure 1: Sample RC circuit for symbolic analysis 2.2. AWE Approximation In lieu of computing an exact symbolic solution for H(s), AWE constructs an approximation [7] by matching moments of the impulse response, where the moments are the coefficients of the Maclaurin series expansion of H(s) about s = 0: H(S) = mo + mls + m2j misi +... (7) The moments can be computed efficiently from a DC circuit related simply to the original system. The moments are used to form a rational Pa& approximation to H(s). AWE is typically more than an order of magnitude or more efficient than traditional circuit simulation techniques, and its results can be applied to the analysis of time and frequency domain problems [7], AWE analysis serves as the cornerstone in AWEsymbolic AWESensi tivi t y Efficient pole-zero sensitivity. using the adjoint method, has been implemented in AWE [4]. Because the poles and zeros completely define the behavior of a system, their sensitivities can seme as a robust mechanism to identify significant elements. If a choice of symbolic elements has not been made, a pole-zero sensitivity analysis is performed using AWE. Elements with large normalized sensitivities are pruned as symbolic elements. Since it is possible to express all behavior of a linear system in terms of the poles and zeros, the pruning mechanism is easily extended to performance measures such as gain, ringing, phase margin, etc... Given that the sensitivities computed by AWE provide only local information, it may be necessary to validate the choice of symbolic elements over the range spanned by the symbolic elements. Once the symbolic functions have been compiled (as will be shown later in this paper) though, the cost of validation is low. I Numeric Partition Figure 2: Partitioning of SymhIic and numeric blocks 2.4. Partitioning AWEsymbolic uses moment level partitioning to separate the numeric and symbolic computations are far as possible. The circuit is split into blocks that contain only symbols (symbolic partitions), and partitions that contain only numeric elements (numeric partitions), as shown in Figure 2. The moments are computed for each partition. In a typical AWE analysis, the time needed to compute the moments far outweighs the time used to form the Pad6 approximation from the moments [13]. Since a mixed numeric-symbolic analysis is inevitably slower than a numeric simulation, separating the symbolic and numeric moment calculation provides the bulk of the execution time improvement. To compute the moments of the overall (composite) circuit, the multi-port admittance parameters are computed for each partition. For a partition with n-ports, the admittance parameters are represented as: 214

3 Each admittance function representation is expanded into a Maclaurin series in s: Ynxn = Ynx, + Ynxn,S + Ynxn2s2 + (9) Given that only one symbolic element is encapsulated in any given symbolic block, its port representation is finite. In fact, when a general formulation technique such as modified nodal analysis (MNA) [3] is used, the expansion has only one term per element. With MNA, impedance elements are handled through auxiliary equations, so the symbolic admittance parameters for resistances and capacitances are stencilled using admittance forms, while inductances are stenciled as impedances. Together, we see that their moment expansion is finite: symbolic = G+s (c+l) (10) where G, C and L are matrices of appropriate dimension. The port parameters from each partition (numeric or symbolic) are stenciled into a global admittance matrix Yglobal. Its dimensions are proportional to the number of ports, which is generally proportional to the number of symbolic elements. Thus the size of Yg/oba/ will generally be much smaller than size of the original circuit, especially when the ratio of numeric elements to symbolic elements is high. The cost of solving this symbolic circuit is much less than that involved in solving a non-partitioned circuit. VYlh..., Each composite moment { Vo, V,,...} can be computed by recursive solution of the resistive Yg/oba/O circuit. Since in general the impulse response is sought, all terms except Io of the Taylor series expansion of I are 0. The moments {Vo, V,,...} are then used to form an AWE approximation. Since the moments are symbolic functions, the result is a reduced order symbolic approximation, where the order is controlled by the number of moments matched [7]. The cost for finding the approximation is small -though somewhat hindered due to the mixed numericsymbolic computation demands. In general, if an nth order approximation is sought, an nxn full matrix is LU factored, and a separate nxn forward substitution performed. 3. Examples This section will illustrate the use of AWE symbolic in the frequency domain, and in the time domain. For each case, it will be seen that a first order analysis can provide a simple intuitive result. Higher order approximations are not as simple, yet are algebraically compact. It is shown that they can be evaluated at a fraction of the cost of a full AWE analysis. Thus, they can be used in applications where a low incremental cost of analysis is required Frequency Domain Symbolic Analysis V.rl)Y Figure 3: 741 Operational Amplifrer The MNA equations for the composite circuit are expressed as: Yg/obo/(s)V(S) = I(s) (11) where V(s) are the outputs of the composite circuit, and I is the vector of independent sources. Each constituent of (11) is expanded into a Maclaurin series: (Yglobo10 + YgIoboI1S+...) (V0+ v,s+...) = (I,+I,s+...) Matching powers of s from the above equation yields: (12) The 741 operational amplifier, shown in Figure 3, is analyzed using AWE symbolic. After linearization, the small signal circuit contains 170 linear elements, 62 of which are energy storage elements. Based on an AWEsensitivity analysis, two most significant elements, goutq14 and Ccompr are chosen to be treated as symbols (due to their contribution to gain and phase of the amplifier s frequency response). A five port representation is used for the numeric partition.this includes the input and output ports, which must be preserved. The first order AWEsymbolic approximation is shown below: 21s

4 1 A,=- (0.28Cc, - (8.75~10 ~g,,,,,, ~ l.27x104g,,,q Cc040,,IQ ~10. ) (1.13~10~g,,~~~+ 1.51x10-~ (7.74~10~g,,,,~~~ ~10 ) A0 1.) 1 The symbolic form is stable for all values of gourq14 and Ccomp, as is the case with the real circuit. Since only two moments are used in the approximation. the symbolic forms are multi-linear in tenns of the symbolic elements. This is the case in general for first order AWEsymbolic approximations. The symbolic expression in (14) is used to generate 3 dimensional plots of performance versus values of the symbolic elements. Figure 4 plots the first pole as a function of Figure 5: Plot of DC gain from first order symbolic form Figure 4: Plot of pi from fist order symbolic form the symbolic parameters. Figure 5 plots the DC gain of the op-amp as a function of the symbolic parameters. The symbolic farms are computed in 3.03 seconds on a DECstation 5000, using Mathematica [ 111 for the symbolic analysis. The time to evaluate the symbolic forms, given numeric assignments for each symbol, is 0.37 microseconds. In contrast, the time needed to do an AWE analysis is 80.4 milliseconds, of which 25 milliseconds is overhead due to parsing and setup. Ignoring the overhead in both scenarios, the per iteration evaluation cost of AWEsymbolic is roughly five orders of magnitude faster. The second order symbolic form is more complex and of course more accurate. In general, four moments are required for a second order approximation, equation (13) is mursively solved four times. However, since the complexity of each successive moment can increase dramatically, a partial F ade approximation, using derivatives is used. This decreases the number of moments needed in the approximation, with little sacrifice in accuracy. The second.order approximation is shown below. The notation P(a!, y ) has been adopted to indicate a polynomial term of ith degree in the symbol x, and@ degree in the symbol y: The symbolic form is not in multi-linear formthat is, our symbolic elements do not have a physical representation in the symbolic form. However, each term can be evaluated very quickly, and the results are identical to those obtained by a numeric AWE analysis (and as accurate), but with a much quicker iterative evaluation time. Table 1 cites run times needed to generate the data directly from AWE, without any symbolic analysis. The run times were generated on a DECstation 5000, and do not include common overhead such as parsing and plotting. As is seen, the symbolic analysis is considerably faster than a pure numeric analysis for multiple evaluations of the system function at multiple symbol values. Per iteration, the symbolic analysis cost is roughly 330 times less than for a numeric AWE analysis. The incremental cost for AWEsymbolic is 0.16 milliseconds, while it is 53.2 milliseconds for AWE. Recall that AWE has also been benchmarked to be at least an order of magnitude faster than SPICE [5] for this class of problem, so AWEsymbolic s speedup over traditional techniques may be quite high. Figure 6 plots the unity gain frequency of the operational amplifier, as a function of the symbolic parameters. Similarly, Figure 7 plots the phase margin of the operational amplifier. The DC gain plot from the second order form is identical to that of the first order form shown above - since the first moment computed by AWE is always an exact form 216

5 Datapoints lo00 AWE AWESymbolic 0.079s 2.27s 5.35s 2.29s 53.2s 2.43s Using AWEsymbolic, a timing model for this circuit is developed. In this case, the driver resistance and the load capacitance are treated as symbols, and the time domain behavior computed. In order to model the non-monotonic nature of the cross coupling response, a second order AWE approximation is used to insure accuracy in the cross talk analysis from one line to the other. A first order approximation suffices to model the direct transmission down each line: Figure 6: Plot from 2nd order symbolic form of the IX gain. Note that each plot was generated by use of the symbolic forms for the poles and zeros. The data is identical to that obtained from a pure numerical AWE analysis. Using the notation from (15), the cross coupling is expressed as: Figure 7: Plot off, from 2nd order symbolic form 3.2. Time Domain Symbolic Analysis A timing model for the circuit shown in Figure 8 is to be Figure 8: Lumped model for coupled lines. constructed. Two coupled lines have been modelled using a lumped RC equivalent, with capacitive coupling along the length of the line. The lines are symmetric. The driver at each line is modeled by a linearized Thevenin equivalent, and the loading is assumed to be purely capacitive. Each line has been approximated with a lo00 segment model. The results of the symbolic analysis coincide exactly with those of a numeric AWE analysis. The difference lies in the run time. A single AWE analysis for this circuit requires on average 1.12 seconds on a DECStation5000, while the AWEsymbolic analysis requires 5.41 seconds (using Mathematica for the symbolic computation) [ll]. However, the incremental cost, which is crucial in iterative applications, is 0.11 milliseconds for AWE symbolic. This is four orders of magnitude faster than a numeric analysis with AWE. Illustrations of the second order symbolic form are shown in Figures 9 and 10. The step response cross talk is plotted from the second order symbolic form, as a function of the symbolic parameters. 4. Conclusion A new method for the symbolic analysis of linear(ized) circuits has been presented. The methodology uses Asymptotic Waveform Evaluation to produce a low order symbolic approximation to the system response of the circuit. The computational cost of evaluating such symbolic forms is orders of magnitude less than that of multiple numeric AWE simulations, as much as five orders of magnitude quicker in one example shown. This marked increase in performance 211

6 Figure 9: Cross-talk transient response as G*iwr is varied Figure 10: Cross-talk transient response, as Clmd is varied results from the compilation of an AWE analysis into a reduced set of operations, which are parameterized in terms of the symbolic elements. In addition, efficient pole-zero sensitivity in AWE allows significant elements to be identified with little additional cost. The encouraging results for AWEsymbolic portend its use for modeling in highly iterative applications, where low incremental cost, robustness and accuracy (in either the time or frequency domains) are required. 5. Acknowledgments The authors thank Eric Bracken and Vivek Raghavan for their strong contributions, especially with AWE partitioning. Financial support for this research was provided by the Semiconductor Research Corporation (Contract #DC ) and the Digital Equipment Corporation. 6. References [l] M.A.Alaybeyi, J.E. Bracken, J.Y. Lee. V. Raghavan, R.J. Trihy and Ronald A. Rohrer. Exploiting Partitioning in Asymptotic Waveform Evaluation (AWE). To appear in Proceedings IEEE Custom Integrated Circuits Conference *a-' (21 G.M. Wierzba and A. Srivastava and K.V. Noren and J.A. Svoboda. Sspice-A Symbolic Spice Program for Linear Active Circuits. 32nd Midwest Symposium on Circuits and Systems, pages [3] C.W. Ho and A.E. Ruehli and P.A. Brennan. The Modified Nodal Approach to Network Analysis. IEEE Tram. Circuit Theory. CAS , June, [4] John Y. Lee and Xiaoli Huang and Ronald Rohrer. Efficient Pole Zero Sensitivity Calculation in AWE. Technical Digest of the IEEE Internatid Conference on Computer-Aided Design, pages [5] L.W. Nagel. SPlCE2. A Computer Program to Simulate Semiconductor Circuits. Technical Report ERL-M520. UC-Berkeley, May, [6] E. S. Kuh and R. A. Rohrer. The State-Variable Approach to Network Analysis. Proceedings of the IEEE. 53(7), July, [7] Lawrence T. Pillage and Ronald A. Rohrer. Asymptotic Waveform Evaluation for liming Analysis. IEEE Trans. Computer-Aided Design. 9(4): April, [8] Georges G.E. Gielen and Herman C.C. Walsharts and Willy M.C. Sansen. ISAAC: A Symbolic Simulator for Analog Integrated Circuits. IEEE Journal of Solid-state Circuits. 24(6). December, [9] Gary E. Alderson and P.M. Lin. Computer Generation of Symbolic Network Functions-A New Theory and Implementation. IEEE Tramactions on Circuit Theory. CT-20(1). January, [lo] P.M. Lin. A Survey of Applications of Symbolic Network Functions. IEEE Tranractions on Circuit Theory. CT-20(6), November, [ll] Stephen Wolfram. Mathematics: A System for Doing Mathematics by Computer. Addison Wesley, [ 121 Kishore Singhal and Jiri Vlach. Symbolic Analysis of Analog and Digital Circuits. IEEE Tramactwn on Circuits and Systems. CAS-24(11), November [13] Xiaoli Huang and Vivek Raghavan and Ronald A. Rohrer. AWEsim: A program for the efficient Analysis of Lmear(ized) circuits. Technical Digest of the IEEE Internatwnul Conference on Computer-Aided Design, pages November, T

Statistical approach in complex circuits by a wavelet based Thevenin s theorem

Proceedings of the 11th WSEAS International Conference on CIRCUITS, Agios Nikolaos, Crete Island, Greece, July 23-25, 2007 185 Statistical approach in complex circuits by a wavelet based Thevenin s theorem