Compact representation and efficient generation of s-expanded symbolic network functions for computer-aided analog circuit design

A graph-based approach is presented for the generation of exact symbolic network functions in the form of rational polynomials of the complex frequency variable s for analog integrated circuits. The approach employs determinant decision diagrams (DDDs) to represent the determinant of a circuit matrix and its cofactors. A notion of multiroot DDDs is introduced, where each root represents a symbolic expression for an individual coefficient of the powers of s in the numerator and denominator of a network function, and multiple roots share their common subgraphs. A DDD-based algorithm is presented for generating s-expanded network functions. We prove theoretically and validate experimentally that the algorithm constructs in O(kl|DDD|) time an s-expanded DDD with no more than kl|DDD| vertices, where k is the degree of the denominator s polynomial, l is the maximum number of devices that connect to a circuit node, and |DDD| is the number of DDD vertices representing the circuit-matrix determinant. For a practical circuit, |DDD| is often many orders-of-magnitude less than the number of product terms. In contrast, previous approaches require the time and space complexities proportional to the number of product terms, which grows exponentially with the size of a circuit. Experimental results have demonstrated that the new approach can produce exact s-expanded-symbolic network functions for μA741 operational amplifiers in several CPU seconds on an UltraSparc-I workstation. The expressive power of multiroot s-expanded DDDs is so remarkable that in one instance, over 10³⁵ symbolic product terms have been represented by a multiroot DDD with less than 17 K vertices. The compactness of DDDs is further demonstrated in the context of symbolic noise evaluation, where potentially many transfer functions, each being used for a noise source in the circuit, can be represented by a single DDD with the size comparable to that for a few transfer functions. This provides a powerful tool for solving many symbolic analysis problems such as deriving interpretable symbolic expressions, dominant pole/zero estimation, and analog testability analysis. We have also demonstrated that repetitive numerical evaluation with the derived s-expanded symbolic expressions for frequency-domain simulation and small-signal noise analysis can be much faster than SPICE-like simulators and the resulting expressions for a circuit block can be used as behavioral models for high-level simulation