By Sachin Sapatnekar
With the arrival of nanometer applied sciences, circuit functionality constraints have gotten ever extra stringent. during this context, automatic timing research and optimization turns into valuable for the layout of high-performance circuits that needs to fulfill a tough set of constraints. Timing overviews the cutting-edge in timing research and optimization, and is meant to function a compendium which could supply an creation to the uninitiated reader, as a prepared reference for a practitioner, or as a resource for the comprehensive researcher. A accomplished assessment of the fundamentals of timing research is supplied, and this is often augmented with ideas that contain actual results bobbing up in deep submicron and nanometer applied sciences. The e-book presents an in-depth remedy of the research of interconnect platforms, static timing research for combinational circuits, timing research for sequential circuits, and timing optimization options on the transistor and format degrees. The meant viewers contains CAD instrument builders, graduate scholars, examine pros, and the only curious.
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Extra resources for Timing
Therefore‚ a vital subproblem in circuit simulation relates to the solution of a system of linear equations‚ Ax = b‚ where A is an matrix‚ and x and b are each vectors. We will briefly overview the LU factorization technique‚ a socalled direct method‚ and some iterative‚ or indirect‚ methods for solving this problem. 1 LU factorization The idea of this method is to decompose the matrix A into the product of two matrices: an lower triangular matrix‚ L‚ and an upper triangular matrix‚ U. To depict this more visually‚ we write 26 TIMING If this could be done‚ the original system could be rewritten as Substituting we have Since L is a lower triangular matrix‚ this may be solved easily to find the value of y using a procedure known as forward substitution.
3 Transfer function approximation using moment matching Having calculated the moments of the transfer function‚ the next step is to find a Padé approximant of order [L /M]. For a physical system‚ we must have for the voltage transfer function. The inequality is strict for a system with zero initial conditions‚ and the case where L = M corresponds to a system with nonzero initial voltages or currents 6 .
Moving the negative terms to the left hand side and rewriting the equations in matrix form‚ we may relate y and b through the relation or simply‚ Ly = b. We now illustrate the procedures of forward and backward substitution: the terms “forward” and “backward” allude to the order in which the variables are solved for in the system. For the case of forward substitution‚ since L is lower triangular‚ one may find the elements of the vector y‚ starting from the first equation of Ly = b‚ which directly yields then proceeding to the second equation‚ which yields (since is known by now)‚ and so on forward 28 TIMING until we reach the equation.