An Accurate and Efficient Computation of Poles and Zeros of Transfer Functions for Large Scale Analog Circuits and Digital Filters

Abstract

The poles and zeros of a circuit transfer function can be efficiently computed solving the generalized eigenvalue problem, which could be transformed into the standard eigenvalue problem to be solved by a suitably modified QR algorithm. In this way, the poles and zeros can be obtained for any linear circuit or a nonlinear circuit linearized at an operating point (using the Laplace transform), or for any digital filter (using the $\mathcal {Z}$ transform). Both the reduction of the generalized eigenvalue problem to the standard form and the iterative procedures of the QR algorithm are very sensitive to the numerical precision of all calculations. The numerical accuracy is especially critical for the two kinds of circuits: the microwave circuits characterized by huge differences among the magnitudes of the poles and zeros, and the large scale circuits, where the errors of poles and zeros are increased by the extreme number of arithmetic operations and frequent multiplicity of the poles and zeros as well. In this chapter, two illustrative examples of the reduction of the general eigenvalue problem (the first for analog and the second for digital circuit) and using the QR algorithm are shown first. After that, four circuits of various sizes are analyzed simpler microwave low-noise amplifier, larger power operational amplifier, more complex example with a 272 integrated operational amplifier, and the most difficult analysis of a distributed oscillator. A meticulous comparison of the obtained results shows that a usage of newly implemented 128-bit arithmetics in GNU Fortran or C compilers with partial pivoting can assure both efficient and enough accurate procedures for computing the poles and zeros of the circuit transfer function.