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In this study the authors introduce an analysis of rms frequency error performance and spurious signals generated by two-point modulators. The analysis is not limited to a constant delay and magnitude imbalance between modulation paths but allows frequency-dependent group delay and amplitude variations as well. Moreover, a discrete time phase frequency detector model is incorporated in Laplace domain analysis that takes into account the sampling nature of a phase-locked loop (PLL). Using the spectrum of pulse width modulated charge pump pulses, the spurious signals at the output of the PLL are evaluated. The proposed formulae are tested on a practical setup and quite accurate results are obtained.

In this study finite part integrals are utilized for evaluation of hypersingular and nearly-hypersingular surface integrals on curvilinear elements. These integrals are related to the second derivative of the free space Green' function and arise in the solution of electric field integral equation (EFIE) via locally corrected Nyström (LCN) method. The curvilinear elements are represented by the Taylor series expansion of the surface function around the observation point. The hypersingular integral, defined on a curvilinear element, is written as a summation of hypersingular and weakly singular integrals which are defined on a flat surface. Numerical studies show that increased accuracy is obtained for hypersingular integrals on curvilinear elements.