On the choice of steps in the piecewise-constant Hill equation model of a quadrupole mass filter

The model of a quadrupole mass filter which uses a piecewise-constant coefficient as an approximation to the sinusoid in the Mathieu equation (leading to so-called phase plane dynamics ) is considered with regard to determining the minimum number of steps required and their values. It is shown that approximations employing as few as 14 steps will lead to mass filter models of high accuracy provided the step values are carefully chosen. Properties of the relevant Fourier coefficient formulae are shown to lead to a technique for assessing the harmonics of the stepped, model waveform a priori, thereby avoiding the need to evaluate harmonic components in a search for suitable approximations. (Auth.)