= Quadrupole mass analyzer
A <quadrupole mass analyzer> is a <mass analyzer> that selects <ions> through stability of motion in combined static and alternating quadrupole <electric fields>. Selected <mass-to-charge ratios> follow bounded transverse trajectories through the analyzer; others grow unstable and fail to reach its exit.<Quadrupole mass analyser in IUPAC Gold Book>[Q04971, stable and unstable trajectory definition]{ref}
For an idealized model, define the electric potential
$$
\Phi(x,y,t)=\frac{U+V\cos(\Omega t)}{r_0^2}(x^2-y^2).
$$
Here $r_0$ is a geometric scale and $U,V$ follow this stated potential convention. An ion with <electric charge> $Q$ and <mass> $m$ then obeys
$$
\ddot x=-\frac{2Q}{mr_0^2}[U+V\cos(\Omega t)]x,\qquad
\ddot y=+\frac{2Q}{mr_0^2}[U+V\cos(\Omega t)]y.
$$
At a given instant the field focuses one transverse direction and defocuses the other. Alternation can nevertheless yield bounded motion in both directions. Rescaling time reduces these equations to forms of the <Mathieu equation>; a <quadrupole stability diagram> summarizes the relevant bounded-motion regions.
The equations also show why the filter selects a ratio. Doubling both $m$ and $Q$ leaves the ideal equations unchanged. With fixed geometry, the dimensionless coefficients scale as $QU/(m\Omega^2)$ and $QV/(m\Omega^2)$. Doubling the drive <angular frequency> while keeping the potentials fixed divides both coefficients by four; multiplying both potentials by four restores them in this model.
A stable mathematical trajectory is only part of instrument performance. Initial ion positions and velocities, finite flight length, field imperfections and detector response also influence transmission and measured <mass resolving power>. The ideal potential is an explicit model, not a guarantee about a particular instrument.
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