We perform 3D finite element calculations of the fields in microwave cavities and analyse the distributed cavity phase (DCP) errors of atomic clocks that they produce. The fields of cylindrical cavities are treated as an azimuthal Fourier series. Each of the lowest components produces clock errors with unique characteristics that must be assessed to establish a clock's accuracy. We describe the errors and how to evaluate them. We prove that sharp structures in the cavity do not produce large frequency errors, even at moderately high powers, provided the atomic density varies slowly. We model the amplitude and phase imbalances of the feeds. For larger couplings, these can lead to increased phase errors. We show that phase imbalances produce a novel DCP error that depends on the cavity detuning. We also design improved cavities by optimizing the geometry and tuning the mode spectrum so that there are negligible phase variations, allowing this source of systematic error to be dramatically reduced.
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