The analytical expressions of the beam-wave coupling coefficients and the beam-loaded conductance in an N-gap coupled cavity are derived based on space-charge wave theory. Through calculating the relations of the beam-wave coupling coefficient and the normalized beam-loaded conductance to the gap number, beam voltage and perveance for 2π mode, the mechanism of the beam-wave synchronization and coupling in the multi-gap coupled cavity are discussed. The results show that, with the increase of N(≥2), the beam-wave coupling efficiency and the normalized beam-loaded conductance vary with beam voltage more rapidly and there is a maximum value for the absolute squared value of the coupling coefficient |MN|2 and a maximum value and a minimum value for the normalized beam-loaded conductance gb. The magnitudes of these extrema increase with the increase of gap number N, and the corresponding voltage is close to the synchronization voltage. The increase of the perveance could make the voltage difference between two extremums of gb increase, the magnitudes of these extrema decrease, and the beam-wave coupling efficiency fall.