Abstract: In view of the requirements for low loss and high power capacity of dual-band/multi-band coaxial feeds, the perturbation method is utilized to derive the formulas of the attenuation constant and power capacity of the TE₁₁ mode in coaxial waveguide based on the vector mode function of the first higher-order TE₁₁ mode. The calculation results of this paper are in good agreement with the simulation results of the commercial software HFSS, which proves the correctness of the method and formula. The relationship between the attenuation constant and power capacity of the TE₁₁ mode and the outer-to-inner diameter ratio of the coaxial waveguide is studied. For several typical dual-band applications, the selection principle for the initial parameters of the outer-to-inner diameter ratio at the input section of the coaxial feed is given based on the minimum loss, maximum power capacity, and overall performance, providing theoretical support for the design of dual-band/multi-band coaxial feeds.