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An analytical method to estimate the scattering of RF power into multiple modes inside a nonuniform waveguide using perturbation techniques is presented. This method is quite general, much faster than numerical methods, and may be applied to a wide variety of nonuniform waveguides. We have used this method to design and analyze a three-section mode transformer that converts a TE01 circular waveguide mode to a TE20 rectangular waveguide mode. This mode converter is specially suitable for use in ultra-high-power applications. Due to the complexity of the problem, we have used the symbolic solver Mathematica to produce most of our analytical results. The results match within reasonable accuracy with High Frequency Structure Simulator field solver simulations. We also present the application of this mode converter to variety of ultra-high-power microwave components and experimentally show the performance of these devices.