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The perturbation in the complex propagation constant of a lossy, nonreciprocal, periodic waveguide produced by the insertion of a rod, which may be cylindrical or periodic, parallel to the waveguide axis is developed. The application to the experimental determination of the interaction impedance and electromagnetic field distribution of waveguides is presented, together with the approximations which are applicable. The perturbation formulas for three particular classes of circuit-lossless, reciprocal, cylindrical circuits; lossy, nonreciprocal, cylindrical circuits; and lossy, nonreciprocal, periodic circuitsÂ¿of interest for traveling wave tubes and other extended interaction microwave tubes, are derived and the limitations discussed. Explicit interaction impedance relations for these circuits in terms of the phase constant perturbation, caused by a cylindrical or periodic rod, are given.