The boundary problem for eigenmodes in coaxial quad-ridged waveguides has been solved by the transverse field-matching technique. The formulas obtained provide possibilities to calculate cutoff wave numbers and electric and magnetic fields distributions of TEM, TE and TM modes in the presence of the ridges either on the inner or on the outer perfectly conducting cylinder. The analysis of the dependences of cutoff wave numbers and electric field distributions convergences on the number of partial modes has been carried out. It has been shown that for calculation of cutoff wave numbers with residual error less than 0,1 % it is enough to utilize 27 partial modes, and for the correct calculation of fields distributions one should utilize more than 30 partial modes.

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