5.3 Axial Mode Matching and Other Methods

The radial mode matching method described in detail in the previous section is based on the partition of the resonator cross section into partial regions in which the permittivity ? _r is independent of the radial coordinate ? (Fig. 5.1). Each partial region is then considered as a section of a dielectrically loaded radial waveguide. In a complementary procedure, which will be referred to as the axial mode-matching method, the resonator is divided into partial regions in which ? _r is independent of the axial coordinate (i.e., z), as illustrated in Fig. 5.3. It is noted that all partial regions so defined may be regarded as sections of dielectrically-loaded (regions II, III, and V) or homogeneous (regions I and IV) cylindrical waveguides. Therefore, the transverse fields in each partial region i, i = I,...,V, can be expanded in terms of cylindrical waveguide modes as [19]

Figure 5.3: Shielded dielectric resonator (III) including substrate (I), support (II), and tuning post (V) (reference [23], 1984 AEU)

(5.92a)

(5.92b)

where are the modal propagation constants, and are the transverse field expansion functions, and and are the field coefficients. At a given frequency, the propagation constants must be found from a transcendental equation involving Bessel functions of order m and their derivatives, obtained by enforcing the continuity of the tangential field components inside each partial region and the condition of vanishing tangential electric field on the metallic enclosure.