2.3.2 Circular waveguide loaded with dielectric and metal discs having different hole radius

This model (model-7) is similar to that of model-6, which has a circular waveguide consisting of alternate dielectric and metal discs, such that the hole-radius of metal discs is lesser than that of dielectric discs [20]. For the purpose of analysis, one may divide the structure into three regions: i) the central disc free region I: 0≤r , rD, 0 , z , ∞; ii) the disc occupied free space region II: rD ≤r , rDD, 0 , z , L � T; and iii) the dielectric filled disc occupied region III: rDD ≤r , rW, 0 , z , L � T; where rDD is the hole radius of dielectric disc. It is considered that the region I (disc free region) supports propagating and regions II and III (disc occupied regions) support standing waves [20].

Figure 7. Circular waveguide loaded with dielectric and metal discs [19].

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide… DOI: http://dx.doi.org/10.5772/intechopen.82124

The relevant (axial magnetic and azimuthal electric) field intensity components in the region I may be given by (1) and (2), and in regions II and III are written as [20]:

In region II:

derivatives with respect to their arguments. q is an integer; and v is a non-negative

constants, respectively. In order to include the effect of azimuthal harmonics due to angular periodicity of the structure, the azimuthal dependence is considered as

In Sections 2.1 and 2.2, we have respectively explored the independent dielectric- and metal-loaded structures. However, in this section we will study the metal

This model (model-6) is formed by alternatively stacking the metal and dielectric discs each of same disc hole radii rD. This is similar to conventional disc-loaded circular waveguide in which the volume between two consecutive metal discs is filled with dielectric of relative permittivity εr. Similar to conventional disc-loaded circular waveguide for the sake of analysis, one may divide the structure into two regions: central disc free region I: 0≤ r , rD, 0 , z , ∞, and disc occupied region II: rD ≤r , rW, 0 , z , L � T (Figure 7). The relevant (axial magnetic and azimuthal electric) field intensity components in the regions I and II may be given by (1), (2), (5) and (6). In (5) and (6), the radial propagation constant in region II is interpreted

2.3.2 Circular waveguide loaded with dielectric and metal discs having different hole

This model (model-7) is similar to that of model-6, which has a circular waveguide consisting of alternate dielectric and metal discs, such that the hole-radius of metal discs is lesser than that of dielectric discs [20]. For the purpose of analysis, one may divide the structure into three regions: i) the central disc free region I: 0≤r , rD, 0 , z , ∞; ii) the disc occupied free space region II: rD ≤r , rDD, 0 , z , L � T; and iii) the dielectric filled disc occupied region III: rDD ≤r , rW, 0 , z , L � T; where rDD is the hole radius of dielectric disc. It is considered that the region I (disc free region) supports propagating and regions II and III (disc occupied

exp ð Þ �jvθ , such that v ¼ s þ qN, where s is also an integer [16–18].

2.3 Metal- and dielectric-loaded circular waveguide

as well as dielectric loading in the circular waveguide.

2.3.1 Circular waveguide loaded with dielectric and metal discs

and β are the radial and the axial phase propagation

integer. <sup>γ</sup><sup>I</sup> <sup>¼</sup> <sup>γ</sup>II <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup>

Electromagnetic Materials and Devices

as γII

Figure 7.

280

<sup>m</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup>

radius

m <sup>1</sup>=<sup>2</sup>

regions) support standing waves [20].

Circular waveguide loaded with dielectric and metal discs [19].

[19].

$$\mathbf{H}\_x^{\mathrm{II}} = \sum\_{m=1}^{\infty} \left[ A\_m^{\mathrm{II}} J\_0 \{ \chi\_m^{\mathrm{II}} r \right\} + B\_m^{\mathrm{II}} Y\_0 \{ \chi\_m^{\mathrm{II}} r \} \right] \exp(j\alpha t) \sin(\beta\_m z) \tag{15}$$

$$E\_{\theta}^{\text{II}} = j\omega\mu\_0 \sum\_{m=1}^{\infty} \frac{1}{\mathcal{I}\_m^{\text{II}}} \left[ A\_m^{\text{II}} f\_0' \{ \boldsymbol{\chi}\_m^{\text{II}} \boldsymbol{r} \} + B\_m^{\text{II}} Y\_0' \{ \boldsymbol{\chi}\_m^{\text{II}} \boldsymbol{r} \} \right] \, \exp(j\omega\boldsymbol{t}) \, \sin(\beta\_m \boldsymbol{z}) \tag{16}$$

In region III:

$$H\_x^{\text{III}} = \sum\_{m=1}^{\infty} A\_m^{\text{III}} Z\_0 \left\{ \gamma\_m^{\text{III}} r \right\} \exp(jwt) \sin(\beta\_m z) \tag{17}$$

$$E\_{\theta}^{\text{III}} = j a \mu\_0 \sum\_{m=1}^{\infty} \frac{1}{\chi\_m^{\text{III}}} A\_m^{\text{III}} Z\_0' \left\{ \chi\_m^{\text{III}} r \right\} \exp(j a \, t) \sin(\beta\_m z) \tag{18}$$

where Z<sup>0</sup> γIII <sup>m</sup> <sup>r</sup> � � <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>III <sup>m</sup> <sup>r</sup> � �Y<sup>0</sup> <sup>0</sup> γIII <sup>m</sup> rW � � � <sup>J</sup> 0 <sup>0</sup> γIII <sup>m</sup> rW � �Y<sup>0</sup> γIII <sup>m</sup> <sup>r</sup> � �; AII <sup>m</sup>, BII <sup>m</sup> and AIII <sup>m</sup> are the field constants in different analytical regions, identified by given superscript, respectively. γ<sup>I</sup> <sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> n � �<sup>1</sup>=<sup>2</sup> h i, <sup>γ</sup>II <sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> m � �<sup>1</sup>=<sup>2</sup> h i, and <sup>γ</sup>III <sup>m</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup> m � �<sup>1</sup>=<sup>2</sup> h i are the radial propagation constants in regions I, II, and III, respectively (Figure 8). The axial phase propagation constants β<sup>n</sup> in region I and β<sup>m</sup> in regions II and III are defined in the same manner as for model-3 [20].

#### 2.3.3 Circular waveguide loaded with alternate dielectric and metal vanes

This model (model-8) is similar to model-5 except the region II filled with dielectric of relative permittivity ε<sup>r</sup> between the two consecutive metal vanes [21]. For the sake of analysis, the structure may be divided into two regions; (i) the central cylindrical vane-free free-space region I: 0≤r , rV, 0 ≤θ , 2π, and (ii) the dielectric filled region II between two consecutive metal vanes: rV ≤r≤rW, , θ , 2π=N (Figure 9). The relevant (axial magnetic and azimuthal electric) field intensity components in the regions I and II may be given by (11)–(14), in which the radial propagation constant is given as: <sup>γ</sup>II <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup> � �<sup>1</sup>=<sup>2</sup> [21].

Figure 8.

Circular waveguide loaded with dielectric and metal discs with the hole radius of metal discs lesser than that of dielectric discs [20].

Figure 9. Circular waveguide loaded with alternate dielectric and metal vanes [21].
