3. Boundary conditions

One of the general boundary conditions comprising all the considered models is due to the null tangential electric field intensity at the inner surface of the metallic waveguide that is represented as [2, 5, 9–21]:

$$E\_{\theta} = \mathbf{0}|\_{r=r\_{W}} \quad \mathbf{0} < \mathbf{z} < \infty \tag{19}$$

The boundary conditions (24) and (25) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rD, between the regions I and II, and the null azimuthal component of electric field intensity at the disc hole metallic surface (Figure 4) [2, 5, 9–13]. Model-4: The boundary conditions (24) and (25) are also true for the model-4 (Figure 5) at the interface, r ¼ rD, between the regions I and II. The additional boundary conditions at the interface, r ¼ rBH, between the regions II and III may be

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

<sup>z</sup> 0≤z≤ ð Þ L � T � TBH =2

The boundary conditions (26) and (27) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rBH, between the regions II and III, and the null azimuthal component of electric field intensity at the disc hole metallic surface (Figure 5) [2, 9, 14, 15]. Model-5: The relevant electromagnetic boundary conditions for model-5

<sup>z</sup> ϕ≤θ ≤ 2π=N

<sup>θ</sup> ϕ≤θ ≤2π=N � �

The boundary conditions (28) and (29) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rV, between the regions I and II, and the null azimuthal component of

Model-7: The relevant electromagnetic boundary conditions (24) and (25) are also true for the model-7 (Figure 8) at the interface, r ¼ rD, between the regions I and II. The additional boundary conditions at the interface, r ¼ rDD, between the

<sup>z</sup> 0 , z , L � T

<sup>θ</sup> 0 , z , L � T

The boundary conditions (30) and (31), respectively, state the continuity of the axial magnetic and the azimuthal electric field intensities at the interface, r ¼ rDD, between the disc occupied free space region II and disc-occupied dielectric region

In general, the field intensity components (1)–(18) contain unknown field constants. In order to establish relations between these unknown field constants, the

Model-8: The relevant electromagnetic boundary conditions for model-8 (Figure 9) are given by (28) and (29), same as for model-5 (Figure 6) [21].

<sup>θ</sup> <sup>¼</sup> 0 0 <sup>≤</sup><sup>θ</sup> , <sup>ϕ</sup>

electric field intensity at the vane tip metallic surface (Figure 6) [16–18]. Model-6: The relevant electromagnetic boundary conditions for model-6 (Figure 7) are given by (24) and (25), same as for model-3 (Figure 4) [19].

EII

� �

> � � � r¼rV

� �

� �

0 ð Þ L � T � TBH =2 ≤z≤ð Þ L � T þ TBH =2

( �

� �

> � � � � r¼rBH

<sup>r</sup>¼rBH (26)

<sup>r</sup>¼rV (28)

<sup>r</sup>¼rDD (30)

<sup>r</sup>¼rDD (31)

(27)

(29)

written as [2, 9, 14, 15]:

EII

<sup>θ</sup> <sup>¼</sup> EIII

HII <sup>z</sup> <sup>¼</sup> <sup>H</sup>III

DOI: http://dx.doi.org/10.5772/intechopen.82124

<sup>θ</sup> 0≤z≤ð Þ L � T � TBH =2

(Figure 6) may be written in the mathematical form as [16–18]:

HI <sup>z</sup> <sup>¼</sup> <sup>H</sup>II

EI

regions II and III may be written as [20]:

III (Figure 8) [20].

283

4. Dispersion relations

HII <sup>z</sup> <sup>¼</sup> <sup>H</sup>III

EII <sup>θ</sup> <sup>¼</sup> EIII

Model-1: The relevant electromagnetic boundary conditions for model-1 may be written in the mathematical form as [7]:

$$\left.H\_x^I = H\_x^{II}\right|\_{r=r\_L} \quad 0 \le z \le \infty \tag{20}$$

$$E\_{\theta}^{l} = E\_{\theta}^{ll}|\_{r=r\_L} \quad \mathbf{0} \le \mathbf{z} \le \infty \tag{21}$$

The boundary conditions (20) and (21) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rL, between the regions I and II (Figure 2) [7].

Model-2: The relevant electromagnetic boundary conditions for model-2 (Figure 3) may be written in the mathematical form as [7]:

$$H\_x^I = H\_x^{II}, \quad \mathbf{0} \le \mathbf{z} \le \infty \Big|\_{r=r\_C} \tag{22}$$

$$E^l\_\theta = E^{ll}\_\theta, \quad \mathbf{0} \le \mathbf{z} \le \infty \Big|\_{r=r\_\mathcal{C}} \tag{23}$$

The boundary conditions (22) and (23) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rC, between the regions I and II (Figure 3) [7].

Model-3: The relevant electromagnetic boundary conditions for model-3 (Figure 4) may be written in the mathematical form as [2, 5, 9–13]:

$$H\_x^I = H\_x^{II} \quad 0 \le x \le L - T \big|\_{r=r\_D} \tag{24}$$

$$E^I\_\theta = \begin{cases} E^I\_\theta & 0 \le \mathbf{z} \le L - T \\ 0 & L - T \le \mathbf{z} \le L \end{cases} \bigg|\_{r=r\_D} \tag{25}$$

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

The boundary conditions (24) and (25) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rD, between the regions I and II, and the null azimuthal component of electric field intensity at the disc hole metallic surface (Figure 4) [2, 5, 9–13].

Model-4: The boundary conditions (24) and (25) are also true for the model-4 (Figure 5) at the interface, r ¼ rD, between the regions I and II. The additional boundary conditions at the interface, r ¼ rBH, between the regions II and III may be written as [2, 9, 14, 15]:

$$H\_x^{\text{II}} = H\_x^{\text{III}} \quad \mathbf{0} \le \mathbf{z} \le (L - T - T\_{\text{BH}}) / 2\Big|\_{r = r\_{\text{BH}}} \tag{26}$$

$$E\_{\theta}^{\mathrm{II}} = \begin{cases} E\_{\theta}^{\mathrm{III}} & 0 \le \mathbf{z} \le (L - T - T\_{\mathrm{BH}})/2\\ 0 & (L - T - T\_{\mathrm{BH}})/2 \le \mathbf{z} \le (L - T + T\_{\mathrm{BH}})/2 \end{cases} \tag{27}$$

The boundary conditions (26) and (27) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rBH, between the regions II and III, and the null azimuthal component of electric field intensity at the disc hole metallic surface (Figure 5) [2, 9, 14, 15].

Model-5: The relevant electromagnetic boundary conditions for model-5 (Figure 6) may be written in the mathematical form as [16–18]:

$$\left.H\_x^I = H\_x^{II}\right.\qquad \phi \le \theta \le 2\pi/N\Big|\_{r=r\_V} \tag{28}$$

$$E\_{\theta}^{l} = \begin{cases} 0 & 0 \le \theta < \phi \\ E\_{\theta}^{l\text{I}} & \phi \le \theta \le 2\pi/N \end{cases}\_{r=r\_{V}} \tag{29}$$

The boundary conditions (28) and (29) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the interface, r ¼ rV, between the regions I and II, and the null azimuthal component of electric field intensity at the vane tip metallic surface (Figure 6) [16–18].

Model-6: The relevant electromagnetic boundary conditions for model-6 (Figure 7) are given by (24) and (25), same as for model-3 (Figure 4) [19].

Model-7: The relevant electromagnetic boundary conditions (24) and (25) are also true for the model-7 (Figure 8) at the interface, r ¼ rD, between the regions I and II. The additional boundary conditions at the interface, r ¼ rDD, between the regions II and III may be written as [20]:

$$H\_x^{\text{II}} = H\_x^{\text{III}} \qquad \mathbf{0} \le \mathbf{z} \le L - T \big|\_{r=r\_{\text{DD}}} \tag{30}$$

$$E\_{\theta}^{\text{II}} = E\_{\theta}^{\text{III}} \qquad \mathbf{0} \le \mathbf{z} \le \mathbf{L} - T \Big|\_{r=r\_{\text{DD}}} \tag{31}$$

The boundary conditions (30) and (31), respectively, state the continuity of the axial magnetic and the azimuthal electric field intensities at the interface, r ¼ rDD, between the disc occupied free space region II and disc-occupied dielectric region III (Figure 8) [20].

Model-8: The relevant electromagnetic boundary conditions for model-8 (Figure 9) are given by (28) and (29), same as for model-5 (Figure 6) [21].

### 4. Dispersion relations

In general, the field intensity components (1)–(18) contain unknown field constants. In order to establish relations between these unknown field constants, the

3. Boundary conditions

Electromagnetic Materials and Devices

Figure 9.

282

waveguide that is represented as [2, 5, 9–21]:

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

written in the mathematical form as [7]:

HI <sup>z</sup> <sup>¼</sup> <sup>H</sup>II z � �

EI <sup>θ</sup> <sup>¼</sup> EII θ � �

interface, r ¼ rL, between the regions I and II (Figure 2) [7].

(Figure 3) may be written in the mathematical form as [7]:

HI <sup>z</sup> <sup>¼</sup> <sup>H</sup>II

EI <sup>θ</sup> <sup>¼</sup> EII

interface, r ¼ rC, between the regions I and II (Figure 3) [7].

<sup>θ</sup> <sup>¼</sup> EII

HI <sup>z</sup> <sup>¼</sup> <sup>H</sup>II

EI

(Figure 4) may be written in the mathematical form as [2, 5, 9–13]:

One of the general boundary conditions comprising all the considered models is due to the null tangential electric field intensity at the inner surface of the metallic

Model-1: The relevant electromagnetic boundary conditions for model-1 may be

The boundary conditions (20) and (21) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the

<sup>z</sup> ; <sup>0</sup> , <sup>z</sup> , <sup>∞</sup>�

<sup>θ</sup> ; <sup>0</sup> , <sup>z</sup> , <sup>∞</sup>�

The boundary conditions (22) and (23) state the continuity of the axial component of magnetic and the azimuthal component of electric field intensities at the

<sup>z</sup> 0 , z , L � T

<sup>θ</sup> 0 , z , L � T 0 L � T ≤z≤L

( �

Model-3: The relevant electromagnetic boundary conditions for model-3

�

�

� �

� � � � r¼rD

Model-2: The relevant electromagnetic boundary conditions for model-2

<sup>E</sup><sup>θ</sup> <sup>¼</sup> <sup>0</sup>j<sup>r</sup>¼rW <sup>0</sup> , <sup>z</sup> , <sup>∞</sup> (19)

<sup>r</sup>¼rL <sup>0</sup> , <sup>z</sup> , <sup>∞</sup> (20)

<sup>r</sup>¼rL <sup>0</sup> , <sup>z</sup> , <sup>∞</sup> (21)

<sup>r</sup>¼rC (22)

<sup>r</sup>¼rC (23)

<sup>r</sup>¼rD (24)

(25)

relevant field intensity components are substituted into the respective boundary conditions. Further, the algebraic manipulations of the obtained relations between these field constants eliminate all the field constants, and it results in a characteristic relation of the model known as the dispersion relation. The dispersion relations of various considered models are:

Model-6 [19]:

Model-7 [20]:

Model-8 [21]:

nrMD <sup>J</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>γ</sup>III m J 0 <sup>0</sup> γII

γII mY<sup>0</sup> γII

5. Dispersion characteristics

obtained using HFSS (Figures 10 and 11).

det Mn m J<sup>0</sup> γ<sup>I</sup>

Figure 10.

285

tion constant in dielectric filled region as γII

DOI: http://dx.doi.org/10.5772/intechopen.82124

0 <sup>0</sup> γII mrD <sup>þ</sup> <sup>ξ</sup>Y<sup>0</sup>

mrDD <sup>Z</sup><sup>0</sup> <sup>γ</sup>III

mrDD <sup>Z</sup><sup>0</sup>

The dispersion relation for model-6 (Figure 7) is same as that for model-3 (Figure 4) and is given by (34) through (35) with interpretation of radial propaga-

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

The dispersion relation for model-8 (Figure 9) is same as that for model-5 (Figure 6) and is given by (38) through (39) and (40) with interpretation of radial

One may clearly observe that the shape of dispersion characteristics of the model-1 (Figure 2) and model-2 (Figure 3) change with change in relative permittivity of the dielectric material (Figures 10 and 11). The cutoff frequency decreases with increase of the relative permittivity of the dielectric material. The increase of relative permittivity of the dielectric material depresses the dispersion characteristics of the model-1 and model-2 and more at higher value of phase propagation constant. The analytical dispersion characteristics are found within 3% of that

Periodic loading a circular waveguide by the metal annular discs (model-3) brings out alternate pass and stop bands with their respective higher and lower

TE01-mode dispersion characteristics of circular waveguide with dielectric lining on metal wall taking ε<sup>r</sup> as the

parameter. The characteristics with ε<sup>r</sup> ¼ 1 (special case) represents the dispersion characteristics of

conventional circular waveguide. Circles represent results obtained using HFSS.

<sup>0</sup> γII mrD

<sup>m</sup> rDD � <sup>γ</sup>II

<sup>m</sup> rDD � <sup>γ</sup>III

� <sup>J</sup>

<sup>0</sup> γIII

propagation constant in dielectric filled region 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>

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

mrDD <sup>Z</sup><sup>0</sup>

mrDD <sup>Z</sup><sup>0</sup> <sup>γ</sup>III

0 <sup>0</sup> γ<sup>I</sup> nrD J<sup>0</sup> γII

<sup>m</sup> J<sup>0</sup> γII

<sup>m</sup> Y<sup>0</sup> <sup>0</sup> γII

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

[19].

mrD

<sup>m</sup> rDD (42)

[21].

 ¼ 0 (41)

mrD <sup>þ</sup> <sup>ξ</sup>Y<sup>0</sup> <sup>γ</sup>II

<sup>0</sup> γIII <sup>m</sup> rDD

Model-1 [7]:

$$\begin{aligned} \gamma^{\text{II}} l\_0' \{ \gamma^{\text{I}} r\_{\text{L}} \} \left( J\_0 \{ \chi^{\text{II}} r\_{\text{L}} \} Y\_0' \{ \chi^{\text{II}} r\_W \} - Y\_0 \{ \chi^{\text{II}} r\_{\text{L}} \} l\_0' \{ \chi^{\text{II}} r\_W \} \right) \\ - \gamma^{\text{I}} l\_0' \{ \chi^{\text{I}} r\_{\text{L}} \} \left( J\_0' \{ \chi^{\text{II}} r\_{\text{L}} \} Y\_0' \{ \chi^{\text{I}} r\_W \} - Y\_0' \{ \chi^{\text{I}} r\_{\text{L}} \} l\_0' \{ \chi^{\text{I}} r\_W \} \right) = 0 \end{aligned} \tag{32}$$

Model-2 [7]:

$$\begin{aligned} \gamma^{\text{II}} l\_0' \{ \gamma^{\text{I}} r\_\text{C} \} \left( J\_0 \{ \gamma^{\text{II}} r\_\text{C} \} Y\_0' \{ \gamma^{\text{II}} r\_W \} - Y\_0 \{ \gamma^{\text{II}} r\_\text{C} \} l\_0' \{ \gamma^{\text{II}} r\_W \} \right) \\ - \gamma^{\text{I}} l\_0 \{ \gamma^{\text{I}} r\_\text{C} \} \left( J\_0' \{ \gamma^{\text{I}} r\_\text{C} \} Y\_0' \{ \gamma^{\text{I}} r\_W \} - Y\_0' \{ \gamma^{\text{I}} r\_\text{C} \} l\_0' \{ \gamma^{\text{I}} r\_W \} \right) = 0 \end{aligned} \tag{33}$$

Model-3 [2, 5, 9–13]:

$$\det \left| M\_{nm} J\_0 \left\{ \boldsymbol{\chi}\_n^I \boldsymbol{r}\_D \right\} Z\_0' \left\{ \boldsymbol{\chi}\_m^{\Pi} \boldsymbol{r}\_D \right\} - Z\_0 \left\{ \boldsymbol{\chi}\_m^{\Pi} \boldsymbol{r}\_D \right\} \boldsymbol{I}\_0' \left\{ \boldsymbol{\chi}\_n^I \boldsymbol{r}\_D \right\} \right| = \mathbf{0} \tag{34}$$

where

$$\mathcal{M}\_{nm} = \frac{\boldsymbol{\chi}\_n^{\mathrm{I}} \boldsymbol{\beta}\_m^{\mathrm{II}} \left(\mathbf{1} - (-\mathbf{1})^m \exp\left[-j\boldsymbol{\beta}\_n^{\mathrm{I}}(L-T)\right]\right)}{\boldsymbol{\gamma}\_m^{\mathrm{II}} \left[\boldsymbol{\beta}\_m^{\mathrm{II}} - \exp\left(-j\boldsymbol{\beta}\_0^{\mathrm{I}}L\right) \left(\boldsymbol{\beta}\_m^{\mathrm{II}} \cos\left(\boldsymbol{\beta}\_m^{\mathrm{II}}L\right) + j\boldsymbol{\beta}\_n^{\mathrm{I}} \sin\left(\boldsymbol{\beta}\_m^{\mathrm{II}}L\right)\right)\right]}.\tag{35}$$

Model-4 [2, 9, 14, 15]:

$$\det\left|M\_{nm}J\_0\{\chi\_n^I r\_D\}\left[I\_0'\{\chi\_m^{\text{II}}r\_D\}+\xi Y\_0'\{\chi\_m^{\text{II}}r\_D\}\right]-J\_0'\{\chi\_n^I r\_D\}\left[I\_0\{\chi\_m^{\text{II}}r\_D\}+\xi Y\_0\{\chi\_m^{\text{II}}r\_D\}\right]\right| = 0\tag{36}$$

where

$$\xi = \frac{\gamma\_p^{\text{III}} J\_0' \left\{ \gamma\_m^{\text{II}} r\_{\text{BH}} \right\} Z\_0 \left\{ \gamma\_m^{\text{III}} r\_{\text{BH}} \right\} - \gamma\_m^{\text{II}} J\_0 \left\{ \gamma\_m^{\text{II}} r\_{\text{BH}} \right\} Z\_0' \left\{ \gamma\_m^{\text{III}} r\_{\text{BH}} \right\}}{\gamma\_m^{\text{II}} Y\_0 \left\{ \gamma\_m^{\text{II}} r\_{\text{BH}} \right\} Z\_0' \left\{ \gamma\_m^{\text{III}} r\_{\text{BH}} \right\} - \gamma\_m^{\text{III}} Y\_0' \left\{ \gamma\_m^{\text{II}} r\_{\text{BH}} \right\} Z\_0 \left\{ \gamma\_m^{\text{III}} r\_{\text{BH}} \right\}} \tag{37}$$

Model-5 [16–18]:

$$
\begin{vmatrix}
P\_{\mathfrak{g}-1} & Q\_{\mathfrak{g}-1,\mathfrak{g}} & Q\_{\mathfrak{g}-1,\mathfrak{g}+1} \\
Q\_{\mathfrak{g},\mathfrak{g}-1} & P\_{\mathfrak{g}} & Q\_{\mathfrak{g},\mathfrak{g}+1} \\
Q\_{\mathfrak{g}+1,\mathfrak{g}-1} & Q\_{\mathfrak{g}+1,\mathfrak{g}} & P\_{\mathfrak{g}+1}
\end{vmatrix} = \mathbf{0} \tag{38}
$$

where

Pv<sup>0</sup> ¼ J 0 <sup>v</sup><sup>0</sup> γIIrV � <sup>J</sup> 0 <sup>v</sup><sup>0</sup> γIIrW Y0 <sup>v</sup><sup>0</sup> γII f g rW Y0 <sup>v</sup><sup>0</sup> γIIrV <sup>2</sup><sup>π</sup> <sup>N</sup> � <sup>φ</sup> � 2π N γII γI J 0 <sup>v</sup><sup>0</sup> γ<sup>I</sup> rV Jv<sup>0</sup> γ<sup>I</sup> f g rV Jv<sup>0</sup> γIIrV � <sup>J</sup> 0 <sup>v</sup><sup>0</sup> γIIrW Y0 <sup>v</sup><sup>0</sup> γII f g rW Yv<sup>0</sup> γIIrV (39) Qv<sup>0</sup> , <sup>v</sup> ¼ J 0 <sup>v</sup> γIIrV � <sup>J</sup> 0 <sup>v</sup> γIIrW Y0 <sup>v</sup> γII f g rW Y0 <sup>v</sup> γIIrV <sup>1</sup> � exp j vð Þ <sup>0</sup> � <sup>v</sup> <sup>φ</sup> j vð Þ <sup>0</sup> � v (40)

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

Model-6 [19]:

relevant field intensity components are substituted into the respective boundary conditions. Further, the algebraic manipulations of the obtained relations between these field constants eliminate all the field constants, and it results in a characteristic relation of the model known as the dispersion relation. The dispersion relations

� <sup>Y</sup><sup>0</sup> <sup>γ</sup>IIrL

� <sup>Y</sup><sup>0</sup> <sup>γ</sup>IIrC

<sup>n</sup>ð Þ <sup>L</sup> � <sup>T</sup>

<sup>m</sup> cos βII

<sup>¼</sup> <sup>0</sup>

0 <sup>0</sup> γ<sup>I</sup> nrD J<sup>0</sup> γII

 J 0 <sup>0</sup> γIIrW

<sup>0</sup> γIIrL J 0 <sup>0</sup> γIIrW

 J 0 <sup>0</sup> γIIrW

<sup>0</sup> γIIrC J 0 <sup>0</sup> γIIrW

<sup>¼</sup> <sup>0</sup> (33)

mrD J 0 <sup>0</sup> γ<sup>I</sup> nrD

mL <sup>þ</sup> <sup>j</sup>β<sup>I</sup>

mJ<sup>0</sup> γII mrBH Z<sup>0</sup>

> 

<sup>m</sup> Y<sup>0</sup> <sup>0</sup> γII mrBH Z<sup>0</sup> γIII

mL : (35)

<sup>n</sup> sin βII

mrD <sup>þ</sup> <sup>ξ</sup>Y<sup>0</sup> <sup>γ</sup>II

> <sup>0</sup> γIII <sup>m</sup> rBH

<sup>N</sup> � <sup>φ</sup> 

j vð Þ <sup>0</sup> � v

Yv<sup>0</sup> γIIrV

<sup>m</sup> rBH

¼ 0 (38)

(37)

¼ 0 (34)

mrD

(36)

(39)

(40)

<sup>¼</sup> <sup>0</sup> (32)

of various considered models are:

Electromagnetic Materials and Devices

Y<sup>0</sup>

Y<sup>0</sup>

nrD Z<sup>0</sup>

nβII

<sup>m</sup> � exp �jβ<sup>I</sup>

mrBH Z<sup>0</sup>

> 

0 <sup>0</sup> γIIrL Y<sup>0</sup>

0 <sup>0</sup> γIIrC Y<sup>0</sup>

det Mnm J<sup>0</sup> γ<sup>I</sup>

Mnm <sup>¼</sup> <sup>γ</sup><sup>I</sup>

γII <sup>m</sup> βII

Model-4 [2, 9, 14, 15]:

nrD J

> <sup>ξ</sup> <sup>¼</sup> <sup>γ</sup>III p J 0 <sup>0</sup> γII mrBH Z<sup>0</sup> γIII

Pv<sup>0</sup> ¼ J 0 <sup>v</sup><sup>0</sup> γIIrV � <sup>J</sup>

, <sup>v</sup> ¼ J 0 <sup>v</sup> γIIrV � <sup>J</sup>

� 2π N γII γI J 0 <sup>v</sup><sup>0</sup> γ<sup>I</sup> rV Jv<sup>0</sup> γ<sup>I</sup> f g rV

Model-5 [16–18]:

γII mY<sup>0</sup> γII

0 <sup>0</sup> γII mrD <sup>þ</sup> <sup>ξ</sup>Y<sup>0</sup>

<sup>0</sup> γIIrW

<sup>0</sup> γIIrW

<sup>0</sup> γIIrW � <sup>Y</sup><sup>0</sup>

> <sup>0</sup> γII mrD � <sup>Z</sup><sup>0</sup> <sup>γ</sup>II

<sup>m</sup> <sup>1</sup> � �ð Þ<sup>1</sup> <sup>m</sup> exp �jβ<sup>I</sup>

<sup>0</sup><sup>L</sup> <sup>β</sup>II

<sup>0</sup> γII mrD

<sup>0</sup> γIII <sup>m</sup> rBH � <sup>γ</sup>III

0 <sup>v</sup><sup>0</sup> γIIrW 

Y0

0 <sup>v</sup> γIIrW 

Y0

<sup>v</sup> γII f g rW

<sup>v</sup><sup>0</sup> γII f g rW

Jv<sup>0</sup> γIIrV � <sup>J</sup>

Y0 <sup>v</sup> γIIrV <sup>1</sup> � exp j vð Þ <sup>0</sup> � <sup>v</sup> <sup>φ</sup>

<sup>2</sup><sup>π</sup>

Y0 <sup>v</sup><sup>0</sup> γIIrV

> 0 <sup>v</sup><sup>0</sup> γIIrW

Y0

<sup>v</sup><sup>0</sup> γII f g rW

<sup>m</sup> rBH � <sup>γ</sup>II

Pg�<sup>1</sup> Qg�1, <sup>g</sup> Qg�1, <sup>g</sup>þ<sup>1</sup> Qg, <sup>g</sup>�<sup>1</sup> Pg Qg, <sup>g</sup>þ<sup>1</sup> Qgþ1, <sup>g</sup>�<sup>1</sup> Qgþ1, <sup>g</sup> Pgþ<sup>1</sup>

� <sup>J</sup>

<sup>0</sup> γIIrW � <sup>Y</sup><sup>0</sup>

Model-1 [7]:

γIIJ 0 <sup>0</sup> γ<sup>I</sup> rL J<sup>0</sup> γIIrL

�γ<sup>I</sup> J<sup>0</sup> γ<sup>I</sup> rL J

Model-2 [7]:

γIIJ 0 <sup>0</sup> γ<sup>I</sup> rC J<sup>0</sup> γIIrC

�γ<sup>I</sup> J<sup>0</sup> γ<sup>I</sup> rC J

where

det Mn m J<sup>0</sup> γ<sup>I</sup>

where

where

Qv<sup>0</sup>

284

Model-3 [2, 5, 9–13]:

The dispersion relation for model-6 (Figure 7) is same as that for model-3 (Figure 4) and is given by (34) through (35) with interpretation of radial propaga-

tion constant in dielectric filled region as γII <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> [19]. Model-7 [20]:

$$\left| \det \left| M\_{nm} I\_0 \left\{ \boldsymbol{\chi}\_n^l \boldsymbol{r}\_{\rm MD} \right\} \right\} \left[ J\_0' \left\{ \boldsymbol{\chi}\_m^H \boldsymbol{r}\_D \right\} + \xi \boldsymbol{Y}\_0' \left\{ \boldsymbol{\chi}\_m^H \boldsymbol{r}\_D \right\} \right] - J\_0' \left\{ \boldsymbol{\chi}\_n^l \boldsymbol{r}\_D \right\} \left[ J\_0 \left\{ \boldsymbol{\chi}\_m^H \boldsymbol{r}\_D \right\} + \xi \boldsymbol{Y}\_0 \left\{ \boldsymbol{\chi}\_m^H \boldsymbol{r}\_D \right\} \right] \right| = 0 \tag{41}$$

$$\xi = \frac{\gamma\_m^{\rm III} I\_0' \left\{ \gamma\_m^{\rm II} r\_{\rm DD} \right\} Z\_0 \left\{ \gamma\_m^{\rm III} r\_{\rm DD} \right\} - \gamma\_m^{\rm II} I\_0 \left\{ \gamma\_m^{\rm II} r\_{\rm DD} \right\} Z\_0' \left\{ \gamma\_m^{\rm III} r\_{\rm DD} \right\}}{\gamma\_m^{\rm II} Y\_0 \left\{ \gamma\_m^{\rm III} r\_{\rm DD} \right\} Z\_0' \left\{ \gamma\_m^{\rm III} r\_{\rm DD} \right\} - \gamma\_m^{\rm III} Y\_0' \left\{ \gamma\_m^{\rm II} r\_{\rm DD} \right\} Z\_0 \left\{ \gamma\_m^{\rm III} r\_{\rm DD} \right\}} \tag{42}$$

Model-8 [21]:

The dispersion relation for model-8 (Figure 9) is same as that for model-5 (Figure 6) and is given by (38) through (39) and (40) with interpretation of radial propagation constant in dielectric filled region 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].

#### 5. Dispersion characteristics

One may clearly observe that the shape of dispersion characteristics of the model-1 (Figure 2) and model-2 (Figure 3) change with change in relative permittivity of the dielectric material (Figures 10 and 11). The cutoff frequency decreases with increase of the relative permittivity of the dielectric material. The increase of relative permittivity of the dielectric material depresses the dispersion characteristics of the model-1 and model-2 and more at higher value of phase propagation constant. The analytical dispersion characteristics are found within 3% of that obtained using HFSS (Figures 10 and 11).

Periodic loading a circular waveguide by the metal annular discs (model-3) brings out alternate pass and stop bands with their respective higher and lower

#### Figure 10.

TE01-mode dispersion characteristics of circular waveguide with dielectric lining on metal wall taking ε<sup>r</sup> as the parameter. The characteristics with ε<sup>r</sup> ¼ 1 (special case) represents the dispersion characteristics of conventional circular waveguide. Circles represent results obtained using HFSS.

#### Figure 11.

TE01-mode dispersion characteristics of circular waveguide with coaxial dielectric rod and taking ε<sup>r</sup> as the parameter. Circles represent results obtained using HFSS.

#### Figure 12.

Pass and stop band characteristics of the infinitesimally thin metal disc-loaded circular waveguide (including higher order harmonics n ¼ 0, � 1, � 2, � 3, � 4, � 5; m ¼ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) [9, 10].

cutoff frequencies [1–3, 5, 9]. The dispersion characteristics taking horizontal axis as normalized phase propagation constant become periodic with the periodicity of βI <sup>0</sup>L ¼ 2π for a given mode (TE01, TE02 and TE03). The normalized passband (krW scale) for the TE02 mode is narrower than that of the TE01 mode. Similarly, the normalized stopband above the TE02 mode is narrower than that above the TE01 mode (Figure 12). The RF group velocity (slopes of dispersion plot) is positive (fundamental forward wave mode, n = 0) for the TE01 or TE02 mode, it is negative (fundamental backward wave mode) for the TE03 mode (Figure 12). The dependencies of the structure dispersion characteristics, for typical mode TE01, on the disc-hole radius (Figure 13), the structure periodicity (Figure 14) and the finite disc thickness (Figure 15) are studied. Further, with the increase of either of the parameters, namely, the disc-hole radius and the structure periodicity, the lower and upper edge frequencies of the passband of the dispersion characteristics both decrease, though not equally. This lead to decrease or increase of the passband

according as the disc-hole radius decreases or the structure periodicity decreases, with the shift of the mid-band frequency of the passband to a higher value for the

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide (solid curve) taking

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide (solid curve) taking the disc-hole radius as the parameter. The broken curve with crosses refers to a smooth-wall circular waveguide

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

DOI: http://dx.doi.org/10.5772/intechopen.82124

Although the disc-hole radius and the structure periodicity tailor the dispersion characteristics, the later one found to be more effective that the former one for widening the frequency range of the straight-line section of the characteristics. Reducing the structure periodicity can increase the frequency range of the straightline section, however it accompanies with shift in waveguide cutoff (Figure 14). This wider straight-line section of the dispersion characteristics may be utilized for wideband coalescence with cyclotron wave (beam mode line) to result a wideband performance of a gyro-TWT. Thus, reducing the structure periodicity (Figure 14)

decrease of both the parameters (Figures 13 and 14) [2, 5, 11].

the structure periodicity as the parameter [2, 5, 11].

Figure 13.

[2, 5, 11].

Figure 14.

287

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

#### Figure 13.

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide (solid curve) taking the disc-hole radius as the parameter. The broken curve with crosses refers to a smooth-wall circular waveguide [2, 5, 11].

#### Figure 14.

cutoff frequencies [1–3, 5, 9]. The dispersion characteristics taking horizontal axis as normalized phase propagation constant become periodic with the periodicity of

Pass and stop band characteristics of the infinitesimally thin metal disc-loaded circular waveguide (including higher order harmonics n ¼ 0, � 1, � 2, � 3, � 4, � 5; m ¼ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) [9, 10].

TE01-mode dispersion characteristics of circular waveguide with coaxial dielectric rod and taking ε<sup>r</sup> as the

<sup>0</sup>L ¼ 2π for a given mode (TE01, TE02 and TE03). The normalized passband (krW scale) for the TE02 mode is narrower than that of the TE01 mode. Similarly, the normalized stopband above the TE02 mode is narrower than that above the TE01 mode (Figure 12). The RF group velocity (slopes of dispersion plot) is positive (fundamental forward wave mode, n = 0) for the TE01 or TE02 mode, it is negative (fundamental backward wave mode) for the TE03 mode (Figure 12). The dependencies of the structure dispersion characteristics, for typical mode TE01, on the disc-hole radius (Figure 13), the structure periodicity (Figure 14) and the finite disc thickness (Figure 15) are studied. Further, with the increase of either of the parameters, namely, the disc-hole radius and the structure periodicity, the lower and upper edge frequencies of the passband of the dispersion characteristics both decrease, though not equally. This lead to decrease or increase of the passband

βI

286

Figure 12.

Figure 11.

parameter. Circles represent results obtained using HFSS.

Electromagnetic Materials and Devices

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide (solid curve) taking the structure periodicity as the parameter [2, 5, 11].

according as the disc-hole radius decreases or the structure periodicity decreases, with the shift of the mid-band frequency of the passband to a higher value for the decrease of both the parameters (Figures 13 and 14) [2, 5, 11].

Although the disc-hole radius and the structure periodicity tailor the dispersion characteristics, the later one found to be more effective that the former one for widening the frequency range of the straight-line section of the characteristics. Reducing the structure periodicity can increase the frequency range of the straightline section, however it accompanies with shift in waveguide cutoff (Figure 14). This wider straight-line section of the dispersion characteristics may be utilized for wideband coalescence with cyclotron wave (beam mode line) to result a wideband performance of a gyro-TWT. Thus, reducing the structure periodicity (Figure 14)

#### Figure 15.

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide taking the disc thickness as the parameter. The broken curve refers to the infinitesimally thin metal disc-loaded circular waveguide [2, 5, 11].

and increasing the disc-hole radius (Figure 13) may widen the device bandwidth. However, such broadbanding of coalescence is accompanied by the reduction of the bandwidth of the passband of the structure as well (Figures 13 and 14) [2, 5, 11]. The decrease of the disc thickness decreases both the lower and upper edge frequencies of the passband such that the passband first decreases, attains a minima and then increases; and the mid-band frequency of the passband as well as the start frequency of the straight-line section of the dispersion characteristics reduces (Figure 15). The shape of the dispersion characteristics depends on the disc thickness, though not as much as it does on the disc-hole radius or the structure periodicity (Figures 13-15) [2, 5, 11].

Similar to the conventional disc-loaded circular waveguide (model-3), both the hole-radii (bigger and smaller) of the interwoven-disc-loaded circular (model-4, Figure 5) waveguide tailor the dispersion characteristics. The lower- and the uppercutoff frequencies decrease with increase in hole-radii, such that the passband increases and decreases with decrease of bigger and smaller hole-radii, respectively (Figures 16 and 17). Similar to the conventional disc-loaded circular waveguide (model-3), the structure periodicity of the interwoven-disc-loaded circular waveguide is the most effective for the increasing the passband and tailoring the dispersion characteristics (Figure 18). Neither, the extent of passband changes nor dispersion tailors with variation of disc-thickness of bigger-hole-disc, however, the mid frequency of the passband shifts to higher frequency with increase of discthickness of bigger-hole-disc (Figure 19). This nature may be used for shifting the operation band in the passive components or in order to optimizing the beam-wave interaction in designing a gyro-TWT with the interwoven-disc-loaded circular waveguide. In addition to tailoring the dispersion characteristics, required for designing a broadband gyro-TWT, the model-4 shows an interesting characteristic. The passband increases with increase as well as with decrease of disc-thickness of smaller- hole-disc with reference to that of bigger-hole-disc, however, the shift of the passband occur towards higher and lower frequency side, respectively, with increase and decrease of disc thickness of smaller-hole-disc with reference to that of bigger-hole-disc (Figure 20). Thus, the structure periodicity (Figure 18) and the

disc-thickness (Figure 19) of bigger-hole-disc of the interwoven-disc-loaded circular waveguide are, respectively, the most and the least sensitive parameter for controlling the passband as well as shape of the dispersion characteristics [14, 15]. Axial metal vane loading to a smooth-wall circular waveguide (model-5, Figure 6) forms an azimuthally periodic structure, which does not shape its dispersion characteristics, however the insertion of the metal vanes in to the circular waveguide shifts the waveguide cutoff frequency to a higher value [16–18] (Figures 21–23). Specifically, the increase of either of the vane angle (Figure 22) and the number of metal vanes (Figure 23) and the decrease of vane-inner-tip

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the smaller disc-

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the bigger disc-

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

Figure 16.

Figure 17.

289

hole-radius as the parameter [9, 14, 15].

hole-radius as the parameter [9, 14, 15].

DOI: http://dx.doi.org/10.5772/intechopen.82124

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

#### Figure 16.

and increasing the disc-hole radius (Figure 13) may widen the device bandwidth. However, such broadbanding of coalescence is accompanied by the reduction of the bandwidth of the passband of the structure as well (Figures 13 and 14) [2, 5, 11]. The decrease of the disc thickness decreases both the lower and upper edge frequencies of the passband such that the passband first decreases, attains a minima and then increases; and the mid-band frequency of the passband as well as the start frequency of the straight-line section of the dispersion characteristics reduces (Figure 15). The shape of the dispersion characteristics depends on the disc thickness, though not as much as it does on the disc-hole radius or the structure period-

TE01-mode dispersion characteristics of the conventional disc-loaded circular waveguide taking the disc thickness as the parameter. The broken curve refers to the infinitesimally thin metal disc-loaded circular

Similar to the conventional disc-loaded circular waveguide (model-3), both the hole-radii (bigger and smaller) of the interwoven-disc-loaded circular (model-4, Figure 5) waveguide tailor the dispersion characteristics. The lower- and the uppercutoff frequencies decrease with increase in hole-radii, such that the passband increases and decreases with decrease of bigger and smaller hole-radii, respectively (Figures 16 and 17). Similar to the conventional disc-loaded circular waveguide (model-3), the structure periodicity of the interwoven-disc-loaded circular waveguide is the most effective for the increasing the passband and tailoring the dispersion characteristics (Figure 18). Neither, the extent of passband changes nor dispersion tailors with variation of disc-thickness of bigger-hole-disc, however, the mid frequency of the passband shifts to higher frequency with increase of discthickness of bigger-hole-disc (Figure 19). This nature may be used for shifting the operation band in the passive components or in order to optimizing the beam-wave interaction in designing a gyro-TWT with the interwoven-disc-loaded circular waveguide. In addition to tailoring the dispersion characteristics, required for designing a broadband gyro-TWT, the model-4 shows an interesting characteristic. The passband increases with increase as well as with decrease of disc-thickness of smaller- hole-disc with reference to that of bigger-hole-disc, however, the shift of the passband occur towards higher and lower frequency side, respectively, with increase and decrease of disc thickness of smaller-hole-disc with reference to that of bigger-hole-disc (Figure 20). Thus, the structure periodicity (Figure 18) and the

icity (Figures 13-15) [2, 5, 11].

Electromagnetic Materials and Devices

Figure 15.

288

waveguide [2, 5, 11].

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the bigger dischole-radius as the parameter [9, 14, 15].

#### Figure 17.

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the smaller dischole-radius as the parameter [9, 14, 15].

disc-thickness (Figure 19) of bigger-hole-disc of the interwoven-disc-loaded circular waveguide are, respectively, the most and the least sensitive parameter for controlling the passband as well as shape of the dispersion characteristics [14, 15].

Axial metal vane loading to a smooth-wall circular waveguide (model-5, Figure 6) forms an azimuthally periodic structure, which does not shape its dispersion characteristics, however the insertion of the metal vanes in to the circular waveguide shifts the waveguide cutoff frequency to a higher value [16–18] (Figures 21–23). Specifically, the increase of either of the vane angle (Figure 22) and the number of metal vanes (Figure 23) and the decrease of vane-inner-tip

#### Figure 18.

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the structure periodicity as the parameter [9, 14, 15].

#### Figure 19.

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the disc-thickness of bigger-hole-disc as the parameter [9, 14, 15].

modes TE01 and TE02, however more significantly for the latter mode (Figure 24).

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the vane-inner-tip

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the disc thickness

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

permittivity in this structure would yield a straightened TE02 mode ω β dispersion characteristics near low value of phase propagation constant for wideband coalescence with the beam-mode dispersion line and consequent wideband gyro-TWT performance (Figure 24(b)). Thus the introduction of the dielectric discs between metal discs in the a conventional metal disc-loaded waveguide, with lower values of relative permittivity for the TE01 mode and with higher values of relative permittivity for the TE02 mode enhances the frequency range of the straight line portion of

Exhibits fundamental forward wave (positive) dispersion characteristics irrespective of the value of relative permittivity (Figure 24(a)), however, the TE02 mode exhibits fundamental forward (positive) and backward (negative) wave dispersion characteristics, respectively, at higher and lower values of relative permit-

tivity. This suggests that an appropriate selection of the value of relative

The TE01 mode of the structure.

radius as the parameter [16–18].

of smaller-hole-disc as the parameter [9, 14, 15].

DOI: http://dx.doi.org/10.5772/intechopen.82124

Figure 20.

Figure 21.

291

radius (Figure 21) increases the waveguide cutoff frequency, and none of the parameters tailors the dispersion characteristics.

For the model-6, the variation of relative permittivity of the dielectric discs changes the lower and upper cutoff frequencies of the passband (Figure 24). Two lowest order azimuthally symmetric (TE01 and TE02) modes are typically considered to study the performance of this model. With the increase of relative permittivity, the passband continuously decreases for the TE01 mode, and first decreases and then increases for the TE02 mode (Figure 24). Also, the variation in relative permittivity shapes of the dispersion characteristics of the structure, for both the

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

#### Figure 20.

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the disc thickness of smaller-hole-disc as the parameter [9, 14, 15].

Figure 21.

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the vane-inner-tip radius as the parameter [16–18].

modes TE01 and TE02, however more significantly for the latter mode (Figure 24). The TE01 mode of the structure.

Exhibits fundamental forward wave (positive) dispersion characteristics irrespective of the value of relative permittivity (Figure 24(a)), however, the TE02 mode exhibits fundamental forward (positive) and backward (negative) wave dispersion characteristics, respectively, at higher and lower values of relative permittivity. This suggests that an appropriate selection of the value of relative permittivity in this structure would yield a straightened TE02 mode ω β dispersion characteristics near low value of phase propagation constant for wideband coalescence with the beam-mode dispersion line and consequent wideband gyro-TWT performance (Figure 24(b)). Thus the introduction of the dielectric discs between metal discs in the a conventional metal disc-loaded waveguide, with lower values of relative permittivity for the TE01 mode and with higher values of relative permittivity for the TE02 mode enhances the frequency range of the straight line portion of

radius (Figure 21) increases the waveguide cutoff frequency, and none of the

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the structure

For the model-6, the variation of relative permittivity of the dielectric discs changes the lower and upper cutoff frequencies of the passband (Figure 24). Two lowest order azimuthally symmetric (TE01 and TE02) modes are typically considered to study the performance of this model. With the increase of relative permittivity, the passband continuously decreases for the TE01 mode, and first decreases and then increases for the TE02 mode (Figure 24). Also, the variation in relative permittivity shapes of the dispersion characteristics of the structure, for both the

TE01-mode dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the disc-thickness

parameters tailors the dispersion characteristics.

of bigger-hole-disc as the parameter [9, 14, 15].

Figure 18.

Figure 19.

290

periodicity as the parameter [9, 14, 15].

Electromagnetic Materials and Devices

Figure 22.

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the vane angle as the parameter [16–18].

#### Figure 23.

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the number of metal vanes as the parameter [16–18].

the ω � β dispersion characteristics, desired for wideband gyro-TWT performance (Figure 25) [19].

(Figure 27) [19]. Similar to a conventional all-metal disc-loaded waveguide

referring to a conventional metal disc-loaded circular waveguide [19].

Dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) for the selected dielectric disc relative permittivity values for the sake of comparison between the modes TE01 and TE02 with respect to the control of the shape of the dispersion characteristics. The broken curves

characteristics).

293

Figure 24.

Figure 25.

conventional metal disc-loaded circular waveguide [19].

DOI: http://dx.doi.org/10.5772/intechopen.82124

(model-4), the structure periodicity is the most effective parameter for tailoring the dispersion characteristics of the structure with dielectric discs between the metal discs (model-6), for the TE01 and TE02 modes, more for the latter. The control of the structure periodicity in straightening the dispersion characteristics, as required for the desired wideband gyro-TWT performance, is enhanced by introducing the dielectric discs in the conventional disc-loaded waveguide, though not enhancing the frequency range of the straight line portion of the dispersion characteristics (Figure 28). In this model, a serious care is required while selecting the dielectric material because a heavily dielectric-loaded structure depresses the dispersion characteristics to the slow-wave region (below the velocity of light line in ω β

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking relative permittivity as the parameter. The broken curves refer to a

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

The lower and upper cutoff frequencies of the passband vary with thickness of dielectric disc ð Þ L � T =rW taking structure periodicity constant such that the passband decreases with an increase in thickness of dielectric disc for both the TE01 and TE02 modes. The thickness of dielectric disc tailors the dispersion characteristics, however, more for the TE02 than for the TE01 mode, and the control is more prominent for thinner dielectric disc (Figure 26) [19]. The less effective parameters, the disc-hole radius, in tailoring the dispersion characteristics of a conventional disc-loaded waveguide [2, 5, 9–13], effectively controls the shape of the characteristics after introducing the dielectric discs between the metal discs, while, the control is more for the TE02 (Figure 27(b)) than for the TE01 (Figure 27(a)) mode, however the characteristics is little irregular for higher disc-hole radius

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

#### Figure 24.

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking relative permittivity as the parameter. The broken curves refer to a conventional metal disc-loaded circular waveguide [19].

#### Figure 25.

the ω � β dispersion characteristics, desired for wideband gyro-TWT performance

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the number of metal

TE01-mode dispersion characteristics of the metal-vane-loaded circular waveguide taking the vane angle as the

The lower and upper cutoff frequencies of the passband vary with thickness of dielectric disc ð Þ L � T =rW taking structure periodicity constant such that the passband decreases with an increase in thickness of dielectric disc for both the TE01 and TE02 modes. The thickness of dielectric disc tailors the dispersion characteristics, however, more for the TE02 than for the TE01 mode, and the control is more prominent for thinner dielectric disc (Figure 26) [19]. The less effective parameters, the disc-hole radius, in tailoring the dispersion characteristics of a conventional disc-loaded waveguide [2, 5, 9–13], effectively controls the shape of the characteristics after introducing the dielectric discs between the metal discs, while, the control is more for the TE02 (Figure 27(b)) than for the TE01 (Figure 27(a)) mode, however the characteristics is little irregular for higher disc-hole radius

(Figure 25) [19].

vanes as the parameter [16–18].

Figure 23.

292

Figure 22.

parameter [16–18].

Electromagnetic Materials and Devices

Dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) for the selected dielectric disc relative permittivity values for the sake of comparison between the modes TE01 and TE02 with respect to the control of the shape of the dispersion characteristics. The broken curves referring to a conventional metal disc-loaded circular waveguide [19].

(Figure 27) [19]. Similar to a conventional all-metal disc-loaded waveguide (model-4), the structure periodicity is the most effective parameter for tailoring the dispersion characteristics of the structure with dielectric discs between the metal discs (model-6), for the TE01 and TE02 modes, more for the latter. The control of the structure periodicity in straightening the dispersion characteristics, as required for the desired wideband gyro-TWT performance, is enhanced by introducing the dielectric discs in the conventional disc-loaded waveguide, though not enhancing the frequency range of the straight line portion of the dispersion characteristics (Figure 28). In this model, a serious care is required while selecting the dielectric material because a heavily dielectric-loaded structure depresses the dispersion characteristics to the slow-wave region (below the velocity of light line in ω β characteristics).

azimuthally symmetric mode in all-metal variants of the axially periodic structure. The increase of the relative permittivity of the dielectric discs in model-7 reduces the lower and upper cutoff frequencies, however not equally therefore shortens the passband for the TE01 mode with shift of the mid-frequency of the passband towards lower value (Figure 29(a)). With the increase of the relative permittivity of the dielectric discs, the lower and upper cutoff frequencies shift to lower value for the TE01 and TE02 modes. The quantitatively the shift in upper cutoff frequency is higher than that of lower cutoff frequency for the TE01, which in turn shortens the passband (Figure 29(a)), however, the shift in lower and upper cutoff frequencies are almost equal for the TE02 mode, effectively the passband does not change (Figure 29(b)). Interestingly for the TE02 mode the introduction of dielectric discs converts the fundamental backward mode (the zero group velocity follows to take negative values then again zero and further positive) into fundamental forward mode (the zero group velocity follows to take positive values then again zero and further negative) (Figure 29(b)). Thus, introduction of dielectric discs into the conventional disc-loaded waveguide turns the negative dispersion into positive. In absence as well as in presence of the dielectric discs, the TE03 mode dispersion characteristics of the disc-loaded waveguide represents the fundamental backward mode, in which the increase of the relative permittivity of the dielectric discs shifts the lower cutoff frequency more than that of upper cutoff frequency, and thus widens the passband for lower relative permittivity. For higher relative permittivity value the lower cutoff frequency remains unchanged and upper cutoff frequency shifts to lower value with the increase of the relative permittivity of the

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

DOI: http://dx.doi.org/10.5772/intechopen.82124

dielectric discs and shortens the passband (Figure 29(c)) [20, 21].

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal discloaded circular waveguide taking relative permittivity of dielectric disc as the parameter [20, 21]. The broken

curve refers to the conventional disc-loaded circular waveguide [2, 5, 9–13].

Figure 29.

295

#### Figure 26.

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking dielectric disc thickness as the parameter for a constant structure periodicity [19].

#### Figure 27.

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking the disc-hole radius as the parameter. The broken curves refer to the special case of a conventional smooth wall circular waveguide [19].

#### Figure 28.

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking structure periodicity as the parameter. The broken curves refer to a conventional metal disc-loaded circular waveguide [19].

We choose first three lowest order azimuthally symmetric modes in the composite (dielectric and metal) loaded structures for exploring the effect of structure parameter on dispersion characteristics while we chose only the lowest order

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

azimuthally symmetric mode in all-metal variants of the axially periodic structure. The increase of the relative permittivity of the dielectric discs in model-7 reduces the lower and upper cutoff frequencies, however not equally therefore shortens the passband for the TE01 mode with shift of the mid-frequency of the passband towards lower value (Figure 29(a)). With the increase of the relative permittivity of the dielectric discs, the lower and upper cutoff frequencies shift to lower value for the TE01 and TE02 modes. The quantitatively the shift in upper cutoff frequency is higher than that of lower cutoff frequency for the TE01, which in turn shortens the passband (Figure 29(a)), however, the shift in lower and upper cutoff frequencies are almost equal for the TE02 mode, effectively the passband does not change (Figure 29(b)). Interestingly for the TE02 mode the introduction of dielectric discs converts the fundamental backward mode (the zero group velocity follows to take negative values then again zero and further positive) into fundamental forward mode (the zero group velocity follows to take positive values then again zero and further negative) (Figure 29(b)). Thus, introduction of dielectric discs into the conventional disc-loaded waveguide turns the negative dispersion into positive. In absence as well as in presence of the dielectric discs, the TE03 mode dispersion characteristics of the disc-loaded waveguide represents the fundamental backward mode, in which the increase of the relative permittivity of the dielectric discs shifts the lower cutoff frequency more than that of upper cutoff frequency, and thus widens the passband for lower relative permittivity. For higher relative permittivity value the lower cutoff frequency remains unchanged and upper cutoff frequency shifts to lower value with the increase of the relative permittivity of the dielectric discs and shortens the passband (Figure 29(c)) [20, 21].

#### Figure 29.

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal discloaded circular waveguide taking relative permittivity of dielectric disc as the parameter [20, 21]. The broken curve refers to the conventional disc-loaded circular waveguide [2, 5, 9–13].

We choose first three lowest order azimuthally symmetric modes in the composite (dielectric and metal) loaded structures for exploring the effect of structure parameter on dispersion characteristics while we chose only the lowest order

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking structure periodicity as the parameter. The broken curves refer to a

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking dielectric disc thickness as the parameter for a constant structure

TE01 (a) and TE02 (b) mode dispersion characteristics of a circular waveguide loaded with alternate dielectric and metal annular discs (model-6) taking the disc-hole radius as the parameter. The broken curves refer to the

special case of a conventional smooth wall circular waveguide [19].

conventional metal disc-loaded circular waveguide [19].

Figure 26.

Figure 27.

Figure 28.

294

periodicity [19].

Electromagnetic Materials and Devices

The increase of dielectric disc radius in the model-7 taking constant metal disc radius shifts the lower and upper cutoff frequencies of the TE01 mode up, and the passband increases due to lesser the shift in lower cutoff frequency (Figure 30). Similarly, the passband of the TE02 mode increases with the increase of dielectric disc radius. Here, it is interesting to note that for the taken structure parameters (rD=rW ¼ 0:6, L=rW ¼ 1:0, TDD=rW ¼ 0:3 and ε<sup>r</sup> ¼ 5:0) the frequency shift is maximum for rDD=rW equal to 0.8–0.9, and minimum for 0.7–0.8 (Figure 30(b)). For the lower and higher values of inner dielectric disc radius, the TE03 mode dispersion characteristics of the model-7 are fundamental forward (positive) and backward (negative) modes respectively. Thus, there is a possibility of getting straight-line dispersion characteristics parallel to phase propagation constant axis, i.e., zero group velocity line (Figure 30(c)) [20, 21]. The increase of periodicity of the alternate dielectric and metal disc-loaded circular waveguide (model-7) reduces the passband and both the lower and upper cutoff frequencies with higher relative reduction in upper cutoff frequency than that of lower (Figure 31) for the chosen three azimuthally symmetric modes TE01, TE02 and TE03. The TE01 and TE02 modes are fundamental forward and the TE03 is fundamental backward [20, 21]. The change of dielectric disc thickness does not much tailor the dispersion characteristics and the passband (Figure 32), however, the decrease of the dielectric disc thickness or the increase of metal disc thickness shifts the passband to lower frequency side for the TE01 and TE02 modes (Figure 31(a) and (b)) and least change occur to the TE03 mode. In very precise observation, the lower cutoff frequency of the TE03 mode is insensitive and the upper cutoff frequency first decreases and then increases with decrease of dielectric disc thickness or with increase of metal disc thickness (Figure 32) [20, 21].

Figure 31.

Figure 32.

297

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal disc-

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

DOI: http://dx.doi.org/10.5772/intechopen.82124

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal discloaded circular waveguide (model-7) taking thickness of dielectric disc as the parameter [20, 21].

loaded circular waveguide (model-7) taking structure periodicity as the parameter [20, 21].

#### Figure 30.

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric- and metal discloaded circular waveguide (model-7) taking inner radius of dielectric disc as the parameter [20, 21].

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

#### Figure 31.

The increase of dielectric disc radius in the model-7 taking constant metal disc radius shifts the lower and upper cutoff frequencies of the TE01 mode up, and the passband increases due to lesser the shift in lower cutoff frequency (Figure 30). Similarly, the passband of the TE02 mode increases with the increase of dielectric disc radius. Here, it is interesting to note that for the taken structure parameters (rD=rW ¼ 0:6, L=rW ¼ 1:0, TDD=rW ¼ 0:3 and ε<sup>r</sup> ¼ 5:0) the frequency shift is maximum for rDD=rW equal to 0.8–0.9, and minimum for 0.7–0.8 (Figure 30(b)). For the lower and higher values of inner dielectric disc radius, the TE03 mode dispersion characteristics of the model-7 are fundamental forward (positive) and backward (negative) modes respectively. Thus, there is a possibility of getting straight-line dispersion characteristics parallel to phase propagation constant axis, i.e., zero group velocity line (Figure 30(c)) [20, 21]. The increase of periodicity of the alternate dielectric and metal disc-loaded circular waveguide (model-7) reduces the passband and both the lower and upper cutoff frequencies with higher relative reduction in upper cutoff frequency than that of lower (Figure 31) for the chosen three azimuthally symmetric modes TE01, TE02 and TE03. The TE01 and TE02 modes are fundamental forward and the TE03 is fundamental backward [20, 21]. The change of dielectric disc thickness does not much tailor the dispersion characteristics and the passband (Figure 32), however, the decrease of the dielectric disc thickness or the increase of metal disc thickness shifts the passband to lower frequency side for the TE01 and TE02 modes (Figure 31(a) and (b)) and least change occur to the TE03 mode. In very precise observation, the lower cutoff frequency of the TE03 mode is insensitive and the upper cutoff frequency first decreases and then increases with decrease of dielectric disc thickness or with increase of metal disc

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric- and metal discloaded circular waveguide (model-7) taking inner radius of dielectric disc as the parameter [20, 21].

thickness (Figure 32) [20, 21].

Electromagnetic Materials and Devices

Figure 30.

296

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal discloaded circular waveguide (model-7) taking structure periodicity as the parameter [20, 21].

#### Figure 32.

TE01 (a),TE02 (b) and TE03 (c) mode dispersion characteristics of the alternate dielectric and metal discloaded circular waveguide (model-7) taking thickness of dielectric disc as the parameter [20, 21].

Although the geometrical parameters do not tailor the dispersion characteristics of the only metal vane-loaded waveguide [16–18], however, it does for composite-loaded structure (model-8). The radial dimensions (Figure 33) are less sensitive in tailoring the dispersion characteristics of the dielectric and metal vanes than their angular dimensions (Figure 34), relative permittivity (Figure 35) and number of vanes (Figure 36). However, the waveguide cutoff (eigenvalue) of the structure depends on all these parameters (Figures 33-36). Thus, one may choose angular dimensions, relative permittivity and number of vanes for tailoring the dispersion characteristics and the radial parameter to control the waveguide cutoff frequency [21].

#### Figure 33.

TE01-mode dispersion characteristics of a circular waveguide loaded with composite alternate dielectric and metal vanes taking the vane inner-tip radius as the parameters. The broken curve refers to a smooth-wall waveguide (free from dielectric and metal vanes) and the star (\*) marker refers representative points obtained using HFSS [21].

A crossover point in the dispersion characteristics appears with varying metal vane angle (Figure 34) or varying number of vanes (Figure 36). This crossover point sifts to another location for another value of relative permittivity. The locus of such crossover points for varying relative permittivity overlaps with the dispersion plot of a smooth-wall circular waveguide of radius equal to the vane tip radius (Figure 34). Below the crossover point, the increase of metal cross-section (either by increasing metal vane angle or by increasing number of vanes) in the crosssection of the waveguide elevates and that of dielectric depresses the dispersion

TE01-mode dispersion characteristics of a circular waveguide loaded with dielectric and metal vanes taking

TE01-mode dispersion characteristics of a circular waveguide loaded with composite alternate dielectric and metal vanes, taking the relative permittivity of dielectric vanes as the parameters, along with the corresponding characteristics of a smooth-wall waveguide (free from dielectric and metal vanes) (broken curve) and the typical representative points of the characteristics obtained by simulation (HFSS) (\* (star) marker) [21]. ε<sup>r</sup> ¼ 1 represents the characteristics of a waveguide loaded with metal vanes alone (Figure 6) [16–18].

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

DOI: http://dx.doi.org/10.5772/intechopen.82124

Figure 35.

Figure 36.

299

number of vanes as the parameter [21].

#### Figure 34.

TE01-mode dispersion characteristics of a circular waveguide loaded with dielectric and metal vanes taking metal vane angle as the parameter. The broken curve represents the locus of the crossover point, which is same as the dispersion characteristics of the smooth-wall circular waveguide of radius rV [21].

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

#### Figure 35.

Although the geometrical parameters do not tailor the dispersion characteris-

tics of the only metal vane-loaded waveguide [16–18], however, it does for composite-loaded structure (model-8). The radial dimensions (Figure 33) are less sensitive in tailoring the dispersion characteristics of the dielectric and metal

(Figure 35) and number of vanes (Figure 36). However, the waveguide cutoff (eigenvalue) of the structure depends on all these parameters (Figures 33-36). Thus, one may choose angular dimensions, relative permittivity and number of vanes for tailoring the dispersion characteristics and the radial parameter to con-

TE01-mode dispersion characteristics of a circular waveguide loaded with composite alternate dielectric and metal vanes taking the vane inner-tip radius as the parameters. The broken curve refers to a smooth-wall waveguide (free from dielectric and metal vanes) and the star (\*) marker refers representative points obtained

TE01-mode dispersion characteristics of a circular waveguide loaded with dielectric and metal vanes taking metal vane angle as the parameter. The broken curve represents the locus of the crossover point, which is same as

the dispersion characteristics of the smooth-wall circular waveguide of radius rV [21].

vanes than their angular dimensions (Figure 34), relative permittivity

trol the waveguide cutoff frequency [21].

Electromagnetic Materials and Devices

Figure 33.

Figure 34.

298

using HFSS [21].

TE01-mode dispersion characteristics of a circular waveguide loaded with composite alternate dielectric and metal vanes, taking the relative permittivity of dielectric vanes as the parameters, along with the corresponding characteristics of a smooth-wall waveguide (free from dielectric and metal vanes) (broken curve) and the typical representative points of the characteristics obtained by simulation (HFSS) (\* (star) marker) [21]. ε<sup>r</sup> ¼ 1 represents the characteristics of a waveguide loaded with metal vanes alone (Figure 6) [16–18].

#### Figure 36.

TE01-mode dispersion characteristics of a circular waveguide loaded with dielectric and metal vanes taking number of vanes as the parameter [21].

A crossover point in the dispersion characteristics appears with varying metal vane angle (Figure 34) or varying number of vanes (Figure 36). This crossover point sifts to another location for another value of relative permittivity. The locus of such crossover points for varying relative permittivity overlaps with the dispersion plot of a smooth-wall circular waveguide of radius equal to the vane tip radius (Figure 34). Below the crossover point, the increase of metal cross-section (either by increasing metal vane angle or by increasing number of vanes) in the crosssection of the waveguide elevates and that of dielectric depresses the dispersion

characteristics, and vice versa above the crossover point. This clearly represents the behavior of model-8 as a smooth-wall waveguide of wall radius equal to the vane tip radius at the crossover point and change of metal cross-section (either by changing metal vane angle or by changing number of vanes) in the cross-section of the waveguide affects the dispersion characteristics away from the crossover (Figures 34 and 36) [21].
