## Abstract

In the present chapter a number of loaded structures are studied with circular cross-section to explore the deviation in their dispersion characteristics from their parent circular waveguide. The dielectric and/or metal loading to the waveguide tailors its dispersion characteristics. In general, the dielectric depresses and the metal elevates the dispersion characteristics from the characteristics of their parent circular waveguide. The axial periodicity results in periodic dispersion characteristics with a lower and an upper cut-off frequency (bandpass). However such a characteristic is not reported for the azimuthal periodic structures. The bandpass characteristic arises due to the shaping of the dispersion characteristics. Therefore the dispersion shaping is only possible with axial periodicity and not with the azimuthal periodicity. The sensitivity of the structure (geometry) parameters on the lower and upper cut-off frequencies, the extent of passband and the dispersion shaping are also included. In the axial periodic structures, the periodicity is found to be the most sensitive parameter for tailoring the dispersion characteristics and the disc-hole radius is the most sensitive parameter for shifting the dispersion characteristics over the frequency axis.

### Keywords

- periodically loaded waveguide
- metal and dielectric loading
- disc-loaded circular waveguide
- dispersion characteristics
- dispersion shaping

## 1. Introduction

Waveguides, due to their low insertion loss and high power handling capabilities at microwave and millimeter-wave frequencies, are the transmission line commonly used for transmitting or propagating or guiding the signals of these frequencies from one point to another. The propagation characteristics of a guiding structure are generally represented by its dispersion characteristics. The dispersion characteristics are the study of structure supported frequency for a given phase propagation constant, and most commonly being plotted as the supported angular frequency

## 2. Structure models

Although, the considered structures being a single conductor structure support TE (

### 2.1 Dielectric-loaded circular waveguide

In this section, we will explore two variants of dielectric-loaded structure: (i) the circular waveguide with dielectric lining on metal wall (Figure 2), and (ii) the circular waveguide with dielectric coaxial insert (Figure 3) for their dispersion characteristics and the tailoring of these characteristics with change of the relative permittivity of the dielectric material.

#### 2.1.1 Circular waveguide with dielectric lining on metal wall

This model (model-1) includes a metallic circular waveguide of inner radius

In region I:

In region II:

where

#### 2.1.2 Circular waveguide with dielectric coaxial insert

Similar to model-1, this model (model-2) also contains a metallic circular waveguide of inner radius

For the sake of analysis, the structure may be divided into two analytical regions, the central dielectric filled region I:

### 2.2 Metal-loaded circular waveguide

In Section 2.1, we have studied the dispersion characteristics of dielectric-loaded structures. In the present section, we will explore three variants of all metal-loaded structure: (i) the conventional annular metal disc-loaded circular waveguide (Figure 4), (ii) the interwoven-disc-loaded circular waveguide (Figure 5), and (iii) metal vane-loaded circular waveguide (Figure 6) for their dispersion characteristics and the effect of change of the geometry parameters on these characteristics.

#### 2.2.1 Circular waveguide with annular metal discs

In this model (model-3) a circular metallic waveguide of inner radius

In region II:

where

#### 2.2.2 Interwoven-disc-loaded circular waveguide

This model (model-4) differs from the conventional disc-loaded circular waveguide due to different additional disc included in between two identical consecutive discs of conventional disc-loaded circular waveguide [2, 9, 14, 15]. Thus, this model is considered with a circular metallic waveguide of inner radius

Similar to the conventional disc-loaded circular waveguide (model-3), it is assumed that the disc free (I) and disc occupied (II and III) regions, respectively, support propagating and standing waves. The relevant (axial magnetic and azimuthal electric) field intensity components in the region I is given by (1) and (2) and in the regions II and III may be written as [2, 9, 14, 15]:

In region II:

In region III:

where

#### 2.2.3 Circular waveguide loaded with metal vanes

This model (model-5) considers a circular waveguide of radius

In region I:

In region II:

where

### 2.3 Metal- and dielectric-loaded circular waveguide

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 as well as dielectric loading in the circular waveguide.

#### 2.3.1 Circular waveguide loaded with dielectric and metal discs

This model (model-6) is formed by alternatively stacking the metal and dielectric discs each of same disc hole radii

#### 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:

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:

In region III:

where

#### 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

## 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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]:

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

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,

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

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,

* Model-3:* The relevant electromagnetic boundary conditions for model-3 (Figure 4) may be written in the mathematical form as [2, 5, 9, 10, 11, 12, 13]:

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,

* Model-4:* The boundary conditions (24) and (25) are also true for the model-4 (Figure 5) at the interface,

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,

* Model-5:* The relevant electromagnetic boundary conditions for model-5 (Figure 6) may be written in the mathematical form as [16, 17, 18]:

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,

* 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,

The boundary conditions (30) and (31), respectively, state the continuity of the axial magnetic and the azimuthal electric field intensities at the interface,

* 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 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-1 [7]:

Model-2 [7]:

Model-3 [2, 5, 9, 10, 11, 12, 13]:

where

where

where

Model-6 [19]:

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 propagation constant in dielectric filled region as

Model-7 [20]:

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

## 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 cutoff frequencies [1, 2, 3, 5, 9]. The dispersion characteristics taking horizontal axis as normalized phase propagation constant become periodic with the periodicity of
_{01}, TE_{02} and TE_{03}). The normalized passband (
_{02} mode is narrower than that of the TE_{01} mode. Similarly, the normalized stopband above the TE_{02} mode is narrower than that above the TE_{01} mode (Figure 12). The RF group velocity (slopes of dispersion plot) is positive (fundamental forward wave mode, * n* = 0) for the TE01 or TE

_{02}mode, it is negative (fundamental backward wave mode) for the TE

_{03}mode (Figure 12). The dependencies of the structure dispersion characteristics, for typical mode TE

_{01}, 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 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 straight-line 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) 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 upper-cutoff 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 disc-thickness 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, 17, 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 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 (TE_{01} and TE_{02}) modes are typically considered to study the performance of this model. With the increase of relative permittivity, the passband continuously decreases for the TE_{01} mode, and first decreases and then increases for the TE_{02} mode (Figure 24). Also, the variation in relative permittivity shapes of the dispersion characteristics of the structure, for both the modes TE_{01} and TE_{02}, however more significantly for the latter mode (Figure 24). The TE_{01} mode of the structure.

Exhibits fundamental forward wave (positive) dispersion characteristics irrespective of the value of relative permittivity (Figure 24(a)), however, the TE_{02} 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 TE_{02} mode
_{01} mode and with higher values of relative permittivity for the TE_{02} mode enhances the frequency range of the straight line portion of the

The lower and upper cutoff frequencies of the passband vary with thickness of dielectric disc
_{01} and TE_{02} modes. The thickness of dielectric disc tailors the dispersion characteristics, however, more for the TE_{02} than for the TE_{01} 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, 10, 11, 12, 13], effectively controls the shape of the characteristics after introducing the dielectric discs between the metal discs, while, the control is more for the TE_{02} (Figure 27(b)) than for the TE_{01} (Figure 27(a)) mode, however the characteristics is little irregular for higher disc-hole radius (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 TE_{01} and TE_{02} 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

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 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 TE_{01} 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 TE_{01} and TE_{02} modes. The quantitatively the shift in upper cutoff frequency is higher than that of lower cutoff frequency for the TE_{01}, which in turn shortens the passband (Figure 29(a)), however, the shift in lower and upper cutoff frequencies are almost equal for the TE_{02} mode, effectively the passband does not change (Figure 29(b)). Interestingly for the TE_{02} 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 TE_{03} 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].

The increase of dielectric disc radius in the model-7 taking constant metal disc radius shifts the lower and upper cutoff frequencies of the TE_{01} mode up, and the passband increases due to lesser the shift in lower cutoff frequency (Figure 30). Similarly, the passband of the TE_{02} mode increases with the increase of dielectric disc radius. Here, it is interesting to note that for the taken structure parameters (
_{03} 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 TE_{01}, TE_{02} and TE_{03}. The TE_{01} and TE_{02} modes are fundamental forward and the TE_{03} 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 TE_{01} and TE_{02} modes (Figure 31(a) and (b)) and least change occur to the TE_{03} mode. In very precise observation, the lower cutoff frequency of the TE_{03} 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].

Although the geometrical parameters do not tailor the dispersion characteristics of the only metal vane-loaded waveguide [16, 17, 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].

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 cross-section 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].

## 6. Conclusion

The waveguide is inherently a high pass filter with a lower cut-off frequency, a signal of above this frequency will be allowed to propagate or travel through the waveguide. Different kind of loading, such as metal and/or dielectric, has been suggested in the published literature for altering the propagation (dispersion) characteristics. In the present chapter a number of loaded structures are studied with circular cross-sections. All the structure parameters are varied to explore the sensitivity over the dispersion characteristics. In general, the axial periodicity results in periodic dispersion characteristics with a lower and an upper cut-off frequency, which makes the guiding structure a bandpass structure. However, such a characteristic is not reported in literature for the azimuthal periodic structures. The bandpass characteristic arises due to the shaping of the dispersion characteristics. Therefore, the dispersion shaping is only possible with axial periodicity and not with the azimuthal periodicity in all metal structure. However, the azimuthal periodicity in all metal waveguide structure shifts the cut-off frequency over the frequency scale. The dispersion characteristics of various loaded structures have been explored. It has been the interest to study the change in the lower and upper cut-off frequencies and in the passband. The sensitivity of the structure (geometry) parameters on the lower and upper cut-off frequencies and the extent of passband are also included. In case of conventional disc-loaded structure, the periodicity is found to be the most sensitive parameter for dispersion shaping and the disc-hole radius is the most sensitive parameter for shifting the dispersion characteristics over the frequency axis. In interwoven-disc-loaded circular waveguide, the structure periodicity and the disc-thickness of bigger-hole-disc are, respectively, the most and the least sensitive parameter for controlling the passband as well as shape of the dispersion characteristics. Insertion of metal vanes to a circular waveguide does not shape the dispersion characteristics but increases the waveguide cutoff frequency. On the other hand, introduction of dielectric discs into the conventional disc-loaded waveguide turns the negative dispersion into positive. In case of composite vane-loaded structure, the angular dimensions, relative permittivity and number of vanes tailor the dispersion characteristics and the radial parameter controls the waveguide cutoff frequency.

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