### I. Introduction

### II. Theory of the C-TCDA

*ρ*,

*φ*,

*z*) are used, with the cylinder being parallel to the

*z*-axis and the direction of wave propagation being along the

*ρ*-axis. Two types of C-TCDA with elements arranged in the

*φ*-direction (horizontal polarization) and the

*z*-direction (vertical polarization) are analytically solved. The cylindrical and planar TCDAs are then compared.

### 1. Horizontally Polarized C-TCDA

*N*elements of the TCDA are arranged in the

*φ*-direction with a period of

*φ*

_{0}= 2π/

*N*and repeated in the

*z*-direction with a period of

*b*. According to Wheeler’s current sheet array theory, a two-dimensionally periodic dipole array located in a plane can be equated to a unit dipole in a hypothetical wave-guide that has a perfect electric conductor (PEC) and a perfect magnetic conductor (PMC) boundary conditions [15]. Similar to a planar array in Fig. 1(a), the cylindrically infinite dipole array is equivalent to the unit dipole inside the hypothetical waveguide, as shown in Fig. 1(b). We assume that the medium inside the waveguide is filled with an isotropic, homogeneous, and lossless dielectric material with a permittivity

*ɛ*and permeability

*μ*

_{0}. The boundaries parallel to the

*z*-axis are PEC, and the other boundaries orthogonal to the

*z*-axis are PMC. Therefore, the boundary conditions are written as

*φ-*direction, and thus the magnetic wave modes transverse to the

*φ-*direction exist. For the magnetic modes transverse to

*φ*that propagate only in the +

*ρ*-direction (radiation direction), the vector potential satisfying the boundary conditions in Eqs. (1) and (2) can be written as

*p*= 0, 1, 2, … and

*m*= 0, 1, 2, …. The variable

*B*

*is the coefficient of the*

_{pm}*pm*mode. The wave number,

*k*, is given by

_{00}transverse to

*φ*, and the corresponding vector potential is

*E*

_{φ}and

*H*

_{z}exist, signifying that the mode is TEM to the

*ρ*mode. As

*E*

_{φ}and

*H*

_{z}are functions of

*ρ*, the hypothetical waveguide can be regarded as a transmission line with characteristic impedance as a function of

*ρ*. The characteristic impedance of the transmission line for the horizontal polarization is given by

_{00}mode is given by [12]

*pm*mode is given by

*m*= 1) when scanning in the

*θ*-direction because

*b*≥

*λ*/2. All modes where

*p*> 0 and

*m*= 0 are possible, but we focus on only the dominant mode for simplicity.

*ρ*

_{g},

*ρ*

_{ant}, and

*ρ*

_{sup}, respectively. The variables

*Z*

_{0h}and

*Z*

_{suph}are the characteristic impedance of free space and the superstrate for the horizontal polarization, respectively, and they are obtained by substituting the vacuum permittivity,

*ɛ*

_{0}, and the superstrate permittivity,

*ɛ*

_{sup}, for

*ɛ*in Eq. (13). The variables

*β*

_{0}and

*β*

_{sup}are the phase constant of free space and the superstrate, respectively, which are obtained by substituting

*ɛ*

_{0}and

*ɛ*

_{sup}for

*ɛ*in Eq. (14). The embedding impedances

*Z*

_{u}and

*Z*

_{d}are the impedances toward the outward direction and inward direction in the dipole antenna, respectively.

*Z*

_{ant}is the antenna impedance, which is composed of the dipole inductance,

*L*

_{dipole}, and the coupling capacitance,

*C*

_{coupling}. Therefore, the input impedance is given by

*Z*

_{in}=

*Z*

_{ant}+

*Z*

_{u}//

*Z*

_{d}=

*jω L*

_{dipole}+ 1 /

*jω C*

_{coupling}+

*Z*

_{u}//

*Z*

_{d}.

### 2. Vertically Polarized C-TCDA

*N*elements of the TCDA are arranged in the

*φ*-direction with a period of

*φ*

_{0}= 2π/

*N*and repeated in the

*z*-direction with a period of

*b*. Similar to the horizontally polarized C-TCDA, the arrays are equivalent to a unit dipole existing in the hypothetical waveguide with PEC and PMC boundary conditions, as shown in Fig. 3(b). We also assume that the medium inside the waveguide is filled with an isotropic, homogeneous, and lossless dielectric material, with permittivity

*ɛ*and permeability

*μ*

_{0}. Contrary to the horizontal polarization case, the boundaries parallel to the

*z*-axis are PMC, and the other boundaries orthogonal to the

*z*-axis are PEC. Thus, the boundary conditions are written as

*z*-direction, and therefore the modes transverse to the

*z*-axis exist. For the modes transverse to the

*z*-direction that propagate only in the +

*ρ*-direction (radiation direction), the vector potential satisfying the boundary conditions in Eqs. (16) and (17) can be written as

*k*

_{ρ}given in Eq. (4), where

*p*= 0, 1, 2, …, and

*m*= 0, 1, 2, ….

*C*

*is the coefficient of the*

_{pm}*pm*mode. The wave number,

*k*, is given in Eq. (5). Thus, the fundamental mode is TM

_{00}transverse to

*z*, and the corresponding vector potential is

*E*

*and*

_{z}*H*

_{φ}exist, signifying that the mode is TEM transverse to

*ρ*. Furthermore, because

*E*

_{z}and

*H*

_{φ}are functions of

*ρ*, the hypothetical waveguide can be regarded as a transmission line, the characteristic impedance of which is a function of

*ρ*. The characteristic impedance of the transmission line for the vertical polarization is

_{00}mode and the cutoff frequencies of the

*pm*mode are the same as those for the horizontal polarization given in Eqs. (14) and (15), respectively. The grating lobe occurs at the first cutoff frequency (

*m*= 1) when scanning in the

*θ*-direction, as

*b*≥

*λ*/2. All modes where

*p*> 0 and

*m*= 0 are possible, but we consider only the dominant mode for simplicity.

*Z*

_{0v}and

*Z*

_{supv}are the characteristic impedance of free space and the superstrate for the vertical polarization, respectively, and they are obtained by substituting

*ɛ*

_{0}and

*ɛ*

_{sup}for

*ɛ*in Eq. (26). The variables

*β*

_{0},

*β*

_{sup},

*Z*

_{u},

*Z*

_{d},

*Z*

_{ant}, and

*Z*

_{in}are identical to those of the horizontally polarized C-TCDA.

### 3. Comparison with the Planar TCDA

*a*and

*d*be the lengths of the hypothetical waveguide of the planar TCDA unit cell that are parallel and orthogonal to the dipole direction, respectively. The characteristic impedance of the waveguide as given by [11] is

*k*is given in Eq. (5). The distance between the ground plane and the antenna is

*h*

_{ant}and that between the antenna and the superstrate is

*h*

_{sup}.

*Z*

_{op}and

*Z*

_{supp}are obtained by substituting

*ɛ*

_{0}and

*ɛ*

_{sup}, respectively, for

*ɛ*in Eq. (17). The variables

*k*

_{o}and

*k*

_{supp}are obtained by substituting

*ɛ*

_{0}and

*ɛ*

_{sup}, respectively, for

*ɛ*in Eq. (5).

*Z*

_{in}=

*Z*

_{ant}+

*Z*

_{up}//

*Z*

_{dp}, is matched for a wide range of frequencies.

*Z*

_{ant}is determined by the dipole antenna shape and the gap size between the neighboring dipoles. However, the embedding impedance,

*Z*

_{u}//

*Z*

_{d}or

*Z*

_{up}//

*Z*

_{dp}, is inherently determined by the shape of the array structure.

*Z*

_{u}//

*Z*

_{d}and

*Z*

_{up}//

*Z*

_{dp}for both polarizations is presented in Fig. 5. For the C-TCDA, the relative distances

*ρ*

_{ant}–

*ρ*

_{g}and

*ρ*

_{sup}–

*ρ*

_{ant}are fixed to 30 mm and 20 mm, respectively. Then,

*Z*

_{u}//

*Z*

_{d}is plotted for

*ρ*

_{g}= 30, 60, and 90 mm. The other specific values are

*ɛ*

_{sup}= 2.2 ×

*ɛ*

_{0},

*φ*

_{0}= π/4 (

*N*= 8), and

*b*= 65 mm. For the planar TCDA,

*h*

_{ant}=

*ρ*

_{ant}–

*ρ*

_{g}= 30 mm,

*h*

_{sup}=

*ρ*

_{sup}–

*ρ*

_{ant}= 20 mm, and

*ɛ*

_{sup}= 2.2 ×

*ɛ*

_{0}are commonly used. The lengths of the hypothetical waveguide are chosen as

*a*= 60 mm ×

*φ*

_{0}≈ 70.7 mm and

*d*= 65 mm for the horizontal polarization and

*a*= 65 mm and

*d*= 60 mm ×

*φ*

_{0}≈ 70.7 mm for the vertical polarization. As a result,

*Z*

_{u}//

*Z*

_{d}when

*ρ*

_{g}= 60 mm and

*Z*

_{up}//

*Z*

_{dp}have similar values for both polarizations. Furthermore, when

*ρ*

_{g}increases, the amplitude of

*Z*

_{u}//

*Z*

_{d}increases for the horizontal polarization and decreases for the vertical polarization. As the C-TCDA has similar impedance characteristics to the planar TCDA, the C-TCDA can achieve a wide bandwidth as the planar TCDA by adjusting the antenna impedance.

### III. Design of Dual-Polarized C-TCDA

### 1. Design

*N*= 8) and has a periodic boundary in the

*z*-direction. A unit cell of the C-TCDA antenna is illustrated in Fig. 6(b). Printed circuit boards (PCBs), which are the horizontal and vertical polarization antennas, are orthogonally soldered to the ground plane. The vertical polarization dipole antenna is located away from the center of the unit cell ground plane for low coupling between both polarizations. The substrate of both polarization antennas is designed using the Rogers RT/Duroid 5880 PCB with a substrate thickness of 0.508 mm, dielectric constant of 2.2, and dielectric loss tangent of 0.001. The superstrate is implemented by polytetrafluoroethylene (PTFE) with a dielectric constant of 2.1 and a dielectric loss tangent of 0.0005. The superstrate is used for impedance matching in a low-profile array antenna [8]. The front and back sides of the horizontal polarization dipole are presented in Fig. 6(c) and (d), and the front and back sides of the vertical polarization dipole are presented in Figs. 6(e) and (f), respectively. The feeding line of the front side of both dipoles is connected to an subminiature version A (SMA) connector and that of the back side of both dipoles is connected with the ground plane (unbalanced feeding). Both dipoles, except the feeding line, are rotationally symmetric and have shorting posts connected with the ground plane to adjust the common mode resonance frequency analyzed in [9]. The specific values are as follows:

*ρ*

_{g}= 60 mm,

*h*= 30 mm,

*h*

_{s}= 20 mm,

*w*

_{sup}= 70 mm,

*b*= 65 mm,

*l*

_{g}= 50 mm,

*v*= 15 mm,

*l*

_{h}= 58.7 mm,

*g*

_{h}= 0.15 mm,

*l*

_{a1}= 12.14 mm,

*l*

_{a2}= 2 mm,

*w*

_{h1}= 1 mm,

*w*

_{h2}=

*w*

_{h3}= 1.5 mm,

*h*

_{h}= 30 mm,

*s*

_{h}= 7.7 mm,

*θ*

_{h}= 35°,

*θ*

_{c}= 22.5°,

*l*

_{v}= 63 mm,

*w*

_{v1}=

*w*

_{v2}=

*w*

_{v3}= 1 mm,

*h*

_{v}= 30 mm,

*s*

_{v}= 12.7 mm,

*θ*

_{v}= 45°,

*s*= 0.1 mm,

*h*

_{b}= 45 mm,

*h*

_{1}= 26 mm, and

*h*

_{2}= 4 mm. The C-TCDA was simulated by CST Microwave Studio.

### 2. Performance

*θ*-direction while maintaining the omnidirectional pattern. Let the phase difference Δ

*θ*be given by

*θ*

_{s}is the desired scan angle. Fig. 8 shows the VSWR versus the frequency graphs for Δ

*θ*= 29.25°, 50°, 70°, and 87.75°. If |

*θ*

_{s}| ≥ 30°,

*f*≥

*c*Δ

*θ*/π

*b*, as obtained from the modified Eq. (28). The minimum frequency corresponding to Δ

*θ*and the upper frequency, 2.25 GHz, are marked by dash-dot lines in Fig. 8. The VSWRs are under 2.05 in both cases, and thus the octagonal C-TCDA can scan in the

*θ*-direction up to ±30° with a low VSWR.

*E*

_{φ}components are predominantly observed when the horizontal polarization dipoles are excited, and the

*E*

_{θ}components are predominantly observed when the vertical polarization dipoles are excited. The C-TCDA has good omnidirectional patterns in the azimuthal plane for both polarizations. In the elevation plane pattern, the C-TCDA can scan in the desired direction up to ±30° for both polarizations. Most of the patterns show a low cross-polarization under −10 dB compared with the copolarization and a sidelobe under −10 dB.

*z*-direction. Most studies on planar TCDAs show that TCDAs using 8 × 8 or more elements perform similarly to the infinite case [10–12]. Therefore, there should be little difference in performance between a practical N × M (≥8) C-TCDA and the N × infinite C-TCDA simulated in this design.