[1] The problem of radiation from narrow axial slots on a conducting circular cylinder with transverse electric (TE_{11}, where the first 1 refers to the number of circumferential variation ad the second 1 refers to the number of radial variation) mode excitation was solved. The Fourier transform and mode matching were used to obtain a set of simultaneous equations. Fast convergent series solutions for radiation, transmission, and reflection coefficients were presented in terms of discrete modal coefficients. A 20-slot antenna was fabricated, and its radiation patterns were measured. Our numerical results agree favorably with measured data.

[2] The problem of radiation from axial slots on a conducting circular cylinder has many practical applications in microwave antennas [Bailin, 1955; Collin and Zucker, 1969; Harrington, 1961; Sensiper et al., 1952; Shafai and Hassan, 1981; Silver and Saunders, 1950; Wait, 1959]. Most existing studies on a single axial slot assumed a constant E_{ϕ} on the slot [Bailin, 1955; Collin and Zucker, 1969; Harrington, 1961; Sensiper et al., 1952; Silver and Saunders, 1950; Wait, 1959]. Meanwhile, there has been relatively little reported research on radiation from multiple axial slots on a conducting circular cylinder [Shafai and Hassan, 1981]. Therefore it is of interest to rigorously study radiation from axial slots on a conducting circular cylinder. This paper presents such a solution by solving the boundary value problem of multiple narrow axial slots on a conducting circular cylinder. Recently, we have investigated radiation from multiple circumferential slots on a conducting circular cylinder by using the Fourier transform, mode matching, and residue calculus [Shin and Eom, 2005]. In this paper, the same technique as used by Shin and Eom [2005] is utilized to obtain rigorous, fast convergent series solutions for radiation, transmission, and reflection with the incident transverse electric (TE_{11}, where the first 1 refers to the number of circumferential variation ad the second 1 refers to the number of radial variation) mode. An antenna is fabricated and its radiation patterns and return loss are measured and compared with the theoretical values.

2. Field Analysis

[3] Consider the problem geometry shown in Figure 1. The conducting circular cylinder extends infinitely along the z direction with an inner radius of a and an outer radius of b. The slots are assumed to have an identical size of length d and angle _{0}. The offset parameters of the mth slot, T^{(m)} and T_{z}^{(m)}, denote the angle from = 0 and the distance from the xy plane, respectively. The time convention e^{−iωt} is suppressed throughout. An incident TE_{11} mode is assumed to propagate in the circular waveguide and its electric vector potential is

where β = , k_{1} = ω, k_{c} = = , and χ′_{11} is defined as the first root of the Bessel function of the first kind of order 1, J_{1}(·).

[4] The scattered electric and magnetic vector potentials for the waveguide region (ρ < a) and the free space region (ρ > b), {F_{z}^{I}(ρ, ϕ, z), A_{z}^{I}(ρ, ϕ, z)} and {F_{z}^{III}(ρ, ϕ, z), A_{z}^{III}(ρ, ϕ, z)}, correspond with those used by Shin and Eom [2005], respectively.

[5] In the slots (a < ρ < b, T^{(m)} < <T^{(m)} + ϕ_{0}, and T_{z}^{(m)} < z < T_{z}^{(m)} + d), the electric vector potential is approximately given as

where k_{ρs} = , k_{2} = ω, a_{s} = , and

Here, N_{0}(·) is a Bessel function of the second kind of order 0 and N denotes the number of slots. The electric field representation using (2) in each slot is valid as long as the width of the slot is narrow compared to the wavelength (aϕ_{0} ≪ λ).

[6] Applying the inverse Fourier transform to E and E_{z} component continuities at ρ = a and integrating with respect to from 0 to 2π respectively, gives

where

Substituting (4) and (5) into the H_{z} component continuity at ρ = a, multiplying by sin a_{q}(z − T_{z}^{(l)}), and integrating dzdϕ yields

where δ_{ml} is the Kronecker delta and

J′_{n}(·) is the derivative of the Bessel function J_{n}(·).

[7] Similarly, E_{z}, Eϕ, and H_{z} continuities at ρ = b yield a set of simultaneous equations

where

The modal coefficients A_{s}^{(m)} and B_{s}^{(m)} can be obtained by solving the simultaneous equations (8) and (11). It is expedient to convert I_{1} and I_{2} into a fast convergent series on the basis of residue calculus and contour integrations [Shin and Eom, 2005].

3. Power Calculations

[8] In the far zone, the electric and magnetic field components are

where

and H_{n}^{(1)′}(k_{3}b sin θ) denotes the differentiation .

[9] The time-average incident power of TE_{11} mode is given as

where

For simplicity, k_{1}a is set to be less than 2.4048 in the derivation of time-average power. Under this condition, only the TE_{11} mode is allowed to propagate, whereas the remaining higher-order modes are evanescent. The time-average powers reflected (z → −∞) and transmitted (z → ∞) in the waveguide are

where

where the symbol Im(·) denotes the imaginary part of (·).The radiation power is the sum of the power radiated through each slot at ρ = b and is represented as

where the symbol (·)* denotes the complex conjugation of (·). The transmission, reflection, and radiation coefficients are defined as (τ = ), (γ = ), and (σ = ), respectively. A power conservation law is given by τ + γ + σ = 1.

4. Numerical and Experimental Results

[10]Figure 2 shows the normalized magnitude of the copolarized component E_{ϕ} in the plane of θ = 90° for a single slot (N = 1) for k_{1}a = 12. A comparison between our results and the experimental data at X band by Sensiper et al. [1952, Figure 5] shows excellent agreement for the axial half-wavelength slot. The number of modes used in our computation is one term (s, q = 1) and 21 terms (n = 0, ±1, ⋯, ±10) in (8) and (11) to achieve numerical convergence.

[11]Figure 3 illustrates the effect of the number of slots N on the normalized power coefficients at 13 GHz. The normalized radiation coefficient exceeds 90% when N > 15, while the power conservation law is satisfied for all N.

[12] A 20-slot antenna has been designed and fabricated (see Figure 4), and its radiation characteristics have been measured. The rectangular-to-circular waveguide transition (Model 18643-VSWR < 1.07, insertion loss < 0.2 dB at 11.9∼18 GHz, Flann Microwave Ltd.) is used to launch the TE_{11} mode into the circular waveguide section. A flexible absorber is attached to the end of the waveguide so as to reduce the reflection. The physical dimensions of the slot antenna are listed in Table 1.

Table 1. Parameters of the Fabricated Antenna

Notation

Value

Inner radius

a

8.05 mm

Outer radius

b

9.55 mm

Slot length

d

10.0 mm

Slot angle

ϕ_{0}

16°

Number of slots

N

20

Spacing of slots

T_{z}^{(m+1)} − T_{z}^{(m)}

11.5 mm

[13]Figures 5a and 5b show the angular behavior of H plane radiation patterns for the fabricated antenna at 11.7 GHz and 13 GHz, respectively. The antenna has been designed to operate between 11 GHz and 14.2 GHz with TE_{11} dominant mode. Reasonable agreement between the measured data and theoretical computation is observed except for near grazing. The discrepancies near grazing are attributed to the theoretical assumption of an infinitely long cylinder whereas the fabricated antenna is finite in length.

[14] The return loss is calculated and compared with the measured data in Figure 6. The discrepancies here are attributed to the influence of mismatches resulting from use of a coax-to-rectangular waveguide adapter as well as a rectangular-to-circular waveguide transition. The fabricated antenna shows broadband reflection with high radiation efficiency.

5. Conclusion

[15] We have analyzed radiation from multiple axial slots arbitrarily placed on a conducting circular cylinder. We have used the Fourier transform, mode matching, and residue calculus to obtain simultaneous equations for the modal coefficients. The theoretical analysis yields very simple yet rigorous expressions, which are amenable to computation. Computations were performed to illustrate the normalized radiation, transmission, reflection coefficients, and radiation pattern when the dominant TE_{11} mode excites the antenna. The antenna was fabricated and measurements were performed at 11.7 GHz and 13 GHz. The numerical results for the H plane radiation patterns agree well with the measured data. The developed theory for antenna radiation will be valuable for the design of slot array antennas comprising multiple narrow axial slots on a conducting circular cylinder.

Acknowledgments

[16] This research was supported by the Ministry of Information and Communication (MIC), Korea, under the Information Technology Research Center (ITRC) support program supervised by the Institute of Information Technology Assessment (IITA-2005-(C1090-0502-0014)).