The proposed slot antennas

The proposed slot antenna, together with its various parameters, is depicted in Fig. 1. According to the figure, the shape of the slot is in the form of a dumbbell. For simple realization, the shape of the end parts is chosen to be circles. The width of the slot in the middle is denoted by W. R denotes the radius of the slot at both ends. Moreover, SL indicates the length of the slot. It is also worthy to note that the slot is placed along the waveguide centerline. For exciting the slot, an iris and a rectangular septum are placed at both sides of the slot. The rectangular septum has offset from the centerline. Furthermore, the symmetry plane of the iris and the septum bisects the slot. The feeding waveguide is also a reduced-height rectangular waveguide with an inside cross-section of 72 mm × 8 mm.

Fig. 1. Dumbbell-shaped slot antenna fed by the rectangular waveguide including its parameters.

Because the dumbbell-shaped slot antennas have many parameters, it is not possible to investigate the characteristics of slot antennas without assigning values to some of their parameters. Please note that there is no frequency limitation on the proposed dumbbell-shaped slot antennas and they can be employed to design different arrays of slots at various frequencies. For this reason, hereafter the listed dimensions in Table 1 are considered to study the characteristics of the proposed dumbbell-shaped slot antennas. Besides, all the following simulations are done at 3 GHz.

Table 1. Dimensions of the proposed dumbbell-shaped slot antennas

It has been found that in the proposed dumbbell-shaped slot antennas, the septum length depends on the iris depth. To derive the appropriate septum length versus the iris depth, the procedure which has been offered in [Reference Moradian49] was followed. Figure 2 shows the applicable septum length versus the iris depth. Overall, only the iris depth and the slot length are selected as variables to control the characteristics of the proposed dumbbell-shaped slot antennas.

Fig. 2. Proper septum length versus the iris depth.

Deriving the design equations and studying their validity

The slotted array antennas consist of the longitudinal slots can be designed by employing two main design equations and following the iterative design procedure proposed by Elliott [Reference Elliott and O'Loughlin40]. The provided design equations in [Reference Elliott1, Reference Elliott and O'Loughlin40] cannot be directly employed to design an array composed of the proposed dumbbell-shaped slot antennas. So, it is crucial to obtain two new design equations similar to the design equations specifically proposed for the round-ended longitudinal slot antennas [Reference Elliott1].

The design equations which have been presented by Elliott relay on one main assumption that the longitudinal slot antennas fed by the rectangular waveguides can be considered as an equivalent normalized admittance on a transmission line. Similarly, to establish the design equations for the proposed dumbbell-shaped slot antennas, first, it has to be shown that considering the antennas as a shunt admittance on a transmission line is valid. Hence, a model of the proposed dumbbell-shaped slot antennas was drawn in Ansys high-frequency structure simulator (HFSS). Two waveguide ports have been assigned to the model. The model has been bounded by some radiated boundaries.

The simulation of the model was done for various depths of the iris and different lengths of the slot. Then, the amplitudes of the forward- and backward-scattered waves were derived. The amplitudes and the phases of the forward- and backward-scattered waves for various iris depths and slot lengths are listed in Table 2. According to the table, for some slot lengths, the difference between the amplitude and the phase of the scattered waves is negligible. So, it is possible to model the proposed dumbbell-shaped slot antennas as an equivalent shunt admittance on a transmission line if one chose the slot length in an appropriate interval. However, for the slot length far from this appropriate interval, the shunt admittance modeling is a weak assumption.

Table 2. Scattered forward- and backward-mode amplitude and phase for various slot lengths and iris depths

As the table shows, for some iris depths, the phase of the scattered waves varies a lot even for small changes of the slot length. Please note that the proposed slot antennas are inherently narrowband. So, it is expected to see a large variation in the amplitude and the phase of the induced propagating waves. In [Reference Stern and Elliott50], a similar large variation in the amplitude and the phase of the scattered waves is reported. It is known that in the slotted array antennas, the slot lengths are a few percents far from their self-resonant lengths [Reference Elliott1]. So, the shunt admittance model can be employed to design different types of arrays consist of the proposed dumbbell-shaped slot antennas without much concern about the invalidity of the considered equivalent shunt admittance model.

It has been found worthy to study the transverse electric field distribution in the slot aperture [Reference Stern and Elliott50]. For simulation, one-watt of electromagnetic power has been delivered to the antenna by one of the Wave-Ports, and the electric field in the slot was derived for two different iris depths. These iris depths were selected such that they lead to strong or weak radiations by the slot. Figure 3 shows the amplitude and the phase of the transverse electric fields in the slot against the normalized slot length for the iris depth equal to 2 and 10 mm and when the slot length was equal to 37.8 mm. The figure shows that the amplitude of the electric fields is almost constant in the middle part of the slot. It starts to drop sharply at the beginning of the circles and approaches zero at the ends of the slot. The phase of the transverse electric fields is almost constant in the most portion of the slot.

Fig. 3. Amplitude and the phase of the transverse electric field distribution against the normalized slot length for iris depths equal to 2 and 10 mm.

Figure 3 also shows that the amplitude of the induced voltage (the integration of the transverse electric field along the width of the slot) for the iris depth equal to 10 mm is much larger than the induced voltage for the iris depth equal to 2 mm. Because for both cases, the amplitudes of the transverse electric fields are symmetrical in the slot aperture while their phases are more or less constant in the slot, it can be concluded again that the proposed dumbbell-shaped slot antennas at least at resonance can be represented as a shunt admittance on a transmission line [Reference Stern and Elliott50].

To derived the design equations, the procedure proposed in [Reference Moradian, Tayarani and Khalaj-Amirhosseini33] has been followed. First, the previously defined model in HFSS is developed by adding a lumped port to the middle of the slot. The model, including its details, is shown in Fig. 4. When the lumped port is excited, an electric field is developed in the slot aperture that in turn, this electric field excites two outgoing propagating electromagnetic waves in the waveguide. The propagating waves and their polarization (although many electromagnetic modes are excited around the slot, it is supposed that only the dominant mode has appeared at output ports) are illustrated in Fig. 4. The developed transverse electric fields have been studied for the iris depths equal to 2 and 10 mm. It has been found that the discrepancy between the characteristics of the developed transverse electric fields along the slot was caused by both the impressed source (current model) and the internally incident wave that passed the slot (previous model) was negligible. The simulation results have also shown that for both cases, the phase of the fictional electric fields (the electric field which is caused by the lumped port) is almost constant. These facts indicate that the model can be employed to find the mathematical relationship between the slot voltage and the amplitudes of the forward- and backward-scattered waves.

Fig. 4. Employed model to find the relation between the slot voltage and the amplitude of the induced waves.

The simulation of the model, shown in Fig. 4, has been carried out when the iris depth and the slot length were swept from 2 to 10 mm and 34 to 47 mm, respectively. Then, the scattering parameters were saved and analyzed. The slot voltage and the forward- and back-ward-scattered wave amplitudes were derived from the scattering parameters of the model [Reference Moradian, Tayarani and Khalaj-Amirhosseini33]. Finally, the following mathematical relation between the slot voltage and the amplitudes of the scattered waves of the nth slot was derived.

(1)$$\begin{align} B_{10}^n & = C_{10}^n = {-}Kf_n( {x_n, \;l_n} ) V_n^s \cr & = 1.493 \times 10^{{-}5}( {0.00247x^2 + 0.06x + 0.157} ) ( 0.0011l^2 \cr & \quad + 0.0202l-0.0813 ) V_n^s \end{align}$$

In the above equation, x represents the iris depth, and l represents half of the slot length. Furthermore, $B_{10}^n$ and $C_{10}^n$ represent the amplitude of the forward- and backward-scattered waves. Besides, $V_n^s$ is the voltage of the nth slot, and f _n is a function of the iris depth and the slot length. In [Reference Moradian51], a design procedure was proposed to design slotted array antennas covered with a dielectric slab. Although the proposed design procedure was mainly aimed to design an array of the dielectric covered slotted arrays, without serious limitation, it can also be employed to design different types of slotted arrays including arrays consist of the dumbbell-shaped slot antennas. The design procedure uses two main design equations [Reference Moradian51]. The first design equation is

(2)$$\displaystyle{{Y_n^a } \over {G_0}} = K_1f_n( {x_n, \;l_n} ) \displaystyle{{V_n^s } \over {V_n}}$$

where V _n is the mode voltage in the equivalent network. $Y_n^a {\rm /}G_0$ is the equivalent normalized active admittance of the nth slot [Reference Moradian51]. Finally, K ₁ can be derived as

(3)$$K_1 = \sqrt {4K^2\omega \mu _0\beta _{10}/G_0} $$

where ω is the angular frequency. μ ₀ is the free space permeability. β ₁₀ is the phase constant of the dominant mode, and G ₀ is the characteristic admittance of the equivalent transmission line. Similarly, the second design equation can be derived from the procedure proposed in [Reference Moradian51]

(4)$$\displaystyle{{Y_n^a } \over {G_0}} = \displaystyle{{K_2f_n^2 } \over {Y_n^{{a}^{\prime}} }}$$

where K ₂ is equal to 4K ²ωμ ₀β ₁₀, $Y_n^{{a}^{\prime}}$ represents the active admittance of the nth loaded slot in the equivalent slotted array embedded in a very large ground plane.

To study the validity of design equations, the equivalent normalized active admittance of the proposed dumbbell-shaped slot antennas cut in the broad walls of two nearby rectangular waveguides are derived by HFSS and compared with the equivalent normalized active admittance which is derived by Eq. (4). Figure 5 shows the model which is used to obtain the equivalent normalized active admittance in HFSS. As seen by the figure, the slots are parallel and non-staggered (i.e. the longitudinal distance between the slots referred to a place bisects the slots is zero). The slot antennas are similar (i.e. all dimensions are the same). The slot lengths are equal to 38.44 mm, and the iris depths are equal to 8 mm. These selected dimensions ensure that each of the slot antennas resonates when the other slot is covered with a conducting tape or when the distance between the slots approaches infinity. The equivalent normalized self-admittances of the slot antennas are equal to 0.47. Four Wave-Ports were assigned to the model, and they have been de-embedded to the middle of the slot antennas. The simulation of the structure has been carried out when the spacing between the slots was swept from 76 mm to 526 mm. From the scattering parameters of the model, it is an easy task to find the equivalent normalized active admittance of the parallel slots [Reference Moradian51].

Fig. 5. Employed model to study the mutual coupling between the parallel slots.

To find the equivalent normalized active admittance from Eq. (4), a model which composed of a large circular conducting plane was drawn in HFSS IE. Two dumbbell-shaped slots were also cut into the conducting plane. Two lumped ports fed the slots at their middles and the model was parameterized. Figure 6 shows the model, including its different parameters. The model was then simulated to find the mutual admittance between the slots while the distance between the slots was swept from 76 mm to 526 mm. It is worthy to mention here that the author has only used HFSS IE to derive the mutual admittance between the slots shown in Fig. 6 and all the other simulations have been carried out using HFSS. For any distance between the slots, the minimum spacing between the edge of the plane and the slots was always greater than seven times of the free-space wavelength (for our case 700 mm). Then, the estimated mutual admittance between the slots and the equivalent normalized self-admittance of the slots were used in Eq. (4) to find the equivalent normalized active admittance. Figure 7 shows the derived equivalent normalized active admittances against the normalized distance between the slots when the slots are parallel and non-staggered. As seen by the figure, the difference between the two approaches is negligible.

Fig. 6. Two parallel dumbbell-shaped slot embedded in the large circular ground plane.

Fig. 7. Variation of the normalized active admittance of the coupled slot antennas versus the normalized distance when the slots are non-staggered.

The defined models have been used again for comparing the normalized equivalent active admittance of the slot antennas when the slots are staggered by 0.5 and 1.0 guided wavelength. Figures 8 and 9 show the derived normalized equivalent active admittances of the slot antennas when they are staggered by one-half and one of the guided wavelength, respectively. As seen by the figures, the agreement between the results associated with two approaches is satisfactory. Based on the performed discussion, it can be concluded that Eq. (4) can be employed to predict the equivalent normalized active admittance of any slot in an antenna array consists of the proposed dumbbell-shaped slots with acceptable accuracy.

Fig. 8. Variation of the normalized active admittance of the coupled slot antennas versus the normalized distance when the slots are staggered by one-half of the guided wavelength.

Fig. 9. Variation of the normalized active admittance of the coupled slot antennas versus the normalized distance when the slots are staggered by one guided wavelength.

Designing the slotted arrays of the dumbbell-shaped slot antennas and the results

To demonstrate the validity and effectiveness of the proposed dumbbell-shaped slot antennas, some slotted arrays have been designed. First, two standing wave slotted arrays consist of four waveguides and each waveguide contains four slots that have been designed and simulated. For the first array, all the antenna elements are centerline round-ended slots [Reference Moradian49]. For the second array, the radiated elements are the proposed dumbbell-shaped slot antennas. For designing the arrays, the equivalent normalized self-admittance of the isolated slots (for both round-ended and dumbbell-shaped) must be known for every combination of the slot length and iris depth. For the round-ended slot, some universal curves have been derived that enable one to easily find the equivalent normalized self-admittance of the isolated slots [Reference Moradian49]. Similarly, For the dumbbell-shaped slot antennas, it has been found that the equivalent normalized self-admittance of the isolated slots can be represented by some universal curves.

To find the universal curves for the dumbbell-shaped slot antennas, the predefined model in HFSS (Fig. 1) has been employed while the slot length and the iris depth in the model were swept from 17 to 23.5 mm and 2 to 10 mm, respectively. Then, the equivalent normalized shunt admittances, the resonant length, and resonant conductance of the dumbbell-shaped slot antennas for every combination of the slot lengths and iris depths were derived [Reference Moradian49]. Figure 10 shows the resonant length times the free space wavenumber versus the iris depth. It is worthy to note that the resonant length of the proposed dumbbell-shaped slot antenna is much smaller than one-half of the free-space wavelength.

Fig. 10. Resonant length multiplied by the free space wavenumber versus the iris depth.

Figure 11 shows the resonant conductance versus the iris depth. As seen from the figure, the resonant conductance is an ascending function of the iris depth. The equivalent normalized self-conductance and susceptance of the proposed dumbbell-shaped slot antennas versus the normalized slot length, and for various iris depths are shown in Fig. 12. For ease of programming, two curves were also fitted over the data.

Fig. 11. Normalized resonant conductance versus the iris depth.

Fig. 12. Normalized conductance and susceptance of the proposed slot antennas versus the normalized slot length.

The arrays have been designed by employing the derived design equations and following the iterative procedure explained in [Reference Elliott1, Reference Lo and Lee4, Reference Moradian51]. A tilted slot in the primary feeding waveguide is used to feed each branch of the antennas. The input power is distributed by the ratio of 1:4.2:4.2:1 between the branches of the antennas. The prescribed power distribution has been achieved by tuning the tilts of the inclined slots while assuming that all the radiating slots are eliminated and the branch lines are terminated to matched loads.

The arrays have been designed to have Dolph–Chebyshev voltage distribution in both E- and H-planes with a sidelobe level of −20 dB in both planes. The top view of the first and the second design arrays are shown in Figs 13 and 14, respectively. At first glance, one can see that the distances between the edges of the coupling slots and edges of the nearest radiated slots are larger for the second array in comparison to the first array. This fact ensures that the internal higher-order modes exciting at the positions of the inclined slots will be suppressed enough before reaching to the neighboring radiating slots.

Fig. 13. Top view of the designed array consists of the round-ended slot antennas.

Fig. 14. Top view of the designed array consists of the proposed dumbbell-shaped slot antennas.

Figure 15 illustrates the reflection coefficients of the designed arrays versus frequency. The figure shows that for the first array, only in a small portion of the frequency band, the reflection coefficient is better than −10 dB. Whereas for the second array, the reflection coefficient is better than −10 dB from 2.95 to 3.05 GHz.

Fig. 15. Reflection coefficients of the designed arrays versus frequency.

The co-polarization and cross-polarization radiation patterns of the arrays in both H- and E-planes are demonstrated in Figs 16 and 17, respectively. As the figure shows, the agreement between the co-polarization radiation patterns is good. The cross-polarization radiation patterns of the designed antennas are also depicted in the figures. One can see from the figures that the discrepancy between the level of the cross-polarizations for the first and the second arrays is maximum at the direction of the main beam. However, the level of the cross-polarizations for both arrays at the direction of the main beam is better than −40 dB which is acceptable for most applications.

Fig. 16. Radiation patterns of the designed antennas in the H-plane.

Fig. 17. Radiation patterns of the designed antennas in the E-plane.

Although, the specifications of the designed slotted array antenna consists of the proposed dumbbell-shaped slots has been satisfactory, it was costly to construct a prototype. Hence, a smaller slotted array antenna has been designed to be implemented. The array is a standing wave array antenna contains eight dumbbell-shaped slot antennas. The array is designed to have Dolph–Chebyshev voltage distribution in H-plane with a sidelobe level of −20 dB. It also operates at S-band with the center frequency equal to 3 GHz. Figure 18 shows the different parts of the designed antenna, together with its various dimensions. As seen by the figure, the antenna has two waveguides, and there are four slots in each waveguide. An SMA connector feeds the main waveguide near one end. Then, the exciting electromagnetic wave feeds the branch line waveguides via the coupling slots. Photographs of different parts of the implemented antenna are shown in Fig. 19. The figure shows the feeding waveguide, the antenna body (branch line waveguides), and the complete assembled antenna.

Fig. 18. Different parts of the designed array together with its dimensions.

Fig. 19. Photograph of the different parts of the implemented antenna.

Figure 20 shows the simulated and the measured reflection coefficient of the constructed antenna. As the figure shows, except for some frequency shift of the measured reflection coefficient, the results are in a good agreement. The measured reflection coefficient is better than −10 dB from 2760 to 3066 MHz. Whereas, the simulated reflection coefficient is better than −10 dB from 2855 to 3125 MHz. In addition, the measured reflection coefficient of the implemented antenna at the design frequency (3 GHz) is better than −12 dB. As the figure shows, an unwanted resonance appears in both the simulated and the measured reflection coefficients. One can see from the figure that there is a frequency shift between the unwanted resonances while the unwanted resonance in the measurement result is deeper.

Fig. 20. Simulated and measured reflection coefficients.

The radiation patterns of the constructed antenna have been measured on the outdoor antenna test range. The simulated and the measured H-plane radiation patterns of the designed antenna are shown in Fig. 21. As seen from the figure, the agreement between the simulated and the measured radiation patterns is good. There is a small asymmetry in the level of the simulated innermost side-lobes, whereas the measured level of the innermost side-lobes is almost symmetrical. The figure also shows that for the innermost side-lobe, the prescribed specification for the side-lobe level is achieved. The simulated cross-polarization radiation pattern of the antenna is added to the figure. As the graph shows, the level of the cross-polarization at the direction of the main beam is approximately equal to −40 dB. The measured cross-polarization level at the direction of the main beam was approximate −26 dB below the co-polarization level. The difference between the simulated and the measured levels of the cross-polarization is due to the limited dynamic range of the available detector, which was used during the measurement.

Fig. 21. Simulated and the measured radiation patterns of the implemented antenna in the H-plane.

The simulated and the measured radiation patterns of the designed antenna in E-plane are illustrated in Fig. 22. Although, the antenna is symmetrical, there is a small asymmetry in the radiation pattern which is due to the unequal power split between the antenna branches. As seen from the figure, the measured radiation pattern fit the simulated radiation pattern well. The simulated cross-polarized radiation pattern of the designed antenna is also added to the figure which is approximately −40 dB below the co-polarization level at the main beam direction. The gain of the constructed antenna was measured, which was roughly equal to 13.5 dBi. The simulated gain of the antenna was equal to 14.1 dBi. The results and the performed discussion indicate that the proposed dumbbell-shaped slot antennas can be employed to design different types of slotted arrays. This enables a designer to design various types of arrays without going through some tedious formulations or relying on trial-and-error procedure. The trial-and-error procedure usually requires the construction of several prototypes and near field measurements.

Fig. 22. Simulated and the measured radiation patterns of the implemented antenna in the E-plane.