# Design of Concentric Ring Antenna Arrays Based on Subarrays to Simplify the Feeding System

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## Abstract

**:**

## 1. Introduction

## 2. Array Pattern with Cophasal Subarrays

_{r}is the number of circular rings in the array, N

_{n}is the number of antenna elements in the nth circular ring, I

_{nm}and α

_{nm}are the amplitude and phase excitations of the mth antenna element for the nth circular ring, respectively, r

_{n}is the radius of nth circular ring, k is the propagation number, θ is the elevation angle and φ is the azimuth angle. The main beam is scanned to (θ

_{0},φ

_{0}) by applying a cophasal excitation, therefore, α

_{nm}is given by:

_{1}= 4, N

_{2}= 6 and N

_{3}= 8. This configuration uses seven subarrays created strategically considering cophasal excitation. The behavior of the cophasal excitation values in the elevation plane based on Equation (2) is illustrated in Figure 4. This figure illustrates the phase value required by each antenna element to scan the main beam in θ

_{0}= [−40°, 0°, 20°, 40°, 60°] in the elevation plane in the cut of φ = π. This behavior for this array geometry illustrates that one same phase value (elements with similar phase values) can be assigned to several antenna elements, as shown in Figure 4b. This can be taken as an advantage for grouping the antenna elements in subarrays and reducing the number of phase shifters employed in the antenna system. Therefore, the groups or subarrays are set considering the cophasal excitation required for scanning the main beam; then, this subarray configuration can be named cophasal subarrays configuration. In the case of Figure 2, the amplifiers can be considered variable or fixed during beam scanning. Figure 3 illustrates the subarray configuration proposed for the case of using a distribution of raised cosine amplitude excitation [2].

_{2}, n

_{4}, n

_{13}, n

_{17}} represent zero phase values for all scanning directions. In this case, these antenna elements do not require a phase shifter (or these antenna elements could be set in a subarray fed by one phase shifter). The next subarrays that present the same phase value (or similar values) are groups of three elements, the cases of {n

_{1}, n

_{6}, n

_{10}}, {n

_{3}, n

_{7}, n

_{9}}, {n

_{5}, n

_{12}, n

_{18}} and {n

_{8}, n

_{14}, n

_{16}}, and finally two groups of one antenna element: {n

_{11}} and {n

_{15}}. The power delivery network can be considered by using a cascaded unequal dividers network [2] (usually with Wilkinson divisors). The power and phase in each element of a same subarray (in our configuration of cophasal subarrays) must be equal. This is achieved by using 1-to-3 and 1-to-4 equal power dividers.

_{0}≤ 40° (with a peak side-lobe level of −15 dB) could be generated maintaining the same subarray configuration, i.e., by just varying the value of the phase shifter for each subarray: α

_{1’}, α

_{2’}, α

_{3’}, α

_{5’}, α

_{6’}, α

_{7’}. This configuration represents the case of using the radii values of r

_{1}= 0.5λ, r

_{2}= 1.00λ and r

_{3}= 1.52λ. These radii values were obtained using our previous experience in antenna arrays optimization [25,26,27,28,29,30]. Although it is obvious that different array configurations and scanning angle (such as different radii values) result in different element phase distribution, the same subarray configuration will be retained for a specific array configuration and scanning angle range. This condition is retained because the phase values for each subarray are assigned in a perpendicular way to the scanning elevation plane (x-z plane), i.e., although the phase value changes for each scanning direction the same groups or subarrays are obtained. As commented previously, this is the reason why we chose concentric rings as the array geometry. The behavior of the cophasal excitation of this structure facilitates the generation of subarrays.

## 3. Results and Discussion

_{1}= 4, N

_{2}= 6 and N

_{3}= 8). Differential evolution was applied to find the values of the amplitude and phase excitations to generate a scannable array pattern over the elevation plane. We followed the literature and our previous results to set the parameters for the optimization algorithm. We considered the design cases of variable and fixed amplifiers. In addition to these design cases, we make a comparison to the case in which a raised cosine distribution (Figure 3) across the antenna array is used.

#### 3.1. Array Factor Results

_{0}= 30°. The minimum value of side-lobe level (in the scanning range) was −20.29 dB found in θ

_{0}= 40°. There was a reduction in the side-lobe level of almost 10 dB with respect to the conventional array of cophasal excitation, and almost 8 dB with respect to the case using cophasal subarrays without optimization. The optimized case of cophasal subarrays provides a scanning range over the elevation plane of [−45°, 45°] for a peak side-lobe level below −15 dB, and a range of [−50°, 50°] for a peak side lobe level of −10 dB.

_{0}= 0°, (b) θ

_{0}= 30° and (c) θ

_{0}= 40°. The method of differential evolution efficiently computes the values of the design variables to generate an array factor with characteristics of low side-lobe level without pattern distortion during beam steering.

_{0}= 30° and θ

_{0}= 40°. The minimum value of the amplitude excitation was 4.46 and the maximum value was 10.98 for the case of variable amplifiers, and 7.18 and 13.66, respectively, using fixed amplifiers. In this case, the amplitude weights could be realized in a real hardware setup rescaling all the complex coefficients before applying them to a real system. In a real system, the amplitudes will be adjusted at every layer in such a way that the conservation law of the energy will be preserved. Furthermore, it is interesting to note that the values of phase excitations obtained follow the behavior of a phase slope, i.e., a progressive phase excitation is obtained similar to phase excitation for linear arrays. This is because the phase values for each subarray are assigned in a perpendicular way to the scanning elevation plane (x-z plane).

#### 3.2. CST Microwave Studio Simulation

_{r}= 4.2, μ = 1 and 0.025 of tangential losses, corresponding to FR4 values. We considered full-wave analysis of the feeding network for the phase and amplitude values found in the optimized design (without post-processing the results of the full-wave analysis of the antenna array).

_{0}= −40°). The active reflection coefficient (or scan reflection coefficient) is the reflection coefficient for a single antenna element in the array antenna, in the presence of mutual coupling. The active reflection coefficient is a function of frequency, scanning direction and the excitation of the neighboring elements [32]. Every scanning direction was examined (of the scanning range) far from the natural radiation response of the array. This was challenging, especially for the farthest directions (from the natural radiation response of the array), but the active reflection coefficient of 18 antenna elements remained below −10 dB for all scanning directions [−40°, 40°] at 2.4 GHz.

_{0}= −40°. As was expected, the elements show a good matching performance. Furthermore, the reflection coefficient of elements is lower than −10 dB in the frequency of interest, which is acceptable for many applications. This behavior remained around the central frequency of the array from 2.37 to 2.41 GHz with 40 MHz of bandwidth.

_{0}= −40°, −30°, 20° and 40°. As shown in Figure 9, the side-lobe level response was maintained at low values, i.e., a side-lobe level of −19.76 dB with CST and −18.75 dB with HFSS at the most distant scanning directions considering the antenna element based in the circular patch with the characteristics mentioned previously. The two pattern responses (obtained using CST and HFSS) are very similar.

_{0}= −40°, θ

_{0}= 0° and θ

_{0}= 40°. These simulation results demonstrate that for the proposed configuration of concentric rings array using cophasal subarrays for grouping the elements, the technique of differential evolution shows that the values of the amplitude and phase excitations feeding each cophasal subarray provide a low side-lobe level.

#### 3.3. Manufacturing Tolerances

_{0}= −40°) considering a tolerance of 5%. Figure 11 illustrates that the error is directly proportional to the value of the amplitude and phase excitation, i.e., we have a higher value of error for higher values of amplitude and phase. The maximum error interval is found in the antenna element n

_{15}for the amplitude excitation (from 10.43 to 11.52, $\pm $0.54), and, in the elements n

_{11}and n

_{15}for the phase ($\pm $0.32 radians $\approx \pm $18°).

_{0}= −40°, and for a scanning direction in θ

_{0}= 20° (Figure 12c) with 5% tolerance. As shown in Figure 12, the side-lobe level response remained below a peak value of −17.04 dB for the most sensitive case.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Subarrays configuration in a concentric ring antenna array (3 rings configuration with r

_{1}= 0.5 λ, r

_{2}= 1.00 λ and r

_{3}= 1.52 λ) to scan the main beam in the elevation plane.

**Figure 3.**Previous subarrays configuration for the case of using a distribution of raised cosine amplitude excitation with fixed amplifiers.

**Figure 4.**(

**a**) Phase values calculated using Equation (2), (

**b**) phase values obtained using cophasal subarrays configuration in ${\theta}_{0}=$ [−40°, 0°, 20°, 40°, 60°]; (

**c**) Array factor comparison using cophasal subarrays.

**Figure 5.**Behavior of the side lobe level versus scanning direction for the concentric ring antenna array using cophasal subarrays with optimization (variable and fixed amplifiers) and its comparison with respect to the other design cases.

**Figure 6.**Array factor optimized with differential evolution and cophasal subarrays using variable and fixed amplifiers and its comparison with respect to the other design cases for scanning directions of (

**a**) θ

_{0}= 0°, (

**b**) θ

_{0}= 30° and (

**c**) θ

_{0}= 40°.

**Figure 7.**Values of the amplitude and phase excitations for the optimized cases using variable ((

**a**) θ

_{0}= 30° and (

**b**) θ

_{0}= 40°) and fixed ((

**c**) θ

_{0}= 30° and (

**d**) θ

_{0}= 40°) amplifiers.

**Figure 8.**Active reflection coefficient of all elements in the array with scanning direction in θ

_{0}= −40°: (

**a**) [S1, S2, …, S6], (

**b**) [S7, S8, …, S12] and (

**c**) [S13, S14, …, S18].

**Figure 9.**Radiation pattern obtained using CST and HFSS simulations for the configuration using cophasal subarrays for the case optimized by differential evolution.

**Figure 10.**3D Radiation pattern obtained using CST Microwave Studio simulation for the configuration using cophasal subarrays (and the circular patch) for the case optimized by differential evolution (scanning directions in θ

_{0}= −40°, θ

_{0}= 0° and θ

_{0}= 40°).

**Figure 11.**Error interval for the values of the amplitude (

**a**) and phase (

**b**) excitations with a tolerance of 5% and considering a scanning direction in θ

_{0}= −40°.

**Figure 12.**Array factor response for 300 iterations obtained by using the amplitude and phase values with an error tolerance of 1% (

**a**) and 5% (

**b**) for a scanning direction in θ

_{0}= −40°, and for a scanning direction in θ

_{0}= 20° (

**c**) with 5% of tolerance.

**Table 1.**Comparison of the CST simulation results (directivity and side-lobe level) with respect to the theoretical results for the analyzed scanning angles.

Main Beam Direction (θ_{0}) | Array Factor | CST | ||
---|---|---|---|---|

Side Lobe Level (dB) | Directivity (dB) | Side Lobe Level (dB) | Directivity (dB) | |

−40° | −20.29 | 11.99 | −19.76 | 12.23 |

−35° | −25.86 | 12.04 | −24.26 | 12.14 |

−30° | −26.87 | 12.22 | −27.83 | 12.28 |

−25° | −25.34 | 12.48 | −26.79 | 12.57 |

−20° | −22.99 | 12.60 | −23.40 | 12.67 |

−15° | −22.68 | 12.63 | −23.38 | 12.68 |

−10° | −23.31 | 12.61 | −23.78 | 12.61 |

−5° | −24.18 | 12.70 | −23.01 | 12.70 |

0° | −25.17 | 12.72 | −25.77 | 12.73 |

5° | −24.18 | 12.70 | −23.01 | 12.70 |

10° | −23.31 | 12.61 | −23.83 | 12.61 |

15° | −22.68 | 12.63 | −23.42 | 12.67 |

20° | −22.99 | 12.60 | −23.46 | 12.67 |

25° | −25.34 | 12.48 | −26.82 | 12.57 |

30° | −26.87 | 12.22 | −28.18 | 12.29 |

35° | −25.86 | 12.04 | −24.52 | 12.14 |

40° | −20.29 | 11.99 | −20.21 | 12.22 |

**Table 2.**A performance comparison of the design case based in cophasal subarrays for the geometry of concentric rings with respect to other array geometries published in state of art.

Geometry | Number of Elements | Number of Phase Shifters | Reduction of Phase Shifters (%) | Scanning Range | Peak Sidel Lobe Level | ||
---|---|---|---|---|---|---|---|

Cophasal subarrays | Concentric rings | 18 | 6 | 66% | ±40° | −20 dB | |

Conventional case | Concentric rings | 18 | 18 | 0% | ±39° | −15 dB | |

Uniformly subarrays [2] | Linear array | 36 | 12 | 66% | ±6° | −15 dB | |

Overlapped subarrays | Planar array | [3] | 80 | 4 | 95 % | ±7° | −20 dB |

[20] | 28 | 14 | 50% | ±24° | −15 dB | ||

28 | 7 | 75% | ±11° | −15 dB | |||

Randomly feeding network [2] | Linear array | 30 | 12 | 60% | ±14° | −15 dB | |

CORPS [33] | Linear array | 10 | 8 | 20% | ±30° | −19 dB | |

10 | 7 | 30% | ±20° | −16 dB |

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**MDPI and ACS Style**

Juárez, E.; Panduro, M.A.; Reyna, A.; Covarrubias, D.H.; Mendez, A.; Murillo, E.
Design of Concentric Ring Antenna Arrays Based on Subarrays to Simplify the Feeding System. *Symmetry* **2020**, *12*, 970.
https://doi.org/10.3390/sym12060970

**AMA Style**

Juárez E, Panduro MA, Reyna A, Covarrubias DH, Mendez A, Murillo E.
Design of Concentric Ring Antenna Arrays Based on Subarrays to Simplify the Feeding System. *Symmetry*. 2020; 12(6):970.
https://doi.org/10.3390/sym12060970

**Chicago/Turabian Style**

Juárez, Elizvan, Marco A. Panduro, Alberto Reyna, David H. Covarrubias, Aldo Mendez, and Eduardo Murillo.
2020. "Design of Concentric Ring Antenna Arrays Based on Subarrays to Simplify the Feeding System" *Symmetry* 12, no. 6: 970.
https://doi.org/10.3390/sym12060970