Energy Patterns of UWB Antenna Arrays with Low Side Lobe Level During Beam-Scanning Marco A. Panduro1 and Heinrich Foltz2 1. Unidad Académica Multidisciplinaria Reynosa-Rodhe, Universidad Autónoma de Tamaulipas (UAT) Carretera Reynosa-San Fernando, Reynosa, Tamaulipas, 88779 México Phone: (52) 899.921.3300, Fax: (52) 899.921.3301, Email:
[email protected];
[email protected] 2. University of Texas-Pan American EE Dept., UTPA, 1201 W. University, Edinburg, TX 78541 Phone: 956-384-5016(W), 956-381-2609(Dept), 956-380-0839(H), 956-381-3527(Fax), Email:
[email protected] Abstract—A synthesis of energy patterns for UWB antenna arrays is presented in this paper. This synthesis regards the optimization of the positions of the antenna elements in a linear geometry. The well-known method of Differential Evolution (DE) is utilized for this optimization problem. A single Vivaldi antenna is simulated by using the commercially available CST Microwave Studio software. The electric field for the array is determined in according to the superposition principle. Simulation results for the optimized non-uniform array are provided. Keywords: energy patterns, linear array, differential evolution.
I.
INTRODUCTION
This paper presents the time domain design of UWB antenna arrays in order to generate energy patterns with minimum side lobe level during beam scanning. This UWB antenna array design considers the optimization of the positions of the antenna elements in a linear geometry. The well-known method of Differential Evolution [1]-[2] is utilized for this optimization problem. A single Vivaldi antenna is simulated by using the commercially available CST Microwave Studio software. The radiated waveform of the signal is calculated on the basis of superposition principle during the optimization process. To the best of the authors’ knowledge the application of an evolutionary optimization technique to obtain energy patterns for an UWB linear array with minimum side lobe level during beam scanning has not been presented previously. II.
VIVALDI ANTENNA CHARACTERISTICS
The Vivaldi antenna that is used as an array element is shown in Fig. 1. The substrate of this antenna is FR4 with relative permittivity r=4.35 and 1.5 mm thickness. According to [3], this Vivaldi antenna presents a |S11| less than -10 dB over the 3-11 GHz frequency band. The fifth derivative of Gaussian pulse that complies with the FCC mask for indoor UWB applications is selected as an excitation pulse.
III.
PROBLEM STATEMENT
Consider N similar Vivaldi antennas with the same amplitude and phase arrayed in the H-plane along the Z axis. The position of the nth element is dn as shown in Fig. 2. For an observation point in the far-field region, each element is visible with the same angle of θ. Therefore, the far-field waveform according to the principle of superposition for a scanned angle at θmax is given by [4]: , ,
,
;
;
1
where c is the light speed and τn is the relative excitation time delay to steer the maximum beam direction [4]. The energy pattern in the θ direction for a scanned angle at θmax is given by [5]: |
,
|
, ,
2
Therefore, the optimization problem can be formulated as minimize the next objective function (of): 3
where SLLmax is the maximum level of the side lobes considering the set of angles where the energy pattern is steered, and θgen is the direction of maximum radiation obtained. In this case of linear geometry, to create an array that will have reduced side lobes during scanning, one must optimize the array when it is steered to the maximum desired scan angle, i.e., the array is steered between broadside and the maximum scan angle, and the side lobes will be bounded by the maximum SLL at the furthest scan angle. x r1
d1
τ1
r2
θ z
r3
rn
r4
d2
d3
d4
dn
τ2
τ3
τ4
... τn
rN
dN
... τN
s(t)
Fig. 1. Vivaldi antenna
Fig. 2. UWB time-delay array for beam scanning in θmax
IV.
SIMULATION RESULTS
The method of DE was implemented to study the behavior of the energy pattern for the linear array. The details of DE as optimization procedure can be followed in [1]-[2]. The parameters for DE are set based mainly on our previous experience in solving similar problems [6]. The population size is set to 50 and the value of F is set to 0.5. The stopping criterion is 500 iterations. In this design case, the FDTD method by CST Microwave Studio software is applied to obtain the electric field (far-field) for the Vivaldi antenna. Probes were set in a range of θ=[0˚, 180˚] with an angular step of 2.5˚ in order to obtain the electric field of the antenna element. The electric field (far-field) for the antenna array is determined considering the linear geometry in according to the superposition principle by applying Equation 1, and the energy pattern is calculated by using Equation 2. The number of antennas was set as N=8. Figure 3 illustrates the case when the linear array is optimized in order to generate an energy pattern steered between broadside and the maximum scan angle of 142.5˚.
Energy pattern (normalized) (dB)
0
-5
-10
-15
-20
-25
-30
0
20
40
60
80
θ
100
120
140
160
V.
180
REFERENCES [1]
0
[2]
-5
[3] -10
[4]
-15
[5] -20
[6] -25
-30
0
20
40
60
80
θ
100
CONCLUSIONS
This investigation reported the design of UWB arrays of Vivaldi antenna in time domain in order to generate energy patterns with low side lobe level during beam scanning. In this case, the optimization of the positions of the antenna elements was considered in a linear geometry. The method of Differential Evolution and the superposition principle were utilized for searching the best element positions. The energy pattern of the non-uniform linear array presented a lower SLL in all scanning range remaining the same length of the array. The maximum SLL generated by the non-uniform linear array was around -6.5 dB in all scanning range, while the energy pattern of the uniform case presented around of -2.5 dB. Although only simulation results are presented in the paper this study could be a basis for designing an antenna array in future work, or more extensive simulations including mutual coupling.
(degrees)
a)
Energy pattern (normalized) (dB)
In this case, the positions of the antenna elements found by DE for the non-uniform linear array were: (d1=0, d2=34.5792, d4=76.5492, d5=96.3492, d6=113.1242, d3=55.8792, d7=127.4792, d8=142.6697) mm. Figure 3 shows the energy pattern obtained by the DE method for the non-uniform case and the uniform case considering the same length obtained by the non-uniform array, i.e., a separation between antenna elements of 20.3813mm to generate a length of 142.6697 mm in order to make a fair comparison. As it can be seen in this Figure, the non-uniform linear optimized by DE generates an energy pattern with a better performance in terms of the side lobe level with respect to the uniform case. The energy pattern of the non-uniform linear array presents a lower SLL in all scanning range remaining the same length of the array. The maximum SLL generated by the non-uniform linear array is around -6.5 dB in all scanning range, while the energy pattern of the uniform case presents -2.5 dB.
120
140
160
180
(degrees)
b) Fig. 3. Energy pattern normalized for the linear array: a) Uniform case, b) Nonuniform case obtained by the DE method.
Kurup, D., Himdi, M., and Rydberg, A.: Synthesis of uniform amplitude unequally spaced antenna arrays using the differential algorithm, IEEE Trans. Antennas Propagation, 2003, Vol. 51, pp. 2210-2217. Feortisov V., Janaqui S.: Generalization of the strategies in differential evolution. Proceedings of the IEEE Conference Evolutionary Computation, 1996. commercially available Antenna Magus software www.antennamagus.com Chao-Hsiang Liao, Powen Hsu and Dau-Chyrh Chang, "Energy patterns of UWB antenna arrays with scan capability," IEEE Transactions on Antennas and Propagation, Vol. 59, No. 4, pp. 1140-1147, April 2011. J. S. McLean, H. Foltz and R. Sutton, "Patterns descriptors for UWB antennas," IEEE Transactions on Antennas and Propagation, Vol. 53, pp. 553-559, January 2005. M. A. Panduro, Carlos A Brizuela, Luz I Balderas and Diana Acosta, “A comparison of genetic algorithms, particle swarm optimization and the differential evolution method for the design of scannable circular antenna arrays”, Progr. in Electro. Re. B, 13, pp 171-186 (2009).
