Yagi-Uda Antenna Optimization by Elipsoid Algorithm

Alisson N. Amaral Dept. Electrical Engineering

CEFET-MG Belo Horizonte, MG, Brazil

alissonamaral2@yahoo.com.br

Ursula C. Resende Dept. Electrical Engineering

CEFET-MG Belo Horizonte, MG, Brazil

resendeursula@des.cefetmg.br

Eduardo N. Gonçalves Dept. Electrical Engineering

CEFET-MG Belo Horizonte, MG, Brazil eduardong@des.cefetmg.br

Abstract— This work presents a mono-objective optimization for a Yagi-Uda antenna applying the ellipsoid algorithm. For problems which the objective function is not known, like function that represents the Yagi-Uda antenna behavior, the ellipsoid algorithm has some features that make it more interesting compared to other optimization methods. To obtain antenna electromagnetic characteristics it is used the electric field integral equation (Pocklington integral equation) numerically evaluated by the Method of Moments (MoM). The ellipsoid optimization leads to antenna geometry with superior directivity than those presented in the available literature.

Keywords-Yagi-Uda antenn; method of Moments; optimization; ellipsoid algorithm.

I. INTRODUÇÃO With the telecommunications development has arisen the

need to seek alternatives for receiving radio frequencies that are efficient and inexpensive. Actually there is not a single criterion to define what kind of antenna is most suitable for reception of a especific radio frequency signal. The project depends on transmission technology and used frequency, availability of airspace where the signal travels, weather conditions, among others.

The Yagi-Uda antenna is a kind of directional antenna constituted of a set of parallel elements arranged along an axis. These elements are classified as: drivers elements, reflectors and directors. This antenna stands out because of its simple constructive caracteristcs, advanced studies and consolidated applications for VHF (Very High Frequency) and UHF (Ultra High Frequency). The main Yagi-Uda radiation caracteristics are: gain, directivity, input impedance, beamwidth, front-to-back ratio (FBR), the magnitude of side lobes, among others. The lengths and diameters of directors and reflectors elements, as well as their spacing, define the optimal values of these quantities. There is not a limit for number of directors. However, after a determinated number there is a reduction in the induced current in the most extreme elements, which justifies the limitation of numerous arrangements [1].

From the 1940s, projects for optimization of the Yagi-Uda parameters were conducted mainly in an experimental way. The first designs emerged from analytical studies. Later, with the advent of high performance computing, several numerical techniques based on analytical formulations have been used to obtain the geometric dimensions with better operational

performance. Ehrenspeck and Poehler in 1959 [2] proved experimentally that reflector element increases the gain. Later, in 1969, G. A. Thiele [3] developed an analytical method based on electric field integral radiation generated by elements of the arrangement. The method was used to describe the complex current distributions in the elements, the phase velocity, and its radiation characteristics. In 1973, Cheng and Chen [4] verified that a Yagi-Uda array can be optimized considering asymmetric spacing between elements. For all experiments in this project the directivity was improved. However the lengths and spacing were not varied simultaneously. In 2003, the autors of [5] proposed a multiobjective optimization for a Yagi-Uda array. By this process it was possivel to take into account various parameters simultaneously (maximum gain, impedance matching and others). In this study, were optimized the FBR, directivity, beamwidth and input impedance for 300 MHz with bandwidth of 5%. The results were higher than others availeble in the literature.

Although the Yagi-Uda antenna has been widely applied, further optimization studies may lead to efficient configuration to meet the requirements for other applications. In this work it is performed the mono-objective optimization for a Yagi-Uda antenna using an ellipsoid algorithm. This optimization method belongs to category methods of deleting half-spaces and for this kind of problem (objective function is not available) has some features that make it more interesting compared to other optimization methods. The optimized parameter is the directivity. To obtain antenna electromagnenétics characteristics it is used de electric field intergral equation (EFIE) (Pocklington integral equation) numerically evaluated by method of moments (MoM).

II. YAGI-UDA ANTENNA ELECTROMAGNETIC MODEL The electromagnetic characteristics of a Yagi-Uda antenna,

as that illustrated in the Fig. 1, are computed using the EFIE [1]:

( )

( )/2

/21 1

E , ,

, cos 2 14

e

e

z

LNE N

enLe n e

dzG

x y z

j zI r r nk L

η ππ −= =

′

=

−� �� �′′ −� �� � � ��

���, (1)

where

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Figure 1. Yagi Uda antenna with de 7 elements

( )1

I cos 2 1N

e enn e

zz I nL

π=

=� �� �′

−� �� � � ��

� , (2)

is the expansion for electric current density in segment e, Le is the segment e length, NE is the number of elements, N is the number of segments used to describe each element (in this work, was adopted for all elements N = 12), a is the dradius of the elements,

( ) ( )( ) ( )225, 1 2 3j k r

Ger r jkr r a kar

r

−� �′ = + − +�

, (3)

( ) ( ) ( )2 2 2r x x y y z z′ ′ ′= − + − + − , (4)

The expansion in (2) is chosen such that Ie(z) go to zero in the end of the segments and the azimuthal variation is being ignored, because the diameter is very small compared to the wavelength. Equation (1) is numerically evaluated by MoM using "point matching". After determining the coefficients Ien, the electric current given by (2) is applied in order to determine the radiation pattern of the antenna and input resistance [1].

III. ELLIPSOID ALGORITHM The ellipsoid method is effective in the treatment of

objective functions whose behavior is not completely known, like as the function that represents the Yagi-Uda antenna. In this case it is not possible to guarantee that there are areas with local or global minimal. The facility offered by the ellipsoid method is regarding the treatment of constraints, it uses an innovative concept to replace the gradient of the objective function by the gradient of the function of the most violated constraint, when a variable value is outside the feasible region. Given the initial values x0 and Q0, the ellipsoid algorithm is described by the following recursive equations [6]:

1

11k k kx x Q m

d+ = −+

� , (5)

2

1 2

21 1

Tk k k k

dQ Q Q mm Qd d+

� �= −� − + �� � , (6)

with

kTk k k

mmm Q m

=� , (7)

where x ∈ ℜd is the vector of optimization parameters and mk is the gradient (or sub-gradient) of the most violated constraint of g(x): ℜd � ℜs, when xk is not a feasible solution, or the gradient (or sub-gradient) of the objective function, f(x): ℜd � ℜ, when xk is a feasible solution. The finite difference method is applied to compute the gradient. The optimization algorithm ends when (fmax � fmin)/fmin � ε, where fmax and fmin are the minimum and maximum values of the objective function at the last Nε iterations and ε is the prescribed relative accuracy.

The optimization algorithm requires the choice of the initial entries of x0 and Q0. The vector x0 can be generated randomly. The matrix Q0 defines the initial ellipsoid size where the optimal solution will be searched. The convergence of the algorithm depends on the size of the initial ellipsoid and the prescribed relative accuracy. Obviously, there is a trade-off between the computational effort and the level of objective function minimization that defines the best choice of Q0, Nε and ε. Fig. 2 shows a block diagram that illustrates the optimization process realized by ellipsoidal algorithm.

Figure 2. Optimization process: ellipsoidal algorithm

IV. YAGI_UDA ANTENNA OPTIMIZTION To optimize the Yagi-Uda antenna, it was developed an

extension of the algorithm presented in [1] which is based on EFIE (Pocklington integral equation) numerically evaluated by MoM. This extension, called yagi_uda.m, was connected to ellipsoid algorithm as illustrated in the Fig. 3.

The optimization was performed in order to maximize directivity (mono-objective optimization). In the optimization process were considered 13 antennas with 3 to 15 elements, respectively. The optimized parameters are the distance (S),and

Le y

z

x

Input of variables and initial options

Startup offmim and fmax

Find the most

violated restriction

Stop test

Newfmim and

fmax

Find new ellipsoid: new x(k) and f(k)

Constraint function gradient

Objective function gradient

Gradient calculus

Optimal x and f(x)

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Figure 3. Connection between ellipsoidal and yagi_uda.m algorithms

the length (L), as illustrated in Fig. 1. The obtained results are presented in Table 1.

As it can be verified, as the number of elements increases, the antenna directivity increases and beamwidth decreases, as expected. The FBR and input impedance values are influenced by several factors, especially by the spacing between the elements. Since in the directivity optimization these parameters varied freely, they do not achieved optimal values. However as it can be seen by Table 1, 2 antennas (08 and 13 elements) presented a directivity and FBR greater than 10 dB, beamwidth less than 60 ° in E and H planes and input impedance between 45 and 55 Ω. These Parameters values are desirable for applications with this kind of antenna.

To evaluate the proposed optimization technique, the results obtained in this work are compared with others available in the literature. For a Yagi-Uda antenna with 6 elements the results are presented in Table 2, for antenna with 15 elements in Table 3, and optimized antennas dimensions in Table 4. For both configurations the obtained directivity by ellipsoid method is higher than those achieved by others optimization techniques. For 13 elements antenna the directivity across the 3% bandwidth is illustrated in the Fig. 4 and it can be observed that this parameter is not significantly modified.

TABLE I. OPTIMIZATION RESULTS

NE

E Plane H Plane Directivity

(dB)

Imput

Impedance (�)

Beam-width

(°)

FBR (dB)

Beam-width

(°)

FBR (dB)

03 52.31 7.19 62.49 7.18 9.86 23.23

04 44.43 7.33 49.94 7.32 11.58 38.01

05 38.24 5.21 41.49 5.20 12.29 38.76

06 36.76 11.19 39.58 11.17 13.87 95.90

07 34.50 12.72 36.79 12.70 14.42 27.59

08 33.52 10.97 35.62 10.95 14.53 46.63

09 27.73 10.84 28.86 10.81 15. 95 27.22

10 27.28 10.98 28.36 10.95 16.32 62.72

11 27.64 9.36 28.73 9.33 15.97 71.48

12 34.95 31.3 37.19 31.32 14.72 66.72

13 33.39 13.25 35.34 13.23 14.18 48.60

14 23.47 12.17 24.14 12.12 16.98 78.28

15 22.18 14.91 22.70 14.85 17.48 57.44

NE = Number of elements

TABLE II. YAGY-UDA ANTENNA WITH 6 ELEMENTS

Work Optimization Technique Directivity(dB)

Ref. [5] Multi objective with genetic algorithm 11.65

Ref. [7] Multi objective with genetic algorithm 12.66

Ref. [8] Spacing and lengths perturbation 13.41

This work Ellipsoidal algorithm 13.87

TABLE III. YAGY-UDA ANTENNA WITH 15 ELEMENTS

Work Optimization Technique Directivity (dB)

Ref. [9] Uncompensated optimization 14.12

This work Ellipsoidal algoritm 17.48

TABLE IV. YAGY-UDA ANTENNAS DIMENSIONS IN λ

6 Elements Antenna 6 Elements Antenna L1 0.4362 S1 0.2950 L1 0.4561 S1 0.3788

L2 0.4256 S2 0.3265 L2 0.4510 S2 0.3921

L3 0.4210 S3 0.3667 L3 0.4490 S3 0.4036

L4 0.4204 S4 0.3653 L4 0.4430 S4 0.4088

L5 0.4891 S5 0.1594 L5 0.4375 S5 0.4249

L6 0.5107 - L6 0.4445 S6 0.4307

- - L7 0.4473 S7 0.4272

- - L8 0.4431 S8 0.4301

- - L9 0.4373 S9 0.4874

- - L10 0.4386 S10 0.3911

- - L11 0.4229 S11 0.4391

- - L12 0.4213 S12 0.3977

- - L13 0.4385 S13 0.4057

- - L14 0.4867 S14 0.2263

- - L15 0.4293

0.97 0.98 0.99 1 1.01 1.02 1.0312

12.5

13

13.5

14

14.5

Dire

ctiv

ity (d

B)

Normalized frequency

Figure 4. directivity across the 4% bandwidth

Startup initial dimensions

of de antenna (x(n))

x and f(x) Optimal

Yagi-uda.m

Ellipsoidal algorithm

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V. CONCLUSION This work presented a mono-objective optimization process

for Yagi-Uda antennas in order to maximize directivity. The optimization was performed by ellipsoidal algorithm. This technique is suitable for objective functions whose behavior is not completely known, like as the function that represents the Yagi-Uda antenna. The directivity obtained by ellipsoidal method was higher than those achieved by others optimization techniques available in the literature.

REFERENCIAS BIBLIOGRÁFICAS [1] C. A. Balanis, “Antenna Theory: Analysis and design” 3ª edição, Wiley,

2005. [2] H. W. Ehrenspeck and H. Poehler, “A New Method for Obtaining

Maximum Gain from Yagi Antennas”, IRE Trans. Antennas Propagat., Vol. AP-7, pp. 379–386, Outubro 1959.

[3] G. A. Thiele, “Analysis of Yagi-Uda Type Antennas”, IEEE Trans. Antennas Propagat., Vol. AP-17, No. 1, pp. 24–31, January 1969.

[4] D. K. Cheng and C. A. Chen, “Optimum Spacings for Yagi-Uda Arrays”, IEEE Trans. Antennas Propagat., Vol. AP-21, No. 5, pp. 615–623, September 1973.

[5] R. M. Ramos, R. R. Saldanha, R. H. C. Takahashi e F. J. S. Moreira, “Otimizaçao Multiobjeto Aplicada ao Projeto de Antenas Filamentares”, Revista Ciencia & Engenharia – SBMag – Edicao Especial , pág. 67-70, 2003.

[6] R. G. Bland, D. Goldfard, M. J. Todd, “The Ellipsoid Method: A Survey Operations Research”, 29(6), pp. 1039-1091.

[7] E. A. Jones, W. T. Jones, “Design of Yagi Uda Antennas Using Genetic algorithm”, IEEE Antennas Propagat. Magazine, vol. 39, pp. 1386-1391, Set. 1997.

[8] D. K. Cheng and C. A. Chen, “Optimum Lengths for Yagi-Uda Arrays”, IEEE Trans. Antennas Propagat., IEEE Trans. Antennas Propag., Vol. AP-23, pp. 8–15, Jan. 1975.

[9] P. P. Viezbicke, “Yagi Antenna Design”, NBS Technical Note 688, December 1976)

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