Experimental Validation of Equi-Area Method for Antenna Directivity, Gain and Efficiency Calculations

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1 Lfe Scence Journal ;(s) Expermental Valdaton of Equ-Area Method for Antenna Drectvty, Gan and Effcency Calculatons Sadq Ullah, Shahbaz Khan, Azz Ahmed, Babar Kamal Antennas and ropagaton Lab, Telecommuncaton Engneerng Department, Unversty of Engneerng and Technology, eshawar (Mardan Campus), Mardan,, akstan. Abstract: In ths paper a detaled analyss of dpole, patch, and array antennas s carred out for samplng ponts and t s expermentally valdated that the equ-area method gves more accurate values of drectvty, and gan as compared to the equ-angle method. The equ-area method uses Leopard s (Leopard, ) algorthm to determne the samplng ponts where as equ-angle method uses equ-angle steps of -degrees to obtan samplng ponts. It s also verfed that the equ-area method s to tmes computatonally effcent than the conventonal equangle method. [Sadq Ullah, Shahbaz Khan, Azz Ahmed, Babar Kamal. Expermental Valdaton of Equ-Area Method for Antenna Drectvty, Gan and Effcency Calculatons. Lfe Sc J ;(s):-9] (ISS: 9-).. 9 Keywords: equ-area method, equ-angle method, samplng ponts.. Introducton The antenna measurements s the one of the most challengng and crucal component of antenna engneerng. Therefore an antenna engneer s requred to have adequate understandng of the theory of antenna desgn and antenna measurements. In the past an extensve contrbuton to the lterature of antenna measurements s gven n (Kumer WH, Gllespe, 9; Kraus, 9; Yaghjan, 9; Balans, 99). The antenna performance parameters lke, gan, drectvty, radaton and antenna effcency may be calculated by analytcal expressons as well as approxmate numercal methods (Kraus, 9; Balans, 99). The approxmate numercal methods are used n the laboratory to analyze the antenna performance parameters. The numercal method uses a grd of samplng ponts as the measurement locatons to gather the antenna radaton nformaton at each pont. Therefore the two man factors nvolved n the numercal analyss of antenna measurements are the samplng crteron and the type of numercal method used. In the past many samplng technques were developed to get the grd of measurement ponts as gven n (Rahmat-Samt and Cheung, 9; Chen and Edwards, 99; Lu et al., ; Yaccarno et al., 99, Wttmann et al., 99; Wttmann et al., ; Oh et al., ; Latnen, ). The accuracy of the measurements depends on the desred samplng technque and type of numercal method. Furthermore the tme requred to analyze the measured results also depends on the type of numercal method. The equ-angle samplng and the numercal method gven n (Balans, 99) gves good results for a range of antennas but ts accuracy may be affected for very narrow-beam antennas. Ths s due to the non-unform dstrbuton of the samplng ponts on the sphercal surface. The smulaton tme for the equ-angle method s also affected due to the weghtng functons nvolved n the far-feld parameter (.e. total radaton power) calculatons. Therefore n ths paper Leopard s algorthm s used as a samplng technque to obtan the unformly dstrbuted grd of samplng ponts. To analyze the antenna, a novel and computatonally effcent method called the equ-area method s used. Secton descrbes the theory of antenna performance parameters (drectvty, gan, radaton effcency and antenna effcency) of a dpole, patch and array antennas. The results are presented n secton ; Secton concludes the paper.. Theory of Antenna erformance arameters In ths secton the theory of antenna performance parameters, such as total radated power, drectvty, gan and effcency s dscussed, usng both analytcal and numercal approaches. To evaluate these parameters two numercal methods, namely, equ-angle and equ-area methods (Ullah et al., ; Ullah et al., ), are used for three types of antennas: () Cylndrcal dpole () Rectangular patch and () two element array antennas. The total radated power of an antenna may be calculated usng ether of the methods.. e., analytcal u(, ) sn dd rad ()

2 Lfe Scence Journal ;(s) L M angle ( ) u(, j )sn( ) L () rad, equ j, equarea R ( c,, c, ) rad u () Where u s radaton ntensty, B s a constant, L, and M=L are the number of samplng ponts along the elevaton and azmuth axs. The angles θ and ϕ j represents the value of the th and jth elevaton and azmuth angles n radans respectvely. The measurement locaton s defned by the par of equ angle sphercal coordnates (θ, ϕ j ) gven by,,,..., L L j j M j,,,..., M () Smlarly A R represents the beam sold angle of the equ-area regons and s nversely proportonal to the total number of radaton pattern samplng ponts (),.e. R () The par of sphercal coordnates (θ c,, ϕ c,j ) shows the th measurement locaton n the equ-area method. The subscrpt c reveals that the samplng pont s the centre of these equ-area regons and s obtaned usng the Leopard s algorthm. If the drecton of the man beam s not specfed then the maxmum value of drectvty s analytcally expressed n the drecton of maxmum radaton ntensty (U max ) as gven by U max D () rad.. Cylndrcal Dpole Antenna The cylndrcal half-wavelength (.9 cm) wred dpole antenna s desgned for a frequency of 9 MHz, wth quarter wavelength balloon (.9 cm) for mpedance matchng as shown n Fgure. Fgure. 9 MHz cylndrcal wre dpole antenna umercal pattern for a dpole antenna s gven n (Balans, 99); therefore the total radated power by dpole can be found usng the equ-area method: u dpole (, ) U sn ( ) () max rad, dpole, equarea U max sn ( c, ) (9). Rectangular atch Antenna The patch antenna (Fgure ) s desgned for a resonance frequency of GHz on a substrate of heght mm and delectrc constant. (RT durod). The length and wdth are equal to. cm (half wavelength). Usng the two-slot transmsson lne model of the rectangular patch antenna, the radaton ntensty of the patch s gven by: V kw u patch ( ) tan ( )sn cos () The total radaton ntensty of an antenna s gven by E E u (, ) u u () Where, E θ and E ϕ are the elevaton and azmuth components of the electrc feld. η s the mpedance of the free space ( ohms). In theory, the numercal radaton ntensty patterns are avalable for the dpole, patch and two-element array antennas (Balans, 99), therefore the two numercal methods (equ-angle and equ-area) can be used to fnd the performance parameters (such as drectvty, gan) of these antennas: Fgure. GHz rectangular patch antenna Where k s the wave number n free space and W s the wdth of the patch antenna. V s a

3 Lfe Scence Journal ;(s) constant whch depend on the thckness of the substrate. The total radaton power of the patch antenna s obtaned by usng the equ-area method: rad, patch, equarea u patch ( c, ) (). Two-Element Array Antenna The normalzed radaton ntensty pattern of the two-element dpole array (Fgure ) s proportonal to the multplcaton of the squares of the element factor and the array factor (Gross, ).. e. u array ( ) [sn( )] [cos(.kd sn )] () rad, array, equarea fd () [sn( c, )] [cos( sn( c, ))] c Where k / f / c the wave number, d s the spacng between the two dpole elements, λ s the free space wavelength. The resonant frequency of the sngle array element (.e. dpole) s 9 MHz. Dpole Fgure. 9 MHz Array antenna n the anechoc chamber Once the total radated power s obtaned, no matter, whch numercal method s used, the maxmum drectvty of a gven antenna can be calculated usng (). The antenna gan can be calculated usng the formula: U max G () n Fxer ower dvder ostoner If we assume that there are no cable losses n the measurement system, then the nput power s equal to the sum of radaton power and reflected power (due to msmatch),.e. n rad r rad n () n rad /( ) () Where r s the reflected power, Γ s the reflecton coeffcent of the antenna under the test (AUT), gven by; / S () Where, S s the return loss of the AUT n db and the factor (- Γ ) s called the reflecton loss factor. Once the gan and drectvty are known, the radaton effcency can be calculated usng the relaton G D () rad / The antenna total effcency s obtaned from the product of radaton and reflecton effcences: ant ( rad ) (9). Results The antenna measurements are carred out n the anechoc chamber and magntude and phases of the elevaton and azmuth component of the electrc feld are recorded. Matlab...s used to analyse ths recorded data. The antennas are analysed usng both equ-angle as well as equ-area methods for samplng ponts. The samplng ponts are obtaned by usng -degrees angular steps n the equ-angle method. Leopard s algorthm s used to obtan the samplng ponts n the equ-area method. The grd of the samplng ponts alone and on a sphercal surface s shown n Fgure. The antennas are analysed one by one as follows: The cylndrcal dpole antenna s analyzed n the frequency range - MHz. The resonance frequency of the dpole s 9 MHz. The antenna drectvty and gan are obtaned from the measured pattern usng both the methods and are compared wth those obtaned va umercal Electromagnetc Code (EC) software. The EC software uses the Method of Moments (MoM) to calculate the gan and drectvty of the antenna. The varatons n gan and drectvty curves are approxmately the same for the dpole antenna for both the methods as shown n Fgure,, and. It s worth notng that the values of drectvty acqured

4 Lfe Scence Journal ;(s) by the equ-angle method are slghtly hgher than that acqured by equ-area method except n the ranges - MHz, and 9- MHz. Ths s due to the fact that the densty of the equ-angle ponts s non-unform on the sphercal surface. Whle measurng the radaton pattern at certan frequency; regons of dstnct radaton ntensty are sampled by non-unformly dense grd of samplng ponts contaned n those regons. Ths non-unformty may results n an ncrease or decrease n the drectvty obtaned by the equ-angle method w. r. t that obtaned by equ-area method, throughout the frequency range. The devaton n drectvty and gan w.r.t the EC based drectvty and gan are plotted n Fgure and Fgure, and t s apparent that the equarea method shows better accuracy w.r.t the equangle method throughout the antenna bandwdth. The patch antenna has a resonance frequency of MHz. The patch antenna drectvty and devaton n drectvty are llustrated n Fgure 9 and Fgure whereas the gan and devaton n gan are depcted n Fgure and Fgure respectvely. In ths case the measured drectvty and gan are compared wth the analytcal drectvty and gan of the patch. The devaton n drectvty and gan w. r. t the analytcal values are less for the equarea method as shown n Fgure and Fgure. The drectvty and devaton n drectvty for the two-element dpole array (wth d= cm separaton between the elements) are llustrated n Fgure, Fgure, whereas the correspondng values of gan and devaton n gan are shown n Fgure and Fgure respectvely. The array confguraton undergoes resonance at 9 MHz. The values of drectvty and gan are compared wth the values obtaned from the EC software and t s proved that the equ-area method gves less devaton throughout the frequency range. Therefore the equarea method s more accurate as compared to the equ-angle counterpart. The two-element array wth d= cm separaton between the elements s analysed and the results are llustrated as shown n Fgs. -. Ths array confguraton resonates at the same frequency,.e. 9 MHz. It s observed that the devaton n gan and drectvty for the equ-area method are less than that for equ-angle method. It s worth notng that at resonance frequency (.e. 9 MHz) the equarea method s sgnfcantly accurate. Fgure. Grd of samplng ponts; Equ area: (a) On sphere (b) Lne-connected; Equ angle: (c) On sphere (d) Lne-connected Max Value (db) Value n (db) D equ angle(meas) D equ area(meas) Fgure. Dpole: drectvty analyss DEV(D) equ area(meas) DEV(D) equ angle(meas) 9 9 Fgure. Dpole: Devaton n drectvty

5 Lfe Scence Journal ;(s) 9.. eak Value (db)... Value n (db) DEV(D) equ area(meas) DEV(D) equ angle(meas) D Analytcal.. G equ angle(meas) G equ area(meas) G EC. 9 9 Fgure. Dpole: gan analyss 9 9 Fgure. atch: Devaton n drectvty G equ angle(meas) G equ area(meas) G Analytcal DEV(G) equ area(meas) Value n (db). DEV(G) equ angle(meas) G EC eak Value (db) Fgure. Dpole: Devaton n gan Fgure. atch: gan analyss DEV(G) equ area(meas) DEV(G) equ angle(meas) G Analytcal Max Value (db). Value n (db) D equ angle(meas) D equ area(meas) D Analytcal. 9 9 Fgure 9. atch: drectvty analyss 9 9 Fgure. atch: Devaton n gan

6 Lfe Scence Journal ;(s). Max Value (db). Value n (db) DEV(G) equ area(meas) DEV(G) equ angle(meas) G EC. D equ angle(meas) D equ area(meas) 9 9 Fgure. Array (d= cm): drectvty analyss 9 9 Fgure. Array (d= cm): Devaton n gan. D equ angle(meas) D equ area(meas) Value n (db) DEV(D) equ area(meas) DEV(D) equ angle(meas) Max Value (db) Fgure. Array (d= cm): Devaton n drectvty Fgure. Array (d= cm): drectvty analyss. DEV(D) equ area(meas) DEV(D) equ angle(meas) eak Value (db). Value n (db). G equ angle(meas) G equ area(meas) G EC 9 9 Fgure. Array (d= cm) gan analyss 9 9 Fgure. Array (d= cm): Devaton n drectvty

7 Lfe Scence Journal ;(s). eak Value (db).. G equ angle(meas) G equ area(meas) G EC Value n (db) Fgure 9. Array (d= cm): gan analyss DEV(G) equ area(meas) DEV(G) equ angle(meas) G EC 9 9 Fgure. Array (d= cm): Devaton n gan The equ-angle method s computatonally slow as compared to equ-area method because of the weghtng functon, snθ. Ths clam s verfed by calculatng the absolute number of seconds n real tme that have elapsed n order to execute the equaton for antenna drectvty. The bult-n functon n Mathematca was used to determne ths absolute tme n order to study the computatonal complexty of both methods. The two methods were tested for dfferent values of samplng ponts (Fgure ). It was found that the equ-area method s approxmately to tmes faster than the equangle method. The computaton tme represents the tme spent by the CU to execute the equaton for drectvty. Ths study was conducted on a -bt,. GHz, Intel entum Dual CU, T, wth a.gb of RAM. Fgure. Computatonal complexty of equ-angle and equ-area methods as a functon of samplng ponts. Dscussons In ths paper the equ-area based samplng ponts were used to measure the antenna radaton characterstcs. Three types of antennas, were expermentally tested, and numercally valdated usng the well-known equ-angle method and the novel equ-area method (based on Leopard algorthm. The two methods were compared on the bass of degree of accuracy n gan, drectvty and effcency calculatons, throughout the frequency range. It was expermentally valdated that the equarea method gves more accurate results at the resonance (desgn) frequency for the gven antennas. Furthermore the computatonal complexty of the equ-area method was low as compared to the equangle method for a gven specfcatons of the system (.e. computer) and number of samplng ponts. As a result the equ-area method can gve to tme faster response than the equ-angle method, to analyse the far-feld parameters of antennas. Acknowledgements: The authors would lke to specally thank the Hgher Educaton Commsson (HEC) of akstan and Loughorough Unversty, Unted Kngdom for conductng ths research Correspondng Author: Dr. Sadq Ullah Telecommuncaton Engneerng Department, Unversty of Engneerng and Technology, eshawar (Mardan Campus), Mardan,, akstan E-mal: sadqullah@nwfpuet.edu.pk References. Leopard. A partton of the unt sphere nto regons of equal area and small dameter. Electronc Transactons on umercal Analyss. ; : 9-.

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