Real array pattern tolerances from amplitude excitation errors

Transcription

1 LETTER IEICE Electroics Express, Vol.11, No.17, 1 8 Real array patter toleraces from amplitude excitatio errors Toshifumi Moriyama 1a), Lorezo Poli 2, Nicola Aselmi 2, Marco Salucci 2, ad Paolo Rocca 2b) 1 Graduate School of Egieerig, Nagasaki Uiversity, 1 14 Bukyo-machi, Nagasaki , Japa 2 ELEDIA Research DISI, Uiversity of Treto, Via Sommarive, Treto, Italy a) t-moriya@agasaki-u.ac.jp b) paolo.rocca@disi.uit.it Abstract: The impact o the omial power patter of radom ad ot a-priori kow errors affectig the excitatio amplitudes of real liear arrays is aalyzed by meas of a aalytic approach based o the iterval aalysis math. Startig from the expressios of the patter bouds as a fuctio of the excitatio toleraces modelig the saturatio i the amplifiers of the array feedig etwork, the effects o the radiatio characteristics ad the patter descriptors are evaluated. Keywords: patter tolerace, atea arrays, liear arrays, excitatio amplitude errors, iterval aalysis Classificatio: Electromagetic theory Refereces [1] E. Brooker: Practical Phased Array Atea Systems (Artech House, Norwood, MA, 1991). [2] R. J. Mailloux: Phased Array Atea Hadbook (Artech House, Norwood, MA, 2005). [3] J. Lee, Y. Lee ad H. Kim: IEEE Tras. Ate. Propag. 53 (2005) DOI: /TAP [4] J. Ruze: Nuovo Cim. 9 [3] (1952) 364. DOI: /BF [5] R. L. Haupt: Atea Arrays A Computatio Approach (Wiley & Sos, Hoboke, NJ, 2010). [6] N. Aselmi, L. Maica, P. Rocca ad A. Massa: IEEE Tras. Ate. Propag. 61 (2013) DOI: /TAP [7] R. Moore: Iterval Aalysis (Pretice-Hall, Eglewood-Cliffs, NJ, 1966). [8] G. Alefeld ad J. Herzberger: Itroductio to Iterval Computatios (Academic Press, New York, NY, 1983). [9] P. Rocca, L. Maica, N. Aselmi ad A. Massa: IEEE Ateas Wirel. Propag. Lett. 12 (2013) 639. DOI: /LAWP

2 1 Itroductio The aalysis of the toleraces o the power patter geerated by a array whe maufacturig errors affect the excitatio coefficiets ad/or the atea cotrol poits is a problem of great iterest i the atea commuity. As a matter of fact, such a problem has bee studied for decades [1, 2] ad several differet methods have bee proposed still recetly [3]. Moreover, the kowledge of the average radiatio performaces ad the potetial deviatios from the correspodig mea values is of iterest to predict the atea behavior whe used i real operatig coditios. O the other had, the availability of fast aalytic techiques for patter tolerace aalysis eables the sythesis of robust ad reliable atea arrays avoidig ad/or simplifyig complex ad time cosumig trial-ad-test calibratio processes that, owadays, are usually madatory. I the past, statistical approaches have bee developed [4, 5] for dealig with the aalysis of the array patter toleraces. By cosiderig radom errors aroud the omial excitatios, the impact o the radiatio characteristics has bee studied as the superpositio of a additioal power patter to the omial/expected oe. By virtue of their itrisic statistical ature ad the fact that a-priori assumptios are required o the error distributios, the arisig tolerace estimatios tur out beig oly probabilistically verified. I [6], a iovative method for the aalytic ad exact computatio of the power patter tolerace for errors o the array amplitude coefficiets has bee proposed. Based o the use of Iterval Aalysis (IA) [7, 8] ad without a-priori hypotheses o the amplitude error distributios, the effects o the arisig power patter of the deviatios of the array coefficiets from the omial oes are expressed through the rules of iterval arithmetic [7, 8]. I this paper, startig from the kowledge of the mid-poit ad width of each error iterval o the excitatio amplitudes, the behavior of the average power patter radiated by a realistic array is evaluated by extedig the prelimiary aalysis for ideal cases proposed i [9] ad further assessig its effectiveess ad reliability. 2 IA-based approach for patter tolerace evaluatio Let us cosider a N-elemet liear array radiatig a referece/omial power patter PðuÞ, u 2½ 1;1Š beig the directioal cosie, by feedig the atea elemets with amplitude weights a, ¼ 0;...;N 1. Sice the amplifiers are affected by ukow error toleraces, each -th excitatio ca assume a value withi the rage where m 1 2 w ; m þ 1 2 w ¼ 0;...;N 1 ð1þ m ¼ asup þ a if 2 ; w ¼ a sup a if ; ¼ 0;...;N 1 ð2þ are the mid-poit ad the width of the -th real amplitude iterval ½a Š¼½a if ;asup Š ( ¼ 0;...;N 1), a if ad a sup beig its lower ad upper bouds supposed give or estimated. 2

3 Accordig to the IA-based approach i [6], the average power patter, mfpðuþg, ad the correspodig patter tolerace width, wfpðuþg, assume the followig closed-form expressios mfpðuþg ¼ MR 2 ðuþþw2 R ðuþþm2 I ðuþþw2 I ðuþ ð3aþ wfpðuþg ¼ 2fM R ðuþw R ðuþgþ2fm I ðuþw I ðuþg ð3bþ whe [M R ðuþþw R ðuþ < 0 or M R ðuþ W R ðuþ > 0] ad [M I ðuþþw I ðuþ < 0 or M I ðuþ W I ðuþ > 0], otherwise mfpðuþg ¼ MR 2 ðuþþw2 R ðuþþ1 2f M IðuÞþW I ðuþg 2 ð4aþ wfpðuþg ¼ 2fM R ðuþw R ðuþgþ 1 2f M IðuÞþW I ðuþg 2 ð4bþ whe [M R ðuþþw R ðuþ < 0 or M R ðuþ W R ðuþ > 0] ad [M I ðuþ W I ðuþ 0 M I ðuþþw I ðuþ], otherwise mfpðuþg ¼ 1 2f M RðuÞþW R ðuþg þ MI 2 ðuþþw2 I ðuþ ð5aþ wfpðuþg ¼ 1 2f M RðuÞþW R ðuþg þ 2fM I ðuþw I ðuþg ð5bþ whe [M R ðuþ W R ðuþ 0 M R ðuþþw R ðuþ] ad [M I ðuþþw I ðuþ < 0 or M I ðuþ W I ðuþ > 0], ad mfpðuþg ¼ 1 2f M RðuÞþW R ðuþg 2 þ 1 2f M IðuÞþW I ðuþg 2 ð6aþ wfpðuþg ¼ 1 2f M RðuÞþW R ðuþg 2 þ 1 2f M IðuÞþW I ðuþg 2 ð6bþ whe [M R ðuþ W R ðuþ 0 M R ðuþþw R ðuþ] ad [M I ðuþ W I ðuþ 0 M I ðuþþ W I ðuþ], beig M R ðuþ ¼ XN 1 m cos ; W RðuÞ ¼ XN 1 w jcos j ð7þ ad M I ðuþ ¼ XN 1 m si ; W IðuÞ ¼ XN 1 w jsi j ð8þ where ¼ kdu, k is the wave-umber ad d is the iter-elemet distace. Startig from the expressios of mfpðuþg ad wfpðuþg, the bouds of the power patter iterval ½PðuÞŠ ¼ ½P if ðuþ; P sup ðuþš tur out to be P if ðuþ ¼mfPðuÞg 1 2 wfpðuþg ad Psup ðuþ ¼mfPðuÞg þ 1 2 wfpðuþg, respectively. I order to quatify the effects of the amplitude toleraces o the power patter, the fuctioal extesios of the mai beam descriptors to the patter iterval ½PðuÞŠ (amely, the sidelobe level iterval ½SLLŠ ¼½SLL if ; SLL sup Š, the mailobe halfpower beamwidth iterval, ½BWŠ ¼½BW if ; BW sup Š, ad the directivity iterval, ½DŠ ¼½D if ; D sup Š) are evaluated as follows SLL if ¼ max u=2 P if ðuþ maxu2 fp sup ðuþg ð9þ SLL sup ¼ max u=2 P sup f ðuþg max u2 P if ðuþ ð10þ BW if ¼ u sup!if 3dB;r BW sup ¼ u if!sup 3dB;r u sup!if 3dB;l u if!sup 3dB;l ð11þ ð12þ 3

4 Fig. 1. Amplitude excitatio tolerace versus the value of the omial amplitude. ( X N 1 2 ) D if X N 1 ¼ if ½a Š j½a Šj 2 ð13þ ( X N 1 2 ) D sup X N 1 ¼ sup ½a Š j½a Šj 2 ð14þ where Ω idetifies the mai-lobe regio, u sup!if 3dB;r=l (uif!sup 3dB;r=l ) is the agular directio o the left (l) or the right (r) side of the mai-lobe peak where P if ðuþ [P sup ðuþ] turs out to be 3 db below the peak of P sup ðuþ [P if ðuþ]. 3 Real-array patter tolerace predictio To ivestigate the behavior of the patter tolerace i real arrays by extedig the aalysis carried out i [6] ad limited to ideal cases i [9], let us cosider the followig bechmark examples. With referece to a liear array of N ¼ 21 elemets uiformly-spaced by d ¼ =2, λ beig the wavelegth at the workig frequecy, let us assume cotrol poits geeratig amplitude excitatios a, ¼ 0;...;N 1 i the rage a if a a sup (Fig. 1) depedig o the omial value. As it ca be observed, a upper threshold for the excitatios, a th ¼ 1:0, has bee set to model the saturatio effect of real amplifiers. Fig. 2. Example 1 - Mea value, m, ad width, w, of the tolerace iterval of the amplitude excitatios ad omial values, a, ¼ 0;...;N 1. 4

5 I the first example (Example 1), the omial excitatios have bee chose to geerate a Dolph-Chebyshev sum patter [2, 5] with sidelobe level 20 db below the peak of the mai beam (SLL ref ¼ 20 db). Fig. 3. Example 1 - Plot of the mea value, mfpðuþg, ad bouds, P if ðuþ ad P sup ðuþ, of the tolerace iterval of the power patter, ½PðuÞŠ, together with the omial beam patter, PðuÞ. By cosiderig the tolerace profile i Fig. 1 ad the omial excitatios, a, ¼ 0;...;N 1, i Fig. 2, the mid-poits ad widths of the excitatio error itervals, ½a Š, ¼ 0;...;N 1, tur out as i Fig. 2. It is iterestig to otice the saturatio of the upper boud a sup of the elemets ¼ 3;...; 17. The average fuctio mfpðuþg of the power patter iterval ½PðuÞŠ ad its bouds, P if ðuþ ad P sup ðuþ, are show i Fig. 3. As expected, the omial beam, PðuÞ, as well as mfpðuþg lay withi the bouds of the power patter iterval [P if ðuþ mfpðuþg P sup ðuþ ad P if ðuþ PðuÞ P sup ðuþ]. The beam descriptors of the average power patter, mfpðuþg, the omial oe, PðuÞ, ad the power patter iterval ½PðuÞŠ (i.e., Eqs. (9) (14)) are give i Table I. As it ca be oticed, the values of the descriptors of mfpðuþg ad PðuÞ are also cotaied withi the iterval descriptors of ½PðuÞŠ (Table I). As for the average deviatio from the omial beam (i.e., mfpðuþg vs. PðuÞ), it turs out that there is a icremet of the SLL of about 3.97 db, while the directivity has a variatio of 0.15 db (Table I). Fig. 4. Example 2 - Mea value, m, ad width, w, of the tolerace iterval of the amplitude excitatios ad omial values, a, ¼ 0;...;N 1. 5

6 I the secod example (Example 2), the same atea geometry is dealt with, but the values of the omial excitatios have bee set to afford a Taylor sum patter [2, 5] still with SLL ref ¼ 20 db ad ¼ 3 (Fig. 4). The values of the excitatio amplitude mid-poits ad widths whe cosiderig the tolerace profile of Fig. 1 result as i Fig. 4. Aalogously to the first example, the upper bouds of the excitatio amplitude itervals a sup, ¼ 0;...;N 1 saturate at the elemets ¼ 3;...; 17. The plots of mfpðuþg, PðuÞ, ad the bouds of ½PðuÞŠ are show i Fig. 5, while the correspodig beam descriptors are reported i Table II. Fig. 5. Example 2 - Plot of the mea value, mfpðuþg, ad bouds, P if ðuþ ad P sup ðuþ, of the tolerace iterval of the power patter together, ½PðuÞŠ, with the omial beam patter, PðuÞ. Table I. Example 1 - Power patter idexes. SLL [db] BW (10 1 ) [u] D [db] mfpðuþg ½PðuÞŠ [ 38.51; 7.39] [0.00;1.37] [9.83;16.49] PðuÞ By comparig the values i Table I ad Table II, the patter idexes tur out quite similar. However, the average power patter of the Dolph-Chebyshev array has a sidelobe level (SLLðmfPðuÞgÞj DC ¼ 16:03 db) lower tha that of the Taylor array (SLLðmfPðuÞgÞj Taylor ¼ 15:74 db). O the other had, the directivity of the average beam of the Taylor patter (DðmfPðuÞgÞj Taylor ¼ 13:17 db) is slightly above that of the Dolph-Chebyshev oe (DðmfPðuÞgÞj DC ¼ 13:14 db). This is caused by the decreasig behavior of the secodary lobes of the omial patter (Fig. 5). It is also worthwhile to poit out that the beam patter descriptors of mfpðuþg ad PðuÞ belog to the itervals of the descriptors of the patter iterval further cofirmig the validity of (9) (14). The last example (Example 3) is cocered with the aalysis of the patter toleraces whe varyig the dimesio of the array, while geeratig the same omial patters of the previous examples. Towards this ed, the umber of array elemets has bee varied betwee N ¼ 10 up to N ¼ 100 elemets. For each array cofiguratio, startig from the distributio of the omial amplitude coefficiets, the correspodig amplitude errors have bee determied accordig to the rules pictorially summarized i Fig. 1. 6

7 Fig. 6. Example 3 Behavior of the bouds of [SLL] together with the omial value. Table II. Example 1 - Power patter idexes. SLL [db] BW (10 1 ) [u] D [db] mfpðuþg ½PðuÞŠ [ 33.34; 7.05] [0.00;1.42] [9.70;16.67] PðuÞ Successively, the iterval power patters ad the itervals of the beam descriptors have bee computed. Figs. 6, 7, ad 8 show the behavior of the lower ad upper bouds of the descriptor itervals [SLL], [BW], ad [D] together with the value of the sidelobe level, half-power beamwidth, ad directivity of the omial power patter as a fuctio of the umber of elemets i correspodece with each patter type (i.e., the Dolph-Chebyshev patter with SLL ref ¼ 20 db ad the Taylor patter with SLL ¼ 20 db, ¼ 3). Fig. 7. Example 3 Behavior of the bouds of [BW] together with the omial value. As it ca be observed (Fig. 6), the values of upper boud of the sidelobe level, SLL sup, turs out beig almost equal betwee the Dolph-Chebyshev beam ad the Taylor oe. As for SLL if, the Dolph-Chebyshev idex rapidly decreases whe elargig the array, while the variatios for the Taylor oe are limited to few 7

8 decibels whatever N. Ulike Fig. 6, Fig. 7 shows that both half-power beamwidth bouds maily deped o the array size ad slightly from the patter type. Fig. 8. Example 3 Behavior of the bouds of [D] together with the omial value. Fially, Fig. 8 gives a idicatio o the depedece of the directivity bouds o the umber of array elemets. As expected [2], the directivity of the omial Taylor patters is higher tha that of the Dolph-Chebyshev oes, especially whe dealig with large arrays. Moreover, the iterval widths of [D] result arrower whe usig Taylor arrays. 4 Coclusio Estimates of the patter toleraces i realistic liear arrays presetig bouded errors i the amplitude excitatios have bee yielded by exploitig IA-derived aalytic expressios. Startig from a realistic model of the amplitude toleraces, umerical bouds for the power patters have bee aalyzed to give the array desiger suitable idicatios o the arisig performaces thus avoidig timeexpesive trial-ad-test umerical predictios. Ackowledgmets This work was supported i part by a Grat-i-Aid for Scietific Research (C) Number