research article beam tilt-angle estimation for monopole...

Research ArticleBeam Tilt-Angle Estimation for Monopole End-Fire ArrayMounted on a Finite Ground Plane

Jia Cao Zhenghui Xue and Meng Cao

School of Information and Electronics Beijing Institute of Technology 5 South Zhongguancun Street Haidian DistrictBeijing 100081 China

Correspondence should be addressed to Zhenghui Xue zhxuebiteducn

Received 2 October 2014 Revised 2 December 2014 Accepted 15 December 2014

Academic Editor Ana Alejos

Copyright copy 2015 Jia Cao et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A modified method for the beam tilt-angle estimation of monopole end-fire array mounted on finite ground plane is proposed Inthe simplified model the monopole array and ground plane are approximated to two line sources of transverse and longitudinalelectric current respectively It is deduced that the beam tilt angle is a function about the length of ground plane in front of array 119871

119892

the length of monopole array 119871119886 and the phase constant 120573

120572 After verifying the optimizing principle of monopole end-fire array

a 10-element monopole Yagi-Uda antenna satisfying Hansen-Woodyard condition is designed and measured for the analysis Bycomparison and analysis the value of 120573

120572is demonstrated to be the key point of the proposed method And a slow wave monopole

array is proved to be able to achieve a low beam tilt angle from end-fire with only a short-length ground plane

1 Introduction

End-fire antennas are useful in all applications that requirelow profile for example in airborne and vehicular envi-ronments After decades of research and developing theperformance of end-fire antennas has been greatly advancedin several fields like broadband gain enhancement andpattern reconfiguration [1ndash4] However those end-fire anten-nas mounted in or on a finite ground plane suffer from aserious problem the main beam is tilted at some angle fromthe horizontal plane by diffraction effects from conductiveground which prevents a true end-fire condition

By image theory the conclusion can be easily obtainedthat the beam tilt angle will disappear when ground planeextends to infinite Meanwhile as a majority of radiationpower distributes in the positive half-space the value of thebeam tilt angle mainly depends on the effective ground planelength in front of end-fire antenna This phenomenon wasfirst described by Elliott in 1954 [5]

Based on Huygensrsquo principle a simplified method fordetermining the effect of finite ground plane was demon-strated by Walter in his book by which the aperture ofantenna and the ground plane were simplified to a uniform

line source of magnetic current and electric current respec-tively As a result the beam tilt 120579119905 was proved to be a functionof both 119871119886 (given antenna length) and 119871119892 (effective groundplane length) For small values of beam tilt (120579119905 le 20

∘) thisfunction reduced to

120579119905 asymp49

[(119871119886 + 119871119892) 120582]12

(1)

where 120579119905 is in degrees [6]Nevertheless this method with high accuracy is not suit-

able for the monopole end-fire array vertical to the groundplane That is not only because of the model abstractedfrom the flush-mounted end-fire antenna which is a kindof aperture antenna parallel to the ground but also due tono consideration about the influence from phase constantrsquosvarying Thus in the former research the beam tilt-angleanalysis of this kind of monopole end-fire array could onlybe obtained by whole model simulation using simulationsoftware far-field measurement or numerical calculationmethod [7] with high cost long time and various operations

A beam tilt-angle estimation method which is modi-fied for the monopole end-fire array mentioned above is

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 351439 8 pageshttpdxdoiorg1011552015351439

2 International Journal of Antennas and Propagation

ZY

X

W

S

h1 h2 h3 hn hN

LaLg

L

middot middot middotmiddot middot middot

Figure 1 119873-monopole end-fire antenna vertically mounted on afinite ground plane

proposed in this paper Firstly a simplified source modelis applied to this kind of antenna The key point of thismodel simplification is evaluating the phase constant 120573119886 ofthe surface wave along monopole end-fire array Secondlymonopole Yagi-Uda antenna is introduced to validate thevalidity and accuracy of this method The estimated tiltangles present a good agreement with CST simulation resultsFurthermore comparison between the proposed methodin this paper and Walterrsquos method shows an advantage ofmonopole end-fire antenna with slow wave characteristicThis advantage that a low tilt angle can be achieved with ashort ground length is demonstrated by the experimentalresults of the 10-element monopole Yagi-Uda antenna

2 Simplified Model for Monopole End-FireArray and Ground Plane

Figure 1 shows the configuration of an 119873-monopole end-fire array mounted on a ground plane with certain size(119882119871) Assume it is working at the frequency point 119891 with awavelength 120582 and 120596 is the corresponding angular frequencyConsulting the original method in [6] the simplified modelshould be composed of two parts one line source approx-imated from the monopole array current distribution andanother one approximated from the current on the groundplane Note that only far-field patterns in119883119885-plane are takeninto account for the beam tilt-angle problem as the mainlobe direction is always in this plane when the ground issymmetric about the119883-axis

21 Simplification for Monopole End-Fire Array Firstly thearray elements electric monopoles with heights ℎ119899 areapproximated by a series of ideal dipoles Ideal dipole isdefined as a Hertz dipole with uniform electric current 119868

on the total length Δ119897 ≪ 120582 and its radiation patternfunction is 119891 = sin 120579 Usually monopole height is lessthan or equal to 025120582 Assuming the current distributionon monopole is 119868(119911) = 119868119898 sin(119896(ℎ minus 119911)) where ℎ is the

0

0

10 20 30 40 50 60 70 80 90

Ideal dipole

minus60

minus50

minus40

minus30

minus20

minus10

h = 01120582

h = 02120582

h = 025120582

120579 (∘)

f = inth0Imsin(k(h minus z))ejkzcos 120579dz

Nor

mal

ized

pat

tern

f(120579

) (dB

)Figure 2 Radiation patterns of 01120582 02120582 and 025120582monopoles andof ideal dipole

zI1Δl zI2Δl zI3Δl zINΔlzInΔl

zJz(x)dxX

Transverse current line source

Ideal dipoleLa = NS

S

Figure 3 Ideal dipole array and transverse current line sourcecorresponding

height of monopole the radiation pattern can be derivedby (2) Figure 2 exhibits the curves of 119864-plane (119883119885-plane)normalized patterns of ideal dipole 01120582 monopole 02120582monopole and 025120582 monopole respectively It is suggestedthat ideal dipolersquos radiation pattern is quite similar to the onesof monopoles with height less than or equal to 025120582 so thisapproximation has no serious loss in accuracy

119891 = int

ℎ

0

119868119898 sin (119896 (ℎ minus 119911)) 119890119895119896119911 cos 120579

119889119911 (2)

Secondly as shown in Figure 3 the array of ideal dipoleswith electric current amplitude and phase offset as (119860119899 120593119899)on each is approximated by a transverse electric currentline source lying on the 119883-axis with amplitude 119860(119909) andphase velocity V(119909) in119883 direction For the line source lengthassume that each ideal dipole 119868119899Δ119897 = 119860119899119890

119895120593119899 lying on 119909119899

corresponds to a segment 119904 (spacing between two adjacentelements) in length so the whole length of line source will be

119871119886 = 119873119904 (3)

International Journal of Antennas and Propagation 3

where119873 is the number of elementsTherefore let the currentsource be

119869119911 (119909) = 119860 (119909) 119890minus119895120573(119909)119909

0 le 119909 le 119871119886 (4)

where

120573 (119909) =120596

V (119909) (5)

is the phase constant on the line source For small elementspacing (one-half wavelength or less) the distinction betweenarray and continuous source tends to disappear So when 119904 le

05120582 this approximation worksFinally simplify the line source expressed by (3) to a

line source with a constant amplitude 119860119886 and uniform phaseconstant 120573119886 For beam-tilting estimation only a relativevalue of 119860119886 is needed and this work will be introduced inSection 23 Meanwhile the value of 120573119886 is evaluated as

120573119886 =sum119873

119899=2(120593119899 minus 120593119899minus1)

119871119886

(6)

120573119886 is just the key point of this new method owing tothe fact that the difference between the value of 120573119886 and thewavenumber in free space 119896 cannot be ignored inmany casesThe further explanation will be given in Section 3

According to the simplification above the line sourceexpression is deduced to

119869119911 (119909) = 119860119886119890minus119895120573119886119909

0 le 119909 le 119871119886 (7)

whose relative far-field in elevation plane (119883119885-plane) is

120579119864119886 = 120579119860119886119890119895119883119886

sin119883119886

119883119886

119871119886 sin 120579 (8)

where119883119886 = (1198711198862)(119896 sin 120579 minus 120573119886)

22 Simplification for Ground Plane In electromagnetic fieldanalysis a ground plane can be replaced by a current sheetwith its induced current components (119869119909 119869119910) distributed onBased on themirror symmetry of end-fire antennarsquos structurein Figure 1 119869119910 on the current sheet has no contribution to thefar-field in119883119885-plane so it can be ignored in the beam tiltingestimation Thus the simplified model for the ground planeshould be a line source of longitudinal electric current on119883-axis

This line source is divided into two parts the partrepresenting the ground plane just under themonopoles (G1)and the part representing the ground plane in front of end-fire array (G2) The current amplitudes are assumed to be119860119892 equal and constant on both parts However the phaseconstant 120573(119909) should not be uniform along the whole linesource The current in G2 is close to the current induced bymonopolesrsquo far-field so the phase constant in G2 is close to119896 Meanwhile the current phase constant in G1 approachesto 120573119886 because of the constraint by monopole arrayrsquos near-field but regresses to 119896 at the end of array because of thecontinuity of 120573(119909) on the boundary between G1 and G2

According to the considerations about both simplicity andaccuracy the phase constant in G1 is valued by the averageof 120573119886 and 119896 The estimation for the phase constant in G1is another special progressing in the novel method becausein previous work only the ground in front of the end-fireaperture corresponding to G2 in our case is taken intoaccount [5 6]

Thus the simplified line source can be expressed as

119869119909

=

119860119892119890minus119895((120573

119886+119896)2)119909

0 le 119909 le 119871119886

119860119892119890minus119895[119896(119909minus119871

119886)+((120573119886+119896)2)119871

119886]

= 119860119892119890minus119895(119896119909+((120573

119886minus119896)2)119871

119886)

119871119886 lt 119909 le 119871119886 + 119871119892

(9)

And its relative far-field is

120579119864119892 = 120579119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(10)

where

1198831198921 =119871119886

2(119896 sin 120579 minus

120573119886 + 119896

2)

1198831198922 =

119871119892

2119896 (sin 120579 minus 1)

1198831198923 = (119871119886 +

119871119892

2) 119896 (sin 120579 minus 1) + 119871119886

119896 minus 120573119886

2

(11)

23 Superposition of Far-Field Patterns The total estimatedradiation is the superposition of the far-fields of the two linesources introduced in Sections 21 and 22 Good results areobtained in practice by letting the sources be in-phase atthe origin and with amplitudes adjusted so that their patternmaxima are equal byWalterrsquos method [6]Therefore the sameassumption is made in this paper

Field 119864119886 will have maximum value at 120579 = 90∘ Setting

|119864119886|max = |119864119892|max gives

119860119886119871119886

sin119883119886

119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

(12)


Thus the total far-field is

119864119905 = 119864119886 + 119864119892

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

times119883119886

sin119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘119890119895119883119886

sin119883119886

119883119886

sin 120579

+ 119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(13)

The main lobe direction 120579119898 can be obtained by solvingthe equation (120597119864119905120597120579)|120579=120579

119905

= 0 and the beam tilt angle 120579119905 has120579119905 = 90

∘minus 120579119898 So 120579119898 is proved to be a function of 119871119886 119871119892 and

120573119886 Moreover for the convenience of calculation numericalcomputing for (12) to find the value of 120579 corresponding to|119864119905|max is also a considerable solution

3 Simulation Analysis and Discussion

A monopole Yagi-Uda antenna is taken as an example ofmonopole end-fire array for the beam tilting analysis and dis-cussion In most conditions conventional Yagi-Uda antennais inherently a slow wave structure and Hansen-Woodyardcondition which is expressed by (14) is usually consideredas the optimization aim in this kind of antennarsquos designfor improving directivity [6] Before antenna designing asimulation experiment was implemented to verify that thisoptimizing principle applies to the monopole end-fire arraycase Consider

120573119886119871119886 = 120587 + 119896119871119886 (14)

31 Optimizing Principle in Monopole End-Fire Array Thesimulation object is a 10-element monopole array like themodel shown in Figure 1 working at 119891 = 3GHz All themonopoles in this array have an equal height ℎ = 025120582 =

25mmand an equal radius 119903 = 2mmThe interspace betweenadjacent elements is 119904 = 20mm so the total length of arrayis 119871119886 = 200mm The finite ground size is 119871119886 = 200mm119871119892 = 30mm and119882 = 60mm To satisfy Hansen-Woodyardcondition the actual phase constant on the arrayrsquos currentshould be 120573119886 = 125119896

Every element was fed with an equal magnitude and auniform-step phase offset By CST MWS simulating the fedsignal phase step Δ120593 between adjacent elements was taken asa variable the arrayrsquos directivities and tilt angles with differentΔ120593were recorded and the surface current data of monopoleswere exported for calculating the phase constant 120573119886 on thisarray

Specifically Figure 4(a) shows the surface current distri-bution onmonopoles with a certain phase stepΔ1205930 Bymeansof processing these current data in Matlab the actual phaseoffset of current source on each element 120593119899 was calculatedand Figure 4(b) shows the results Substituting 120593119899 in (6) theactual phase constant 120573119886 was derived

Table 1 Parameter for monopole Yagi-Uda antenna

Variables Values119903 (mm) 2

119904 (mm) 20

ℎ1 (mm) 23

ℎ2 (mm) 215

ℎ3 (mm) 19

ℎ4 (mm) 17

ℎ5 (mm) 156

ℎ6 (mm) 155

ℎ7 (mm) 145

ℎ8 (mm) 1067

ℎ9 (mm) 683

ℎ10 (mm) 3

Thus through this simulation experiment the relation-ships between 120573119886 and arrayrsquos directivity and 120573119886 and arrayrsquos tiltangle were both obtained The black solid curve in Figure 5illustrates that the directivity119863 asymp 1143 dB when 120573119886 = 125119896and it is almost the optimum directivity for this 10-elementarray It demonstrates Hansen-Woodyard condition is stillsuitable for the monopole end-fire array case

To be worthy of attention the red solid curve in Figure 5reveals the trend that beam tilt angle is decreasing while 120573119886 isincreasing

32 BeamTilting Estimation to aMonopole Yagi-UdaAntennaA 10-element dipole Yagi-Uda antenna has been designed fol-lowing Hansen-Woodyard condition which has been verifiedabove at 119891 = 3GHz And the corresponding monopoles inthe positive-119885-space are mounted on a finite ground plane toconstitute a monopole Yagi-Uda antenna The configurationis shown in Figure 6 and parameters of the monopole arrayare given in Table 1

According to (3) the monopole arrayrsquos length is 119871119886 =

200mm Substituting it in (14) the solution is 120573119886 asymp 7854whose distinction with 119896 = 2120587120582 asymp 6283 cannot beignored Thus only the length of ground plane in front of119871119892 is the variable in this analysis A 120579119905-119871119892 curve is simplycalculated by the estimation method proposed in Section 2Meanwhile there are two other 120579119905-119871119892 curves calculated byWalterrsquos method and simulated by CST Microwave Studiorespectively All the three curves are shown in Figure 7 forcomparison The estimated result by Walterrsquos method has aclear distinction to the simulation one Meanwhile a goodagreement is achieved between the modified estimation andCST simulation The accuracy of the proposed estimationmethod is illustrated to bemuch better thanWalterrsquos for beamtilt angle of monopole end-fire array like this monopole Yagi-Uda antenna

Furthermore the estimated value of this new method isin general lower thanWalterrsquos when the finite ground plane isshort and two curves trend to approximation and the lengthof ground plane increases


Number 1 Number 10

10

909818727636545455364273182

09090

z

x3D maximumFrequencyPhase

91830

Surface current (f = 3) [simulation_1] (peak)

(a)

minus180

minus120

minus60

0

60

120

180

0 2 4 6 8 10

Monopole number n

Phas

e offs

et120593n

(∘)

(b)

Figure 4 The current distribution on the 10-element monopole end-fire array with a finite ground 119871119886= 200mm 119871

119892= 30mm and 119882 =

60mm at 119891 = 3GHz for a certain phase step Δ1205930 (a) current vector on the surface (b) actual current phase offsets of elements

Phase constant 120573a (k)09

75

80

85

90

95

100

105

110

115

10

15

20

25

30

35

40

45

50

Dire

ctiv

ity (d

B)

Directivity Tilt angle

10 11 12 13

120579t

(∘)

Tilt

angl

e

Figure 5 The relationships between 120573119886and directivity 120573

119886and tilt

angle for 10-element monopole end-fire array with a finite ground119871119886= 200mm 119871

119892= 30mm and119882 = 60mm at 119891 = 3GHz

33 Discussion on the Significance of 120573119886 in Estimating Basedon the phenomenon shown above we will explain thesignificance of wavenumber 120573119886 in the beam tilting estimationof monopole end-fire array

Setting 119871119886 = 2120582 the 120579119905-119871119892 curves when 120573119886 = 119896 11119896 12119896

and 13119896 are respectively shown in Figure 8 compared withthe curve of Walterrsquos The distinction between the condition120573119886 = 119896 and Walterrsquos estimation caused by the differentvector current elements (119869119911 119869119909) and (119870119910 119869119909) [6] is not quiteremarkable However as 120573119886 increases the distinction isgetting larger and 120579119905 with the same 119871119892 is getting lower It issuggested that 120579119905-119871119892 relationship is sensitive to 120573119886 and an

2r

h1 h2 h3 h4 h5 h6 h7 h8 h9

Feeding point

ZY

X

h10

S

Figure 6 Geometry for the 10-element monopole Yagi-Udaantenna

appropriate high value of120573119886 can result in a low beam tilt anglein the range where ground plane is short

By changing the arrayrsquos feeding situation or structureparameters the phase constant of monopole end-fire arraycan be adjusted in a wide range Therefore the evaluation of120573119886 will determine the validity and accuracy of estimation

4 Experiment for 10-Element MonopoleYagi-Uda Antenna

According to the analysis in Section 3 for a 10-elementmonopole end-fire array when the total length of finiteground plane is about 119871 = 119871119886 + 119871119892 = 23120582 the beam tiltangle is about 27∘ when 120573119886 = 119896 For the same length when 120573119886

is satisfyingHansen-Woodyard condition the tilt angle can be


0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Simulation by CSTProposed method in this paper Proposed method in this paperSimulation by CST

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

Walterrsquos method Walterrsquos method

Figure 7 Beam tilt angles of the 10-element antenna when 119871119886= 200mm got by the proposed method in this paper Walterrsquos method and

CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k

Walterrsquos method120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k

Figure 8 Estimated beam tilt angles of the 10-element monopole end-fire antenna when 119871119886= 2120582 120573

119886= 119896 11119896 12119896 and 13119896 and Walterrsquos

Method


Figure 9 Photo for the 10-element monopole Yagi-Uda antenna mounted on a finite ground plane with size 119871 = 230mm119882 = 60mm

030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)

Figure 10 Measured radiation pattern for the 10-element monopole Yagi-Uda antenna shown in Figure 9

lowered to about 20∘ which needs a ground plane whose totallength 119871 = 119871119886 + 119871119892 ge 45120582 when 120573119886 = 119896

The 10-element monopole Yagi-Uda antenna which sat-isfies this condition as mentioned in Section 3 is fabricatedand measured to prove this deduction Parameters for the 10-monopole array are totally the same as those given in Table 1The size of finite ground plane is 119871 = 230mm 119882 = 60mmThe photo of the fabricated antenna is shown in Figure 9

Simulation and measured results for the elevation (119883119885-plane) and azimuth (119883119884-plane) radiation patterns at 3GHzare shown in Figure 10 This antenna achieves about 1156 dBgain in practice and themain beam point at tilt angle of about21∘ from end-fire which is very close to the estimated tiltangle 20∘ at the beginning of this section This experimentalresult demonstrates not only the accuracy of the estimationmethod proposed in this paper but also the conclusionsdeduced at the end of Section 3 the beam tilt angle is sensitiveto 120573119886 and an appropriate 120573119886 can lower the angle with a shortground plane

5 Conclusion

Amodifiedmethod for the beam tilt-angle estimation of end-fire antenna mounted on finite ground plane is proposedand studied This method focuses on analyzing the case ofmonopole end-fire array The monopole array and groundplane are simplified into two line sources of transverseand longitudinal current source respectively Superpositionresults of two line sources suggest the beam tilt angle is afunction about the length of ground plane in front of array119871119892 the length of monopole array 119871119886 and the phase constant120573119886 Evaluating the value of 120573119886 correctly is demonstrated to bethe key point of this estimationmethod A low beam tilt anglecan be achieved by a monopole array satisfying Hanson-Woodyard condition with only a short-length ground planeA 10-element monopole Yagi-Uda antenna is designed andmeasured for the analysis It produced a tilt angle about 21∘which is close to the estimation value with a total ground size119871 = 230mm119882 = 60mm


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W Cao B Zhang A Liu T Yu D Guo and Y Wei ldquoGainenhancement for broadband periodic endfire antenna by usingsplit-ring resonator structuresrdquo IEEE Transactions on Antennasand Propagation vol 60 no 7 pp 3513ndash3516 2012

[2] X Ding and B Z Wang ldquoA novel wideband antenna withreconfigurable broadside and endfire patternsrdquo IEEE Antennasand Wireless Propagation Letters vol 12 pp 995ndash998 2013

[3] Y H Sun G J Wen H Y Jin P Wang and Y J HuangldquoGain enhancement for wide bandwidth endfire antenna withI-shaped resonator (ISR) structuresrdquo Electronics Letters vol 49no 12 pp 736ndash737 2013

[4] J Liu and Q Xue ldquoMicrostrip magnetic dipole yagi arrayantenna with endfire radiation and vertical polarizationrdquo IEEETransactions on Antennas and Propagation vol 61 no 3 pp1140ndash1147 2013

[5] R Elliott ldquoOn the theory of corrugated plane surfacesrdquo IRETransactions on Antennas and Propagation vol 2 no 2 pp 71ndash81 1954

[6] C H Walter Traveling Wave Antennas McGraw-Hill NewYork NY USA 1965

[7] H Nakano Y Ogino and J Yamauchi ldquoBent two-leaf antennaradiating a tilted linearly polarized wide beamrdquo IEEE Transac-tions onAntennas and Propagation vol 58 no 11 pp 3721ndash37252010

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ZY

X

W

S

h1 h2 h3 hn hN

LaLg

L

middot middot middotmiddot middot middot

Figure 1 119873-monopole end-fire antenna vertically mounted on afinite ground plane

proposed in this paper Firstly a simplified source modelis applied to this kind of antenna The key point of thismodel simplification is evaluating the phase constant 120573119886 ofthe surface wave along monopole end-fire array Secondlymonopole Yagi-Uda antenna is introduced to validate thevalidity and accuracy of this method The estimated tiltangles present a good agreement with CST simulation resultsFurthermore comparison between the proposed methodin this paper and Walterrsquos method shows an advantage ofmonopole end-fire antenna with slow wave characteristicThis advantage that a low tilt angle can be achieved with ashort ground length is demonstrated by the experimentalresults of the 10-element monopole Yagi-Uda antenna

2 Simplified Model for Monopole End-FireArray and Ground Plane

Figure 1 shows the configuration of an 119873-monopole end-fire array mounted on a ground plane with certain size(119882119871) Assume it is working at the frequency point 119891 with awavelength 120582 and 120596 is the corresponding angular frequencyConsulting the original method in [6] the simplified modelshould be composed of two parts one line source approx-imated from the monopole array current distribution andanother one approximated from the current on the groundplane Note that only far-field patterns in119883119885-plane are takeninto account for the beam tilt-angle problem as the mainlobe direction is always in this plane when the ground issymmetric about the119883-axis

21 Simplification for Monopole End-Fire Array Firstly thearray elements electric monopoles with heights ℎ119899 areapproximated by a series of ideal dipoles Ideal dipole isdefined as a Hertz dipole with uniform electric current 119868

on the total length Δ119897 ≪ 120582 and its radiation patternfunction is 119891 = sin 120579 Usually monopole height is lessthan or equal to 025120582 Assuming the current distributionon monopole is 119868(119911) = 119868119898 sin(119896(ℎ minus 119911)) where ℎ is the

0

0

10 20 30 40 50 60 70 80 90

Ideal dipole

minus60

minus50

minus40

minus30

minus20

minus10

h = 01120582

h = 02120582

h = 025120582

120579 (∘)

f = inth0Imsin(k(h minus z))ejkzcos 120579dz

Nor

mal

ized

pat

tern

f(120579

) (dB

)Figure 2 Radiation patterns of 01120582 02120582 and 025120582monopoles andof ideal dipole

zI1Δl zI2Δl zI3Δl zINΔlzInΔl

zJz(x)dxX

Transverse current line source

Ideal dipoleLa = NS

S

Figure 3 Ideal dipole array and transverse current line sourcecorresponding

height of monopole the radiation pattern can be derivedby (2) Figure 2 exhibits the curves of 119864-plane (119883119885-plane)normalized patterns of ideal dipole 01120582 monopole 02120582monopole and 025120582 monopole respectively It is suggestedthat ideal dipolersquos radiation pattern is quite similar to the onesof monopoles with height less than or equal to 025120582 so thisapproximation has no serious loss in accuracy

119891 = int

ℎ

0

119868119898 sin (119896 (ℎ minus 119911)) 119890119895119896119911 cos 120579

119889119911 (2)

Secondly as shown in Figure 3 the array of ideal dipoleswith electric current amplitude and phase offset as (119860119899 120593119899)on each is approximated by a transverse electric currentline source lying on the 119883-axis with amplitude 119860(119909) andphase velocity V(119909) in119883 direction For the line source lengthassume that each ideal dipole 119868119899Δ119897 = 119860119899119890

119895120593119899 lying on 119909119899

corresponds to a segment 119904 (spacing between two adjacentelements) in length so the whole length of line source will be

119871119886 = 119873119904 (3)



119869119911 (119909) = 119860 (119909) 119890minus119895120573(119909)119909

0 le 119909 le 119871119886 (4)

where

120573 (119909) =120596

V (119909) (5)




120573119886 =sum119873

119899=2(120593119899 minus 120593119899minus1)

119871119886

(6)



119869119911 (119909) = 119860119886119890minus119895120573119886119909

0 le 119909 le 119871119886 (7)


120579119864119886 = 120579119860119886119890119895119883119886

sin119883119886

119883119886

119871119886 sin 120579 (8)

where119883119886 = (1198711198862)(119896 sin 120579 minus 120573119886)





119869119909

=

119860119892119890minus119895((120573

119886+119896)2)119909

0 le 119909 le 119871119886

119860119892119890minus119895[119896(119909minus119871

119886)+((120573119886+119896)2)119871

119886]

= 119860119892119890minus119895(119896119909+((120573

119886minus119896)2)119871

119886)

119871119886 lt 119909 le 119871119886 + 119871119892

(9)


120579119864119892 = 120579119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(10)

where

1198831198921 =119871119886

2(119896 sin 120579 minus

120573119886 + 119896

2)

1198831198922 =

119871119892

2119896 (sin 120579 minus 1)

1198831198923 = (119871119886 +

119871119892

2) 119896 (sin 120579 minus 1) + 119871119886

119896 minus 120573119886

2

(11)



|119864119886|max = |119864119892|max gives

119860119886119871119886

sin119883119886

119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

(12)



119864119905 = 119864119886 + 119864119892

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

times119883119886

sin119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘119890119895119883119886

sin119883119886

119883119886

sin 120579

+ 119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(13)


119905






120573119886119871119886 = 120587 + 119896119871119886 (14)







119904 (mm) 20

ℎ1 (mm) 23

ℎ2 (mm) 215

ℎ3 (mm) 19

ℎ4 (mm) 17

ℎ5 (mm) 156

ℎ6 (mm) 155

ℎ7 (mm) 145

ℎ8 (mm) 1067

ℎ9 (mm) 683

ℎ10 (mm) 3








Number 1 Number 10

10

909818727636545455364273182

09090

z


91830


(a)

minus180

minus120

minus60

0

60

120

180

0 2 4 6 8 10

Monopole number n

Phas

e offs

et120593n

(∘)

(b)


119892= 30mm and 119882 =



75

80

85

90

95

100

105

110

115

10

15

20

25

30

35

40

45

50

Dire

ctiv

ity (d

B)


10 11 12 13

120579t

(∘)

Tilt

angl

e


119886and tilt


119892= 30mm and119882 = 60mm at 119891 = 3GHz




2r

h1 h2 h3 h4 h5 h6 h7 h8 h9

Feeding point

ZY

X

h10

S








0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30


Lg120582Lg120582

120579t

(∘)

120579t

(∘)



CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k


120573a = 12 k120573a = 13 k



Method



030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119869119911 (119909) = 119860 (119909) 119890minus119895120573(119909)119909

0 le 119909 le 119871119886 (4)

where

120573 (119909) =120596

V (119909) (5)




120573119886 =sum119873

119899=2(120593119899 minus 120593119899minus1)

119871119886

(6)



119869119911 (119909) = 119860119886119890minus119895120573119886119909

0 le 119909 le 119871119886 (7)


120579119864119886 = 120579119860119886119890119895119883119886

sin119883119886

119883119886

119871119886 sin 120579 (8)

where119883119886 = (1198711198862)(119896 sin 120579 minus 120573119886)





119869119909

=

119860119892119890minus119895((120573

119886+119896)2)119909

0 le 119909 le 119871119886

119860119892119890minus119895[119896(119909minus119871

119886)+((120573119886+119896)2)119871

119886]

= 119860119892119890minus119895(119896119909+((120573

119886minus119896)2)119871

119886)

119871119886 lt 119909 le 119871119886 + 119871119892

(9)


120579119864119892 = 120579119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(10)

where

1198831198921 =119871119886

2(119896 sin 120579 minus

120573119886 + 119896

2)

1198831198922 =

119871119892

2119896 (sin 120579 minus 1)

1198831198923 = (119871119886 +

119871119892

2) 119896 (sin 120579 minus 1) + 119871119886

119896 minus 120573119886

2

(11)



|119864119886|max = |119864119892|max gives

119860119886119871119886

sin119883119886

119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

(12)



119864119905 = 119864119886 + 119864119892

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

times119883119886

sin119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘119890119895119883119886

sin119883119886

119883119886

sin 120579

+ 119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(13)


119905






120573119886119871119886 = 120587 + 119896119871119886 (14)







119904 (mm) 20

ℎ1 (mm) 23

ℎ2 (mm) 215

ℎ3 (mm) 19

ℎ4 (mm) 17

ℎ5 (mm) 156

ℎ6 (mm) 155

ℎ7 (mm) 145

ℎ8 (mm) 1067

ℎ9 (mm) 683

ℎ10 (mm) 3








Number 1 Number 10

10

909818727636545455364273182

09090

z


91830


(a)

minus180

minus120

minus60

0

60

120

180

0 2 4 6 8 10

Monopole number n

Phas

e offs

et120593n

(∘)

(b)


119892= 30mm and 119882 =



75

80

85

90

95

100

105

110

115

10

15

20

25

30

35

40

45

50

Dire

ctiv

ity (d

B)


10 11 12 13

120579t

(∘)

Tilt

angl

e


119886and tilt


119892= 30mm and119882 = 60mm at 119891 = 3GHz




2r

h1 h2 h3 h4 h5 h6 h7 h8 h9

Feeding point

ZY

X

h10

S








0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30


Lg120582Lg120582

120579t

(∘)

120579t

(∘)



CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k


120573a = 12 k120573a = 13 k



Method



030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864119905 = 119864119886 + 119864119892

= 119860119892

1003816100381610038161003816100381610038161003816100381610038161003816

(1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 1205791003816100381610038161003816100381610038161003816100381610038161003816max

times119883119886

sin119883119886

10038161003816100381610038161003816100381610038161003816120579=90∘119890119895119883119886

sin119883119886

119883119886

sin 120579

+ 119860119892 (1198901198951198831198921

sin1198831198921

1198831198921

119871119886 + 1198901198951198831198923

sin1198831198922

1198831198922

119871119892) cos 120579

(13)


119905






120573119886119871119886 = 120587 + 119896119871119886 (14)







119904 (mm) 20

ℎ1 (mm) 23

ℎ2 (mm) 215

ℎ3 (mm) 19

ℎ4 (mm) 17

ℎ5 (mm) 156

ℎ6 (mm) 155

ℎ7 (mm) 145

ℎ8 (mm) 1067

ℎ9 (mm) 683

ℎ10 (mm) 3








Number 1 Number 10

10

909818727636545455364273182

09090

z


91830


(a)

minus180

minus120

minus60

0

60

120

180

0 2 4 6 8 10

Monopole number n

Phas

e offs

et120593n

(∘)

(b)


119892= 30mm and 119882 =



75

80

85

90

95

100

105

110

115

10

15

20

25

30

35

40

45

50

Dire

ctiv

ity (d

B)


10 11 12 13

120579t

(∘)

Tilt

angl

e


119886and tilt


119892= 30mm and119882 = 60mm at 119891 = 3GHz




2r

h1 h2 h3 h4 h5 h6 h7 h8 h9

Feeding point

ZY

X

h10

S








0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30


Lg120582Lg120582

120579t

(∘)

120579t

(∘)



CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k


120573a = 12 k120573a = 13 k



Method



030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Number 1 Number 10

10

909818727636545455364273182

09090

z


91830


(a)

minus180

minus120

minus60

0

60

120

180

0 2 4 6 8 10

Monopole number n

Phas

e offs

et120593n

(∘)

(b)


119892= 30mm and 119882 =



75

80

85

90

95

100

105

110

115

10

15

20

25

30

35

40

45

50

Dire

ctiv

ity (d

B)


10 11 12 13

120579t

(∘)

Tilt

angl

e


119886and tilt


119892= 30mm and119882 = 60mm at 119891 = 3GHz




2r

h1 h2 h3 h4 h5 h6 h7 h8 h9

Feeding point

ZY

X

h10

S








0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30


Lg120582Lg120582

120579t

(∘)

120579t

(∘)



CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k


120573a = 12 k120573a = 13 k



Method



030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30


Lg120582Lg120582

120579t

(∘)

120579t

(∘)



CST simulating

0 5 10 15 20 25 305

10

15

20

25

30

0 2 4 6 8 1010

12

14

16

18

20

22

24

26

28

30

Walterrsquos method

Lg120582Lg120582

120579t

(∘)

120579t

(∘)

120573a = k120573a = 11 k

120573a = 12 k120573a = 13 k


120573a = 12 k120573a = 13 k



Method



030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











030

60

150

120

180210

270

300

330

CST Mea

CST Mea

30

6090

90

120

150

180

210

240

240

270300

330

0

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

0

0

minus40

minus30

minus30

minus20

minus20

minus10

minus10

Nor

mal

ized

pat

tern

f(120579

) (dB

)

Nor

mal

ized

pat

tern

f(120579

) (dB

)

120579t (∘) 120579t (∘)





5 Conclusion





References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article beam tilt-angle estimation for monopole...

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