Antenna Field Transformation with Directive Sources

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1 Forum for Eectromagnetic Research Methods and ppication Technoogies (FERMT) ntenna Fied Transformation with Directive Sources LPC Loughborough ntenna & Propagation Conference Thomas F. Technica University of Munich November

2 Copyright The use of this work is restricted soey for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author. November

3 bstract Inverse equivaent source soutions are powerfu approaches for antenna fied transformations and they provide in particuar very detaied diagnostic insight into the radiation behaviour of the considered antenna. Commony such techniques work with eectric and/or magnetic surface current densities, where the fied observation equations are sometimes augmented by a nu-fied condition for the region externa to the Huygens surface. The surface current densities obtained with such a nu-fied condition are known as Love currents and they are directy inked to the tangentia eectric and magnetic fieds on the Huygens surface. so, it is cear that these currents radiate in genera (for convex surfaces) ony away from the antenna under test (UT) and are, therefore, very suitabe for antenna diagnostics. s an approximate and inexpensive aternative to the nu-fied condition, we propose to work with directive surface sources, which are obtained by an impedance boundary condition inking eectric and magnetic surface currents densities and by shifting the equivaent surface current densities into compex space. These directive sources are integrated into the fast irreguar antenna fied transformation agorithm which achieves ow numerica soution compexity by utiizing mutieve fast mutipoe ike principes. Fied transformation resuts based on rea measurement data are investigated to iustrate the performance of the directive surface sources. Keywords: antenna measurement, near-fied measurement, fied transformation, antenna diagnostic, integra equation, fast mutipoe method November

4 Outine Introduction: Near-Fied ntenna Measurements and Transformations The fied transformation probem with equivaent surface sources Variations of the Huygens principe Directive source modeing: Huygens radiators and sources in compex space Resuts with measured and syntheticay generated near-fied data Gaussian beam based pane wave transation operators Concusions November

5 ntenna Measurement Chamber at Technica University of Munich Near-fied and far-fied measurements 500 MHz up to 24 GHz UT Probe November

6 ntenna Measurement Chamber at Technica University of Munich Panar near-fied measurements 26 GHz up to 110 GHz Source: C. Koenen, et. a.: voumetric Near-Fied Scanner for Miimeter-Wave ntenna Measurements November

7 Portabe ntenna Measurement Systems (PMS) a project with the European Space gency (ES) fied transformation agorithms by TUM HFT PMS gondoa PMS gondoa Laser tracker Laser tracker November

8 utomobie ntenna Measurements above PEC Ground Cooperation with ntenna Technoogy Center (Europe) TC GmbH Hemispherica near-fied antenna measurement faciity Metaic ground pane November

9 ntenna Near-Fied Measurements and Transformations UT representation by equivaent sources probe... probe b1 b2... Measure UT near-fied for sufficient number of sampe ocations Representation of UT by equivaent sources: eigenmode expansions equivaent currents pane waves (far-fied) Near-fied sampes reated to equivaent sources (radiation operator) minimum sphere a1 a2... inverse probem... b1 b2... Equivaent sources determined by inverse probem soution Far-fied/near-fied obtained from equivaent sources vaiabe transformation agorithms differ in their equivaent sources and radiation operator representations! November

of measured fied Probe correction mandatory Fitering in spatia domain: U ( r ) =

r ) = w ( k ) ( k ) dkˆ wprobe ( r ) wprobe( k ) Er Ek ( ) ( ) M Pro be E spatia

10 Why Inverse Probem Soution? Infuence of the Measurement Probe Rea probe finite sized Spatia/spectra fitering of measured fied Probe correction mandatory Fitering in spatia domain: U ( r ) = w ( r r ) Er ( ) dv M V Probe Probe M ( ) Ur M Fitering in spectra domain: 2 U ( r ) = w ( k ) ( k ) dkˆ wprobe ( r ) wprobe( k ) Er Ek ( ) ( ) M Pro be E spatia probe weighting function spectra probe weighting function spatia near-fied spectra near-fied of UT at probe ocation November

11 Radiation Mode with Equivaent Surface Current Sources Consider the measured or simuated near-fieds of an UT or a scatterer (DUT). ( ) Ur M ( ) Ur M Virtua Huygens surface ˆn J M E E U ( rm ) = wprobe( r rm ) GJ ( r, r ) J ( r ) + GM ( r, r ) M ( r ) da dv V Probe Huygens surface on object surface ˆn J M Key question of this presentation: What are the most appropriate surface sources? November

12 The Standard Form of the Huygens Principe ˆn ( ) Ur M Eectric and magnetic surface currents reated to tangentia fieds: J n H = M = n E Virtua Huygens surface J M Mosty attributed to Schekunoff/Love Excusion voume free of any fied (nu-fied) Wi we obtain this soution if we just sove the inverse probem as it is? S. Schekunoff, Some equivaence theorems of eectromagnetics and their appication to radiation probems, Be System Technica Journa, vo. 15, no. 1, pp , E. H. Love, The integration of the equations of propagation of eectric waves, Phiosophica Transactions of the Roya Society, vo. 197, no , pp. 1 43, November

Equivaent Surface Current Discretization Eectric and magnetic

p ( ) M ( r ) = M β r q q q ( ) β n : vector basis functions Derive

RWG: Low order 1 st order Quasi Cur-Free 3 basis functions per

13 Equivaent Surface Current Discretization Eectric and magnetic surface current densities on trianguar meshes 3 J ( r ) = J β r p p p ( ) M ( r ) = M β r q q q ( ) β n : vector basis functions Derive inear equation system by Method of Moments procedure and sove it. RWG: Low order 1 st order Quasi Cur-Free 3 basis functions per triange ( ) β = ˆn λ λ λ λ Rao Witon Gisson November

14 Near-Fied Measurement HF907 by Rohde & Schwarz Probe UT HF907 spherica near-fied measurement at 18 GHz measurement distance 2.73 m performed in the anechoic chamber at TUM November

15 Transformed Far-Fieds HF907 Reative far-fied pattern in db Copo E-pane FIFT curr. FIFT sph. NSI Diff. sph. Reative far-fied pattern in db Crosspo E-pane FIFT curr. FIFT sph. NSI Diff. sph ϑ in degree ϑ in degree FIFT: Fast irreguar antenna fied transformation agorithm November

16 Surface Current Densities J/M Without Further Condition HF907 J in / m Rea J in / m Imag J in / m y x z undesired M currents in V/ m Rea M in V/ m Imag M in V/ m y x z November

17 Eectric Near-Fied Due to J/M Without Further Condition HF907 Undesired radiation through the UT voume! Eectric fied magnitude in dbv/m Eektrische Fedampitude in dbv/m Eectric fied magnitude in dbv/m Eektrische Fedampitude in dbv/m x in m undesired radiation y in m undesired radiation z in m z in m xz-pane yz-pane November

18 Surface Current Densities J/M with Love Condition HF907 J in / m Rea J in / m Imag J in / m y x z M in V/ m Rea M in V/ m Imag M in V/ m y x z Undesired currents/radiation avoided, however,... November

19 Variations of the Huygens Principe What kinds of surface current densities aow correct representation of the radiation fieds? i.e are suitabe for the Huygens principe... Virtua Huygens surface ˆn J M ( ) Ur M Standard form, Schekunoff/Love J n H = M = n E expicit enforcement required (surface integra equation) arge computationa effort, M without further condition J works fine but redundant Either of or M aone J works but bad operator Linear combination of and M 0 ( e.g., M = Z n J ) : imped. bound. cond. approximate Love condition so-caed Huygens radiator J S. Schekunoff, Some equivaence theorems of eectromagnetics and their appication to radiation probems, Be System Technica Journa, vo. 15, no. 1, pp , E. H. Love, The integration of the equations of propagation of eectric waves, Phiosophica Transactions of the Roya Society, vo. 197, no , pp. 1 43, November

20 Variations of the Huygens Principe The goa is to have surface sources, which radiate predominanty to the outside! Love condition perfect, but with arge computationa effort (surface integra equation) Impedance boundary condition RWG Huygens radiator 0 ( M = Z n J ) gives directive basis functions J Ipβ p M IpZ0 ( = = n β p ) Shifting the sources into compex space gives further directivity to the outside! Huygens principe sti vaid appicabe to a types of sources as discussed before we known within the framework of compex source beams/gaussian beams r = r + jn rea T. B. Hansen and G. Kaiser, Generaized Huygens principe with pused-beam waveets, Journa of Physics : Mathematica and Theoretica, vo. 42, no , pp. 1 33, T. B. Hansen and G. Kaiser, Huygens principe for compex spheres, IEEE Transactions on ntennas and Propagation, vo. 59, no. 10, pp , November

4λ p eectric current eectric current eectric current x in m J = 1 β p M = 1 Z n = 0.

21 x in m Radiation Fieds of Rao-Witon-Gisson-Radiators Normaized eectric fied magnitude in db J = 1 = 0.0λ β p Normaized eectric fied magnitude in db J = 1 β = 0.2λ p J Normaized eectric fied magnitude in db = 1 β = 0.4λ p eectric current eectric current eectric current x in m J = 1 β p M = 1 Z n = 0.0λ 0 ( β p) J = 1 β p = 1 0 ( β p) M = 1 Z0 ( n β p) M Z n = 0.2λ J = 1 β = 0.4λ p g Huygens radiator Huygens radiator Huygens radiator y in m y in m y in m November

22 Transformed Far-Fieds HF907 Reative far-fied pattern in db Copo E-pane Source mode: Eectric and Huygens radiators with imaginary shift 0 J, =0.2λ JH, =0.2λ JM, =0.0λ Error wrt. JM, =0.0λ ϑ in degree Reative far-fied pattern in db Copo H-pane = 0.2λ J, =0.2λ JH, =0.2λ JM, =0.0λ Error wrt. JM, =0.0λ ϑ in degree November

2λ Eectric fied magnitude in dbv/m Eectric fied magnitude in

23 Eectric Near-Fied Due to Directive Sources HF907 Source mode: Eectric and Huygens radiators with imaginary shift = 0.2λ Eectric fied magnitude in dbv/m Eectric fied magnitude in dbv/m x in m y in m z in m z in m xz-pane yz-pane November

24 Surface Current Densities HF907 J/M: no constraint, no imaginary shift Huygens radiators: = 0.2λ z z J in / m J in / m November

25 Eectric Near-Fied Due to Directive Sources HF907 Near-fied aong the UT main beam axis dependent on the imaginary shift Eectric fied magnitude in dbv/m eectric currents J, =0.0λ J, =0.1λ J, =0.2λ J, =0.3λ J, =0.4λ z in m Eectric fied magnitude in dbv/m Huygens radiators JH, =0.0λ JH, =0.1λ JH, =0.2λ JH, =0.3λ JH, =0.4λ z in m November

26 U ( r ) = w ( r r ) E( r ) dv M Probe M V Probe E E Er ( ) = GJ ( rr, ) J ( r ) + G M( rr, ) M ( r ) da E E U ( rm ) = wprobe( r rm ) GJ ( r, r ) J ( r ) + GM ( r, r ) M ( r ) da dv V Probe The Spectra Domain Near-Fied Operator : spatia representation E 1 e GJ ( rr, ) = jωµ I+ 4 - jk r r 2 k π r r - jk r r E 1 e GM ( rr, ) = I 4π 4π r r : Green s functions - jk X + d L - jk d jk (2) ˆ 2 e (- ) (2 + 1)h ( )P( ) 4π = 0 e j kx kˆ X d kˆ X + d T ( k, X) : FMM transation operator L ˆ2 Ur ( ) = ωµ M j wproe b ( k) TL( k, rm) J ( k) + M ( k) dk 4π : spectra representation November

27 The Spectra or Pane Wave Representation ωµ 2 Ur ( ) ( ) (, ) ( ) ( ) ˆ M = j wprobe k TL k r M J k + M k dk 4π jk ( r - rm ) w Probe( k ) = wprobe( r - rm )e dv V Probe Probe radiation pattern jk T k r j kr k r L (, ) (- ) (2 1)h (2) ( )P( ˆ L = + ˆ) 4π = 0 ˆˆ jk ( r rs ) J ( k ) = ( I kk) J ( r ) e da k ( ) ( ) ( ˆ jk r rs M k = j k M ( r )) e da 4π UT radiation pattern Incident pane waves at fied probe Pane waves radiated from UT November

28 Fast Mutipoe Near-Fied and Far-Fied Transation Operators FMM-Standard Transation Operator FMM-Far-Fied Transation Operator jk T k r j kr k r L (2) (, ) (- ) (2 1)h ( )P( ˆ ˆ L = + ) 4π = 0 FF T L e jkr r ( ) ( ) = δ ϑ ϑ0 δ ϕ ϕ0 r somehow directive, but with strong side obes a pane waves need to be considered very directive just one pane wave to be considered appicabe in far-fied distance good accuracy hard to achieve Fied transformation probems have reativey arge transation distances due to separation of sources and observation ocations! This is in contrast to MLFMM soution of radiation and scattering integra equation soutions. November

29 Fast Mutipoe Near-Fied and Far-Fied Transation Operators Transation operator with ess side obes than standard operator? T. J. Cui, W. C. Chew, G. Chen, and J. Song, Efficient MLFM, RPFM, and FFF agorithms for EM scattering by very arge structures, IEEE Transactions on ntennas and Propagation, vo. 52, no. 3, pp , R. J. Burkhoder and D.-H. Kwon, High-frequency asymptotic acceeration of the fast mutipoe method, Radio Science, vo. 31, no. 5, pp , Ray propagation fast mutipoe method / windowed FMM Summation over the mutipoes with a windowing function Fast far-fied approximation fast mutipoe method FF T L e jkr r ( ) ( ) = δ ϑ ϑ0 δ ϕ ϕ0 r November

30 Gaussian Beam Based Transation Operators Observation: Sources in compex space are directive. T. B. Hansen, Transation operator based on Gaussian beams for the fast mutipoe method in three dimensions, Wave Motion, vo. 50, pp , T. B. Hansen, O. Bories Gaussian transation operator in a muti-eve scheme, Radio Science, vo. 50, pp , jk X + d L - jk d jk (2) ˆ 2 e (- ) (2 + 1)h ( )P( ) 4π = 0 e j kx kˆ X d kˆ X + d T ( k, X) By choosing - jk X + d L L X + d = X + d X ( j X ˆ ) ( + j ˆ) a highy directive transation operator with very ow side obes can be constructed: e - ( ˆ ˆ jk d jk k kx ) (2) ˆ ˆ 2 e e (- ) (2 1)h ( )P( ) ˆ j + kx jk k X d k X + d 4π = 0 T ( k, X, ) L November

31 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 1λλ November

32 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 2λλ November

33 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 3λλ November

34 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 4λλ November

35 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 5λλ November

36 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 6λλ November

37 Gaussian Beam Based Transation Operators ( ˆ ˆ L jk ) (2) (,, ) (- ) (2 1)h ( )P( ˆ ˆ ) 4 e k kx TL k X = j + kx jk k X π = 0 Consideraby fewer pane wave sampes needed Extended transation ranges of benefit Fewer MLFMM eves needed Faster fied transformation with ess memory Fuy error controabe Carefu choice of parameters needed Δ = 0 Δ = 7λλ November

Rohde & Schwarz HF907 ntenna Transformation Performance Mesh with 188 090 triange 393 452 Rao-Witon-Gisson unknowns (one set) 45 451 near-fied sampe points with two poarizations measured at every

38 Rohde & Schwarz HF907 ntenna Transformation Performance Mesh with triange Rao-Witon-Gisson unknowns (one set) near-fied sampe points with two poarizations measured at every point CPU: one core of Inte i7, 3.6 GHz One iteration: 90 s (standard FMM transator) 12 s (Gaussian transator) 7.5 times speed-up! Typica iteration counts: 25 to 200 (Dependent on the mode and accuracy) November

39 25 m (500λ) Refector ntenna Method of Moments Soution RWG unknowns 145 GByte RM 6 GHz 4-Eement Yagi Feed with deta gap votage source J in / m November

25 m (500λ) Refector ntenna Synthetic Near-Fied Data reguar samping, z = 7 m (1 m above aperture) λ/2 sampe distance 5 764 801 measurement ocations (two

40 25 m (500λ) Refector ntenna Synthetic Near-Fied Data reguar samping, z = 7 m (1 m above aperture) λ/2 sampe distance measurement ocations (two orthogona poarizations each) Normaized E in db in scan pane obtained from MoM soution November

13 m (sighty above aperture) Copo, E-pane Eectric far-fied in dbv/m Error in db 40 20 0-20 -40-60 -80-100 MoM ref FIFT Error J in /

41 25 m (500λ) Refector ntenna Transformation Resuts Reguary samped near-fied data Equivaent surface currents from FIFT transformation z = 6.13 m (sighty above aperture) Copo, E-pane Eectric far-fied in dbv/m Error in db MoM ref FIFT Error J in / m nge with aperture norma in degree surface current unknowns CPU time on one core of Inte i7, 3.6 GHz: about 1378 sec per iteration (99 iterations, fina resut) RM about 106 GByte (eectric currents) November

42 25 m (500λ) Refector ntenna Synthetic Near-Fied Data strongy irreguar samping, z = 7 m (1 m above aperture) λ/2 sampe distance on average measurement ocations (two orthogona poarizations each) y in m z in m x in m x in m November

43 25 m (500λ) Refector ntenna Transformation Resuts 4-Eement Yagi Feed with deta gap votage source Reative far-fied pattern in db Copo, E-pane Error wrt. MoM Irreguar Reguar MoM ϑ in degree Directive sources very hepfu for irreguar samping on order to achieve ow error! November

44 Concusions Near-fied transformations with directive sources: Two ways: Huygens radiators and imaginary shift into compex space Huygens principe sti fuy appicabe Improved diagnostic capabiities Improved conditioning of resuting equation systems Improved transformation performance with Gaussian beam transators November

45 References. D. Yaghjian, n Overview of Near-Fied ntenna Measurements, IEEE Transactions on ntennas and Propagation, vo. 34, no. 1, pp , C. H. Schmidt, M. M. Leibfritz, T. F., Fuy Probe-Corrected Near-Fied Far-Fied Transformation Empoying Pane Wave Expansion and Diagona Transation Operators, IEEE Transactions on ntennas and Propagation, Vo. 56, No. 3, pp , March T. F., C. H. Schmidt, Mutieve Fast Mutipoe cceerated Inverse Equivaent Current Method Empoying Rao-Witon-Gisson Discretization of Eectric and Magnetic Surface Currents, IEEE Transactions on ntennas and Propagation, Vo. 57, No. 4, pp , pri C. H. Schmidt, T. F., Mutieve Pane Wave Based Near-Fied Far-Fied Transformation for Eectricay Large ntennas in Free-Space or bove Materia Hafspace, IEEE Transactions on ntennas and Propagation, Vo. 57, No. 5, pp , May T. F., Ismatuah, E. Kaiyaperuma, C. H. Schmidt, Inverse Equivaent Surface Current Method with Hierarchica Higher Order Basis Functions, Fu Probe Correction and Mutieve Fast Mutipoe cceeration (Invited Paper), Progress in Eectromagnetics Research (PIER), Vo. 106, pp , T. F., Emre Kiic, C. Lopez, R..M. Mauermayer, O. Neitz, G. Schnattinger, Eectromagnetic Fied Transformations for Measurements and Simuations (Invited Paper), Progress in Eectromagnetics Research (PIER), Vo. 151, pp , S. Schekunoff, Some equivaence theorems of eectromagnetics and their appication to radiation probems, Be System Technica Journa, vo. 15, no. 1, pp , E. H. Love, The integration of the equations of propagation of eectric waves, Phiosophica Transactions of the Roya Society, vo. 197, no , pp. 1 43, T. B. Hansen, G. Kaiser, Generaized Huygens Principe with Pused-Beam Waveets, Journa of Physics : Mathematica and Theoretica, vo. 42, no , pp. 1 33, T. B. Hansen and G. Kaiser, Huygens principe for compex spheres, IEEE Transactions on ntennas and Propagation, vo. 59, no. 10, pp , T. F., D. Vojvodic, T. B. Hansen, Fast Inverse Equivaent Source Soutions With Directive Sources, IEEE Transactions on ntennas and Propagation, Vo. 64, No. 11, November T. B. Hansen, Transation Operator Based on Gaussian Beams for the Fast Mutipoe Method in Three Dimensions, Wave Motion, vo. 50, pp , T. B. Hansen, O. Bories, Gaussian Transation Operator in a Muti-Leve Scheme, Radio Science, vo. 50, pp , November

Physics 506 Winter 2006 Homework Assignment #6 Solutions

Physics 506 Winter 2006 Homework Assignment #6 Solutions Physics 506 Winter 006 Homework Assignment #6 Soutions Textbook probems: Ch. 10: 10., 10.3, 10.7, 10.10 10. Eectromagnetic radiation with eiptic poarization, described (in the notation of Section 7. by