ANTENNA SPHERICAL COORDINATE SYSTEMS AND THEIR APPLICATION IN COMBINING RESULTS FROM DIFFERENT ANTENNA ORIENTATIONS

Transcription

1 NTNN SPHRICL COORDINT SSTMS ND THIR PPLICTION IN COMBINING RSULTS FROM DIFFRNT NTNN ORINTTIONS llen C. Newell, Greg Hindman Nearfield Systems Incorporated rd St. Bldg. 524 Carson, C 9745 US BSTRCT Te results of teoretical calculations or measurements on antennas are generally given in terms of te vector components of te radiated electric field as a function of direction or position. Bot te vector components and te direction parameters must be defined wit respect to a sperical coordinate system fixed to te antenna. long te principal planes tere is no ambiguity about te terms suc as vertical or orizontal component, but off te principal planes te definition of directions and vector components depends on ow te sperical coordinate system is defined. Tis paper will define four different sperical coordinates tat are commonly used in measurements and calculations, suggest a terminology tat is useful to distinguis between tem, and define te matematical transformations between tem. One important application of tese concepts arises wen comparing or combining measurement results from two antenna orientations. In tis case, te axis of rotation dictates te preferred coordinate system and vector components. Measured results will be sown to illustrate te proper coice of coordinates for eac situation. accomplised, te - - and -axes remain fixed to te antenna and do not cange for any of te sperical coordinate systems tat will be discussed. q 1. INTRODUCTION n antenna coordinate system is implicit in almost every antenna measurement. Terms suc as pattern, main and crosscomponent, beam pointing and peak gain, imply te definition of directions and/or vector field components wic require a coordinate system. Reference is often made to te crosscomponent of an antenna as if tere is a unique definition of suc a quantity wen in fact te cross-component as well as te main component will depend on wic coordinate system is used and ow it is oriented wit respect to te antenna. mbiguity and confusion can be avoided by using precise definitions and a terminology tat distinguises te different coordinates. Ludwig 1 proposed suc a terminology, and it is widely used. It does not clearly define all te coordinate systems commonly used owever, and te following discussion will propose additions to te Ludwig terminology. 2.THT PHI SPHRICL COORDINTS We begin by defining te -- axes of te antenna coordinate system suc tat te main beam is approximately along te -axis. Te - or -axes are defined approximately coincident wit te major polarization axis as sown in Figure 1. Te precise definition of te axes location and orientation must be accomplised by using fiducial marks, alignment mirrors or optical telescopes. Once tis definition is Figure 1 q-f Sperical Coordinates Te sperical coordinates sown in Figure 1 are te usual - coordinates wit te -axis as te polar axis. Te spere surrounding te antenna sould be muc larger tan te antenna since it represents te spere on wic te far-field vector components are measured or represented. For convenience it is sown just enclosing te antenna. Wit tis coordinate system, directions are specified by te angles (,) and vector components by te unit vectors q and f. Te rotator system used for tis coordinate system is te roll over azimut positioner sown in Figure 1. In a far field measurement, a vertically polarized source antenna illuminates te UT wit a field tat is polarized in te f-direction and te received signal will ten correspond to te -component pattern. orizontally polarized source will produce te -component pattern. It is important to note tat wen te antenna is rotated, te spere defining te sperical coordinates and components stays fixed to te UT and also rotates. Te spere is used to define te pattern radiated by te UT as a function of coordinates fixed to te antenna. Tis spere is not used to specify te directions te main beam points relative to axes fixed in space as te angles are canged.

2 v (, ) = (, )cos (, )sin (, ) = (, )sin + (, )cos v Corresponding transformations will be sown for all te coordinate systems. 4.LUDWIG-2, IMUTH-LVTION COMPONNTS (1) Tere are actually two coordinate systems tat come under te Ludwig-2 definition. Tey correspond to different types of far-field rotators and te corresponding orientations of te polar axis of te sperical coordinates. Te - components are used wit te zimut over levation rotator sown in Figure 3 were te polar axis is coincident wit te -axis. Figure 2 Ludwig-3 or -v vector components. If te UT is linearly polarized, te vectors q and f sown in Figure 1 are not te most appropriate for specifying te field vectors since in te region of te main beam te vectors cange direction relative to te antenna as a function of. Te orientation of te -axis relative to te antenna could be canged to define new sperical coordinates, but tere are some situations were te current definition is preferable, and it also makes it muc easier to transform between te oter sperical coordinates if we retain tis definition for te q f sperical coordinates. x 3. LUDWIG-3 OR H-V VCTOR COMPONNTS modification of te - components overcomes te polarization problem noted above, and is widely used in anecoic camber measurements. Te q and f unit vectors are rotated about te radial direction by te angle to obtain te vector components referred to as Ludwig-3 components. We ave used te notation and v to refer to tese components since tey define vectors tat are respectively approximately orizontal (parallel to te x-axis) and vertical (parallel to te y-axis) over most of te emispere. Te resulting coordinates and vectors for tis coordinate system are sown in Figure 2. In a far-field measurement using te Ludwig-3 components, te source antenna is rotated about its axis by te angle in syncronization wit te UT -rotation. If te magnitude and pase of two components in one system are known, te components in te oter one are found from te following transformations. Figure 3 z-l Coordinate system and azimut over elevation rotator. Wit tis rotator, te origin of te UT coordinate system is not centered on te antenna. It is at te intersection of te two axes of rotation, wic is inside te mecanical structure. Tis will not effect te far-field amplitude pattern, but will cange te pase pattern. Te rotation of te spere wit te UT for tis coordinate system is apparent in Figure 4 were te UT as been rotated in bot zimut and levation. Te point on te spere wit coordinates (,) is now along te line from te origin to te source antenna, and te and components at tis location will be measured by orizontally and vertically polarized source antennas respectively. Te transformations between - components and - components is given by,

3 cos (, ) cos sin = (, ) cos cos (, ) (2) were cos sin cos (, ) = (, ) + cos cos (, ) 2 cos = 1 (sin sin ). (3) Te angles in te different coordinate systems are related by, sin cos = cos sin = sin α, sin sin = sin = sin ε, (6) cos = cos cos = cos ε. e a Figure 5 a-e coordinate system wit associated elevation over azimut rotator. Figure 4 Rotated azimut over elevation rotator. 5. LUDWIG-2 LPH-PSILON COMPONNTS Te oter Ludwig-2 components are associated wit an elevation over azimut rotator were te polar axis is coincident wit te -axis sown in Figure 5. Te transformations for tese components are, cos cos sin Ε ( α, ε) = Ε (, ) Ε α sin cos cos Ε ( α, ε) = Ε (, ) + Ε ε cos sin Sin Εα ( α, ε) = Ε (, ) Ε sin Sin cos Ε ε ( α, ε) = Ε (, ) + Ε (, ) (, ) (, ) (, ) (4) (5) long te xz principal plane, Ε (, ) = Ε ( α, ) = Ε (, or) = ± Ε (, or) Ε (, ) = Ε ( α, ) = Ε (, or) = ± Ε (, or) α ε v nd along te yz-principal plane, Ε (, ) = Ε α (, ε) = Ε (, ± ) = mε (, ± ) 2 2 (8) Ε (, ) = Ε ε (, ε ) = Ε v (, ± ) = ± Ε (, ± ) 2 2 Terefore along te principal planes, tere is little or no difference between te components, but as we move off te principal planes, te differences increase. Tis means for example tat if a far-field measurement using an levation over zimut rotator is compared to te results of a near-field measurement, te comparison will depend on wic components are used in te near-field program. ll te components will sow good correlation along te principal planes, owever in directions off te principal planes, only te α-ε components will sow good agreement wit te L/ far-field measurements. If measured results from near-field, (7)

4 far-field or anecoic measurements are compared to teoretical predictions, te vector components must be te same for measurement and teory or te results will not agree. Suc disagreement can mistakenly be interpreted as indication of problems wit measurements or antenna construction wen te actual cause is an inconsistent coice of coordinate systems. 6. COMBINING RSULTS FROM DIFFRNT NTNN ORINTTIONS noter measurement application tat requires te proper coice of bot te vector components and te coordinate system used to define directions as recently been developed at Nearfield Systems Inc. Tis application uses two or more planar near-field measurements on te same antenna were te antenna is rotated about one axis between eac measurement. Te results of tese measurements can be used to extend te angular coverage of te planar near-field measurements or to estimate te magnitude of specific errors in a measurement. Bot of tese applications will be illustrated. Relative mplitude in db -4 Deg Beam Figure 7 Illustrating te increase in angular coverage by combining steered beams. +2 Deg Beam -2 Deg Beam zimut ngle in Degrees system defines a direction or location in space and te vector components define two ortogonal components of te electric field. In a sense te "natural" coordinate system is te one tat corresponds to te vector components since te field vectors are along te lines of te corresponding coordinates. In te specific case of a rotated coordinate system, tere is a compelling reason to use a specific coordinate system and a specific set of vector components as te "initial" or "natural" set. For rotation about te z-axis,, coordinates and vector components are te natural coice. For rotation about te x- axis, α, ε coordinates and vector components are te natural coice. For rotation about te y-axis,, coordinates and vector components are te natural coice. If tese natural coordinates and components are cosen tere will not be any cange in te sape of te pattern due to rotation, only an offset. Once te offset as been taken care of, we can always convert to any oter components or coordinates. Figure 6 Slot rray test antenna in initial unrotated orientation. Te angular coverage of planar near-field measurements is limited by te size of te scan plane, and te region of validity is defined by te angle between te edge of te UT and te edge of te scan plane. In some applications, results are required over a larger angular region tan is possible wit te available scanner. Te angular coverage can be increased by rotating te antenna and repeating te measurement. Te results of te two measurements are ten combined. Successful combination depends on using bot te coordinate system and vector components tat are appropriate for te antenna rotation. In general for a single antenna orientation, any coordinate system suc as (x, y, z); (k x, k y, k z ), (, ); (α, ε); or (, ); can be used wit any vector components since te coordinate Figure 6 sows te antenna tat was used for te following tests in its initial orientation wit te main beam normal to te scan plane. It is an -Band slotted array approximately 14 inces in diameter. Wit a measurement lengt of 6 inces in x and y, and a z-distance of 8 inces, te resulting far-field pattern is valid over ±7 degrees in bot azimut and elevation. To increase te angular coverage in te azimut direction, te antenna was rotated +2 degrees about te y- axis using a rotator tat was aligned wit its axis of rotation parallel to te y-axis of te scanner. Tis corresponds to te zimut angle of te coordinate system sown in Figure 3. Near-field measurements were repeated, for tis antenna orientation and for a similar rotation of 2 degrees in zimut. Figure 7 sows te results of tese tree measurements along te orizontal principal plane. Te +2 degree and 2 degree patterns ave been sifted to put te peak on-axis for tis display. Te az imutal coverage as now been increased to ± 9 degrees and it is apparent tat te patterns sow very good agreement witin teir common regions of validity.

5 Relative mplitude in db Degrees + 2 Degrees - 2 Degrees levation ngle in Degrees Figure 8 V-cut troug beam peak for tree zimut rotations of UT. L coordinates and vector components used. Relative mplitude in db Degrees + 2 Degrees -2 Degrees levation ngle in Degrees Figure 9 V-cuts troug te beam peak for tree zimut rotations of te UT. lpa psilon coordinates and vector components. Tese used. patterns ave used coordinates and vector components tat are te natural coice for rotation about te y-axis. Wit tis coice, te patterns sould be sifted in zimut and te sape of te patterns sould not cange. Tis is illustrated in Figure 8 were vertical cuts troug te peaks of te beams are plotted for te degree, -2 degree and +2 degree antenna orientations. Tese patterns sow excellent agreement, tat is in contrast to Figure 9 wic is a similar comparison of te V-cuts using α-ε coordinates and components. Te comparison ere is especially poor at angles far off axis. Te differences in te patterns in Figure 9 do not mean tat tere is an error in te measurement or tat it is incorrect to use te α-ε coordinates and components. ll tree curves are a correct and valid representation of te field for te given situation. Te patterns are different because te sape of te pattern canges wen using tese coordinates and components wit a rotation about te y-axis. If te sifted patterns are going to be combined, or if te sifted patterns are going to be compared and te differences used to estimate errors in te measurement, coordinates and vector components must be used wit a rotation about te y -axis. Relative mplitude in db Deg. +2 Deg. -2 Deg. rror Level levation ngle in Degrees Figure 1 Comparison of V-Cuts troug beam peak for tree rotations of UT after averaging 4 results for eac beam rotation. 7. STIMTING RRORS IN PLNR NR-FILD MSURMNTS Comparing te results of two different near-field measurements can be an effective tool for estimating errors if te measurement system is canged in a known way between measurements 2. Te cange in te measurement system is designed to identify one or more sources of error wile leaving oter sources uncanged. Rotation of te antenna about one axis can identify some sources of error tat are difficult to quantify wit oter tests as illustrated by te following examples. If te correct coordinates and components are used for a given rotation, if te rotation is precisely known, and if tere are no errors in te measurement system, te original and rotated patterns sould agree exactly. For te azimut rotations previously described, te rotations were carefully controlled and known, and te differences seen in Figure 8 are due to measurement errors. Multiple reflections between te UT and te probe produce some errors in every near-field measurement. Tis error can be identified and partially corrected by taking measurements at a series of 4 z-distances in increments of λ/8. Tis was done for all tree of te azimut beam rotations and te four far-field measurement

6 results for eac beam rotation were averaged to reduce te effect of te multiple reflections. Te average far-fields for te tree beam rotations were ten compared as sown in Figure 1 and a residual error signal calculated tat would cause te differences in te patterns. In te main beam region, were te multiple reflection error is te largest, te residual error level is approximately 5 db wile in te sidelobe region it as been reduced to approximately 6 to 65 db. Te error sources tat sould be identified by te steering of te beam in zimut are -position errors, pase linearity and room reflections in te zimut directions. From te results sown in Figure 1, te combined effect of tese error sources is less tan -6 db below te main component peak. Te beam rotation tests were repeated for levation rotations Relative mplitude in db L = Deg L = Deg. rror Signal levation ngle in Degrees Figure 12 Results of comparing veraged patterns for degrees and degrees levation beam rotations. Near-field amplitude wit Beam L = 24.5 Deg Figure 11 ntenna rotated in levation 24.5 degrees. of 8.5 and 24.5 degrees. It was not possible wit available equipment to control te rotations as precisely for tese measurements, and so te comparisons were not as good as for te zimut rotations. Te results do illustrate te identification of a different source of error as sown in Figure 12. Te large error signal of approximately -35 db for levation angles between 3 and 7 degrees is due to scattering from eiter te metal support beind te antenna or te floor. Te effect of tis scattering is evident in te nearfield data were te amplitude sows local peaks at te bottom of te scans as indicated by Figure 13. Wit tis information, additional absorber could be added beind te antenna or on te floor to furter reduce te effect of tis scattering. 8. CONCLUSIONS Four different sperical coordinate systems and te associated vector components associated wit tese coordinates ave been defined. Far-field or Near-field measurements can be presented using any of tese systems, and eac is a valid representation of te antenna pattern and polarization. If comparisons are made between different measurements or between measurements and teoretical calculations, te coordinates and components must agree. lso if te antenna is rotated about one axis and measurements are combined, te proper coordinates and components must be used tat will not cange te sape of te pattern for te rotation. mplitude (db) Figure cut -2-15near-field -1-5 amplitude for 2 beam 25 3 (in) steered 24.5 degrees in levation. 9. RFRNCS 1. C. Ludwig, Te definition of cross polarization, I Trans. ntennas Propag. Vol 21, No. 1 pp , January C. Newell, rror nalysis Tecniques for Planar Near-Field Measurements, I Trans. ntennas Propag., Vol 36, No. 6, pp , June 1988.