Near-field Computation and. Uncertainty Estimation using Basic. Cylindrical-Spherical Formulae

Transcription

1 Nea-field Computation and Uncetainty Estimation using Basic Cylindical-Spheical Fomulae Poject... IEC 63 Subject... Basic Nea-Field Computation Document numbe... N/A Revision... 3 Autho... F.D. du Plessis F.J.C. Meye Date... 7 Ap 11

2 Basic Nea-Field Computation CONTENTS 1. INTRODUCTION Intoduction Layout of document SUMMARY OF THE BASIC COMPUTATION METHODS Cylindical equations Spheical equations Boundaies fo computation Calculation fo multiple souces THE SIMPLE/ADJUSTED SPHERICAL FORMULAE Deiving of the simple/adjusted spheical fomulae Reduced fa-field bounday Veification of the adjusted spheical fomulae UNCERTAINTY ESTIMATIONS Antenna paamete anges Adjustment tables CONCLUSIONS REFERENCES... 3 Page of 3

3 Basic Nea-Field Computation 1. INTRODUCTION 1.1 Intoduction Simple cylindical-spheical (C-S) fomulae can be used to pedict the adiofequency (RF) field intensities emitted fom panel and omni-diectional cellula base station antennas. These fomulae allow compliance assessments to be pefomed without knowing the specific antenna apetue dimensions usually concealed in the antenna adome. The fomulae can be applied in the eactive and adiating nea-field egions as well as the fa-field egion of an antenna. In the eactive and adiating nea-field egions in font of an antenna the popagating fields show a cylindical chaacte. The cylindical fomulae by Faaone et al [1] [] is used in this document to calculate the powe density in this egion. When moving into the fa-field of an antenna the popagating fields stat showing a spheical chaacte. The classical spheical equation S P 4π can be used in the fa-field of an antenna whee S is the powe density P is the tansmitted powe is the gain of the antenna and is the distance fom the antenna. Fo the egion between the spheical fa-field bounday and the cylindical nea-field egion an adjusted spheical equation was fomulated. The accuacy of the above mentioned fomulae is investigated in the diffeent egions fo mobile base station antennas in the 9 18 and 1 MHz bands. Tables ae povided so that the fomulae can be applied with a specified cetainty with o without extenal paametes such as eflections and cable loss. 1. Layout of document The basic fomulae fo powe density calculation and the applicable egions ae defined in Chapte of this document. In Chapte 3 the adjusted spheical equations ae fomulated and validated against a ange of benchmak antenna simulations. In Chapte 4 the diffeent fomulae ae compaed to the benchmak simulations with o without extenal paametes to detemine the calculation accuacy. The esults ae given in the fom of adjustment tables so that each fomula can be used with a specified accuacy. Page 3 of 3

4 Basic Nea-Field Computation. SUMMARY OF THE BASIC COMPUTATION METHODS A summay of the computation methods used in this document is given below. All the methods allow the calculation of powe density using basic antenna paametes nomally supplied by the antenna manufactue. Table 1 Definition of symbols. Vaiable Unit Desciption Ŝ W m - Maximum.m.s. equivalent plane wave powe flux density value ove the longe peiod of eithe the modulation o the pulse wavefoms at an evaluation point. S W m - Linea atio Linea atio Time aveaged equivalent plane wave powe flux density at an evaluation point. Peak antenna gain at the fequency unde evaluation. Antenna gain in diection specified in spheical coodinates at the fequency unde evaluation. Linea atio ain of the maximum in the vetical gain patten at the fequency unde evaluation. 3dB Radians Azimuth 3 db beam width of the adiation patten at the fequency unde evaluation. 3dB Radians Vetical 3 db beam width of the adiation patten at the fequency unde evaluation. Radians Electical down-tilt angle of the antenna main beam fom x-y plane at the fequency of evaluation. Down is positive and up is negative. L metes Physical antenna length at the fequency unde evaluation. metes Radius in spheical coodinates metes Radius along boesight of the antenna λ m Wavelength adians The angle between the positive z-axis and the line fomed between the oigin and the point of inteest adians The azimuth angle between the positive x-axis and the line fom the oigin to the point of inteest pojected onto the xy-plane.1 Cylindical equations The efeence fame elative to an aay antenna axis and the elevant analytical notations employed in the analytical pediction fomulae fo the spatially-aveaged and the spatial-peak powe density ae illustated in Figue -1. Page 4 of 3

5 Basic Nea-Field Computation Page 5 of 3 Figue -1 Refeence fame and notations employed fo cylindical fomulae fo calculation at a point (left) and on a line pependicula to boesight (ight)..1.1 Aveage cylindical equations The pediction fomulae fo the spatially-aveaged equivalent powe density ae [1]: Omni-diectional aays ( ) π cos 1 1 ) ( cos + L L P S av (1) Panel-coveage aays ( ) 3dB 3dB cos 1 1 ) ( cos 3 + Φ L L P S db av ().1. Peak cylindical equations The coesponding pediction fomulae fo the spatial-peak equivalent powe density ae []: Omni-diectional aays ( ) π cos 1 1 ) ( cos ˆ + L L P S av (3)

6 Basic Nea-Field Computation Page 6 of 3 Panel-coveage aays ( ) 3dB 3dB cos 1 1 ) ( cos ˆ 3 + Φ L L P S db av (4) whee - π < Ø < π. The maximum azimuth gain of the antenna is at Ø adians. The distance is taken fom the cente of the antenna. The individual linea aay antenna lengths fo dual-band and ti-band antennas must be used.. Spheical equations The efeence fame used fo the spheical fomulae is illustated in Figue -. The deiving of the simple and adjusted spheical fomulae is shown in Chapte 3. Ø z x y Obsevation point Figue - Refeence fame and notations employed fo spheical fomulae...1 Simple spheical equations The coesponding simple spheical pediction fomulae fo both the spatially-aveaged and spatial-peak equivalent powe densities ae: Omni-diectional aays ( ) ( ) 3 4 ˆ + db P S S av π π (5) Panel-coveage aays ( ) ( ) ˆ + db db P S S av π π (6) whee in Figue - is the vetical angle of otation towads the obsevation point in adians.

7 Basic Nea-Field Computation.. Adjusted spheical equations The coesponding adjusted spheical pediction fomulae fo both the spatially-aveaged and spatial-peak equivalent powe densities ae: Omni-diectional aays Sˆ ( ) S( ) kpav 4π π k Θ 3dB k + (7) Panel-coveage aays Sˆ ( ) S( ) kpav 4π k + π kθ kφ 3dB 3dB (8) with 1. 6 k k 1 k 1. 9 and k 1. Θ Φ of otation towads the field point in adians as shown in Figue -..3 Boundaies fo computation (explained in Chapte 3). is the vetical angle a) b) Figue -3 a) Antenna -D view and b) equivalent 3-D view illustating the thee valid zones fo calculation. The 3-D view shows pie-slice sections of the same thee zones which ae symmetically equivalent aound the z-axis. Table gives the fomulae fo calculating the zone boundaies applicable to the C-S fomulae. A minimum distance of 3.6 λ along the antenna boesight is equied to ensue that the benchmak ay-tacing Page 7 of 3

8 Basic Nea-Field Computation computations ae at least.5 λ fom the edge of the antenna at the maximum electical tilt of 1 and a maximum antenna length of 1λ. Table Definition of boundaies fo selecting the zone of computation Point of inteest in zone: A B C D Method of calculation Peak/Aveage Cylindical equations Aveage/Peak Adjusted spheical equations Aveage/Peak Simple spheical equations Aveage/Peak Adjusted spheical equations Bounday estictions 3.6 λ L max λ L +.5λ Used within height of antenna (See NOTE 1). Panel aays: π π Omni-diectional aays: b± b 4ac max L a +.5λ and L whee λ λ a b 4.8 c 3 ( L ) ( L ) sin sin cos L max λ L +. 5λ L +.5λ and b± max b 4ac a Used above and below height of antenna (See NOTE 1). π π Used above and below height of antenna (See NOTE 1). NOTE 1: In the case of electical downtilt the condition in boesight within height of antenna is defined by: L L illustated in the ight hand of Figue -1. Sin z Sin.4 Calculation fo multiple souces Fo multiple souces the powe density value at the evaluation point can be detemined by adding the contibutions fom each souce (9) whee I denotes the numbe of souces. I S Total S i 1 i (9) Page 8 of 3

9 Basic Nea-Field Computation 3. THE SIMPLE/ADJUSTED SPHERICAL FORMULAE The gain pattens of a typical panel aay antenna ae shown in Figue 3-1. These pattens can usually be obtained fom antenna manufactues and can be used to accuately calculate the powe density in the fa-field egion of an antenna. When these calculations ae pefomed in the nea-field calculation eos ae intoduced. The gain in a given diection can howeve be adjusted to compensate fo the calculation eos. ain (dbi) Angle ( ) ain (dbi) Angle ( ) Figue 3-1 Hoizontal (top) and vetical (bottom) gain pattens of typical panel aay antenna. A line indicates the maximum gain in the vetical patten. 3.1 Deiving of the simple/adjusted spheical fomulae The spheical equations can be used to calculate the powe density close to panel and omni-diectional aay antennas and ae deived fom the fa-field powe density equation (1). P is defined as the input powe in Watts the antenna gain in a given diection is defined as in metes and S is the powe density in W/m². is the distance fom the antenna P S 4π (1) Equations ae defined fo modelling the antenna gain in a given diection fo panel and omni-diectional antennas. The maximum antenna gain is used in the diection of the main beam and the maximum gain is used in the off-main beam diections. The gain fo a panel antenna is indicated in Figue 3-1 by a hoizontal line. The spheical coodinate axis used fo the gain equations is Page 9 of 3

10 Basic Nea-Field Computation shown in Figue -. Antenna boesight is defined in the diection of the positive x-axis and the antenna cente is positioned at the oigin. The modelling of the bell-shaped tansition (seen in the top section of Figue 3-1) between the main and off-main beam egions is done using the aussian pofile (11). This equation was used by the authos of [] to epesent the bell-shaped hoizontal gain patten of a panel aay antenna. is defined as the 3dB specified hoizontal beamwidth of the antenna in adians. ψ ( ) given azimuth angle. epesents the amplitude of adiation at a ψ ( ) 3dB (11) The gain equation fo linea omni-diectional aays (1) is defined by combining the gain with the aussian equation (11). The gain equation is a function of. Two constants wee included so that the gain can be adjusted. The fist constant constant k is used to incease the pedicted - gain and the second k Θ to incease the pedicted vetical beamwidth. and ae the unit-less main beam and gains espectively. (adians) is used to offset the main beam fo antennas with electical tilt and a constant offset of π/ is used to set the boesight of the antenna to the hoizontal x-y plane. omni ( ) k + π k Θ 3dB (1) The gain equation fo linea panel aays (13) is a function of both and Φ. An additional constant intoduced to incease the hoizontal beamwidth simila to both the vetical and hoizontal beams. k Φ is k Θ. The aussian equation (11) is applied fo π kθ kφ 3dB secto( ) k + 3dB (13) Equation (1) multiplied by k in combination with the gain equations (1) and (13) defines the spheical equations fo powe density calculation (7) and (8). Fo a neutal bias (simple spheical fomulae) k Φ must equal and k and k Θ and k must equal 1. This allows calculation in the fa-field egion of an Page 1 of 3

11 Basic Nea-Field Computation antenna. In ode to use the spheical equations consevatively up to a educed fa-field bounday close to the antenna the constants must be set to 1. 6 k k 1 k 1. 9 and k 1. Θ Φ. The educed fafield bounday is descibed in Section 3.. The biased spheical fomulae ae efeed to as the adjusted spheical fomulae. The values of the constants wee detemined iteatively while pefoming the simulations descibed in chapte 3.3. These constants ensue that the powe density calculations using the adjusted spheical equations ae consevative fo 95% of the ange of known antennas. 3. Reduced fa-field bounday Because some of the spheical equation paametes ae obtained fom the fa-field pattens of the antenna the magnitude of the pediction eo of the spheical equations inceases as the obsevation point appoaches the antenna. This incease is believed to be mostly a esult of the inceasing phasechange between the individual signals aiving fom the elements of the linea aay. In ode to pefom calculations close to the antennas a bounday was defined whee a constant maximum phase eo is maintained. The maximum phase eo is measued between the cente and end elements of a linea aay descibed in [4]. The phase eo occus when the assumption that the paths of popagation fom each aay element is no longe valid. The taditional fa-field appoximation (14) assumes a maximum phase eo of.5 [4] whee L is the length of the antenna and ff the fa-field distance in metes. A new fa-field appoximation bounday was defined whee the maximum phase eo is 75. This inceased phase eo allows calculations to be pefomed significantly close to antennas. L ff (14) λ The distance fo the 75 fa-field appoximation can be calculated by finding the maximum oot of (15) whee L is the length of the antenna λ the wavelength and α the phase eo in adians. The deivation of equation (15) can be seen in [4]. λα π ff ( L ) sin ( L ) ff 3 sin cos (15) Additionally it is equied that the obsevation point is at least.5λ fom any pat of the antenna so that a fa-field condition is met fo the individual aay elements. This allows the basic equiements fo fa-field conditions [4] to be satisfied fo the maximum aay element length which is typically λ/ fo base-station antennas. The fa-field bounday nomalized to wavelength is shown in Figue 3-. Page 11 of 3

12 Basic Nea-Field Computation 5 L 4λ L 6λ L 8λ ff z/λ x/λ Figue 3- The fa-field boundaies nomalized to the electical size of thee antennas viewed fom the. Each bounday is symmetical in both the hoizontal and vetical diections. 3.3 Veification of the adjusted spheical fomulae A statistical appoach was used to validate the use of the spheical equations in the powe density calculation of a set of known cellula base station antennas. A lage numbe of antenna models wee ceated with antenna paametes vaying ove the known anges fo panel and omni-diectional aay antennas. Due to the lage numbe of simulations equied the fast ay tacing method descibed in [3] was used to pefom the benchmak calculations. It is shown that the calculated values ae consevative in the valid egion fo at least 95% of the antennas coned. The calculation eo is detemined by compaing calculated powe density values fom the spheical equations to the ay tacing method. The maximum non-consevative eo is calculated pe antenna fo a lage numbe of antenna models by stoing the maximum eo fo each model. This is used to detemine the statistical eo Antenna paamete anges Ray tacing antenna models ae used to calculate the benchmak powe densities. The model paametes ae vaied within the known anges fo mobile base station antennas in the 9 18 and 1 MHz bands. A set of antenna models was andomly selected within the paametes anges. This set is egaded as a fai epesentation fo antennas commonly found on mobile cellula base stations. The pobability densities fo the ay tacing model paametes ae plotted in Figue 3-3 to Figue 3-6. Page 1 of 3

13 Basic Nea-Field Computation A symmetical Dolph-Chebyshev distibution fo -5 db s was used to calculate the voltage amplitude distibution fo the aay elements. This voltage amplitude distibution is believed to be a easonable epesentation fo the tape applied to actual antennas. Electical tilts ae implemented though linea phase tape Pobability density Pobability density Numbe of elements Element spacing in λ Figue 3-3 Numbe of elements (left) and element spacing (ight) pobability densities fo the epesentative antenna model list Pobability density Pobability density ain (dbi) Vetical beamwidth ( ) Figue 3-4 ain (left) and vetical beamwidth (ight) pobability densities fo the epesentative antenna model list. Page 13 of 3

14 Basic Nea-Field Computation Pobability density Hoizontal beamwidth ( ) Pobability density SLL (db) Figue 3-5 Hoizontal beamwidth (left) and level (SLL) (ight) pobability densities fo the epesentative antenna model list Pobability density electical tilt ( ) 3.3. Results and compaisons Figue 3-6 Electical tilt pobability density. The paametes fo the spheical equations wee extacted fom the fa-field pattens calculated using the ay tacing antenna models. Fo each antenna model the powe density was calculated fo both the aytacing and spheical equation methods. This was done fo a numbe of points on the 75 fa-field appoximation bounday up to the taditional.5 bounday in 1.5 steps. The points wee spaced in spheical coodinates with steps fo and 5 steps fo as seen in Figue 3-7. The distance fom the oigin ( ff ) was calculated by finding the maximum oot in (15) fo each α and. Page 14 of 3

15 Basic Nea-Field Computation Figue 3-7 Coodinate points fo compaing the powe density calculated using the ay tacing and spheical equation methods. The maximum eo e (db) between the powe density calculations of the two methods (ay tacing and spheical equations) was computed fo each antenna using (16) whee s denotes the vecto of calculated powe densities in dbw/m² fo each method. This was done in ode to detemine the pobability density function (PDF) shown in Figue 3-8 of the spheical equations ove the ange of antennas. e max( s aytacing s spheical ) (16) Powe density values below -4 dbw/m² wee not coned in the uncetainty analysis. This is at least 45 times less than the ICNIRP [5] public efeence level ove the fequency ange and is coned small enough to ignoe. This pevents lage vaiations on vey small numbes fom playing a ole duing the eo calculation. The constants k k Θ k Φ and k wee adjusted iteatively until the spheical equations poduced consevative calculations fo at least 95% of the antenna models. Page 15 of 3

16 Basic Nea-Field Computation Pobability density Powe density eo (db) Figue 3-8 Pobability density of the spheical equation eo fo the ange of antennas with at least 95% of the pobability density suface below zeo. Page 16 of 3

17 Basic Nea-Field Computation 4. UNCERTAINTY ESTIMATIONS Detailed uncetainty analyses wee pefomed to detemine the accuacy of the C-S fomulae. Table 4 to Table 6 ae pesented to allow the adjustment of calculated powe density to each a desied calculation cetainty. The extenal uncetainty paametes in Table 3 have been included with the calculations of the values in Table 5 and Table 6. To each a specified uncetainty the appopiate table value must be added to the powe density calculated using the C-S fomulae. Table 3 Estimated extenal uncetainty paametes. Paamete Name Bief Desciption Unifom Uncetainty in output Powe Density (dbw/m²) Max Input Powe Tansmitte maximum powe is usually specified within this ange. ± Cable Loss Long cables connect the tansmittes to the antennas. -4 to -.5 Reflection and Scatteing Scatteing and eflections on top of buildings can ceate hotspots whee fields add in-phase to 4.8 The adjustment tables ae povided fo each of the peak and aveage cylindical equations. The tables povide adjustments in futhe divided zones A.1 A. and A.3 shown in Figue 4-1. In ode to skip complex bounday calculations the wost case esults fo a combination of applicable zones can be used. Figue 4-1 Definition of sub-zones in the hoizontal plane. Omni-diectional antennas fall unde zone A.1 while panel aays can fall unde zones A.1 A. o A.3 accoding to Φ. Page 17 of 3

18 Basic Nea-Field Computation 4.1 Antenna paamete anges The calculation eo is detemined by compaing calculated powe density values fom the applied fomulae to the ay tacing method. The maximum non-consevative eo is calculated pe antenna fo a lage numbe of antenna models by stoing the maximum eo fo each model. This is used to detemine the statistical eo. Ray tacing antenna models ae again used to calculate the benchmak powe densities. A set of antenna models was andomly selected within the paamete anges. The pobability densities fo the ay tacing model paametes fo the spheical equation uncetainty analysis ae plotted in Figue 3-3 to Figue 3-6. Some of the pobability densities fo the cylindical equation uncetainty analysis ae slightly diffeent because the electical tilt ange is fom to 1 and not to 15. They ae plotted in Figue 4- to Figue Pobability density Pobability density ain (dbi) Vetical beamwidth ( ) Figue 4- ain (left) and vetical beamwidth (ight) pobability densities fo the epesentative antenna model list fo the cylindical equation analysis Pobability density Pobability density SLL (db) electical tilt ( ) Figue 4-3 Side level (SLL) (left) and electical tilt (ight) pobability densities fo the epesentative antenna model list fo the cylindical equation analysis. Page 18 of 3

19 Basic Nea-Field Computation 4. Adjustment tables Table 4 Powe density (db) adjustment table fo cyl-sph fomulae no extenal uncetainty paametes. Aveage Cylindical fomulae Peak Cylindical fomulae Adjusted spheical fomulae CI-95% Simple spheical fomulae A A A B C D +5.8 CI-8% A A A B C D +4.6 CI-5% A A A B C D -1.7 CI-% A A A B C D -4.8 CI-5% A A A B C D -7.3 Page 19 of 3

20 Basic Nea-Field Computation Table 5 Powe density (db) adjustment table fo C-S fomulae excluding cable loss but including input powe and eflection as extenal uncetainty paametes. Aveage Cylindical fomulae Peak Cylindical fomulae Adjusted spheical fomulae CI-95% Simple spheical fomulae A A A B C D +8.4 CI-8% A A A B C D +4.8 CI-5% A A A B C D -. CI-% A A A B C D -5.3 CI-5% A A A B C D -9. Page of 3

21 Basic Nea-Field Computation Table 6 Powe density (db) adjustment table fo C-S fomulae and extenal uncetainty paametes in Table 3 fo diffeent confidence levels. Aveage Cylindical fomulae Peak Cylindical fomulae Adjusted spheical fomulae CI-95% Simple spheical fomulae A A A B C D CI-8% A A A B C D CI-5% A A A B C D CI-% A A A B C D CI-5% A A A B C D Page 1 of 3

22 Basic Nea-Field Computation 5. CONCLUSIONS Basic cylindical-spheical (C-S) fomulae wee given fo the calculation of powe density fo panel and omni-diectional base station antennas. The deiving and validation of the spheical fomulae was shown as well as the definition of a educed fa-field bounday. The educed fa-field bounday allows the use of adjusted spheical equations in a egion close than the taditional fa-field egion. It was shown that adjusted spheical fomulae can be used up to a educed fa-field bounday to consevatively calculate the powe density fo 95% of a specified ange of panel and omni-diectional linea boad aay antennas. This was done by compaing powe densities calculated using the spheical equations to powe densities calculated using a ay tacing method. Futhe uncetainty analyses wee pefomed on the C-S fomulae in the applicable zones by compaing powe densities to a ay tacing method. Adjustment tables wee povided that stipulate values that can be added to the calculated powe densities so that the calculation accuacy falls unde a specific cetainty. Fo calculations in zone B and D combined the adjusted spheical fomulae can be used with k The additional calculations fo the educed fa-field bounday ae then not necessay. This facto was calculated by adding the +5.8 db adjustment fom zone D in Table 4 to the existing linea k 1. in equations (7) and (8). k 4. 6 allows the calculations in zone B and D to be consevative fo at least 95% of the antenna models in an envionment without extenal uncetainties. Page of 3

23 Basic Nea-Field Computation 6. REFERENCES [1] Cicchetti R. Faaone A. and Balzano Q.: A Unifom Asymptotic Evaluation of the Field Radiated fom Collinea Aay Antennas IEEE Tansactions on Antennas and Popagation Vol. 51 No. 1 pp Jan. 3. [] Renato Cicchetti and Antonio Faaone Estimation of the Peak Powe Density in the Vicinity of Cellula and Radio Base Station Antennas IEEE tansactions on Electomagnetic compatibility Vol.46 No. May 4 [3] Fancois du Plessis Field Assessment Calculation and Uncetainty Estimation EMSS Consulting (Pty) Ltd Tech. Rep. Document Numbe 8 [4] Waen L. Stutzman ay A. Thiele Antenna Theoy and Design nd edition. [5] Intenational Commission on Non-Ionizing Radiation Potection (ICNIRP) uidelines fo limiting exposue to time-vaying electic magnetic and electomagnetic fields (up to 3 Hz). Health Phys 74: Page 3 of 3