Figure 1-1 Sample Antenna Pattern

Transcription

1 1.0 ANTENNAS 1.1 INTRODUCTION In EE 619 we discussed antennas fom the view point of antenna apetue, beam width and gain, and how they elate. Moe specifically, we dealt with the equations and 4 Ae G (1-1) 5,000 G AZ, EL in degees. (1-) AZ EL In this couse we want to develop equations and techniques fo finding antenna adiation G, whee and ae and gain pattens. That is, we want to develop equations fo othogonal angles such as azimuth (AZ) and elevation (EL) o angles elative to a nomal to the antenna face (as would be used in phased aay antennas). We want to be able to poduce plots simila to the one shown in Figue 1-1. We want to use the ability to geneate antenna pattens to loo at beam width, gain, sidelobes and how these elate to antenna dimensions and othe factos. Figue 1-1 Sample Antenna Patten M. C. Budge, J., 01 mev@thebudges.com 1

2 We will begin with a simple two-element aay antenna to illustate some of the basic aspects of computing antenna adiation pattens and some of the popeties of antennas. We will then pogess to linea aays, and then plana phased aays. 1. TWO ELEMENT ARRAY ANTENNA Assume we have two isotopic adiatos sepaated by a distance d, as shown in Figue 1-. In this figue, the ac epesents pat of a sphee located at a distance of elative to the cente of the adiatos. Fo these studies, we assume that d. The sinusoids epesent the electic field geneated by each adiato. Figue 1- Two Element Aay Antenna Since the adiatos ae isotopic the powe of each adiates is unifomly distibuted ove a sphee at some adius (ecall the ada ange equation). Thus the powe ove some small aea, A, due to eithe adiato is given by PA K Ps (1-3) 4 Since the electic field intensity at is popotional to P s we can wite E E s. (1-4) If we ecall that the signal is eally a sinusoid at a fequency of we can wite the electic field (E-field) at A as E o, the caie fequency, E s j o e (1-5) whee is the time the signal taes to popagate fom the souce to the aea A. M. C. Budge, J., 01 mev@thebudges.com

3 Fo the next step we invoe the elations c, o fo, and fo c whee c is the speed of light and denotes wavelength. With this, we can wite the electic field at A as E E s j e. (1-6) Let us now tun ou attention to detemining the E-field at some A when we have the two adiatos of Figue 1-. We will use the geomety of Figue 1-3 as an aid in ou deivation. We denote the uppe adiato (point) of Figue 1-3 as adiato 1 and the lowe adiato as adiato. The distances fom the individual adiatos to A ae 1 and. We assume that the electic field intensity of each adiato is equal to P. Whee P is the total powe deliveed to the adiatos. The facto of is included to denote the fact that the powe is split evenly between the adiatos (unifom weighting). In the above, we have made the tacit assumption that the adiation esistance is 1 ohm. The othe tems that we will need ae shown on Figue 1-3. Figue 1-3 Geomety fo two element adiato poblem We can wite the E-fields fom the two adiatos as and P j 1 E1 e (1-7) 1 E P j e. (1-8) Fom Figue 1-3, we can wite 1 and as and x y d d d (1-9) 1 o o 4 sin M. C. Budge, J., 01 mev@thebudges.com 3

4 x y d d d. (1-10) o o 4 sin As indicated ealie, we will assume that and d d 1 sin 1 sin d. With this we can wite (1-11) d d sin 1 sin. (1-1) Whee we have made use of the elation N 1 x 1 Nx fo x 1. (1-13) We note that since 1 and ae functions of, the E-fields will also be functions of. With this we wite and P d E1 exp j sin 1 sin d d (1-14) P d E exp j sin. (1-15) 1 sin In Equations 1-14 and 1-15, we can set the denominato tems to since d 1. We can t do this in the exponential tems because phase is measued modulo. o The total electic field at 1 A is E E E (1-16) P P d d E exp j sin exp j sin P e e e P j j d sin j d sin d j e cos sin. (1-17) At this point we define an antenna adiation patten as (note: late we will define a diective gain patten) E R. (1-18) P M. C. Budge, J., 01 mev@thebudges.com 4

5 With this, we obtain the adiation patten fo the dual, isotopic adiato antenna as cos sin R d We will be inteested in. (1-19) R fo. We call the egion visible space. Actually, physical visible space extends fom to but values of coespond to the bac of an antenna, which is usually shielded. Figue 1-4 contains plots of R fo thee values of d: d, and 4. Fo d thee ae thee peas in the adiation patten: at 0,, and. The peas at ae temed gating lobes and ae usually undesiable. Fo d 4 the adiation patten doesn t etun to zeo and the width of the cental egion is boad. This is also a geneally undesiable chaacteistic. The case of d is a good compomise that leads to a faily naow cente pea and levels that go to zeo at. In the design of phased aay antennas we find that d is usually a desiable design citeion. Figue 1-4 Radiation Patten fo a two element aay with vaious element spacings The cental egion of the plots of Figue 1-4 is temed the main beam and the angle spacing between the 3-dB points (the points whee the adiation patten is down 3 db fom its pea value) is temed the beamwidth. It will be noted that the beamwidth is invesely popotional to the spacing between the adiatos. This coesponds to ou findings in the fist ada couse whee we found that thee was an invese elation between beamwidth and the antenna diamete. The poblem we just solved is the tansmit poblem. That is, we supplied powe to the adiatos and detemined how it was distibuted on a sphee. We now want to conside the evese poblem and loo at the eceive antenna. This will illustate an impotant poblem called ecipocity. Recipocity says that we can analyze an antenna eithe way and get the same adiation patten. Stated anothe way, in geneal, the adiation patten of an antenna is the same on tansmit as on eceive. M. C. Budge, J., 01 mev@thebudges.com 5

6 Fo this case we conside the two adiatos of Figue 1- as eceive antennas that ae isotopic. Hee we call them eceive elements. We assume that an E-field adiates fom a point that is located at a ange fom the cente of the two eceive elements. The eceive elements ae sepaated by a distance of d as fo the tansmit case. The equied geomety is shown in Figue 1-5. The outputs of the eceive elements ae multiplied by and summed. The voltage out of each element is popotional to the E-field at each element. It is epesented as a complex numbe to account fo the fact that the actual signal, which is a sinusoid, is chaacteized by an amplitude and phase. Figue 1-5 Two element aay, eceive geomety The E-field at all points on a sphee (o a cicle in two dimensions) has the same amplitude and phase. Also, since d the sphee is a plane (a line in two dimensions which is what we will use hee) at the location of the eceive elements. The line is oiented at an angle of elative to the vetical. We tem this line the constant E-field line. is also the angle of the point fom which the E-field adiates to the hoizontal line of Figue 1-5. We tem the hoizontal line the antenna boesight. In moe geneal tems, the antenna boesight is the nomal to the plane containing the elements and is pointed geneally towad the point fom which the E-field adiates. We define the vetical line though the elements as the efeence line. Fom Figue 1-5, it is obvious that the spacing between these two lines and the efeence line is d sin. If we define the E-field at the cente point between the elements as E (1-0) j Ee then the E-field at the elements will be and j d sin E E e (1-1) 1 j d sin E E e. (1-) M. C. Budge, J., 01 mev@thebudges.com 6

7 Since the voltage out of each element is popotional to the E-field at each element, the voltages out of the elements ae and j d sin V V e (1-3) 1 j d sin V V e. (1-4) With this, the voltage at the summe output is 1 V j d V V1 V e cos sin. (1-5) We define the adiation patten as which yields V R, (1-6) V cos sin R d. (1-7) This is the same esult that we got fo the tansmit case and seves to demonstate that ecipocity applies to antennas. This will allow us to use eithe the eceive o tansmit appoach when analyzing moe complex antennas. We will use the technique that is easiest fo the paticula poblem that we ae addessing. 1.3 N-ELEMENT LINEAR ARRAY We now want to extend the esults of the pevious section to a linea aay of elements. A dawing of the linea aay we will analyze is shown in Figue 1-6. As implied by the figue, we will use the eceive appoach to deive the adiation patten fo this antenna. The aay consists of N elements with a spacing of d between the elements. The output of each element is weighted by a facto of a and the esults summed to fom the signal out of the antenna. In geneal, the weights, a, can be complex (in fact, we will find that we can stee the antenna beam by assigning appopiate phases to the a ). M. C. Budge, J., 01 mev@thebudges.com 7

8 Figue 1-6 Geomety fo N-element linea aay The slanted line oiginating at the fist element depicts the plane wave that has aived fom a point taget that is at an angle of elative to the boesight of the antenna. In ou convention, is positive as shown in Figue 1-6. Fom Figue 1-6 it should be obvious that the distance between the plane wave and the th element is d d sin 0 N 1. (1-8) This means that the E-field at the th element is j j d j j d sin E E e e E e e (1-9) and that the voltage out of the th element is j j d sin V ave e. (1-30) With this, the voltage out of the summe is N 1 N 1 N 1 j jd sin j jd sin. (1-31) V V a V e e V e a e Fo convenience, we will let d sin and wite N 1 j j 0 V V e a e. (1-3) As befoe, we define the adiation patten as V R (1-33) V M. C. Budge, J., 01 mev@thebudges.com 8

9 which yields R N 1 0 j a e. (1-34) We now want to conside the special case of a linea aay with unifom weighting. Fo this case a 1 N. If we conside the sum tem we can wite 1 1 A a e e e N 1 N 1 N 1 j j j 0 0 N N 0. (1-35) At this point we invoe the elation to wite N 1 N 1 x x (1-36) 1 x 0 A N 1 1 j 1 1 N 0 e 1 e jn jn jn j j j Finally, we get R 1 e jn j 1 sin N sin jn N 1 e e e N e e e N e. (1-37) Nd 1 sin N 1 sin sin A d. (1-38) N sin N sin sin Figue 1-7 contains a plot of R vs. fo 0 N and d, and 4. As with the two element case, it will be noted that gating lobes appea fo the case whee d. Also, it will be noted that the width of the mainlobe vaies invesely with element spacing. As indicated ealie, this is expected because the lage element spacing implies a lage antenna which tanslates to a smalle beamwidth. It will be noted that the pea value of R is 0, o N, and occus at 0. This value can also be deived by taing the limit of R as 0. M. C. Budge, J., 01 mev@thebudges.com 9

10 Figue 1-7 Radiation patten fo an N-element linea aay with diffeent element spacings Fo the geneal case whee R N 1 0 j a C, whee C is a constant, one must use a e, d sin (1-39) to compute the adiation patten. With moden computes, and softwae such as MATLAB, this doesn t pose much of a poblem. It will be noted that the tem inside of the absolute value is of the fom of a discete Fouie tansfom. This suggests that one could use the FFT to compute R. In fact, when we conside plana aays we will discuss the use of the FFT to compute the antenna pattens. 1.4 ANTENNA GAIN PATTERN The antenna adiation patten is useful fo detemining such antenna popeties as beamwidth, gating lobes and sidelobe levels. Howeve, it does not povide an indication of antenna gain. To obtain this we want to define an antenna gain function. As its name implies, this function povides an indication of antenna gain as a function of angle. To be moe specific, it povides an indication of the diective gain of the antenna as a function of angle. This is the gain we used in the ada-ange equation and is also the gain we detemine fom Equations 1-1 and 1-. Accoding to Chapte 6 of Solni Rada Handboo 1 and Jasi s antenna handboo we can wite the antenna (diective) gain patten 3 as 1 Solni, M.I., Rada Handboo, 3 d ed, McGaw-Hill, New Yo, 008. Johnson, R. C. and H. Jasi (eds.), Antenna Engineeing Handboo, nd ed., McGaw-Hill, New Yo, We distinguish between diective gain and diective gain patten hee. As we will discuss below, diective gain is a numbe. The diective gain patten is a function of and. M. C. Budge, J., 01 mev@thebudges.com 10

11 o Radiation intensity on a sphee of adius at an angle, G,, (1-40) Aveage adiation intensity ove a sphee of adius G, 1 4 sphee R, R, R, d R (1-41) whee d is a diffeential aea on the sphee. To compute the denominato integal we conside the geomety of Figue 1-8. In this figue, the vetical ow of dots epesents the linea aay. Fom the figue, the diffeential aea can be witten as cos d du ds d d (1-4) and the integal in the denominato becomes 1 R R, cos d d 4 0. (1-43) Fo the linea aay we have R, R and 1 R R cos d. (1-44) Fo the special case of a linea aay with unifom weighting we get 1 1 N d sin sin R cos d d N sin sin 1 N d sin sin cos d d N sin sin 0 whee the last equality is a esult of the fact that the integand is an even function. Afte consideable computation, it can be shown that 1 l1 (1-45) N1 R 1 sinc ld. (1-46) N M. C. Budge, J., 01 mev@thebudges.com 11

12 Figue 1-8 Geomety used to compute R As a sanity chec we want to conside a point-souce (isotopic) adiato. This can be consideed a special case of an N-element, unifomly-illuminated, linea aay with an element spacing of d 0. Fo this case we get R 1. Futhe, fo d 0, sinc ld 1 and This give7 N 1 N 1 R N N N. (1-48) 1 l1 1 R N G 1. (1-48) R N It can also be shown that, fo a geneal, N-element, unifomly-illuminated, linea aay with an element spacing of d, and weights of a 1 N, R 1 and G 1 N d sin sin R. (1-49) d N sin sin Fo the case of a geneal, non-unifomly illuminated, linea aay, R must be computed numeically fom the Equation 1-44 We want to now conside the diective gain, G. We will define this as the maximum value of G. Fo the unifomly illuminated, linea aay consideed above, 0 G G. Figue 1-9 contains a plot of nomalized G vs. d fo seveal values of N. In this plot, the nomalized G is G N. M. C. Budge, J., 01 mev@thebudges.com 1

13 Figue 1-9 Nomalized Diective Gain vs. element spacing The shapes of the cuves in Figue 1-9 ae vey inteesting, especially aound intege multiples of d. Fo example fo d slightly less than 1, G N is between about 1.7 and 1.9, wheeas, fo d slightly geate than 1, G N is about 0.7. In othe wods, a vey small change in element spacing causes the diective gain to vay by a facto of about 1.8/0.7 o 4 db. The eason fo this is shown in Figue 1-10 which contains a plot of R fo d values of 0.9, 1.0 and 1.1 and a 0-element linea aay with unifom weighting (unifom illumination). In this case bette illustate the widths of the gating lobes (the lobes not at 0). R is plotted vs. sin to Fo the case whee d is 0.9 (blue cuve), the adiation patten does not contain gating lobes. This means that all of the tansmitted powe can be focused in the main beam. Fo the cases whee d is eithe 1.0 o 1.1, the adiation patten contains gating lobes. This means that some of the tansmitted powe will be taen away fom the main lobe (the cental egion) to be sent to the gating lobes. This will thus educe the diective gain of the antenna elative to the case whee d is 0.9. Futhemoe, since thee ae two gating lobes fo d 1.1 and only one gating lobe fo d 1.0 (½ lobe at sin 1 and ½ lobe at sin 1), the gain will be less when d is 1.1 than when it is 1.0. Fom Figue 1-9, we would expect simila behavio of the diective gain fo values of d nea othe intege values. Howeve, the vaiation in gain as d tansitions fom below to above an intege values deceases as the intege value of d inceases. M. C. Budge, J., 01 mev@thebudges.com 13

14 Figue 1-10 Radiation Pattens fo d close to BEAMWIDTH, SIDELOBES AND AMPLITUDE WEIGHTING Figue 1-11 contains a plot of GD fo a 0 element aay with an element spacing of d 0.5 and unifom weighting. In this case, the units on the vetical scale ae in dbi. The unit notation dbi stands fo db elative to an isotopic adiato, and says that the diective gain is efeenced to the gain of an isotopic adiato, which is unity. Figue 1-11 Diective Gain fo a 0-element linea aay with a unifom tape As discussed ealie, the lobe nea 0 is temed the main beam. The lobes suounding the main beam ae the sidelobes. The fist couple of sidelobes on eithe side of the main beam ae temed the nea-in sidelobes and the emaining sidelobes ae M. C. Budge, J., 01 mev@thebudges.com 14

15 temed the fa-out sidelobes. Fo this antenna, the diective gain is 13 db (10log(0)) and the nea-in sidelobes ae about 13 db below the pea of the main beam (13 db below the main beam). The fa-out sidelobes ae geate than 0 db below the main beam. The beamwidth is defined at the width of the main beam measued at the 3-dB points on the main beam. Fo the patten of Figue 1-11 the beamwidth is 5 degees. The nea-in sidelobe level of 13 db is often consideed undesiably high. To educe this level, antenna designes usually apply an amplitude tape to the aay by setting the a to diffeent values. Geneally, the values of a ae vaied symmetically acoss the elements so that the elements on opposite sides of the cente of the aay have the same value of a. One usually ties to choose the a so that one achieves a desied sidelobe level while minimizing the beamwidth incease and gain decease usually engendeed by weighting. The optimum weighting in this egad is Chebychev weighting. Up until ecently, Chebychev weights wee vey difficult to geneate. Howeve, ove the past 10 o so yeas, standad algoithms have become available. Fo example, the MATLAB Signal Pocessing Toolbox has a built-in function (chebwin) fo computing Chebychev weights. An appoximation to Chebychev weighting is Taylo weighting. Taylo weights ae a computed fom a Taylo seies appoximation to Chebychev weights. An algoithm fo computing Taylo weights is given in Appendix A. In space-fed phased aays and eflecto antennas the amplitude tape is ceated by the feed and is applied on both tansmit and eceive. Since the amplitude tape is ceated by the feed, the type of tape is somewhat limited in because of feed design limitations. In copoate o constained feed phased aays the tape is contolled by the way that powe is deliveed to, o combined fom, the vaious elements. Again, this limits the type of amplitude tape that can be obtained. In solid state phase aays, one has consideable flexibility in contolling the amplitude tape on eceive. Howeve, it is cuently vey difficult to obtain an amplitude tape on tansmit because all of the tansmit/eceive (T/R) modules must be opeated at full powe fo maximum efficiency. The afoementioned types of antennas and feed mechanisms will be discussed again late. Figue 1-1 contains a plot of G fo a 0-element linea aay with d 0.5 and Chebychev weighting. The Chebychev weighting was chosen to povide a sidelobe level of 30 db, elative to the main beam. As can be seen fom Figue 1-1, the sidelobe level is, indeed, 30 db below the pea main beam level. It will be noted that the diective gain at 0 (the diective gain o, simply, gain) is about 1.4 db athe than the 13 db gain associated with a 0-element, linea aay with unifom illumination. Thus, the amplitude tape has educed the antenna gain by about 0.6 db. Also, the beamwidth of the antenna has inceased to 6. degees. M. C. Budge, J., 01 mev@thebudges.com 15

16 Figue 1-1 Diective Gain fo a 0-element linea aay with Chebychev weighting 1.6 STEERING Thus fa, the antenna pattens we have geneated all had thei main beams located at 0 degees. We now want to addess the poblem of placing the main beam at some desied angle. This is temed beam steeing. We will fist addess the geneal poblem of time delay steeing and then develop the degeneate case of phase steeing. The concept of beam steeing, as discussed hee, applies to phased aay antennas. It does not apply to eflecto antennas. To addess this poblem, we efe to the N-element linea aay geomety of Figue 1-6. Let the E-field fom the point souce be j fot E pt t ect e p t whee p is the pulse width, f o is the caie fequency and ectx is the ectangle (1-50) function. We will futhe assume that the point souce adiato is stationay and located at some ange, R. The voltage out of the th antenna element (befoe the weighing, a ) is t v t e p j fo t ect (1-51) whee is the time delay fom the point souce adiato to the th element and is given by R R d sin R d. (1-5) c c M. C. Budge, J., 01 mev@thebudges.com 16

17 In the above, we have assumed that the voltage magnitude V is unity (see page 9). Instead of teating the weights, a, as multiplication factos we teat them as opeatos on the voltages at the output of the antenna elements. With this we wite the voltage out of the summe as N 1 0 V a v t,. (1-53) We want to detemine how the weighting functions, focus the beam at some angle o. Figue 1-13 contains a setch of the magnitudes of the vaious a v t,, must be chosen so as to v t. The main thing illustated by this figue is that the pulses out of the vaious antenna elements ae not aligned. This means that the weighting functions, a v t,, must effect some desied alignment of the signals. Moe specifically, the a v t, must be chosen so that the signals out of the weighting functions ae aligned (and in-phase) at some desied o. To accomplish this, the a v t, must intoduce appopiate time delays (and possibly phase shifts) to the vaious amplitudes of the vaious v v some angle o is temed time delay steeing. t. They must also appopiately scale the t. This intoduction of time delays to focus the beam at Figue 1-13 Setch of v t If we substitute fo into the geneal v t we get t t o o R d j f t R d j f t v t ect e ect e p p. (1-54) To time align all of the pulses out of the weighting functions, the weighting function must intoduce a time delay that cancels the d tem in v t at some o. Specifically, a v t, must be chosen such that, V t a v t a v t (1-55) do M. C. Budge, J., 01 mev@thebudges.com 17

18 whee do d sin o c. Using this with the v t above esults in j f t t R d do Vt a e p ect o R d do It will be noted that at o, d do and t V t a e R j fo t R ect p. (1-56). (1-57) In othe wods, the pulses out of the weighting functions ae time aligned, and popely weighted. Time delay steeing is expensive and not easy to implement. It is needed in adas that use compessed pulse widths that ae small elative to antenna dimensions. This can be seen fom examining Figue If p is small elative to N 1 d then, fo some, not all of the pulses will align. Stated anothe way, the pulse out of the fist element will not be aligned with the pulse out of the N th element. Howeve, this implies eithe a vey small p o a vey lage antenna (vey lage N 1 d ). Fo example, if p was 1 ns and the antenna was m wide we would have N ns and time delay steeing would be needed. Howeve, if p was 1 µs all of the pulses would be faily well aligned and time delay steeing would not be necessay. Figue 1-14 contains a plot that gives an idea of the bounday between when time delay steeing would and would not be necessay. The cuve on this figue coesponds to the case whee the antenna diamete, D, is 5% of the compessed pulse width. The choice of 5% is somewhat abitay but is pobably epesentative of pactical situations whee the beam is steeed to a maximum angle of 60 degees. d p M. C. Budge, J., 01 mev@thebudges.com 18

19 Figue 1-14 Antenna diamete vs. compessed pulsewidth tade The two egions of Figue 1-14 indicate that the altenative to time delay steeing is phase steeing. Indeed, if we assume that the pulses ae faily well aligned then we can wite o that V t a v t, t R j fo t R d do j fo do a ect e a e v t p a a e j fo do phases of the vaious v. This says that the weights, a, modify the amplitudes and t. This is why this technique is called phase steeing. Substituting fo do in the phase tem esults in j d sin a o a e (1-58). (1-59) 1.7 ELEMENT PATTERN In the equations above it was assumed that all of the elements of the antenna wee isotopic adiatos. In pactice antenna elements ae not isotopic but have thei own adiation patten. This means that the voltage (amplitude and phase) out of each element depends upon, independent of the phase shift caused by the element spacing. If all of the elements ae the same, and oiented the same elative to boesight, then the dependence voltage upon will be the same fo each element (again, ignoing the phase shift caused by the element spacing). In equation fom, the voltage out of each element will be t v t A e j fo t elt ect p and the voltage out of the summe (assuming phase steeing) will be N 1 elt elt aay 0 (1-60) j jd sin j V A e a e A e A. (1-61) The esulting adiation patten will be R V R R (1-6) elt aay and the esulting diective gain patten will be G G G. (1-63) elt aay M. C. Budge, J., 01 mev@thebudges.com 19

20 In othe wods, to get the adiation o gain patten of an antenna with a non-isotopic element patten one multiplies the aay patten (found by the afoementioned techniques) by the adiation o gain patten of the elements. As a closing note, in geneal, the element patten is not steeed, only the aay patten. 1.8 PHASE SHIFTERS In the above discussions, a tacit assumption is that the phase of each weight, a, can tae on a continuum of values. In pactice, the phase can only be adjusted in discete steps because the devices that implement the phase shift, the phase shiftes, ae digital. Typical phase shiftes use 3 to 5 bits to set the phase shift. If B is the numbe of bits used in the phase shifte then the numbe of phases will be N B. As an example, a 3-bit phase shifte will have 8 phases that ange fom 0 to 7 8. As we will see when we conside plana aays, this phase quantization caused by the phase shiftes can have a deleteious effect on the sidelobes when the beam is steeed to othe than boesight. Solni s Rada Handboo has a discussion of phase shiftes in Chapte COMPUTING ANTENNA PATTERNS USING THE FFT whee Ealie we showed that we could wite R A A (1-64) N 1 0 sin j d a e. (1-65) Equivalently, we could wite whee R B B (1-66) N 1 0 It will be noted that sin j d a e. (1-67) B has the fom of the Discete-Time Fouie Tansfom (DTFT). Indeed, if we wee to conside the a to be a discete-time signal then we would wite its DTFT as M. C. Budge, J., 01 mev@thebudges.com 0

21 whee and B f N 1 jtf ae (1-68) 0 t is the time spacing between the a and f denotes fequency. When we compae the above two equations we can mae the following analogies t d (1-69) f sin. (1-70) We now that we can use the FFT to compute B f. Thus, by analogy, we can B. The tic is to popely intepet the FFT output. also use the FFT to compute Fo a time-fequency FFT, the fequency extent of the FFT output is F 1 t (1-71) and the spacing between the FFT output taps is f 1 M t (1-7) whee M is the numbe of FFT taps, o the length of the FFT (usually a powe of ). Fo a esponse centeed at 0 Hz, the fequencies associated with the M FFT taps ae M M 1 f, 1. (1-73) Mt By analogy, the total extent of the FFT output fo the antenna case is sin u 1 d (1-74) whee we have used u sin. The spacing between FFT output taps is sin u 1 Md. (1-75) The u s associated with the FFT taps ae M M u sin, 1 As we incease M we obtain Md we incease M by zeo-padding the input to the FFT.. (1-76) B s at fine angle scales. As with time-fequency FFT s In futue discussions we let N elt be the numbe of elements in the phased aay. We let M N be the numbe of points in the FFT. To zeo-pad, we load the fist N elt taps of the FFT with the a and set the emaining M Nelt taps to zeo. M. C. Budge, J., 01 mev@thebudges.com 1

22 We want to examine the total extent of sin at the FFT output. Fom the above, we note that u sin. (1-77) d If d we have that sin 1 which means that the output taps cove all angles between and. If d we note that d 1 and thus that some values of sin can have a magnitude geate than 1. This means that some of the FFT output taps do not coespond to eal angles and that we need to ignoe these taps when we plot the antenna patten. Said anothe way, we eep the M FFT output taps that satisfy sin 1. If d then we have d 1 and thus that the ange of values of sin does not extend to 1. This means that the FFT does not geneate the full antenna adiation patten. To get aound this poblem we place fae elements between the eal elements so as to educe the effective d, d eff, so that it satisfies deff. We give the fae elements amplitudes of zeo. If d we must discad some of the FFT outputs as discussed above. eff With the above we wite the adiation patten, in sine space, as B u R u. (1-78) The antenna gain patten, in sine space, is given by whee G u R u R (1-79) R R u (1-80) Md and the sum is taen ove the M FFT output taps that mae-up the antenna patten (see the above discussions). The fom of R given hee is an Eule appoximation to Equation 1-44 with the substitution: u sin. The facto of in the denominato of the above equation comes fom the facto of ½ in Equation To plot R o G one 1 would mae the substitution sin u Algoithm Given the above, we can fomulate the following algoithm If the antenna has an element spacing of d, inset fae elements between the actual elements until the element spacing satisfies d. Set the amplitudes of the fae elements to zeo. The amplitudes of the egula M. C. Budge, J., 01 mev@thebudges.com

23 elements will be the a fom above (1 s fo unifom weighting, Hamming coefficients fo Hamming weighting, phases fo steeing, etc.). Choose an FFT length, M, that is 5 to 10 times lage than N elt and is a powe of. If you had to add fae elements, the N elt used in this computation must include both the actual and fae elements. Tae the FFT and pefom the equivalent of the MATLAB FFTSHIFT function to place the zeo tap in the cente of the FFT output data aay. Call the esult B. Compute u sin M : M 1 Md whee d is the element spacing fom the fist step. The notation hee and in the following is MATLAB notation. Compute u find abs 1 (find is the MATLAB find function). Compute R B and, if needed, G R sum R* Md MATLAB sum function) Compute asin u.. (sum is the Plot G o R vs. u o. If you plot G o R vs. you will plotting the adiation o gain patten in angle space. If you plot G o R vs. u you will plotting the adiation o gain patten in sine space. M. C. Budge, J., 01 mev@thebudges.com 3

24 1.10 PLANAR ARRAYS We now want to extend the esults of linea aays to plana aays. In a plana aay the antenna elements ae located on some type of egula gid in a plane. An example that would apply to a ectangula gid is shown in Figue Figue 1-15 Example Geomety fo Plana Aays The aay lies in the X-Y plane and the aay nomal, o boesight, is the Z-axis. The intesections of lines with the numbes by them ae the locations of the vaious elements. The line located at the angles and points to the field point (the taget on tansmit o the souce, which could also be the taget, on eceive). The field point is located at a ange of that, as befoe, is vey lage elative to the dimensions of the aay (fa field assumption). In the coodinate system of Figue 1-15 the field point is located at x f, y f sin cos, sin sin. (1-81) The 00 element is located at the oigin and the mn th element is located at md x, nd y whee d x is the spacing between elements in the x diection and d y is the spacing between elements in the y diection. With this and Equation 1-81 we can find the ange fom the mn th to the field point, mn as x md y nd md sin cos nd sin sin (1-8) mn f x f y x y whee we have made use of the fact that is much geate than the dimensions of the antenna. M. C. Budge, J., 01 mev@thebudges.com 4

25 If we invoe ecipocity and conside the eceive case (as we did fo linea aays) we can wite the voltage out of the mn th element as j j Vmn, Vamn exp exp md x sin cos nd y sin sin (1-83) whee a mn is the weighting applied to the mn th element. Given that the outputs of all mn elements ae summed to fom the oveall output, V,, we get V N M, V, n0 m0 mn N M j j V exp amn exp md x sin cos nd y sin sin n0 m0 (1-84) whee M 1 is the numbe of elements in the x diection and N 1 is the numbe of elements in the y diection. If we divide by V and ignoe the phase (see the sections on linea aays) we can wite N M j A, amn exp md x sin cos nd y sin sin n0 m0. (1-85) At this point we adopt a notation that is common in phased aay antennas, and consistent with the notation we used fo linea aays: sine space. We define and u sin sin cos (1-86) v sin sin sin. (1-87) With this we wite o N M j Asin,sin amn exp md x sin nd y sin n0 m0 (1-88) N M j Au, v amn exp md xu nd yv n0 m0. (1-89) Consistent with ou wo on linea aays we wite the adiation patten as, Au, v R u v and the diective gain as (1-90),, G u v R u v R (1-91) whee R will be discussed late. M. C. Budge, J., 01 mev@thebudges.com 5

26 When we plot G u, v we say we ae plotting the diective gain patten in sine space. When we plot G, we say that we ae plotting the diective gain patten in angle space. The usual pocedue fo finding G, is to fist find G u, v and then pefom a tansfomation fom uv, space to., To deive the tansfomation, we can solve the uv, equations above fo and. Doing so esults in and 1 sin u v (1-9) 1 tan vu (1-93) whee the actangent is the fou-quadant actangent. With the above 0 and, which coves the entie sphee. An obvious constaint fom the equation fo, and the definition of u and v, is u 1, v 1 and u v 1. This is woth noting because, as we found fo the linea aay, we will compute values of Au, v fo uv, values that violate this constaint. As befoe, ou solution will be to ignoe Au, v fo uv, values that violate the above constaint Elevation and Azimuth Cuts As a final note, if one wants to plot an elevation cut of the diective gain patten G, G u, v u0. If one wanted an azimuth cut one would plot one would plot 1 1 sin v0 vsin G,0 G u, v u. This deives fom the fact that an elevation cut is a plot of the diective gain in the Y-Z plane of Figue 1-15 and an azimuth cut is a plot of the diective gain in the X-Z plane of Figue Weights fo Beam Steeing In the equation fo Au, v, the a mn ae the weights that ae used to povide a pope tape and to stee the beam. They ae of the geneal fom j a a exp md u nd v mn mn x o y o whee u, v ae the desied beam angles in sine space. o o (1-94) M. C. Budge, J., 01 mev@thebudges.com 6

27 Use of the FFT to Compute Plana Aay Pattens As with the linea aay, we ecognize that Au, v has the fom of a Fouie tansfom, albeit a two-dimensional Fouie tansfom. Analogous to the technique fo A u, v. The basic technique is the linea aays, we can use the -D FFT to compute same. Namely: Put the antenna on a ectangula gid with spacings of d and d. This could equie adding dummy elements as you did fo the linea aay. We will discuss this futhe shotly. This essentially equies specifying all of the amn, even the dummy elements (whee amn 0). As indicated above, this is also whee the beam steeing is pefomed Tae the -D FFT. As befoe, the FFT should be a powe of and should be 5 to 10 times lage, in each diection (X and Y), than the aay. The lengths in the X and Y diection do not need to be the same. Set Au, v to zeo fo u 1, v 1 o Find R u, v and, u v 1 G u v and mae the appopiate plots. x y Aay Shapes and Element Locations (Element Pacing) The wo above was developed fo the case of a ectangula aay with the elements placed on a ectangula gid. This is not a common method of constucting antenna. Many antennas ae non-ectangula (cicula o elliptical) and thei elements ae not placed on a ectangula gid (ectangula pacing). In both cases the deviations fom ectangula shape and/o ectangula pacing ae usually made to conseve aay elements and incease the efficiency of the antenna (the elements at the cones o ectangula aays do not contibute much to the antenna gain and cause the idges you will note in you homewo. You will loo at the effects of non-ectangula aays as pat of the homewo poblems. The most common element pacing scheme besides ectangula pacing is called tiangula pacing. The oigin of the phases will become clea in the ensuing discussions. Figue 1-16 contains setches of sections of a plana aay with ectangula and tiangula pacing. The dashed elements in the tiangula pacing illustation ae dummy elements that must be included when one uses the -D FFT to compute the adiation patten fo an aay with tiangula pacing. The need fo the dummy elements stems fom the fact that the FFT method must use ectangula pacing. The amplitudes of the dummy elements ae set to zeo (as was done when we wanted to analyze aays with elements spacings that wee geate than. In the tiangula pacing, the elements ae aanged in a tiangula patten. Thus the oigin of the pacing nomenclatue. M. C. Budge, J., 01 mev@thebudges.com 7

28 Figue 1-16 Illustation of ectangula and tiangula element pacing Amplitude Weighting As with linea aays we can use amplitude weighting to educe sidelobes. We use the same types of amplitude weightings as fo linea aays (Taylo, Chebychev, Hamming, Gaussian, etc.). The diffeence is that we now need to be concened with applying the weightings in two dimensions. Thee ae two basic ways to do this: 1. multiplicative weighting and. ciculaly symmetic weighting. Fo the multiplicative method we would wite the magnitudes of the weights as amn am an. (1-95) This method of detemining the weights is the easiest of the two discussed hee. It will povide pedictable sidelobe levels on the pincipal planes (u cut and v cut) but not in the sidelobe egions between the pincipal planes. To achieve pedictable sidelobe levels ove the entie sidelobe egion one must use ciculaly symmetic weighting. To do this one can use the following pocedue, which wos well fo cicula aays and easonably well fo elliptic aays. 1. Geneate a set of appopiate weights that have a numbe of elements that is equal to Nwt Lmax dmin whee L max is the maximum antenna dimension and d min is the minimum element spacing. Define an aay of numbes, x w that goes fom -1 to 1 and has N wt elements.. Find the location of all of the antenna elements elative to the cente of the aay. d and d ymn be the x and y locations of the mn th element elative to the Let xmn cente of the aay. Let D x and D y be the antenna widths in the x and y M. C. Budge, J., 01 mev@thebudges.com 8

29 diections. Find the nomalized distance fom the cente of the aay to the mn th element using x mn d xmn ymn D x D y d. (1-96) 3. Use x mn to intepolate into the aay of weights vs. x w to get the a mn FEEDS An antenna feed is the mechanism by which the enegy fom the tansmitte is conveyed to the aay so that it can be adiated into space. On eceive, it is used to collect the enegy fom the aay elements. Thee ae two boad classes of feed types used in phased aays: space feed and copoate, o constained, feed. These two types of feed mechanisms ae illustated in Figues 1-17 and In a space feed the feed is some type of small antenna that adiates the enegy to the aay, though space. The feed could be a hon antenna o even anothe, smalle phased aay. In a space fed aay, the feed geneates an antenna patten, on tansmit, which is captued by small antennas on the feed side of the aay. These ae epesented by the v-shaped symbols on the left side of the aay of Figue The outputs of the small antennas undego a phase shift (epesented by the cicles with in them) and ae adiated into space by the small antennas epesented by the v-shaped symbol on the ight of the aay. On eceive, the evese of the above occus: The antennas on the ight of the aay captue the enegy fom the souce The phase shiftes apply appopiate phase shifts The antennas on the left of the aay focus and adiate the enegy to the feed The feed sends the enegy to the eceive. M. C. Budge, J., 01 mev@thebudges.com 9

30 Figue 1-17 Space-Feed Phased Aay The phase shiftes povide the beam steeing as indicated in pevious discussions. They also pefom what is called a spheical coection. The E-field adiated fom the feed has constant phase on a sphee, which is epesented by the acs in Figue This means that the phase at each of the phase shiftes will be diffeent. This must be accounted fo in the setting of the phase shiftes. This pocess of adjusting the phase to account fo the spheical wave font is temed spheical coection. Geneally, the feed poduces its own gain patten. This means that the signals enteing each of the phase shiftes will be at diffeent amplitudes. This means that the feed is applying the amplitude weighting, a, to the aay. Geneally, the feed gain patten is adjusted to achieve a desied sidelobe level fo the oveall antenna. Feed pattens ae typically shaped lie pat of one lobe of a sine/cosine function. In ode to obtain a good tadeoff between gain and sidelobe levels fo the oveall antenna, the feed patten is such that the level at the edge of the aay is between 10 and 0 db below the pea value. This is usually temed an edge tape. A feed that povides a 0 db edge tape will esult in lowe aay sidelobes than a feed that povides a 10 db edge tape. Howeve, a space-fed phased aay with a 10 db edge tape feed will have highe gain than a space-fed phased aay with a 0 db edge tape feed. mn M. C. Budge, J., 01 mev@thebudges.com 30

31 Figue 1-18 Copoate Feed (Constained Feed) Phased Aay In a copoate feed phased aay the enegy is outed fom the tansmitte, and to the eceive, by a waveguide netwo. This is epesented by the netwo of connections to the left of the aay in Figue In some applications the waveguide netwo can be stuctued to povide an amplitude tape to educe sidelobes. The phase shiftes in a copoate feed aay must include additional phase shifts to account fo the diffeent path lengths of the vaious legs of the waveguide netwo. Geneally, space feed phased aays ae less expensive to build because they don t equie the waveguide netwo that is equied by the copoate feed phased aay. Howeve, the copoate feed phased aay is smalle than the space feed phased aay. The space feed phased aay is geneally as deep as it is tall o wide to allow fo pope positioning of the feed. The depth of a copoate feed phased aay is only about twice the depth of the aay potion of a space feed phased aay. The exta depth is needed to accommodate the waveguide netwo. Finally, the copoate feed phased aay is moe ugged than the space fed aay since almost all hadwae is on the aay stuctue. A sot of limiting case of the copoate feed phased aay is the solid state phased aay. Fo this aay, the phase shiftes of Figue 1-18 ae eplaced by solid state tansmit/eceived (T/R) modules. The waveguide netwo can be eplaced by cables since they cay only low powe signals. The tansmittes in each of the T/R modules ae faily low powe (10 to 100 watt). Howeve, a solid state phased aay can contain thousands (3000 to 1,000) of T/R modules so that the total tansmit powe is compaable to that of at space feed phased aay. M. C. Budge, J., 01 mev@thebudges.com 31

32 1.1 POLARIZATION Thus fa in ou discussions we have played down the ole of the electic field (Efield) in antennas. As ou final topic we need to discuss E-fields fo the specific pupose of discussing polaization. As you may have leaned in you electomagnetic waves couses, E-fields have both diection and magnitude (and fequency). In fact, an E-field is a vecto that is a function of both spatial position and time. If we conside a vecto E- field that is taveling in the z diection of a ectangula coodinate system we can expess it as,,, E t z E t z a E t z a (1-97) x x y y whee a x and a y ae unit vectos. This fomulation maes the assumption that the electic field is nomal to the diection of popagation, z in this case. In fact this is not necessay and we could have been moe geneal by including a component in the a z diection. A gaphic showing the above E-field is contained in Figue In this dawing, the z axis is the LOS (line of sight) vecto fom the ada to the taget. The x-y plane is in the neighbohood of the face of the antenna. The y axis is geneally up and the x axis is oiented so as to fom a ight-handed coodinate system. This is the configuation fo popagation fom the antenna to the taget. When consideing popagation fom the taget, the z axis points along the LOS fom the taget to the antenna, the y axis is up and the x is again oiented so as to fom a ight-handed coodinate system. ` When we spea of polaization we ae inteested in how the E-field vecto, E t, z, behaves as a function of time fo a fixed z, o as a function of z fo a fixed t. To poceed futhe we need to wite the foms of E t, z and E, simplified fom of sinusoidal signal. With this we get, sin sin x y t z. We will use the E t z Exo fot z ax Eyo fot z a y. (1-98) In the above E xo and E yo ae positive numbes and epesent the electic field stength. f o is the caie fequency and is the wavelength, which is elated to f o by c fo. is a phase shift that is used to contol polaization oientation. M. C. Budge, J., 01 mev@thebudges.com 3

33 Figue 1-19 Axes convention fo detemining polaization If E t, z emains fixed in oientation as a function of t and z the E-field is said to be linealy polaized. In paticula If 0, E 0 and E 0 we say that the E-field is hoizontally polaized. xo yo If 0, E 0 and E 0 we say that the E-field is vetically polaized. yo xo If 0, and E E 0 we say that the E-field has a slant 45 polaization. xo yo If 0, and E E 0 we say that the E-field has a slant polaization at some xo yo 1 angle othe than 45. The polaization angle is given by tan Eyo Exo If and Exo Eyo 0 we say that we have cicula polaization. If the polaization is left-cicula because E t, z otates counteclocwise, o to the left, as t o z incease. If the polaization is ightcicula because E t, z otates clocwise, o to the ight, as t o z incease. If is any othe angle besides, 0 o and/o E E 0 we say that the polaization is elliptical. It can be left ( ) o ight ( ) elliptical. As a note, polaization is always measued in the diection of popagation of the E-field to/fom the antenna fom/to the taget. This is usually also the boesight angle. Howeve, if one is looing at a taget though the antenna sidelobes the diection of popagation is not the boesight. When polaization of an antenna is specified, it is the polaization in the main beam. The polaization in the sidelobes can be damatically diffeent than the polaization in the main beam. xo yo. M. C. Budge, J., 01 mev@thebudges.com 33

34 1.13 REFLECTOR ANTENNAS Olde adas, and some moden adas whee cost is an issue, use eflecto types of antennas athe than phased aays. Reflecto antennas ae much less expensive than phased aays (thousands to hundeds of thousands of dollas as opposed to millions o tens of millions of dollas). They ae also moe ugged than phased aays and ae geneally easie to maintain. They can be designed to achieve vey good gain and vey low sidelobes. The main disadvantages of eflecto antennas, compaed to phased aay antennas, ae that they must be mechanically scanned. This means that adas that employ eflecto antennas will have limited multiple taget capability. In fact, most taget tacing adas that employ eflecto antennas can tac only one taget at a time. Seach adas that employ eflecto antennas can detect and tac multiple tagets but the tac update ate is limited by the scan time of the ada, which is usually on the ode of 5 to 0 seconds. This, in tun, limits the tac accuacy of these adas. Anothe limitation of adas that employ eflecto antennas is that sepaate adas ae needed fo each function. Thus, sepaate adas would be needed fo seach, tac, and missile guidance. This equiement fo multiple adas leads to inteesting tadeoffs in ada system design. With a phased aay it may be possible to use a single ada to pefom the thee afoementioned functions. Thus, while the cost of a phased aay is high, elative to a eflecto antenna, the cost of thee adas with eflecto antennas may be even moe expensive than a single phased aay ada. When one also accounts fo factos such as cost of opeatos, maintenance, and othe logistic issues, the cost tadeoff becomes even moe inteesting. Almost all eflecto antennas use some vaiation of a paabolid. An example of such an antenna is shown in Figue 1-0. The feed shown in Figue 1-0 is located at the focus of the paabolic eflecto. Since it is in the font, this antenna would be temed a font fed antenna. The lines fom the eflecto to the feed ae stuts that ae used to eep the feed in place. Solni s ada handboo and Jasi s antenna handboo have dawings of seveal vaiants on the paabolid type antenna. In almost all of these, the eflecto is fomed by cutting off the top and/o bottom of the eflecto, and sometimes the sides. Thus, the eflectos ae potions of a paabolid. A paabola is used as a eflecto because of its focusing popeties. This is somewhat illustated by Figue 1-1. In this figue it will be noted that the feed is at the focus of the paabola. Fom, analytic geomety we now that if ays emanate fom the focus and ae eflected off of the paabola, the eflected ays will be paallel. In this way paabolic antenna focuses the divegent E-field fom the feed into a concentated E-field. Stated anothe way, the paabolic eflecto collimates the feed s E-field. M. C. Budge, J., 01 mev@thebudges.com 34

35 Figue 1-0 Example of a Paabolic Reflecto Antenna As with space fed phased aays, the feed patten can be used to contol the sidelobe levels of a eflecto antenna. It does this by concentating the enegy at cente of the eflecto and causing it to tape off towad the edge of the eflecto. The pocess of computing the adiation patten fo a eflecto antenna, whee the feed is at the focus is faily staight fowad. Refeing to Figue 1-1, one places a hypothetical plane paallel to the face of the eflecto, usually at the location of the feed. This plane is temed the apeatue plane. One then puts a gid of points in this plane. The points ae typically on ectangula gid and ae spaced apat. The bounday of the points will be in a cicle that follows the edge of the eflecto. These points will be used as elements in a hypothetical phased aay. We thin of the points, pseudo aay elements, as being in the x-y plane whose oigin is at the feed. The z axis of this coodinate system is nomal to the apetue plane. If we daw a line, in the x-y plane, fom the oigin to the point (x, y) the angle it maes with the x axis is 1 y tan x (1-99) whee the actangent is the fou-quadant actangent. The distance fom the oigin to the point will be x y. (1-100) Now, one can daw a line fom the point, pependicula to the apetue plane, to the eflecto. Examples of this ae the lines dl 11 and dl in Figue 1-1. The next step is to find the angle,, between the z axis and the point on the eflecto. M. C. Budge, J., 01 mev@thebudges.com 35

36 Figue 1-1 Geomety Used to Find Reflecto Radiation Patten Fom Figue 1-1 it will be noted that d l f (1-101) whee f is the focus of the paabola. Also, l d. (1-10) With this we can solve fo d to yield and 1 d f 4 f (1-103) 1 sin d. (1-104) The angles and ae next used to find the gain of the feed at the point whee the ay intesects the paabolic eflecto. This gain gives the amplitude of the pseudo element at (x, y). The above pocess is epeated fo all of the pseudo elements in the apetue plane. Finally, the eflecto antenna adiation patten is found by teating the pseudo elements in the apetue plane as a plana phased aay. Thee is no need to be concened about the phase of each element since the distance fom the feed to the eflecto to all points in the apetue plane ae the same. This means that the vaious ays fom the feed tae the same time to get to the apetue plane. This futhe implies that the E-fields along each aay will have the same phase in the apetue plane. M. C. Budge, J., 01 mev@thebudges.com 36

37 If the feed is not located at the focus of the paabolid the calculations needed to find the amplitudes and phases of the E-field at the pseudo elements become consideably moe complicated. It is well beyond the scope of this class. M. C. Budge, J., 01 37

38 APPENDIX A AN EQUATION FOR TAYLOR WEIGHTS The following is an equation fo calculating Taylo weights fo an aay antenna. It is simila to the equation on page 0-8 of Antenna Engineeing Handboo Thid Edition by Richad C. Johnson, with some claifications and coections. whee The un-nomalized weights fo the a n1 n x 1,, cos K th element of K element linea aay is F n A n (A-1) n1 and F n, A, n 1 R n1 n n 1! 1 m1 A m 1 1! 1! n n n n, (A-) cosh A, (A-3) A n n 1 (A-4) 0 10 SL R. (A-5) SL is the desied sidelobe level, in db, elative to the pea of the main beam. It is a positive numbe. Fo example, fo a sidelobe level of -30 db, SL 30. This says that the sidelobe is 30 db below the pea of the main beam. n is the numbe of sidelobes on each side of the main beam that one desies to have a level of appoximately SL below the main beam pea amplitude. The x can be computed using the following MATLAB notation z z K : K ; x : : end ; Finally, one needs to nomalize the weights by dividing all of the a by max a. M. C. Budge, J., 01 mev@thebudges.com 38

39 APPENDIX B CALCULATION OF R FOR A PLANAR ARRAY In Section 1.4 we deived an equation fo R fo a linea aay. We futhe extended this equation in Section 1.7 to allow us a method of calculating R when we had the adiation patten of a linea aay expessed in sine space. In this appendix, we want to deive an equation that will allow us to compute R fo a plana aay when we have the adiation fo a plana aay expessed in sine space. The geomety we need to solve this poblem is shown in Figue B-1. This figue is simila to Figue 1-8 whee we have eplaced the linea aay (the column of dots) by a plana aay (seveal columns of dots). We have also edefined the angles. We had to do this so as to not confuse the angles used to deive R with those used in deiving the R u, v. When we deived Equation 1-43 we did not mae this distinction. equation fo As a esult, we used the same angle definitions ( and ) to deive the equation fo R as we did when we deived the equation fo Ru, v. We should not have done this since the angles ae defined diffeently. We will e-deive Equation 1-4 and Equation 1-43 using the angles ( and ) defined in Figue B-1. I chose the angles and because, if the aay is vetical as shown, is azimuth and is elevation. Figue B-1 Geomety Used To Compute R fo a Plana Aay We stat with the definition 1 R R, d. (B-1) 4 sphee Fom Figue B-1 we wite cos d dw ds dd. (B-) M. C. Budge, J., 01 mev@thebudges.com 39