Figure Geometry for Computing the Antenna Parameters.

Transcription

1 Spheical Coodinate Systems Definitions Figue Geomety fo Computing the Antenna Paametes. Antenna Radiation Patten: The distibution of adiated enegy fom an antenna ove a suface of constant adius centeed upon the antenna. Fa-Field: Field egion of the antenna whee > 2D 2 /λ, povided D > λ o, moe geneally, the antenna adiation patten at =. Nea-Field: Field egion of antenna whee < 2D 2 /λ. Click hee fo a supplemental undestanding of antennas - a quick oveview. In this lectue, we ae going to begin to study antenna fundamentals and some vey impotant definitions. EE 438: Antenna Engineeing 1

2 Figue shows a spheical coodinate system. At the oigin thee is an antenna. On the spheical coodinate system we have a sphee of adius,. Imagine this as a hypothetical spheical suface enclosing the antenna. We can identify a point on the sphee by adius, angle 2, and angle φ. At the indicated point, f, thee is some electic and magnetic field intensities ceated by the antenna. At the point of inteest, maked as f, we want to measue the electic field intensity o the magnetic field intensity as a function of position on the sphee, that is, as we vay 2 and φ keeping the distance,, to the cente of this antenna constant. A plot of the magnitude of the electic field intensity o the magnetic field intensity gives what we call the adiation patten. The adiation patten is a measue of how the electic o magnetic field intensities vay with angula positions 2 and φ fo a fixed ange,. This will be a mathematical calculation, as we will show when we study a dipole. We will be able to compute exactly how the field vaies ove the sphee. In a pactical system, one can actually measue the electic field intensity o magnetic field intensity as a function of position. This would be done in an antenna test ange. If we vay 2 by keeping φ constant, we ae moving along a line which would look like the longitudinal lines on a globe. If we ae keeping 2 constant and vay φ we ae moving along what looks like latitudinal lines on the globe. So we have an option of vaying 2 by keeping φ constant o vaying φ by keeping 2 constant. When we study a simple antenna, like a dipole, we will see that the electic field is made up of seveal tems. One is called the fa-field; the othe is called the nea-field. When you ae close to the antenna, the nea-field tem will dominate the fa-field tem. When you ae fa away fom the antenna, the fa-field tem will dominate the nea-field. In othe wods, the nea-field will die off vey apidly as is made lage, leaving pimaily the fa-field component. But close to the antenna, the nea-field is much stonge then the fa-field and it dominates. Now, what is the dividing line when the nea-field ends and the fa-field begins? We need to define a fa-field egion and we need to define a nea-field egion. This is impotant when we measue antenna pattens; we measue the antenna patten in the fa-field. Now it can be shown that the fa-field of an antenna is whee the ange, exceeds twice D 2 divided by the opeating wavelength. As the fomula states hee, must be geate that two times D 2 divided by 8. When we make some mathematical calculations of the antenna adiation patten in the fafield, we will actually mathematically let become vey lage, appoaching 4. It tuns out that ou calculation becomes vey easy thee, and this will be evident late when we study an actual antenna. But suppose you want to make a measuement of the adiation patten of this antenna. You have to make sue that you ae in the fa-field. Let us take a pactical example. In the pevious lesson you wee asked to click on the ICON which showed a view of the Aecibo spheical eflecto antenna. That antenna was appoximately 1, feet in diamete, o exactly 3 metes. The antenna is used fo adio astonomy puposes and its beam is essentially upwad. If you wee EE 438: Antenna Engineeing 2

3 going to fly an aicaft ove some imaginay sphee enclosing this antenna, how fa should we be to make sue we ae in the fa-field? Well, D would be the lagest dimension of the antenna stuctue which is given as 3 metes. Fo adio astonomy puposes that facility sometimes opeates at 43 Mhz, theefoe, the fee space wavelength is.698 metes. You can check that out youself. Plugging some numbes into the fomula tells us that to be in the fa-field must exceed 257,879 metes o appoximately 16 miles. 16 miles is the dividing egion between fafield and nea-field; actually we should be much futhe than 16 miles to be consideed in the fa-field. Thee is no aicaft that we can fly ove that antenna that can each the altitude of 16 miles, so we ae peplexed on how to measue the adiation patten of this antenna. Unfotunately, we ae confined to make the measuements in the nea-field, that is, when is less than 2 times D 2 divided by 8. Fotunately, we have mathematics which allows us to tansfom nea-field data into fa-field data, and that is exactly how it can be done. Thee ae many instances whee antennas ae tested in antenna test anges in the nea-field because one can always convet nea-field data to fa-field data. With the Aecibo antenna thee is anothe possibility. Conside a adio sta as a souce. Radio stas emit vast amounts of electomagnetic enegy in cetain adio bands. So instead of thinking of the Aecibo antenna as a tansmitting antenna, and tying to measue the field intensity ove a sphee, let s view it as a eceiving antenna. The enegy fom the distant sta is in the fa-field. We can stee this antenna. You notice fom the pictue of the Aecibo Reflecto that thee ae feeds above the eflecto dish and they ae moveable, theefoe we can stee the antenna to scan though the adio sta and make a plot of the measuement of the eceived antenna signal as a function of 2 and φ. So we do have a way, at least with the Aecibo obsevatoy, of measuing tuly a fa-field. EE 438: Antenna Engineeing 3

4 Pola Plot Figue 1.2.2a Pola Plot of a Radiation Patten. Usually the nomalized *E* field patten is plotted fo the adiation patten. P " *E* 2 powe patten *E* db = 2 log *E* PdB = 1 log P Now that we will be able to detemine the electic field intensity ove the sphee and that we ae in the fa-field, how do we pesent this data? Antenna enginees plot data in pola plots o in ectangula plots. So shown on Figue 1.2.2a is a pola plot and we see that we have plotted the magnitude of the field intensity, as you would in a pola plot, along the adial diection as a function of 2. This is how we constuct a pola diagam. We note a geneal chaacteistic of antennas; they have a main beam, some side lobes, mino lobes, possibly a back lobe, and thee ae seveal nulls. EE 438: Antenna Engineeing 4

5 Figue 1.2.2b Rectangula Plot of a Radiation Patten. Figue 1.2.2b is a ectangula plot of the patten instead of a pola plot. We plot the field intensity along the z axis vetically, and we plot the angle 2 along the hoizontal axis. Both plots display the same infomation. We see a vey clea cut main beam, diminishing side lobes, in this case, a back lobe and seveal distinct nulls. We have a choice of plotting the magnitude of the electic field, which was measued, o as some enginees do, the squae of the magnitude which is popotional to the Poynting vecto, and that we would call a powe patten. You must be vey caeful when you see a adiation patten. Is it a plot of the electic field o a plot of the electic field squaed? Is it a field patten o is it a powe patten? Usually we nomalize the patten. You can see the main beam has a maximum, in this case a 2 =. Whateve the value of the electic field intensity was at that paticula point, we divide all measuements by the maximum, so now the maximum would have stength unity. This is a nomalized plot. Antenna enginees pefe to plot not only the magnitude of the electic field, but the decibel equivalent. In othe wods, 2 times the log of the magnitude of the electic field intensity. O 1 times the log of the powe patten. Both ae acceptable foms. Now one impotant paamete on a adiation patten is the half-powe beamwidth, identified by the symbol HPBW. This is whee the electic field intensity equals.77 of the mainbeam maximum field intensity.also shown is the angula diffeence between the nulls on each side on the main beam giving the beamwidth measued between the fist nulls, FNBW. EE 438: Antenna Engineeing 5

6 Figue 1.2.2c Half-Powe Beamwidth Definition. Look at Figue 1.2.2c which shows two points on the adiation patten identified as having stength one ove the squae oot of two. These ae the half-powe points. If the electic field is one ove the squae oot of two, the powe is ½. We have shown the patten as nomalized, when 2 is ; the main beam is identified as unity, that is full powe density. Thee ae two angles on eithe side of the main beam at which the electic field will diminish to one ove the squae oot of two of the maximum and the maximum was one. These ae called the half-powe points because powe density is popotional to the squae of the magnitude of the electic field intensity. The angula spead between those two maked half-powe points, defines the halfpowe beamwidth. This convenient measue tells us how naow the beam is. Is the beamwidth 4 degees wide o is it 1/1th of a degee wide? If we made some calculations on the Aecibo spheical eflecto we would find that the beamwidth is about a few tenths of a degee. It is like a seach light which could have a vey naow beam of enegy, o the aveage flash light which could have a wide beam of enegy. The beamwidth is then a measue of how well the enegy has been concentated in a fixed egion, that is the half-powe egion. Within the half-powe beamwidth, the powe density vaies between one and down to ½. Outside the half-powe beamwidth, the powe density will be ½ o less, so half-powe is a convenient dividing point. Sometimes it is easie to calculate the beamwidth between the fist nulls, identified at FNBW. Thee is a null on each side of the main beam. Obviously, FNBW exceeds HPBW but sometimes it is moe convenient to calculate FNBW, as you will see late on. EE 438: Antenna Engineeing 6

7 Powe Density and Radiated Powe The Poynting Vecto 1 P = E H 2 P * is defined as: which is a powe density with units of W/m 2. (1.2.1) P ds Figue The w/m 2 Vaies with Position on the Suface of a Sphee. The total complex powe flowing out though a closed suface S is: * ( ) Pc = P ds 1 = E H ds 2 s s = nds $ whee ds $n = unit nomal diected outwad fom the suface. (1.2.2) Click on the icon fo explanation by D. Feao. Text of audio is povided below. We now continue to calculate the total adiated powe fom an antenna; we have aleady eviewed the Poynting vecto, which is epeated in Equation (1.2.1). It is the numbe of watts EE 438: Antenna Engineeing 7

8 pe squae mete that happens to be at a given point and the diection of the vecto is the diection of the powe flow. We show in Figue 1.2.3, that the antenna is suounded by this imaginay o hypothetical enclosue, a spheical suface, with an element of aea, ds, on the sphee. We want to compute the total complex powe which flows out though the entie enclosed suface, S. As we have aleady discussed on Figue , this equies one to integate the dot poduct of the complex powe density, P a vecto, with the element of aea as shown by Equation (1.2.2). The element of aea ds, a vecto, is diected outwad fom the sphee. The unit nomal is outwad. EE 438: Antenna Engineeing 8

9 Powe Density and Radiated Powe Note that P { } { P } Re = Aveage eal powe density Im = Aveage eactive (stoed) powe density. In the fa-field of an antenna the powe density is mostly eal. Hence, the aveage powe adiated by an antenna is P ad = 1 { } 2 Re P ds s 1 { } 2 Re * E H ds (1.2.3a) = (1.2.3b) s Fo a spheical suface ds Whee n$ = a$ = nds $ n$ = $ a ds Figue Element of Aea fo Powe Calculations. EE 438: Antenna Engineeing 9

10 Click on the icon fo explanation by D. Feao. Text of audio is povided below. We continue with what we have aleady discussed that the eal pat of the Poynting vecto is the aveage eal powe density and the imaginay pat of the Poynting vecto is the aveage eactive powe density. We will show, and you will have to wok on faith at this paticula time, that when you ae in the fa-field of an antenna, the powe density is mostly eal. Hence, the aveage total powe adiated by an antenna is found by integating the eal pat of the powe density ove this lage sphee because we ae in the fa-field. EE 438: Antenna Engineeing 1

11 Assignment #1 On the spheical suface of Figue show that * ( ) 1 E H a$ = E + E η θ 2 2 φ Wheeη is the intinsic impedance of fee space. EE 438: Antenna Engineeing 11

12 Powe Density and Radiated Powe 2π π 1 2 P = Re ( E H * ) a d d ad $ sinθ θ φ 2 Hence, (1.2.4) The esults of lesson assignment #1 may be used to show that 2 2π π P = ( E + E ) 2 2 ad θ dθ dφ θ φ sin 2η =( Pad ) + ( Pad ) whee ( P ) θ 2 2π = 2η φ E ad θ θ ( P ) π 2 2π = 2η E ad φ φ π 2 2 sinθdθdφ sinθdθdφ Now suppose we define the adiation intensity fo a given antenna to be 2 U( θφ, ) = 2 2 Eθ + E φ Watts 2η = U + U θ φ 2π π ( ) P = U d d ad θ, φ sinθ θ φ Then we may wite: : (1.2.5) We wish to pefom the integation shown in Equation (1.2.4). You fist assignment was to show that the E H dotted into the unit vecto, which is in the adial diection, is made up of two pats, involving the tems *E 2 * 2 and *Eφ * 2. Remembe, in geneal can have 2 and φ H E E components and can have φ and 2 components because the and fields ae othogonal in the fa-field. is the intinsic impedance of fee space. Assuming that you have shown what the assignment equested, we will use that infomation in Equation (1.2.4) to obtain H EE 438: Antenna Engineeing 12

13 the expession fo the powe adiated by the antenna, which is witten as two tems (P ad ) 2 and (P ad )φ. The powe adiated due to the 2 component of the field and the powe adiated due to the φ component of the field can be computed as shown in the expession fo (P ad ) 2 and (P ad )φ. We make a change of vaiable hee. Instead of woking diectly with the Poynting vecto, some authos pefe to wok with the adiation intensity, U. The adiation intensity as shown is found by taking the Poynting vecto tems and multiplying by the squae of the adius. Theefoe, thee ae two tems to the adiation intensity, a 2 value and a φ value, and. The adiation intensity is a convenient way of witing some of the tems which appea in the integals. Theefoe, we can wite the total powe adiated in tems of the adiation intensity as shown in Equation (1.2.5). We ae now to a point that if we know the electic field components, that is E 2 and Eφ adiated by a given antenna stuctue, and expess the adiation intensity, the integal in Equation (1.2.5) gives us the total adiated powe though a hypothetical sphee enclosing an antenna. Whee did that powe come fom? It came fom the antenna. Theefoe, we have an expession fo the total powe adiated by an antenna. U θ U φ EE 438: Antenna Engineeing 13