Estimating Directivity and Gain of Antennas
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1 Estimating Directivity and Gain of Antennas Warren L. Stutzman Antenna Group Center for Wireless Telecommunications Bradley Department of Electrical Engineering Virginia Polytechnic Institute and State University Blacksburg, VA Tel: +1 (540) Fax: +1 (540) WWW: Keywords: Antenna theory; antenna measurements; antenna directivity; antenna gain 1. Abstract Gain is the most important performance parameter of an antenna. However, in many practical situations it is not possible to measure or calculate the gain of an antenna. Also, recent interest in wireless applications has increased the need of system engineers to accurately estimate antenna gain. Many simple formulas are available for estimating gain. But, each formula has a range of applicability, and inappropriate use of these formulas will result in inaccurate gain values. This paper explains the process of gain evaluation, and gives several simple gain-estimation formulas, together with recommendations for their use. 2. Introduction For applications such as point-to-point or point-to-multipoint communications, the gains of the transmit and receive antennas have a significant impact on system performance. In fact, fractionsof-a-db deviation in antenna gain values can impact greatly on both system performance and cost. Although it is desirable to measure the gain of an antenna, this is frequently not practical. Also, antenna gain determination by analytical or numerical methods is often difficult. In addition, in system design trade-off studies, it is very important to have simple and accurate antenna gain expressions. This need to estimate gain has motivated the introduction of many gain approximation methods. There are a number of simple formulas for directivity and gain that can be found in texts, journals, and trade magazines. Proponents of certain formulas at times criticize the use of other formulas. In reality, one representation is not better than another, but each representation has an appropriate use. In this paper, we present several simple formulas, along with guidelines for their use. Basic principles of gain and directivity are reviewed. Widelyused simple formulas are presented, as well as the associated underlying theory that leads to rules for determining under what conditions are they appropriate. Section 3 defines directivity and gain. The simple directivity and gain formulas are presented in Sections 4 and 5, including a handy table for future reference. Array-antenna gain is considered in Section 6. The topic of this article is part of the new material added to the second edition of Antenna Theory and Design by W. L. Stutzman and G. A. Thiele, John Wiley, Directivity and gain Directivity is the increase in power density in a given direction at a fixed distance from a transmitting antenna, relative to the power density with the same transmit power distributed equally in all directions. Usually, the maximum value of directivity is of pimary interest, and here we use directivity to mean maximum directivity. So, directivity is a measure of the increase in maximum power density at a fixed distance in W/m2 (or equivalently, radiation intensity in W/m2/steradinn), compared to that from a hypothetical isotropic radiator. Directivity, D, is a unit-less power ratio, usually expressed in db, using 10 log D. Often, the unit dbi is used to clarify the fact that the reference is an isotropic standard. Although the definition is for transmitting antennas, it also applies to receiving antennas, based on reciprocity. Losses are represented by radiation efficiency e,. (0 < e,. 4 1). Gain, G, is directivity reduced by losses on the antenna structure: G=e,D. (1) For electrically small antennas, the radiation efficiency can be very poor. For other antennas, such as hom antennas, the radiation efficiency approaches 100%. Exceptions are antennas that include lossy devices, such as radomes, cable/waveguide runs, filters, etc. Losses due to impedance mismatch are not included in the definition of gain, but are often unavoidable. The term realized gain is used when mismatch effects are included. Usually, such effects can be estimated and removed to obtain gain. In many wireless applications, gain is expressed in units of dbd. Gain in dbd equals gain in db reduced by the 2.15 db gain of the half-wave dipole reference antenna. Directivity can be evaluated if the radiation pattern, F(B,+), is known for all angles, using where 477 D=-, QA 2K K QA =beam solid angle = jif(b,c)lzsinbdbd+, 0 0 F(B, 4) = (field) radiation pattem normalized to unity peak. Occasionally, the integral in QA can be evaluated analytically, but IEEE Antennas and Propagation Magazine, Vol. 40, No. 4, August /98/$ IEEE 7
2 more often, it is performed numerically. For many antennas, complete pattem data are not available, and are too tedious or expensive to obtain. The directivity of aperture antennas, such as homs or reflectors, can be found by analytical techniques, using the aperture distribution [ 1, Section However, the aperture distribution is usually not known. When there is incomplete knowledge of the pattem or aperture distribution, the directivity must be estimated based on limited information. Such techniques are presented in the following sections. 4. Simple directivity formulas It is important to understand how simple directivity formulas are developed. This reveals the restrictions on their use, which if overlooked, can lead to significant error. In this section, we derive simple directivity formulas for several practical situations. The directivity of a rectangular aperture, with a separable aperture distribution, for broadside operation (Le., the beam peak direction is 8, = 0 for the aperture in the xy plane) can be expressed as [2] D=nD, Dy, (3) where D, and D, are the directivities of line source (or lineararray) distributions associated with the x and y aperture positions, E,,(x) and Ea2(y). The directivities of the linear current distributions are related to the aperture extents, L, and L,, as 2LX 2L Dx=cxy, D Y =C Y2 l (4) where R is the operating wavelength. The directivity factors c, and cy are constants that vary slightly with the distributions E,,(x) and E,,(y). For uniform line sources, c, =cy =I. The half-power beamwidths associated with the x and y distributions, EUl(x) and E,,(y), are given by R R HP, = k, -, HPY = k, - [radians], LX LY where and k, and k, are the beamwidth factors. For uniform line sources, k, = k, = Using Equation (4) in Equation (3) along with Equation (5) gives This can be expressed more generally as D=- DB (7) HpXHP, where DB is the directivity-beurnwidth product. The numerator of Equation (6) provides an approximate value for DB as 180 DB = 4 n cn cy k, ky ( T) [deg2]. DB is similar to the gain-bandwidth product that is commonly used to characterize circuit devices. It is relatively constant under a variety of operating circumstances. This is true because as the amplitude distribution is tapered, the constants c, and cy decrease due to aperture-taper efficiency reduction, but the constants k, and ky increase due to beam broadening, and nearly cancel the decrease of c,.and cy. If the value of DB could be determined, Equation (7) would be a very simple expression for directivity that depends only on a knowledge of the principal-plane pattem beamwidths. We now examine DB for various common aperture distributions. A uniform rectangular aperture (Le., constant amplitude over a rectangular area) has a distribution composed of uniform line sources in x and y. Equation (8) becomes DB =4zc,cY kx ky Then, Equation (6) is = 4 z(1) (1) (0.886) (0.886) (9) =9.86 rad2=32,383 deg2. D= HPEo HPEo uniform rectangular aperture, (IO) where HPEo and HPHo are the principal-plane beamwidths in degrees. While this is for a uniform rectangular aperture, it produces fairly accurate results for a variety of pattems with moderately narrow major lobes and with minor lobes present. For a rectangular aperture with a uniform amplitude distribution in the E- plane and a cosine amplitude taper in the H-plane, such as encountered with an open-ended waveguide, DB is 35,230 deg2 [I, Section Equation (7) can be used to find DB based on known directivity expressions. For example, the uniform circular aperture with ( / ) radius a has directivity D = 4n l2 zu2, and half-power beam- widths of l.o2(a/2a) [l, Section Then, the directivitybeamwidth product is ; ( ;T(y = 33,709 deg2 uniform circular aperture. (1 I) DB =D HP2=-za2, This is close to the directivity-beamwidth product of 32,383 deg2 ofequation (10) for a uniform rectangular aperture. As the amplitude taper from the center to the perimeter of a circular aperture is increased, DB varies from 33,707 to about 39,000 deg2 [3]. Some simple mathematical models for pattems have no minor lobes. Examples are Gaussian and cosq B beams. All radiation is contained in the main lobe, and there are no minor lobes. One no-sidelobe pattern is the rectangular beam, which approximates the beam solid angle as the product of the principalplane beamwidths, Le., QA = HPE HPH. Then, the directivity is 8 IEEE Antennas and Propagation Magazine, Vol. 40, No. 4, August 1998
3 4n 47l D=--k- - 41'253 rectangular beam. (12) ClA HPE HPH HPE. HPp This formula can work well for low-directivity antennas. For example, a half-wave dipole has HPEo = 78, HPHo = 360, and from Equation (12), D = 1.47, which is close to the correct value of For a short dipole, with HPE = 90" and HPH = 360, Equation (12) gives 1.27, which compares to the exact value of 1.5. If HPE = HPH, a ''conical-'' beam approximation can be used that takes the main beam to be constant from the center to HP/2 and drops to zero, This leads to QA = cos(hp/2)] z nhp2/4. Then, D=-=--- 4z l6-52'525 =~p2 HP2 Hpd2eg conicalbeam. (13) 4 This may have been first proposed by Schelkunoff [4]. We have presented several simple formulas for estimating directivity. Many other simple formulas can be found in the literature [5-91, but they are limited to special cases. Formulas of the form in Equation (7) assume only knowledge of the beamwidth values and, at best, knowledge of the general form of the pattem. If the principal-plane pattems are known, more accurate directivity estimation is possible, as we now explain. One of the most wideiy used analytical models for a radiation pattem is the cosq 6' pattem. When generalized to include unbalanced E- and H-plane beamwidths, it can be used to approximate a variety of antenna pattems. The attraction of this model is that there is a simple analytical formula for the directivity. For an analytical model of this type, the normalized electric field, radiated by an antenna linearly polarized parallel to the x axis, is F(6',q5) = COS' 6' 6'<90". (15) These unidirectional patterns are zero for 6' > 90". The q values in Equations (14) and (15) are determined by matching the pattem at one angle, B', such as the -3 db or -10 db down points, using log[ F(B')] 4= log[cos8'] The pattems of Equations (14) and (15) are especially useful in modeling the element pattem in an array, and for an antenna used to feed a reflector. The directivity associated with the pattem of Equation (1 4) is found using the following simple formula [ 1, Section 7.71: For the rotationally symmetric pattem of Equation (15), this reduces to An additional benefit to having the exact directivity expressions of Equations (17) and (18) is for use in determining the accuracy of approximate formulas. For example, a short dipole is modeled with Equation (14) using qe = 1 and qh = 0, and Equation (17) gives D = 3. This is twice the directivity of a short dipole, as it should be, since Equation (17) assumes that the back half-space of the pattem is zero. 5. Simple gain formulas where The most direct gain expression is G=-E 4z A2 ap P A =-A,, 47c P A = operating wavelength [m] gap = aperture efficiency Ap = physical aperture area [m2] Ae = cap Ap = effective aperture [m*]. This formula applies to all antennas, since it serves as a definition for effective aperture. It is very useful for aperture antennas with physical aperture area A,, which is easily measured. Then, gain is found very easily from Equation (19) if,zap is known. Aperture efficiency includes losses on the antenna structure. For some antennas, such as parabolic reflectors with high-quality wideband feeds,.sap is nearly constant with frequency, and is usually 55% or greater. Note that for high-gain antennas, deviations in the value of cap give only small gain deviations. For example, a 30-dB gain antenna with,zap = 55% that is evaluated by estimating,zap to be 60% will yield only a 0.38 db overestimate of gain. A cautionary note with the use of Equation (19) is that for many antennas, gap is a strong function of frequency. For example, an optimum-gain pyramidal-horn antenna has E, ~ of about 51%. [l, Section Homs are usually designed for optimum operation somewhat below the middle of their operating band, and E, ~ will be much higher than the optimum value at the low end of the band, and much lower at the high end of the band. Thus, use of Equation (19) requires some knowledge of the antenna's performance at the desired operating frequency, in order to establish an accurate value for,zap. We found in the previous section that DB D= HPEo HPH, This is a very popular directivity approximation, since the principal-plane half-power beamwidths are often known. If losses are small, as we assume in most of this paper, then D -k G. The beamwidths can be found from measurement, analytic evaluation, or from simple formulas such as Equation (5). As we showed in the previous section, DB remains relatively constant for a variety of situations. Proper use of Equation (20) then reduces to the choosing the directivity-beamwidth product. These values are given in Table 1, for a variety of situations. Cases 1 and 2 are often cited in the literature without comment. They apply to situations where the IEEE Antennas and Propagation Magazine, Vol. 40, No. 4, August
4 main beam can be approximated as rectangular or conical and there are no sidelobes. In other situations, they give optimistic directivity values. Cases 3 to 6 in Table 1 are for patterns with perfect nulls, and sidelobes that decrease away from the main beam. These cases are most appropriate for theoretical pattems. This is to be contrasted with practical situations, where implementation details cause nulls to be filled in and sidelobes to rise, leading to reduced directivity. Thus, in practice, DB will be lower than that given by theoretical models such as Cases 3 to 6. This leads to the general formula for approximating gain, which is Case 7 in Table 1 : 26,000 G= HPEo HPHo Gain is used here instead of directivity, not because of losses; that is, er is near unity. Instead, gain is used to indicate that the formula is appropriate to real antenha hardware, where gain is the parameter used in performance descriptions. If the complete principal-plane pattems are available and they can be approximated using the pattem models of Equations (14) or (15), and then gain (directivity) is well approximated by Equations (17) or (18). Table 1. Directivity-beamwidth product for use in directivity DB estimation: D = HPEo HPp 5 7 beam; see Equation (1 2) see Equation (1 0) 4 I Cosine-uniform rectangular 1 35,230 aperture, such as open-ended waveguide Uniform circular aperture, 33,709 see Equation (11) 6 1 Circular aperture with a parabolic- I 38,933 on-a -1 2-dB-pedestal distribution General use for practical antennas 26, Example: The gain of a 40-GHz horn antenna [l, p.2991 A popular standard-gain horn antenna, operating from 33 to 50 GHz, has a measured gain of 24.7 db at 40 GHz. This example compares the gain values for two formulas we have presented. Using the 51% aperture efficiency associated with optimum gain, and the aperture dimensions of 5.54 cm and 4.55 cm, in Equation (19) at 40 GHz, gives a gain of 24.6 db. Since the pattem has filled nulls, Equation (21) is the appropriate formula for gain estimation based on the beamwidths. The principal-plane half-power beamwidths of 9 and lo, measured at 40 GHz, in Equation (21) give G = 24.6 db. These two estimated gain values agree with each other exactly, and are in excellent agreement with the measured value. 6. Array antenna directivity estimation The following simple formula for the directivity of an array often appears in the literature: Equations (12) or (13) can be used. For theoretical pattems with perfect nulls and sidelobes that decrease away from the main beam, Cases 3 to 6 of Table 1 are appropriate. If the complete principal-plane patterns are available and can be approximated with the analytical pattem models of Equations (14) or (1 5), the directivity formulas of Equations (1 7) and (1 8) should be used. If losses on the antenna structure are low, gain equals directivity, and can be estimated using the approaches in Table 1. The appropriate one to use is based on what is known about the pattern and the antenna. If aperture efficiency is known, Equation (19) should be used. If only the principal-plane half-power beamwidths are known for a real antenna, Equation (21) is an excellent approximation for general use. Array antennas can be treated by these techniques, if feedline losses are not large. Approximate methods, such as in Equation (22), must be used with caution. where D, Di = directivity of one element in the array = directivity of the array with isotropic elements This formula is for large arrays without grating-lobe effects. But the reader is cautioned that its accuracy varies greatly with array geometry [9]. The estimation formulas presented above can be used for array antennas and are recommended over Equation (22). 7. Summary If only the half-power beamwidths in the principal planes are available, directivity must be estimated using a formula of the form in Equation (7). If no sidelobes are present, approximations such as 8. References 1. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, (second edition), New York, John Wiley, R. C. Hansen (ed.), Microwave Scanning Antennas, Volume 11, New York, Academic Press, 1964, p A. F. Sciambi, Instant Antenna Patterns, Microwaves, 5, June 1966, pp S. A. Schelkunoff, Electromagnetic Waves, New York, Van Nostrand, 1943, Chapter R. J. Stegen, The Gain-Beamwidth Product of An Antenna, IEEE Transactions on Antennas and Propagation, AP-12, July 1964, pp IEEE Antennas and Propagation Magazine, Vol. 40, No. 4, August 1998
5 6. C. T. Tai and C. S. Pereira, An Approximate Formula for Cal- Editor s Comments Continuedfiompage 6 culating the Directivity of an Antenna, IEEE Transactions on Antennas and Propagation, AP-24, March 1976, pp N. A. McDonald, Approximate Relation Between Directivity and Beamwidth for Broadside Collinear Arrays, IEEE Transaclions on Antennas and Propagation, AP-26, March 1978, pp classes on E&M, led to the decision to re-print the whole article in this issue, rather than trying to simply provide a list of corrections. We ve never done that before and1 doubt we ll ever do it again, but it seemed appropriate in this one instance. If you reference or. use the article, please use the version in this issue. The Magazine apologizes for the problem. 8. D. Pozar, Directivity of Omnidirectional Antennas, IEEE Antennas and Propagation Magazine, 35, October 1993, pp B. J. Forman, of scannable planar hays,^, IEEE Transactions on Antennas and Propagation, AP-20, May 1972, pp Introducing Feature Article Author Warren L. Stutzman received his BS in electrical engineering and AB in mathematics degrees from the University of Illinois, in 1964, and received the MS and PhD degrees in electrical engineering from Ohio State University, in 1965 and 1969, respectively. In 1969, Dr. Stutzman joined the electrical engineering faculty of Virginia Polytechnic Institute and State University, where he is currently the Thomas Phillips Professor of Engineering. He is director of the Satellite Communications Group and the Antenna Laboratory at Virginia Tech, which are part of the Center for Wireless Telecommunications. In 1983, he was a Visiting Professor at the Physical Science Laboratory of New Mexico State University. He works in several areas of antennas and propagation. His research activities include antennas for wireless, wideband elements and arrays, phased-array antennas, reflector antennas, personal communication systems, and atmospheric effects on earth-space communication links. He is co-author, with Gary A. Thiele, of the textbook Antenna Theory and Design (John Wiley, 1981, 1998), and is author of the book Polarization in Electromagnetic Systems (Artech House, 1993). He was co-author of two papers that received the Wheeler Prize for the best applications paper to appear in the IEEE Transactions on Antennas and Propagation, in 1992 and He has held the following offices in the IEEE Antennas and Propagation Society: Administrative Committee ( ), Chair of the AP-S Meetings Committee ( ), Vice President (1991), and President (1992). AU There s more. In the last issue, the SABOR software, for rapid Of horn and reflector antennas, was introduced in John Volakis EM Programmer s Notebook. In this issue, the methods used to carry out the analysis are explained, and examples of the analysis of several typical antennas are given. This appears to be a powerful for a small donation, and the description of the methods used is interesting and valuable, as well. In Tom Milligan s Antenna Designer s Notebook, John Mahony presents a method for estimating the directivity of an omnidirectional pattem, centered on the horizon, from the beamwidth. That s a timely contribution, given the feature article by Warren Stutzman, also in this issue. The results are quite simple, and Tom went on to add a nomograph for easy hand calculation. I mentioned that we had reports on several meetings from around the world in this issue. These include the Intemational Conference on Applied Electromagnetics and Communications (Dubrovnik, Croatia); the Fourth Intemational Workshop on Finite Elements in Microwave Engineering (Poitiers, France); another report on the VIIth Intemational Conference on Mathematical Methods in Electromagnetic Theory (MMET*98, Kharkov, Ukraine); and a report on a Russian seminar on an unusual, large antenna measurement system. I am very pleased that we have these, and thank the organizers and attendees who have provided them. This is an excellent way for our readers to find out what is going on in the rest of the world, and to share in meetings that only a few are able to attend. Most of these have proceedings available, if you re interested in the details of any of the papers presented. I ve seen the proceedings for the meetings held in Dubrovnik and Kharkov, and they contain some very interesting and useful papers. If you re organizing a meeting, give a little thought to having one or more reports on the meeting for the Magazine. The photos from the awards banquet at the 1998 AP- S/USNC/URSI Symposium in Atlanta, Georgia, appear in this issue. Sad news. Again, we have lost some of our family. Thomas Taylor, whose name we remember whenever we speak of the Taylor distribution, and George Monser, who made major contributions to defense- and aircraft-related antenna s, are memorialized in this issue. As this issue was going to press, I leamed that Jim Wait, one of the most prolific authors in our field, had just died. We will have remembrances of him in the next issue. Jim was so very active in propagation, antennas, electromagnetics, AP-S, and URSI. Telephone numbers. A member of the URSI community recently asked why the Magazine inoludea tho oountry oode for telephone and fax numbers located outside of North America, but does not include the country code for numbers inside North America. He went on to say that he felt that this special treatment was not consistent with the intemational (the IEEE would prefer the term transnational ) nature of the publication. Continued on page 44 IEEE Antennas and Propagation Magazine, Vol. 40, No. 4, August
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