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This vol- ume was printed and bound by the Columbia University Press. Distribution of the Summary Technical Report of NDRC has been made by the War and Navy Departments. Inquiries concern- ing the availability and distribution of the Summary Technical Report volumes and microfilmed and other reference material should be addressed to the War Department Library, Room 1A-522, The Pentagon, Washington 25, D. C., or to the Office of Naval Research, Navy Department, Attention: Reports and Documents Section, Washington 25, D. C. Copy No. 644 This volume, like the seventy others of the Summary Technical Report of NDRC, has been written, edited, and printed under great pressure. Inevitably there are errors which have slipped past Division readers and proofreaders. There may be errors of fact not known at time of printing. The author has not been able to follow through his writing to the final page proof. Please report errors to: JOINT RESEARCH AND DEVELOPMENT BOARD PROGRAMS DIVISION (STR ERRATA) WASHINGTON 25, D. C. A master errata sheet will be compiled from these reports and sent to recipients of the volume. Your help will make this book more useful to other readers and will be of great value in preparing any revisions. SUMMARY TECHNICAL REPORT OF THE COMMITTEE ON PROPAGATION, NDRC VOLUME 3 THE PROPAGATION OF RADIO WAVES THROUGH THE STANDARD ATMOSPHERE __ DATA LIBRARY 17 ame b> ry if 3 Xa pit, AN WIRE OF OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT VANNEVAR BUSH, DIRECTOR NATIONAL DEFENSE RESEARCH COMMITTEE JAMES B. CONANT, CHAIRMAN COMMITTEE ON PROPAGATION CHAS. R. BURROWS, CHAIRMAN WASHINGTON, D.C., 1946 NATIONAL DEFENSE RESEARCH COMMITTEE James B. Conant, Chairman Richard C. Tolman, Vice Chairman Roger Adams Frank B. Jewett Karl T. Compton Army Representative 1 Navy Representative 2 Commissioner of Patents 3 Irvin Stewart, Executive Secretary 1 Army representatives in order of service: Maj. Gen. G. V. Strong Col. L. A. Denson Maj. Gen. R. C. Moore Col. P. R. Faymonyille Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine Col. E. A. Routheau 2 Navy representatives in order of service: Rear Adm. H. G. Bowen ~—_ Rear Adm. J. A. Furer Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren Commodore H. A. Schade Commissioners of Patents in order of service: Conway P. Coe Casper W. Ooms NOTES ON THE ORGANIZATION OF NDRC The duties of the National Defense Research Committee were (1) to recommend to the Director of OSRD suitable projects and research programs on the instrumentalities of warfare, together with contract facilities for carrying out these projects and programs, and (2) to administer the technical and scientific work of the contracts. More specifi- cally, NDRC functioned by initiating research projects on requests from the Army or the Navy, or on requests from an allied government transmitted through the Liaison Office of OSRD, or on its own considered initiative as a result of the experience of its members. Proposals prepared by the Division, Panel, or Committee for research contracts for performance of the work involved in such projects were first reviewed by NDRC, and if approved, recommended to the Director of OSRD. Upon approval of a proposal by the Director, a contract permitting maximum flexibility of scientific effort was arranged. The business aspects of the contract, including such matters as materials, clearances, vouchers, patents, priorities, legal matters, and administra- tion of patent matters were handled by the Executive Secretary of OSRD. Originally NDRC administered its work through five divisions, each headed by one of the NDRC members. These were: Division A — Armor and Ordnance Division B — Bombs, Fuels, Gases & Chemical Problems Division C — Communication and Transportation Division D — Detection, Controls, and Instruments Division E — Patents and Inventions In a reorganization in the fall of 1942, twenty-three administrative divisions, panels, or committees were cre- ated, each with a chief selected on the basis of his outstand- ing work in the particular field. The NDRC members then became a reviewing and advisory group to the Director of OSRD. The final organization was as follows: Division 1— Ballistic Research Division 2— Effects of Impact and Explosion Division 3— Rocket Ordnance Division 4— Ordnance Accessories Division 5 — New Missiles Division 6— Sub-Surface Warfare Division 7 — Fire Control Division 8 — Explosives Division 9 — Chemistry Division 10 — Absorbents and Aerosols Division 11 — Chemical Engineering Division 12 — Transportation Division 13 — Electrical Communication Division 14 — Radar Division 15 — Radio Coordination Division 16 — Optics and Camouflage Division 17 — Physics Division 18 — War Metallurgy Division 19 — Miscellaneous Applied Mathematics Panel Applied Psychology Panel Committee on Propagation Tropical Deterioration Administrative Committee NDRC FOREWORD gs EVENTS of the years preceding 1940 revealed JAK more and more clearly the seriousness of the world situation, many scientists in this country came to realize the need of organizing scientific research for service in a national emergency. Recom- mendations which they made to the White House were given careful and sympathetic attention, and as a result the National Defense Research Com- mittee [NDRC] was formed by Executive Order of the President in the summer of 1940. The members of NDRC, appointed by the President, were in- structed to supplement the work of the Army and the Navy in the development of the instrumentalities of war. A year later upon the establishment of the Office of Scientific Research and Development [OSRD], NDRC became one of its units. The Summary Technical Report of NDRC is a conscientious effort on the part of NDRC to sum- marize and evaluate its work and to present it in a useful and permanent form. It comprises some seventy volumes broken into groups corresponding to the NDRC Divisions, Panels, and Committees. The Summary Technical Report of each Division, Panel, or Committee is an integral survey of the work of that group. The first volume of each group’s report contains a summary of the report, stating the problems presented and the philosophy of attacking them, and summarizing the results of the research, development, and training activities under- taken. Some volumes may be ‘state of the art” treatises covering subjects to which various research groups have contributed information. Others may contain descriptions of devices developed in the laboratories. A master index of all these divisional, panel, and committee reports which together con- stitute the Summary Technical Report of NDRC is contained in a separate volume, which also includes the index of a microfilm record of pertinent technical laboratory reports and reference material. Some of the NDRC-sponsored researches which had been declassified by the end of 1945 were of sufficient popular interest that it was found desirable to report them in the form of monographs, such as the series on radar by Division 14 and the monograph on sampling inspection by the Applied Mathematics Panel. Since the material treated in them is not duplicated in the Summary Technical Report of NDRC, the monographs are an important part of the story of these aspects of NDRC research. In contrast to the information on radar, which is of widespread interest and much of which is released to the public, the research on subsurface warfare is largely classified and is of general interest to a more restricted group. As a consequence, the report of Division 6 is found almost entirely in its Summary Technical Report, which runs to over twenty vol- umes. The extent of the work of a division cannot therefore be judged solely by the number of volumes devoted to it in the Summary Technical Report of NDRC: account must be taken of the monographs and available reports published elsewhere. Though the Committee on Propagation had a comparatively short existence, being organized rather late in the war program, its accomplishments were definitely effective. That so many individuals and organizations worked together so harmoniously and contributed so willingly to the Committee’s efforts is a tribute to the leadership of the Chairman, Charles R. Burrows. The latest information in this field was gathered from the four corners of the earth, organized, and dispatched to the points where it would aid most in the prosecution of the war. Much credit must be given, not only to the mem- bers of the Committee and its contractors, but also to the many other individuals who gave so gener- ously of their time and effort. This group included a number of our Canadian and British allies. In addi- tion to the assistance given the war effort, a consider- able contribution has been made to the knowledge of short-wave transmission and especially to the inter- relation of this phenomenon with meteorological conditions. Such information will be most valuable in weather forecasting and in furthering the useful- ness of the whole radio field. VaNnnevAR Busu, Director Office of Scientific Research and Development J. B. Conant, Chairman National Defense Research Committee FOREWORD He suCCEsS Of the propagation program was the ee: of the wholehearted cooperation of many individuals in the various organizations concerned, not only in this country but in Mngland, Canada, New Zealand, and Australia. The magnitude of the research work accomplished was possible only because of the willingness of the workers in many organizations to undertake their parts of the overall program. Jn fact, the entire program of the Com- mittee on Propagation was carried out without the necessity of the Committee exercising directive authority over any project. Dr. Hubert Hopkins of the National Physical Laboratory in England and Mr. Donald KE. Kerr of the Radiation Laboratory at the Massachusetts Institute of Technology, who were working on this phase of the war effort when the Propagation Com- mittee was formed, were instrumental in giving 4 good start to its activities. The largest single group working for the Committee was under Mr. Kerr. The existence of a common program for the United Nations in radio-wave propagation resulted from the splendid cooperation given the Propagation Mission to Hngland by Sir Edward Appleton and his Ultra Short Wave Panel. Later, through the cooperation of Canadian engineers and scientists, Dr. W. R. McKinley of the National Research Council of Canada and Dr. Andrew Thomson of the Air Services Meteorological Division, Department of Transport, Toronto, Canada, undertook to carry on 4 part of the program originally assigned to the United States. The program was further rounded out by the willingness of the New Zealand Govern- ment to undertake an experiment for which their situation was particularly favorable. Dr. F. E. 8. Alexander of New Zealand and Dr. Paul A. Anderson of the State College of Washington initiated this work. Needless to say, the labor of the Committee on Propagation could hardly have been effective without the cooperation of the Army and Navy. Maj. Gen. H. M. McClelland personally established Army cooperation, and Lt. Comdr. Ralph A. Krause and Capt. Lloyd Berkner were similarly helpful in organizing Navy liaison and help. Officers and scientific workers of the U.S. Navy tadio and Sound Laboratory at San Diego, Cali- fornia, altered their program on propagation to fit in with the overall program Capt. David R. Hull, Bureau of Ships, understand- ing the importance of the technical problems, paved the way for effective cooperation by this laboratory. Dr. Ralph Bown, Radio and Television Research Director, Bell Telephone Laboratories, integrated the research programs undertaken by Bell Telephone of the Committee. Laboratories for the Committee on Propagation. This joint research program included meteorological measurements on Bell Telephone Laboratories property by meteorologists of the Army Air Forces working with Col. D. N. Yates, Director, and Lt. Col. Harry Wexler of the Weather Wing, Army Air Forces. The accomplishments of the Committee on Propagation are a good example of the effective- ness of cooperation — all parts were essential and none more than the rest. I want to thank Dr. Karl T. Compton, President of Massachusetts Institute of Technology, who was always willing to discuss problems of the Committee and who helped me to solve many of the more difficult ones, and also, Prof. S. S. Attwood, Uni- versity of Michigan, whose continual counsel throughout my term of office was in no small way responsible for the success of our activity. Credit is also due Bell Telephone Laboratories, which made my services available to the Govern- ment and paid my salary from August 1943 to September 1945, and to Cornell University, which has allowed me time off with pay to complete the work of the Committee on Propagation since September 1945. Cuas. R. Chairman, Committee on Propagation 3URROWS fa PREFACE HE MATERIAL presented in this book was pre- pared by the Columbia University Wave Propagation Group at the request of the Committee on Propagation of the National Defense Research Committee. The International Radio Propagation Conference, meeting at Washington in May 1944, recommended that a book be prepared dealing with problems of radio wave propagation in the standard atmosphere at frequencies above 30 megacycles. The importance of these higher frequencies is apparent when it is recalled that most radars operate in this range and that an increasing number of communication circuits are being equipped for operation above this frequency. A certain amount of evidence from operational theaters indicates that lack of familiarity with the underlying theory of propagation and calculations based thereon not infrequently has resulted in ineffective installation and operation of radar and communication sets. This is ascribable, in part at least, to the lack of publications which give a clear picture of the problems of propagation or show how the important factors may be evaluated. A considerable volume of basic material on propagation had appeared in technical journals before World War II, and during it a great quantity of classified material has come from laboratories and operational theaters illustrating new applications of old principles, giving valuable information on propa- gation problems as well as on characteristics of radar and communication sets, antennas, targets, siting problems, ete. But this information has not been coordinated under one cover for practical use by signal personnel in the field. The Columbia University Wave Propagation Group was asked to undertake this task and it is hoped that this book will, in some measure, answer the need. Our effort, then, has been to provide a book de- signed for men with college training in radio, physics, or electrical engineering, which states the basic laws of propagation, that is, shows how the characteristics of the earth and the atmosphere control the propaga- tion of radio waves; gives the fundamental properties of the basic types of antenna systems, particularly their directivity and gain; gives the reflecting properties of targets such as airplanes for use in detection by radar; teaches the reader how to calcu- late field strength or obtain the coverage diagrams given a particular set, power, and site; gives the fundamental information required in the above calculations for application to the radar and com- munication sets used in operational theaters; and provides illustrative material and sample calcula- tions which show how the laws of propagation may advantageously be used in the location and operation of radar systems, communication sets, and counter- measure equipment designed to deceive the enemy and to prevent jamming of equipment by enemy action or by mutual interaction of our own sets. The members of the group chiefly responsible for the preparation of the manuscript were Drs. §. Rosseland, W. M. Elsasser, P. Newman, and Prof. S. Fich. Others who contributed special sec- tions were Messrs. E. R. Wicher, M. Ettenberg, M. Siegel, and Capt. E. J. Emmerling, on special assignment from the Signal Corps. The editor wishes to acknowledge also the courtesy of the Radiation Laboratory Wave Propagation Group, under Mr. Donald E. Kerr, in supplying the universal coverage charts given in Chapter 6, and the steady interest and assistance rendered by Dr. Chas. R. Burrows, Chairman, NDRC Com- mittee on Propagation. STEPHEN S. ATTwoop Editor Oo 7 / Oe OS Se 7 CONTENTS CHAPTER 1 Propagation of Radio Waves: Introduction and Objectives 2 Fundamental Relations . 3 Antennas . 4 Factors Influencing Transmission 5 Calculation of Radio Gain 6 Coverage Diagrams 7 Propagation Aspects of Equipment Operation 8 Diffraction by Terrain 9 Targets 10 Siting Glossary Bibhiography* OSRD Appointees Contract Numbers . Service Project Numbers Index *Refer to Bibliography of C. P. Summary Technical Report, Volume 1. Xi Rees ' : a at . fi + = < 7 } : : : * Chapter 1 PROPAGATION OF RADIO WAVES: INTRODUCTION AND OBJECTIVES 11 FACTORS INFLUENCING PROPAGATION HE PROPAGATION of radio waves through the A Wy atmosphere and around the curve of the earth, at frequencies above 30 me, is influenced by so many factors that it is desirable to give first an overall survey of the problem. This chapter presents the problem of propagation in broad perspective in contrast with many of the later chapters which are devoted to detailed consideration of special phases and methods of calculation. as Assignment The International Radio Wave Propagation Con- ference recommended that a book be prepared deal- ing with problems of radio wave propagation in the standard atmosphere at frequencies above 30 me. The importance of these higher frequencies is appar- ent when it is recalled that most radars operate in this range and that an increasing number of com- munication circuits are being equipped for operation above this frequency. A certain amount of evidence from operational theaters indicates that lack of familiarity with the underlying theory of propagation and calculations based thereon not infrequently has resulted in inef- fective installation and operation of radar and com- munication sets. This is ascribable, in part at least, to the lack of publications which give a clear picture of the problems of propagation or show how the important factors may be evaluated. A considerable volume of basic material on propa- gation had appeared in technical journals before World War II, and during the war a great quantity of classified material came from laboratories and operational theaters illustrating new applications of old principles, giving valuable information on propa- gation problems as well as on characteristics of radar and communication sets, antennas, targets, siting problems, etc. But this information has not been coordinated under one cover for practical use by sig- nal personnel in the field. The Columbia University Wave Propagation Group was asked to undertake this task, and it is hoped that this volume will, in some measure, answer the need. Purpose Our effort then has been to provide a book, de- signed for men with college training in radio, physics, or electrical engineering, which: 1. States the basic laws of propagation, that is, shows how the characteristics of the earth and the atmosphere control the propagation of radio waves; 2. Gives the fundamental properties of the basic types of antenna systems, particularly their direc- tivity and gain; 3. Gives the reflecting properties of targets such as airplanes for use in detection by radar; 4. Teaches the reader how to calculate field strength or obtain the coverage diagrams, given a particular set, power, and site; 5. Gives the fundamental information required in the above calculations for application to the radar and communication sets used in operational theaters; 6. Provides illustrative material and sample cal- culations which show how the laws of propagation may advantageously be used in the location and operation of radar systems, communication sets, and countermeasure equipment designed to deceive the enemy and to prevent jamming of equipment by enemy action or by mutual interaction of our own sets. fe FUNDAMENTAL PROBLEMS AND LIMITATIONS rah Meaning of Propagation By propagation is meant the movement of radio waves through the atmosphere, and the transfer, by a wave mechanism, of radiant energy from a transmit- ting antenna to a receiving antenna. The problem of propagation requires an understanding of the _manner in which the wave energy is emitted and received as well as of the manner in which it flows through the atmosphere. The radio engineer must 1 2 PROPAGATION OF RADIO WAVES understand this general mechanism, be able to eval- uate the factors which play contributory roles, and, for a given amount of power emitted from a given transmitter, be able to compute the strength of the radiation field at any point in space or to locate all the points in space where a given field strength occurs. The problem divides naturally into two parts, (1) the one-way transmission or communication problem, and (2) the two-way transmission or radar problem. In the former the prime requisite is to calculate the amount by which the wave and its field strength are attenuated in passing from the transmitter to a receiver and yet permit a field at the receiver suffi- i Atmospheric Layers The atmosphere from one point of view is com- posed of two layers, the troposphere and the strato- sphere. The former is a layer adjacent to the earth which extends upward approximately 10 km, in which the temperature decreases about 6.5 C per kilometer with increasing altitude to a value, at the upper boundary, of about — 50 C. Above this is the stratosphere in which the temperature remains approximately constant at — 50 C. The ionosphere, as its name implies, is a layer (or series of layers) composed of ions and free electrons lying at an elevation of approximately 100 km. See Ficure 1. Transmission along and reflection from the ionosphere occurs primarily with frequencies below 30 mc. At higher frequencies useful transmission is primarily concerned with the nearly horizontal rays in the troposphere; higher angle radiation passes through the ionosphere and is lost. cient at least to produce the minimum detectable signal. In the latter problem the attenuation must be calculable for the two-way journey from trans- mitter to the target and back to the receiver, which frequently uses the same antenna as the transmitter. In this type of problem, due account must also be taken of the reflecting and reradiating properties of the target. Knowledge of these factors is indispensable for the design, installation, and successful operation of both communication and radar systems. Figure 1. These layers play an important role in the transmission of waves at frequencies below 30 me and are responsible for transmission over very long distances. At the higher frequencies, which are the concern of this volume, the portion of the waves which penetrate the ionosphere is not useful for transmission. From this it follows that propagation at the higher frequencies (above 30 me), to be useful, must occur entirely in the troposphere. This volume therefore is concerned only with tropospheric propagation. FUNDAMENTAL PROBLEMS AND LIMITATIONS 3 1.2.3 Standard Atmosphere Propagation of radio waves in the troposphere is materially influenced by the distributions of tem- perature, pressure, and water vapor. The condition most nearly approximated in the Temperate Zone has been accepted as the so-called standard atmosphere, and propagation under this condition has been stud- ied and calculations made thereon. In the standard atmosphere specified by the National Advisory Committee on Aeronautics [NACA] the temperature is assumed to decrease with altitude at the rate of 6.5 C per kilometer, starting from 15 C at sea level, with a sea level dry air pressure of 1013.2 millibars, which is equivalent to 760 mm Hg pressure (see Table 1). TABLE 1. the total pressure and moisture vapor pressure, re- spectively, in millibars, at height h above sea level. In the moist standard atmosphere, n decreases lin- early with height A at the approximate rate of 0.039 X 10-° units per meter. There are several reasons why this book concerns propagation in the moist standard atmosphere. 1. The atmosphere in certain places (particularly the temperate zones) and over considerable periods of time is substantially standard in character. 2. Calculations based on the standard atmosphere serve as a standard against which propagation in nonstandard atmospheres may be compared. 3. A great deal of propagation information now available in the field is based on propagation cal- culated for standard conditions. Properties of the atmosphere. NACA standard dry atmosphere Moist standard atmosphere h p—e e Saturated Altitude t Dry air Index of Water vapor water vapor Per cent Index of Tem pressure refraction pressure pressure relative refraction Feet Meters Cc millibars (n — 1)108 millibars millibars humidity (n — 1)108 0 0 15.0 1,013.2 278 10 17.1 58.5 318 500 152 14.0 995 274 9.5 16 59.4 312.4 1,000 305 13.0 977.1 270 9 15 60.0 309 1,500 457 12.0 960 266 8.5 14 60.7 304 2,000 610 11.0 942.1 262 8 13.1 61 295.6 3,000 915 9.1 908.1 254 7 11.6 60.3 284 4,000 1,220 Coll 875.1 247 6 10.1 59.4 273 5,000 1,525 5.1 843 240 5 8.8 57 262 1.2.4 To simulate the actual atmosphere of the temper- ate zones it is necessary further to specify a water vapor pressure. The value chosen is 10 millibars at sea level, decreasing with altitude at the rate of 1 millibar per 1,000 ft up to 10,000 ft. With this addition the conditions for a moist standard atmos- phere are specified in Table 1. This value of water vapor pressure yields a value of relative humidity approximating 60 per cent. Listed also in Table 1 is the index of refraction n. The gradient of this quantity, dn/dh, controls the curvature of the rays for a wave moving in the ap- proximately horizontal direction; n is given by the formula 79 = Wilts | Spe (n — 1) EG BaP 4800e : , () where 7’ is the absolute temperature, p and e are Propagation in the Moist Standard Atmosphere The radiation energy emitted by a transmitter is a wave spreading out in three dimensions, which may be represented by a series of concentric spherical wave fronts or by a system of lines called rays. The velocity at any point on the wave front is given by Ch 3 e108 v= -—— n n meters per second. (2) Since n decreases with height, the upper portions of the wave front move with higher velocities than the lower portions, and the wave paths as represented by the rays are curved slightly downward, as shown in Figure 2. The radius of curvature of the rays p is given by 4+ PROPAGATION OF RADIO WAVES Es > Ce + 0.039 * 10-6 units per meter (3) p dh and p, therefore, is equal to 25.5 X 10° meters, which is approximately four times the radius of the earth (p = 4a). As a result the distance to the radio horizon is some 15 per cent greater than the geometrical line- of-sight distance from the transmitter to the horizon. This curvature of the rays by the atmosphere is called refraction. 0=6,37 x10°m Fiagure 2. Curvature of rays in the standard atmos- phere. For the purpose of calculating wave propagation, only relative curvature of the earth and the rays is of interest. We can be compensated for the effects of refraction by replacing the actual earth with a radius a by an equivalent earth with a radius ka and replacing the actual atmosphere (in which the index of refraction n decreases with height) by a homogeneous atmosphere (constant m) in which the rays are straight lines. Since 1/a is the curvature of the earth and 1/p the curvature of the rays, we may set their difference equal to 1/ka, the curvature of the equivalent earth. Thus and (4) h l 1 1—(a/p) l+a% Since p = 4a, k = 4/3, and ka, the radius of the equivalent earth, equals 8.49 X 10° meters. See Figures 4 and 5 in Chapter 4. 125 Propagation in Nonstandard Atmospheres Though this subject is beyond the scope of this volume it is desirable to present a brief discussion of the salient features. Not infrequently the lower atmosphere is stratified in horizontal layers in which the variations with height of the temperature and moisture content are nonstandard. Of particular interest is a sharp rise in temperature with increasing height (tempera- ture inversion), or a sharp decrease in water vapor content, or a combination of the two. If these varia- tions from the standard distribution are sufficiently great, horizontal radio ducts may be formed in the atmosphere. In this event an appreciable fraction of the wave energy (only that fraction moving in the nearly horizontal direction) may be constrained to propagate along the duct to distances far beyond the horizon and the field strength may be large compared with that obtainable under standard conditions. This phenomenon produces a marked bending of the wave paths or rays and is known as super-refraction. To take fullest advantage of this phenomenon the antennas should be located in the duct. Ducts are of various types: 1. Overland. These are surface ducts formed at night by the radiation cooling of the earth. 2. Oversea. In the trade-wind belt there appears to be a continuous duct of the order of 50 ft thick starting at sea level. 3. Land to sea. Warm dry air flowing from land out over cooler water often yields surface ducts 100 or more feet thick. 4. Elevated. Caused by subsidence of large air masses, these ducts may be found at elevations of perhaps 1,000 to 5,000 ft and may vary in thickness from a few feet to 1,000 ft. They are common in Southern California and certain areas in the Pacific. Depending upon the strength and the thickness of the duct, there is a limiting frequency below which the duct cannot trap the wave energy. Though trapping does at times occur at 200 me, it is more likely to occur at the higher frequencies such as 3,000 me. Ability to calculate performance under standard conditions is necessary if performance under non- standard conditions is to be evaluated. 1.2.6 Radio Gain The basic problem to be solved is that of com- puting the radio gain of a transmitting-receiving system. The radio gain of a transmitting-receiving system FUNDAMENTAL PROBLEMS AND LIMITATIONS 5) is defined as the ratio of received power Ps, delivered to a load matched to the receiving antenna, to power P;, supplied to the transmitting antenna, with both antennas adjusted for maximum power transfer. Thus 2 : Pp» e Radio gain = ~, (5) which is equal, in the decibel scale, to P» : Ret : — = radio gain in decibels. (6) uk 10 logio The attenuation is the reciprocal of the gain. Since P2/P, <1, the gain in decibels is necessarily a nega- tive quantity. The attenuation in decibels is a positive quantity equal in magnitude to the gain in decibels. The radio gain can be taken as the product. of physically significant factors. Among these are the gains G, and G_ of the transmitting and receiving antennas respectively; and A? which accounts for all other influences modifying the transmission of power. A is called the gain factor. IPR, Radio gain = — = G(2,A°. 1 (7a) The radar problem involves double transmission over the path as well as the reradiating properties of the target, 1670/9). EGC (a) P “ON where o is the radar cross section of the target and \ is the wavelength. Radar gain = ce 2s Su = a GO Radiation ansmitter f Circuit ae adjusted for maximum power transfer. Ao = 3d/8ad where d is the distance between doublets. A, is the path gain factor which includes all additional influ- ences modifying the transmission of power. These factors may also be related to the field strength, #, at any point in space by = E)VG,A, (9) and eee oe. (10) Ay ENVG Here Ep is the free-space field at a point in space set up by a doublet transmitter and EoVG) is the free-space field of a transmitter with antenna gain (7. The primary function of this book is to show how the factors A and A, may be calculated, taking into account all contributory influences which modify their magnitudes. 127 Radio Gain of Doublet Antennas in Free Space This is the fundamental and simplest case of transmission of radiant energy, against which other transmitting combinations may be compared. Two doublet antennas (for which the gains, by definition, are unity) are set up in free space in a manner which insures the maximum transfer of power to the re- ceiver circuit, i.e., the doublets are parallel to each other, have a common equatorial plane, and the Reapadioticn Eo wes SS To Receiver Circuit A ee Figure 38. Doublet antennas in free space. The gain factor, A, may also be split into two factors, so that A = AoAy. (8) Here Ao is the free-space gain factor for doublet antennas (see Sections 1.2.7, 2.1.3, 2.2.2, and 5.1.2) receiver circuit impedance is matched to that of the receiving antenna (see Figure 3). Then for free space, Free-space gain = ae Ay = (Sy), (11) 1 Sad (=>) PROPAGATION OF RADIO WAVES and the free-space field strength at distance d from the transmitter is _ 3V5 VP, d . P, is the power radiated by the doublet transmitting antenna (see Section 2.1.1). Ey (12) 1.3 SURVEY OF PROPAGATION eset Outline We will first consider those factors which are instrumental in modifying the transmission or the attenuation that arises from the presence of the earth, then give typical curves of both vertical and horizontal variation of field strength, and lastly, consider the problem of coverage. 132 Factors Modifying Transmission The important factors which affect the distribu- tion of field strength are the following: 1. Antenna characteristics. For many applications the most important feature is the gain which is a measure of the directivity of the antenna. From a pattern gives the relative amount of power per unit area radiated in that direction. CO. , ee Doublet Ant i i antennes enna With High Directivity Reflector Figure 4, Antenna radiation patterns. 2. Polarization. The wave is said to be polarized horizontally or vertically according to whether the electric vector FH is parallel to the earth’s surface or is in a plane perpendicular thereto. A horizontal electric doublet (axis parallel to the earth’s surface) radiates horizontally polarized waves, whereas a vertical doublet radiates vertically polarized waves. Too many factors are involved to make it possible to state in general which type of polarization should be used in a particular case. 3. Refraction. As explained in Section 1.2.4, refraction in the standard atmosphere can be taken into account by using an equivalent earth with a radius equal to */; that of the actual earth and a homogeneous atmosphere in which the rays traverse straight line paths. s Recelver Transmitter T R Tq Direct Ray Interference Region hy ¥ Reflection Radio line bial Point Horizon iy. Ficure 5. Geometrical relations for rays. value of 1.09 for the half-wave dipole, the gain may increase to several thousand for the highly directive parabolic antennas used in the microwave range. Antennas with high gains which concentrate the radiated energy into beams of small angles require less power to produce a detectable signal. This is particularly important in radar where the attenua- tion of the two-way path is pronounced. Qualitative radiation patterns for the doublet antenna and an antenna with high directivity are illustrated in Figure 4. The radial distance to the 4. Reflection. Well above the line of sight (see Figure 5) the field at the receiver R is the vector sum of the fields radiated along the paths of the direct and reflected rays. The contribution from the reflected ray path depends primarily on the manner in which the earth (or sea) acts as a reflecting body. Upon reflection, the angle of incidence (90° — y) is equal to the angle of reflection, irrespective of the polarization of the wave, but the strength of the field in the reflected ray relative to that in the inci- dent ray depends upon (a) the grazing angle y, SURVEY OF PROPAGATION i | (b) the type of polarization, (¢) the reflecting proper- ties of the earth or sea, and (d) the divergence factor. The incident beam or bundle of rays, in general, is partially absorbed by the earth, while the reflected portion is reduced in strength and suffers a phase shift relative to the incident beam. (In the case of sea water with horizontal polarization, the earth acts substantially as a perfect reflector, for which the reflection is 100 per cent complete and the phase shift is 180 degrees for all grazing angles. This is true for vertical polarization only at zero grazing angle.) The divergence factor is introduced to account for the fact that an incident bundle of rays striking a spherical surface diverges upon reflection and pro- duces a further decrease in strength of the reflected beam. Reflection from hills, trees, and other obstacles must frequently be taken into account, particularly in the siting of very high frequency [VHF] com- munication sets. 5. Diffraction. The mechanism by which radio waves curve around edges and penetrate into the shadow region behind an opaque obstacle is called diffraction. The explanation usually given is based on Huyghens’ principle. This, in effect, states that every elementary area on a wavefront (see P in Figure 6) is a center which radiates in all directions Wave ea | Tronsmitter \ Figure 6. Diffraction around an obstacle. on the forward side of the wavefront; the intensity of radiation is a maximum in the direction per- pendicular to the wavefront and depends on angle 6 according to the function (1 + cos @). The field at any point, either inside or outside the shadow zone, is obtained by summing the contributions from all the elementary areas comprising the wavefront. As a result of these calculations, the field along the line AA’ in Figure 6 varies approximately as indi- cated in the curve. Unity represents the field value if the obstacle were removed. It is seen that the field strength rises from a minimum at point A to 0.5 at the edge of the shadow zone and thereafter oscil- lates about unity. The field outside the shadow zone, therefore, at certain points is stronger and at other points is weaker than it would be if there were no obstacle. The curve, of course, varies with the position of the line AA’, the size and shape of the obstacle, the wavelength of the radiation, and the type of polarization. The diffraction of radiant energy into the shadow zone increases with increas- ing wavelength. Of prime importance for propagation of radio waves is the diffraction of these waves into the diffraction region below the line of sight (see Figure 5). But it should be noted that the influence of diffraction is not confined to this region but extends well above the line of sight. [In general, the influ- ence extends upward far enough to affect the shape of the lower part of the first lobe in a coverage diagram (see Figures 25 and 26 of Chapter 5). In this region the diffraction contribution must be added to the contributions of the direct and reflected rays to give the correct value of the field strength at R in Figure 5.] Of importance in communication problems is diffraction of waves around obstacles such as hills, trees, houses, etc. This is illustrated in Figure 6. Again diffraction is important in problems involving propagation above two different earth conditions. An especially important case is that of a radar set well inland and searching far out over the sea. Here the shore line is treated as a diffracting edge for the radiation from the image antenna. 6. Absorption and scattering. No account is taken in this book of the absorption and scattering of radio waves by the various constituents of the atmos- phere. Oxygen, water vapor, water droplets, and rain all contribute to absorption. Their influence, however, is important only in the microwave range and in general tends to increase with frequency. UES General Nature of the Radiation Field In Section 1.3.2, reference has been made to the role of reflection by the earth. The resultant of the direct and indirect rays at points in the region above the line of sight gives rise to the lobes of an inter- co PROPAGATION OF RADIO WAVES ference pattern (see Figures 9 to 12). The maximum number of lobes is the largest integral number of times that the quarter wavelength is contained in the transmitter height. In the case of horizontal polarization over a smooth surface, e.g., a calm sea, the reflected and direct rays are comparable in strength, so that at certain points (on lines for which the points corre- spond to a path difference of a half wavelength) where the reinforcement is a maximum, the field may be as much as twice the free-space field. More exactly, the free-space field is multiplied at points of maxima by (1 + FD), where D is the value of the divergence factor for the point and F gives the rela- tive strength of the reflected and direct rays attribut- able to the antenna beam pattern. At points of minima (the nulls) the field is (1 — FD) times the free-space field. In general the magnitude of the reflected wave is reduced both by the increased divergence resulting from reflection from the convex surface of the earth but also because the electrical properties of the earth are such that only part of the incident energy is reflected. The magnitude of the reflection coeffi- cient is then pD instead of the D used in the preced- ing paragraph, where p is the magnitude of the reflection coefficient for plane waves impinging on a plane surface. The field strength, then, lies be- tween (1+ pFD) and (1 — pFD). As a result of the smaller value of pFD for vertical polarization the maxima of the interference pattern are reduced and the nulls strengthened. At low heights (see Figure 3 of Chapter 5) the effect of diffraction is important, so that when refer- ence is made to the optical interference region, it should be understood that the portion of the optical region near the earth is not included. [It must be considered instead as part of the diffraction region. The diffraction region, accordingly, designates a layer in the optical region as well as the region below the line of sight (see Figure 3 in Chapter 5). Below the line of sight the field falls off exponentially. Within the diffraction region, fields are strengthened by raising the receiver or transmitter antennas. 134 Typical Radio Gain Curves Three types of graphical representation of radio gain in a vertical plane through the transmitter antenna are possible, namely, (1) at a specified distance, radio gain against height; (2) at a specified height, radio gain against distance; and (3) a set of contour lines representing constant radio gain. 10,000 FREQUENCIES --- 3000MC —100,200,500MC HORIZONTAL POLARIZATION TRANSMITTER HEIGHT 9 DISTANCE 8OKM SEA WATER 1000 100 RECEIVER HEIGHT he IN METERS -200 = -180 -I60 -140 -120 -100 -80 20 LOG A IN DB Fiaure 7. Radio gain vs receiver height for horizontal polarization. In Figure 7, curves of type (1) are exhibited for various frequencies. The transmission is over sea water with horizontally polarized waves. It may be observed that the higher the frequency the lower the first maximum and the narrower the lobe. Figure 8 gives similar information for vertically polarized 10,000 FREQUENCIES | ---- 3000MC | — 100,200,500MC 500 VERTICAL POLARIZATION TRANSMITTER HEIGHT 9M DISTANCE 80 KM 5 SEA WATER on fe) fe) fo) LINE OF SIGHT 100 RECEIVER HEIGHT— h, IN METERS 10 -240 -200 -180 -160 -I40 20 LOG A IN OB -120 -100 -80 Ficure 8. Radio gain vs receiver height for vertical polarization. waves. Note that the minima are not so deep with vertical polarization. Curves of type (2) exhibit similar characteristics (see Figure 6 of Chapter 5). In Figures 9 to 12", vertical coverage diagrams of a Figures 7 to 12 have been adapted from Radiation Lab- oratory Report C-6. SURVEY OF PROPAGATION 9 type (3) are given. These illustrate the effects of frequency, polarization, and transmitter height. A comparison of Figures 9 and 10 shows the effect of frequency. As the frequency increases, the lobes become more numerous, narrower, and_ lower. Another effect is exhibited along the surface. For the higher frequency the corresponding decibel lines come in closer to the transmitter. This illustrates the fact that for the higher frequency the shadow effect is more pronounced along the surface of the earth. A comparison of Figures 10 and 11 shows that for horizontal polarization the nulls are deeper but the lobes extend out farther. Along the line of sight, vertical polarization gives the higher field strength, while well within the diffraction region the field Kilometers pb oO nm oO ; so Kilometers 100 Frequency l!00Mc Horizontal Polarization Antenna Height 9.14 meters k=4/3 Figure 9. Kilorneters strength is about the same. The last observation holds for all frequencies greater than 300 me, the greater the frequency the less difference in the diffraction region between the two polarizations. A comparison of Figures 9 and 12 shows the effect of the height of the transmitting antenna. As the antenna height is increased, the lobes are narrower and depressed toward the horizon. The range is improved. However, there are broad nulls for the higher antenna in which detection will fail. Below the horizon, the corresponding decibel contours are pushed to the right so that point-to-point com- munication is improved. It should be observed that the effect of height upon the lobe structure is similar to that of frequency. Contours of constant radio gain factor for horizontal polarization on 100 me over sea water. Kilometers Frequency 3000Mc Horizontal Polarization Antenna Height 9Qmeters K= 4/3 “305 Figure 10. Contours of constant radio gain factor for horizontal polarization on 3000 me over sea water. 10 PROPAGATION OF RADIO WAVES Which contour actually represents the limit of detection for a given radar depends on the power output of the transmitter, the minimum power detectable by the receiver, the antenna gains, and the radar cross section of the target. For com- munication sets, the same quantities, except the target cross section, apply. 14° QRGANIZATION OF THIS VOLUME tad Arrangement of Material This volume is composed of two classes of material. One class, comprising Chapters 2, 5, and 6, is de- voted primarily to the major problem of calculating Kilometers Kilometers Frequency 3000 Mc Vertical Polarization Antenna Height 9 meters K=4/3 Ficure 11. Kilometers Kilometers Frequency !OOMc Horizontal Polarization Antenna Height 152.4meters K=4/4 Figure 12. the field strength, while the other class discusses collateral problems of importance if these calcula- tions are to be utilized for obtaining the most effective use of radar and communication sets. Chapter 2 is devoted to a presentation of basic relationships such as the definition of radio gain, the transfer of power between doublet antennas in free space, antenna gain, receiver sensitivity and noise, and the definitions of radar gain and cross section. The problem of computing the field strength or radio gain at any point in the atmosphere is given at length in Chapter 5, and Chapter 6 extends this material to the calculation of coverage diagrams. In Chapter 7 these calculations are related to the Contours of constant radio gain factor for vertical polarization on 3000 me over sea water. Contours of constant radio gain factor for horizontal polarization on 100 me over sea water. UNITS AND FREQUENCY RANGES 11 performance characteristics of particular radar and communication sets. In Chapter 3 will be found a discussion of the radiating properties of a wide variety of antennas. Particular attention is devoted to a consideration of the shape of the radiation patterns, methods for improving directivity of antennas, and computation of the gains. A general discussion of the factors which modify the manner in which radio waves are transmitted through the atmosphere is given in Chapter 4. Here also is given the reflecting properties of sea water and various types of soil. An important practical problem is that of diffrac- tion of waves around obstacles such as hills and trees. A simplified treatment of this problem is given in Chapter 8. Chapter 9 is devoted to a presentation of the reflecting problems of targets and their bearing on the operation of radar sets. Successful operation of these sets is dependent, in no small measure, on the proper choice of siting. Factors bearing on the siting problem are evaluated in Chapter 10. 15° UNITS AND FREQUENCY RANGES “abst Units In this book the units used are those of the mks rationalized system, in which distances are ex- pressed in meters, masses in kilograms, and time in seconds; and the formulas have been rationalized so that the factor 47 appears in equations involving point sources, 27 in equations involving line sources, and is generally absent from equations for uniform or unidirectional fields. The Coulomb formula, for illustration, for point sources in classical electrostatic units, pot (13) er- with fin dynes, q; and qin statcoulombs, 7 in centi- meters and e referred to unity in free space, is transformed to 192 = ee (14) Here force f is given in newtons (1 newton = 10° dynes), q: and q2 in coulombs, r in meters, ¢ is the dielectric constant relative to that of free space ¢. Similarly, the Coulomb formula for magnetic poles, a (15) with m; and m in unit poles in the electromagnetic system and yu equal to the permeability, transforms to myM2 , mame Tene (16) where mm, and m: are now given in webers, pu, is the permeability relative to that of free space po. In the mks rationalized system, the free-space values of ¢ and wo must carry the burden of the change of units and the inclusion of 47, and thus take on the values Sis54renl Op Be 10° farads per meter, (17) 367 po = 4r- 10" & 1.257 - 10° henries per meter. (18) With these values, c, the velocity of light in free space, is equal to 1 ll €0 c = —— = 2.998- 10° 3 - 108 meters per second ay (19) and the impedance of free space is P= 376.7 ohms. (20) i) This system of units has been chosen because it is unified, free from numerical factors required in equations using arbitrary choices of units, and has been adopted by the International Electrotechnical Commission. Since the various Armed Services use differing sets of units for their operational instructions, it would have been impossible to choose any one that would have been satisfactory to all; hence the choice of using the only system which is generally recognized and scientifically sound. 152 Symbols for Frequency Ranges The following symbolism has been adopted for various ranges of frequency. Taste 2. Symbols for frequency ranges. Frequency | Frequency Wavelength Symbol name me meters LF | Low 0.03-0.3 10,000-1,000 MF | Medium 0.3-3 1,000—100 HF | High 3-30 100-10 short waves VHF | Very high 30-300 10-1 _\ ultra-short UHF | Ultra-high | 300-3,000 1-0.1 { waves SHE | Super-high >3,000 <.1 microwaves Chapter 2 FUNDAMENTAL RELATIONS eed! THE ELECTRIC DOUBLET IN FREE SPACE 21-1 Radiation of an Electric Doublet T IS CONVENIENT to present the basic relation- if ships of radiation and reception by antennas in their simplest form, that of the radiation and recep- tion of electric doublets in free space. The resulting formulas will later be generalized to include other types of antennas and their positions relative to the earth. An electric doublet is a rectilinear antenna, which is symmetrical about the point or points of connec- tion thereto and is so short that its directive proper- ties are independent of its length. The field of such an antenna does not depend on the distribution of current along the wire, because the wire is so short x Fiaure 1. Polar coordinate system. that there is no phase difference between waves reaching a point in space from different portions of the wire. In symbols, 1 << \ where 1 is the length of the antenna and ) is the wavelength of the radia- tion. To facilitate the analysis of the field of the doublet antenna, the spherical polar coordinate system shown in Figure 1 is introduced. The upper half of the doublet is shown in the figure. The distance from the center of the antenna to a point in space is here denoted by r. Elsewhere in this volume this quantity is written d. 12 s Let dl be an infinitesimal portion of 1, the length of the doublet, and let the current in this portion be the real part of [e’°""”), Let dE,, dE», dE and dH,, dH,, dH, be the components of the electric and magnetic field strengths at any point P(r, 0,@) due to the current in dl. A straightforward solution of the fundamental equations of electromagnetic theory gives the following values for these components, valid at distances large compared with the length of the doublet: dE, = 601 E eee ja cos 6 e1?" A) (ct—r) 7) oar? volts per meter, L ae ye | dl sin 6 e2(2" ‘A)(et—7) 4rrs dE, = war 2 =e Ar 2a? volts per meter, (1) dE, = 0, dH, = 0, dH, = 0, dH, = aR Be =| disin 6 ef 2m/Met-) ; Oya) Mr 2a amperes per meter, where c = velocity of ight = 3 X 10° meters per second j=vV-1, and all distances are measured in meters. Electric field strengths are in volts per meter and magnetic field strengths are in amperes per meter. Unless otherwise explicitly stated, the mks rationalized units are used throughout this volume. Equation (1) can be simplified at once. Since the time variation of the field is assumed sinusoidal, e 7) may be omitted. The term ¢??""”) gives the phase, and it too can be omitted when only the amplitude is required. From here on, unless other- wise stated, it is understood that root-mean-square (rms) values will be used for dH,, dE4, dHy, and I. The field in the neighborhood of the doublet is called the induction field and is given by the terms in equation (1) which include the highest powers of r in the denominators. This field is important when mutual effects between closely spaced antennas, or antennas and reflectors or directors, are involved. The radiation field, of greater interest for most of the purposes of this volume and the only important field at large distances (r>>), is given by the THE ELECTRIC DOUBLET IN FREE SPACE 13 terms in equation (1) containing 7r!. Thus the radiation field for the element dl of the doublet may be written: dE, = (edenleteunt itoa os volts per meter, \r (2) dH Idlsin@ — dE, ¢ np 1207 amperes per meter. The other components are relatively negligible except near the antenna or near ground for low antennas. The electric field dH, is perpendicular to the radius vector r and lies in the r,z plane, and the magnetic field dH, is perpendicular to r and to dE». It will be noted that H/H = 1200 & 376.7 ohms. This is the impedance of free space in the mks rationalized system of units, the ohms of the electrical engineer. Equation (2) describes the radiation field of a differential element of the doublet. To get the radiation field of the whole doublet, these equations must be integrated over the length 1. This gives 1/2 arsine [ Tdl E = 1/2 : Mr H, = E,/1207 amperes per meter. volts per meter, (3) Equation (3) may be written in exactly the form of equation (2) by introducing the effective length, L, of an antenna, which is defined as the length that a straight wire carrying current constant over its length would have if it produced the same field as the antenna in question. Calling the current meas- ured at the input point J;, ib Tdl ib aes: meters, (4) i and hence 6071,L sin 6 Ey = ar aaa volts per meter, (5) Ei = Be LA 7 t ~ ’ 1207 Umberes per meter, so that equations (5) are the same as equations (2) with J;L replacing i Idl. or a short dipole or doublet the current varies linearly from J; at the midpoint to zero at each end so that from equation (4) L = 1/2 for a dovblet. The power per unit area, W (that is, the power flowing through a unit area normal to the direction of propagation), is represented by Poynting’s vector and is given by the product E,H, times the sine of the angle between H, and H,. This angle is 90 degrees. Consequently, W = EH watts per square meter, Kk? We — =—_— Ui 1207 E = V1207W To find P, the power output of the doublet, W is integrated over a large sphere concentric with the source. Using equations (5), watts per square meter, (6) volts per meter. | atl watts 45 es and (7) ad NT where d is written in place of r. The subscripts @ and @ have been dropped at this point because the FE and H referred to in equations (7) are the fields in the equatorial plane, where sin 6 = 1. As the antenna is part of a circuit, it is often con- venient to think of the radiated power as being dissipated in a fictitious resistance called the radia- tion resistance, defined by Ike = it where P is the radiated power and J; the rms input current. For the doublet, R, = 807° (2) ON where L is the effective length given by equation (4). ohms, (8) ohms, (9) 212 Reception by an Electric Doublet When an electromagnetic wave falls upon an antenna, a current is induced in the antenna and power is abstracted from the wave. If the antenna is connected to a load, the power abstracted is dissipated in two ways: (1) by absorption in the load (reception), and (2) by reradiation from the antenna (scattering). In this classification, the power dissipated by the antenna itself (due to its ohmic resistance) is ignored because this loss is likely to be negligible compared with the power dissipated through reradiation. Hereafter, power absorbed by the load will be called received power and power reradiated by the antenna 14 FUNDAMENTAL RELATIONS will be called scattered power. The sum of these is equal to the power abstracted from the wave. The calculation of the received and scattered power may be carried out by means of the equivalent circuit of Figure 2. In this figure, Z, is the impedance VOLTAGE GENERATED IN ANTENNA Ficure 2. Equivalent circuit of antenna and load. of the doublet and Z, is the impedance of the load, that is, the impedance connected across the terminals of the antenna when it is acting as a receiver. V is the voltage generated in the antenna. The load is supposed to be tuned, which means that the reactance part of Z, is set equal and opposite to the reactance part of Z,, so that Z, + Z, = Ra + R) that is, the total impedance is simply the sum of the resistance parts of the impedances of the antenna and the load. Hence V «Rat Ri gives the current. But P, = R)J? is the power ab- sorbed by the load and hence is equal to _ VR, (Ra + R,)?’ where P, is called the received power. In the same way (10) (11) r _ Wola (lite ae h,)? is the power scattered by the doublet. It is easy to show that the maximum power is delivered to the load if Rg = R,. In this, the matched load case, (12) Ss mlee® ke 4R, 4k, Now the resistance of the doublet, neglecting its low ohmic resistance, is only the radiation resistance [equation (9)] and the potential or voltage across the terminals is equal to Hol, where Ep is the field strength of the incident plane wave and L is the P,=P,= (13) effective length of the doublet. quantities into equation (13), Ee 3” 1207 ; Sr” In these equations it has been assumed that the line of the doublet has been oriented parallel to the electric vector of the incident wave in order to obtain maximum power absorption. The factor ,?/1207 will be recognized from equation (6) as the power per unit area of the inci- dent wave. The formula thus says that all the power crossing an area 3\?/87 is received, and that all the power crossing an equal area is scattered. The area 3/87 is therefore called the absorption cross section or scattering cross section of the matched doublet. Since the antenna has been placed parallel to the polarization of the incident wave, this is the maximum absorption cross section. Moreover, this formula holds only when the doublet has been matched to its load, and consequently 3\?/8z is the maximum absorption cross section. It will be noted, however, that 3?/8z7 is not the maximum scattering cross section. This maximum is achieved by shorting out the load, that is, setting R, = 0. In this case, P, = 0 Inserting these P, =P; = (14) and (15) ins Hence the scattering cross section of the shorted (dummy) doublet is four times the scattering cross section of the matched load doublet. It should be noted in passing that the cross sections introduced here should not be confused with the radar cross section which is discussed in Section 2.4. 21-3 Transmission between Doublets in Free Space Assume that two doublets, one to function as a transmitter and the other as a receiver, a distance d >> apart, are adjusted for maximum power transfer. This means that the axes of the doublets are parallel and lie in their common equatorial plane and that each is matched to its connected circuit. Then the power radiated by the transmitting doublet, from equation (7), is equal to E,d? 45 J = (16) watts, OsI- POWER TRANSMISSION. RECIPROCITY 15 d (KILOMETERS) —_— Ww n\) = ono p w i) Fa} wn Ss w N ro} & o fe} o Oo ro) ° 8 8 ee) ° ° ro) , r s 5 & o 4 3 a a be Pe) i) = 3 3 3 5 3 co} oO fo} fe) to} fo} See GAIN IN DECIBELS 20 LOG ‘Sadbs 20 LOG Ay 9 oO 2 5 8 oe S © Oe iS S "4 Rn © 2 Ww w bh oW = tw w S&S w \(METERS) Ficure 3. Free-space gain for doublets P2/P; = and from equation (14) the power delivered to the load circuit of the receiving doublet is given by Lie watts. (17) 1207 8r Hence the ratio of the received power (to the load circuit) to the output power for maximum power transfer is Pr _ =) = A. Py Sad The ratio P:/P; = Ao? (as used here) is called the free-space radio gain for matched doublets or for short the free-space gain, since all objects, including the earth, are supposed remote from both doublets. This free-space gain, Ag = 3\/8rd. On the decibel scale, it takes the form 10 logio aie 20 log Ao 1 (18) = 1325 = oon, < decibels (19) The nomogram, Figure 3, gives a convenient means of calculating the free-space gain for doublets adjusted for maximum power transfer. 22 POWER TRANSMISSION. RECIPROCITY a2 Radio Gain The formulas in Section 2.1 apply to doublets in free space. This section considers the modifications (3A/87d)? = A*%). (Adjusted for maximum power transfer.) that must be made in the formulas when the re- striction of free space is removed. In actual trans- mission problems, ground reflection, reflection from elevated layers of the atmosphere, diffraction by earth curvature and by obstacles, and refraction by the atmosphere must be considered. In Chapters 5, 6, and 7 special forms of gain are discussed and separate gain factors are introduced to take care of each effect. For the present a factor that will be called the path-gain factor, A», representing the product of all these special factors will be used. A, is defined by E = EA, (20) where E is the absolute value of the actual field strength and Ep is the absolute value of the free- space field strength that would exist at the same distance d from the doublet transmitter in free space. Replacing Ey with EyAy, equation (17) for the received power, becomes EvAy | 3’ 1207 87 while the power output as given by equation (16) remains unchanged, so that P2 3X = US ty ae AvA,)2 = A? Ie (>) Cha) replaces equation (18) as the ratio of received power to output power for maximum power transfer between doublets. The quantity defined by (22) is the free-space gain and A is the gain factor. P, (21) (22) Q9- ol 16 FUNDAMENTAL RELATIONS The general relation between the input voltage at the receiver and the received power is V; = VPR), where R; is the resistance of the receiver load circuit (which is equal to the radiation resistance for maxi- mum power transfer) and V;, is the input voltage. Hence, using equation (14), V; = 0.0178E AVR, volts (23) 1 = Kyb Se NN Gea jue ee 2 22.2 Antenna Gain. Polarization The equations of Section 2.1 may be further generalized to apply to any type of antenna through the introduction of a quantity called the antenna gain. The term gain, as applied to an antenna, is a measure of the efficiency of the antenna as a radiator or receiver as compared with that of a doublet antenna, with all antennas located in free space. Quantitatively, the gain, Gi, of a directive trans- mitting antenna is the ratio of the power P,’ radiated by a doublet antenna to the power P; radiated by the antenna in question to give the same response in a distant receiver, with both transmitting antennas adjusted for maximum transfer of power. Hence Ga Jes The gain G of a directive receiving antenna is the ratio of the power P;” radiated by a transmitting antenna, which produces a certain response in the matched load circuit of a distant doublet receiving antenna, to the power P; radiated by the same transmitting antenna to produce the same response in the matched load circuit of the receiving antenna in question, with both receiving antennas adjusted for maximum transfer of power. Hence See Pe (24) Gs From the definitions given above it follows that for a transmitting and receiving antenna combina- tion in free space, with gains G, and G2 and adjusted for maximum power transfer, the power ratio is equal to Po ‘ 2 2 aa.( 2) S P, Sad where Pi, G, are the power output and gain of the transmitter and P, is the power delivered to the matched load of a receiving antenna of gain Go. (26) Gy nA 0°; If the antennas are not in free space, equation (26) becomes P, : — = G1G> (2) A - P, Sad where A is the gain factor and A, is the path-gain factor. Note that for highly directive antennas A, may depend upon the directivity characteristic of the antennas, e.g. when the antenna discriminates between the direct and reflected waves. Since power is proportional to the square of field strength, equation (20), for any transmitting an- tenna, becomes = G1G2(AoA,)?, (27) = GGA, E = E\VGiA). (28) In defining gain, the electric doublet is selected here as the comparison antenna in place of the iso- tropic radiator (that is, a hypothetical antenna which radiates equally in all directions) which is sometimes used in the literature. Since the gain of an isotropic radiator relative to a doublet is 24, the gain of any antenna referred to an isotropic radiator is 3/2 the value referred to a doublet antenna. (29) The chief objections to the isotropic radiator are that it does not occur in practice and cannot be produced experimentally, even approximately. In experimentally measuring the gain of an an- tenna, a half-wave dipole is often used as a reference antenna. While the gain of a half-wave dipole rela- tive to a doublet is approximately unity, being 1.09 for a very thin dipole, it depends somewhat on its actual dimensions so that it is better to express the experimental gain in terms of the doublet antenna even though a longer antenna is used as a reference antenna in making the measurements. When antennas are oriented so that the directions of polarization make an angle y with each other G (isotropic) = il 5G (doublet) 5 DIRECTION OF MAX RADIATION > es I RANSEIRECE =| ee ee == TRANSMITTER DIRECTION OF RECEIVER END VIEW MAX RE-RADIATION Fiaurr 4. Relation of antenna axes and wave polariza- tion. (while the maxima of their angular patterns still point toward each other), the formulas for power transfer, equations (18), (22), and (27), are multi- plied by a factor cos? y (see Figure 4). RECEIVER SENSITIVITY be 228 The Reciprocity Principle aa Thermal Noise So far in this chapter the radiation and reception Thermal noise is generated by the random of power by antennas have been treated separately. Actually, many of the properties of an antenna are the same for either reception or radiation; in partic- war, the current distribution, the effective length, and the gain are unchanged. The reciprocity prin- ciple, from which these propositions may be proved, may be stated as follows: If an electromotive force V, inserted in antenna 1 at a point 7, causes a cur- rent I to flow at a point v2 in antenna 2, then the voltage V applied at x, will produce the same cur- rent J at 2. From this principle the statement of the equiva- lence of current distribution, effective length, and gain follow readily. This theorem does not hold when the propagation of a wave takes place in an ionized medium in the presence of a magnetic field (the ionosphere), but it does hold for all cases of transmission discussed in this volume. 28) RECEIVER SENSITIVITY The sensitivity of a radio receiver is that charac- teristic which determines the minimum strength of signal input capable of causing a desired value of signal output. In high-frequency receivers the limiting factor for reception is usually set noise, that is, noise produced in the tubes or other ele- ments, such as crystals, of the receiver itself. At frequencies below about 100 me, atmospheric dis- turbances sometimes exceed the set noise in intensity, but at higher frequencies atmospheric static is negligible. Man-made noise (automobiles, ete.) may be a source of serious trouble, but such inter- ference can often be eliminated by proper siting. Consequently, for high-frequency receivers, sensi- tivity may be expressed, at least approximately, in terms of set noise only. Although set noise has an important bearing on sensitivity of radar receivers, there are other factors which must be considered for this type of equipment. There are several types of set noise. Though all noise sources in a well-designed receiver are mini- mized with the exception of the thermal noise whose magnitude is independent of equipment construction, the total set noise is usually several times the purely thermal noise. (temperature) raotion of electrons in a conductor; it is, therefore, a universal property of matter and independent of the design features of the receiver. The rms thermal-noise voltage that appears across the terminals of any circuit element is a function of the frequency interval (receiver bandwidth) over which the noise is averaged; it is given by V, = V4kT Af. R, (30) where FR is the resistance across which the noise voltage is measured, Af the bandwidth in cycles per second, 7 the absolute temperature, and k, the Boltzmann constant, is equal to 1.38 x 10° watt-second per degree. The noise voltage is inde- pendent of the reactance components in the circuit. Consider now, for the purpose of definition, a receiver Without internal noise, that is, let all the noise be generated in the receiving antenna of re- sistance R,. If R, designates the load resistance (that is, the resistance of the receiver exclusive of its antenna), the average noise power delivered to the receiver will be V PR, (Ra + R,)?’ where V,, is the rms value of the noise generated in the antenna. The noise power is maximum when the receiver is. matched to its input; this maximum is J ( 2 am (31) P= =kTAf watts (32) a by equation (30). Assuming equivalent temperature T = 290 degrees absolute, and measuring Af in megacycles, P,=4X10 Af watt. (33) This result means that in an idealized receiver, noise is the thermal noise of an antenna of equivalent temperature J’ = 290 degrees absolute, and the minimum detectable signal would be approximately equal to 4 X 10° Af watt. ae Noise Figure The sensitivity of a set cannot be described in terms of the thermal noise alone, because the set noise is usually several times the purely thermal noise. For this purpose another quantity called the 18 FUNDAMENTAL RELATIONS noise figure is used. The noise figure of a system (taken here to be a receiver, for definiteness) is defined as Prof Pai 34 ecole : n where P,; = noise power (k7TAf) from the antenna which is being delivered to the receiver. Pyo = noise power at the output of the re- ceiver, that is, the noise after the amplifications and additions arising in the receiver circuit. P;; = signal power from the antenna which is being delivered to the receiver. = signal power at the output of the receiver, that is, the signal power after detection and amplification have taken place. The ratio P,,/P.; is called the receiver gain. This quantity is called g and must not be confused with antenna gain G. Using equation (32), equation (34) may be written Ne at (35) The bandwidth Af is measured by finding the area under a curve of power-gain versus frequency and equating this area to the area of a rectangle whose width is interpreted as Af and whose height corre- sponds to the gain at the frequency at which the gain is a maximum. ao8 Receiver Sensitivity Frequently receiver sensitivity is defined by the assumption that a received signal can be discrim- inated when its output power is equal to the noise output power. This assumption, while true for a large class of receivers, is too rough for radar re- ceivers. The method given here will explain the procedure used for calculating the minimum dis- cernible power of receivers for which the assumption is true. The sensitivity of radar receivers is con- sidered in Section 2.3.5. Referring to equation (34), the assumption that signal output power is equal to noise output power means that P;, = Py». Hence (36) But P,; is, on the assumption discussed above, just the minimum discernible signal power, Pmin, at the receiver input, that is, before amplification. Hence, using equations (32) and (33), Puig] hl Afe Pes X10 Af. Fy watt Oe) 23-4 Vieasurement of the Noise Figure Remembering that the cases under discussion are those for which the minimum discernible signal is equal to the noise output power, equation (37) gives an estimate of the minimum detectable power from a measurement of the noise figure /, which may be obtained as follows. An antenna (or other signal generator) whose impedance is matched to the receiver is connected to the receiver. With the signal output reduced to zero (so that the antenna furnishes only noise power to the receiver), the receiver gain is increased until the noise gives a measurable output and the output noise power is measured with a power meter. Now a signal is impressed on the antenna and increased to a point where the receiver output power is doubled, and tbe input signal power is measured. Thus, referring to equation (34), and F,, = eae is Ps = = A P,; kTAf leera so that the measurement of the impressed signal power indicated here gives F,,. If the receiver consists of several elements in caseade, including attenuators, amplifiers, and con- verters, the overall noise figure can be compounded from the noise figures and gains of the individual components by means of the following equation: joes iere Fy2—1 a valle (38) gi Gigz where F,, = overall noise figure, I, = noise figure of the kth element, g. = gain of the kth element. In using this equation it is understood that the sue- cessive stages are matched. It is clear from equation (38) that most of the noise comes from the early stages of reception; in high-frequency radar sets, it comes from the crystal mixer and the first intermediate-frequency (i-f) stage. This means of course that noise picked up at RADAR CROSS SECTION AND GAIN 19 later stages is much less amplified by the system than the noise from the early stages. In equipment specifications, the noise figure is usually expressed in the decibel scale as decibels above thermal noise. Actual noise figures vary from a few decibels above thermal noise in the very high- frequency [VHF] region (receivers built a few years ago often have appreciably higher noise figures) to larger values for microwave receivers. 235 Sensitivity of Radar Receivers It is by no means true for radar receivers that Prin= Po; a8 a matter of fact, Pmin>>Pyo. That is, the minimum discernible power considerably exceeds the noise level. The largest single additional loss in radar recep- tion is scanning loss which is relate to the rotation of the antenna (one or several revolutions per minute). As an example, for one particular radar which has a bandwidth Af = 2 me, this loss is from 10 db to 12 db. In case the antenna does not rotate, there is no scanning loss. This fact would seem to be of limited operational importance, since it would usually be necessary to locate the target (a plane, for example) by scanning. Another loss, closely connected with scanning loss, is sweep-speed loss. This loss is due to the fact that practical targets, such as airplanes, reflect rapidly varying amounts of power to the radar receiver, these amounts depending on the precise orientation of the target at the moment when the radar beam sweeps over it. Consequently, sweep- speed loss will depend on the speed of rotation of the antenna, on the distance of the target from the antenna, and, to some extent, on the beamwidth and the nature of the target. The overall figure for this loss on the same radar used to illustrate scanning loss is about 4 db for targets 200 miles from the radar. In addition to these losses, careful experiments with the radar used as an example above have indicated that there is an operator loss of about 4 db for even experienced operators. This might be thought of as a loss due to the difference between laboratory and field conditions. Statistical consideration about the extent of noise fluctuation and about the fact that a target need not be seen on every sweep lead to further small losses which total, for the radar under discussion, 2 db. Summarizing for the case of the radar of the above example, the minimum detectable power is about 34 db above kTAf or about 8 X 107'*° watt, not 12 db above kTAf or 8 X 10° * watt, as would be indicated from the noise level alone. This amounts to 22 db or a factor of 166; that is, the actual min- imum discernible power is 166 times that calculated from noise alone. It will be seen in the results of the next section that the maximum range of a radar set varies with the inverse fourth root of the min- imum discernible power. Consequently, a calcula- tion of the maximum range of the radar of the example, which assumed that the minimum dis- cernible power was equal to the noise power, would give a range too great by a factor of 166 = 3.59. Since this would be a serious error, it shows the importance of a very careful consideration of radar receiver sensitivity in calculations of this type. 24 RADAR CROSS SECTION AND GAIN a Radar Cross Section The total scattering of a target may be described by the use of a parameter (having the dimensions of an area) called a scattering cross section. This concept has already been presented in the latter part of Section 2.1.2, where both scattering and absorp- tion cross sections of doublets were discussed. The scattering cross section S is defined by where P, is the total power scattered by the target irrespective of its angular distribution and W; is the incident power per unit area. The scattering cross section S, which gives in- formation about the total scattered energy, is not directly useful in radar work because in such applica- tions one is interested only in that fraction of the total scattered power which is scattered in the direc- tion of the radar; that is, one wants a parameter involving the scattered power per unit area at the receiver instead of the total scattered power. If the target is an isotropic scatterer, Ps 4rd?’ W, = 20 FUNDAMENTAL RELATIONS where W, is the scattered power per unit area at the receiver, d the distance from the target to the receiver, and P, the total scattered power. This gives, using equation (39), : W S = 4rd? — (40) 1 as a formula for the scattering cross section of an isotropic scatterer which involves scattered power per unit area at the receiver W, instead of total scattered power P,. For targets other than isotropic scatterers, how- ever, this procedure fails since one cannot say that W, = P,/4rd?. Nevertheless, it is useful to define a parameter ¢ which is called the radar cross section, by o = 4nd? W; in analogy with equation (40). Here W, is the actual power per unit area at the receiver. From the pre- ceding discussion it is apparent that « may be thought of as the scattering cross section which the target in question would have if it scattered as much energy in all directions as it actually does scatter in the direction of the radar receiver. For a target scattering isotropically, o = S, but for any other type of target o does not, in general, equal S. A radar gain formula analogous to the radio gain but applicable to two-way transmission can be de- veloped from equation (41) by replacing W, and W; witb the directly measurable quantities P: (power output) and Py», (received power). From equation (6), W; = E®/1207 in which F is the field strength incident on the target. Substituting this value of # into equation (7) gives W; = 3P/87d? for a doublet transmitter in free space. Including the gain of any type of transmitting antenna, this takes the form (41) Wo (42) Further, the power received by a doublet with a matched load, equation (17), may be written oN WW, P, = (43) 81 if H?/1207 is replaced by W,, where here H is the field at the receiver. If the receiver is not a doublet, equation (43) may be replaced by 3 8r P, = WG: (44) where Gs is the gain of the receiver. Substituting the values for W; and W,, given by equations (42) and (43), into equation (41) yields BENE (2). (45) IPS 4rd? \8rd This is the radar gain for two-way transmission in free space. By means of it, ¢ may be measured, or if o and P;/P, are known, it may be used to calculate ranges. Generalizing equation (45) we have s JZ = GGs o (2 yay P; 4ad? \Sad where A, is the path gain factor (see Section 2.2.1). It may be observed here that some writers call oA,', not o, the radar cross section. These writers call their c, for the case A, = 1 (free space), the free- space radar cross section oo. Since, in this volume, the complicated terms appearing in A, are treated separately and not as part of the cross section, this distinction is not made here. For some simple targets, « may be calculated. The following are a few of the values. (46) Radar cross Targets Condition section ¢ | Conducting sphere, radiusa | a >> Ta? Metallic plate, area = ab a>>,b >> _ | 47a*%b?2/d2 Cylinder, diameter = d, Axis of cylinder mdl2/® length =1 | parallel to field andd >>, SSX Matched load doublet | Oriented parallel 92/167 | to field Shorted doublet (dummy) | Oriented parallel 92/470 | to field Objects of tactical interest (ships, airplanes) have very complicated radar cross sections. In particular, a strong dependence on the aspect of these unsym- metrical targets is observed. For ships the situation is still further complicated by the variability of the incident field over the target area. Some writers on the subject of targets use a characteristic length L (sometimes also called a scattering coefficient) which is related to o by op Eb (47) Babe: Radar Gain It is possible to write equations for two-way transmission which bear a formal resemblance to corresponding equations for one-way transmission by RADAR CROSS SECTION AND GAIN 21 introducing a quantity Gp, called the gain of the target. Gp is the gain of a target in the direction of the radar receiver relative to a shorted (dummy) doublet. By writing formulas connecting the radar gain with the power per square meter incident on the target. and the power per square meter scattered back to the receiver, it is possible to establish a connection between radar gain and the radar cross section defined in the last paragraph, and from this to calculate a gain formula involving Gz instead of o. Applying equations (15) and (6), 3” Qa P, = W; (48) for the case where the target is a shorted doublet. P, is the total seattered power and W; is the power per square meter incident on the target. For a target with a radar gain Gp it follows that 1 3 eee (49) Tv In a similar way a formula for W,, the scattered power per unit area at the receiver, can be developed. A target which scattered equally in all directions would scatter an amount Pf = 4n@W,. (50) But ie = 5 GaP, (51) where P, is the amount scattered by an actual target with gain Gp. [The factor 3/2 appears be- cause the gain of the target relative to an isotropic radiator is (3/2)Gz.| Hence Do, GaP: (52) Eliminating P,; from equations (49) and (52), Wry _ 9NGn* —, 53 W; = 167°d? ey Putting this value of W,/W; in equation (41), go Uae (54) dar which is the required general formula connecting target gain and radar cross section. It will be noted that the factor 9/47 is just the radar cross section of the shorted doublet. Inserting the value of o given by equation (54) into equation (45), Pr, = AG GG R* (2) ) Py Sid which is the radar gain formula for free space in terms of the gain of the target relative to a dummy doublet. The reasonableness of the factor 4 in the above equation may be made apparent by the following analogy. Compare the doublet antenna with a generator whose internal resistance corresponds to the radiation resistance of the antenna. When the generator is shorted all the power is dissipated in the internal resistance. When the doublet is shorted all the power is reradiated. The maximum power that can be extracted from either the generator or the antenna occurs when the load resistance equals the internal generator, or antenna radiation, re- sistance. It is 14 the above short-circuit power. This is the 4 that occurs in the above equation. Equation (55), in the nonfree-space case, takes the form Py» >{ 3r i — = 46,42. | — ) A,', P en = 4 1 (55) (56) where A, is the path gain factor defined by equation (20). Chapter 3 ANTENNAS 34 FUNDAMENTALS eine Function of Antennas TRANSMITTING ANTENNA converts the power A delivered to it into electromagnetic radiation (neglecting losses); a receiving antenna abstracts power from an incident electromagnetic wave and delivers to the receiver that part which is not re- radiated or lost in the antenna. In the short and microwave region the power conversion is effected with a very small loss so that for most practical purposes the power loss inside the antenna may be disregarded. Apparent losses caused by reflection owing to mismatch between the antenna and its input circuit are of a different nature and are not included herein. For many purposes it is desirable to concentrate the power radiated into a beam of comparatively small angle as in this way the field strength in the preferred direction is enhanced. The gain of a directional antenna is defined by means of a com- parison of the given antenna radiation pattern with that of an electric doublet. The gain of an antenna is the ratio of power that must be supplied to a doublet to the power that must be supplied to the antenna considered in order that, at a given large distance, the electric field at the maximum of the antenna pattern is equal to the field at the same distance in the equatorial plane of the doublet. From the reciprocity principle it is found that the gain of a receiving antenna is equal to the gain of the same antenna used as a transmitter. A discussion of antenna gain and reciprocity is given in Chapter 2, Section 2.1. sae Directive Antennas Polar plots of antenna radiation patterns are of two kinds: either the relative magnitude of the Poynting vector (power per unit area) is plotted along the radius vector, or the relative magnitude of the radiation electric field strength is plotted in the same way. Usually the value of the radius vector at 22 the maximum of the pattern is taken equal to unity. The Poynting vector plot is obtained from the field strength plot by squaring the radial distances (Figure 1). Antenna radiation patterns. FIGURE 1. If an antenna system is designed so that most of its power is concentrated into a comparatively small cone, the corresponding part of the radiation pattern is called the main lobe. Commonly there are a num- ber of secondary maxima (side lobes) much smaller than the main lobe. The width of the main lobe is measured by the angle between half-power points. Half-power points are those points in the polar diagram of the antenna pattern where the power per unit area is equal to one-half that at the maxi- mum, the field strength being /v2 = 0.707 times that at the maximum. This angle is also referred to as the beam width. The beam width varies from a degree or less for some specialized radar antennas to very large angles such as 50 to 60 degrees, depending on the design and purpose of the antenna. The larger the beam width the smaller the gain. It should be noted that an antenna radiation pattern may have high directivity with respect to one plane going through the antenna and little or no directivity in another plane. Thus a doublet antenna (for definition see Section 2.1.1) is directive in a plane which contains the antenna itself but is nondirective in the equatorial plane perpendicular to the antenna (see Figure 11). FUNDAMENTALS 23 oles Antenna Pattern Factors in Ground Reflection With highly directive antennas the magnitude of the direct wave may differ appreciably from that of the ground-reflected wave owing to their differ- ence in angle of emergence from the antenna (Fig- ure 2). This must be taken into account by using the oT T RAY FIELD. str PIRES (a5 AY G Ls Ci FLA. “7 Ficure 2. Antenna pattern factors. antenna pattern factors F; and F, in computing the interference pattern above the line of sight. This subject is dealt with in Section 5.2.6. 4 Standing-Wave Antennas An important class of antennas is that in which standing waves of the currents and the voltages are set up. In a transmitting antenna of this type, for instance, a progressive or traveling wave is supplied from the connected source of power. This is re- flected from the end of the antenna and the inter- action of the two sets of waves moving in opposite directions results in a standing-wave system. In this event the current amplitude is zero at the ends of the antenna and assumes differing values at the other positions on the antenna. The dis- tribution of current amplitudes is usually assumed to vary sinusoidally with the distance from the end of the antenna. This is a good approximation where the diameter of the antenna wire is small compared with the length, but may be seriously in error for thick-wire antennas. The simplest, and one of the most commonly used, standing-wave antennas is the half-wave dipole antenna, discussed in Section 3.2.3. $8 Resonant Antennas Many antennas are operated at or near resonance, which means that the reactive component of their impedance vanishes or is very small. Two types of resonant antenna may be dis- tinguished: either (1) the radiating element as a whole is resonant, as in the case of the half-wave dipole, shortened the right amount; or (2) the an- tenna system is made resonant by adding suitable reactive components to the radiative elements. To illustrate, the center-fed half-wave dipole of exactly half-wavelength, assuming sine distribution of cur- rent, has an inductive reactance; it may be made resonant by the addition in series of a capacitive reactance. This is known as antenna loading and is common at the longer wavelengths where half-wave dipoles would be too cumbersome. Another example is that of a dipole radiator shorter than the half- wave dipole and having the form of a metallic tube; this is combined with a tunable cavity resonator eLESS THAN RESONANT LENGTH INNER COAXIAL LINE TUBE <=METAL SUPPORT INPUT Figure 3. Antenna tuned to resonance by a shunt impedance. inside the tube that acts as a shunt impedance, the whole system being tuned to resonance (Figure 3). Although the actual antenna impedance is made up in a complicated way of distributed capacitances and inductances, the input impedance of the simpler types of antennas for a limited frequency band containing the resonance frequency is essentially that of an ordinary series resonant circuit [the resist- ance at resonance being essentially the radiation resistance of the antenna (see Section 3.1.7)]. The input impedance of certain other antennas is essen- tially that of parallel-resonant circuits with very large shunt resistances at resonance (see Section 3.2.2). For illustration, see Figure 6. ae Traveling-Wave Antennas In this type of antenna there is no standing-wave system set up since the progressive or traveling wave of current fed into the antenna is absorbed, without reflection, by a terminal resistance placed at the end of the antenna, which is equal to the characteristic impedance of the antenna regarded as a transmission line. Such antennas are necessarily nonresonant. The traveling-wave antenna radiates most strongly in the general direction of the wave motion. The major lobe makes an angle a < 90 degrees with this 24 ANTENNAS direction as indicated in Figure 20. Here we have a long-wire antenna with the input at the left and the characteristic impedance (resistance) at the right. A traveling-wave V antenna uses two of these elements (see Section 3.3.2) and a rhombic is com- posed of four elements (see Section 3.3.3), with the elements arranged at angles which produce maximum directivity of the combinations. Antennas of the nonresonant or traveling-wave types are used both for longer and for very short waves. (However, there is an intermediate fre- quency region extending from about 100 to 3,000 me where the half-wave dipole is of such convenient size that standing-wave dipoles or dipole arrays are most frequently employed.) In the microwave band where transmission is effected by wave guides it is possible to terminate a wave guide with a horn which “matches the imped- ance of the wave guide to that of free space” and acts as a directive antenna (Section 3.7). A slot or a series of slots in the side of a wave guide may also act as an antenna at these frequencies. Sil Radiation Resistance The radiation resistance R, of an antenna is the ratio P, of the total power radiated in all directions to the square of the current at the point of measure- ment. The power may be computed by integrating the radial component of the Poynting vector over a spherical surface surrounding the antenna. Then if I; is the effective value of the input current, eee (1) I? The radiation resistance of the doublet antenna is stated in equation (9) in Chapter 2 to be R, = 807 @) eine! (2) i Influence of Near-by Conducting Bodies The impedance of an antenna is affected by the presence of conductors in the vicinity and depends upon the mutual impedances between the conductors and the antenna. The mutual impedance decreases with increasing distance so that for conducting bodies of comparable size the effect is negligible for distances greater than, perhaps, 2 to 3 wavelengths. But for conductors set less than a wavelength apart, such as an antenna and reflector (or director) combination or as antenna arrays, the mutual effect plays an important role and modifies the input impedance of the antenna. For an antenna set near a large conducting body, such as a large metallic sheet or the earth, the mutual effect is cared for in a different way. If the earth, for instance, is assumed plane and perfectly conducting, its effect is the same as that of the mirror image of the antenna in the ground. As shown in Figure 4, t 7 f ANTENNA h PLANE EARTH od Ze fy PERFECT CONDUCTOR 1 if IMAGE | VERTICAL HORIZONTAL Fiaure 4. Method of images. the image of a vertical antenna is a similar antenna with current in the same direction, while the current is reversed for a horizontal antenna. The radiation field at any point above ground is obtained by summing the radiation fields of antenna and image. ae STANDING-WAVE ANTENNAS aa Linear Antennas A linear antenna is a straight thin rod supplied with alternating current. According to whether the connection to the antenna is made at the middle or a) = END FED ALTERNATE CURRENTS CENTER FED CO-PHASED CURRENTS Fiaure 5. Distribution of current amplitudes with linear antennas. STANDING-WAVE ANTENNAS 25 at the end, center-fed and end-fed antennas are distinguished. Center-fed linear antennas are also called dipole antennas. Typical current amplitude distributions are illus- trated in Figure 5. The amplitude is always zero at the open end while the amount at the input point depends on the position of the input connection. For thin wires, compared with the length, the distribu- tion of amplitudes is approximately sinusoidal. 3.2.2 Half-Wave Antennas Figure 6 illustrates two types of half-wave dipole or center-fed antennas and one end-fed antenna, together with their lumped-circuit analogues. The ACTUAL CIRCUIT {I LOW Z Vegan LL HIGH Z CURRENT-FED OR OIPOLE VOLTAGE-FED OR CENTER-FED DIPOLE END FED SCHEMATIC CIRCUIT A B c LUMPED-CIRCUIT ANALOGUE aE Ficure 6. Three methods of exciting half-wave an- tennas and their analogues in lumped-constant resonant circuits. SERIES RESONANCE PARALLEL RESONANCE input current required varies with the position of the input point. The voltage distribution in general has a maximum at the points of current zero and has a minimum where the current is maximum. a3 Half-Wave Dipole The half-wave dipole, shown in A and B of Figure 6 and in Figure 7, is the type most frequently used in the 100 to 3,000 me range. In this range the length /2 lies between 1.5 and 0.05 meters. In this section it is assumed that the current distribution is sinusoidal. 1. Radiation field. The radiation field at point P, Figure 7, where d >> X, is obtained by dividing the half-wave current distribution into an infinite Ficure 7. Half-wave dipole. number of infinitesimal doublets, using equation (2) in Chapter 2 and taking into account the differences in phase at P introduced by the differences in the distances which the radiation from the various doublets must travel. The net result, using d in place of 7, is _ 601; cos [(a/2) - cos 0)] He d volts per meter, (3) d sin 6 Hy, = ae amperes per meter. (4) 1207 The normal part of the field, HZ, (difference), pre- scribes the antenna pattern factor (measured in relative field strength) and is plotted in Figure 11. The corresponding pattern for a doublet is Hy ~ sin 8, which is a circle in polar coordinates. These patterns are circularly symmetric about the antenna axis. Squaring the radial lengths in the above patterns gives the pattern in terms of relative power per unit area in the same angular direction. The radial component of radiated power per square meter (Poynting’s vector) is given by [ cos(Eese) f 2 | (5) watts per square meter. E? _ 3012 W, = E.H, = 1207 T awd? sin 0 In the equatorial plane, 601; E, = ma (6) 26 ANTENNAS 2. Gain of half-wave dipole. The gain of the dipole relative to a doublet is the ratio of the power supplied to the doublet to the power supplied to the dipole to produce the same field strength at the same distance in the direction of maximum radiation (here the equatorial plane, 6 = 90 degrees). For equal maximum fields, comparing equations (3) in Chapter 2 and (6) in this chapter, faa aie. (7) . TT The power per unit area for the doublet, using equation (3), in Chapter 2, is E? 307? sin? 6 Ww oublet ~~ — ‘ ) (8) noun’ 120m ne and for the dipole the power per unit area is given by equation (5). The dipole gain is then : Waubiet 1A _ Power radiated by doublet G = 2 : . Power radiated by dipole’ W dipole dA where the integration is carried out over spheres surrounding the antennas. Carrying out this opera- tion, Garscie = 1.09 (or 0.4 db) . (9) 3. Radiation Resistance. The radiation resistance of the half-wave dipole is i IEE 4. Impedance of an Infinitely Thin Dipole. § The formulas given here are valid only for a half-wave ite = [Wee dA = 73.1 ohms. (10) CYLINDRICAL COORDINATES Figure 8. Half-wave dipole field components. dipole composed of wire of vanishing thickness. For wire of finite dimensions, see Section 3.2.7. Here (type A in Figure 6) it is necessary to calcu- late the voltage V; required at the input to establish a current distribution I; cos [(27/\)z], as shown in Figure 8. To do this, the total field of the dipole must be known, including the induction field which is significantly large at short distances as well as the radiation field. In cylindrical coordinates, the total field is given by e—d/4 -1(22 4 -5¢27 B, = + jp0r,| PMA) 4 EM, 2, ar br (11) _ j(2%2 amb B= — por[ tC $1 AGI], (12) a b Hy = — jt [EO EEG i (13) 4dr Lr r By the reciprocity theorem a small current length T,dz = I; cos [(2a/d)z]- dz induces a voltage (— dV;) at the input point which is equal to the voltage dV, = E,dz induced in dz by a small current length T,dz taken at the input point. Hence Exdz _ —dV,; ee I; cos (= :) - dz and the total input voltage is 2=A/4 or ve=2f — 8, eos (2 2). ae. z=0 r Carrying out the operation indicated and dividing by I; gives the impedance of the half-wave dipole as Z = 73.1 + 742.5 ohms. (14) The dipole thus has an inductive reactance of 42.5 ohms if a sine distribution of current amplitudes is assumed. The reactance can be altered by changing the length of the wire. Increasing the length increases the inductance; decreasing the length decreases the inductance, first to zero for resonance, and then for still shorter lengths to a capacitive reactance. Changes in length of only 4 to 5 per cent will pro- duce large changes in the reactance. 3.2.4 Modifications of the Half-Wave Dipole Two modifications will be given. 1. Quarter-wave dipole with artificial ground. A convenient device for doubling the effective length STANDING-WAVE ANTENNAS bo ~I of a dipole is to use an artificial ground plane. It usually takes the form of a number of grounded rods spreading radially from the base of the antenna FicureE 9. Quarter-wave dipole with artificial ground. (Figure 9). If the antenna is a quarter-wave dipole the effect of the artificial ground is to produce an image quarter-wave dipole; the radiation resistance and the radiation pattern of the system are those of a half-wave dipole. d/2 pod ee t INPUT Figure 10. Folded dipole. 2. Folded dipole. Another variant of the dipole antenna is the folded dipole, shown in Figure 10. Y \ a HALF-WAVE \ ! \ DIPOLE BU ! Lo *¢ WV I \ DOUBLET (CIRCLE) FIGureE 11. It is essentially a center-fed half-wave dipole with a parasitic counterpart “dummy”’ (see Section 3.5) in its immediate neighborhood and connected to the latter at the ends of the dipole. The induced current in the dummy has the same distribution as, and is in phase with, that of the primary dipole. Hence the radiation pattern is essentially that of a simple half- wave dipole. The radiation resistance is four times that of the ordinary dipole. 3.2.5 Multiple Half-Wave Long Antennas For an antenna of length equal to an integral number, 7, of half wavelengths, the radiation field is given by: 1. nis odd: cos (‘2 ) ( 208 (GO }s} 5 AL ND Gs) Eo : d sin @ 2. nis even: in (‘eo 6 ( 8 — cos , GOI, \2 16) ig = - ’ d sin 6 where d is the radial distance te a field pomt and I; is the input current at the center of one of the half-wave elements. The radiation patterns are illustrated in Figure 11 for the doublet, n = 1 (the half-wave dipole), and ee Le LS | 138 | | / { \ a: i \ | Ie} Antenna radiation patterns (relative field strength). 28 ANTENNAS The radiation resistance, both for integral and nonintegral numbers of half wavelengths, is plotted in Figure 12. Figure 12. Radiation resistance for linear antennas. In Table 1 the radiation resistances and the power 3.2.6 Cophased Half-Wave Dipoles The directivity and gain of linear antennas may be increased considerably by the suppression of alter- nate current loops, leaving therefore only loops in which the currents are all cophased. The suppressed loops are contained in either (1) quarter-wave stubs or (2) short inductive elements, as indicated in Figure 13. The suppressed loops are practically nonradiative. The radiation field at distance d is the vector sum of the fields from the n half-wave elements. The contribution from each element lags that of the next element above by an angle ins integra If-wav S ar r 2a z gains for integral half-wavelength antennas are feos be = 7 cos 6 radians, (17) listed. r Taste 1. Comparison of alternate and cophased half-wave dipoles. Rn Em, n Radiation Relative major lobe amplitudes ; Gn n resistance-ohms for same current input Gain (power) Half waves Alternate Cophased Alternate Cophased Alternate Cophased currents currents currents currents currents currents 1 73.1 73.1 ; 1 1.09 1.09 2 93 199 20) 2 1.19 1.47 3 105 | 317 | 8 3 132 2.09 4 113 439 4 1.46 2.67 fi 121 560 2 5 1.58 3.26 G,= Rdoublet (2 =" = R doublet ( Em,n ) is Re i ~ — R, ~ \ doublet Fiacure 13. Cophased half-wave dipoles. STANDING-WAVE ANTENNAS 29 determined by the extra distance [(\/2) cos 6] which it must travel. The radiation field is then equal to the radiation field of one half-wave element (as a function of angle 6) multiplied by the vector 1.0 2.0 n=! n=2 5,0 n=5 The radiation patterns for various values of n are plotted in Figure 14. Table 1 gives the radiation resistances, relative lengths of major lobes, and the gains, with comparative figures for the doublet and 6.0 n=6 Fiaure 14. Cophased half-wave dipoles (relative fields). resultant for the n elements. Thus Tv 601, cos (3 cos ‘) D a sin 6 [e? + i 4 P= 444 eX da] (18) the multi-half-wave antennas of Section 3.2.5. 327 Effects of Finite Diameter on Center-Fed Linear Antennas Figure 15 shows the input reactance, and Figure 16 the input resistance of a center-fed antenna of arbitrary length. The input impedance is a series combination of the two components. The important een ee DIAMETER: \J500_/ B ‘eer ZZ \|_ ez" REACTANCE IN OHMS FIGuRE 15. HALF-LENGTH OF ANTENNA Reactance at input of a center-fed antenna of arbitrary length. 30 ANTENNAS regions of the curves correspond to antenna half- lengths near \/4 and near \/2. The former repre- sents a center-fed half-wave antenna, whereas the latter represents a pair of end-fed half-wave antennas excited in phase. The half-length of the antenna was used in plotting, because in these terms the reactance curves resemble those for an open-ended trans- mission line. In the regions of principal interest the reactance curves are nearly straight lines whose slopes depend on the diameter of the antennas expressed in wave- lengths. The slopes of the reactance curves decrease as the antenna diameter increases. This feature is important in radar antennas which need to be in- sensitive to small changes in frequency. The curves 10,000 9000 8000 7000 6000 5000 4000 3000 RESISTANCE IN OHMS 2000 1000 \/4 Ficure 16. show that antennas of large diameter present less than a specified amount of reactance, say one ohm, over a greater range of antenna length than slender antennas do. In terms of frequency, this means that a given length of antenna has less than one-ohm reactance over a wider range of frequency when the antenna has a large diameter than when it has a small diameter. Radar antennas are commonly made of tubing and frequently have diameters in excess of 4/20. Figure 16 shows that the input resistance also depends on antenna diameter. This dependence is more pronounced when the half-length is about \/2 than when the half-length approximates \/4, as it does for a single center-fed antenna. The values for an antenna whose half-length is \/4 is not readable on the curve, but the component representing radia- tion ranges from 73 ohms for infinitely thin antennas, through 64 ohms for a diameter of 0.0001 A, 55 ohms for a diameter of 0.01 \, to less than 50 ohms for certain large-diameter radar antennas. The change is mainly due to a decrease in the resonant length of the thicker antennas. A feature of Figure 15 which is not easily readable is that the lengths at which the reactance is zero are less than 4/2 and X. The amount by which an an- tenna with zero reactance is shorter than these lengths depends on the antenna diameter. For very 3d/4 ry HALF-LENGTH OF ANTENNA 5\/4 Resistance at input of a center-fed antenna of arbitrary length. slender antennas the shortening is slight, but for large-diameter antennas or for special shapes as shown in Figure 17, a resonant length may be as COAXIAL LINE Figure 17. Non-cylindrical half-wave antenna. much as 20 per cent shorter than \/2. Special shapes, such as the one shown in Figure 17, have the ad- vantage of being insensitive to small changes in frequency and at the same time are not so subject to corona (breakdown of the air because of large poten- tial gradients) as slender antennas are. 328 Standing-Wave V Antennas This type of antenna (Figure 18) utilizes the directive properties of the multi-half-wave antenna. STANDING-WAVE ANTENNAS 31 Two such elements are combined in a V arrange- ment so that the major lobe of each (at angle @ with each element) is parallel to the axis of the V. By feeding the two halves of the V with currents 180 degrees out of phase the lobe structure is reversed to produce maxima, forward and backward, along the axial direction, while the field in the plane perpendicular to the axis is greatly reduced. The xIS IN PLANE OF THE V n=16, ¢=I7.5 IN PLANE -L TO V n=16, a=17.5° Fieure 19. Power distribution for standing-wave V antenna. (Courtesy of IRE ) 32 ANTENNAS value for angle a is equal to the angle between each element and its maximum lobe (see Figure 11). Figure 19 gives the (power) radiation pattern for n = 16 half-wavelengths. The directivity of this antenna system may be improved by adding one or more reflectors (see Section 3.4.6). The reflector is a V antenna of identical type. The legs of the reflector are placed parallel to those of the primary V and lie in the same plane as the original V. The reflector is set approxi- mately \/4 behind the primary V. 3.3 TRAVELING-WAVE ANTENNAS set Field and Pattern A traveling-wave antenna is one in which only progressive (or traveling) waves are allowed. Re- flected waves are eliminated by terminating the end opposite the input point in the characteristic imped- ance. See Figure 20. 90 80 70 60 The major lobes given by this equation are plotted in Figure 21, and the major lobe angles with the wire 6, are plotted in Figure 22. Angle 9@,,, it will be noted, decreases with increasing wire length. $32 'Traveling-Wave V Antenna As in the case of the standing-wave antenna a pair of lines arranged at a suitable angle with each CHARACTERISTIC IMPEDANCE Lobe structure for L = 2) traveling-wave Ficure 20. antenna in free space. °o ° 50 40 FIGURE 21. The equation of the radiation field, neglecting wire losses, is 60T, L as 1 E (1 — cos | . (19) tt sin 6 d 1—cos@ QIN DEGREES Major lobes (relative field strength) for traveling-wave antenna. other, and carrying traveling waves, can be made to produce a directional pattern with fairly high gain. The traveling-wave V antenna can be designed so that the plane of the V is horizontal and the maximum lies in the direction of the axis of sym- ANTENNA ARRAYS 33 metry, as in Figure 18. In this case the radiation is horizontally polarized. It can also be used as an inverted V in a vertical plane with the point of the V directed upwards; the radiation is then vertically polarized. This antenna, also called a semi-rhombic, is represented by the upper half of Figure 20. Om IN DEGREES Ficure 22. Angles for major lobes for traveling-wave antenna, Cocke) Rhombic Antenna This type of antenna is based on the same prin- ciple as the traveling-wave V antenna. The rhombic antenna consists of four wires arranged in the form of a rhomboid or diamond (Figure 23). The reflec- tionless termination of the wires is achieved by connecting the two wires at the end opposite the input to a resistance equal to their characteristic impedance. CHARACTERISTIC AXIS IMPEDANCE DIRECTION OF I. MAX RADIATION c 3 PATTERN OF? EACH LEG Figure 23. Rhombic antenna. As in the case of the V antenna, the rhombic antenna can be used both horizontally and vertically; at the longer waves the horizontal arrangement is usually more practical. The optimum tilt angle of the rhombic (angle @ of Figure 23) is not very critical provided the legs are not less than two wave- lengths long. The radiation pattern is not very sensitive to frequency and the rhombic antenna can therefore be used over a fairly wide frequency range (of the order of 2 to 1). Rhombie antennas have appreciably higher gains than V antennas. The field in the axial direction is equal to gO cose sin'| 2 (1. — sin 6) |, (20) d 1i—snod oN Effect of Perfectly Conducting Ground. If the rhombic is placed in a horizontal plane, height H above ground, the effect of the image rhombic must also be considered. The net result is that the direction of the resultant lobe maximum is tilted up by an angle e. It can be shown that, for a given angle « and wavelength A, to point the major lobe at vertical angle e the following relations for H, L, and @ must hold: JDreten = oN i - 4 sine 0.371X ae me sin? € d = 90° — « Figure 24 illustrates the radiation pattern (relative field strength) for a particular case. ANGLE IN DEGREES 50 40 30 20 O PLAN VIEW =41X = 0.83) = 7/2.5° =17,5° aer 7 90 70 60)5350 40 30 T 20 €=175° ELEVATION Figure 24. Rhombic antenna above ground (relative field strength). (Courtesy of Bell System Technical Journal.) 3.4 ANTENNA ARRAYS 3.4.1 Principle of Arrays An antenna array is a combination of several antennas, usually of equal strength and equally spaced in any one given direction. One-, two-, and three-dimensional arrays may be distinguished. The 34 ANTENNAS spacings in different directions may be different for two- or three-dimensional arrays. The use of arrays permits great increases in the amount of power radiated, in directivity, and gain. Although the most common array element is a half-wave dipole, the elements of an array may be radiators of any type; in particular, the elements may themselves be arrays. In this way it is possible to interpret a two-dimensional array as an array of arrays. A vertical curtain may be considered either as a horizontal array of elements which, themselves, are vertical, or it may be considered a vertical array of elements which, themselves, are horizontal arrays; similarly for three-dimensional arrays. In most arrays the elements radiate very nearly equal power, but in the binomial array the elements, although identical in structure, differ in the amount of power radiated because of differing current dis- tributions. In most arrays there is a constant phase shift (which might be zero) between adjacent ele- ments. By suitable phasing a great variety of an- tenna patterns can be produced. 342 Basic Types of Dipole Arrays There are three basic types of dipole arrays. 1. Broadside array. The centers of the elements are arranged in a line, with the axes of the elements parallel to each and perpendicular to the line. With the currents adjusted all in phase, the maximum radiation is broadside to the plane of the elements. 2. End-fire array. The geometric arrangement is the same as in the broadside array, but through appropriate phasing of the currents in the elements the maximum radiation can be directed primarily along the line joining the centers. 3. Colinear array. Here the axes of the antenna elements are arranged along the line of centers with the currents all in phase. The radiation is a max- imum in the equatorial plane perpendicular to the line of centers. To illustrate the principles most simply, two half- wave dipole elements are considered first, and later extension is made to arrays composed of a larger number of elements. 343 Two-Dipole Side-by-Side Array Two half-wave dipoles are placed side by side with spacing s and the currents J; and J, are equal but differ in phase by angle y (see Figure 25). If J, lags I, by time angle y, the field of the second element at P lags that of the first by angle @ where a is EQUATORIAL PLANE, 9=7//2 Figure 25. Two dipole side-by-side array. composed of y and the time delay caused by the extra distance traveled, (27/X)s cos @ sin 8, a= wt (27/\) s cos¢@ sin 6. (21) For equal currents, I, = I, = I, the field is equal to E 60L | ei 4. go ie ] [ cos G cos s) | " - ad | aes cor] ]f Ge) | | sng || | The first bracket gives the directional characteristic of an array of two elements, while the second bracket gives the directional characteristic of the,element itself. og ane ZERO OS / \ Z \ / \ \ IT ot, e \ By Di 1, if / \ / (22) sin @ BROADSIDE END-FIRE UNI-DIREC TIONAL s=)h/2 S=)/2 S=)/4 y= 0° =180° W=90°C IZ LAGS [,) EQUATORIAL PLANE 6=90° Fiaure 26. Radiation patterns (field strength) for two dipole side-by-side array. Three special cases are particularly to be noted. The field patterns for the equatorial plane (@ = 90°) and | I, | = | I; | are plotted in Figure 26. ANTENNA ARRAYS 35 1. Broadside. Here s = d/2, the currents are in phase (YW = 0°). The maximum field is broadside and twice that of each dipole. ce eon 2) Eg x < I 1 Ne EQUATORIAL PLANE a 7 Ss Yoo all aN Q= AN ig ican FIGURE 28. 2. End-fire. Again s = d/2, but the currents are out of phase (Y = 180°). The maximum field is found in both directions along the line of centers. 3. Unidirectional couplet. Here s = \/4 and Ip lags I, by y = 90°. The result of this combination is to produce a maximum field along the line of centers in the direction looking from the leading to the lagging current and zero field in the reverse direction. 3.4.5 3.4.4 Two-Dipole Colinear Array For two equal currents in time phase (see Figure 27), the field is equal to Tv 601 cos C cos i) sin a | By = =", (083) d : . a sin @ sin = where 2 | a= a s cos @. (24) The field is circularly symmetrical about the axis. Its variation with @ is plotted in Figure 28. This is, of course, equivalent to a vertical antenna with center height at distance s/2 above a perfectly conducting flat earth. Two half-wave dipole colinear array. One-Dimensional Array Two geometrical arrangements will be considered. 1. Broadside. Consider n elements with equal co- phased currents equally spaced (see Figure 29). Ue sin — 2 J Eo= Ez (8, >) (25) sin w]e ANTENNAS PLAN VIEW Ficure 29. Broadside array, elements perpendicular to paper. Effect of array length for broadside with (From Radio Engineers’ Figure 30A. element spacing s = \/2. Handbook by Terman.) where 2 a =— scos®@. (26) dr For center-fed half-wave dipoles, from equation (3) 601 cos (= cos a) d sin 6 Ea = (27) The patterns for the equatorial plane are illustrated in Figures 30A and 30B. Figure 30A shows the in- crease in directivity with increasing number of elements. Figure 30B illustrates the effect of element spacing on the production of side lobes. Figure 31 gives the gain for various spacings and array lengths. From this it appears that s = 5\/8 is approximately the optimum spacing. Spacing infinitely close Eee Spacing =— p g 2 7elements 3 Spacing=4 5 elements Spacing=A 4 elements Apaiy yo aun Ficure 30B. Effect of element spacing for broadside for array length L = 3. (From Radio Engineers’ Handbook by Terman.) & °o ET Tift HH Hs: 8 35 isa} CLS es Wy RN] ol fo) Gain Over One Element Expressed As Power Ratio 4 6 8 10 12 14 16 Array Length In Wave Lengths Figure 31. Gain for broadside array of doublets as a function of array length and spacing. (From Radio Engineers’ Handbook by Terman.) ANTENNA ARRAYS 37 2. Broadside: pattern factor and beam width. For illustration, suppose that the antenna consists of a vertical array of m horizontal center-fed dipoles spaced s apart with all fed in phase to give a broad- side beam strongly directive in the vertical plane. For this arrangement the field strength in the horizontal plane is given by m times equation (27), that is, TT cos e cos 6 Eorizontal Sie 601; = d sin 6 (28) with angle 6 measured from the dipole axis. See also equation (15) and Figure 11 (for n = 1). In the vertical plane, the beam is much narrower. If @ is the angle from the vertical and B = 90° — @ is the angle from the (horizontal) broadside direc- tion, the field in the vertical plane (@ = 90°) is given by . ma Or Ssin— , 6 FE vertical = ae — (29) sin — 2 with Tv a a= scos@ = — gs sin B. Line of Array ~< «< «< Ep) or asa director (Hp > Ep). 1. Parasite as a reflector. For good reflector per- formance, the spacing s/\ should lie between 0.15 and 0.25 with the parasitic element made slightly longer (perhaps 5 per cent) than d/2 in order to increase its inductive reactance. A few of the equatorial field patterns are shown in the lower row of Fieure 37. To obtain the strongest field in the R direction, it is necessary to lengthen the parasite to a particular length (obtained by trial). If this is done, Figure 38 indicates that the field Hp is a maximum for s/\ = 0.15 and that the ratio of Ep to E for the antenna alone is 1.83; for s/A = 0.25 it is 1.65. This does not, however, give the best front-to-back ratio. PARASITIC REFLECTORS AND DIRECTORS Al 2. Parasite as a director. Good director perform- ance is obtained when s/A = 0.1 and the parasitic element is cut slightly shorter (perhaps 4 per cent) than \/2 to produce a capacitative reactance. See Figure 37, upper row, for the field patterns and Er/Eantenna alone Ep/Eantenna alone Reflector —--=— Director Fieure 388. Adjustment of parasite for strongest fields Hp and Ep. (Courtesy of I, R. E.) Figure 38 for the best ratio of Hp to EL for the antenna alone. The latter, again, does not give the best front-to-back ratio. Oa Multiple Parasites. Yagi Antennas By using several parasites, rather pronounced directive effects can be achieved. Figure 39 shows a typical example. This antenna uses three parasitic FIELD 3 PARASITES PAT TERN _? = Ae) b a=.248X ee rae b=.588\ 4 C=.535X DRIVEN ELEMENT Figure 39. Antenna with three parasite elements. (From Radio Engineers’ Handbook by Terman.) dipoles arranged in a triangle or parabolic curtain. In order to obtain the most favorable pattern in such cases, careful tuning of the parasites is required. The most commonly used of the multiparasitic arrays of half-wave dipoles is the Yagi antenna (Figure 40). It has one reflector and several (usually 2 to 5) directors. Since the voltage at the center of a dipole is always zero, it is possible to weld all the DRIVEN ——9/e—— ELEMENT \ REFLECTOR ,,4! iy ul INPUT Ficure 40. Yagi antenna with three directors. parasites to a central sustaining rod, as shown. By increasing the number of directors, it is possible to obtain highly directive patterns. The spacings between the elements of a Yagi array are not uniform. They are determined so that the phase difference of the currents in adjacent ele- ments is equal to their distance expressed in wave- lengths. If this condition is fulfilled, the elements are in phase with respect to radiation in the D direction. In practice, the spacing is determined experimentally rather than by calculations, which become very cumbersome when several directors are employed. 364 Reflecting Screens A plane-conducting screen placed behind a radiat- ing dipole has a similar effect in the forward direction as an image dipole whose distance s from the primary dipole is twice that of the screen and which has a phase shift of 180° from the primary dipole. Radia- tion in the backward direction is confined to the weak fields leaking around the edges of the screen. The pattern in the forward direction is given by the array formula of equation (22), and end-fire array with wy = 180° and s =/2. Good results are achieved when the distance from the screen to the dipole is small (less than \/4) but larger spacings are also used. The change in input impedance of the primary element caused by the presence of the screen is appreciable. Reflecting screens are used primarily in connection with broadside arrays (curtains) to eliminate one 42 ANTENNAS of the two main lobes in opposite directions. An adequate screening effect is produced by a set of wires parallel to the direction of the radiating dipole with spacings somewhat less than a tenth of a wavelength. 3554 Corner-Reflector Antenna A simple directional device that gives an appre- ciable power gain (of the order of 10 to 20) is a corner reflector, which is essentially a combination of two reflecting sheets and a dipole. In the case shown in Figure 41 where the angle subtended by ® DIPOLE o IMAGES if CROSS SECTION Figure 41. Corner reflector antenna. the corner is 90°, the corner is equivalent to the combined radiation of three image antennas. The reflector can also be made of wires parallel to the direction of the radiating dipole. The reflecting wires do not, however, act as parasitic antennas but are taken so long that they are practically equivalent to conductors of infinite length. These should not be confused with corner re- flectors which are extensively used as targets and consist then of three mutually perpendicular con- ducting planes (see Section 9.2.4). 3.6 PARABOLIC ELEMENTS S6t Parabolic Reflectors These reflectors are the devices most commonly used to produce highly directive radiation patterns in the microwave region. The three main types are shown in Figure 42; they are the parabolic cylinder, the paraboloid of revolution, and the truncated paraboloid, the latter being a rectangular section cut from a paraboloid of revolution. If the parabolic cylinder is relatively short and provided with flat metallic covers at the top and bottom, its shape and DIPOLES —™ es / \ N \ / / Ne > PARABOLIC B PARABOLOID Cc TRUNCATED CYLINDER OF REVOLUTION PARABOLOID Ficure 42. Types of parabolic reflectors. its electrical properties resemble those of a sectoral horn (see Section 3.7.2). The directive action of the parabolic reflector depends on two geometrical properties of the parab- DIRECTRIX | Ficure 43. Properties of a parabola, ola (Figure 43). A ray coming from the focus is reflected into a direction parallel to the axis of the parabola, and the distance from any point P on the parabola to the line called the directrix is equal to the distance from P to the focus. Consequently, the effect of the parabola in the forward direction is equivalent to that of a distribution of sources in the directrix that all oscillate in phase (but usually have varying intensities over the directrix). The parabolic cylinder produces a directive pattern only in a plane perpendicular to the generating line of the cylinder (horizontal plane in A of Figure 42). In order to concentrate the beam in a plane parallel to the generating line of the cylinder (vertical plane in Figure 42),an additional directive device must be employed. Usually this is a colinear array of dipoles, as shown; the direction of polarization is parallel to the focal axis. In microwave work this type of antenna offers advantages over the two-dimensional curtain of dipoles employed in VHF directional antennas. PARABOLIC ELEMENTS 43° For the paraboloid of revolution or the truncated paraboloid, a simple source of radiation at the focus is used. Often this is a half-wave dipole, sometimes combined with a parasitic dipole which acts as a reflector (Section 3.5.2). In other types, the energy is brought to the focal point by a wave guide and is then reflected back onto the parabolic surface. If the wavelength is small compared with the dimensions of the parabolic reflector, the following approximate formula holds for the radiation pattern produced by a parabola: s fwd 2 sin} —- sin — E een (40) 5 ( wD sin 0/X , E = constant where D is the aperture of the reflector and 6 the angle from the axis. The half-power points corre- spond approximately to 9 ain) 4 Gay SS (41) These formulas correspond to the case of nearly uniform illumination of the reflector from the source at the focus. In practice a source that concentrates the field toward the center of the parabola is used in order to reduce the magnitude of the side lobes, ELECTRIC FIELD WAVE-GUIDE Figure 44. Sectoral horn. Fiaure 45. Radiation pattern for a sectoral horn hay- ing various flare angles. The half-power angle is then more nearly equal to 6 = 0.6X/D. The maximum gain of a parabolic reflector is a= (yt Ney ae For D = 2 meters and \ = 0.1 meter, the gain is approximately 1,000. (42) Optimum design a > Power Gain for A. =1 (0) 10 20 30 40 50 Flare Angle ,Degrees power Gain for 94, =1 6 8 10 15 20 30 Horizontal Aperture in ) Figure 46. Gain of sectoral horn with 71,0 wave. (These curves are for a vertical aperture ratio a /A = 1. For other ratios the gain given should be multiplied by a/\.) (From Radio Engineers’ Handbook by Terman.) 44 ANTENNAS a2 HORNS sa Types of Horns Many of the antennas previously described are used in the high-frequency [HF] and very high- frequency [VHF] bands of frequencies. Horns cannot readily be used at these frequencies because the sizes required would be excessive. But at the microwave frequencies, the size of the horn is small and it is easy to feed energy to it through a wave guide. In this arrangement the horn acts as transition between the impedance of the wave guide and the 377 ohms impedance of free space and thus reduces to a minimum the reflection of energy backward into the guide (such as would occur if the wave guide ended in an open pipe). Common types of horns are sectoral (discussed in Section 3.7.2), pyramidal, conical, biconical, ete. Only the first type is discussed in this section. 372 Sectoral Horn with TE,, Wave For this case the horn is flared only in width and is an extension of the wave guide of width b and depth a. For the TE,» wave the electric field is parallel to the dimension a and varies in strength cosinusoidally across the wave guide and horn open- ing, as shown in Figure 44. The length of the horn is R and the flare angle is %. Figure 45 illustrates the pattern shapes in the plane parallel to dimension 6 for various flare angles. For this type of wave, the cutoff frequency of the wave guide is _ 3X 108 2 ” AE (43) with b in meters. The operating frequency should be near but not greater than twice this value. The gain depends upon the length R and the flare angle @, and is plotted in Figure 46. Chapter 4 FACTORS INFLUENCING TRANSMISSION 41 REFRACTION 411 Survey Rees is caused by the variation of the dielectric constant (square of refractive index) of the atmosphere. Although the atmosphere is tenuous and the variations of refractive index are small, the effect of refraction upon the field- strength distribution of waves is considerable. As will be shown, refraction under average conditions may be taken into account by using an earth with a modified radius. A representative average value of modified earth radius commonly used is ka with k = 4/3. Under certain conditions, especially in warmer climates, a slightly higher value of k might be preferable. The case where a is replaced by 4a/3 is referred to as standard refraction. It corresponds to a linear variation of refractive index with height in the atmosphere. In recent years, more complicated variations of refractive index in the atmosphere have received considerable attention and have proved to be of great operational interest. This volume, however, is restricted to consideration of standard atmosphere propagation. othe Snell’s Law Let mo and m; denote the refractive indices of two media separated by a plane boundary. The ordinary law of refraction known as Snell’s law is then usually stated (see Figure 1), as mo Sin Bo = m1 sin Bi, where $) and §; are the angles which the ray makes with the perpendicular to the boundary. It is con- venient to use the complementary angle a, so that No COS Ap = Ny COS QQ. For several plane-parallel boundaries, Snell’s law generalizes to No COS Ao = NM COS Ay = Ne COSA] = eee, In the atmosphere, the refractive index is a con- tinuous function of the height. Again, it is usually legitimate to consider the atmosphere as horizon- tally stratified, so that the refractive index is a function of height only. The case of a continuously variable refractive index is readily obtained by (n)) BOUNDARY Ficure 1. Refraction at between two media. boundary passing to the limit of an infinity of parallel bound- aries infinitely close together, Snell’s law remaining the same; thus n(h) + COS @ = No COS ao, where now n and @ are continuous functions of the height. In place of a discontinuous change in direc- tion, there will now occur a bending of the rays (Figure 2). et as o w I DISTANCE ALONG EARTH Figure 2. Refraction in the atmosphere with variable nh). If the boundaries are not plane but spherical, Snell’s law must be modified. Analysis shows that over a spherical earth surrounded by an atmosphere in which the refractive index n is a function of the distance r from the earth’s center, the law of re- fraction becomes n(r) + T COS a = NP COS ao, (1) where a is the angle between a ray and the horizontal (see Figure 3). 45 46 FACTORS INFLUENCING TRANSMISSION Refraction is of practical importance only when the angle between the rays and the horizontal is small. In the determination of gain as given in later TO CENTER OF EARTH Ficure 3. Refraction over curved earth. chapters, the effect of refraction becomes com- pletely negligible when a is more than a few degrees. For small angles, cosa may be replaced by 1 — a?/2. In this case equation (1) is well approx- imated by 5 (a? — ay?) =n — M+ ue (2) 2 a where h is the height above the ground, so that ™m™ =aandr=a+h. This is the practical form of Snell’s law for the atmosphere above a curved earth. The reference level (see Figure 3) is here taken at the surface of the earth where mo is the index of refraction. 4.1.3 Modified Refractive Index In place of the sum (7 + h/a) that appears in equation (2), it is customary to define and use a quantity MV given by M= le = | 10°. (3) a M is called the modified refractive index. It gives a unit that is convenient for practical use. The modi- fied index is then said to be expressed in W units, values of which commonly le in the range of 300 to 500. Using this definition, equation (2) becomes 1 =o? — a0") = (M — Mp). 105° (4) An important special case is that in which the refractive index decreases linearly with height, nm — NM = constant X h. Then equation (2) may be written in the form ii E h 2 5 ai) ==, (5) where k& is the factor mentioned in Section 4.1.1 which determines the modified earth’s radius ka. Comparing the above expression with equation (2), and differentiating, it follows that Limit ka dh a or (6) an a M 1+a— dh Proof of the fact that refraction is negligible unless the angle is very small may readily be deduced from the preceding formulas. Thus, on differentiating equation (4), —6 dane ie (ae and in the standard linear case, by equation (5), da = dh/kaa ~ 1.2- 10°" dh/a for k = 4/3. Taking @ = 0.05 radians (3°) and dh = 100 meters, one finds da = 0.00024 radians (50 seconds of arc), a very small change in angle. This is the standard deflection which is accounted for by replacing a by ka. The deviations from this value experienced with nonstandard refraction are even smaller. The larger the angle a with the hori- zontal at which a ray issues from the transmitter, the less the angular deviation. In communication work and for certain radar problems, however, angles of less than one degree are of importance, and da may then become comparable to a. pe Graphical Representation Figures 4 to 6 show three different ways of repre- senting rays subject to refraction. Figure 4 gives a true picture apart from the exaggeration of heights. In the case of standard refraction, the curvature of the rays is always concave downwards, the center of curvature being below the surface of the earth. The middle ray shown is the horizon ray and to the lower right is the diffraction region into which rays do not penetrate. Figure 5 shows a diagram with REFRACTION 47 modified earth’s radius, ka, in which the rays are straight lines. Figure 6, finally, is a plane carth diagram; the rays are here curved upwards. TRANSMITTER V/EART Yy SS papius2ka Fiaure 5. Rays in a homogeneous atmosphere (equivalent radius ka). These diagrams may be considered as resulting from each other by changing the earth’s curvature by an arbitrary factor. From this viewpoint Figure 6 TRANSMITTER, INTERFERENCE REGION DIFFRACTION REGION Ficure 6. Rays in a plane earth diagram. (Radius of curvature of rays is — ka.) h'! | ! ! amany a re x tl2 Z 3i3 212 pe Oy als Banta “le Cc £4n \¢ Th | Figure 7. Equivalent parabolic earth diagram. represents the limiting case of an infinite earth’s radius. The plane earth diagram is widely used for problems of nonstandard propagation. In drawing diagrams for a curved earth of equiva- lent radius ka, it is customary to replace the spher- ical earth outline by an equivalent parabola (see Figure 7). The equation for the surface reduces from the circular form, re + (hs + ka)? = (ka)?, to the parabolic form, 1 9 hs SS Oka xs", for hs<< a ,. The height 2 measured from the surface of the earth, instead of from the x axis, is given by DS h=h,+ aie Ca. in which h is laid off perpendicular to the x axis and not to the earth’s surface. For clarity in drawing rays or field-strength diagrams, the vertical scale is expanded by an arbitrary factor p, whence 1 aes ; h=p (i. + ae ) : (7) This distortion of vertical distances, it can be shown, does not distort angles. The parabolic representa- tion to be reasonably accurate must be restricted to heights in the atmosphere small compared with the extent of the horizontal scale. sey Curvature Relationships The curvature of a ray is defined as the reciprocal of the radius of curvature p. Let yw be the angle between the ray and a nearly horizontal 2 axis. x A B Ficure 8. Angular relationships of rays. By Figure 8, p = — ds/dy, and since w is a small angle we may, to a sufficient approximation, put ds = dx, so that AS FACTORS INFLUENCING TRANSMISSION Here the curvature has been defined so that it is positive when the ray curves in the same direction as the earth; with this system the curvature of the earth itself is positive. Referring to Figure 8B, -=— = oS (8) 1/a, and since a is a small angle But do/dx = da da = dh da _ 1 d(a’) =—— + =a = és dx dh dx dh 2 dh Consequently, by equation (2) 1 1 d(a?) 1 dn SS oe == = ‘ - te dh (9) From this, the curvature of the ray is equal to the vertical rate of decrease of the refractive index. Notice that dn/dh is usually negative, so that the true curvature of a ray is usually concave downwards. A simple relationship exists between m = p/a, the ratio of the radius of curvature of a ray to the radius of the earth, and &. Combining equations (6) and (9) gives to iL (10) m Consider again the special case where dn/dh = constant, so that m is a linear function of the height (standard refraction). Consider the plane earth diagram of Figure 6. The angles between corre- sponding curves are the same as in the true diagram, Figure 4. Hence, for the plane earth diagram, equation (8) becomes 1/p’ = — da/dx, where p’ is the radius of curvature of the ray in the plane earth representation. It is readily found that are re ees Colas -(@+4) al p a 2 dh dh a (11) __iM, 107° = Ze dh ka Since V7 usually increases with height, the curvature of rays is concave upwards in this diagram. Again, equation (11) shows that when the modified earth’s radius ka is introduced (Figure 5), this amount of upward curvature is just canceled and the rays appear as straight lines. 4.1.6 Alternate Method Instead of taking account of refraction by chang- ing the earth’s curvature, another method is some- times more convenient. It may be shown that the ratio of the field F to the free-space field Eo trans- forms in the same way, whether (1) radius a is replaced by ka, or (2) the horizontal distance x is replaced by xk" ?’? and at the same time all elevations hare replaced by hk7!/*. An angle a must then be replaced by ak !/*. Method (2) is usually less con- venient than method (1) because it involves a change of horizontal distance which makes it necessary to transform the ratio E/E> rather than the field itself. In method (1) where only curvatures are changed, this difficulty does not appear as the distance x and hence £> remains unaltered. Method (2) may be used to advantage to account for deviations of k from the standard value of k = 4/3. Coverage diagrams are usually drawn for this value; the deviations owing to a change in k may then be estimated by multiplying distances, heights, or angles with the appropriate powers of k/(4/8). 417 Computation of Refractive Index The following equation gives the dependence of the refractive index on temperature, pressure, and humidity : (n — 1)- 10° = 2(» + te (12) where J is the absolute temperature, p is the total pressure, and e the water-vapor pressure, both the latter in millibars. Introducing W from equation (3), the modified refractive index, for use on a plane earth diagram, is equal to if = 2( + 2) + 0.157h, where the height 2 is in meters. If h is in feet, the last term is 0.048h. Tables have been prepared by means of which J can be computed rapidly from meteorological data, namely temperature, humidity, and pressure given as a function of height. For this purpose M is the sum of three terms which are computed independ- ently: (18) M=M,+4M,+ M,. (14) The dry term M, is obtained from Table 1 as a function of temperature and height in meters above the ground. (If the pressure at the ground pp is substantially different from 1,000 millibars all values of MW, should be multiplied by po/1,000. In TaBLe 1 A M, t(°C) h(t) h(m) —20 —18 —16 —14 —12 —10 —8 —6 —4 —2 +0 0 | 312.3 309.8 307.4 305.0 302 7 300.4 298.1 295.9 293.7 291.5 289.4 0.0 10 | 311.9 309.4 307.0 304.6 302.3 300.0 297.7 295.5 293.3 291.1 289.0 32.8 20 | 311.5 309.0 306.6 304.2 301.9 299.6 297.3 295.1 292.9 290.8 288.7 65.6 30 | 311.0 308.6 306.2 303.8 301.5 299.2 297.0 294.8 292.6 290.4 288.3 98.4 40 | 310.6 308.2 305.8 303.4 301.1 298.8 296.6 294.4 292.2 290.1 288.0 131-2 50 | 310.2 307.8 305.4 303.0 300.7 298.4 296.2 294.0 291.8 289.7 287.6 164.0 75 | 309.2 306.8 304.4 302.1 299.8 297.5 295.2 293.0 290.8 288.7 286.6 248.1 100 | 308.1 305.7 303.4 301.1 298.8 296.5 294.3 292.2 290.0 287.9 285.8 328.1 150 | 306.0 303.6 301.3 299.0 296.8 294.6 292.4 290.3 288.2 286.1 284.0 492.1 200 | 303.9 301.6 299.3 297.1 294.9 292.7 290.5 288.4 286.3 284.2 282.2 656.2 250 | 301.9 299.6 297.3 295.1 293.0 290.8 288.7 286.6 284.5 282.5 280.5 820.2 300 | 299.9 297.7 295.4 293.2 291.1 288.9 286.8 284.7 282.7 280.7 278.7 984.3 350 | 297.8 295.6 293.4 291.2 289.1 287.0 285.0 282.9 280.9 279.0 277.0 1,148.0 400 | 295.8 293.6 291.5 289.4 287.3 285.2 283.2 281.1 279.1 277.2 275:8° | 131220: 450 | 293.8 291.7 289.6 287.5 285.4 283.3 281.3 279.3 277.3 275.4 273.5 | 1,476.0 500 | 291.9 289.8 287.7 285.6 283.5 281.5 279.5 277.6 275.6 273.7 271.8 1,640.0 600 | 288.0 285.9 283.8 281.8 279.9 277.9 276.0 274.1 272.2 270.3 268.5 1,969.0 700 | 284.1 282.1 280.1 278.1 276.2 274.3 272.4 270.5 268.7 266.9 265.1 2,297.0 800 | 280.3 278.3 276.4 274.5 272.6 270.7 268.9 267.1 265.3 263.6 261.8 2,625.0 900 | 276.5 274.6 27250 270.9 269.0 267.2 265.4 263.7 262.0 260.3 258.6 2,953.0 1,000 | 272.8 271.0 269.1 267.3 265.6 263.8 262.1 260.4 258.7 257.0 255.3 3,281.0 1,500 | 255.0 253.4 251.8 250.2 248.7 247.2 245.7 244.2 242.8 241.3 239.9 4,921.0 2,000 | 238.3 237.0 235.7 234.3 233.0 231e0 230.4 229.1 227.8 226.5 225.3 6,562.0: —4.00 —0.40 +3.20 6.80 10.4 14.00 17.6 21.2 24.8 28.4 32.0 h(ft) iF) TaBLE 1 (Continued) M, h(m) t(°C) h(ft) =E0 2 4 6 8 10 12 14 16 18 20 0 | 289.4 287.3 285.2 283.2 281.1 279.2 277.2 275.3 273.4 271.5 269.6 0.0 10 | 289.0 286.9 284.8 282.8 280.8 278.9 276.9 274.9 273.1 271.2 269.3 32.8 20 | 288.7 286.6 284.5 282.5 280.5 278.5 276.6 274.6 272.8 270.9 269.0 65.6 30 | 288.3 286.2 284.1 282.1 280.1 278.2 276.2 274.3 272.4 270.5 268.7 98.4 40 | 288.0 285.9 283.8 281.8 279.8 277.8 275.9 274.0 PH PLN 270.2 268.4 131.2 50 | 287.6 285.5 283.4 281.4 279.5 201-9 275.6 273.7 271.8 269.9 268.1 164.0 75 | 286.6 284.7 282.6 280.6 278.7 276.7 274.8 272.8 271.0 269.1 267.3 248.1 100 | 285.8 283.8 281.7 279.7 277.8 275.8 273.9 272.0 270.2 268.3 266.5 328.1 150 | 284.0 282.0 280.0 278.0 276.1 274.2 272.3 270.4 268.6 266.8 265.0 492.1 200 | 282.2 280.2 278.3 276.3 274.4 272.5 270.7 268.8 267.0 265.2 263.4 656.2 250 | 280.5 278.5 276.6 274.7 272.8 270.9 269.1 267.2 265.4 263.7 261.9 820.2 300 | 278.7 276.8 274.9 273.0 PY Alea 269.2 267.4 265.6 263.8 262.1 260.3 984.3 350 | 277.0 275.1 PHP 271.3 269.4 267.6 265.8 264.0 262.2 260.5 258.8 1,148.0 400 | 275.3 273.4 271.5 269.6 267.8 266.0 264.2 262.5 260.7 259.0 257.3 1,312.0 450 | 273.5 271.7 269.8 268.0 266.2 264.4 262.6 260.9 259.2 257.5 255.8 1,476.0 500 | 271.8 270.0 268.1 266.3 264.6 262.8 261.1 259.4 257.7 256.0 254.4 1,640.0 600 | 268.5 266.7 264.9 263.1 261.4 259.6 258.0 256.3 254.7 253.0 251.4 1,969.0 700 | 265.1 263.4 261.7 259.9 258.2 256.5 254.9 253.2 251.6 250.1 248.5 2,297.0 800 | 261.8 260.1 258.4 256.7 255.0 253.4 251.8 250.3 248.7 247.2 245.6 2,625.0 900 | 258.6 256.9 255.2 253.6 252.0 250.4 248.8 247.3 245.8 244.3 242.8 2,953.0 1,000 | 255.3 253.7 2525 250.5 249.0 247.4 245.9 244.4 242.9 241.4 239.9 3,281.0 1,500 | 239.9 238.5 237.0 235.6 234.3 232.9 231.6 230.3 228.9 227.6 226.3 4,921.0 2,000 | 225.3 224.1 222.9 221.7 220.5 219.3 218.1 217.0 215.8 214.7 213.5 6,562.0 32.0 35.6 39.2 42.8 46.4 50.0 53.6 57.2 60.8 64.4 68.0 REFRACTION 49) 50 FACTORS INFLUENCING TRANSMISSION TaBLeE 1 (Continued) M t(°C) h(m)\ 99 29 24 26 28 30 32 34 36 38 40 h{ft) 0 | 269.6 267.8 266.0 2642 2625 260.7 2590 257.5 255.7 254.0 2524 0.0 10 | 2693 2675 2657 2639 2622 2604 2587 2572 2554 253.7 2521 32.8 20 | 2690 2672 2654 2636 261.9 260.1 2584 2569 2551 2534 2518 | 656 30. 268.7 2669 2631 2633 2616 2599 2582 2565 2549 2532 2516 | 984 10 | 2684 ©2666 ~=— 2648 «26306132596 «=—257.9 256.2 «= 2546 = 2529 =. 251.3 | 131.2 50 | 2681 2663 2645 2627 2610 2593 2576 2559 2543 2526 251.0| 1640 75 | 2673 2655 2638 2620 2603 2586 2569 2542 2536 251.9 2503 | 2481 100 | 2665 2647 2630 2612 2595 2578 2562 2545 2529 2513 249.7 | 3281 150 | 2650 2632 2615 2307 2580 2563 2547 2531 2515 2490 2483] 4921 200 | 2634 2617 260.0 2583 2566 2549 2533 251.7 250.1 2485 2469 | 656.2 250 | 261.9 2602 2585 2568 2551 2535 251.9 2503 2487 247.2 245.6 | 38202 300 2603 2586 2369 2553 2536 2520 2504 2489 2473 2458 2443 | 9843 350 | 2588 2572 2555 2539 2522 2506 2490 2475 2459 244 2429 | 1,1480 100 2573 42557 2540 2524 2508 2492 247.7 2462 2446 2431 241.6 | 113120 450 | 2558 2542 2526 2510 2494 2478 2463 2448 2433 2418 2403 | 1476.0 500 | 2544 2528 2W51l1 2495 2480 2464 2449 2434 2419 2404 2390 | 1.6400 600 | 2514 2498 2483 2467 2452 2437 2422 2407 2392 2378 2364 | 1.9690 700 2485 2470 2455 2439 2424 2409 2395 2380 2366 235.2 2338 | 22970 800 2456 2441 2426 2412 2397 2383 2369 2354 2340 2327 231.3 | 26250 900 2428 2413 2398 2384 2370 2356 2342 2328 2314 2301 2288 | 2.9530 1,000 | 2399 2385 2371 2357 2343 232.9 2831.6 230.3 2289 2976 2263 | 3,281.0 1'500 | 2263 2251-2238 = 2226 = -2214~=—«220.2- 2190 ITB-—-2NSGH ~©—«-NS.S.—«-214.3-| 4,921.0 2000 | 2135 2124 = 211321032092 = 281 ~—«- 207.1 -—«-206.0 205.0 +=. 203.9 ~—.202.9 | 6,562.0 86.0 896 932 968 1004 1040 896 932 968 1004 1040 hi(ft) t(°F) general this correction may safely be neglected un- less the difference between po and 1,000 is quite large, corresponding to an elevation of the ground level of several thousand feet above sea level.) The wet term M, is obtained from Table 2 as a function of temperature and relative humidity. Finally, MZ, = 0.157h, if his in meters, or M, = 0.048h, if his in feet, may readily be computed by means of a slide rule. M is then obtained by addition. 448 Atmospheric Stratification Ordinary weather data give comparatively little information about the atmospheric stratification near ground level. Radiosonde data are too widely spaced (vertical distances of the order of 100 meters between successive readings) for reliable determina- tion of the variation of M with height at low levels. Special instruments have therefore been developed in recent years for low-level soundings. Such instru- ments contain temperature and humidity measuring elements that are relatively free of lag; they are attached to airplanes or dirigibles or they are carried aloft by means of captive balloons or kites. The above tables are for use in connection with such measurements. Two main cases must be distinguished. First, the refractive index or M is very nearly a linear function of the height in the lower layers (at heights above about 500 to 800 meters the variation of refractive index with height will deviate from linearity only in very exceptional instances). This is the standard case where dM/dh is independent of h, and by using equation (6), k is conveniently obtained from the slope of the MM —h curve. It is found that the vertical temperature gradient has a comparatively small influence on k, while fairly small variations of the humidity gradient will affect k appreciably. The other case is that of nonstandard refraction. Here JM is not a linear function of the height. The most important special case is that of superrefrac- tion, which occurs when, in certain height intervals, M decreases with height instead of following the usual increase with elevation. Such a decrease of M in certain layers of the atmosphere is caused by a steep negative moisture gradient or steep positive temperature gradient, or even more by a combina- tion of both influences. With superrefraction, propagation conditions are greatly different from those encountered with standard refraction and the methods to be given later for the determination of the transmitted power REFRACTION 51 TABLE 2 M w r.h. rh t(°C) 10 20 30 40 50 60 70 80 90 100 t(F) —20 0.6 2 1.8 2.5 3.1 3.7 4.3 4.9 5.5 6.130 — 40 —18 0.7 : 1.5 2.2 2.9 3.7 4.4 5.1 5.8 6.6 7.309 — 0.4 —16 0.9 1.7 2.6 3.5 4.3 5.2 6.1 6.9 7.8 8.678 + 3.2 —14 1.0 2.1 3.1 4.1 5.1 6.2 ee, 8.2 9.3 10.287 + 6.8 —12 1.2 2.4 3.6 4.8 6.1 7.3 8.5 9.7 10.9 12.124 +10.4 —10 1.4 2.9 4.3 5.7 Hal 8.6 10.0 11.4 12.9 14.284 +14.0 —8 Tee 3.4 5.0 6.7 8.4 10.1 11.7 13.4 15.1 16.754 +17.6 —6 2.0 3.9 5.9 7.8 9.8 ILLEz/ oud 15.7 17.6 19.568 22 —4 2.3 4.6 6.9 9.1 11.4 13.7 16.0 18.3 20.6 22.871 24.8 —2 Pho 5.3 8.0 10.6 13.3 16.0 18.6 Zilles 23.9 26.589 28.4 +0 3.1 6.2 9.3 12.4 15.5 18.5 21.6 24.7 27.8 30.911 32.0 2 3.5 7.0 10.5 14.1 17.6 Pale 24.6 28.1 31.6 35.139 35.6 4 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 35.9 39.939 39.2 5 4.3 8.5 12.8 17.0 ZieES 25.5 29.8 34.0 38.3 42.533 41.0 6 4.5 9.1 13.6 18.1 22.6 27.2 31.7 36.2 40.8 45,280 42.8 7 4.8 9.6 14.5 19.3 24.1 28.9 Bont 38.5 43.3 48.175 44.6 8 5.1 10.2 15.4 20.5 25.6 30.7 35.9 41.0 46.1 51.215 46.4 9 5.4 10.9 16.3 21.8 27.2 32.6 38.1 43.5 49.0 54.394 48.2 10 5.8 11.6 17.3 23.1 28.9 34.7 40.5 46.2 52.0 57.793 50.0 11 6.1 12.3 18.4 24.5 30.6 36.8 42.9 49.0 55.1 61.270 51.8 12 6.5 13.0 19.5 26.0 32.5 39.1 45.6 52.1 58.6 65.094 53.6 13 6.9 13.8 20.7 27.6 34.5 41.4 48.3 55.2 62.1 69.010 55.4 14 7.3 14.6 22.0 29.3 36.6 43.9 51.2 58.5 65.9 73.169 57.2 15 7.8 15.5 23:3 31.0 38.8 46.5 54.3 62.0 69.8 77.505 59.0 16 8.2 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.9 82.060 60.8 M w r.h. 20 30 40 50 60 70 80 90 100 t(°F) 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.9 82.060 60.8 17.4 26.0 34.7 43.4 52.1 60.8 69.5 78.1 86.813 62.6 18.4 27.6 36.7 45.9 55.1 64.3 (ae) 82.7 91.866 64.4 19.4 29.1 38.9 48.6 58.3 68.0 Wath 87.4 97.127 66.2 20.5 30.8 41.1 51.4 61.6 71.9 82.2 92.4 102.71 68.0 PALES 32.5 43.4 54.2 65.1 75.9 86.8 97.6 108.46 69.8 22.9 34.4 45.8 57.3 68.7 80.2 91.6 103.1 114.56 71.6 24.2 36.3 48.3 60.4 72.5 84.6 96.7 108.8 120.87 73.4 25.5 38.3 51.0 63.8 76.5 89.3 102.0 114.8 127.53 75.2 26.9 40.3 53.8 67.2 80.7 94.1 107.6 121.0 134.44 77.0 28.3 42.5 56.7 70.9 85.0 99.2 113.4 127.6 141.74 78.8 29.9 44.8 59.8 74.7 89.6 104.6 119.5 134.5 149.39 80.6 31.5 47.2 62.9 78.7 94.4 110.1 125.9 141.6 157.34 82.4 33.1 49.7 66.2 82.8 99.4 115.9 132.5 149.1 165.62 84.2 84.9 52.3 69.7 87.1 104.6 122.0 139.4 156.8 174.27 86.0 36.7 55.0 73.3 91.7 110.0 128.3 146.7 165.0 183.32 87.8 38 5 57.8 ie 96.4 115.6 134.9 154.2 173.5 192.74 89.6 40.5 60.8 81.0 101.3 121.5 141.8 162.0 182.3 202.56 91.4 42.6 63.9 85.1 106.4 WAGE 149.0 170.3 191.6 212.86 93.2 44.7 67.1 89.4 111.8 134.1 156.5 178.8 201.2 223.50 95.0 46.9 70.4 93.9 117.3 140.8 164.2 187.7 211.2 234.63 96.8 49.2 73.9 98.5 123.1 147.7 172.3 197.0 221.6 246.20 98.6 51.6 77.5 103.3 129.1 154.9 180.8 206.6 232.4 258.23 100.4 54.2 81.2 108.3 135.4 162.5 189.6 216.6 243.7 270.80 102.2 56.8 85.2 113.6 142.0 170.4 198.8 227.2 255.5 283.94 104.0 IN 52 FACTORS INFLUENCING TRANSMISSION do not apply. This is especially true for the field near or below the optical horizon. The discussion of nonstandard propagation is beyond the scope of this volume, High-angle coverage is generally independent. of refraction and is therefore also unaffected by the variations of Jin the lowest layers of the atmosphere. ae Direct Determination of k In Figure 9 the reciprocal of k is plotted as ordinate against the vertical gradient of relative humidity as abscissa. The values shown refer to a standard temperature gradient of — 0.65 degrees Centigrade per 100 meters; unless the temperature gradient differs greatly from this value, the corresponding values of k will not be much affected. The curves given refer to 100 per cent relative humidity at 4.2 GROUND REFLECTION *21 Ground Reflection and Coverage The reinforcement of the direct ray by the ground- reflected ray is of great importance both in radar and in communication work. In favorable cases, the reflected ray may be of comparable intensity to the direct ray and thus the received intensity may be approximately doubled in places where the two rays have the same phase. This means, in many cases, the possibility of an appreciable increase in range relative to that in free space. 422 Complexity of Reflection Problem In order to facilitate the discussion of the problems encountered in reflection, it is necessary to analyze ground level, and an auxiliary table is provided the complex phenomena into simpler constituents. 12 (RH)g-30C -25C -20C -15C -10C_-5¢ _0C_+5C +100 +15C¢ +200 +250 +300 +350 +40 ; 10% 3039) 1055950 TOSOMIEUS RS LOL ORS OLN Seo WEN 20% -034 .049 .069 .080 .105 .134 .172 .214 .267 .329 .405 30% -030 .043 .060 .070 .092 .118 .151 .187 .234 .288 .354 0 40% -026 .037 .052 .060 .079 .101 .129 .161 .200 .247 .304 OK 35 50% -021 .031 .043 .050 .066 .084 .108 .134 .167 .206 .253 0 60% -017 .024 .034 .040 .052 .067 .080 .107 .134 .165 .202 BS 70% +013 .018 .026 .030 .039 .050 .065 .080 .100 .123 .152 9X25 80% -009 .012 .017 .020 .026 .034 .043 .065 .067 .082 .101 {5 90% -004 .006 .009 .010 .013 .017 .021 .027 .033 .041 .051 8 ——- 1 [=30 — =36 7h — 2 | zl |_> ~ = Le | ° (=) = 6+ = + | eee se | sed mt > | | w | = } EE | 4.0 rs a SCS) a | | | 3-2) —— == } Lx | a to} is | | : “ ze | | i id oa) Ea eel i | AP he | REL HUMIDITY GRADIENT 40° 5° 301 | 25° 20° aa % PER 100 METERS ————> camer emma rc ae eae a T- uay= Van=7'S m enrE Va [ESR UPY. [mat Ye Temes pe =P FIGURE 9. Graph 1/k versus RH (relative humidity) gradient and temperature for 100 per cent RH at ground. Add correction tabulated to obtain 1/k for RH at ground less than 100 per cent. at the top of the graph which gives figures to be added for other values of the relative humidity. The sensitivity of k to moisture gradients in warm climates is obvious from these data. First of all, the incident radiation field is resolved into nearly plane wave components, each forming a narrow pencil of rays striking the reflecting surface within a small area which we shall call the reflection GROUND REFLECTION 53 point. There are two types of such rays depending on their state of polarization. If the electric vector is parallel to the reflecting plane, the rays are said to be horizontally polarized and if the electric vector is parallel to a vertical plane through the rays, they are said to be vertically polarized. When consider- ing a very irregular surface, the reflected field may show extreme complexity even though the incident wave is linearly polarized. Increasing roughness may result in diffuse reflection which is ineffective in reinforcing the direct wave. The existence of diffuse reflection depends primarily on the size of the irregularities of the surface in comparison with the wavelength of the incident radiation and on the grazing angle of the incident field. This problem will be discussed in more detail later. a2? Plane Reflecting Surface Consider first the simplest case, when a plane wave strikes a plane surface such as that of an absolutely calm sea. The incident ray is then split into two parts. One is the reflected ray, which is returned to the atmosphere, and the other is the refracted ray, which is absorbed by the sea. At the point of reflection, the ratio of any scalar quantity in the reflected wave to the same quantity in the incident wave is defined as the reflection coefficient of the sea for plane waves of given frequency. Thus defined, the reflection coefficient can and will be different for the various components of the field. For simplicity, let us assume the reflecting plane to be the zy plane of a rectangular coordinate system, the xz plane to coincide with the plane of incidence, and the reflection point to be the origin of the coordinate system. PLANE OF INCIDENCE t REFLECTED RAY. LL i REFRACTED RAY N ——'| Figure 10. Geometry of reflection and refraction. For horizontal polarization, the electric vector for the incident wave then is jee Eye?! [¢-(1/c) (x cosy —z sin W)] uv (15) where y is the grazing angle, f the frequency of the radiation and ¢ the velocity of light in free space. The electric vector of the corresponding reflected field is given by the similar expression = Ey pei? ~J2nf [t-(1/c) (xcos y +zsin y)] (16) where p and @ are real constants. The ratio of the reflected to the incident field at the reflection point 2 = x = 0) is seen to be Boar (17) By definition this is the reflection coefficient for horizontally polarized waves. Thus the reflection coefficient is a complex quantity, the amplitude of which is the reflection coefficient of the wave amplitude and the phase is the lag in phase of the reflected wave with respect to the incident wave at the point of reflection. The reflection coefficient for vertically polarized radiation is defined in the same way. It is found, however, that the expression of p and ¢ in terms of the grazing angle y and the ground constants are quite different for the two types of polarization. For an arbitrary position of the plane of polariza- tion, the wave must be separated into its vertically and horizontally polarized components, and the proper reflection coefficient applied to each compo- nent separately. The quantities p and @ are determined by the boundary conditions for the electric vector at the reflecting surface, namely, that the tangential components of the electric vector on the two sides of the boundary surface shall be equal. This brings in the ground constants, that is, the conductivity and dielectric constant of the reflecting body. How these boundary conditions are applied may be illustrated by the simple example of horizontally polarized rays reflected from a surface of infinite conductivity. In the surface itself, the sum of the incident and reflected field strength must always be such that the currents set up in the body just suffice to produce the reflected field. Within a reflecting body of infinite conductivity, an infinitely weak field is sufficient, and hence the boundary condition is such that the reflected field, at the reflection point, shall be equal in magnitude and opposite in phase to the incident field, so that the resultant field is zero. Hence for infinite conduc- tivity and horizontal polarization R=—=1, p=1, = 180° (18) 54 FACTORS INFLUENCING TRANSMISSION eet! Fresnel’s Formulas For finite conductivity, the reflection coefficient may assume a variety of values. The general formu- las, as derived from electromagnetic theory, are given in equations (19) and (20). For horizontal polarization sin y — Vv €, — cosy (19) sinw + Ve, — cos’) ’ R= and for vertical polarization e,sinw — Ve, — cos? ; Vv (20) e.siny + Ve, — cosy’ R= where e¢, is the complex relative dielectric constant of the reflecting ground which is given by (21) eS Seb A material that acts like a good conductor for DIELECTRIC CONSTANT Ficure 11. low-frequency waves may act as an approximately pure dielectric for microwaves. The case e; = 0 is therefore of considerable practical importance. When e; = 0, and R consequently is real, the phase lag is 180° for horizontal polarization. For vertical polar- 2 14 1000 Mc Dielectric constant of sea water at 17 C. ization the phase change is 180° from y = 0 up to an angle Yo determined by tan po =1/Ve,. Here, the coefficient is zero. For larger angles the phase change is zero. As e, increases indefinitely, Yo approaches zero and for infinite «, the phase shift is zero everywhere, except for Y = 0 where it is indeterminate. The angle Y is called the Brewster angle. For y = 0 the amplitude is unity, and for y = 90° _vVve-1 [i ieee (22) for both cases of polarization. When e; is no longer zero, the amplitude p will show a deep minimum for a certain value of y instead of the zero found for e; = 0. The angle corresponding to the minimum is called the pseudo-Brewster angle. These various points are illustrated by examples in Figure 16. 16 30 42° The Complex Dielectric Constant of Water As much of the available radar and communica- tion equipment is either shipborne or erected along the coast, reflection from sea water is one of the GROUND REFLECTION 55 principal problems to be discussed here. For micro- waves the salt content in sea water makes little difference, so that it may be assumed that the dielec- tric constant and conductivity are the same over all oceans at the same temperature. With increasing temperature, the real part of the dielectric constant diminishes roughly by one unit per 5° C. Figure 11 gives the dielectric constant ¢, of ordinary sea water at 17 C asa function of frequency. The dielectric constant also diminishes with in- creasing salinity, but in the UHF-SHF region, normal variations of salinity have much smaller influence than changes in temperature and frequency. The imaginary part ¢; of the dielectric constant is, for frequencies less than say 1,000 me, related to the conductivity o as follows: €; = + 600d (23) (c in mhos per meter and \ in meters). At 25 C the average conductivity of sea water is usually given as 10O0Mc. ERTICAL ptt bl tet — Ie 20° 30 40° 50 60° 70° 80° 90° Figure 12. Amplitude of the reflection coefficient p versus angle of reflection YW for sea water. (From Radiation Laboratory Report C-11.) 4.3 mhos per meter. The temperature dependence is given by o = o25[1 + 0.02(¢ — 25) |, where ¢ is temperature in degrees centigrade. The conductivity of fresh water is much smaller. An average for inland lakes is ¢ = 10°? mho_ per meter. For wavelengths shorter than about 10 em, the dielectric constant is influenced by the fact that water is built up of polar molecules. The maximum effect is found for wavelengths of the order of 1 em. For this region, there is no appreciable difference between salt and fresh water. The probable run of e; for sea water is shown in Figure 11. The part of the e; curve extending from 6,000 me to 30,000 me corresponds essentially to results obtained at Clarendon Laboratory, Oxford. Other investigators give results which are markedly different. This curve is thus affected by consider- able uncertainties and should be used with caution. In Figures 12 to 15 are shown amplitude and phase of the reflection coefficient for a smooth sea for 166 9 140 | sole ee ARIZATION i Ficure 13. Phase of the reflection coefficient @ versus angle of reflection Y for sea water. Reflected wave E, lags incident wave E; by @. (From Radiation Labora- tory Report C-11.) various frequencies. Figure 12 gives the amplitude p of the reflection coefficient for both kinds of polarization, for reflection angles up to 90 degrees, and for 100 and 3,000 me. Figure 13 gives the phase @ under the same conditions. The next two figures give p and @ on a greatly enlarged scale of y, in the interval y = 0 tow = 5.5°, which embraces all cases of practical interest. 56 FACTORS INFLUENCING TRANSMISSION ToOMe a HORIZONTAL POLARIZATION 200Mq,_§ VERTICAL POLARIZATION = ° ’ s° fhe US 2g" gis) 3° SS at ais 5" 95° Figure 14. Amplitude p of the reflection coefficient versus reflection angle V from Y = 0 to W = 5.5° for sea water. ° VERTICAL POLARIZATION 3000Mc ° 160 i °| 120 26 =| a 0 se? 15° 2° 25° 3° 35° 4° 45° 5° 55° yr Ficurp 15. Phase @ of the reflection coefficient versus reflection angle VY from ¥ = 0 toW = 5.5° for sea water. Ey lags Ej by @. (From Radiation Laboratory Report C-11.) peed Overland Transmission Conditions over land are very different from those found over the sea. Land as a reflecting surface has larger irregularities and their effect is more pro- nounced. Therefore in selecting a radar site, it is preferable to choose a location which is surrounded by relatively smooth ground. The electrical proper- ties of the earth vary considerably for different localities, so that it is necessary to study the ground conditions for each particular case. Experimental data concerning reflection of very short waves from ice- or snow-covered ground seem to be lacking. Precise information might be of operational interest, particularly in Arctic regions. Laboratory exper- iments indicate that ground covered by ice or snow will influence the propagation of short waves some- what in the same way as very dry ground. 4.2.7 Conductivity of Soil Extensive investigations have been made on the conductivity of different types of soil, particularly on low and medium frequencies. For 10 me, the observed values range from 6-107? mhos per meter for chalk to 0.13 mhos per meter for blue clay. The conductivity increases with increasing moisture HORIZONTAL POLARIZATION VERTICAL AG POLARIZATION a \ VS Ficure 16. Amplitude of the reflection coefficient for moist and dry soil. content, so that marked seasonal changes may be anticipated for a given locality. It also varies with frequency. Under field conditions it will not be possible to measure the conductivity in individual cases and one will have to assume a value of about GROUND REFLECTION 57 10°? mhos per meter for poorly conducting ground like chalk or very dry soil and take a value of about 107! for good conductors like blue clay or water- bogged marshy land. Fortunately, the amplitude of the reflection coefficient is not very sensitive to minor changes in conductivity when the frequency is sufficiently high, say 200 me or higher. Then the real part of the dielectric constant is the most important factor. 428 Dielectric Constant of Soil It is not possible to give a standard table of dielectric constants of various types of soil, because the variation with the moisture content is consider- able. For very dry ground ¢, is likely to be about 4, but this value may rise to 25 when the ground is thoroughly soaked with water. The dielectric con- stant of ground will normally decrease with increas- ing frequency. Above 200 me, the dielectric constant will dom- inate the conductivity term, and for field conditions the ground may be assumed to be a pure dielectric. This is illustrated in Figure 16 for ¢ = 7; 6, =3 HORIZONTAL POLARIZATION _ i a MOIST |SOIL €,=25,€,=19.2 N \ §=7, €=3 ORY SOIL NG 120 6 40 x = \ N 1 SS 20 — ™~™ SS — =a 2 Ye | aap | eae ° 1o 20° 30 40° 50° 60° 70° 80° 90° Ficure 17. Phase of the reflection coefficient for moist and dry soil. and e; = 0; and for ¢«, = 25, e; = 19 and e; = 0. Except for values close to the Brewster angle, the zero conductivity curves give a usable approxima- tion. In Figure 17, the phase, @, of the reflection coefhi- cient corresponding to the above values of ¢, and e, is also given. bait The Divergence Factor The preceding considerations apply only to reflection from plane surfaces. For reflection from a sphere like the earth, the divergence of a bundle of rays is increased when it suffers reflection, and the plane earth reflection coefficient, R, must be multi- plied by a divergence factor denoted by D, which accounts for the earth curvature. This factor ranges from unity at close range where the earth can be considered plane to zero at points just above the tangent line. [Note: When the divergence factor approaches zero at grazing angles less than the last minimum, other components of the wave must be R r. : ‘ aS . aN Ficure 19. Reflecting pattern of an airplane. tion by terrain features, the main difference being that the angle of scattering is nearly 180 degrees instead of approximately 0 degrees. In the case of target diffraction, theory is less DIFFRACTION (GENERAL SURVEY) 59 useful than in many other problems of wave propa- gation. This is due to the fact that radar targets such as airplanes or ships have an extremely complex structure; the scattered intensity will therefore often change by many decibels as a result of only a small tilt of the target. Figure 19 shows a typical reradiating or reflecting pattern for an airplane. Numerous measurements of the average radar cross section of planes and ships have been made. A phenomenon of great importance in the micro- wave region is the scattering of radiation by water drops in the atmosphere. Small droplets such as are found in fogs and most types of clouds do not give reflection visible on the scopes. Only drops large enough to produce actual precipitation give appre- ciable radar echoes. However, this does not neces- sarily mean that rain is falling at the locality indi- cated by the scope; frequently vertical updrafts of air will maintain drops afloat that in still air would fall to the ground; moreover, drops falling from a comparatively high cloud can evaporate before reaching the ground. Especially in tropical regions, the last-named phenomenon is more common than is ordinarily thought. Chapter 5 CALCULATION OF RADIO GAIN 5.1 INTRODUCTION mi Objectives HIS CHAPTER is devoted to the definition and af renee of the various factors which enter into a computation of the field strength of radio waves propagated in the standard atmosphere above the earth. In Chapter 2, particularly in Sections 2.1 and 2.2, are given the basic definitions of path-gain factor and radio gain for transmission between doublets and other antenna types in free space. The present chapter shows how these quantities must be modified to account for the influences introduced by the curvature and electrical properties of the earth. The methods of computation are presented in considerable detail to enable the interested reader to apply them to his particular problem, and sample calculations are given which should assist in reducing to a minimum the time required for obtaining the answers in a given case. 5.12 Definitions Relative to Radio Gain The radio gain is defined as the ratio of received power Ps, delivered to a load matched to the receiver antenna, to transmitted power P;, with both an- tennas adjusted for maximum power transfer. For doublet antennas in free space this ratio is given by (3\/87d)? and is denoted by A,”, that is, IE 3h \? P, a ( ) a P, ——\8nd and the free-space gain factor is given by 3X Ao = (2 : 8rd ) in which d denotes the distance from the transmitter to the receiver measured in the same units as the wavelength 2. When the radiation is emitted and received by directive antennas and the propagation takes place through a refracting and absorbing atmosphere, and 60 reflection and diffraction effects of the ground are taken into account, the expression for the radio gain becomes a very complicated affair. For the general case of one-way transmission, the radio gain is given by 5 GiGeA*, (3) where (7; and G, are the gains of the transmitting and receiving antennas, respectively, and A, the gain factor, is equal to A=A,A, (4) with A, equal to the path-gain factor. [See equation (27) in Chapter 2.] For radar or two-way transmission, the radar gain is decreased because the energy traverses the path both ways and is influenced by the radiating proper- ties of the target as given by the radar cross section o. Combining equations (46) in Chapter 2 and (2) in this chapter, the radar gain equals Po 1670 1670 eG AtAs] = GG At. (5 poe (2) [Atty] , (S=) ©) 2 1 Comparing equations (3) and (5), it is seen that the gain factor A may be used also for two-way trans- mission, provided the additional term 1670/9)” is included in the formula. Later on it will be shown how to split up A and A, into a product of various factors, represented by graphs which make it possible to carry out computa- tions in specific cases. The gain factor A may also be expressed in terms of the field strength E and free-space field strength of a doublet transmitter Ho. From equation (28) in Chapter 2, E = Ey VGA). (6) Combining this equation with equation (4) A _ E 1 7 > > (7) Ay EovG, where E,VG; is the free-space field of the trans- mitting antenna with gain G4, INTRODUCTION 61 The free-space field, in terms of the transmitted power P), is given by ae V5 VP, — VGiE> = 3V5 VPi VG (8) for a point in the direction of maximum radiation. In terms of the power P» delivered to the load circuit of a receiving antenna, with matched load and oriented for maximum pickup, the field at any point in space is equal, from equation (17) in Chapter 2, to 8rV5 [Ps d Go It is sometimes convenient to express F in terms of the (radiation) field at one meter from the trans- mitter, whence Ey = E,/d and, from equation (6), I pas E i 7 VGAp: i = (9) (10) 513 Factors Affecting Attenuation and Gain The above definitions are quite general. In the absence of the earth, there remains only the free- space attenuation which results from the spreading out of the radiated energy as it moves away from the transmitter. At a distance which is several times larger than the wavelength, the field strength varies inversely as the distance from the antenna. The presence of the earth affects the field through two sets of quantities. One set is geometric and includes the heights of the antennas and their dis- tance apart, the curvature of the earth, and shape of terrain features. The other set is electromagnetic and depends on the dielectric constant and con- ductivity of the earth and of its atmosphere, the polarization and the wavelength of the radiation. vie Simplifying Assumptions The present chapter is mainly concerned with the computation of the field-strength distribution of a transmitter for certain idealized standard condi- tions, so chosen as to give a fair average picture of propagation conditions for very high-frequency radiation. The reasons for this limitation are stated in Chapter 1. In substance, the limitations are imposed by the great complexity of the general problem, which makes it necessary to proceed in successive steps. The first step is to consider propa- gation under standard conditions, which will be defined farther on. Successive steps take into account diffraction by terrain, that is, by trees, hills, mountain ranges, or shore lines, or by non- standard propagation effects in the atmosphere. The fundamental importance of a knowledge of propagation under standard conditions is first of all due to the fact that in a large number of cases condi- tions do not differ significantly from standard. On the other hand, when they do deviate significantly, the standard solution sets up a criterion for the discoy- ery of deviations and the evaluation of the influence of the nonstandard conditions upon propagation. The basic assumptions which define what we have been calling standard propagation conditions will now be given. 1. Standard atmosphere. It is assumed that the index of refraction of the atmosphere has a uniform negative gradient with increasing elevation. As has been pointed out in Chapter 4, the influence of such an atmosphere upon propagation is equivalent to that of a homogeneous atmosphere over an earth of radius ka, where k is a constant that usually is taken equal to 4/3. 2. Smooth earth. The earth is assumed to be perfectly smooth. It can be considered sufficiently smooth if Rayleigh’s criterion is satisfied, that is when the height of surface irregularities times the grazing angle (in radians) is less than X/16 (see Section 4.2.10). 3. Ground constants. The dielectric constant and conductivity of the earth are assumed uniform. For wavelengths less than one meter this assumption is particularly valid since in this case propagation is largely independent of the ground constants. In the VHF (1 to 10 m) range, the same is true with the important exception of vertically polarized radiation over sea water. For the VHF range, the assumption of uniform earth constants is unsatisfactory for paths partly over land and partly over sea water, or over sea water with large land masses near-by (see Chapters 8 and 10). 4. Doublet antenna and antenna gain. For the formulas of this chapter, the radiating system is assumed to be a doublet antenna (1.e., a straight wire, short compared to the wavelength). Actual antennas have radiation patterns different from that of a doublet, usually having greater directivity. The antenna gain of a half-wave dipole is 1.09 times (or 0.4 db greater than) that of a doublet, the field maximum being the same in the two cases. This 62 CALCULATION OF RADIO GAIN gain is insignificant in practice. For other types of antenna systems and for microwave frequencies, the gain may be many times larger. The propagation problem, thus limited, has been solved mathematically ; but the explicit mathematical formulas are far too complicated to be of much use to the practical computer. Much additional work has been done, however, to bring the solution into a form suitable for practical use. This involves reducing the computations to the use of graphs, nomograms, and tables, and it is this final stage of the problem which is the subject of subsequent parts of this chapter as well as of Chapter 6. als Curved-Earth Geometrical Relationships Let fy and hy denote the heights of transmitter and receiver above the earth’s surface, respectively, and let d denote the distance from the base of the transmitter to the base of the receiver, measured along the earth’s surface. For a number of cases concerned with high-frequency radiation over the earth’s surface, it is sufficient to identify the straight- line distance from transmitter to receiver with the distance d between the bases measured along the curved earth. But when path differences are of importance, as they are in interference problems of reflection and in diffraction, it is necessary to com- pute distances to a higher order of accuracy. Throughout this chapter, the earth will be assumed to have the equivalent radius ka, and the atmosphere to be homogeneous, and radiation to travel along straight lines. The straight line from the transmitting antenna and tangent to the earth’s surface (the so-called line of sight) touches the earth along a circle which constitutes the radio horizon of the transmitter. The distance measured along the earth from the transmitter to the radio horizon will be denoted by dr; and the horizon distance of the receiver by dp. These geometrical illustrated in Figure 1. From this figure, it follows that ka cos (dp/ka) © relations are kath = (11) Inasmuch as ka nee ka cos (dp/ka) 1 dy? = ka 5) e 2 ka te - (d/Iea)? equation (11) assumes the form h = Z aie a (12) or dp = V2 kah,. (13) Similarly, the horizon distance of the receiver is dp = V2 kahy. (14) The sum of the two horizon distances is given by dz, where dy, = dp a dp. (15) °° Optical and Diffraction Regions The points visible from the transmitting antenna (on an earth of equivalent radius ka), i.e., the points above the line of sight, constitute the optical region (Figure 1). The rest of space lies beyond the trans- mitter horizon and below the line of sight and is called the shadow or diffraction region. OPTICAL REGION OPTICAL HORIZON TRANSMITTER EINE OfesIGHT babel ES SHACCW REGION Figure 1. Geometry for radio wave propagation over curved earth. It is frequently necessary to know whether a receiving antenna lies in the optical region or the diffraction region of a given transmitter. This evidently is equivalent to knowing whether the distance d of the receiver from the transmitter is smaller or larger than the combined horizon distance dy. By equations (13), (14), and (15), it follows that in the optical region d < V2ka (Why + Vhe), (16) and in the shadow region d > V2ka (Wh, + Vho). (17) INTRODUCTION 63 A graphical representation of the equation dy = V2ka (Vk, + Vig) (18) is given in Figure 2. For k = 4/3, a = 6.37 X 10°m, this takes the form dy, = 4120 (Vi, + Vig) meters, (19) where hy, he are given in meters and d; in meters. h, METERS du km ha METERS fe) ° fe) 25 es 100 100 200 Ise 200 300 380 400 600 200 600 800) 800 1000 1000 300 2000 2000 400 3000 3000) 500 4000 4000 5000 600 5000 6000 6000 7000 700 7000 8000) 8000 9000 800 9000 10000 10000 900 15000 1000 15000 1100) 20000 20000 1200 d. =4.12 (hy +{Ma)= de+dp Figure 2. Sum of transmitter and receiver horizon distances for standard refraction. To change scale: Divide h; and hz by 100, and divide dz, by 10. 517 Nature of the Radiation Field in the Standard Atmosphere The mathematical solution for the radiation field takes various forms for particular cases. The treat- ment for low antennas, for instance, differs from that for high antennas, and similarly the equations must be handled differently for the two types of polarization. These and related problems are dis- cussed in general in the following. 1. General form of field variation. The mathemat- ical expression for the radio gain of the radiation field of a doublet under standard conditions is given as the sum of an infinite number of complex terms or modes. (See Section 5.7.6.) Disregarding the phase factor, a representative term (mode) of this series: has the form F(d) « fi(hi) « fa(ha) . These modes are attenuated unequally. Well within the diffraction region, the first mode contributes practically all of the field so that the effects of dis- tance and height are separable. In this region, the problem of numerical computation is simplified, since it is possible to use separate graphs for the dependence on height and distance. As the receiver is moved toward the transmitter, the number of modes required for a good approximation increases. For low antennas, the addition of the modes is practicable and the graphical aids are useful for short distances. These conditions are illustrated in Figure 3 for horizontal polarization or ultra-short waves. In the optical region, the methods of geometrical optics give a result equivalent to that of the rigor- ous solution at points which are not close to the line: of sight. The field is then the sum of a direct and a reflected wave, resulting in an interference pattern. The preceding discussion is illustrated by Figure 4 which shows the variation of field strength with distance for fixed antenna heights, for propagation over dry soil with a wavelength of 0.7 meter on vertical polarization. The numbers refer to the number of modes required for a better than 99 per cent approximation. The interference pattern is illustrated by the oscillatory nature of the curve. It will be observed that beyond the first maximum, the points found by geometric optics give a value of the field which is slightly too low. (See dots in Figure 4.) In fact, as the line of sight is approached, the optical formula approaches zero whereas the exact solution does not. The geometric-optical method breaks down in the optical region as the line of sight is approached. It may be noted that Figure 4 has been drawn for k = 1 rather than for the cus-- tomary value of k = 4/3 corresponding to standard atmosphere conditions and is for a hypothetical! isotropic radiator. (20) 64 CALCULATION OF RADIO GAIN TRANSMITTER Low OPTICA L-INTERFERENCE REGION LINE OF SIGHT TRANSITION TRANSMITTER ELEVATED OPTICAL-INTERFERENCE REGION LINt OF SIGHT TRANSITION REGION Figure 3. Field regions as related to transmitter heights for horizontal polarization or ultra-short waves. Low antenna region = Ad > 2hiho; hi, he 150 140 < 30A2/3. 130 107 N10 aoe DUE TO MINIM 3 E OF SIGHT = 100 x 10° = 5 «90 Ge i w TF oi 5 = = 4 $0 10* > 3 x ar w 70 w 3 hizh>=!00 m Q meh DRY o =10°2mhos/m 103 = =4 3 o peu rs * OPTICAL FORMULA a e NES X DIFFRACTION FORMULA z Z 50 k= - + ff 1 KW OUTPUT VERTICAL POLARIZATION 2 29 ISOTROPIC RADIATOR is) 30 | 7 -Oly Al Ficure 4. Variation of fie i) 2 10 100 1000 d IN KILOMETERS d strength with distance for propagation on vertical polarization with a wavelength of 70 em over dry soil. The point “due to minimum” results from minimum in reflection coefficient at the pseudo-Brewster angle. INTRODUCTION 65. If the earth were flat and perfectly reflecting, the envelope of the maxima of the curve in Figure 4 would coincide with the line 2K, twice the free- space field, corresponding to the in-phase addition of the direct and reflected waves. An envelope of the minimum points would be # = 0, corresponding to the destructive interference of the direct and reflected waves. The curvature of the earth, resulting in increased divergence of the waves (see Section 5.2.5), and the lack of perfect reflection (see Section 5.2.4) cause the maximal and minimal envelopes to differ from 2H and 0, respectively. In the neighbor- hood of the first maximum in Figure 4 (1.e., when the direct ray makes small angles with the earth), the reflection coefficient tends to be unity in magnitude for both polarizations except for the increase in diver- gence which results in the deviation of the maxi- mal and minimal lines from 2# and 0, respectively. At a smaller distance, for vertical polarization, as shown in Figure 4, the deviation is caused principally by the smaller magnitude of the reflection coefh- cient. The virtual meeting of the maximal and minimal lines corresponds to the minimum value of the reflection coefficient at the pseudo-Brewster angle. (For horizontal polarization at small dis- tances, the envelope of maxima would virtually coincide with 2) and the minima would be closer to zero. As the distance is increased, the difference between the envelopes for vertical and horizontal polarization gradually decreases.) 2. Both antennas low; h < h,. In a discussion of the height function, it is convenient to distinguish between high and low antennas. The critical height separating the two cases for horizontal polarization or ultra-short waves is given by he = 30”? meters (21) where } is expressed in meters. For \ = 0.1 meter, h, = 6.46 meters, and for \ = 10 meters, h, = 139.5 meters. If both antennas are at elevations less than h,, the height-gain functions f(h), to a first approx- imation, are the same for all the modes, so that the complete solution fiChi) + filha) + Pi(d) + fo(ha) + fo(he) - Po(d) + + + can be written in the form ile) oUa))o Cia ty 3058 Vy f(a) + fe) F@), (23) where f(h1) replaces fi(h1), f(hz) replaces fi(he), ete., while F(d) stands for the sum F, + FPF, +--+. (22) or The distance function F(d) can be calculated for particular cases. This has been done for high fre- quencies and is represented graphically in Section 5.7, the results being valid for low antennas for all distances in the optical as well as in the diffraction region such that 2hih, << dd (see Figure 3). The condition 2/yh, << dd assures that the antennas are below the interference pattern. At the ground, f(h) = 1, so that if both antennas are close to the ground, the distance dependence is given by F(d) only. 3. One or both antennas elevated; h > h,= 30?’?. For elevated antennas, h > h, and the height-gain functions of f(h) vary with the modes. Conse- quently, it is not possible to separate the height and distance effects as in the previous paragraph. In the optical-interference region, it is more ad- vantageous to use the method of combining the direct and reflected waves. This is equivalent to the rigorous solution which is illustrated by the dots in Figure 4. Simple graphical aids can be given for points well within the diffraction region where the first mode predominates. The range of usefulness of the first mode can be extended by plotting the field strength given by the first mode as a function of height (or distance) and plotting a similar curve by using the ray method as far as the lowest (or first) maximum (see Figure 7). Then by joining these partial curves into a smooth overall curve, a fairly good value of the field can be obtained for intermediate points. There is a further possibility occurring with the transmitting antenna clevated, the receiver low and lying below the interference pattern, and the dis- tance short. In this event, none of the previous methods apply. However, the reciprocity principle (see Chapter 2) can be applied to find the radio gain at the receiver by interchanging the role of receiver and transmitter. Suppose the original transmitter height is 100 meters, the original re- ceiver height 15 meters, and the wavelength 1 meter. Now let the transmitter height be 15 meters. If the receiver height is low (hz < 30 meters), values of the gain can be found (Section 5.7); if the receiver height is in the interference region, the gain can be found by the ray method. Now suppose a curve be drawn for these results, giving the attenuation versus receiver height. From this graph, the value of the gain factor A at h, = 100 meters can be read. This value of A by the principle of reciprocity is the gain factor for the original heights. ‘66 CALCULATION OF RADIO GAIN 4. Ultra-short waves in the diffraction region. Dielectric earth. For \ < 10 meters (f > 30 mc) and for either polarization, land acts as a dielectric earth or absorbing earth in contradistinction to a conducting earth. Propagation over a dielectric earth is practi- 2 =2i 1 ! o=!0 mhos/m = €r=4 0.5 = Pa b= 100m fe) tN! =Om val K21 or] ~9O| = 0.05 i cE = E, | *! 0.01 + i + f 0.005 | £ 27m 0.001 = + 0.0005 if 2:0}! =7mMA=7em A=0.7m 0.0001 : 10 20 30'40 50 60 70 80 90 100 d IN KILOMETERS —> Ficure 5. Field strength ratio versus distance for vertical polarization over dry soil for h; = 100 meters and hy = 0. 2 o= cea | €,=4 hizhs00 sei fe) 10 20 30 40 50 60 70 80 90 100 » d IN KILOMETERS Ficure 6. Field strength ratio versus distance for vertical polarization and heights h; = he = 100 meters. cally independent of earth constants. For a given type of polarization, the chief variables affecting gain are then the heights of the antennas, their distance apart, and the wavelength. Within the diffraction region, the effect of increasing wavelength is to increase the field strength. This is illustrated by the curves in Figures 5 and 6. The dielectric earth is characterized by a value of 6 >> 1. 6 is given by equation (193). While sea water has a relatively high conductivity, radio wave propagation over it is the same as that over a dielectric earth in the case of horizontal polar- ization for \ < 10 meters, and in the case of vertical polarization for \ < 1 meter. Consequently, verti- cally polarized radiation of wavelength range 1 to 10 meters over sea water is given special treatment in Section 5.7.4. In the same range, 1 to 10 meters, for vertically polarized radiation and for distances less than those given in Table 3, propagation conditions over land also deviate slightly from those corresponding to a dielectric earth. 5. Optical region. In the optical-interference region, the lobes for the shorter waves are more numerous, narrower, and lower, as can be seen from the oscillatory part of the field strength versus distance curves of Figure 6. The dependence of reflection coefficients upon polarization, wavelength, and ground constants is discussed in Section 4.2. 6. Horizontal versus vertical polarization. In the optical region, for rays at small grazing angles, there is not much difference between the two types of polarization. For larger grazing angles, the differ- ence is more marked (see Section 4.2 on reflection coefficients, and see Section 5.2.4). FIRST MAXIMUM SHADOW ZONE nw OPTICAL REGION —— ELEVATED ANTENNA 20 LOG ( oot hy)+8 (APPLIES ONLY WELL BELOW LINE OF SIGHT) rie / 20 LOG gy ala oO fo} 4 4 : Sls o|o Zziw | dijo LOW HS ANTENNA | | | 2 | hg= 30A 13 i | B= RADIO GAIN AT noe B | FOR £,SEE TABLE 4 | | ow ue hp IN METERS Ficure 7. Gain versus height at distances beyond the radio horizon. It has been pointed out in the previous paragraph that within the diffraction region for \ < 10 meters PROPAGATION FACTORS IN THE INTERFERENCE REGION 6 N and propagation over land, there is practically no difference in intensity between a horizontally and a vertically polarized radiation field. For \ < 1 meter there is, similarly, no difference for propagation over sea water. When there is a difference, as for low antennas, horizontal polarization gives a smaller gain; but as the antennas are raised, the two cases approach equality (see Section 5.7.4). okt PROPAGATION FACTORS IN THE INTERFERENCE REGION 524 Propagation Factors The factors affecting gain in the region where the methods of geometrical optics may be applied are discussed in Sections 5.2 to 5.5 inclusive. 5.2.2 Spreading Effect From the formula of equation (1) in Chapter 2, for the field intensity components of the radiation field, it follows that for distances from the transmit- ter large in comparison to the wavelength, the domi- nant term falls off inversely as the distance from the transmitter, or E E Sia (24) where F; is the field strength at unit distance. This means that the power per unit area in the radiation field varies inversely as the square of the distance. This spreading effect is the consequence of the fact that the energy of the wave is distributed over larger and larger areas as the wave progresses away from the transmitter. Sek) Interference When a wave travels over a conducting surface, constructive and destructive interference occurs between the direct wave from the transmitter and the wave reflected by the surface. This is illustrated in Figure 8, which is drawn for a plane earth. If there is no energy lost in reflection, the direct and reflected waves are of equal intensity, and their resultant varies from zero to twice the free-space value, depending upon the phase difference between the two components. The reflected wave lags the direct wave by an angle 6 + ¢, where 6 is the phase retardation caused by the greater path length Figure 8. Geometry for radio propagation over a plane earth. traversed by the reflected wave and @¢ is the phase lag occurring at reflection. Figure 9 shows the vector diagram for the case where the phase shift at reflection is @ = 180 degrees. Er Ficure 9. Vector diagram showing the addition of the direct and reflected waves for@ = 180° and p = 1. This condition holds for horizontally polarized radiation of frequency above 100 megacycles, re- flected from sea water at grazing angles of less than 10 degrees. The resultant electric field is equal to B =) Ee? + EP — 2E.E, cos 6 oo | = 4) (Eo — E,)? + 4B 0B, sin? (25) 2 . If the reflection is complete, as from a conductor of infinite conductivity, E, =, E = 2E, sin : (26) aa Imperfect Reflection In general, the strength of the reflected wave E, is less than that of the incident wave E;, partly be- cause of diffuse reflection and partly because some energy is refracted into the surface and absorbed. 68 CALCULATION OF RADIO GAIN Furthermore, the phase lag usually differs from 180 degrees, depending upon the frequency and grazing angle. This is especially true for vertical polarization where the reflection coefficient is a critical function of both the grazing angle and fre- quency. The ratio _ Er ie 7 R E pe (27 ) is a complex number and defines the reflection co- efficient R, which has an absolute value p and a phase angle @. In equation (27), a lagging angle is considered positive. Writing @ = 7+ o’, equation (27) may be expressed as Pie i, = pe aC) ae Ei; In equation (27), the lag angle ¢ is measured with respect to zero-degree phase shift at reflection. For horizontal polarization, @ varies from 180 degrees to 183 degrees, and from 180 degrees to 3 degrees for vertical polarization at 3,000 me over sea water. In equation (28), lag angle ¢’ is measured with respect to a 180-degree phase shift (that is, from E, reversed), and varies from 0 degree to 3 degrees for horizontal polarization and from 0 degree to —177 degrees for vertical polarization. The resultant field intensity is iF = 1B, ae pEye ** a) oat Ries ee = Ey(1 — pe ce Ey mL (28) (29) where Q=6+¢'=54+6-7. The absolute value of the received field | E| is given by | B| = Boa] — pe) (1 — po) |E|= Ey A 1 + = 20 cos @ (30) Equation (30) shows that the received field intensity has a maximum of 2) when =1 p ) een ye ep The value of # is a zero when p=1, QO = 217: (32) In equations (31) and (32), 7 includes all integral values and zero. Equation (30) may be written as E B= Ela pet dos’, (33) where £ is the field at distance d from the trans- mitter, and F, is the field strength at unit distance. From equation (33), Ey - a d= la — p)? + 4p sin’ 5 : (34) In free space where there is no reflecting earth, p = 0, and (35) where dy is the equivalent free-space distance from the transmitter at which the field strength EH would be found. Hence equation (34) may be written in the form d= ala = p)? + Ap sin? 5 (36) a2 Divergence The divergence factor D is introduced to account for the decreased gain produced by the spreading Increased divergence resulting from re- flection by a sphere. Ficure 10. of a wave reflected from a spherical surface. Re- ferring to Figure 10, Bs _ [dos _ [dt don _ 1% [dan — "0, Ey das N day daz; "3 N da; "3 (37) GENERAL SOLUTION 69 In calculating the field intensity reflected from a spherical earth, the inverse distance attenuation factor 1/rg used for the direct wave must be multi- plied by the divergence factor D, which is always less than unity. As a result of this divergence the reflection coefficient for a spherical surface is less than that for a plane surface as given by pD = p’ (38) where p’ is the spherical! earth reflection coefficient. Equations (30) and (36) may then be written as |Z|=£) Ja — p')?+ 4p’ sin? (39) and d= dy “| (1 — p’)? + 4p’ sin?—- (40) Dols 526 Antenna Gain and Directivity The effects of antenna gain and directivity are expressed by means of the gain factor G, defined in Section 2.2.2, and the antenna pattern factors Fy and F», which are the fractions of the maximum radi- ation amplitude in the direction of the direct and reflected rays respectively. The maximum ampli- tude for a transmitting antenna with gain G is Ea NG Ey, where Ey is the free-space field strength radiated by a doublet. When the antenna pattern factors are taken into account, the resultant field intensity, following equation (30), is E = VG, Ey V Fe + F2p2D? — 2F\FopD cos 2. (41) HORIZONTAL — — REFLEC TED Ray = EARTH TTTTTTTTT Fieure 11. Antenna pattern factors Ff; and F». The factors F,; and F, are functions of the angles ¥1 — a and y+a which the direct and reflected rays make with the axis of the beam. Figure 11 is a typical antenna pattern. 5.3 GENERAL SOLUTION 53.1 Generalized Reflection Coefficient Equation (41) may be simplified by introducing a generalized coefficient which includes the effects of reflection, divergence, and directivity. The ampli- tude of this coefficient will be denoted by K and is given by ie K =—pD. 42 Pr? (42) Substituting fF. = F,K/pD in equation (41) gives E =VG\F,\E) V1 + K?—2Keos2 (48) or nA r\9 pop NY i = VEGF, Bun| 1 — K)?+ 4K sin? oe (44) ii If 608 lp 1 eles sel be EH I (I-K)° +4k Sin’ oO i Ficure 12. (1 — K)? + 4K sin? 2 as a function of 2 K and Q. If the transmitting antenna is pointed so that the direct ray lies in the direction of maximum gain, F, = 1, and EB =VGiE a (ako? ein? ; (45) and d = VGid va — K)? + 4K sin? 5 : (46) 70 CALCULATION OF RADIO GAIN Figure 12 shows Va — K)' + 4K sin? ; as a function of K for various values of sin 2/2 and may be used to calculate E. The value of G, to be used in equation (46) may be found from the antenna specifications for a given set. The free-space field Hy at distance d from a transmitting doublet with power output P; is equal to 3V5 VP, Eo SS ee (47) d 4 Ww e Ww = a S a w > ® 80 a o a Zz 60 uw cael eh L} 10? 10 50 100 500 1000 5000 10,000 d IN METERS Ficure 13. Free-space range d as a function of field strength EH) and transmitter power P,. Figure 13 shows Ep in decibels above 1 microvolt per meter as a function of d for various values of transmitted power. The free-space field at distance of d meters expressed in decibels above 1 microvolt per meter for P; watts radiated is Decibels = 20 logio al (48) 10° °d 5.4 PLANE EARTH 541 Use of Plane Earth Formula It will be shown in Section 5.5.5 that under cer- tain conditions calculations based upon the assump- tion of a plane earth yield satisfactory results. Cal- culations for propagation over a plane earth are given in this section. 5.4.2 Path Difference It follows from equation (29) that when p = 1 and @ = 180 degrees, the received field depends only upon the phase lag caused by path difference. Referring to Figure 8, rel @+ Ca— ht = anf + (oy, 1 hy = 2) 1 (" a _) | (eres ae ‘| eal d eda a (49) Ve + (hy + In)? = agit (BEhy, i a) a) ] 1} 1 = ere || il +3( d Scares a ll ll (50) Qhyho Iyho(h? 4 he?) | s=r-n=al — +...]; ; & di an | h?? + he? | Le gee 51 5 pe ta (51) If hy? a he ae ZQhyhz A=> = q (52) The phase lag caused by the path difference A is equal to a 22( 2h) = Salata (53) A\ d dd where \ is the wavelength of the radiation. 5.4.3 Field Strength Equations When p = 1 and @ = 180°, equation (26) may be used. Substituting equation (53) for 6 into equation (26) gives = onan Gam). (54) Ad If 6/2 < 10°, sin (6/2) — 6/2 and 4ahyhs i ye 55 oer (55) When hy or hy equals 0, equation (55) indicates that the received field intensity is equal to zero, which is contrary to fact. For this case the diffraction field must therefore be calculated and included as ex- plained in Section 5.1.7. In the general case (p < 1), equation (44) may be applied with K = (F2/F,) pD. A refinement may be added to the calculation by taking into account SPHERICAL EARTH 71 the fact that the image source (Figure 8) of the reflected wave is at a distance r + A from the re- ceiver. The reflected wave is attenuated more than the direct wave, according to the free-space attenua- tion ratio (r + A)/r. If this is taken into account the ordinary reflection coefficient is replaced by ka m( z ) pp. Fy r+aA The correction is not necessary when 2hih, << d@ [see equation (52)]. (56) 5.5 SPHERICAL EARTH Sool Measurement of Distance The difference between the slant range rz and the distance measured along the surface of the earth and designated by d in Figure 14 is usually negligible. For a transmitter height of 1,500 meters, the error in assuming rg = d is 0.04 per cent at a distance of 161 km and height of 6,900 meters, and 1 per cent at the same distance but at a height of 22,500 meters. As the transmitter height is increased, the error is increased. In order to express the slant range rz in terms of the curved distance d to a higher order of accuracy, the cosine law is applied to the triangle, transmitter- receiver-earth center. This gives the equation Te = (ka + hy)? — (ka + he)? 1 — 2(ka + hy) (ka + he) cos (4) : ka Selecting the relatively important terms of the order @Phyh, and d*(hy + hz), as well as powers of d higher than the fourth, the above equation reduces to ad? a ) _ (87) 12ka rr = 1P + (he — hy)? + = @ + ho J Solving equations (13) and (14) for A; and hz gives ka o3 Equivalent Heights hy — aa 2ka 2 hy = art . 2ha These results may also be expressed by saying that the distance from the surface of the earth to a plane which is tangent at a distance d; from the trans- mitter is d;?/2ka. ko Ficure 14. Geometry for radio wave propagation over a spherical earth. (Vertical dimensions greatly exaggerated.) “I bo CALCULATION OF RADIO GAIN Hence for a transmitter of height h; above the ground the height above the tangent plane at the reflection point, the so-called equivalent height is, to a first approximation, dy, 2 hy = hy — (58 1 AE ak a \ ) and for the receiver the equivalent height is dy? Ti => ho = —- 59 2ka oo) The equivalent heights are shown in Figure 14, which illustrates the geometry of the spherical earth having an effective radius of ka. 5.5.3 Angles Referring to Figure 14 and remembering that the angles are greatly exaggerated in the figure, it is seen that ei SS = SS 60 ¥ dy, ds, ( ) lis d tan y = —- —_, 61 Wee ie OVE hy — hy d tan = ——_ —- —_ , 62 ¥a d 2ka 2) 1 ypeyts (63) ka Angle y must be evaluated in order to determine the reflection coefficient. Angles ¥z and v determine the antenna pattern factors Fy and F., which are shown in Figure 11. The angle y is significant in coverage calculations and angular approximations. 55-4 Determination of Reflection Point (d,) Inasmuch as several equations of Sections 5.5.2 and 5.5.3 depend upon dj, it is necessary to be able to determine this distance when the transmitter and receiver heights are given and the distance between them is known. Let = 5 (+d), (64) and m="FMq4o, (65) so that d d, =-—(1—b), (66) 2 ond oe as Gey (67) where b= fie ; (68) d, + dy and ay F (69) hy + he Assume hy > hy and d; > ds, so that b and ce will always be positive. This is always possible because of the principle of reciprocity. From equation (60), hy’ he! hy dy hy ds =— or = =—— F (70) d, ds d, 2ka ad, 2ka Substituting for d; and d, from equations (64) and (66), hy + he (1+ *) _ d+) d 1+ 56 Aka white ta (l= cy, db) d ( —b 4ka Simplifying, hi th, 2(¢e—b)_ bd d 1—b? ka Solving for c, c=b+bdm(1 — 6’), (71) where d? re oe (72) 4Aka(hy + he) To determine d;, equation (71) must be solved for b. This is a cubic equation, which is easily solved when 7 is small in comparison to unity. However, for m values comparable to unity, or larger, it is easier to plot a series of curves showing ¢ as a func- tion of m for assigned values of b ranging from 0 to 1. These are straight lines with a slope of b(1 — Bb?) and are given in Figure 15. The procedure for calculating d; when fy, he, and d are given is as follows: (ae ho 1. Compute ¢ = ‘ : hy — ho SPHERICAL EARTH 73 dz 2. Compute m = ——____.. Aka(hi + he) From equation (70) : hy = sd = he tae dl —s)d 3. Read b from Figure 15. ad Qa i — sd Sean d 4. Calculate d, = 5 (1 + 6). Sumpliyine: | It should be noted that hi may be the height of — .s _ 3 ee o| a, Lips +] ans kaha 0. (74) either the transmitter or receiver. The only re- 2 ?? 2 a 10 10 0.90 0.80 = 0.70 ) - <= ss i 0.60 0.8 eae) Este 0.30 ah ce ae La 07 eee = 0.40 [es eee c = = 0.30 b 05 Es iS —= 04 a za a Lear Ieee aes eet eal ae fae ee are Lee | ake = a ae = ae | 0.3 oe el ee ae i =] eee Ze2on08 ee eee ca 0.2 a Se + 0.1 Jie) i =F 0.1 et IL al fe i fF ie =i sia =I ie) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a8 0.9 1O m Ficure 15. Graph for obtaining 6 for given values of ¢ and m. strictions are that hi > he and d; represents the distance from hi. Another method will now be given for calculating d;. This method will be particularly useful when d,/d << 1 and will be applied in Section 5.5.8 on generalized coordinates. Let them 6> iy, so that the reflection point is much closer to the transmitter than to the receiver, a good approximation to A is obtained by replacing hy’ by hy and hy! by hy — d/2ka in equation (78). A (Ap)max = 5.33 X 10* aa ——.0 20 15 ie 1a he/hyzto | 47 | t | ae | = i = 6 a5 a 1 1 = S gre Eee [ | 3 | 20] | { 2 1 Lo he/hn2 4 = a dod. d. 2xalh+{h, ah 4 4 1 4 4 n n io ° a 2 3 A S 6 a 2 3 70 Figure 16. Plane earth correction factor vant , (Radiation Laboratory.) dy, Az 72 (hy = a d 2ka which, to the same approximation, means that A = 2hi tan +. Then (82) In general, equation (82) is an improvement over the plane carth approximation except close to the transmitter and at low heights where fz is not much greater than /y. SPHERICAL EARTH 75 The dependence of path difference upon distance and height may be seen by considering the path difference parameter _ kad eG Since ho! = hy'd2/d, it follows from equations (78) and (59) that (83) (i) ne Dhy’ he! _ 2(h;!)2de - 2Qdo hy? Qkahy/ . d dd, d dy Hence, using dy? = 2kahi, _ ht? () [1 — (di/dr)*2 ee dr \d Cite and pat ye Ua h/dr) d d,/dp a ( a ae) [1 — (d/dr)??? (85) d/dy d,/dy i The form of this expression suggests the introduction of two new dimensionless parameters i) = oh and v = g 5 dr dp (86) In terms of these parameters, equation (85) for R assumes the turm — m2)\2 r=(1-2)¢ aa (87) and in terms of s and v — e2y2\2 Re q-)So (88) S58 Divergence Factor The reflection of a beam of radiation from the spherical earth increases the divergence of the beam and reduces the intensity of the reflected wave by spreading, as explained in Section 5.2.5. This is taken into account by introducing a divergence factor D, less than unity, which appears in the formu- las as a multiplier of the plane earth reflection coeffi- cient. Expressions for D are 1 D= i V1 + 2hy'he! /kad tan? p (89) Using equation (60), D becomes 1 D= 90 Vi + 2d2d2/kahy'd - oo) If d,— d, 1 ° D= @<3)) Ol) V1 + 2hi//ka tan? p and if y is small, so that tan ~y—y, 1 (92) Dy —<———————— ay hae 2hy' /kay* par Parameters p and q Useful expressions for the divergence factor, path difference, and receiver height may be obtained by use of the dimensionless parameters, = ae 93 O Aeiin 0 dhe ee) and = 5 (94) or dy = (1 — g)d=sd. (95) The divergence factor may be expressed directly in terms of p and qg by modifying equation (90) as follows: 1 D = V1 + 2d,2d2/kahy'd 1 A 4 Mas/d) (d2/2kah;) 1 — (d2/2kah,) where fy’ has been replaced by its equivalent ex- pression, given in equation (58). The above form of D shows that it can be expressed in terms of p and gq only: iD 1 V1+ 4p?q/(1— p’) Figure 17 shows contours of constant D as a function of p and q. The path difference A may be written in terms of p and q by substituting into equation (78): at! Qhy! he! 2(hy’)? ds 2d hee (1—d?2/2kah,)? = = i . d dd, d dy (96) 76 CALCULATION OF RADIO GAIN Go lies ret la 5 | 6 PPE | 8 |.85/.9 | .95 i: i =a = | erie aE ee + As} aes mo] ~ 5 Lal Figure 17. where D = 0.) Note: S = 1 —q =d,/d. Hence, on using equations (93), (94), and (95), Qh (1 — p?)? | ——— i el g————— ‘< dp p A (97) Figure 18 shows (1 — p?)?/p asa function of p. The receiver height hy may also be expressed in terms of fi, p, and g. This will be found useful in drawing coverage diagrams in which both hy and d ad a 21 Hoy t Al L | ia 3 ) ine ) Divergence factor D as a function of p and gor pand s. (Radiation Laboratory.) (p = 1 is the line of sight are unknowns. From equation (60) or SPHERICAL EARTH 77 Hence it follows that dy? ) dy d.? | hg = h 1- ae : 3 : I( 2Qkah, dy 2Qhkah, Since dy Ce eg aaa =— 2 I= haf d )(1= +) (98) Lo U0 = il Tr Mt L Ke { re) 8] | 1 7 -py §(0) f (p) = OFT m iP 7 67 ae =) a] +2] [ | al 2 oe) 4 ob5 6 a7 8 9 LO Figure 18. (1 — p?)*/p as a function of p. (Radia- tioa Laboratory.) Se Generalized Coordinates The distance from the transmitter to the reflection point d; and the ratio p may be eliminated by using the dimensionless coordinates he =—, 99 of (99) d ==, 100 are (100) The advantage of this substitution lies in the fact that the coefficients of s = d,/d in the cubic equation (74) may be expressed as functions of w and v only. Se — of Min thy - 4] 440% =0, 3 — 3 oo = s3| 0 + No/hy) 1 ae Se 20) 2 2 L d?/(2kah) | 2d?/(2kahy) In terms of w and 2, 3 So —$(1 4 ao 2 2 a 2v? Figures 19 and 20 show contours of constant s plotted in w, v coordinates. The curves are parab- olas. (101) 6 s CURVES s=02 sda a al 60) spt-ut fol ee s2| = - aa ; 3503 : | ane 40 T las =] | | + | | 492.04 36) —— — =a 32) Lt $=,06 28) SS oo ell [eal 24 t43=-08—4 20 Bez | + [_s=.1 Ss | | a Ls a 12 Saeed es 8 if 4 i 0 =t = 2 7 3 a ‘ vas Figure 19. sasafunction of wandv. (See Figure 14 for definition of lengths.) 0) T ae 2d “dr Fiaure 20. s as a function of uw and »v. 14 for definition of lengths.) (See Figure 78 CALCULATION OF RADIO GAIN The divergence factor D is given by 1 f Vie A4p?q/(1 — p?) ; oe where ji) = and en eS ee ee 103) d d d Since s is a function of u and v only, it follows that D may be plotted in u,v coordinates. This may be accomplished by solving equation (96) with respect to q, which gives ph) 4p°D° ; Equation (103) gives s = 1—q; v= p/s, and u may be read from Figure 19 or 20. Contours of constant D are shown in Figures 21, 22, and 28. The grazing angle y is important in calculations for vertical polarization since it determines the magnitude and phase of the reflection coefficient (104) 50 450D:954.0 for a particular frequency and reflecting surface. The grazing angle y may be expressed in terms of s, u, and v, as follows. From equation (60), Tat ts ( dy? ) tany = = 1-— : 105 ¥ dy dy 2kah, ( ) Hence tany = Se (1 — sv) = (0 foes ) dp(sd/dr) dp \sv or tan y 1 ( 1 ) (106 = Sv], Vii V2ka \sv ) and for k = 4/3, tan By, 210. (4 — s). (107) Viy sv From Figure 24, ¥ may be obtained for given values of fy and sv = p. The generalized coordinates described in this section will be found highly useful both in field strength and coverage calculations. 35 30 Da9 27 25 D:99908.0 70 60 ts a DIVERGENCE AND PATH DIFFERENCE CONTOURS DIVERGENCE D — — — PATH DIFFERENCE R fe] <2 4 6 8 wade Figure 21. Contours of constant divergence factor D and path difference variable R. 20 (Radiation Laboratory.) CALCULATIONS FOR OPTICAL-INTERFERENCE REGION DIVERGENCE AND PATH DIFFERENCE CONTOURS DIVERGENCE DO — — — PATH DIFFERENCE R 15 140799 FL Vie (1312 Bay ACS ! ——— | 2 5 VvV=— dy Figure 22. 5.6 TLLUSTRATIVE CALCULATIONS FOR THE OPTICAL-INTERFERENCE REGION 5.6.1 Introduction The general expression for the gain factor A in the interference region is obtained by combining equa- tions (44) and (7). Then A = AFi Va — K)?+ 4K sin? z. (108) The value of the radio gain is then given by equa- tion (3) and the value of radar gain is given by equa- tion (5). The value of the radical which defines the interference pattern has a range of values between 0 and 2. The extreme values can occur only when K =1 (@ =1, D =1, F.2/F, = 1); the value 0 (nulls) is then given by sin? (Q/2) = 0 and the value 2 (maxima) is given by sin? (2/2) = 1. In general, the value of A lies between the two extremes A = AoFy (1 se K), Contours of constant divergence factor D and path difference variable R. | (Radiation Laboratory.) the positive sign giving a maximum and the negative giving a minimum. At any other point, the value lies between these two extremes. For range calcula- tions (which involve maxima), the variation in A is from | to 2 times the free-space value, according to the value of K, so that in practice a quick rule of thumb for range may be devised. Assume (1 + KK) equal to 1.9 for favorable conditions (sea water, horizontal polarization, or, in the case of vertical polarization, small grazing angles) down to (1 + K) = 1 or K = 0 for propagation over rough terrain. The problem of finding the range is thus reduced to a problem for free space. In range calculations, P./P, is given by the ratio of minimum detectable power to power output. A is then determined by equations (3) or (5), and the range is given by find- ing d from the relation (writing Ay = 3d/87d) “ee (*) (1+ K). (109) TC More detailed calculations are presented in this 80 CALCULATION OF RADIO GAIN 220 15 - D=99 Wa y 7 3 200 ae Z la sae | Z I D> 180) as aes Biv a 10 v ( a eee | 7 9 8 140 u=he é 7 Pol a 120|—™ ae, 7 a 6. uae Y T AK KR = go eee ea ac D=95 100 Z Sle ex 55. | TZ” PS - Dele a e 80 A Sa Ee = aS - 5 x ay ee Cee 4 ae ewe e390 e —_— = _ \ is 60 | 4A C A 30 20 10 2 4 A 10 3 = Nn ala 2 .4 _ lefS =|N o£} 5 =I : : 10 ae}, 6 < " 7 > 2 ol 8 A 05 We 02 2 Ol 10 .005 002 .001 98 1 Ficure 24. yas a function of h, and p=sv=d,/d7. [See equation (107).] (See Figure 14 for definition of lengths.) by n\/2 tive value. Substituting V2kah, for dr, equation (83) assumes the form , where n (see below) may assume any posi- R= nr, (114) where 1 [ea » r=—-\J——5 115 Dose 2 ee) (for k = 4/3). or r = 1030 we hy fe A graphical representation of r is given in Figure 15 in Chapter 6. Then for a reflection coefficient of p = 1,@ = 180: degrees (i.e., 6’ = 0), equation (29) gives GaAs Rea oN (116) If r is fixed, a complete pattern of contour lines (along which A is constant) is determined. Take as independent variables p and n (rather than wu and v). A given choice of p and n determines R by equation (114), 2 by equation (116), v by equation (118), s by equation (118), vu by equation (121), D by equation (117), and finally 20 log A by equation (110). By varying 7, new patterns are obtained. Accordingly, 7 may be called a pattern or chart parameter (see Section 6.8.3). The lobes on the charts depend on n, in accordance: with equation (116). Accordingly, for p= 1, o = 180 degrees, n is the lobe variable. For the first (lowest) lobe, n = 0 gives the first null, n = 1 gives the first maximum and n = 2 the second null. For the second lobe, » varies from 2 to 4, with a maximum at n = 3, and so on. It should be remem- bered that if n <1, corresponding to the lower side of the lowest lobe, the value of the field (or of A) given by the optical formula is too low. A more accurate value can be obtained by joining the curves found in the optical and diffraction regions into a smooth overall curve. Combining equations (102), (103), and (113) gives the divergence factor D, 3 =i/p) 2 g]-12 D= [: aL ARp | a E fe 4Rp | ; ay Li aye (117) The variable s = d,/d is ‘Si—p/0s (118) and, repeating equation (101), e—3e-s(1F¥_i)4 <0 (119) 2 2 v Qu’? In terms of p, equation (119) becomes 2p? — 3p'v + pw? —w—1)+v=0. (120) Equation (120), resolved for v and wu, gives r= Map—t4al(l—r) +00 p) +4], (121) U2 Da 3p + — —140. p 82 CALCULATION OF RADIO GAIN Some useful approximations : For v large, ¢g— land R = 4 (1 — p?)? ap- p proaches the value Rw~ ae 20; (122) p which, solved for p, becomes a —R+ VRP + 8 (123) 4 IGE De (124) Ihe ay << A 3—-f ~—- (125 pr 125) For R > 3, Dw~ 1. (126) The calculations will be divided into four types. Type I. The direct calculation of the radio gain (or field) when the heights and distance apart of the antennas and the wavelength are given. Type I. The calculation of the radio gain as a function of the receiver height hz when the trans- mitter antenna height fy, distance d, and wavelength d are given. Type Ill. The calculation of the radio gain as a function of the distance d when the transmitter antenna height /,, receiver height fz, and wavelength are given. Type LV. The calculation of the possible positions of the receiver in space (he,d) when the radio gain will have the given value for given values of the gain factor A, the transmitter antenna height /, and the wavelength . Special cases, such as the re- ceiver antenna height, hs for given d, or d for given he, can be solved by use of the curves in Type IT and ‘Type III, in Sections 5.6.3 and 5.6.4. This type of problem is of importance in estimat- ing the range of a set when the minimum detectable power of the receiver and power output of the trans- mitter are known. 5-62 Problem of Type I. Radio Gain for Fixed Heights and Distance The radio gain at a given receiver is to be found, their heights, as well as the wavelength being given. The polarization is assumed horizontal, the effective earth’s radius 40/3. The following data are assumed. Transmitter height: h; = 50 meters Receiver height: yz = 1,500 meters Distance apart: d = 100 kilometers Wavelength: \ = 1 meter (f = 300 me) Gains (over doublet): G, = Gz = 100 (20 db) ll OnzE-Way TRANSMISSION 1. dz, = 188 kilometers (Figure 2). d < dz, so that the receiver is in the optical region. 2. The u,v coordinates of the receiver are U= Ia = omy = 30, hy 50 dp = V2kah, = 29,100 meters ri ne CD dp 29.1 3. Referring to Figure 19, s (= p/v) is estimated to be 0.05. Since the result is very sensitive to slight changes in s, it is desirable to improve the value of s. In Newton’s method, the next approximation, using equations (101) or (119), is eee ( wo | 2v" a Ks) - " 5; , (127) Me) ae — 3-1] tuna] 2 v s' = 0.04794. The next approximation gives the same result. 4. Using the above value of s’ and the relation p = sv (v = 3.48), pis equal to p = 0.1645. With the value of p = 0.1645, equation (117) and Figure 22 give the value D = 0.95. 5. The path difference variable F is obtained from equation (112) or Figures 21, 22, 23, as R = 5.477. 6. The number 7, from equation (115), is 2.91, so that the lobe number 7 is fee 5 n= Hence the receiver is on the upper part of the first lobe close to a null. CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 83 Ue Q = nr = 5.9 [see equation (116)], DE OG 2 Q sin? — = 0.0353. 2, 8. To use equation (110), the value of the free- space gain factor Ao is needed. Figure 3 in Chapter 2 gives 20 log Ay = Substituting in equation (110), 20 log A = 20 log Ap + 10 tog (0.0025 + 0.1342) —118 —8.642 —127 By equation (3), using gains of 20 db, 10 log = = —127+ 40 = —87. 1 Accordingly the radio gain is —87 db and the received power is given by P, = P,10°°". Suppose the receiver has a minimum detectable power of 107° watt, then the required minimum power output under the given conditions would be Py Jey <0 = 10°" x 10°7= — 118. 1071? watts. RADAR Suppose that, instead of a receiver, there is a target at the same position with a radar cross section of o = 50 square meters. The value of P:/P; at the radar receiver can be found from equation (5) using _ the value of A found above and the given values of o and 2X. If the radar uses the same antenna for transmitting and receiving, G; = G2, which in this case is 100 or 20 db, and the radar gain 10 log = = 20 log Gy, + 10log + 10 log « 1 +40 log A — 20 logy, = 40+ 7.5+ 17 — 254 — 0, = — 189.5 db This gives P; = 10°° watts, which obviously is unat- tainable. EFrrect oF VERTICAL POLARIZATION The general value for K in equation (108), when the reflection coefficient p differs from — 1 and F,/F, = 1 is (see Section 5.3.1) IK = wD (128) Q in equation (108) is no longer given by equation (116) but is the sum of two phase shifts, one caused by path difference, (R/r)a = nz, while the other is gd’ = o — o, the difference between the phase of the reflection coefficient and that for perfect reflection. Hence Me eek (129) r The lobe variable (for imperfect reflection) is now N, defined by Q@=Nr (130) rather than n = R/r. The relation between N and n is derivable from equation (129), giving «—®@ wT N=n- (131) The propagation is assumed to take place over sea water. The angle between the reflected wave and the earth is given by equation (107) or Figure 24, and is W = 0.582°. From Figures 14 and 15 in Chapter 4, od = 168°, p = 0.76. The lobe variable N, in terms of the old lobe variable (for p = 1,@ = 180°), by equation (181), is The fact that N = 6 db. < Hence 20log A = — 115db+ 6db = — 109 db, and 10 log = = — 109db + 10 log GG, = — 95.5 db. 1 p is approximately 1/r, since R = nr > 2. Accord- 7. The foregoing values, together with results ingly, we begin with p = 0.1. obtained with other values of p, are listed in Table 1. TaBLeE 1* Radio Gain Radar Gain a D fotos {olor 2 } zZ = Of a p R : s he - n T 8 Pp, EP, 0.15 6.16 0.0337 1,396 0.655 1.03 0.96 5 —96.5 —172 0.1 9.58 0.0225 1,861 1.019 1.60 0.98 6 —95.5 —170 0.05 19.68 0.0112 3,216 2.09 3.29 0.995 —11 —112.5 —204 0.04 24.70 0.00898 3,885 2.63 4.13 0.997 4 —97.5 —174 0.03 33.05 0.0067 5,005 3.51 5.52 0.998 3 —98.5 —176 0.02 49.74 0.0045 7,235 5.29 8.31 0.999 i} —96.5 —172 0.01 99.76 0.0022 13,917 10.61 16.67 1.000 4 —97.5 —174 See also Table 5. t 20 logy (1-D)? + 4D sin? (@/2). CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 85 The value of P:/P; [equation (3)] is represented graphically in Figure 25. Tos i Fixed Distance 100 Km bh, =, 30m i of E eS 1.5m | |} 1 | S 1 IE k L v3 sh IL G, = 224 (13.508) i | i 6, = ( 0 os) i \ | f f = 200MC | } ~ ae 1 2 | Free Spoce i 1 ' Radio Gain 4 | \ | o? —+ + + fj f) Al ' \ ; 4 | \ 1 l 452 ; iS Line of Sight * \ 1 Ee ec caieasee nea | oeee | eee! | eens | Wes Chasen Wee Log poocentbeaccd « Z ale if 4-7}From Dittraction y Formula b= ie Hea gic =n0 =100 =30 =i =o =130 =120 10 log P2/P, Figure 25. Radio gain in decibels versus height he. Horizontal polarization. Rapar Gain: Two-Way TRANSMISSION Knowing the values of 20 log A, the corresponding values of P:/P; are given by equation (5). Taking Gy = Gy = 13.5 db, 1627/9 = 5.58, = IlBay 10 log = = 27+ 7.5+ 17+ 40 log A — 20 log 1.5, 1 = + 40log A + 48. 564 Type III. Radio Gain Versus Distance for Given Antenna Heights A radar used over the sea has a wavelength of 1.5 meters. The transmitter is 30 meters above sea level and the target is at an altitude of 1,000 meters above the sea. The power gain of the trans- mitting antenna is 13.5 db, the polarization hori- zontal. The one-way radio gain is to be found as a function of distance. Also, the radar gain at the radar set by echo from the target of cross section ¢o = 10 square meters is to be calculated. Rapio Gain:ONE-Way TRANSMISSION (see Figure 26). 1. The number r from Figure 15 in Chapter 6 or equation (115) is found to be 9.403. 2, th = laf S B33 3. For r > 2 and n = 1, =: is approximately equal to 1/r. Hence we start with p = 0.1. 4. Equation (121), with p = 0.1 and wu = 33.3, givesv = 2.76. 5. From equation (112), R = 9.445. 6. n = R/r = 1.005. Hence the target is on the first (i.e., lowest) lobe. Moving in the direction of increasing distance, the target soon approaches the first maximum. To get points beyond, ie., n < 1, we need greater values of p but it is not necessary or desirable to go below about n = 0.8, since the curve beyond n = 0.8 is generally in the transition region in which the curve is more easily and more accurately obtained by joining the optical and diffractive curves. The diffractive part of the calculation is given in Section 5.7. Accordingly, in Table 2, the values of p are taken only slightly above p = 0.1 (correspond- ing to n = 1) and are diminished to find points at the nearer distances. 7. To find A, we need first the free-space value Ao, which is given in Figure 3 in Chapter 2 as a function of d. Since dp = 4.12 V30 = 22.6 km, the value of d corresponding to v = 2.76 is d = vdp = 62.3 km, and 20 log Ay = — 111. 8. To find the value of Q 10 low 4) ( — Dy + 4D sin? » we need D. Since n is practically unity, correspond- ing to the first maximum, sin? (Q/2) may be taken as unity. Hence the radical reduces to 1 + D. Caleula- tion using equation (103) and Figure 17 gives D = 0.98, and hence 1+ D = 1.98, 20 log (1 + D) = 6. Since the transmitting antenna gain G; is 13.5 db, and assuming the receiving antenna gain to be 0 db, 10 log (P2/P:) has the value 20 log A + 13.5 or Py 10 log P, = Ae Va — D)?-+ 4D sin? = Gz + 0.0db — 111 db + 6db + 13.5 db = — 105+ 13.5 = — 91.5 db. 86 CALCULATION OF RADIO GAIN kK = 4 /3 » = 15m (f= 200MC) h = 30m hy = 1looom / Free Space : — Radio Gain G, = 22.4 (13.508) / < G, = 1} 0DB Ath | Horizontal Polarization laa i / / a zal = ' -80 -90 -100 “10 -120 -130 -140 -150 =160 70; -180 =190 -200 10 log Po/p 1 FiaureE 26. Radio gain (in db) versus distance. in Kilometers ° d Distance CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 87 TaBLE 2. 10 log G, = 13.5 db; 10 log G2=0.0 db; \ = 1.5 meters; h; = 30 meters; h: = 1,000 meters; ¢=10 m*. RadioGain Radar Gain 2 101 P P» Dp v d(km) R n D 2 - 20 log Ay 20 log A Oe3 P, ADS P, 0.12 3.098 70.49 7.78 0.83 0.97 1.30 5.6 —112 —106.4 —93 —172 0.11 2.934 66.75 8.54 0.91 0.98 1.43 6 —111 —105 —92.5 —170 0.10 2.755 62.68 9.45 1.005 0.98 1.58 6 —110 —104 —91.5 —169 0.09 2.561 58.25 10.55 1.22 0.98 1.76 6 —110 —104 —91.5 —169 0.08 2.349 53.45 11.92 1.27 0.99 1.99 5.2 —109 —104 —90 —167 0.07 2.119 48.21 13.68 1.45 0.99 2.29 3.5 —109 —105.5 —92 —170 0.06 1.870 42.54 16.02 1.70 0.99 2.68 —1 —107 —108 —94.5 —175 0.055 ~=-1.738 39.22 17.50 1.86 0.99 2.92 —7.5 —106 —113.5 —100 —186 0.05 1.600 36.4 19.28 2.05 1.00 3.22 —16 —106 —122 —108.5 —203 0.045 1.457 32.89 21.45 2.28 1.00 3.58 1.5 105 106.5 —93 —172 0.04 1.311 29.59 24.16 2:57 1.00 4.04 +3.8 —104 —100 —87 —159 0.0385 1.158 26.14 = 27.64 2.94 1.00 4.62 6 —103 — 97 —79.5 —153 0.03 1.002 22.62 32.28 3.43 1.00 5.39 +2 —102 —100 —86.5 —159 0.025 0.841 18.98 38.67 4.12 1.00 6.48 —8 —100 —108 —94.5 —165 0.02 0.678 15.30 48.49 5.16 1.00 8.10 5.7 — 98 —104 —91.5 —167 0.01 0.345 7.79 97.08 10.33 1.00 16.22 0 — 92 — 92 —78.5 —143 * Lobe pattern factor = 20 log VY (1—D)? + 4D sin? (22/2). Repeating the process for other values of p, a table of 10 log (P:/P:) versus d is obtained. These points have been plotted in Figure 26, together with the points in the diffraction region obtained with the same data in Section 5.7.3. For sketching in the optical part, the value of n is kept in mind, since this indicates on which lobe and where on the lobe the point lies. Rapar Gain: Two-Way TRANSMISSION To change from 20 log A = — 105 to 10 log (P2/P1) at the radar receiver, using equation (5) and Gy = G, = 13.5 db, 10 log (P2/P1) = 27 + 7.5 + 10 log « — 20 log 1.5 ll + 40 log A, = 27+ 7.5 +10 —3.5+4+ 40 log A, = 41+ 40log A = — 169 db. °°5 Type IV. Determination of Con- tours Along which the Gain Factor 4 Has a Given Value, the Transmitter Height and Wavelength Being Given This is the so-called coverage problem which is treated in greater detail in Chapter 6. While the usual coverage diagram is derived on the basis of one-way or communication formulas, the diagram is still useful for radar since a target wil! return more or less energy to the receiver according to its position on the coverage diagram. The contour gain factor A is readily converted to P2/P; for either one-way or radar by means of equations (3) and (5). The method described here is more accurate than the graphical methods given in Chapter 6. It is best suited for finding maximum lobe ranges correspond- ing to a given radar gain. If A is given, and hy for a given distance d is wanted, a curve can be drawn as in Section 5.6.3 and then h. found for the given value of A. PROBLEMS A radar set operating over the sea has a trans- mitter with antenna height of 30 meters and a wave- length of 1.5 meters. As in the previous problems, a receiver with an antenna of 0 db gain is assumed in place of a target and the gain of the transmitter is again assumed as 13.5 db. The polarization is hori- zontal. The gain factor A, for illustration, is chosen as —1380db. Positions of the receiver are to be found at which the gain factor takes that value. In Chapter 6, purely graphical methods of de- termining the contours (lobes) are given. Here we are concerned with finding individual points on the contour. Thus, for example, n = 1 if the tip of the lowest contour is wanted (as in range determination). Points near the tip require values of n near 1, such as n = 1.2 or n = 0.9. For the next higher lobe, the tip of the lobe corresponds to n = 3 and so on. The nulls are at n = 0, 2, 4,.... However, while the optical formula gives a null at n = 0, that is, near the line of sight, the true value of the field, or of A, 88 CALCULATION OF RADIO GAIN is greater. To obtain the correct result, the contours for the diffraction field (Section 5.7.3) and those obtained for the optical region should be merged into smooth overall curves. For values of r > 3, R = nr> 3, the tip and upper part of the lowest lobe, and the higher lobes, by equation (126), have a value of D close to 1. Con- sequently, equation (110) reduces to A= 2Aosin >. (132) If n is given, sin 2/2 is determined and the calcula- tion can be performed as a free-space calculation. In terms of d, the equivalent free-space distance do, corresponding to A is given by Figure 3 in Chapter 2 and is equal to gaan, sin 5. (133) In the stated problem r > 3, so that the free-space calculation suffices. This is given in paragraph (1). In paragraph (2), the method including the diver- gence is given. 1. Free space. In the given problem, 7 = 9.4, so that the simplified calculations should be sufficient. The value of dp corresponding to 20 log A = —130 is found from Figure 3 in Chapter 2 to be 566 km. Hence by equation (133) for n = 1, the true distance d for complete reinforcement by the reflected wave is twice as much and d = 2-(566) = 1,132 km. For n = 1.2, 2/2 = 0.67 [from equation (116)], giving d = 2-(566) sin (0.67), = 1,076 km. To find the corresponding height he, a curve of the type in Section 5.6.3 is needed for —20 log A versus height. From this, the height corresponding to 20 log A = —130 can be read. Alternatively, the calculation given later in (2) will determine both d and he. If instead of A, either the radio gain or radar gain is given, A is first found from equations (3) or (5). 2. Calculation including divergence, n = 1. As in the previous problem, r = 9.4 (from Figure 15 in Chapter 6). A convenient value of p, approximately equal to 1/r, is selected. In this case, we take p = 0.1. This value is then improved by applying f(p) Newton’s method p; = p — 7"(p) to the equation 1 1 OD) at ag Vo Vv Va — D)?+ 4D sin 5 (134) remembering that D is a function of p as given in equation (117), where vo is the value of v which corre- sponds to the distance dy at which the given value of A would occur in free space. If n = 1, equation (134) reduces to eer ee) Vo Vv (135) The correction formula corresponding to equation (134), denoting by p; the improved value of p, is Vv — U9 Ja — D)? + 4D sin? (Q/2) 92 31 -— zl a a +7) eee i] =D aeatinak Det 4pD ( Misa 1—p? (136) The correction formula corresponding to equation (135) (n = 1) is Diss Dee) Vo v =p 137 SE ey ee ae - (1 — D’) p?bD? = 2pv \1l — p Taking n = 1, d) from A = 3d/8rdp (or Figure 3 in Chapter 2) is 566 km, and Taking p = 0.1 as an initial value as explained above, then D is found to be 0.9788 [equation (117)]. Using R = nr = 9.4 and equation (113), v = 165.6. Substituting these values in equation (135) gives pi = 0.10385. Repetition of this procedure requires no change in these five figures. Accordingly, for n = 1 and A = —130db, p = 0.10385. The corresponding values of D and v are D = 0.9789 v = 49.38, d = 1,110 km. In (1) d was found to be 1,132 km. Now s = p/v = 0.0021. From equation (121), ho = 86,940 meters. For n = 1.2, the calculation proceeds as for n = 1 except that equation (134) rather than equation (135) is used: R = 11.283. CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 89 The value of p by successive approximations in equation (136) is found to be p = 0.08127 v = 47.35, d = 1,065 km D = 0.9851 s = 0.00184 he = 2,772 hy ho = 83,160 meters 3. Vertical polarization, p # 1, @ # 180° (sea water). The procedure given here is first to find the position of the point for a given n assuming the reflection coefficient to equal —1 and then find the shift caused by the change in the reflection coeffi- cient. For the most important case, n = 1, the new maximum distance is given by d = dy (1+ pD), (138) where do is the free-space distance corresponding to the given value of A. pis found from the value of y, the grazing angle at the reflection point given by p found above in the calculation for p= 1 and @ = 180°. For the new contour tip p changes, but the angle y does not change sharply, so that p and @ as found for the p obtained by the simplified calcula- tion is a close approximation. For p #1, @ # 180°, Q = 6+ ¢ — a [see equa- tion (29)] consists of two parts. 6 = rR/r = (A/X) 2a =nrand @d’ = @ — cz. Writing 2 = Nz, N the lobe number for the case p ¥ 1, @ ¥ 180°, the require- ment for the new lobe tip is N = 1; for the first maximum 2 = 7 = Rr/r +o — zw where R is the path difference variable at the new lobe tip. Hence the value of R at the new tip is given by R=r(2—$/n). (139) Using the results in (2), p = 0.10385, D = 0.9788, r= 9.4, dy 566, we find from Figure 24 or equation (107), ~ = 0.725°; from Figures 14 and 15 in Chapter 4, p = 0.76, @ = 170°. Substituting in equation (138), d = 566 (1.744) = 987 km. From equation (139), R = (9.4) (1.056) = 9.92. Also The above value of p can now be improved by substitution in equation (137) with D replaced by pD which gives p = 0.0985. In the calculation just made, p has been assumed constant. This can be checked by finding yw as de- termined by the new value of p. The result is yw = 0.766° and the corresponding values are p = 0.74 (as against 0.76 previously) and@ = 169° (as against 170°), which is good enough. We now find aoe i = 0100224 v 43.9 he _ 9.359 hy h, = 70,770 meters. Hence the new maximum point of the lobe is at a distance of 987 km and at an elevation of 70,700 meters, as compared with a distance of 1,110 km and height 86,900 meters for perfect reflection. 566 Maximum Range Versus Receiver (or Target) Height If the value of A has been determined by using the minimum detectable power in equation (3) or (5), the corresponding contour is a curve of maximum range versus receiver height for com- munication or maximum range versus target height for radar. Generally, the lower part of the lowest curve (n <1) is of greatest interest. If A is suffi- ciently small (i.e., 20 log A numerically large and negative), the complete contour has points below the line of sight. If the transmitter antenna is low (hy < 30°? meters), the lower points are likewise given by the diffraction formula, discussed in Sec- tions 5.1.7 and 5.7.3. Several such curves, the lower part of the lowest lobe corresponding to various transmitter heights for \ = 1 meter and 20 log A = — 130, are given in Figure 27. Consider, for example, the curve for hy = 2 meters. The uppermost points of the curve correspond to the tip of the lowest lobe and were found by the procedure used in Section 5.6.5, putting n = 1. The lowest points were found from the diffraction formula by the method of Section 5.7.3 with the aid of Figures 31 to 36. It has been pointed out that for n < 1, the optical interference formula is inadequate. To locate a point between the upper extreme (n = 1) and the diffraction points, a curve of A 90 CALCULATION OF RADIO GAIN 100000 a 10000 [ +—— —! +—+ jer HI 1 saa as "I rH =e laa) sete =: 8 8 888 fo} o Oo 8 ha IN METERS | h=2m_ | J 16 32 | | 100 | 100 al ee a [| ace | 50 a 46 | 30 L [ a (eo seas ea [J] FOR 20 log A=-130 d=I1 meter el | HORIZONTAL POLARIZATION tani = eae oo = = r eae] 1 1 1 2 g 456 810 100 200 400 600 1000 d iN KiLOMETERS—> Ficurs 27, Maximum range versus receiver height A: for given values of transmitter height /,. BELOW THE INTERFERENCE REGION 91 versus hy for some distance is constructed. In Figure 28, a set of such curves is given for the dis- tance 100 km and for various transmitter heights. By taking the intersection of 20 log A equal to — 130 -100 = Daversa -125) ry -130 | T= 100 HH Bal T oH -150 a 2 ball Lf 8 > 1 and for distances greater than 1.5/s, the shadow factor is By == 2:507(sd) ee E 1 | S = coccmet d (ka)? ’ = 443. ory 3k Equation (149) gives the value of the shadow factor for the first mode only. In Figure 32, the curve marked ‘dielectric earth’ is a plot of the shadow factor evaluated by using all terms or modes. However for sd > 1.5 only the first mode is impor- tant. Consequently, equation (149) represents this curve accurately for all values of sd larger than 1.5. (149) where (150) Hercut-GaIin For antennas at zero height, the height-gain functions f are equal to unity, so that equation (148) represents the actual value of the first mode for both antennas at zero height. When the antennas are raised above the ground, it is convenient to distinguish between low and high antennas, the division between the two cases being given by the critical height h, = 3007/8. 1. Low antenna; h Figure 29. Height-gain as a function of height. (See Figures 7, 25, and 47.) Inspection of Table 4 shows that, except for the case of sea water at wavelengths above 1 meter, the approximate equation (153) is good for heights above about 50 meters. Tasie 4. Values of 4/1 for different wavelengths (vertical polarization). Ninmeters |01 12 3 4 5 6 7 8 9 10 115 133 160 235 285 300 Sea water 0.06 10 28 50 80 Fresh water 6 1117 22 30 33 40 44 53 57 Moist soil a ¢/ ill ay 13 Pal OA) PRS GR} By) Fertile ground 3 5 8 10) 13 16 17 (20) 25 27 Very dry ground 23 46 8 9 10 12 15 16 To a first approximation, the value of f for low antennas is the same for all modes, so that the height- gain functions may be factored out, as was pointed out in Section 5.1.7. The gain factor for the case when both antennas are low can then be represented by A = 2AAF, (H,):(Hz)o, (154) where F’, is sum of the shadow factors of all the modes. F, has been plotted in Figure 32. Equation (154) is also valid in the optical region, provided a = 2Qhihs : (155) This condition is added to insure that the point is well below the center of the lowest lobe. 2. Elevated antenna; h > 30d”/°. In this case the height-gain function f increases exponentially. Rep- resenting the increase over lh by g, we have f =glh. (156) For the dielectric case 6 >> 1, refer to equation (193) to define 6. (See Sections 5.1.6 and 5.7.3.) g = 0.1356 Gaye 10-248 Veh (157) where 2\1/3 = (=) ; (158) Wha 5 aN\1/3 fy (4) (159) 60 3k See Figures 35 and 36. For the case of sea water, vertical polarization and wavelength in the VHF (1 to 10 meters) range, g requires a correction factor g’ which depends on X. A table of g’ is given with the graph of g in Figure 36. The change of f with height h is represented schematically in Figure 29 (see also Figures 7, 25, and 47). FORMULA FOR THE DreLectric HarTH. (See Sections 5.1.7 and 5.7.3.) 6>> 1. The first mode has the form ,(d) - f(h1) + f(he). If the value of the gain is given substantially by the first mode, and substitution from equation (148) is made, it is found that A = [2A0AiF si] f(hi) + fhe). (160) For the dielectric case equation (160), using equa- tions (140), (147), (151), and (156), becomes F, Ae =" (gh)s (gh)s. “ (161) The same formula holds for sea water, vertical polarization, wavelengths in VHF range, d > 50/p’, 94 CALCULATION OF RADIO GAIN and h > 4/1, as described above, except that Fs or F., is represented by various curves in Figure 32, according to the value of \, and g is modified shghtly by a correction factor g’. FormMuLA For Sea Warrer. VERTICAL POLARIZA- tion. VHF. Equation (160), the general formula for the first mode, in the VHF (1 to 10 meters) range, becomes A = [2A, Ay Fy |(Az99’)1(H199’)2, (162) where Hz is the low antenna height-gain function whose formula is given by equation (152), and gg’ = 1 for low antennas. As pointed out above, if h > 4/l and d > 50/p’, equation (162) reduces to equation (161). g’, the correction factor for g, 1s given in Figure 36. For a more extended discussion, see Section 5.7.4. 572 Effect of Changing the Value of k In the optical region, the effect of a linear gradient of refractive index has been shown to be equivalent to replacing the radius of the earth a by an effective radius ka and then treating the atmosphere as homogeneous (see Chapter 4). In view of the equivalence of the sum of modes to the optical formula in the optical region, it follows that the modes should be changed in the same way in the optical region, i.e., a should be replaced by ka. These same modes supply the solution of the wave equation in the diffraction region, so that in both regions the substitution of ka for a will take care of an atmosphere with a linear variation of refractive index with height for all values of k. For given transmitter and receiver antenna heighis, hy and he, the first maximum of the field-strength versus distance curve (see Figure 4) will frequently represent the limit of detection. The first maximum occurs not far from the line of sight which, for a standard atmosphere (k = 4/3), is given by dy, = | 2(*) a(Wiy + Vio). (163) If 4/3 is changed to k, the more general form is C= V2ka (Vhy + Vio). The first maximum will now be near (164) dy’ = NE dy. Suppose next that the point at distance d,’ for the given heights was originally well within the diffrac- tive region. The exponential part of the gain due to change of k from 4/3 is equal to oa 1.607s (dy ’— dz) or e-1.607 (V3k/4 - 1)sdz_ The above formula is obtained by combining equa- tion (149) for F,, and s is obtained from Figure 31 (k = 4/3). This corresponding gain in decibels is 20 logio e ~ 1-607 (W3k/4 - 1)sd 7 20| - 1.607 X 434 (# = 1) wis |, ie wd ( - - 1) db, below the free-space value. Since a maximum is now near dz’ as a result of changing 4/3 to k, ie., the field has a value of 6 db above the free-space value, the gain is approximately the exponential value — 14 sd, (J — '), as a consequence of refraction giving a value k greater than 4/3. For k = 12, and sd; = 5, the gain would be about 140 db at some point near d;’ = 3dr, at heights fy and he such that dy! = V2ka (Why + Viz) and such that hz is well within the diffraction region. For given transmitter height hy and distance d, the effect. of increasing k is to lower the lowest lobe roughly by the amount by which the line-of-sight elevation at the distance d is lowered in changing from 4a/3 to ka. At this distance the height of the line of sight is hz, and ll Viz, = ms Vii (fork = 4/3) (165) a N83 becomes = l = Vie ie 166 2 V2ka : ee) so that there is a downward shift of approximately 3a? 4 wa ( Ey hy — hy’? = 1 een | a ( 4/3), with a consequent substantial gain in signal strength at some points and loss at others. The chances of detection in the region, however, have improved. If both antennas are low, a change of 4a/3 to ka gives a field strength change such that the field (168) where ll d! (170) ll Q aN al i] bo where i may now have any value. A itself will change to A’ where OA ey” 4 Figure 30 illustrates the values of (8k/4)" for various values of k and n used in equations (169), (170), and (171) 50 100 5000 10,000 k—e Ficure 30. Values of (3 k/4)”. Note: Values shown between 0.1 and 0.5 of the ordinate scale are minus values. which was formerly at a point d well within the diffraction region will now be found at a distance of approximately SD © 4 except for the decrease in free-space gain (20/3) log 3k/4, occasioned by the increase in distance. If, for instance, k = 12, the free-space gain is equal approximately to —7 db, and the ratio of the new distance to the old is equal to (3k/4)°? & 9° & 4. More generally, if A/Ao is known at a point (fo,d) for a transmitter height h, and for k = 4/3, the same value of A/Ao will be found at hy’, he’, d (171). Figure 43 gives the same information in nomographic form. Hence if a coverage diagram is known for k = 4/3, then the same diagram can be used for k # 4/3 if the diagram is interpreted in terms of h’, d’, and A’. ae Graphs for the Case of the Dielectric Earth (6>>1) 1. Fundamental formula for gainfactor. Thisformula is (see Sections 5.1.7 and 5.7.1) Me an (dn De (172) where g = 1 when h < 300”. 96 CALCULATION OF RADIO GAIN FREQUENCY IN MEGACYCLES 30 50 70 100 200 300 500 700 1000 2000 3000 5000 10,000 20000 30,000 1x107 sf (8) 0.1 WAVELENGTH IN METERS Ficure 31. sf(6) versus wavelength and frequency. For dielectric earth and k = 4/3, f(6) = 1, ands = 4.43 X 10°1/9, (See Figure 57 for f(6).) BELOW THE INTERFERENCE REGION 4 1 \ 4 OT T i EFFEGT OF EARTH CURVATURE \ 20 LOG F, -10 , inal \ \ \ \ \ -20 aT \ a os ) -30 \ fh (e) ' aD =a = 4 2 oe LL ' oO oO DIELECTRIC EARTH Pr |, SEA-WATER, V P \< 10, OVER-LAND, V P LAND ann SEA,HP FOR SEA-WATER, V P USE CURVE WITH APPROPRIATE VALUE OF i i [o} 20 LOG F, © ° © i) DIELECTRIC EARTH -100 “110 \ | \ -120 \ \ ale \ | LL | 2 3 8 9 ie) tl VW 14 5 sd ——> 20 log F, /(sd)?, is the same as curve O in Figure 58. Kei Ure Figure 32. Shadow factor F's, 20 log F, (evaluating all modes), or 20 log F,/(sd)? versus'sd. Curve for dielectric earth, 98 CALCULATION OF RADIO GAIN 20L0G Fy a ° ie} 20 30 40 50 60 70 80 Fictre 33. Shadow factor for dielectric earth. i | Le | | “1 20 LOG x ou °o | \ aE ° 5 =| 40 | he all L toh] 0.2 0.3 04 05 1 2345 10 20 30 40 50 100 500 1000 5000 10000 x Ficure 34. 20 log x versus z. BELOW THE INTERFERENCE REGION FREQUENCY IN MEGACYCLES .20 30 2000 3000 5000 FOR USE WITH ELEVATED | { me AKiE 4/3) HEIGHT -GAIN Ne IN METERS 06 0.5 0.3 0.2 O15 A ~| 0 [=1|-2-4] 0 @ o {2} fe} to) 10 20 30 40 50 60 70 80 90 100 NO 120 eh (5) Figure 36. g andy versus ehg(6). 100 CALCULATION OF RADIO GAIN FIGURE 387. eh: = eu versus sd = gv. Points eh,sd, lying to right of dotted line corresponding to eh; for a given trans- mitter indicate that the points lie in the diffraction region of the transmitter, ord > dy, Diffraction region dielectric earth — curve parameter: 20 log A = 20 log A — 20 log hi — 20 log gi. BELOW THE INTERFERENCE REGION 101 [ le] ie 60 40 20 — 3 5 fo) 2 Los 23 24 25 15 i6 17 18 19 2 sdor sy! Figure 88. eh, = eu versus sd = sv. 1 2/3 BOTH ANTENNAS LOW nf <20% 2 2010g A= 20 log A-20 log h, 5 ” eh.4 or eu gs 9 g Thaf 4 2 ! Ome a ey oe oo 7 8 9 10 sdor sv Figure 39. eh, = eu versus sd = sv. DIELECTRIC EARTH Ol 03- 204 10 CALCULATION OF RADIO GAIN sd d (km) Ol 2 10 02 68 \ The d and sd scales may - be multiplied by any suitable power of 10 to accommodate values of d 0S greater than 10 km S = A 4 4 Zz 3 iS} & oN 3% < 2 = 9 3 i aZ 5 VY = a uJ > 2 Relation of A, sd and d 02 Ticurr 40, Relation of A, sd [representing sdf(6)], and d. See Figure 31. f(6) = 1ford >> 1. DIELECTRIC EARTH BELOW THE INTERFERENCE REGION 02 2 d(m) eh 20L0Gg h(m) Ol 1000 300 250 200 200 02 150 100 os 500 ie 400 04 50 ao O05 40 a0 60 300 50 , ao 40 : 200 30 10 Ze 20 6 = 2 an 5 15 oc | 5) ES Gs a 100 A= 6 > 1. See equation (193). If both antennas are low (h < 30d"/*) equation (172) and the accompanying figures (Figures 31 to 41) are valid for all distances d such that ies 2Qhyhe Xr (173) If one or both antennas are elevated, equation (172) is valid only well within the diffraction region of the transmitter, 1.e., for The following quantities required to find 20 log A are given in Figures 31 to 41: sas a function of dis given in Figure 31. 20 log F, versus sd in Figures 32 and 33. 20 log d can be found by using Figure 34. eas a function of A by Figure 35. 20 log g versus eh is given by Figures 36 and 41. When one antenna is low, h < 30d"/*, and the other quite elevated, h > 1,200d?/*, a result valid for hy near the line of sight can be found from the formula and graphs in Section 5.7.5, obtained by summing several modes. A more general method of finding the gain near the line of sight is to use equation (172) well below the line of sight to obtain a curve of A versus /, and by constructing a similar curve for the optical region d eh 20109 g h 20logh NOTE h scale is being multiplied by 10, therefore _* value given by he=1520 ~ eh, must be . multiplied by 10 * eh, =12.3 —Y (SEE NOTE) «— 20 log h=413 20 log g=1,5 Figure 42. represents ehg(6)); lehy = 12.3]. Illustrating use of Figure 41. Note: (eh 12.3] represents [ehog(6) = by the method of Chapter 6, “Coverage Diagrams.” By joining the two curves into a smooth overall curve, it is possible to estimate A in the transition region near the line of sight. For the case of short distances and receiver below the interference region, see Section 5.1.7. 2. For h <4/l. Vertical polarization. A more accurate result can be obtained by replacing the height-gain (gh) by H,/I or in decibels by 20 log Hz, — 20 log l. Hy is given by Figure 47 and | by Figure 46 (see Table 3). 3. Graphical aids (continued). A. Definition of A. Figures 31 to 36 can be com- bined into a form more convenient for numerical computation. In Figure 37, a curve parameter A is introduced, defined by 7 A A hig where g; is a function of eh. This may also be ex- pressed in the form 20 log A = 20 log A — 20 log higs. (176) (For hi < 4/1, 20 log A = 20 log A — 20 log H;, + 20 log J.) Equation (172) can be written as (175) A177 X10" Be y (177) (sd)? or z = P, we 20 log A = — 135+ 20 log + 20logy, (177a) (sd)? where v= (ehz) Ja; with g. a function of eh. Note that F,/(sd)* is a function of sd only and is independent of height. While h; usually represents the transmitter antenna height and h, that of the receiver, the role of /; and hy. in equations (176) and (177) may be interchanged. To facilitate the use of Figure 37, three nomograms have been added (Figure 40 gives sd when \ and d are given. Figure 41 gives eh, 20 log h and 20 log g when \ and h are known. Figure 43 gives the modi- fied height h’ and distance d’ for given h, d, and k.) To find sd for a value of d which is not on the nomo- gram, say 120 km, find sd corresponding to a distance 100 times smaller (i.e., 1.2 km) and multiply result- ing sd by 100. Proceed similarly for eh. B. Both antennas low. In the case of both an- tennas low, hy and h, < 30”, the contours 20 log A BELOW THE INTERFERENCE REGION 105 h d eo i 10—==—100 h d 20 109/31) %s 10 100 4 fe) 4/3 50 2 40 50 30 P 3 40 20 4 4 5 30 5 10 10 5 10 20 =) 2 4 eee 3 15 20 2 m, 30 3 i] i] 56 40 50 5 4 5 3 G 25 100 2 4 2 4 3 | 200 3 30 300 5 05 is wy 400 oe e 500 02 i 1 1000 i Ol Ficure 43. Relation of h,d to h’,d’ as a function of k. 106 CALCULATION OF RADIO GAIN are given by Figure 39. If both antennas are so low, say, h, and hy < 4/l (see Table 4) that it is desired to use Hy, for greater accuracy for vertical polariza- tion (Figure 47), then for 20 log A we take 20 log A + 20 log 1 — 20 log Hy, (see Section 5.7.3) and instead of eha in Figure 39, we use as ordinate eH72/l. Then for given sd, Figure 39 would give the value of eHz./l. If the frequency is given, e/! is known (Figures 35 and 46), and we must find the value of Iho which corresponds to a known value of Hy» (Fig- ure 47) for the appropriate value of Q = «,/600. From Iho, he is found by dividing by 1. If only one antenna height is less than 4/1, then de- fine 20 log A as 20 log A + 20 log 1 — 20 log Hy, for that antenna and /, then refers to the other antenna. C. Non-standard atmosphere. k # 4/3. The pre- ceding graphs are all based on k = 4/3. Uk # 4/3, hy, ha, d, and A should be replaced by hy’, he’, d’, and A’, where ih oh rs, + Bk\? i — al 3k\e? = Lay A 1 (2) \2/3 20 log A’ = 20 log A + 20 log (#) ; (178) or The change of h,d, A to the primed values can be made with the aid of Figure 43, ie., if h,d are known, change to h’,d’, then Figure 37 will give A’, which in turn will give A’, and this with the aid of Figure 43 will give A. D. Change to dimensionless coordinates. In the op- tical region, convenient coordinates are (see Section 6.5) d d v= = V2kahy dp ho u=— hy For these coordinates, equation (175) becomes ge aaa (179) hig(e) Writing e=eh, 7 (180) s = sV2kah,, it follows that ehy = eu, sv = sd, and, using equations (150) and (159), Ss? = 2e. Consequently, Figure 37 can be used with sv replacing sd, ew replacing eho, and A is defined in equation (179). Caution: In using the graphs, care must be exercised when one or both antennas are elevated to see that the receiver antenna ts well within the diffraction region, 1.€., d >> dy. (181) EK. Illustrative problems; diffraction formula; di- electric earth. In Section 5.6, four types of problems were considered for the optical-interference region. The same four types are given here, for a receiver below the optical-interference region. A dielectric earth is assumed so that the figures in Section 5.7.3 are applicable. These require supplementing by equations (3) and (5). For one-way transmission the radio gain is 10 log - = 20 log A + 10 log (GiG2). (182) 1 For two-way transmission the radar gain is 10 log Polos Ane 1010s Cas) : +7.5+10log>—20logd. (183) Type I. The heights and distance apart of the transmitter and receiver antennas and the wave- length are known. The radio gain is to be found. An early-warning set has a horizontal antenna, located 118 meters above sea level. A receiver is located in an airplane 1,520 meters above sea level, at a distance of 300 km. The wavelength is 3 meters. The gain of the radar antenna is 96 db and its power output 100 kw. (a) The power received by the air- plane receiver, assuming a gain of 10 db, is to be found. (b) The power returned to the radar by the airplane, assuming that the airplane has a radar cross section o of 40 square meters, is to be found. One-way: From Figure 2, dz, = 205 km. Hence the receiver may be assumed well within the diffrac- tion region. From Figure 40, sd = 9.3, with f(6) =1. From Figure 41, eh: = 12.3, with g(6) = 1. From Figure 37, 20 log A = — 2138. BELOW THE INTERFERENCE REGION 107 To convert 20 log A to 20 log A by equation (175), we need 20 log hy and 20 log g;, which are given by Figure 41: 20 log hy = 41.3, 1.5. ll 20 log gi Hence 20 log A = and by equation (182) = ko), 10 log = = —170+ 96+ 10 = —64. 1 Since P; is 10° watts, 12 = MOP SC WO Oy Radar: Substituting in equation (183), the radar gain is given by 10 log (P2/P1) —340 + 2(96) + 7.5 + 16 — 9.5, — 134 db ll or P, = 125 x< 107 = The power output P; is 10° watts, so that the max- imum received powe1 P, = Ome Ww. The minimum detectable power of the set is given as 1.6 X 10° = 107° watt, so that under the given conditions the power returned by the target would be slightly below the threshold of detection. Type II. Gain versus receiver (or target) height is to be found for given distance, given wavelength, and given transmitter height: A radar has an antenna height of h, = 30 meters, a wavelength of \ = 1.5 meters and a distance from a receiver (or target) of d = 100 km. Assuming a receiver antenna gain G2 = 1, the variation of P:/P; at the receiver with receiver height is to be found. Also, assuming a target of cross section o = 50 sq meters, the varia- tion of P2/P,; at the radar receiver with target height is to be found. The radar antenna has a gain of 13.5 db. One-way: sd = 3.9 (from Figure 40). From Figure 37, for the fixed value of sd = 3.9, we find a correspondence between values of 20 log A and ehs, listed in Table 5 below. By means of Figure 41 or equation (159), ehs is changed to he and by means of equation (176), A to A. From Figure 41, it is seen that 20 log h; = 29.5 and 20 log g, = 0. To change A to P2/P;, the transmitter gain of 13.5 db and the receiver gain of 0 db must be taken into account, according to equation (182). The result is given in Table 5. The values of 10 log (P:/P,) are plotted in Figure 25, together with the results found with the same data in Section 5.6.3 for the optical-inter- ference region. TaBLE 5* ho | Radio Radar Meters | ehe | 20 log A | 20 log A Gain Gain in db in db 63 0.8 —190 —160.5 —147 —273 142 1.8 —180 —150.5 —137 — 253 259 3.3 —170 —140.5 —127 —233 417 5.3 —160 —124.5 —117 —208 * See also Table 1 and Figure 25. Radar: 10 log = 40 log A + 27+ 7.5 + 10 log o . —20logts, = 40log A + 274+ 7.5+ 17 — 3.5, = 40 log A + 48. Type III. Gain versus distance is to be found, with antenna heights and wavelength given: Using the same data given in Section 5.6.4, the gain as a function of distance in the diffraction region is to be found. The result has been plotted in Figure 26. The polarization is horizontal. G, = 22.4 (13.5 db) G's (one-way) = 1 (0 db) Gz (radar) = 22.4 (13.5 db) ao = 10 square meters hy = 30 meters hg = 1000 meters \ = 1.5 meters From Figure 41, eho = 11225; Referring to Figure 37, we find a correspondence between A and sd. Values of A are to be assumed. To change sd to d, use Figure 40. To change A to A, use equation (176). From Figure 41, 20 log 30 = 29.5, 20 log gi = 0, 20 log A = 20log A + 29.5 + 0. The radio gain is then given by: One-way: 10 log = = 20 log A + 13.5+ 0; 1 The radar gain is then given by: Radar: 10 log = = 40 log A + 7.5 + 27 + 10 — 3.5 1 = 40log A + 41. 108 CALCULATION OF RADIO GAIN These equations are evaluated in Table 6 and the one-way values are plotted in Figure 26. TABLE 6 Radio Gain | Radar Gain 20 log A} sd d |20log A in db in db —170 6.2 | 159 | —140 —127 —239 —180 6.9 | 177 —150 —137 —259 —190 | 7.6 | 193 | —160 —147 —279 —200 | 8.3] 213 | —170 —157 —299 —210 | 9.0 | 231 | —180 —167 —319 —220 | 9.7} 249 | —190 —177 —339 —230 | 10.3 | 263 | —200 —187 —359 —240 | 11.0 | 282 | —210 —197 —379 —250 | 11.8 | 305 — 220 —207 —399 —260 | 12.5 | 320 | —230 —217 —419 —270 | 13.2 | 340 | —240 —227 —439 —280 | 13.8 | 357 | —250 —237 —459 Type lV. The determination of contours along which the radio gain (or A) is constant (the coverage problem): A radar has a wavelength of 0.107 meter and a power output of 750 kw. Assume a receiver in space with a minimum detectable power of 10° watt. The maximum possible distance between the radar transmitter whose elevation is 100 meters and the receiver for varying heights of the receiver is to be found. The gain of the radar antenna is 10,000 (or 40 decibels), the gain of the receiver will be assumed to be 30 decibels. For the radar problem, a target of radar cross section o = 50 square meters is assumed to take the place of the receiver. The minimum detectable power of the radar is taken as 10° '° watt; the range of the set for varying altitudes of the target will be calculated. One-way: The radio gain sought is the ratio of the minimum detectable power to the power output, or ie 10e2 a x 1018 P, 750X110? 3 or 10 log (Pe/P:) = —160 + 1 = —159 db. From equation (3), 20 log A = — 159 — 40 — 30, = — 229 db. From Figure 41, 20 log g: = 18.5, 20 log hy = 40. Therefore 20 log A = —229 — 40 — 18.5, = — 287.5 db. Referring to Figure 37, the pairs of values of sd and eh, along the contour 20 log A = —287 are given in Table 7. By means of Figures 40 and 41, sd and ehg are changed to d and he. The points found are to the right of the curve for eh = 7.3, so that they correspond to points in the diffraction region. TABLE 7 dim sd eh hy meters 118 11 1.3 18 129 12 3.5 48 140 13 7.0 96 151 14 iil {5) 158 161 15 16.5 226 Radar: The value of 10 log (P2/P;) is the same as for the one-way calculation, —159 db. This must be changed to 20 log A by equation (5), 20 log A = —141 db, and, as above, 20 log A = —141 — 40 — 18.5, = —199.5 db. Referring to Figure 37, we see that the contour 20 log A = —199.5 is to the left of eh: = 7.3 [see caution in equation (181)]. Therefore it is not possible to get the necessary power return P2 from the given target so long as it is below the line of sight. The desired contour would he above the line of sight. The determination of the contour is discussed in Section 5.6.5. 5.7.4 Sea Water, VHF, Vertical Polarization 1. Graphical Aids. Graphical aids are given in this section, which, as in Section 5.7.8, are valid for all practical distances when both antennas are low [h < h, (see Figure 35)! and 2hyhg << dd. If one or both antennas are elevated, they will give the value of the radio gain for the first mode, which is a good approximation for the result found by summing all the modes when the receiver is well within the diffraction region, i.e., when d > d,. [If one antenna is low (h 40h,), the result found by using several modes is given in Section 5.7.5. | BELOW THE INTERFERENCE REGION 109 Referring to equation (162) and Section 5.7.1, A = 2A0AiF (A 799’): Ax99')2, (184) with gg’ = 1 forh < h,. h, is given by Figure 35, 20 log gg’ is given by Figure 36, 20 log Ao is given by Figure 3 in Chapter 2, F, is given by Figures 31, 32, and 33. 2. Plane earth factor A,. This has been discussed in Section 5.7.1. A; is a function of p’d; p’ is given in Figure 44 and 20 log A, in Figure 45. The curve shift of A; with \ in the VHF band is less than 1 db. When p’d > 50, Ai = 1/p’d, as in the dielectric ease. 3. The low height-gain Hy, is a function of lh (see Section 5.7.1). lis given by Figure 46, H;, or 20 log Hy, by Figure 47. Hy, depends on the curve parameter H= H,—lh for lh > 4, 20 log Hz; can be found from Figure 34. 4. hy, hp > 4/land d > 50/p’. As in Section 5.7.1 [noting especially equations (151) and (153)], equation (184) reduces, when h > 4/landd > 50/p’, to e,/60c0X which for sea water is = Since Ae oe ams 185 2 @ 8) (gg'h)i(gg’h)o TaBLe 8. Values of 4/1 and 50/p’ for various wavelengths. Se lene 2a csh 4 Se 6) i) 7) 18) 9) 108m : 3 | 27 | 50 | 80 | 110} 148 | 174 | 222) 267 | 308; m — | 2 | 17) 17 | 80 | 50) 71) 100} 125) 175 | 200) km Equation (185) may be used generally, provided (gg’'h) is replaced by H;,/l when h < 4/l, and the right-hand member is multiplied by Aip’d or, in decibels, 20 log A, + 20 log p’d are added, when d < 50/p’. In this formula d < 50/p’ is given in meters. 5. Horizontal versus vertical polarization. It is of interest to compare the gain of vertically and hori- zontally polarized waves over sea water at VHF. (It has been pointed out earlier that there is no marked difference in attenuation between horizon- tally and vertically polarized waves for wavelengths less than one meter.) Equation (184) is valid for horizontally polarized waves also by using the appropriate /, curve and putting g’ = 1 and can be made the basis of a comparison between vertical and horizontal polarization. See, for comparison, equation (160). p’ FOR USE WITH Ay SEA WATER VERTICAL POLARIZATION Az1-10m 01 .005 .004 .003 Pp’ 002 001 0001 eS 3 4 5 6 it 8 9 10 1 WAVELENGTH )} METERS Ficure 44. p’ versus ) for sea water, vertical polariza- tion. aca = PLANE EARTH GAIN FACTOR Ay SEA- WATER VERTICAL POLARIZATION ° oO | | I I [ | LC] | | | fl | \\ | | [ssf | | ea} - | } FOR P'd >140 USE FIG. 34 hp PUTTING x= p'd | | | [| | oO on + | | ot | ef BIE I j t T | |e | | | 30 | 30 | | | [ a] | [ | ii [ | | i | (EI [| | (al | [ [zal [ | 40 1 - ] 7 T {a0 — 10 20 30 40 $0 60 70 60 90 100 no 20 130 pd Figure 45. Plane earth factor A,, for sea water, vertical polarization. Note: All db values are negative. 110 CALCULATION OF RADIO GAIN FREQUENCY IN MEGACYCLES 4900 2000 400 200 40 40 5000 3000 300 a i see BPAPAS. Saeki LA aa a 7a Za) Sth SEA WATER | L 2% Ver-+(60on? ~ VE? + (60oA)? 004 VERTICAL POLARI: 1 3 WAVELENGTH IN METERS Fraure 46. Parameter / versus frequency for vertical polarization. BELOW THE INTERFERENCE REGION 111 For antennas at, or very close to, zero height, the gain-factor ratio depends on Ai/’,. While F, gives greater attenuation (lower gain) for horizontal than for vertical polarization, the difference between the two lies principally in the values of Ai. For \ = 1 meter, the ratio is 64,000 to 1 in favor of vertical polarization. For \ = 10 meters, the ratio is about 8.6 X 10°. However, as the antennas are raised above the ground, the strength of a horizontally polarized field increases much more rapidly than does the corresponding vertically polarized field, for a given which, as in Section 5.7.3, can be written A=1.77X 107] Fe | ap'ay (188) (sd)? or o F, 20 log A = —135 + 20 log —* + 20log A (sd)? +20 log p’d + 20logy. (189) Since Ff, has a graphical representation which de- pends on the wavelength, it is necessary to assume a particular value of \; equation (188) then becomes 20 aa | 25 10 Stat + 20 Saas al iE 8 | c imal 15 He + | HORIZONTAL ») | POLARIZATION LI 105 O fe) =) 2 Lee | a 5 Oa Q= VERTICAL 6002 OLARIZATION i] B t (e} —o 0.6 fea [ 006 0.1 0.2 0.4 06 1 2 4 6 10 20 th Ficure 47. Height-gain function H;, versus lh, for low antenna heights. [See equation (152.)] wavelength up to a certain height above which the field is substantially independent of polarization. For example, above a height of 3 meters for \ = 1 meter, and above 77 meters for \ = 10 meters, the two fields are practically equal. 6. Parameter A. As in Section 5.7.3, curves can be drawn in terms of the parameter A where as for hy > 4/1, a (hgg')1 A= (186) — forh, < 4/1. ib Equation (185), including the correction for d < 50/p’, becomes 3 F, y 7 / A = ~—(Ajp'd) (gg’h)i(gg’h)o, 187 an (187) a relation between A, d;, and fo, and Figures 48, 49, and 50 for \ = 1, 3, 6 meters are in terms of these coordinates. The height-gain function of the trans- mitter g; can be found from Figure 41. 7. Illustrative example: Communication. A com- munication set used in ship-to-ship work has a wave- length of 1 meter, a receiver sensitivity of 10 micro- volts with a resistance of 50 ohms across the input terminals and a transmitter power output of 100 watts. The transmitter and receiver antennas are vertical half-wave dipoles at an elevation of 30 meters. The range is to be found. To produce a voltage of 10 microvolts across 50 ohms, a power of - -12 4 P, = ees watt = 2 x 10” watt, R 50 BBB 2222 : Hee . ty PEM | 7 hy Mag CL ; th mae ET \LOGh, 20 LOG gg: * = Rn oO = w 8 ro) = ro 8 a ed D & m iv 2) <3 3) 2 i w oO fe) 10,000 MIEEEUAGAACAAE. T My - Bo WAN AA AAG Aan HEL TELAT EU TEE | 25 50 75 100 «=l25 150 175 200 225 250 275 300 325 3 d IN KILOMETERS Ficure 49. Maximum range for \ = 3 meters (sea water vertical polarization — 20 log g’ = 1). See equation ( ae EES GSS 50: 186)- 100000 20000 wm 1000 fannr eegeacr car a areitienice mil eal FINITE ECCI cs yee - LES gaceee WEEE LV PIL VVVIAAA ZV le IN KILOMETERS BELOW THE INTERFERENCE REGION 115 so that this is the minimum detectable power. The value of P2/P; for the given power output of 100 watts is —12 BS WO 52 10 100. and Toon (140) 21372 P, This is to be changed to 20 log A by equation (3). The gain of a half-wave dipole over a doublet is 1.09 (see Sections 2.2.2 and 3.2.3), so that G; = Gz. = 1.09 or 0.4 db, 20 log A = —137 —0.8 = —138 db. In changing from 20 log A to 20 log A it must be determined which of the relations in equation (186) is required by comparing the transmitter height of 30 meters with 4/1. The value of J as given by Figure 46 is 0.4. Hence the value of 4/1 is 10, which is less than 30. Then 20 log A = 20 log A — 20 log 30 — 20 log gg’, = — 138 — 30-0, = — 168. Referring to the chart for \ = 1, Figure 48, we and that for h: = 30 meters and 20 log A = — 168, the distance d is 53 kilometers. maximum theoretical range between the two sets. °75 Radio Gain Near the Line of Sight For d much greater than d;, the first mode is sufficient, as given in Sections 5.7.3 and 5.7.4. For d nearly equal to dz, i.e., the receiver near the line of sight, a formula [equation (190)] can be given which takes into account several modes and still permits the use of graphical aids. This formula is valid only when the elevated antenna is very high, ie., h > 1,200°" and the other antenna is low, ice., h < 30°. (Otherwise the transition curve near the line of sight must be sketched in graphically, as indicated by the broken portion in Figure 7.) Denoting by Hz; the height-gain of the low antenna at height fi, A = 2AgH is M(8) EAs 2ehs (190) where hz and F(A) refer to the elevated antenna. This then is the 1. sd and eh are given in Figures 40 and 41. 6 is given in Section 5.7.6. 2, 4= Vq(8) (sd — V2eh). 3. g(6) = polarization, over sea water. Table 9. 1, except for the VHF range, vertical The values are given in TaBLe 9. Values of V (6) for VHF (sea water). 10 meters N ie ie Bot Si Ora SEO vg(6) 0.98 0.97 0.95 0.94 0.98 0.91 0.90 0.88 0.86 0.84 [| HE EHH at “: sooo MITTIN CHUI A Figure 51. F(A) [representing /’,(A)] versus A for use in equation (190). 4. F(A) is given by Figure 51. 5. M(6) for vertical polarization is given by Figure 52 as a function of 6 which can be found from Figures 53 and 54. For horizontal polarization 1/3 =( d ) A Lee E (101) 2rka Ve = = 1p + (GOod)? 116 CALCULATION OF RADIO GAIN 576 General Solution for Vertical (or Horizontal) Dipole Over a Smooth Sphere 1. Field strength of dipole. The vertical component of the electrical field of a vertical dipole radiating in a homogeneous atmosphere over a sphere of radius ka (or horizontal coniponent in the case of a horizontal dipole) is given by equation (192). The solution is valid provided the distance between For horizontal polarization, \ 2/3 res (C) Caan IN _ 14.2 x 10! SUE d. ¢ = sd, where recelver and transmitter and the radius of the 10 +20 +10 FROM FIG 54 | to) 0.5 0.4 MII os =10 0.2 DB Out -20 se FOR |§|> 1000 .04 m= |8|-/2 .03 = -30 02 201 -40 0,01 0.02 0.05 Of 0,2 05 #10 2 5 10 20 50 100 200 500 1000 | Ficure 52. sphere are much greater than a wavelength, condi- tions which are fulfilled in any practica! application of short waves. 20 pI TS E = 2K)(2 gi? > eee nh n (he) . (192 TS ~ 5. She ) a ) a. hy and hy are antenna heights, b. Eois the value of # for a doublet in free space, c. 6is the ground parameter which depends on the complex dielectric constant €, = ¢,—j600r. For vertical polarization, 142 10'e« —1, 4 = — ——_ ee 2/3 2 (193) M(6) versus | 6 | for use in equation (190), vertical polarization. OF a 4 2/3 s = 4.43 X 107°N" (4) ; 3k e. 7, are complex numbers which characterize the individual terms (modes) in equation (192). They are a function of 6. f. f,(Ai) and f,(h2) are height-gain functions for the nth mode. (194) 2. The complex ground parameter 6. 6 depends on the wavelength and the electrical constants of the ground. (The dielectric constant is referred to air as unity.) 6 is large for horizontally polarized waves, but for vertically polarized waves it may vary con- siderably, as may be seen from Figure 53. In Figure 54, the phase of 6 is given. For wavelengths less BELOW THE INTERFERENCE REGION LINZ Ft4,= 4, O=.001 || —— Ne -OO| a —- = SCT SSesos OSS FS @ERn0 ee . R= > a INN ia LY Bezel , Nw NU NN NT \ aH Tt] AOL Ti ae ii Ht a | | ei = ieee NK uN Lae oath AWN ——=_ Sevava aati [ oh 98 SANS O. EW NUN At =| "sO [==] 107! 2a 46 Hee a ie r ie | tI Cry + s co eZ ag Ae ‘lies | aun I asec Spek ANN aN aN \ at UUM FLT, INANE 10 | 10 102 ‘o3 104 \ IN METERS Figure 53. | 6 | versus \ for vertical polarization (see Table 10). Dielectric earth, 6— ; perfect conductor, 6— 0. Phase of 6 is 45° for intersection of two asymptotes for any particular ground. 118 CALCULATION OF RADIO GAIN than 10 meters | 6 |, as given by Figure 53 for vertical polarization, is large except in the VHF range over es () eel sea water, €, The ground constants ¢«, and o for water and various types of earth are given in Table 10. For » < X, the ground material is a dielectric earth. Tasie 10. Ground constants. Type of ground hee cs : = = A0e Sea water er = 80 o = 4mbhos per meter 0.33 m Fresh water e =80 o =5 X10 267 Moist soil e- = 30 o = 0.02 25 Fertile ground «¢=15 o« =5X1073 50 Rocky ground «,=7 o =10°3 117 Dry soil e=4 o =10- 6.7 Verydrysoil e =4 o«a=107% 66.7 PHASE OF 6 IN DEGREES (61) [o) °o f\ eae 10 el mail / way vs Zaalll 3. The mode numbers r,,. below for the two limiting values for 6 = conductivity orA\— o), andforé6 = o,Le., tric earth. These numbers are given 0 (infinite the dielec- Ne | 6= 0 | 6 =o ary: | aay 1 |71,0 = 0.885 e77/8 | 71 = 1.856 27/3 2 |t29 =2.577 et/8 | 72 oo = 3.245 e 77/3 3 | 73.9 = 3.824 7/3 73.0 = 4.382 e27/3 >4| lin = 4, | lin > 4, 7,0 =3[3m(n+3)]? SE ee =3[8m(n+2)]? sedis 102 io? XIN METERS Figure 54. Phase of 6 for vertical polarization. See Table 10. BELOW THE INTERFERENCE REGION ILLS) From these limiting values of 7 the value of 7 in general for any given 6 can be found from the follow- ing two series, of which only the first is of interest in short-wave work. 6 large, =- 9) 2, —3 /2 1 —2 4 9 ce) tn = tap 0 NEA =Trj9 We SS iy Sen) ee 3 2 5 seco 0% (195) 6 small, One 5 Tn Tn = zi 4 27m 0 871.0 wil : (1 4 bee 1270 47 n'0 4. The height-gain functions f(h). (a) For low antennas (h < 30d”), the height-gain functions, to the first approximation, are inde- pendent of n. Ve,— 6) =e | for vertical polarization, € (196) B27: jee i nite fh) =1+9 ne Ve,—1 |forhorizontal polarization Note that the magnitudes of the bracketed quantities are equal to lh. The magnitude of f has been denoted by Hy, and is represented in Figure 47 as a function of lh. The phase of the bracketed quantities in equation (196) is taken into account by using differ- ent curves with the parameter Q = ¢,/600d. For large values of lh, Hy — lh. (b) For elevated antennas (h >> 30d°’*). function f, can be represented by The = 3 1 exp {+79 — jy [(eh) a ime a V2zx (2eh)!/4 J /3(2) + J-1/3(2) (197) where, from equation (159), 9) 1/3 -2/3 / 4\1/3 eats = (+) (198) (2ka)*/ 60 \8k and where the argument of the two Bessel functions is ay 3 ( as Deer On For the nth mode, if (eh) >> 27, the magnitude of f, can be written 3 1 exp(jr» V2ch) |V2m (2eh)! 4 Jy /3(a) + J-1/3(2) (199) For large 6, using the first two terms of equation (195), and writing x, for x when 7, is replaced by —27 u/2 ae -( 3) : 6 Substituting this in J, /3(v) + J_1/3(x), writing down the first two terms of the Taylor expansion, making use of the fact that the 7,,, are roots of Jy/3(a) + J_4)3(%) = 0 and of the relation given by a prop- erty of the Bessel function, Jy /3'(x) + J-1/3'(t) = — 1/82) [Ji /3(2) + J-13(2)] + J_2/3(x) — J2/3(2), Ue) 4b SS we have J, 13(@) ab Jy 3(2) 1/2 ie ic 22) [J-2/3(t,,) — J2,3(z,,)]. (200) If these results are substituted into equation (192) for both antennas, the factor (6 + 27,) becomes 1 + 27,,,/6, which approaches unity for large 6. This means that 7zf both antennas are sufficiently elevated, short-wave propagation is practically inde- pendent of ground constants. The value of f given by equation (199) can be written as glh so that g represents the gain over lh, the value approached by H;, when lh > 4. The value of g for 6— = is represented in Figure 36. If 6 is not very great, as in the case of vertically polarized VHF over sea water, the effect of 6 can be taken into acocunt by changing e to eg(6) and g to gg’. The functions g(6) and g’ are given by Figure 55. 5. Plane earth gain factor and shadow factor. The field near the ground over a plane earth with infinite conductivity is equal to 2H , twice the free-space field. For an imperfectly conducting ground, the field for antennas at zero height over a plane earth may be written B= 2EyA,. (201) A, is represented in Figure 56 as a function of p’d, where , _ 2n|le—1| _ 24 V(e — 1)? + (600d)? r | €, [2 oN e + (600d)? p (202) for vertical polarization. For horizontal polarization, 1 . = e- -1|=~V(e, — 1)? + (600d)*. (203) n aan The curve parameter is Q = €,/60on. 120 CALCULATION OF RADIO GAIN Comparing equations (202) and (203) with height [ie., f(0) = 1] and equation (205) is now of equations (193) and (194), we find that the form p'd = | 6| ie (204) E = 2EvAiF, (208) Hence, equation (192) may be written as If 6 is large (e.g., \ small), 27,,/6 in equation (207) rg = may be neglected and 7, replaced by 7,,,,80 that the E = 2E,(20)1” Ne ee fs (du)fg(ha) (205) shadow factor is practically independent of ground pd == toe a constants. The shadow factor is represented graphi- 0,001 0,0! 0.1 Figure 55. If p’d is large, we see from Figure 56 that 1 pd’ oe A,= so that the physical significance of 1/p’d in equation (205) becomes apparent. The factor represents the effect of earth curvature in increasing the attenuation over that of a plane earth at zero imaginary part of Sie Le ite OECIBELS V g(6) and g’ as functions of magnitude and phase. cally in Figures 32 and 58. Where the factor 27,/6 cannot be neglected, as in the VHF range, vertical polarization over sea water, the dependence of F, on 6 is accommodated by changing the abscissa from ¢ = sd to 7 = 8s'd where s’ = sf(6), and by representing /’, by a family of curves in Figure 58, whose parameter is given by dotted lines in Figure 57. f(6) is represented also in Figure 57. For” < 0.4, F, is less than 1 db below unity. This corresponds to a distance over which the earth may be considered plane, ie., d < 10’, as given in Section 5.7.1. The greater the wavelength, the smaller the effect of the earth’s curvature. BELOW THE INTERFERENCE REGION $1381930 ooo! 25 100'0 a ose c S000 GbE 7 BS FOO Ove ral ilexe) Qb aD 4 ce x i?) b Zz oe n Lb g <2 Sy He 02 a 0 _ m gle fo) Ol a s‘0 (6) Ol ‘09< pd 103 (p,d)/{ = 'y ‘p,d snsiaa ty IOpoR] ures YIvI oUB[G “9G MUNDI NOILVZINV10d IWLNOZINOH HOS 3(X.0 09)+ a(t-43)/\(x/p uz)=Pd 3(¥.009)+ 479 (x.0 09) + 73 NOILVZINV10d WOIMLYSA YO TD Ae dal =pd 2X0 09)* ,(1-79) X7P Lz Pi 00s 00! os ol g o1 s'0 0 Soo (exe) 100°0 X009_ "| 1E a5 poet La | | E if in | s00'0 : 10°0 fa] | aealae eee foe) ot ryt t ike) O=0 dH @=0 dH dA OPE | | Bx oho) f oO" MYOLOVS NIVD HLYV3 3NV1d ie) bo CALCULATION OF RADIO GAIN < Perfect Conducto’ Dielectric Eart h—y Figure 57. (6) versus 6. Solid lines correspond to phase of 6. Dotted lines indicate curve in Figure 58. For horizontal polarization f(6) = 1. -20 “ 6 2) Q : < to -400 oO 3 “ fe} < PA ” 10 -60 -4 10 -80 ON 0.5 1,0 5 10 n=o (6) Figure 58. Shadow factor I’, versus 7 = (¢f(6) = sdf(6). See Figures 32, 33, and 57. Curve +10 corresponds to a perfect conductor. BELOW THE INTERFERENCE REGION 123 If the antennas are elevated, F, can still be used for the first mode for great distances where the first mode gives most of the field. Sample Calculation for Very Dry Soil The general solution given in Section 5.7.6 is here illustrated for the case of doublet antennas, either function of distance d for doublets at zero height. Figure 60 gives the first mode height-gain factors for transmitter and receiver heights, i; and he, respec- tively. To obtain the radio gain under different conditions it is merely necessary to add the decibel gains of the transmitter and receiver antennas, the radio gain for zero height (Figure 59), the height-gain factor (Figure 60) for the transmitter, and a similar figure HA “seep UL 20 I] = ae +H = : —- 5 : My Pate (aco ss SSS SS SS Es et et a | (Sh ares = al PLL VAViAY, H Contd ey STR SCT i Sa GAIN -300 | RADIO =a =z HEE ERS eae = 4 -500 a Ss Ba Be Ss BS eee 100 100,000 1,000,000 10,000,000 DISTANCE d IN METERS Figure 59. Free-space radio gain 4) (—~—) and radio gain A, in decibels, for propagation over very dry soil with doublet antennas on the ground ( ~ for horizontal polarization and —— for vertical polarization). Numbers on the curves give the wavelength A. Note: Radio gain is independent of the radiated power. horizontally or vertically polarized, placed at various heights over an earth assumed to be very dry sozl for which the constants are «, = 4, and o = 0.001 mho/meter. The following graphs cover, in decimal steps, the frequency range of 30,000 to 0.03 me or wavelengths \ = 0.01 to 10,000 meters. Figure 59 gives the free space radio gain A» and the radio gain A decibels over very dry soil, as a for the receiver (Figure 60). This process, however, is subject to the restriction mentioned in the next paragraphs. The addition of the factors given in the preceding paragraph is valid all the way up to the maximum of the first lobe, where the field is given by the sum of the direct and reflected rays, provided that the antennas have comparable heights. 124 CALCULATION OF RADIO GAIN The radio gain can therefore never be more than 6 db greater than the free-space gain with the same antennas. If, however, one antenna is low, h < h,, and the other is very high, h > 40h,, the method discussed 210 = BALI og a four tables of computations are given. Table 11 gives the values of certain quantities, for a wide range of frequencies and for very dry soil, which are independent of polarization. Those quantities which are dependent on polarization are given in Table 12. —— 180 LTT TT = = 0,001 MHOS/METER |_| VERY ORY SOIL €,=4 ) BASED ON FIRST MODE ONLY Use @- MARKS HEIGHT h, ABOVE WHICH 140 FIELD INCREASES EXPONENTIALLY DECIBELS fo) ao 100 1,000 HEIGHTS h, OR hy IN METERS Ficure 60. fails since the height-gain factors are based on the first mode only. In this event, either the methods outlined in Section 5.7.5 must be employed or the radio gain at low elevations must be connected graphically with the value obtained in the optical region for the first maximum. As an aid to the computer in checking his results, First mode height-gain factors for transmitter and receiver. Table 13 gives detailed calculations for ground level radio gain for doublets for a wavelength of \ = 1.0 meter, while Table 14 gives the first mode height- gain factors for the same wavelength. Similar charts and tables may be prepared easily for transmission over other types of earth for a similar range of frequencies. 125 BELOW THE INTERFERENCE REGION “910'09E GZ | S10'009 (00¢/— T)€ | e-O1 X €0Z°0| an Ng HEIGHT IN KILOMETERS n VERTICAL COVERAGE DIAGRAM ANTENNA HEIGHT 32.3 WAVELENGTHS FOR 106 MC ' he 914m Ve2° 192 224 256 d DISTANCE IN KILOMETERS FIGURE 3. 4 THE p-q METHOD (HORIZONTAL POLARIZATION) ads Outline of Method This method consists in plotting the locus of points having a constant range d and locating those points on this curve which are at such a distance from the transmitter that the phase shift caused by path difference corresponds to the required range. The range corresponding to a total phase shift is given by =f 9 d = VGidy VD) 2) ee (12) Vertical coverage diagram. hy. The difficulty of the problem consists in the fact that equation (12) provides an extremely complicated relation between hy and d which cannot be solved explicitly for either coordinate. Under such circumstances, the natural procedure is to introduce new coordinates which make the handling of equation (12) easier. The method de- scribed in the following makes use of the variables and q= aati ae d eid discussed in Sections 5.5.5 and 5.5.7, and the pro- cedure will be called the p-g method. THE p-q METHOD 133 It may be recalled that expressed in coordinates p and q Qa. oth, (13) = An? -1/2 D -|1 pee | (14) (il = 7) and the total phase shift, by equations (97) and (29) in Chapter 5, is o a Arh al = P') +’. (15 dr p Y ) then proceed in the. following manner. To a fair approximation, we may assume the extreme range of the lobes to correspond to sin?(Q/2) = 1, so that by equation (12) the corresponding distance dyax 18 given by Ohrveres = VG, do(1 + D). (16) Expressing dma,and D by p and q, the above equation determines the envelope of all lobe maxima. The practical way of doing this is to use a graphical representation of D in p and q coordinates (Figure 17 in Chapter 5) and to start by selecting a particular 15 140299 50 DIVERGENCE D — — — PATH DIFFERENCE R ————: =~ 7 i} SAIL| CO || PL 13 Tod eat 10 = ua “ae a { aa — we : 1D=.40 SS SS So °o NF d V=— dy Figure 4. Curves of constant-divergence factor D and path difference parameter R. For horizontal polarization, @ > 0, so that for this case (which is the one under consideration) all variable quantities in equation (12) have been expressed in terms of p and q. 642 Construction of Range Loci Suppose to start with that we want to compute the position of the extreme range of a lobe. We may | (Radiation Laboratory.) value of D, say D = D,. Inserting this in equation (16) gives a corresponding value of d,,ax, and insert- ing this value of dmax for din equation (13) determines a straight line in the p,q plane, since dr = V2kah is known. Whatever is the value of dyax, this line passes through the point g = 1, p = 0. In order to determine the position of the line, only one more point is needed. A convenient point to choose is to take q = 0.9 and compute the corresponding p from 134 p = 0.1dmax/dz. The point of intersection of this line with the selected D,-contour then gives the desired p,q combination that corresponds to the given values of dmax and D,. From this pair of values (p,q), the corresponding receiver height hy may be calculated by the relation {equation (98) in Chapter 5] q [r- + BL | I (0) 1G, hs — hy oe 180 160 140 y=he hy 120 fA 100 80 60 eZ Z s 40 20 A ee eC. = —— ——— 0) SS ————— Ss SaaS DIVERGENCE D — — — PATH DIFFERENCE R \ ce) iS COVERAGE DIAGRAMS FIGurReE 5. Now both coordinates of the desired point are known, and the point may be plotted. Plotting a series of different points by the same method yields a smooth curve, which is the envelope of all lobe maxima. The locus of minima may similarly be plotted by using sin? aa 0 and (thai ma VGydo(1 a D) (17) instead of equation (16). Intermediate range curves are found by assigning nonintegral valuesto m in the equation 2 = m7 and substituting in equation (12). oe Construction of Path-Difference Loci An assumed value of 2 in equation (12) determines 6, the phase shift caused by the path difference, as 6 = 2+ 2an, (18) where n assumes all integral values and zero. This value of 6 determines the path difference A = r — rq, (19) Curves of constant-divergence factor D and path difference parameter R. (Radiation Laboratory.) But from equation (97) in Chapter 5 Aye ai — p’)? _ 2h ofp) (20) dp dr p where f(p) is given in Figure 18 in Chapter 5. In this calculation, gq may be taken as the inde- pendent variable. The assumed values of ¢ together with A from equation (19) determine f(p) in equa- tion (20). For given values of f(p), the correspond- ing values of p may be read from Figure 18 in Chap- ter 5. The coordinates h. and d on the path-differ- THE u-v METHOD 135 ence loci are found from equation (98) in Chapter 5 and equation (13) in Chapter 6, giving pq a) Pp d=d 4 (2) Intersections of the path-difference loci with the range curves determine points on the lobes. and (46) in Chapter 5] are constructed and their inter- sections with path-difference contours corresponding to the assumed values of Q/2 determine points on the lobes. Ose Construction of Lobes (Horizontal Polarization) In this method, the divergence factor D is con- sidered to be the independent variable. Dividing 30 D=9 27 cs DIVERGENCE DvD — — — PATH DIFFERENCE R Friaure 6. THE u-v METHOD 6.5 Oe Outline of Method This method makes use of the generalized coordi- nates w = he/h; and v = d/dp described in Section 5.5.8. The curves of constant-divergence factor D and path-difference parameter R are plotted on the same sheet in Figures 4, 5, and 6. The divergence lines are shown in full and the path-difference curves are dotted. Envelopes of constant sin?(Q/2) [equation Curves of constant-divergence factor D and path difference parameter R. (Radiation Laboratory.) both sides of equation (46) in Chapter 5 by dz gives Ce oar See Fite De \ ( — K)?+ 4K sin 5. ae Rae es =VGid ae —K)?+4Ksin?—, (21) 2 where d, = by, dd T 136 COVERAGE DIAGRAMS If p = 1, and the effect of antenna directivity is neglected, K = D, and d dr v= = 2 V Gide Vil — D)?+ 4D eee (22) | Ms t0 BRB. D-v LOC! AND LOBES FIGURE 7. The following discussion illustrates how one con- tour of a coverage diagram, corresponding to a particular value of radio gain, may be plotted on Figure 4 or its equivalent, Figure 7. The result is a curve similar to Figure 2, but plotted in w,v coordi- nates instead of hy and d. See also Figures 16 to 39. For illustration, let the transmitter gain, G; = 1 and let the radio gain be such a value that do = do/dp = 2. Further, let \ = 0.1 meter and h; = 20 meters. From Figure 15 it is seen that 7 = 1,030A/h.?? = 1.2. Select one of the curves for sin? (Q/2) in Figure 12, Chapter 5, say sin? (Q/2) = 1, for which Q = 7, 37, 52, etc. These values correspond to tips of the lobes for which n = 1, 3, 5, ete., since, for perfect reflec- tion, 2 = na by equation (116) in Chapter 5. Next select values of K = D and note the corre- sponding value of the radical \ (1 — K)? + 4K sin? ia) 50 Gano ceo wee ed “Coe oe ets given by Figure 12 in Chapter 5. Equation (22) then gives the value of v. These quantities together with R = nr may conveniently be tabulated, as in Table 2. Corresponding values of D and v are plotted as crosses on Figure 7. The line drawn through these 7 <|es - my) R=10 - 2 a oS ate a gia EL ap Pa | cal wu JE - = | B l= ao bed! = N a \ \ — n ve) ‘ AY \ \ \ V7 p-v/Locus uo \ \] ] \7 D, van Hh} /\ fe V\ \ \ i" at \ Oo “ o ° " @ npaud o9u0 000 9 “on D-v loci and lobes. Table 2. Values of v and F for sin?(Q/2)=1, dy =2. | 2 ang 8 v (r = 1.2) D NG — D)?+ 4D sin Bl(eduation i: er (Figure 12, Chapter 5) | (22)] ||" a mY | maxima 0.2 117) 2.4 0.3 iL-8} 26 1 1.2 |1st lobe 4 | : 2.8 AE | i : 30 3 3.6 | 2d lobe 6 6 Peale ae : : oy 5 6.0 | 3d lobe 0.8 | Ls 3.6 | 0.9 1.9 38 || 7 | 8.4 |4th lobe | 0.95 1.95 3.9 | 1.0 2.0 40 | points is the locus of the tips of the lobes. The actual position of each lobe tip is then marked with a circle where the corresponding value of R crosses the locus in Figure 7. THE u-v METHOD 137 Additional lobe points are located by choosing some other value of sin? (Q/2), say sin? (Q/2) = 0.7. Each value of sin? (Q/2) now gives two points on each lobe, one on the upper branch and the other on the lower. Again choose values of K = D, obtain the corresponding values of the radical <| (Ql — K+ 4K sin’ S from Figure 12 in Chapter 5, and calculate new values for v. The D-v values are plotted as crosses on Figure 7 and the line through them is the locus of points for which sin? (Q/2) = 0.7 (see Table 3). TaBLe 3. Values of v and R for sin? (2/2) = 0.7, do = 2. Ri a} (r = 1.2) = 2 2 D Ja D)? + 4D sin | » : (Figure 12, Chapter 5) K | n | R = nr | Lobe 02 11 2.2 || 0 |0.63| 0.756 0.3 1.15 2:3 0 (1.387 1.644 1 0.4 1222 2.44|) 1 |2.63 3.16 2 0.5 1.28 2.56 || 1 |3.387| 4.04 2 0.6 1.36 PPA 72 5.56 3 0.7 1.43 2.86 || 2 |5 37| 6.44 3 | 0.8 1252, 3.04|| 3 |6.63] 7.96 4 0.9 1.6 3.2 || 3 /7.37| 8.84 4 i} 0.95 1.64 3.28 1.0 1.6 3.36 For sin? (Q/2) = 0.7, sin (Q/2) = +0.836, 0/2 = 0.3157 or 0.6857. Then Q = na = 0.632 + 2kr and 1.377 + 2k, in which k is an integer. Then n = 0.63 + 2k and 1.37 + 2k, and R = nr. Values of n and R are listed in Table 3. The intersections of the R values and the locus for sin? (Q/2) = 0.7 are plotted as circles on Figure 7. The entire lobe structure for one contour may be drawn by choosing additional values of sin? (Q/2). A large number of contours have been calculated by the Radiation Laboratory and are plotted in Figures 16 to 39. In order to construct one contour of a coverage diagram, it remains to find the intersection between the curves giving values of wu for constant sin? (2/2) and the corresponding path-difference contours. The equations relating R to Q/2 are given below. From equation (18) 6=2+21n (d = 180°, d’ = 0), (23) and from equation (19) 2-20) From equation (83) in Chapter 5 An assigned value of © fixes two values of 6 for each lobe, as explained in the previous paragraph. All values of sin? (Q/2) other than 1 or 0 determine two intersections with the lobe. When sin? (Q/2) = 1, the envelope of maxima is obtained, while sin? (2/2) = 0 corresponds to the envelope of minima. By selecting several values of sin? (Q/2) in Figure 12, Chapter 5, and following the method outlined above, a coverage diagram may be constructed in generalized (u,v) coordinates. The actual values of h, and d are h (24) (25) = hu, [Sy Q = dv. OR Construction of Lobes (Vertical Polarization) Problems involving vertical polarization or cases where the ratio of the antenna-pattern factors F,/F, cannot be neglected, may be solved by suc- cessive approximation. As a first approximation the method of Section 6.5.2 is applied to determine points (he,d) on the lobe. The corresponding values of w and v determine s in Figure 19 or Figure 20, in Chapter 5, and tany may be found from Figure 24 in Chapter 5 for the given transmitter height h;. An alternate method is to calculate tan y from equations (73), (58), and (60) in Chapter 5, which are dy = sd, d ON tany = fas dy The anglesy, and v required to calculate the antenna pattern factors /; and F, are found from. equations (62) and (63) in Chapter 5, fan Vi = he hy a re dy Pa i 138 COVERAGE DIAGRAMS The values of p, @ (or @’) may now be read from the reflection curves in Chapter 4. Equation (46) in Chapter 5 may now be applied with K = (F/F;)pD where D assumes the same values as in the approximate solution. Equation (21) determines the value of d from which wu = d/dr may be calculated. This value of wu = d/dr is laid off on the original divergence contour in Figures 4, 5, or 6. This determines v. The assumption under- lying this procedure is that the divergence factor is not appreciably affected by the change in coordinates caused by imperfect reflection and an unsymmetrical antenna pattern. The corrected phase difference 6’ is found from sf =2-¢' (26) and the path difference A’ and parameter R’ from rn (6 A’ =- («) 27 ats (27) Re oo (28) hdr The intersections between the path-difference con- tours and the distance envelope determine points on the coverage diagram. The above method should be applied even for horizontal polarization when the directivity of the antenna is such that F./F; # 1. This follows from the concept of generalized reflection coefficients of Section 5.3.1. 6-6 LOBE-ANGLE METHOD (HORIZONTAL POLARIZATION) 6.6.1 Outline of Method In this method the angles of lobe maxima are determined by modifying the plane earth formula, equation (2). In this equation, fy is replaced by hy’, which is the equivalent height above a plane tangent. to the earth’s surface at the reflection point, as shown in Figure 8. The value of h,’ is given in equations (58) and (60) in Chapter 5. The maximum and minimum distances from the transmitter base to a point on the lobe are calculated by equation (46) in Chapter 5, using a modified divergence factor to be descr*bed in Section 6.6.5. e022 ° ° oe Basic Relations Referring to Figure 8 and assuming d, << dp, vy’ < 10° andy & vy’, the following relations hold. Ficure 8. Lobe angles corrected for earth curvature. mr tan y’ > 7’ = — 29 4h,’ ( ) tan iy oN : (30) dy Hence hy’ nd dy 4h,’ and 1\2 pe eS (31) md where, from equation (58) in Chapter 5, d; hy’ =h —- : : ‘oka The basic equations of the lobe-angle method are ,_ mr md yah dz 62) : 4 @ = a ka and lh ~ 3) he ee (33) 6.6.3 Reflection-Point Curves The elimination of d; from equations (82) and (33) is most conveniently accomplished by graphical aids, which may be used in the following way. 1. From equation (33), a curve may be plotted showing d, as abscissa and n as ordinate for a given transmitter height and wavelength. This is illus- LOBE-ANGLE METHOD 139 trated in Figure 9 for two stations A and B with heights equal to 146.5 meters and 302 meters and where wavelengths } = 1.50 meters and \ = 1.42 meters i) = dy (35) respectively. ka 2. From equation (58) in Chapter 5, a second curve ; : “ ee - Hence by equation (32) may be plotted with d; as abscissa and the equivalent z height /,’ as ordinate as shown in Figure 10. To illus- " trate, computed data for station A are given in a VS ae dy (36) Table 4, for a free-space range of dy) = 100 km. d is A ( ha a ka? calculated from equations (16) and (17). 2ha Tasie 4. Data for station A of Figure 9.* Max. range dy hy’ y' 6 =’ — ot n (km) (meters) (rad) (rad) (rad) D (km) 0 50.1 0 0 0.00789 — 0.00789 0 1 28.0 99.6 0.00362 0.00440 — 0.00078 0.502 150.2 2 20.2 122.5 0.00590 0.00317 0.00270 0.740 26.0 3 15.9 131.7 0.00827 0.00250 0.00575 0.817 181.7 4 12.8 137.0 0.01058 0.00200 0.00858 0.864 13.6 5 10.9 139.7 0.01300 0.00170 0.00130 0.910 191.0 6 9.35 141.4 0.01540 0.00145 0.01395 0.930 7.0 7 8.05 143.0 0.01760 0.00128 0.01632 0.943 194.3 8 7.25 143.6 0.02000 0.00112 0.01888 0.960 4.0 9 6.45 144.0 0.02260 0.00100 0.02160 0.964 196.4 10 5.80 144.2 0.02510 0.00090 0.0242 0.970 3.0 11 5.32 144.7 0.0275 0.00083 0.02667 0.973 197.3 12 4.98 145.0 0.03000 0.00078 0.08022 0.980 2.0 13 4.51 145.2 0.03240 0.00071 0.03169 0.982 198.2 14 4.19 145.7 0.03480 0.00066 0.03414 0.986 1.4 15 3.87 145.7 0.03730 0.00061 0.03669 0.889 198.9 16 3.70 145.7 0.04000 0.00058 0.03942 0.990 1.0 17 3.48 146 0.04220 0.00053 0.04167 0.991 199.1 18 3.22 146 0.04450 0.00050 0.04400 0.993 0.7 19 3.06 146.2 0.04700 0.00048 0.04652 0.994 199.4 20 2.90 146.2 0.04960 0.00045 0.04915 0.995 0.5 * Antenna gain and directivity factors have been omitted from the above calculations. +See Section 6.6.4. 3. For any n, including integral and fractional values, d; may be found from Figure 9 and the corresponding h;’ from Figure 10. The angles y’ corresponding to lobe maxima may then be calculated from equation (32). 664 Lobe Angles with Horizontal The angle y’ given by equation (32) is measured with respect to the tangent plane through the reflec- tion point shown in Figure 8. This plane is inclined at an angle 6 with the horizontal at the base of the transmitter. The true angle y which the lobe-center line makes with the horizontal is Ye, Os (34) where odd values of n give maxima and even values minima, provided the reflection phase shift is z radians. The angle may be either positive or nega- tive, as shown by equation (86). °°5 Use of Modified Divergence Factor The value of the divergence factor must be de- termined in order to calculate the maximum and minimum lobe lengths by equation (46) in Chapter 5. A convenient formula for the divergence factor at the angles of lobe maxima is obtained by substitut- ing y’ for Y in equation (92) in Chapter 5. The errors involved in this assumption have been given in Section 6.2.3. Substituting y’ = y in equation (92), 140 COVERAGE DIAGRAMS in Chapter 5 yields (37) Hence Zl 1 20 HORIZONTAL POLARIZATION ve! [ Is =5 NULL N THIRD MAX NULL or TION B h,=302 M SECOND MAX dA=L42M w Pe arn @os NULL 2 LOBE NUMBER 7 STATION b= 196.5 X15 M Zp FIRST MAX nt Ait ) | * 8 __| af ——————EE A ° 10 20 30 40 50 60 70 REFLECTION POINT d, IN KILOMETERS Ficure 9. Location of reflection point d; as a function of lobe number n. Contours of constant y’ are shown in Figures 11 and 12 where y’ is a function of D and na. Hoe Construction of Lobes For horizontal polarization, the distance dmax from the base of the transmitter to the lobe max- imum is calculated from equation (46) in Chapter 5 by substituting K = (F2/F\)pD and sin? (Q/2) = 1. For horizontal polarization p = 1. Thus, dmax = VG@ido NE = = by + 42D (40) 1 1 or max = VGido E a * | (41) 1 Here F, and F; arecomputed from the angles yaand » given by equations (62) and (63) in Chapter 5. The dis- tance dmin from the transmitter base to the minimum REFLECTION POINT IN KILOMETERS (STATION 8) aa 32 as 64 80 96 | | aXe VARIATION OF EFFECTIVE HEIGHT hy WITH REFLECTION POINT d, FOR Ar h,=146.5 M DELS M FOR Bt h,=302M AtL42M HORIZONTAL POLARIZATION 100 STATION A o ° STATION B o ——————— ETERS (STATION &) ao}- — ° EFFECTIVE HEIGHT IN METERS (STATION A) EFFECTIVE HEIGHT IN mt v ° | | | ry ° ° 6 16 2a 32 40 a6 REFLECTION POINT IN KILOMETERS (STATION A) Fiaurr 10. Variation in effective height Ay’ with reflection point dy. point is obtained by substituting sin’ (Q/2) = 0 and K = (F2/F,)D in equation (46) in Chapter 5. Thus (42) The values of D to be used in equations (40), (41), and (42) may be read directly from Figure 11 or Figure 12, or calculated by equation (39). LOBE-ANGLE METHOD 141 Intermediate points in the lobe may be formed by assigning fractional values to n. The corresponding path-difference angle 5 may be calculated in the following manner. Suppose it is desired to find intermediate points on the fourth lobe. The values of n for this lobe range from n = 6 to n = 8, with the maximum at n = 7. It follows that a change of or as given in equation (46) in Chapter 5, in which . = (F2/F)pD and p 1. The value of D may be read from Figures 11 and 12, using the assigned value of m\ and the relation y’ nd/4hy’. The proper value of d, to be used in equation (63) in Chapter 5 to determine the antenna-pattern angle v may be read from Figure 9. Loe NGEANGINGEN INES aN N DIVERGENCE FACTOR 1 Ficure 11. Contours of y’ 2 in n corresponds to a change of 27 in 6. Thus if n = 6.5, 5 = (0.5/2)(27) = = 90°. For hori- zontal polarization Q ( 6 + ©’) reduces to 6, since @’ & 0. The distance from the transmitter base to this intermediate point in the lobe is equal to at Bn aie d = VGido ql (2a Resin: (8), (43) 0.8 D as a function cf D and nX. °°7 Correction for Low Angles The method outlined in Sections 6.6.2 to 6.6.6 depends upon the assumption that y’ > y or d; > 0. This assumption gives good results when n is a large number but serious errors are involved for small values of n. The method described in this paragraph is designed to avoid this difficulty. The procedure 142 COVERAGE DIAGRAMS consists in plotting point by point the lobe-center Hquations [(44a to e)] are obtained by inspection of lines and angles of lobe maxima. The points will be Figure 13. Equations (44f) and (44g) are derived as located by polar coordinates with the pole at the follows: ea — a NAN CEEEEERETA UULEEEN Ah \ epics Dee Dae] 1.0 0.5 0.2 oO. 05 02 01 Ficurer 12. Contours of ¥’ as a function of D and nX. transmitter. The coordinates are shown in Figure sin 2y ma 2Qy 13 as rq and y.. sn(@+va) Bo wtya’ Referring to Figure 13, the following relations OwB hold when the angle y is less than 10 degrees: YVtw= M : ‘a ; hy’ hy’ (a ee ee _ M A dy (f) av( 8) =v #). Ta ra (b) = a ka ka ' RECEIVER — — .% Ra (c) V1 = va — 8. fy as (d) ee ee (44) Ze yw = h - — 2ka ay | AN (e) fps AH Bi dete ; = (f) ba _ fe = ay Ta nr/2)d ; : (g) BS = ( = 2) = Ficure 13. Geometry for radio propagation over 2d? — nd/2 spherical earth. See Figure 14 in Chapter 5. LOBE-ANGLE METHOD 143 The path difference A is given by mdr Be A ab 1 Sp Gr a802)s Hence | x < md A+ B-/A? + BY + 2AB cos 2y = >: Squaring, B(2A — nd — 2A cos 2) = nA — @e mA 1 (2) BX? nr A(1 — cos 2y) — — ( cos 2p) 5 Therefore, B= For small angles, Since the angle y is of the order of a few degrees only, it is permissible to write A = d; in the above equation. Neglecting further the term \/4 in comparison with d;, which is permissible for short waves and small values of n, the above expression for B reduces to equation (44g). The method of determining the locus of points having a path difference of \/2 (ie., n = 1) is as follows. Assume a value of d; and calculate the corresponding values of hy’ by equation (44d), y by equation (44a), B by equation (44¢), ra by equation (44e), Ww, by equation (44f), 6 by equation (44b), yi by equation (44c). The assumed values of d; are limited to those which will give positive values of B in equation (44g). Select as many values of d; as are necessary to plot a smooth path-difference locus. Repeat for n = 3, etc. This method of determining the angles of lobe maxima is of particular value in constructing the first few lobes, since the approximations of Sections 6.6.2 to 6.6.6 may cause considerable error for low angles. These path-difference loci will intersect a vertical line drawn from the antenna to the ground below at heights equal to n\/4. For short waves, this height is negligible for the lower lobes. 6.7 LOBE-ANGLE METHOD (VERTICAL POLARIZATION) idee Angles of Lobe Maxima The values of n corresponding to the angles of lobe maxima are determined exactly as in Section 6.2.3 for the case of a plane earth. The values of n in the expression y’ = nd/4h;’ are increased above those for horizontal polarization by an amount (An) to compensate for the reduced phase shift on reflec- tion. In other words, the path difference must be greater than integral multiples of \/2 to compensate for the reduced phase shift. The expression for this compensation, from equation (5), is An = ol (45) Tw Hence y' = (n+ An) as 5 (46) 4h,’ [See Figure 8 and equation (32).] or Construction of Lobes As a first approximation, the angles of lobe maxima are calculated on the basis of horizontal polarization. A table is constructed giving values of m and y’ for maxima and minima. The next approximation is to let Y = y’. This assumes that di << d,. The values of @’ and p may then be found from reflection curves, and (An) calculated from equation (45). The corrected values of y’ may be determined from equation (46). It is simpler to find y’ by interpolat- ing between integral values of n in the n versus y’ table previously constructed. The values of dmax and din are dinar = VGido(1 ae Im). Ginin = VG ydo(1 a kK), (47) (48) 144 COVERAGE DIAGRAMS where K = (F2/F)pD. The divergence factor may be found directly from Figure 11 for the corrected values of n and y’. It will be found that for the higher lobes, the effect of (An) upon the value of the divergence factor is negligible. Table 5 shows calculations of the corrected values of n and y when the radiation from antenna A of 6.6.3 and Table 4 is vertically polarized. Trans- mission over sea water is assumed. Tables 6 and 7 illustrate the effects of vertical polarization on reduc- ing the maxima and increasing the minima. of n and minimum ranges. The free-space range is 100 km, as in Section 6.6.3. The last column shows the ratio of ranges for horizontal and vertical polar- ization. 68 GENERALIZED COVERAGE DIAGRAMS (HORIZONTAL POLARIZATION) Obi! Basic Parameters The u-v method applied to generalized coordi- nates which was given in Section 6.5.2 may be TABLE 5 g* g* Ay’ te Vom Ve, n (in degrees) (in degrees) An (rad) (rad) (rad) 0 180.0 0 0 0 0 — 0.00788 1 175.0 5.0 0.0278 0.000064 0.00369 — 0.00071 2 171.5 8.5 0.0472 0.000112 0.00602 0.00284 3 168.0 12.0 0.0664 0.000155 0.00843 0.00593 4 164.7 15.3 0.085 0.00204 0.01080 0.00878 5 160.6 19.4 0.108 0.000250 0.01325 0.01153 6 157.3 22.7 0.126 0.000290 0.01569 0.01423 7 153 27.0 0.15 0.000374 0.01807 0.01681 8 149 31.0 0.172 0.000413 0.02061 0.01947 9 144.5 35.5 0.197 0.000491 0.02309 0.02208 10 140.0 40.0 0.222 0.000532 0.02563 0.02472 11 135.2 44.8 0.249 0.000623 0.02812 0.02735 12 130.5 49.5 0.274 0.000685 0.03068 0.02991 13 125.8 54.2 0.301 0.000722 0.03312 0.03241 14 120.8 59.2 0.329 0.000790 0.03559 0.03493 15 116.0 64.0 0.355 0.000886 0.03819 0.03778 16 110.2 68.8 0.388 0.000968 0.04077 0.04019 17 105.3 74.7 0.415 0.000995 0.04320 0.04177 18 101.1 78.9 0.437 0.001091 0.04569 0.04519 19 96.1 83.9 0.466 0.001130 0.04823 0.04775 20 91.8 88.2 0.490 0.001224 0.05082 0.05037 *% corresponds to y’in Table 4. ¢’ = 7—@. TABLE 6 TABLE 7 dmax(HP) dmax(VP) —-dmax (VP) dmin(HP) dmin(VP) —_dmin (VP) n K (km) (km) dmax (HP) n K (km) (km) dmin (HP) 1 0.904 150.2 145.5 0.968 2 0.835 26.0 38.2 1.47 3 0.765 181.7 162.5 0.895 4 0.725 13.6 37.4 2.75 5 0.670 191.0 161.0 0.844 6 0.625 7.0 41.9 5.98 u 0.585 194.3 155.2 0.80 8 0.548 4.0 44.8 11.2 9 0.516 196.4 149.8 0.762 10 0.486 3.0 52.9 17.6 11 0.458 197.3 144.6 0.733 12 0.436 2.0 57.3 28.6 13 0.415 198.2 140.8 0.710 14 0.40 1.40 60.6 43.3 15 0.385 198.9 138.0 0.695 16 0.375 1.0 62.9 62.9 17 0.362 199.1 135.7 0.681 18 0.360 0.70 64.3 91.9 19 0.360 199.4 135.8 0.680 20 0.355 0.50 64.7 129.4 Tables 6 and 7 show the effect of a reflection coefficient which is less than unity upon the max- imum and minimum ranges of station A in Section 6.6.3. The first table gives odd values of nm and maximum ranges; the second table gives even values extended to all transmitter heights and wavelengths. In this method, points on the lobe are located by the intersection of the path-difference locus with the normalized distance envelope. The basic parameters are do and R. GENERALIZED COVERAGE DIAGRAMS 145. In constructing a coverage diagram for a doublet transmitter, the transmitter height, the wavelength and the radio gain are known. It will be shown in Section 6.8.2 that the normalized free-space distance, dy = do/dr7, is related to the gain factor A by the equation 1 = cae 49 # ON (49) The path-difference parameter R has been expressed in equation (114) in Chapter 5 in terms of a height- wavelength factor r which is defined by enn (50) where ot Ne XS 2 | Fy ROP . The first maximum, which for horizontal polariza- tion occurs when A = \/2, corresponds to n = 1, the second minimum to n = 2, ete. Recalling the discussion in Section 6.5.2, it follows that it is pos- sible to construct coverage diagrams in generalized coordinates with r the pattern or chart parameter and do the curve parameter on a chart for which r is fixed. 8:2 Determination of d It is possible to express dy = do)/dp in terms of E/E,, the ratio of the field strength H corresponding to the lobe, to the free-space field HE, at unit distance from the transmitter. Since E hy = ae it follows that d= OE (52) dr dp E The ratio H;/E may be expressed in terms of the gain factor A through the following relationships. By equation (16) in Chapter 2 PRP, = ar . 45 When d = 1, this gives EY? Py =—. 53 aii (53) For a doublet receiver with matched load and ad- justed for maximum power transfer, equation (17) in Chapter 2 gives P, = : o aN . 1207 8r Hence Jy BN PA 3n teed = : (54): i, 7 Sa a, 0 Substituting the value of £,/E from equation (54) into equation (52): d (2) (i) 3 —— ) ‘aa dp SrA 1 81 aoe ay he db (=) Vata (° =) Vin. Equation (56) shows that if log h; is plotted against log A for fixed values of do and X, a straight line: results. These straight lines are shown in Figure 14.. (55). and (56) 19900 Sete rs —— 1000 h, IN METERS Nu ° 3° 100) 19\30 0 =100 =90 = 60 =70 =60 = 50 20 LOG A— 20LO0G A Ficure 14. Values of do as functions of h; and 20 log A —20 log X. (See equation 56. The letters refer to cover- age diagrams plotted in Figures 16 to 39.) If h; and L/E, or jy, \, and A are known, do may be calculated from equations (52) or (55). The: 146 A _ METERS -O1 02 w igure 15. COVERAGE DIAGRAMS -001 002 -005 Ol 02 O® OnNf UW CHART NUMBER 20 50 100 200 500 400 300 200 50 40 30 20 lo- Chart number and 7 as a function of X and fi. (See Figures 16 and 17.) GENERALIZED COVERAGE DIAGRAMS 147 value of do determines the range of the lobe for specified values of r. In Figure 14, the various values of do used in constructing the charts are specified as A, B, C,--+ , M, N, and are shown as functions of fy as ordinate and 20 log A — 20 log X as abscissa. OH Determination of r Figure 15 shows r for various values of transmitter height h; and wavelength \ where 3/2V2ka 372 [9 hy 2Qka i Qhy (57) dka nN ka The values of 7 determine how the path-difference curves intersect the envelopes corresponding to assigned values of sin? (9/2) in the equation Qhidr es ka ys s ST Gl (aD): WP OS ap = do Gy \ al = Nz + 4D sin* 2 6 (58) The generalized coverage charts are designated as 1,2,--- , 11, 12 in Figures 16 to 39, with the chart number being given by Figure 15. ae Use of Charts Figures 16 to 39 give 24 basic charts developed by the Radiation Laboratory and presented in report 702. On each chart are complete Jobes or lobe outlines labeled A, B, C,---, M, N. In the following description these letters are referred to as lobe letters and the numbers 1 to 12 as chart num- bers. It will be noted that each chart number is plotted to two scales, giving 24 charts in all. The problem of constructing coverage diagrams resolves itself into finding the chart number and lobe letter corresponding to given values of gain factor A, transmitter height h:, and wavelength ». As stated in Section 6.8.1, the basic parameters of the general- ized coverage diagrams are R and do or 7 and do. The value of r is given in Figure 15 as a function of dX and hy. Figure 15 was constructed from equation (51) which, after the substitution of numerical values, becomes: (ior k= +) 3 1030X ee The value of r determines the chart number. The lobe letter is found from the do corresponding to the given gain factor, transmitter height, and frequency. Figure 14 gives the lobe letters A, B, C,---, M,N as functions of 20 log A — 20 log \ and the trans- mitter height. The relationship for plotting these lines is given by equation (56). As an illustration of the use of the generalized coverage diagrams, assume 20 log A = —83, hy = 33 meters, and f = 200 me (A = 1.5 meters). If a straight line is drawn connecting h; = 33 and \ = 1.5 in Figure 15, it will intersect the 7 scale at r = 8. Thus the chart number is 5. Now the lobe letter to be used in chart 5 must be found. For this case, 20 log A — 20 log \ = —86.48. The coordinates 20 log A — 20 log \ = —86.48, and h; = 33, de- termine the lobe letter to be # in Figure 14. Figure 24 shows lobe # on chart 5. The first lobe is shown completely, together with the lower half of the second lobe. It must be noted that the coordinates of these charts, v = d/dp and uw = he/hi, are dimensionless. To convert to height hy and range d, the vertical distances must be multiplied by h, and the hori- zontal distances by dy. In this case hy = 33 meters and dp = V2kah, = 23.6 km. The actual coordi- nates of the position of maximum range are hy = 375 X 33 = 12.4km andd = 15:4 X 23.6 = 365 km. The charts given in Figures 16 to 39 may be used for drawing coverage diagrams where the reflection coefficient is assumed equal to —1 and when the directivity factor F/./F, is equal to unity. Each chart may be used for values of 7 near that for which the chart is drawn. For intermediate values, interpolation between charts is necessary. Errors inherent in interpolation limit the accuracy attained. (59) 148 149 GENERALIZED COVERAGE DIAGRAMS Ae q Mi ALD +} . HAY y a iat ] LY CHART 2 N N a bz ' 4 =n ty m0 COVERAGE DIAGRAMS ‘OR Ou (Ol Cin Oran OC Ga Oe Olu O. fe} fo} fo} fey oO oO ¢ sg. o ao ww eos 150 5.) Figure 1 32. (See »verage diagram for r CHART 3 TI Face CHART 3. GENERALIZED COVERAGE DIAGRAMS 151 . IN 7 S TEN IK NY Ty ~ NG |\ TANGENT, Ray Neen 1 \ == ( \ } TIA \ \ \ » ‘ \ \ } \ ‘ Ke NN \ } CONSTANT GRADIENT OF REFRACTIVE INDEX \ ON AS XX NENG ING I \ it 1 ! 1 1 1 L SI Nae ° 5 ~ 1 15 20 Ve xn oa may 4) CHART 4 R EFLECTION COEFFICIENT =—! Figure 23. Generalized coverage diagram for r = 16. (See Figure 15.) 152 COVERAGE DIAGRAMS nas LV? y L) 7 Ficgure 25. Generalized coverage diagram for r = 8. (See Figure 15.) 153 GENERALIZED COVERAGE DIAGRAMS E Vi RAL KY I 6 I 4. (See Figure 15.) !\ QY V mi SRR \ q ZA L Xk (ee E ae SAQA Fy Po | SN HE Be i] \ \\ z 0 ef] a = é | IN = Ee oS #8 ‘ONE OC le Onn O fa O et © an One One On © ln Oar) a oF oF wo StF NN Oo D9 wo ¢ wD ao ao - - = FF = = 4. (See Figure 15.) Figure 27. Generalized coverage diagram for r COVERAGE DIAGRAMS 154 \ \ an ann IN igo 7 A e D Deexh\ p 6 oR XK K2 Sp p< > ae = 2. (See Figure 15.) ge diagram for r CHART 7 GENERALIZED COVERAGE DIAGRAMS 155 > = : X <= A PT \ K (DBA LAs I EA \ OES ' > ESS ss KS SS a SNUNNNNINN ° 5 10 \ i IN MR AN \ i \ i\ tA CHART 8 REFLECTION COEFFICIENT =~! CONSTANT GRADIENT OF REFRACTIVE INDEX Ficure 31. Generalized coverage diagram for r = 1. (See Figure 15.) 156 COVERAGE DIAGRAMS Louw ee RSioh 1 \d A AA SS wiz | “Unease ie REFLECTION COEFFICIENT =—1t CONSTANT GRADIENT OF REFRACTIVE INDEX Fiaure 32. Generalized coverage diagram for r = 14. (See Figure 15.) Friaure 33. Generalized coverage diagram for r = 14. (See Figure 15.) GENERALIZED COVERAGE DIAGRAMS K NK A et VAY { na i 2 ASAT 7 CG i : A ey =e ——————— =o \ A val vA OTE AW 4 ata —L_] Ea | | | —— ——T A \ ANN A\ ) ] OF Generalized coverage diagram for FACE SUR Ficure 34. CONSTANT GRADIENT OF REFRACTIVE INDEX REFLECTION COEFFICIENT=—) REFLECTION COEFFICIENT = —-1 CHART 10 (See Figure 15.) yy. Figure 35. Generalized coverage diagram for r COVERAGE DIAGRAMS 158 \\( V wT Way \\) NX eae (See Figure 15.) HAA | EA He MeL / ES ole Generalized coverage diagram for r = 14. co w a z \\ w \} = A / pe / Hise 2 aye \ Z /- [aes \ 7 ou / / tu mi > bt fa i z ub 7@ own wu Oo et ‘ea , 7 f& COM CP aean u 2 - o ce} Ww re oD woia us Oo 4 3 io og ome fo oO uw 2 p 2 u & °o cae Ti, 1 oO Ook GES je) 32) m= Sc ° a jee o <4 zo) 4 oo Eeaon az 52 x 8 p<@ wy a (re) i io oO £8 ees Co fo} ° fe) fo} [o) fo) fo] fo) fe) ° fo} fo} a 3° © © t a ° Cs) o t a n a = = = iin = sen zy (See Figure 15.) Generalized coverage diagram for r = 14. Ficure 37. 159) GENERALIZED COVERAGE DIAGRAMS Ts Put / ci oe et | A NM AA Mi " OS Wea \ \\WAT UA a \\\\\ \ i, Mi) d d NW \\ AN (\\\ 2 \\ X SAUNA) \ Mi 2 Vy \ i rT ¢ SATAN 2 “ i me : iz A : Hy Kt Ss 7 We i e i i aE aaa a= a i = = : CONSTANT GRADIENT OF REFRACTIVE INDEX Generalized coverage diagram for r FIGURE 38. REFLECTION COEFFICIENT =-! CONSTANT GRADIENT OF REFRACTIVE INDEX REFLECTION COEFFICIENT=—) (See Figure 15.) ‘16. y Figure 39. Generalized coverage diagram for r Chapter 7 PROPAGATION ASPECTS OF EQUIPMENT OPERATION TA GENERAL PROBLEM AEM Introduction Fe A STANDARD atmosphere and with the basic assumptions set forth in Section 5.1.4, the relation between the factors affecting the power of a set and the gain factor A is given in equations (3) and (5) in Chapter 5. The problem of computing A depends on the set in the sense that some sets are designed to operate in free space, others with the aid of reflection from the sea, as in low-angle and surface coverage. The characteristics of a set as given in the manu- facturer’s description or in Tables 3, 4, and 5 at the end of this chapter may not represent the true values for a set in field use. Expected set performance, such as maximum range and coverage, can be calcu- lated on the basis of the set’s rated characteristics. Such performance can be termed “normal.’? If a set is behaving abnormally, it may be that it is not functioning most efficiently. Unfortunately, the problem is complicated by the possible presence of atmospheric ducts and by the variability and diffi- culty of finding accurately the radar cross sections of aircraft and ships. Ducts are especially important for low antennas in surface search. In the case of communication sets, the most important item of information from a propagation standpoint is the maximum range. In the case of radar, not only is knowledge of maximum range wanted but also the ability to estimate the size and type of the target. ae The Performance Figure and Efficiency The maximum range of a set depends on the peak power output P, of the transmitter, the minimum detectable power Prin (see Sections 2.3.1 to 2.3.5) of the receiver, and the antenna gains G; and G). These can be grouped to give a performance figure. For communication, this figure is (P)/Pmin)GiG@e. For radar, the gains G; and G are generally equal. ‘The performance figure is then (P,/Pmin)G?. The 160 ratio of the actual performance figure to the max- imum possible value, or the difference in decibels, gives the efficiency of the set. In field use, it is generally impossible to measure the working per- formance figure with any precision and methods for obtaining a rough measure must be employed. 7.1.3 Effect of Reflection It has been pointed out in Chapters 5 and 6 that reflection may increase the maximum range of a radar up to twice the free-space value. This aid 7000'—SeAM UP 8° e 6000 2 5000 ° --4 4000 ° 3 3000 aoa : 2000) a 1000 _}—'° ) 60 6000 e BEAM UP 2.5 5 5000 4 BeOS a . © +209 ae = 4000 2 B w 3000 fae 2 : FREE SPACE S ° zZ 2000 L—2 bE Z ° ° = 1000-8 of} 1 = () 10 20 30 50 60 S000/BEaM DOWN4? | | 4000 | 3 3000 t 2 2000 ie) 10 20 30 40 50 60 DISTANCE — KILOMETERS © Experimento! Points Ficure 1. Effect of beam tilt on coverage for a radar. the early detection of aircraft at low angles. How- ever, the minima which occur in the resulting inter- ference pattern prevent the continuous tracking of an airship coming in. GENERAL PROBLEM 161 This effect. can be counteracted in several ways. One way is to employ microwaves whose inter- ference lobes are narrow and close together. Vertical polarization is another means of filling in the nulls while gaining in maximum range at low angles. Another device is to tilt the antenna beam upward so that some radiation (but substantially less than half) falls upon the sea. The result is a gain in low- angle coverage while the high-angle coverage is that of free space, without minima. The effect of various percentages of specular reflection in comparison with the free-space pattern is shown in Figure 1 for various beam tilts of a radar with a comparatively narrow beamwidth (11 degrees between half-power points). The experimen- tal data, shown by small circles, illustrate the increase in detection range at 0 degree while at 5 degrees elevation angle there is little gain over free space. The roll of a ship, by varying the beam tilt, results in a shift in coverage, as can be seen from Figure 1. a Signal-to-Noise Ratio The visibility of a signal on a scope depends on its relation to the noise. In early work with radar, the maximum range was defined by a ratio of signal voltage S to noise voltage N of unity, 1.e., S —=1. 1 a (1) However, as pointed out in Section 2.3.5, the min- imum detectable signal is greater than N. Since the pip on a scope includes noise, equation (1) is equiva- lent to Saou oO N 2. (2) On an A scope, this relation signifies that the height of the signal is twice that of the noise grass. Since the visual signal in a set functioning properly varies linearly with the signal voltage, the size of targets can be estimated by means of the size of the visual signal. The ratio S/N gives a means of meas- uring a signal in terms of the noise. To change (S + N)/N to S/N, the value of (S + N)/N is expressed with unity as denominator. For example, if (S + N)/N is estimated to be 8/2 from the scope, the equivalent fraction is 4/1. The value of S/N is (4 — 1)/1 = 3/1. The relation of receiver power, Ps, to noise power, NP, and the signal-to-noise ratio, S/N, is given by P, s 10) are 2 S ory ye 3 NP oa 8) 715 Calibration of an A Scope On an A scope, the ratio (S + N)/N can be estimated roughly by eye. To improve upon this, a calibration is employed. One method is to mark the A scope to facilitate the reading of heights. Another method goes beyond this and calibrates the gain control. A turn of 5 db is equivalent to a raito of 1.8/1. A datum line 1 em above the time line and 10 OB LINE 5 OB LINE 18.cm DATUM LINE 4¢mTIME BASE fe) 10 20 30 RANGE SCALE 70 80 90 100 Ficure 2. Method of calibrating an A scope. Figure 3. Calibration of gain control of an A scope. (Dotted lines are uncorrected calibration.) another at 1.8 em are drawn on the A scope. The noise is brought up to the datum line by means of the gain control. The position is marked 0 on the gain control (see Figure 2). A steady signal is found (permanent echo, large boat, or signal generator) which produces a signal height of 1.8 em above the time line. The gain control is then turned until the signal is reduced to the datum line. The position 162 PROPAGATION ASPECTS OF EQUIPMENT OPERATION on the gain control is marked 5 db (see dotted lines in Figure 3). Keeping the 5-db position on the gain control, another signal is obtained which comes up to 1.8-em line. The gain control is turned until the signal is brought down to the datum line. The new setting is marked 10 db. This is repeated until markings up to about 70 db are obtained. This calibration of (S + N)/N must be corrected to S/N (Figure 3) which can be done by means of Table 1. For 25 db and above, the values of (S + N)/N can be taken as equal to S/N. Any signal voltage can then be measured in decibels above the noise voltage (V) by turning the gain control until the signal height is 1 em. By equation (3) this is also the signal power received, in decibels, above the noise power which is discussed in Sections 2.3.1 and 2.3.2. TaBLe 1. Correction of (S + N)/N to S/N. Corrected Uncorrected (4) db ie a5 uy db N N 0 6 5 9 10 12.5 15 6.5 20 21 In the calibration just described, the pip on the scope is supposed to be proportional to the received signal strength. In a set functioning normally, this is justified, but occasionally defects in the set may destroy the linearity. The existence of a linear rela- tion can be tested by means of a signal generator. es FREE SPACE — HIGH-ANGLE COVERAGE 721 Maximum Range Formulas In free space, the gain factor A has the value 3\/8rd (for maximum power transfer between doublets). Equations (3) and (5) in Chapter 5 then take the simple forms: One-way, radio gain: P= outn( >) 4) P. 8 Two-way, radar gain: Py = G . . Te 9d? If P: is replaced by the minimum detectable power of the receiver, Prin, and P; by the peak power P, of the transmitter, the maximum range is given by: One-way, Crna aa eh a GiGy ) (6) 81 min Two-way, 4 P = drnax = Pay = e ul \ ba 2567 (7) 722 Deviation from Maximum of Beam For an antenna whose direction is fixed, the equations in Section 7.2.1 apply only to points on the axis of the beam. Denoting by f(z) the ratio of gain in a direction at an angle 7 from the axis of the beam to the gain at the axis and by 27) the beam width between half-power points, then f(r) = exp — 0.692(7/10)?. (8) Accordingly, for points off the axis, G must be multiplied by f(7) before substitution in the formulas of Section 7.2.1. IN DB SCANNING LOSS n 5 10 SCANNING RATE RPM Ficure 4. Scanning loss as a function of scanning speed and beamwidth. chic Performance Figure Equations (6) and (7) for dmax depend on the performance figure defined in Section 7.1.2. The quantities which appear in the performance figure can be measured. The one which offers most diffi- culty is Pmin, the minimum detectable power, which has been discussed in Section 2.3. In Tables 3 and 4 at LOW-ANGLE AND SURFACE COVERAGE 163 the end of this chapter, noise figures and bandwidths of various sets are given. An important correction to P,p;,, a8 determined from the noise figure and band- width is the scanning loss. This loss for various scanning speeds and beamwidths is represented in Figure 4. Another source of loss is deviation of the product of bandwidth B (mc) and the pulse width t (microseconds) from the optimum value of 1.2. The losses for various values of the product are tabu- lated in Table 2. TasieE 2. Loss resulting from band- and pulse widths. Bt * Loss (db) 0.1 5.0 0.3 1.5 0.7 0.5 1.2 0.0 2.5 0.8 5.0 3.0 10.0 5.0 20.0 8.0 *B = i-f bandwidth (mc); ¢= pulse width (microseconds). A field measure of the performance figure of a radar can be determined by the use of a target of known radar cross section, such as a silvered balloon and equation (7). A check on variability of per- formance can be made by finding the maximum range on a plane (using a constant aspect, such as nose or tail). ees Radar Cross Section An important but troublesome factor in calculat- ing dmax Of a radar from a knowledge of the perform- ance figure is a, the radar cross section (see Section 2.4.1 and Chapter 9). The value of ¢ can be found by: 1. Laboratory measurement of the factors which constitute the performance figure and field determi- nation of the maximum range dy,ax. The value of o is then given by equation (7). 2. Measuring the signal returned by the target at a convenient distance on a calibrated A scope or by direct comparison with a pulsed signal generator. In this method neither Pj, nor dmax enter. The equations involving o assume a point target. Since an airplane intercepts a small solid angle over which the beam strength varies little, the assumption of a point target is adequate for aircraft. ues LOW-ANGLE AND SURFACE COVERAGE sas Maximum Range Since the gain factor A for this case is more complicated than for free space, the relation be- tween d,,,, and the performance figure cannot be given in general by a simple expression, as can be seen from equations (172) and (184) in Chapter 5. For ranges such that the shadow factor F, ~ 1, ie., distances d less than 10*A'*, \, and d in meters, and both antennas low, or antenna and target low (Ii,ho< 30°/*), the form of A is simplified so that a simple relation can be given for day. Otherwise, the methods of Sections 5.6, 5.7.3, and 5.7.4 must be employed, especially of Sections 5.6.5 and 5.7.3. 732 Ducts and Set Performance It has been found from field tests that atmospheric ducts are likely to be found close to the surface of the sea. The consequent increase in range may mask subnormal set performance. If the antenna is tilted upward so that no radiation reaches the earth, then the free-space discussion of Section 7.2 applies to the field determination or check of the per- formance figure. Otherwise field testing, when condi- tions are normal, can be accomplished by the use of permanent echoes or the use of a ship of known target cross section (see Section 7.3.7). 733 Low Heights and Plane Earth Ranges For these conditions, the following relations must hold: Ci AOS A where h, d and ) are in meters (see Section 5.7.1). In the dielectric case (see Section 5.7.3), generally applicable to radar, the value of A [equation (172) in Chapter 5] for F, = 1 andg = 1, becomes 3 hyhe DGB ~ (9) Since A = AoA, and Ay = 3d/8rd, the preceding equation is equivalent to a path-gain factor value of A, = 47 —_. (10) [See equation (55) in Chapter 5.] Instead of equa- tions (4) and (5), we now have [from equations (3) and (5) in Chapter 5] for a dielectric earth, 164 PROPAGATION ASPECTS OF EQUIPMENT OPERATION One-way, radio gain: eee Gg ee (11) Py 4 ds Two-way, radar gain: 4 ( ) % q %, Q k A X 13 November 1943 @ 50 c¥ @ @ 30 November 1943 12) [e) Oo w fo) 4 _30 Nov. 1943 2, 2 £ @ These a3 Measurements showed 740 that the set was about 2 20db below its ied ee performance of |3 Nov. i ay ' wo rex a 30 9 _ é ' @ |e \e e 20 10 10) 10 20 30 40 (e) 60 70 80 RANGE IN THOUSANDS OF METERS Fieurs 9. Range in thousands of meters. (From Coast Artillery Experimental Establishment, England.) DATA ON COMMUNICATION AND RADAR EQUIPMENT 169 7.4 The tables given in this section are not intended DATA ON COMMUNICATION AND RADAR EQUIPMENT to be exhaustive. These are the best available data but should be used with caution, since specification TasLe 3. Communication equipment. changes may change rated characteristics of sets. Power Antenna Communication Frequency output Receiver Polarization Gain Beamwidth equipment (me) (watts) sensitivity (db) Horizontal Vertical SCR-608 30 25 0.2 pv VA Whip SCR-300 44 0.5 2.5 wv V Whip AN/TRC-1 70-100 50 25 wv H 6 (4 channel) SCR-522 125 6 4 pv V Whip AN/TRC-8 240 12 30 LV H 7 (4 channel) AN/TRC-5 1,400 400 NF* 12 db H 14 AN/TRC-6 4,600 2 NF* 19 db H 33 TBS 70 50 5 pv V 1 AN/ARC-1 125 10 5 pv V Whip * Noise figure. TaBie 4. Radars. Receiver Power sensitivity Frequency output Noise figure Bandwidth Antenna Beamwidth Radars (me) (kw) (db) (me) Polarization Gain (db) Horizontal Vertical SCR-271DA 106 100 6 1 Jef 19.8 Ihe 12% AN/TPS-3 600 200 11 1.8 Jaf 23.4 1p ie SCR-584 3,000 300 15 lee H 30.8 ie ha AN/MPG-1 10,000 60 17 10 40.8 0.6° 3° SC-4 200 200 6 0.5 H 13.5 20° 60° SQ 2,500 1 13 2 Vv 20 8° 152 SG-1 3,000 50 18 H 28 6° IGS? ASB 515 10 15 1.4 H 30 60° Tape 5. IFF or beacons. Antenna Frequency Power Receiver Polarization Gain IFF or beacons (me) output sensitivity (db) AN/TPX-4 170 0.5 kw 15 My V 6 AN/UPN-1,2 3,000 50 kw 0.05 Kv H 6 AN/UPN-3,4 9,320 300 kw 0.02 LV Ff 12 Chapter 8 DIFFRACTION BY TERRAIN oe OUTLINE OF THEORY ootL Introduction HE EFFECTS of diffraction around natural ob- Ahk stacles of complicated shape are difficult to analyze. Theory offers two lines of approach to diffraction problems, both based on the substitution of contours of simple shape in place of the natural obstacle. The first and oldest method, known as the Fresnel- Kirchhoff method, is an approximate procedure for calculating the diffraction by a flat screen. It yields comparatively simple formulas for the diffracted field; the present chapter is concerned with a presentation of this method. The second method is based on the fact that the wave equation can be solved for obstacles of very simple geometrical shape, especially cylinders and spheres. If the curvature of a hill is fairly constant, so that its shape can be approximated by a cylinder or sphere, the field behind the hill can be obtained by the use of this method. Observations on diffraction by obstacles in the short wave and microwave region are very sparse. It is, therefore, not possible to present a consistent body of results that could be utilized in radio prac- tice. It seems, however, rather certain from the observations that when the shape of the obstacle approaches one of the special shapes dealt with by the theory, the latter gives a fair account of the facts. Such cases will not be found too frequently in practice. The hope is nevertheless justified that the right order of magnitude is obtained by a judi- cious application of the theory. The main applica- tion is in the lower frequency band (30 to 200 me); for higher frequencies, the diffracted field is rela- tively unimportant. 81.2 The Fresnel Diffraction Theory The Fresnel-Kirchhoff approximate theory was originally developed to account for the diffraction of beams of light when cut off by diaphragms, slits, and similar optical devices. In applying this theory to 170 the propagation of radio waves over the earth, only one basic problem is usually encountered, namely that of diffraction around a straight edge. In the present section, the general method of handling this problem and obtaining numerical results is given. On applying the method to actual cases certain accessory problems arise which will be dealt with in Sections 8.3 and 8.4. The most important of these complications is caused by ground reflection. Figure 1. Diffraction around straight edge. In Figure 1, the area CAPBD forms an opaque screen bounded by a straight edge BPA. The width AB of the screen is assumed infinite in the mathe- matical theory, but is here shown finite for simplicity. The line connecting the transmitter 7 to the re- ceiver R intersects the plane of the opaque screen in the point MW whose distance from the edge is PM = ho. The shortest unobstructed path of the radiation is TPR. In a purely geometrical theory, the point R would be in the shadow of the screen and would receive no radiation. If the wave nature of radiation is taken into account, it is found that an electromagnetic field is generated in the shadow of the screen; the waves are bent around the obstacle. The mathematical derivation of the diffraction formulas will not be given here as it is rather intri- cate; however, the problem is discussed in Section 8.2. The discussion here is limited to a qualitative visualization of the mechanics of diffraction (Sec- tion 8.1.3); the final formulas used for computa- OUTLINE OF THEORY ib/Al tions will then be written down at once (Sections 8.1.4 to 8.1.6). B13 Mechanism of Diffraction The physical idea underlying the Fresnel-Kirch- - hoff diffraction theory may be presented as follows. At points visible from 7 the field, to a first approx- imation, is equal to the free-space field Hy. This applies in particular to all points of the plane HCDF containing the screen. The receiver R receives radiation from the open part HAPBF of the plane, while there is no radiation incident upon FR from the opaque surface CAPBD. In order to compute the field at R, it is assumed that in the open part of the plane the field is Hy while on the opaque screen the field vanishes. Such a field distribution may be realized physically by assuming that there is a con- ducting sheet in the open region HAPBF with suitably chosen oscillating charges or currents such that the field Hy is produced on the side of the sheet facing the receiver. The total radiation received at R from such a current-sheet will be equivalent to the radiation from 7 bent around the diffracting screen. The fictitious sheet HAPBF forms a system of secondary sources of radiation whose effect is equivalent to that of the primary source for all points on the far side of the plane ECDF (side of the receiver), but not on the near side (side of the transmitter). It is evident that most of the radiation received at R comes from the area near the point P above the line APB. The relative importance of contributions of areas more or less removed from P is discussed in Section 8.2. When the primary source at 7 is replaced by a distribution of secondary sources in the plane of the screen, an essential approximation is made. It is assumed that there are no secondary sources in the opaque region CAPBD. In reality, the screen is a physical body and, whether it is a conductor or a dielectric, there is an electromagnetic field in its surface layers, especially near the edge APB, and this field makes a contribution to the radiation received at R. In the approximate theory, it is assumed that the field on the surface of the opaque screen is negligible. In the terminology of optics, this implies that the screen is black; in radio termi- nology, it means that the surface of the screen is rough (Section 8.3.2). 8.1.4 The Straight Edge Formula The physical picture just described can be put into mathematical language. When the rather intricate derivations are carried through, a rela- tively simple formula results. The symbols and designations used are illustrated in Figure 2. In accordance with practice it is assumed that the line 7'R is nearly horizontal. The trace of the opaque screen on a vertical plane through T and FR is assumed perpendicular to the line TR ELEVATION R M | | | | | AN g | I, | a + —_——__——* PLAN VIEW \ \8 Fiagure 2. Diffraction around straight edge. (upper part of Figure 2). The trace of the screen on a horizontal plane may, however, make an angle @ with the line TR (lower part of Figure 2). In view of the approximate nature of the theory explained in the preceding paragraph, the following conditions must be fulfilled in order to obtain reliable results: dy ,d2 > >ho > >x. (1) That is, the distances from the transmitter and receiver to the obstacle must be large compared to the height of the latter above the line 7'R, and this height must be large compared to the wavelength. The second of these conditions is likely to be ful- filled in the short-wave and microwave bands, and the first will be fulfilled when the angles of elevation a, and a», of the rays, drawn from the transmitter and receiver to the edge, are small. A second condition for the validity of the diffrac- tion formula refers to the horizontal extension of the screen. The formulas are derived for a screen of infinite horizontal extent, but in practice it will usually suffice if the horizontal extension of the screen is large compared to the height ho. 172 DIFFRACTION BY TERRAIN If these conditions are fulfilled, the field at the receiver is given by pm /4 v —_ 2 V2 J-« @) where FE is the free-space field at the receiver in absence of the screen and Dream ho, sy nm th? (5 +2) = 2a ta). (3) In the last formula, use is made of the fact that a, and az are small angles ‘so that approximately Qa, = ho/ dy and ae = ho/de. peas err? dy Ri> The Fresnel Integrals An integral of the type appearing in equation (2) is known as Fresnel’s integral; its properties will now be briefly discussed and numerical data given. The standard Fresnel integral is usually defined as Civ) — jS(v) = | a 72) dy (4) 0 where C(v) = [cos (< “) dv, S(v) =) sin (Z °) dv. If this function is plotted in the complex plane, with C and S as abscissa and ordinate, respectively, for all values of v, a curve is obtained that is known as Cornu’s spiral (Figure 3). C — jS is represented, in magnitude and phase, by a vector from the origin to a point on this spiral. It may be shown that the length of are along the spiral, measured from the origin, is equal to v. In the graph, values of v, counted positive in the first quadrant and negative in the third quadrant, are indicated along the spiral. As v approaches infinity, the spiral winds an infinity of times around two points lying at the distance 1/ V2 from the origin on a 45-degree line. C and S for the end points are See) ee) il aay 2 T i ese ealeae Ficure 3. Cornu spiral. OUTLINE OF THEORY 173 316 Application to Straight Edge Since ert 14g V2 os equation (2) may be rewritten, on using equations (4) and (5), as i 1a E +0) -£- iso), (6) It will be noticed that the quantities 1 aff dk : have a simple geometrical meaning. They are the real and imaginary components, respectively, of a vector drawn from the lower point of convergence (point —1/2, —j/2) of the Cornu spiral to a point on this spiral. The bracket in equation (6) is equal to this vector in magnitude but with opposite phase. The Fresnel formulas and Cornu spiral as given above will assist the reader in establishing the rela- tions of our equations with the classical theory of diffraction as found in all textbooks on the subject. For practical purposes the field behind a diffracting straight edge given by equation (6) will be denoted by DS hh (7) Ky In Figure 4, the modulus z is plotted as a function of v. In Figure 5, the phase lag ¢ is plotted in a similar way. (With the above choice of the sign, ¢ is positive in the shadow.) The variable v is given by equation (3). On account of the square root, there is an ambiguity in sign. Closer inspection shows that v must be taken positive when the receiver is in the illuminated region, above the line of sight; v must be taken negative when the receiver is in the shadow zone. When »v tends to —o , the line APB (Figure 1) moves far upwards relative to the line 7'R; the rece:fver lies deep in the shadow and F approaches zero by equation (6). When v tends to + o , the line APB moves far downward, and the screen ceases to orm an obstruction, # approaches Eo. At the line of sight (when the point P in Figure 1 coincides with M), v = F(v) = 0 and FE = E)/2. Clearly, the effects of diffraction are not confined to the shadow region but extend considerably into the illuminated zone. If the receiver is sufficiently deep in the shadow, about v > —1, the following approximate formula holds: E B) _ 0.225 Eo : v 2.= ABOVE LINE OF SIGHT V——> MAGNITUDE OF RELATIVE FIELD STRENGTH E/Eo SHADOW ZONE Pe ) = =2 =) -4 = Figure 4. Magnitude of relative field strength E/E versus 0. 174 DIFFRACTION BY TERRAIN $1.7 Polarization. Large Angles It may be noticed that in the preceding equations no reference is made to the state of polarization of the diffraction field. The results of the approximate Fresnel-Kirchhoff theory are independent of the state of polarization in agreement with observation. 360° 270 Vi ABOVE LINE OF SIGHT Figure 5. Phase lag (ordinate) of relative field strength (#/E ) versus v (abscissa). If the angles of diffraction (a; and a,, Figure 2) become large, larger than a few degrees, for instance, the approximate theory no longer applies. The deviations from the Fresnel formulas then go in opposite directions for the two states of polarization. If the electric vector is parallel to the diffracting edge, the field in the shadow at large angles is slightly diminished as compared with that given by the Fresnel formulas; if the electric field is per- pendicular to the diffracting edge, the diffracted field in the shadow at large angles can become appre- ciably larger than the calculated one and, in the case of very large angles, the excess may reach the mag- nitude of, say, 6 to 15 db. In the region above the line of sight, the sign of the polarization effect is reversed (slight increase for polarization parallel to the edge, appreciable decrease for polarization perpendicular to the edge). These effects are entirely analogous to those that are observed when the currents induced in the surface of the obstacle cannot be neglected (Section 8.1.3), and they have the same physical origin. 82 DIGRESSION ON FRESNEL’S THEORY Sat Fresnel Zones The concept of the Fresnel zone has played an important role in the development of diffraction theory. As it is frequently referred to in papers on the subject, it may be useful to digress briefly on it. Fresnel’s original construction is based on the con- ception that any small element of space in the path of a wave may be considered as the source of a secondary wavelet, and that the radiation field can Relations of Fresnel zones and diffracting Ficure 6. slots. be built up by the superposition of all these wave- lets (Huyghens’ principle). In particular, consider the field produced by the transmitter in the open part of the plane containing the diffracting screen (EAPBF in Figure 1) and let each element of this plane be the source of a secondary wavelet. This may be achieved by distributing a suitable ficti- DIFFRACTION BY HILLS tious system of oscillating currents (or a system of elementary doublets of proper strength) over the surface of the plane. The field at the receiver is then the superposition of all the fields produced by the wavelets. Now let (Figure 6) S be a plane perpendicular to the line 7'R and let M be the point in which the line TR intersects the plane S. Let Q be a point on the plane S such that the difference in path between TQR and TMR is just 4/2. The locus of these points is a circle about M. Similarly we can con- struct other circles so that the corresponding path differences are integral multiples of \/2. The area within the first circle is called the first Fresnel zone, the subsequent. ring-shaped areas are called the second, third, ete., Fresnel zones. The secondary wavelets originating in the first, third, fifth, ete., Fresnel zones are in phase with each other and rein- force each other by constructive interference at R, while the secondary wavelets originating in the second, fourth, etc., zones are in phase with each other but out of phase with the former group and tend to cancel the field produced by this group. Hence if the plane S is opaque except for a round hole centered on M, the intensity of the radiation field at R will depend on the number of Fresnel zones that fall inside the hole. If we start out with a very small hole and progressively increase its size, there will be a maximum of intensity at R (nearly twice the free-space field Ho) when the hole just comprises the first Fresnel zone. If the size of the hole is further increased, the destructive interference of the second zone comes into play, decreasing the intensity, and a minimum (very nearly zero) is reached when the hole contains just the first two zones. On continued increase of the hole size, further maxima and minima appear. The amplitude of these oscillations decreases very gradually until eventually the field at R approaches the free-space value. 8.2.2 Diffraction by a Slot The preceding considerations indicate that only a comparatively small area of an opening, of the order of one Fresnel zone, is required to produce an illumination that is comparable in order of magni- tude to the free-space field. It is also seen that the simple geometrical construction of the Fresnel zones is more suitable when dealing with the diffraction by round openings than with screens bounded by straightedges. Qualitatively, however, the conditions are similar. As an example, consider the case of a slot bounded by parallel edges at distances ho and ho’ from the point of intersection WZ between the plane of the slot and the direction from the observer to the distant light source (see Figure 6). The diffracted field H will obviously be equal to the free-space field / if the slot is infinitely wide on both sides of M, which corresponds to a vector joining the two foci of the Cornu spiral. However, there is an infinite number of other finite openings of the slot which also will give the free-space field. Suppose, for instance, that ho = Ao’ in Figure 6 and that the slot width is gradually increased from zero. A glance at the Cornu spiral (Figure 3) shows that when v = 0.75 and v’ = —0.75, the vector representing the diffraction field is approximately equal to the free-space field. This width represents, for a slot, the analogue of the first Fresnel zone for a circular opening. che DIFFRACTION BY HILLS en Introduction The formula for diffraction by a straight edge may be applied in radio practice to determine the diffrac- tion field behind a ridge. The ridge need not be perpendicular to the transmission path, but the condition given in equation (1) must be approx- imately fulfilled. The distance from transmitter and receiver to the ridge should be large compared to the height of the latter above the straight line TR; and that height should be large compared to the wavelength. Moreover, as pointed out in Section 8.1.3, the diffraction formula applies in principle only to the case where the effect of the currents induced on the surface of the ridge upon the field at the receiver ean be neglected. This is the case (1) when the ridge has the shape of a steep and narrow knife-edge protruding from the surrounding countryside; or (2) when the surface of the ridge is rough (see Sec- tion 8.3.2). Experience shows that so long as the profile of the ridge is reasonably compact and its surface reasonably rough, the diffraction formula will give the magnitude of the field behind the ridge to within a few decibels. 176 DIFFRACTION BY TERRAIN If the ground near the transmitter or receiver is smooth, however, it becomes necessary to take ground reflection into account. This may be done by introducing an image transmitter and receiver. The field is then the sum of four components whose relative phase must be calculated (see Figure 11). Earth curvature will be neglected throughout the present section. betas Criterion for Roughness It is difficult to establish a quantitative criterion for the roughness of a surface. From the viewpoint of radiation theory, the effect of a rough surface is to scatter incident radiation diffusely in all directions with no preference for the direction of regular reflec- tion, whereas a smooth surface will reflect the inci- dent radiation according to Snell’s law. In radio work, the effect of diffuse reflection is to weaken the radiation scattered in the direction of the receiver so much that its intensity may be neglected com- pared to the direct ray. A moderately rough surface will give a coefficient of reflection intermediate between zero and unity. A surface will be optically smoother as the incident radiation appreaches grazing, and even surfaces that are comparatively rough geometrically may then give partial reflection. A rule taken from optics and known there as Rayleigh’s criterion has been used successfully in radio practice. Assume that the roughness is pro- duced by numerous small elevations above a level surface and let H be the typical height of such an elevation. The difference in path between a ray Figure 7. Geometry for Rayleigh’s criterion for rough ground. reflected from the ground and a ray from the top of the elevation is 2A B in Figure 7, which is equal to 2H siny or 2Hy approximately for small angles y. The difference in phase between the two rays is 2HY(27r/d). The criterion now requires that the surface be considered as rough when this phase difference exceeds 45 degrees, or 7/4 radians. Hence the critical value of H is given by SEN SE rm (8) nN 4 16y with y in radians and 3.60 ay (9) y with y in degrees. The surface is considered smooth or rough according to whether H is smaller or larger than this value. Sometimes it is convenient to refer to the field pattern that would be present over a reflecting surface. This is done by introducing a new variable, the lobe number 1 aN (A, transmitter height above the ground), where n = 1,3, 5, ete., correspond to the angle of the first, second, etc., maxima in the lobe pattern and n = 0, 2, 4, etc., to the nulls of the lobe pattern. Introduc- ing ” into equation (8), the criterion assumes the form (10) nal, 4n Although the criterion is approximate and gives no more than an order of magnitude estimate, it is rather surprisingly well fulfilled in radio practice. Experi- ence has shown that when the differences in level which constitute roughness are of the order indicated by these equations, the reflection coefficient is reduced to a small fraction (about one-fifth) of the value calculated for an ideal surface. (11) 833 Diffraction by a Straight Ridge Assume that the ground intervening between the transmitter and receiver is everywhere rough, so Ficure 8. Diffraction by a straight ridge. that all ground reflection may be neglected. For the sake of computation, the ridge is replaced by a vertical screen of height fo above the line TR. The DIFFRACTION BY HILLS (eve top of the ridge forms the diffracting edge (P in Figure 8). If the profile of the ridge is somewhat more complicated, the effective diffracting edge might be a purely mathematical line, as shown in the lower part of the figure. The height ho is conveniently determined from a profile of the transmission path obtained from a topographic map. If the heights hy, he, and h of transmitter, receiver, and obstacie, respectively, above a given reference level such as sea level are given, we have ae dyhy + shy _ dy + ds where the signs have been chosen so that ho is nega- tive when the receiver is in the shadow of the ridge and positive when it is in the illuminated region. Now, by equation (3), h, (12) Radio if | ho : in (G+ 3) X In these equations, give the angles a; and az the same sign as ho and give v the same sign that ho has in equation (12). The ratio of the field to the free-space field at the receiver is now given by | E/E | = 2(v), defined by equation (7) and plotted in Figure 4. In Figure 9, this ratio is given in decibels as a function of the a + as). (13) quantity «= —ho/Vdd (all lengths in meters). The successive curves in Figure 9 correspond to different values of the ratio di/d: or ds/d, (choose whichever one is the smaller). Only the field below the line of sight is shown. 834 Field Near the Line of Sight The fact that just above the line of sight the field increases above its free-space value may sometimes be used to obtain a favorable site (Figure 10). The maximum value of the field is about 1.17 times the free-space value (Figure 4), equivalent to 1.36 db. On the other hand, there are advantages in avoiding a line TR that is too close to grazing the top of an intervening obstacle, as this will substantially reduce the signal. At the line of sight, the signal is 6 db below free space. In order to get approximately the free-space value of the field, the crest of the obstacle should be sufficiently below the line TR so that v > 0.8 where v is given by equation (18), ho being the clearance between the line TR and the obstacle. In cases where the heights and distances are not quite certain, it is therefore preferable to. select a higher and definitely unobstructed site rather than to try to utilize the small gain that might possibly be had from the diffraction field. FIELD IN SHADOW BEHIND DIFFRACTING RIDGE OB BELOW FREE SPACE O15 0.2 0.3 0.4 0.6 #08 1 2 3 4 6. Fiaure 10. Diffraction field above the diffracting edge. 835 Diffraction with Reflecting Ground When the ground near the transmitter or receiver is smooth and reflecting, the diffraction problem. becomes very complicated. It can be solved by the- 178 DIFFRACTION BY ‘TERRAIN method of images on assuming that the radiation reflected on the transmitter side of the obstacle issues from an image transmitter and that the radia- tion reflected on the receiver side is incident upon Diffraction of both direct and reflected Fiaure 11. rays. an image receiver (Figure 11). The total field at the receiver may be written a Ey = Bp ns where each term on the right-hand side is of the form of equation (6), £, corresponding to the direct radiation, H, to the radiation from the image trans- mitter to the receiver, EZ; to the radiation from the transmitter to the image receiver, and F; to the radiation from one image to the other. These four terms differ in the value of v assigned to each of them; the effective height ho computed by equation (12) and the path lengths being different in each case. ue Example Assume that from a topographic map the profile shown in Figure 12 has been drawn. The horizontal scale is in kilometers and the vertical scale in meters above sea level. From this profile, combined with inspection of the terrain, it has been found that the ground is so rough that the reflected rays may be METERS ABOVE SEA LEVEL 90 60 ; ng R cole 10 J = 5 Oh Toa D275” KILOMETERS Figure 12. Assumed profile. disregarded. The heights above sea level of trans- mitter, receiver, and obstacle, are respectively hy = 24 meters, ho = 33 meters, h = 69 meters. Since d; = 9,000 meters, dz = 5,400 meters, d = 14,400 meters, we find from equation (12) that ho = —39 meters. Assume a wavelength of 1 meter: —h a c= —— = 0.325 with = 06. Vid dy From Figure 9, the diffracted field is found to be 14 db below the free-space field at the same distance. Ghee DIFFRACTION BY COASTS Bice! Introduction Diffraction occurring at coast lines is significant for coverage problems of coastal radars. It becomes particularly important when the sets are used for height-finding purposes where an accurate knowledge of the lobe angle and possible deformation of the lobes is required. The diffraction might be due either to the fact that the radar is sited on a cliff or to the sudden change in surface properties. Reflection from rough ground is diffuse, so that there is no interference between direct and reflected rays when the reflection point lies on this type of terrain, but interference does occur when the reflection point lies on the sea surface from which regular reflection is obtained. A situation commonly occurring is that of a search radar sited on rough terrain a few miles inland from the coast. Here coastal diffraction may result in an appreciable deformation (shortening or lengthening) of the lobes. More generally, diffraction occurs with level ground whenever there is a change, especially a sudden change, of ground properties along the trans- mission path. The formulas developed for coast- line diffraction may equally be applied to the case where rough ground suddenly changes into smooth, reflecting ground. Similarly, the effect of patches of smooth ground in rough surroundings, such as a lake in wooded country and, vice versa, rough patches in smooth terrain, may be treated by means of the Fresnel-Kirchhoff theory. Here, attention will be confined to the case of a straight boundary, applying the diffraction theory developed in Sec- tion 8.1. eg Level Site Near Coast Assume a transmitter sited on rough ground near acoast. If diffraction were disregarded, the coverage pattern would appear as follows. When the reflec- tion point is on the land, the reflected ray is diffusely scattered and its field at the receiver is negligible. Again, if the reflection point falls on the sea, the reflected ray will be present and will interfere with DIFFRACTION BY COASTS 179 the direct ray with its full or nearly its full intensity. The ray leaving the transmitter at an angle yo (Figure 13), such that its reflected counterpart undergoes reflection right at the shore line, divides the coverage diagram into two parts. For angles of VERTICAL SECTION’ IMAGE =~ tly Tee fw 1% T 7 d, — \ Y PLAN VIEW do A cere Figure 13. Diffraction by a coast line. elevation larger than Y = Yo the field will be essen- tially the free-space field; for angles of elevation less than Y = y the familiar lobe pattern, for complete reflection, will appear with maxima equal to twice the free-space field. When diffraction by the coast line is taken into account, the discontinuity expressed by this rough picture is replaced by a smooth transi- tion of the field from one region to the other. The land surface may be considered as an opaque screen for the image transmitter from which the reflected rays seem to come (Figure 13). This prob- lem is somewhat different from the diffraction prob- lem treated previously since the trace of the screen in the vertical plane through 7’ and R is no longer perpendicular to the line 7’R as it was, for instance, in Figure 2, upper part. In the present case, the effective height ho of the diffracting edge for any given ray is the perpendicular projection from the coast line upon this ray, as shown in Figure 13. The slant distance of the coast from the trans- mitter is d,. Assuming that the receiver (target) is far distant, a condition usually fulfilled in radar practice, d, >> d, and the angle y between the direct ray and the horizontal will be equal to the angle between the image ray and the horizontal. Then approximately, since the angles are small, ho — day = dio = y), where d; and aq; have the significance given them in Section 8.1.4. Here the signs have again been chosen so that ho is negative when the receiver (14) (target) is in the shadow of the screen with regard to the image transmitter. The distance from the transmitter to the diffract- ing coast depends on the azimuth (Figure 13). Therefore, with the designations of the figure, wo (15) cos y dy = 843 Equation for Field Strength The expression for the diffracted field of the image transmitter is given by the straightedge formula, equation (6), with v given by equation (3). Since 1/d. is assumed negligibly small compared to 1/d,, we find, on using equation (14), 2dy v= o-WNZ° cs) This may be further simplified by introducing (as in Section 8.3.2) a new variable, the lobe number aes Ah aN (17) (hy = transmitter height). This quantity is equal to 1,3,5---at the interference maxima and equal to 0,2,4,--- at the interference minima but is here taken as a continuous variable, defined for any value of ¥. In particular for Y = Yo we put n = Nm. Since Yo = hi/di, we have by equation (15) re Aho a 4h? cos ie (18) nN Ado Equation (16) may now be written No — n y= Roars (19) The diffraction formula will again be written, in the form of equation (7), as BE = ze, Ko where z and ¢ are the functions of v shown in Figures 4 and 5. The total field obtained by the interference of the direct and reflected ray is E = B,(1 — ze ™"-*), (20) where the negative sign in front of the second term in parentheses accounts for the 180-degree phase shift at reflection, and the phase lag mn corresponds 180 DIFFRACTION BY TERRAIN to the path difference between the reflected and direct rays. The absolute value of the field is Ey Figure 12 in Chapter 5 may be used for the numer- ical evaluation of this equation. The formula can readily be generalized to the case where the reflected ray is weakened by (1) a reflection coefficient, R, different from unity, and (2) the effect of the earth’s curvature expressed by the divergence factor, D, (Chapter 5). If, moreover, the phase lag at reflection is not + but + + ¢’, the equation becomes E E | = V(1 — 2RD)?+ 42RD sin? % (an +’ + §). 0 (22) = V(1 — 2)?4+ 4zsin 14(an + O). (21) soa Example Assume the following conditions. A radar set of 200 me (A = 1.5 meters) is sited ata height hy = 15.3 meters (about 50 feet) and at a distance to a straight shore line of d) = 195 meters (about 0.12 mile). The ground between the radar and the seashore is level but can be considered as rough for prac- tically any angle of elevation, on applying the criterion of Section 8.3.2. The coverage diagram will first be determined in the azimuth perpendicular to the coast line, where d,; = do, or cos y = 1. Then by equation (18), m = 3.20. With this value of the variable v is determined by equation (19). We shall confine ourselves to integral values of n, that is, to those angles which, in the presence of simple reflecting ground, correspond to lobe minima and maxima. Having obtained v, one then deter- mines z and ¢ from Figures 4 and 5. The field in terms of the free-space field is then obtained from equation (21), either by direct computation or by means of Figure 12 in Chapter 5. The numerical data for the first five lobes are summarized in Table 1. The last column of this table contains the values of #/Eo which would be obtained if the magni- tude of the reflection coefficient were assumed to be zero over land and unity over the sea and if diffrac- tions were neglected. The same calculations are carried out for an azimuth inclined by an angle y = 45° with respect to the coast line. Then, from equation (18), mo = 4.5. The results are given in Table 2. It is seen from these data that the lobes near the critical ray (ray whose reflection point is at the coast line) undergo very considerable deformation. The Taste 1. (vy = 0°). |E/E,| with | E/E>\ without n v z ¢ diffraction diffraction (degrees) 0 1.27 lave 0 0.17 0 t 0.87 1.05 —12 2.05 2 2 0.48 O80 —15 0.75 0 3 0.008 0.54 — 4 1.54 2 4 —0.32 0.36 24 0.69 a 5 —0.71 0.26 iit 1.26 1 6 —1.11 0.19 145 1.16 1 7 -—1.51 0.14 242 0.94 i 8 —1.90 0.12 5 0.88 1 9 —2.30 0.10 158 0.92 1 10 —2.70 0.08 339 0.94 i TaBLE 2. (y = 45°). |E/Eo\ with — |E/Eo| without n v z ¢ diffraction diffraction (degrees) 0 1.50 1.07 6 0.10 0 al 117) «#117 -— 3 2.17 2 2 0.83 1.03 —138 0.21 0 3 0.50 O82 —15 1.80 2 4 0.17 O59 — 8 0.42 0 5 -—O017 0.42 it 1.42 1 6 —0.50 0.31 42 0.80 1 7 —0.83 0.23 90 0.91 1 8 -—1.17 0.18 157 0.88 1 9 —1.50 0.14 240 0.99 1 10 — 1.83 0.12 338 1.12 1 coverage pattern corresponding to Table 1 is shown graphically in Figure 14. ™ —————-WITH DIFFRACTION SS — — — WITHOUT DIFFRACTION HEIGHT 777 Tae 7 FLEA LE. DISTANCE | Figure 14. Coverage diagram (relative field strength). (Heights exaggerated 3.5 to 1.) In the problem considered here, the angles of elevation are comparatively large (for n= 1, y = 1° 24’). If the effects of diffraction occur at DIFFRACTION BY COASTS 181 lower angles, the divergence factor D must be taken into account (see Chapter 5). This is done by com- puting D for the angles desired and replacing z by 2D in equation (21). 8.4.5 Cliff Site If the radar is sited on a cliff and if the land inter- vening between the radar and the reflecting plane (ocean) is rough, the equations of Section 8.4.3 apply. We shall now consider the case where the radar is sited at some distance from the cliff edge and where the ground between the radar site and the cliff edge is reflecting. There are then two reflecting planes, the lower of which might be the ocean, or Diffraction from a cliff site. Fiaure 15. might be a reflecting land surface. In Figure 15, this surface has been designated as ocean. The upper plane is at a height H above the lower plane and the transmitter at a height h; above the upper plane. Assume that the azimuth chosen is perpendicular to the direction of the cliff edge; the distance of the radar to the cliff edge is do. For any other azimuth (angle y in Figure 13), replace dy by do/cos y in the following equations. Two images are shown in Figure 15 and two fictitious opaque screens, one corresponding to each image. The corresponding variables are distinguished by single and double primes. The lobe numbers are given by n= Shay oN ee AH + hy)y — nH + In) r hy The critical angle, Yo = h1/do, is the same for both image transmitters. Thus Pes Mah No! a 4(H + hi)hy a no! (H + hy) Ado hy Further Was ny’ — n! VIN! gi Oe ele ee V2n9!" 1+ H/hy Again, the field is given by 1 SA GOS eS (23) The expression for the field strength [equation (23)] is in a form where all the quantities involved may be evaluated for any given height of transmitter, height of the cliff, and any wavelength by using graphs and tables given in earlier paragraphs. Chapter 9 TARGETS oo SCATTERING PARAMETERS as Radar Cross Section N DETERMINING the coverage to be expected of I radar systems, it is important to know what fraction of the power incident upon a target will be returned to the receiver. A parameter involving the dimensions and orientation of the target, and usually also the wavelength, and which measures the propor- tion of power returned, is called a scattering para- meter. The most generally used of these parameters is the radar cross section introduced in Section 2.4.1. It is denoted by o and is defined by Ser) (1) where W, is the scattered power per unit area at the receiver and W; is the incident power per unit area at the target. In terms of o, the radar gain is Fm Ga (2S Vast 2) yee 4rd? \8rd This equation may also be written in the form Pe = GG Eons By 2 where 3d A= (==) A) = AA (>) : Oe is the gain factor introduced earlier (see Section 5.1), A» is the free-space gain factor and A, is the path- gain factor. Target Gain Another scattering parameter is Gp, the target gain, discussed in Section 2.4.2. It is the gain of the target in the direction of the receiver relative to a shorted (dummy) doublet antenna. The target gain is connected with o by the relation V4ro 3 3h 8) Gp rs The corresponding radar gain is P, P, = AG GG p? At. (4) The factor 4 is due to the calculation of Gp relative to a shorted doublet rather than to a matched load doublet. If the calculation of Gp were made relative to the matched load doublet, the factor 4 would be replaced by 1. Ss Echo Constant The echo constant, denoted by K, is defined by _ WwW, a ; ae | * Wy, (c 9) and is related to o by kK=—~—. 6 4a (6) The corresponding power ratio is P, = sr): = KG,G.{ — ) At. 7 pa ( 3 we Except for the factor 47, K is just ¢ measured in square wavelengths. 9.1.4 Equivalent Plate Area A plate of area S placed normal to the direction of propagation has a radar cross section given by go =47r—, (8) provided the linear dimensions of the plate are large compared with \. Any target may be supposed to scatter (in the direction of the radar) an amount of energy equal to the amount that a plate of area S would scatter in this direction. This area S is called the equivalent plate area of the target. The corre- sponding radar gain is Py — 1G Ear (9) RADAR CROSS SECTION OF SIMPLE FORMS 183 ae Scattering Coefficient or Characteristic Length This parameter has also been called the radar length of the target. The definition is , (10) where #, = field strength at the receiver, E, = field strength incident on target. It is evident that o Pat i fe (11) connects 1 with radar cross section. The radar gain becomes seule) 20 (12) 1 3X 2.2 RADAR CROSS SECTION OF SIMPLE FORMS 9.24 Spheres The radar cross section of any large curved con- ducting surface having principal radii of curvature p, and p2 at the reflection point is given by (13) This formula applies if the surface is sufficiently large and sufficiently curved to contain many Fresnel zones. For a sphere of radius a, where a>>r, go = TPpip2.- (14) Thus, in the case of a large conducting sphere, the radar cross section is equal to the geometrical cross section and is independent of wavelength. The result for small spheres (a < < ) is C= Ta". = o = 1447° aa (15) There is no simple formula for the radar cross section in the region a ~ 2. 9.2.2 Cylinders The radar cross section of a cylinder whose length is large compared with the wavelength is 2ral? gat r , (16) where a = radius, L = length (LZ >>). This formula assumes that the direction of inci- dence is normal to the cylindrical surface. If the cylinder is tilted so that there is a small angle 6 between the normal to the cylinder and the direction of incidence, the result is ox 2rLé |? tbs 2raL? f oN (17) r 27L0 r This result holds for small angles of tilt 6 such that sin 6 = 6. mane Plates A flat plate of area S with all dimensions large compared with and oriented so that the normal to the plate is in the direction of incidence, has a radar cross section given by S? o=4r—, 2 (18) regardless of shape. For a circular plate (a disk) of radius a, whose normal is at an angle @ with the direction of incidence, o= 7a? [cots - Jy (e sin ) S (19) where J; is the first-order Bessel function. The maximum value is at @ = 0, where Z 374 poe, (20) Mw This agrees with equation (18), since at normal incidence S = za’. The peculiar feature of equation (19) is that the maximum at @ = 0 is very sharp. For example, if d/a = 1/10, o is only 1/10 of its maximum value when 6 = 1.25°. The average value of o over all orientations is (21) = — 107 2 This result is independent of wavelength and sug- gests that a large number of flat plates oriented at random will have a cross section independent of }, or that a few surfaces of rapidly changing orienta- tion may have this property. re) zx 2 = Zz oN OSE a C3 x < = +72" 4 — 0.000766), Bx! (25) where L = length of edge of reflector, 6 = angle between direction of incidence and the axis of symmetry in degrees (0 < 26°). As a function of 6, o has a broad, flat maximum. Consequently, the return to the radar receiver from such a target is not sensitive to the precise orienta- tion of the axis of symmetry. ae AIRCRAFT 9.3.1 Variation with Aspect Diagrams showing the dependence of o on orienta- tion indicate very large and irregular fluctuations. Radar cross section o can change from values of nearly 1,000 square meters to a few square meters as a result of a change of aspect of a few degrees. These instantaneous values of the radar cross section are dependent on wavelength, polarization, details of plane design, areas of specular reflection, propeller rotation, etc. Reflection patterns have been meas- ured for a few simplified models by laboratory means (see Figure 1 as an example). It would be difficult to calculate instantaneous values of o by theoretical methods. In practice, however, an airplane is in motion and is affected by air currents. These factors cause the airplane, in a short interval of time, to present many widely different instantaneous values of o to the radar, so that the signal actually seen on the scope by the observer is in effect a time average, where the most violent fluctuations of instantaneous values of « have been smoothed out. Hes Measurement of o The radar equation for free space, equation (45), in Chapter 2, may be used for the computation of average values of o from observed instantaneous values, provided conditions are such that ground reflections are unimportant. The received power P2 is determined by matching the signal from the plane with the measured signal from a signal generator. The procedure followed in work at the Radiation Laboratory is to measure the maximum value of P2 186 TARGETS for each of a series of 3-second intervals. A plot is made of Pz against range d on log log coordinates. As might have been anticipated from equation (45), in Chapter 2, it is found that a line with a constant slope of —4 passes through the average of the 3- second interval maximum points, although the individual points fluctuate widely. The value of ¢ corresponding to this line is calculated. The resulting value of o still cannot be called an average value because the maximum value of o has been used for each point. Consequently these values of o, substituted into equation (45) in Chap- ter 2, cannot be expected to give the average value of Pe, or to give observed maximum ranges. How- ever, it is found that if the values of o thus computed are reduced 40 per cent, they give correct results. These empirical cross sections are relatively inde- pendent of wavelength. This result may be inter- preted to mean that a plane in motion behaves more or less like a collection of specularly reflecting sur- faces oriented at random, as equation (21) indicates. Attempts have been made to develop formulas giving operational cross sections as a function of some large feature of plane design, such as wing span or length of fuselage, but these attempts have not been successful. Chapter 10 SITING 10.1 GENERAL HO Introduction ITING REFERS to the selection and utilization of local terrain features which affect. propagation and the performance of equipment. From a pre- liminary analysis, the general location, type of equipment, and height may be determined. The specific sites available may, however, profoundly alter performance in several ways. Careful analysis and tests may then be necessary to determine the best use of the facilities at hand and for an under- standing of the limitations due to the terrain. pent 2 Siting Requirements With communication equipment, the siting prob- lems are principally concerned with visibility and, in wooded areas, absorption by vegetation. When siting direction-finding equipment, it is important to realize that reflections from mountains or other irregularities may cause serious angular errors which should be avoided by proper choice of the location. Both direction-finding and radar equipment require orientation. Radar siting requirements are rather different and depend on whether ground reflection is of importance or not. The siting of radars operating mainly on the direct ray is relatively easy and is principally concerned with permanent echoes and _ visibility. The most exacting site requirements are presented by the VHF early warning and height-finding radars, which to a large extent depend on ground reflection for successful operation. The siting problem then requires the consideration of terrain effects such as limited reflection areas, cliff edges, obstacles, etc., which involve diffraction problems of considerable complexity. Recommendations for specific sets are given in instruction manuals furnished with the equipment. 10.2 TOPOGRAPHY OF SITING 10.2.1 Maps Radar and direction-finding systems, which may cover a large area and involve many services, use a grid for plotting purposes. The grid location, height, and orientation of each station must be known with reasonable accuracy. Topographic maps of a scale of one or two miles to the inch and contour intervals of not more than 100 feet, preferably 20 feet, should be secured. These may be supplemented by aerial photographs and surveys. 10.2.2 Profiles In a complicated terrain, it is usually necessary to have profiles on several azimuths to determine the effective height above the reflecting surface. The accuracy required decreases with the distance from the transmitter. In most cases sufficient detail is not available on maps, so that a personal inspection of the terrain should be made to become familiar with the nature of the soil and degree of roughness. Special attention should be given to ridges, flat areas, bodies of water, distance to the shore, hills to the rear, obstacles in the operating area and at the boundaries. Me Orientation Where long distances and directive beams are involved, fairly accurate orientation of the order of one-half degree is required. Care must be taken when using compasses because of local attractions or inadequate information on declinations. Observa- tions on Polaris give the greatest precision but this star is not always visible and it is often inconvenient to use a transit at night. Caution must be used in aligning on permanent echoes, as they may be diffi- cult to identify. In general, several methods should be used to obtain independent checks. Solar azimuths, correct to the nearest quarter of a degree, may be determined from the date time to the nearest minute, and the latitude and longitude to the nearest degree. Two methods will be given for obtaining the azimuth of the sun: (1) by calculation, (2) from tables. A third method gives true south only. 187 SITING The azimuth of the sun may be calculated from the formula, tan B = — ——— aie ey) ; (1) cos@ tan 6 — sing cos (HA) bearing of the sun. The bearing is east or west of south when @ — 6 is positive. The bearing is east or west of north when o — 6 is negative. The bearing is east in the morning (6 will be negative) and west in the afternoon (8 will be positive). hour angle of the sun. During the morn- ing hours when the hour angle is greater than 12 hours, its value should be sub- tracted from 24 hours for use in the formula. = latitude of the place of observation. declination of the sun at the time of observation. The signs of @ and 6 are important and each is positive when north of the equator and negative when south. The hour angle HA is the local apparent time (LAT) minus 12 hours. To convert the observed time into LAT, the civil time at Greenwich (GCT) must be found and combined with the equation of time to correct for the apparent irregular motion of the sun. This gives Greenwich apparent time GAT, which is converted to LAT, by allowing for the longitude. The equation of time and the decli- nation of the sun are plotted for 1945 in Figure 1. The annual change is small and these curves may be used for most orientations without regard to the year. Standard time meridians are given every 15 degrees east or west of Greenwich, each zone corresponding to one hour. Care should be used to take daylight saving or other changes from standard into account correctly. The calculations may be illustrated from the following data: date, 16 March; time, 1345 hours PWT; latitude, 40° north; longitude, 118° west. The HA is computed first. Observed time (PWT) where B = 13 hr 45 min Zone difference + 7hr Greenwich civil time 20 hr 45 min Equation of time (Figure 1) — 9 min Greenwich apparent time 20 hr 36 min Longitude difference (for 118° W) — Thr 52 min Local apparent time (LAT) 12 hr 44 min LAT —12 hours = HA —12hr Hour angle of sun + Ohr 44 min HA in are +11° Latitude b + 40° Declination of sun 6 (Figure 1) eg Substituting in equation (1), sin 11° tan 6 = — cos 40° - tan (—2°) — sin 40°- Be= 16710! Since @ — 6 is positive, B is the bearing from the south. The bearing is west of south, since HA is positive (p.m.). The azimuth of the sun is 180° + 16°10’ = 196° 10’.* The equal altitude method is less convenient but requires no calculation. This method consists in measuring the horizontal angles between the sun and a mark taken when the sun is at the same alti- tude on both sides of the meridian of the observer. The bisector of the horizontal angle between the two equal altitude positions of the sun during the observations is very close to true south, and the azimuth of the mark may be determined. cos 11° 10.3 GEOMETRICAL LIMITS OF VISIBILITY Be: Horizon Formula It is assumed throughout that the earth radius is ka (see Section 4.1). Whenever numerical examples are given, the standard value, k = 4/3, is used. The alternate method of accounting for refraction given in Section 4.1.5 may also be used in connection with the following equations if k # 4/3. When a horizontal ray, tangential to the earth, is drawn, the earth slopes away (Figure 2) at the rate of d? ee (2) 2ka Hence the horizon distance dr for a transmitter at a height h above level ground is equal to dp = V2kah. (3) Numerically, when all the lengths are in meters _ 4 dr = 4,120 VA fork =<. (4) 3 With h in feet and dr in statute miles, by a curious numerical coincidence, = 4 dp = V2h fork = 3 (5) 2 This result could have been obtained directly from Azimuths of the Sun, HO71, U.S. Naval Department, Hydro- graphic Office. The equation of time may be obtained from a current copy of The American Nautical Almanac, U.S. Naval Observatory, Washington, D.C. GEOMETRICAL LIMITS OF VISIBILITY When both terminals of a path are elevated above the ground (Figure 3), the horizon distance is dz, = V2ka (Vi; + Via), (6) 10.3.2 Height of Obstacle As a first case, consider a smooth earth and two terminals at the ground. The earth itself forms an SS he i] BEa te iz gia ae H+ - DECLINATION OF ERS H+ ECE EERE oa is aaea He oe iN are V v2 ia Ss es \ =o a i SF ae a gS 08 io Zz NEES | | eee o a5 SNES BEB uw ae EN paola IN el (eg Mo el Bes LD (ols | V | — +4 SES aSANSea Ala a es sO [LN nS 24 fir 21 31 10.20,2 12221 M2) 4 Ml 2 8) 10 203010 20309 19290 18280 18287 7277 17 27, JAN FEB MAR APR MAY JUN JUL AUG SEP oct NOV DEC SUN DATA FROM NAUTICAL ALMANAC 1945 Fiaurn 1. Calculation of solar azimuth. + ka Figure 2. Geometry for horizon distance for zero Ficure 3. Geometry for horizon distance with ele- height transmitter. where again V2ka = 4,120 in the metric system. If dis in statute miles and h in feet, dy, = V2h, + V2h. fork = (7) The relation between hi, hy and d;, is graphically presented in Chapter 5 in the form of a nomogram Figure 2. ? vated transmitter. obstacle which reaches its maximum height h,, in the middle of the path (Figure 4). By equation (2) hn = — ~~ Ska’ (8) A point P on the ground at distances d’ and d’’ from the two terminals (see Figure 4) has an elevation 190 SITING above the straight line connecting the terminals given by (ae i 20 Bie A - — — (9) 2kha 2ka or, after a simple reduction, ih jhe a : h = ——- = 5.9-10°a'a”, (10) 2ka d+d 2 foe. Ficurs 4. Height of earth as an obstacle. where h, d’, and d”’ are given in meters. If h is in feet and d’ in statute miles, nee fork = 4/38. (11) Secondly, assume that the terminals are elevated (Figure 5). The elevation of the straight line con- necting the terminals, for a flat earth, is equal to d’hy — d'’hy ag? (12) where d’, d’’ are again the distances to the terminals, and hy, ho are the corresponding elevations. In order to account for the effect of the earth’s curvature, Figure 5 may be considered as a plane earth diagram on which a ray will appear curved, the deviation from a straight line being downward and given by equation (10). This is indicated by the dashed line in Figure 5. h le. d' Ficure 5. Height for elevated terminals. Hence, the total height above the theoretical ground is d'hg —d"'hy d’d’’ d’'—d" —— ka * When the heights are expressed in feet and the dis- tances in miles, the first term remains unaltered, j= (13) while the second term again becomes d‘d‘’/2 for k = 4/3. Equation (13) is used to decide whether and by how much an obstacle such as a hill will obstruct a given transmission path. heed Extended Obstacle When the obstacle is of appreciable horizontal extension, it may not possess a single peak to which equation (13) can be applied without ambiguity. The case of twin mountains is shown in Figure 6 for straight rays (earth’s radius ka). Ficure 6. Height of equivalent diffracting edge. The optical peak P of the obstacle for radio or radar transmission is the point from which both terminals are just visible. For a given profile, the limiting rays to the terminals may be found by trial and error by applying equation (13) to those points of the profile which are most likely to represent limiting elevations. In the theory of diffraction given in Chapter 8, P marks the position of the equivalent diffracting edge. 10.3.4 Degree of Shielding As a measure of the degree of shielding, the angle between the two limiting rays drawn from the terminals to the (actual or equivalent) peak of the obstacle of height h, may be used. Since all angles considered are small, the sine or tangent of the angle may be replaced by the angle in radians. Consider first the ray going from the first terminal to P (Figure 7). The angle of the ray with the horizontal at the terminal is = t= lon as a = (14) d’ ka and its angle with the horizontal at P is d’ h,—h d’ ae =aq+—=~- . 15 a, ka d’ 2ha a) The angle of the ray going from the second terminal to P is determined correspondingly. The angle between the tio rays is then equal to 1 hp hy cme ) (16) hea = Bogie: (i: dia" da’ a where d = d’ + d’’, and (ka)! = 1.18- 10°‘ (meter) '. When d is measured in miles and h in feet, equation (16) becomes Bi + Bo = 1.89- 10a] Fe ee [ (17) d’d!’ ae ah! Figure 7. Shielding between transmitter and receiver. 10.4 PERMANENT ECHOES 10.4.1 Introduction Permanent echoes are caused by reflections from terrain features such as mountains or even smooth surfaces near the antenna (ground clutter). With radars, the indicator is obscured by the strong echoes from hills and the minimum detection range is increased by ground clutter. With direction finders, erroneous indications are caused by the spurious reflections. Permanent echoes are among the princi- pal problems involved in siting, as many otherwise excellent sites are rendered worthless by excessive fixed echoes. Several methods are available for determining the suitability of sites in this regard without actual field tests. A number of factors combine to make permanent echoes more troublesome than might be expected. 1. Hills and land surfaces are so much greater in extent than the target which the equipment is de- signed to detect that strong echoes may be obtained from distances where an ordinary target would give an echo far below normal detection levels. 2. The low elevation of the land surfaces places them in regions most subject to nonstandard propagation effects where extreme ranges and large responses are frequently obtained. 3. Side lobes of the horizontal pattern of the an- tenna cause permanent echoes to appear at several other azimuths in addition to that of the main lobe. PERMANENT ECHOES 19] 4. Strong permanent echoes from mountains to the rear may be caused by back radiation from the antenna. The low intensity of the back radiation may be compensated by the size of the mountains. Such echoes are especially harmful as they obscure ihe operating sector. 5. Objects appear wider because of the antenna beamwidth and of greater extent in range as a result of the pulse width. 6. Diffraction over intervening ridges may be sufficient to nullify their screening action so that objects behind a ridge are visible. 7. The use of a permanent echo as a standard target may be very misleading. A decrease in per- formance that seriously affects echoes from small targets may not have any noticeable effect on the response from large targets. An echo used for a standard target should be weak and near by. 042° Permanent Echo Diagrams The permanent echoes associated with a radar station may be plotted on a polar chart and their extent, location, and strength represented. Such diagrams should be prepared for each unit of a radar system, using a standard procedure for taking and presenting the data. Permanent echo data should be taken under average conditions with the gain set at some stand- ard level. At intervals of azimuth such as 5 de grees, the ranges of the permanent echoes are recorded. These data are then plotted on a polar chart and the points are connected to indicate obscured areas. The skill and judgment of the operator are important factors. In most cases the amplitudes of the echoes are so far above that of ordinary target echoes that the actual amplitudes need not be noted. In Figure 8 is shown an observed permanent echo diagram for a VHF radar. This was selected for purposes of illustration rather than as an example of a good site. The mountains to the north are un- shielded and cause extensive echoes. The large echo at 200 degrees is due to a mountainous island 260 miles away and appears only during times when propagation is nonstandard. Care must be exercised in identifying the cause of an echo. Antenna side lobes cause spurious echoes and distant echoes may come in on the second or third sweep on the scope after the main pulse. These latter echoes may be checked by changing the pulse: repetition rate and observing the shift of the echo. 192 SITING Permanent echo diagrams are useful for: hills are screened by a local obstruction. This local 1. Indicating blind areas in a station’s coverage. echo at, say, three miles, is combined with the main 2. Assigning the operating area of a station. pulse or ground return and the distant echo is 3. Checking the range and azimuth accuracy. weakened or eliminated entirely. 4. Checking the performance. Shielding causes a loss of coverage, which in 5. Estimating nonstandard propagation. operating regions may be more serious than the 6. Planning test flights. permanent echoes. Rear areas which are not scanned Ficure 8. A typical permanent echo diagram for a VHF radar. a Shielding should be well shielded so that back and side echoes The principal device in the field for the control of do not interfere with targets in important tactical permanent echoes is shielding. This means that the regions. Operation over such shielded sectors would antenna must be sited in such a way that distant be limited to high targets. PERMANENT ECHOES 193, 0-4-4 Prediction of Permanent Echoes Permanent echoes may be determined by several methods: (1) tests with the radar at the site; (2) radar planning device [RPD]; (3) supersonic method; and (4) profile method. The feasibility of moving the radar to the site to determine the permanent echoes is dependent on the portability, accessibility, ete. Echoes obtained with one type of equipment may be very different from those of another type of radar with a different an- tenna directivity, frequency, and range. The RPD technique requires construction of a relief model of the terrain considered. A small light source is used to simulate the radar transmitter and the echoes are plotted as a result of a study of the areas illuminated. This method is useful for short ranges and microwaves where the diffraction and side and back lobe radiation are small. Construction of a fairly difficult relief model may take a crew of specially trained men several days to a week, as the model should be accurate. Once completed, all possible sites or aspects from a plane or ship may be readily examined. Models of enemy areas may be used to predict the coverage of possible enemy sites and evasive action may be planned. The RPD is well suited for training and briefing of air personnel. GO 200 METERS 1 fo) 2 4 6 8 10 12 14 16 KILOMETERS FiaureE 9. Typical profile. Kits are provided containing the light source, sup- ports, etc. Photographic and darkroom facilities are also required. The supersonic method uses a relief model under water. Supersonic gear is used to send out pulses which are reflected like radar pulses and an echo is picked up and presented on a plan position indi- cator [PPI] scope. Photos may be taken of the scope or it may be used directly to train operators and for briefing. Considerable equipment is re- quired, but the construction of the models is com- paratively simple. The profile method involves a study of topo- graphical maps and plotting of the echoes according to their visibility and the amount of diffraction. A fairly difficult site may be handled in perhaps eight man-hours. This method is adapted to long-range, low-frequency radars where diffraction and side and back lobe radiation are important. On microwave equipment, prediction of permanent echoes is sim- pler and the profile method may be worked out in a few hours. 045 Prediction by the Profile Method The discussion here refers chiefly te VHF (1 to10 m) radars in a mountainous terrain, but the methods have general application. The principal requirements are topographic maps of the surrounding area with a seale of one or two miles to the inch and a contour interval of 20 feet, although intervals up to 100 feet may be used. Regional aeronautical maps with a seale of about 1 inch to 16 miles and 1,000-ft contours are suitable for checking distant echoes. From the maps, profiles are prepared for various azimuths about the radar station. The first mile or so should be plotted accurately; at greater distances, the critical points, such as hills and breaks, should 1eceive the most attention. On each profile is drawn the tangent line from the center of the antenna to the point on the profile which determines the shield- ing, as in Figure 9. The angular elevation a of this line of sight is marked on the diagram. If a@ is nega- tive, the profile should be checked out to the radar horizon to obtain the correct shielding angle. On a plane earth diagram, the line of sight is actually curved, but for distances up to 10 miles it may be taken as straight with small error. The height difference, with a in radians, is then equal to (18) When the distance is larger so that earth curvature has to be taken into account, to the above expression for the height difference hy, — h; must then be added the amount by which the earth is sloping eway over the distance d. This amount is d?/2ka, and the complete expression for h, — h; becomes hg — hy = dtana. @& hg — hy = dtana + a (19) a For easier handling of this equation, a set of curves may be drawn where hz — hz is plotted against d for various constant values of a. These curves may then be used to determine the height of the shielded region at any range. Thus all other moun- tains along the profile in Figure 9 might be checked for visibility by comparing the height of the moun- tain with the value of hz — h; read from the curve a = 0.5°. Any desired allowance for diffraction may be made by using a different curve such as a = 0. When the shield consists of several ridges Ficure 10. close together, an equivalent shield should be used. This is derived by enclosing the ridges in a triangle, whose apex is taken as the shield (see Figure 6). The general procedure to be followed in preparing a prediction of permanent echoes will now be out- SITING lined. From an examination of the map, the azi- muths at which profiles should be prepared are determined. This will normally be about every 10 degrees. Where the shielding is obviously good, the interval may be 20 degrees, but where the terrain is questionable, such as in a region of low hills, the profiles should be taken at 5-degree intervals. Theoretical permanent echo diagram. An overlay of the region is then prepared, showing important geographical features and a polar-grid system. On this chart is drawn the coverage contour lines (broken lines in Figure 10). These lines repre- sent the limits of the heights of the shielded regions. EFFECTS OF TREES, JUNGLE, ETC. 195 Targets or mountains below and beyond the cover- age contours will not be visible except by diffraction. These contours may be drawn for several heights. Where they are close together, the shielding is good but the coverage is poor. Where the lines are widely separated, as toward the sea, there is little or no shielding except that due to earth curvature. With the coverage contour diagram superimposed on a map, the peaks exposed to radiation may be noted. The extent of the echoes due to these peaks de- pends, besides the size of the peak, on the horizontal radiation pattern, the pulse width, and the power and sensitivity of the radar. It should be noted that the half-power beamwidth is only a rough measure of the width of an echo and some greater angle between the half-power points and the nulls will usually be obtained for the echoes. The extension of the echo in range will be at least as great as the pulse width in miles as represented on the scope. This is about 0.1 mile per micro- second of pulse width. Actual echoes are thicker than this, since all the exposed hill sends back echoes. After a careful inspection of the profiles, taking into account the various factors mentioned above, the echoes are sketched in on the chart. In doing this, judgment and experience are important factors, but the following rules may be used as a guide. 1. Shade in a circle for the main pulse several miles wide, depending on the pulse width and local return. 2. Check each profile in turn and for each peak or hillside in front of the shielding ridge or mountain plot an echo for the main and all side lobes of the antenna. 3. A series of sharp hills within the shielding part of the terrain should be plotted as a single large echo. 4. The inner edge of an echo should be at the same range as the hill. 5. Peaks beyond the shield may be in the diffrac- tion region and the relative strength of the echo may be estimated from a diffraction curve. 6. In general, the echo strength varies as the inverse square of the distance and is roughly pro- portional to the target area. 7. Where there is any doubt, the echo should be plotted. Experience is an essential factor in permanent echo prediction, regardless of the method used. The methods described here have been used successfully in many areas and are capable of accuracy adequate for most purposes. 0.5 EFFECT OF TREES, JUNGLE, ETC. The Effect of Trees 10.5.1 Trees form very effective obstacles for high-fre- quency radio waves. A single tree may cause a drop in signal strength of several decibels. The attenua- tion is less for horizontal polarization than for vertical polarization for frequencies below 300 to 500 megacycles. For higher frequencies, the polarization is not an important factor. With the transmitting antenna sited in a moderately wooded area, repre- sentative values for the losses are given in Table 1. TasLE 1. Decrease in gain for transmitting antenna situated in a moderately wooded area. Horizontal Vertical Frequency polarization polarization 30 me Negligible 2-— 3 db 100 me 1-2 db 5-10 db When both antennas are in the woods these losses should be doubled. Measurements at 200 me for transmission through a grove of trees 100 feet wide show losses of 21 db for vertical polarization and 6 db for horizontal polarization. When the antennas are in clearings, so that each is more than 200 or 300 feet from the edge of the woods, the decrease in gain is small. With vertical polariza- tion, there may be large and rapid variations of field intensity within a small area, due to reflections from near-by trees. nF 2 The Effect of Jungles In jungles or heavy undergrowth, an exponential absorption is to be expected. Tests made of trans- mission through heavy jungles, such as are found in Panama or in New Guinea, show that the limit of transmission for ordinary field sets is 1 mile. An increase of power of several hundred fold is needed for a range of 2 miles. The decreases in gain en- countered are of the order of 50 to 60 db per mile. If the antennas are elevated above the jungle or located in clearings, the effect of the jungle may be minimized. Antennas should be 10 or more feet, away from trees to avoid a change in antenna impedance. The best solution is sky-wave transmission even for distances as short as 1 mile. Due consideration should be given to the selection of optimum fre- 196 SITING quencies based on ionosphere predictions. For distances up to 100 or 200 miles, a half-wave hori- zontal wire antenna should be used and the fre- quency range is about 2 to 8 mc. The decrease in gain for the short path up through the trees is negligible at these frequencies. 5.3The Effect of Trees and Obstacles on Microwaves At 10 em, the absorption is so great with most objects that the diffracted energy is the principal portion transmitted. Only windows, light wooden walls, or branches of leafless trees show less than 10 db loss. Opaque objects include: 1. Rows of trees in leaf if more than two in depth. 2. Screens of leafless trees if so dense that the skyline is invisible through them. 3. Trunks of trees. 4. Walls of masonry. 5. Any but the lightest wooden buildings, espe- cially if there are partitions. Losses of a brick wall may be increased from 12 db to 46 db by wetting. In computing diffraction over treetops, the diffracting edge may be taken to be 5 feet or so less in height. In a 1.25-cm test, the transmission loss through two medium-sized bare trees increased 18 db after leaves appeared. C,C(v). GLOSSARY 1) Radius of the earth 2) Radius of scattering plate, sphere, or cylinder Gain factor = A Ap Gain factor for doublet antennas in free space, adjusted for maximum transfer of power = 3\/87d Plane-earth factor Path-gain factor Gain-factor curve parameter Bandwidth 1) Velocity of light in free space Dees hy — hy mma Real part of Fresnel’s integral Distance from center of transmitting antenna to a point in space measured along the surface of the earth Free-space distance for field of strength # . : d Normalized free-space distance = cam T Distance from transmitter, receiver to reflecting point measured along the earth’s surface Distance from transmitter, receiver to the radio horizon measured along the earth’s surface Line of sight distance measured along the earth’s surface = dp + dr Maximum radar range 1) Divergence factor for spherical earth 2) Aperture of reflector 1) Water-vapor pressure 2) Coefficient for height-gain function Electric-field strength Maximum free-space field strength of a doublet transmitter at distance d Radiation field strength at one meter from trans- mitter 1) Frequency 2) Focal length of paraboloid reflector Cutoff frequency of a wave guide Height-gain function Height-gain function for the n*® mode Fraction of maximum radiation field strength in the direction of direct, reflected rays Noise figure Shadow factor for the first mode Sum of shadow factors for all modes 1) Receiver gain 2) Exponential factor of height-gain function for elevated antennas Correction to g(2) Transmitting antenna gain Receiving antenna gain m, Radar gain of a target Height above ground Height of transmitter, receiver above ground Height of transmitter, receiver above tangent plane at point of reflection Critical height distinguishing high and low antennas located in diffraction region = 30A2/3 Virtual height of obstructing screen 1) Magnetic field strength 2) Height of a reflecting or diffracting obstruction Height-gain function for low antennas Hour angle of the sun RMS current Input current to antenna or circuit s/o 1) Boltzmann’s constant 2) Factor multiplying earth’s radius to account for atmospheric refraction 1) Amplitude of generalized reflection coefficient 2) Echo constant of a target 1) Length of a doublet 2) Height coefficient to include effect of earth’s constants and wavelength 1) Effective length of a doublet 2) Characteristic length or scattering coefficient of a target 3) Radar length of a target 1) Ratio of radius of curvature of a ray to the radius of the earth = p/a a ~ 4ka(hy + he) Modified index of refraction 2) m 1) Index of refraction 2) Number of elements in an antenna array Lobe numbers Lobe variable for imperfect reflection Noise figure 1) Total pressure of the atmosphere 2) Dimensionless parameter = d,/d7 Distance coefficient to include earth constants Power Power output of a transmitting doublet Power delivered by a receiving doublet to a matched load Minimum power detectable by a receiver Noise power Power received by load circuit of receiving antenna Scattered power Dimensionless parameter = d2/d Parameter determining phase of beam reflected by &r 600d the earth = 197 198 GLOSSARY ie 1) Distance from center of antenna to a point in 6. 1) Declination of the sun space (usually replaced by d in applications) 2) Angle of phase retardation due to path-length 2) Height wavelength factor difference between direct and reflected rays 3) Pattern or chart parameter 3) Ground parameter depending on complex dielec- 4) Path length of reflected ray tric constant db Path length of direct ray A. Path-length difference between direct and reflected R. 1) Resistance ; rays =71 —7q 2) Plane-earth reflection coefficient = pe’? Ap. = (Qhyhe) /d 3) Path-difference parameter = (kaA)/(hid7) A(Ap). Correction factor for Ap I r litee Resistive component of antenna impedance Af. Bandwidth oes ls RH. Relative humidity in per cent N- Variable used in diffraction region Ri. Resistive component of load impedance Dielectri ff Rr. Radiation resistance of an antenna as He ale aE ee Space 8. 1) Spacing between dipoles in an antenna array = 8.854 X 10°? =——_ 10-9 2) Coefficient of distance for shadow factor P : : ols : 3) Dimensionless coordinate = d/d So Complex dielectric constant = ¢, — je; S i) Seatterine crosalsection ’ €i. Imaginary part of dielectric constant = 600A , 5) Aten 3 : er. Real part of dielectric constant S,S(v). Imaginary part of Fresnel’s integral ¢ 1) Phase-angle lag due to diffraction YF L 1) Time 2) Dimensionless distance variable for curved-earth 2) Pulse width diffraction 3) Degrees centigrade 6. 1) Angle between horizontal at transmitter base T. Absolute temperature and horizontal at point of reflection ue Dimensionless coordinate = ho/h1 2) Angle of tilt of scattering cylinder v 1) Velocity of wave propagation = ¢/n r Wavelength 2) Argument of Fresnel’s integral jy Permeability of free space = 4710-7 3) Dimensionless parameter = d/dp Ur. Permeability relative to free space V. Voltage V Nopeeyelies v. Angle between reflected ray and horizontal at trans- i oi : mitter W. Power per unit area : Wi. Incident power per unit area p. 1) Radius of curvature Wy. Scattered power per unit area at the receiver 2) Amplitude of reflection coefficient ee Amplitude of ratio of diffracted field to free-space 7 1) Conductivity field 2) Radar cross section Z. Impedance Ts 1) Complex mode numbers Lae Antenna impedance 2) Half beam-width angle ZI. Load impedance ; ; Q, 1) Attenuation constant, real part of propagation p. 1) Phase angle of reflection coefficient constant +/ 2) Latitude ste i ce alien eet ‘ ae 2) Angle made by ray with the horizontal g’. = Z fe T ' ee ee i reflection coefficient QQ Angle of elevati f diffracting edge as seen from Seca teal shes ahs mee ea ah kee a ®,. Distance function for the first mode Yu transmitter, receiver 1) Bearing of the sun 2) Phase constant, imaginary part of propagation constant y Angle between ray from transmitter, receiver, and horizontal at diffracting edge 1) Propagation constant = a@ + 76 2) Angle between horizontal at base of transmitter and line joining transmitter base to receiver Angle between the direct ray and the horizontal at the transmitter Wa. 1) Phase-angle difference between currents in dipole-antenna array 2) Angle between direct or reflected ray and the horizontal at the reflection point 3) Height variable in the diffraction region Angle between direct ray and horizontal at reflec- tion point Angular velocity = 2af Total phase lag between direct and reflected rays =$'+5=¢-7+6 H. H. BEVERAGE T. J. CARROLL J. H. DELLINGER S. S. Arrwoop A. F. Murray OSRD APPOINTEES COMMITTEE ON PROPAGATION Chairman Cuas. R. Burrows Members Martin Karzin D. E. Kerr J. A. STRATTON Consultants J. A. STRATTON C. E. BUELL Technical Aides S. W. THomMAsS R. J. Hearon 199 CONTRACT NUMBERS, CONTRACTORS, AND SUBJECT OF CONTRACTS Contract No. Contractor Subject OEMsr-1207 OEMsr-728 CEMsr-1497 OEMsr-1496 OEMsr-1502 Columbia University New York City, New York State College of Washington Pullman, Washington Humble Oil and Refining Co. Houston, Texas University of Texas Austin, Texas Jam Handy Organization, Inc. Detroit, Michigan Correlation, analysis, and integration of data on radio and radar propagation. Develop meteorological equipment and conduct meteor- ological soundings in the Southwest Pacific and correlate it with radio-propagation data. é Development and construction of microwave field- strength measuring sets. Development of equipment for, and making measure- ments of, time and space deviations in radio-wave propagation, Preparation of a General Outline of Training Material and the preparation of manuals, films, and other train- ing aids for use in instructing technical and other per- sonnel in radio-weather and radio propagation. SERVICE PROJECTS The Committee on Propagation did all of its work under Project Control SOS-9, which was originally set up through the request of the Combined Chiefs of Staff following recommendations submitted by the Combined Meteorological Committee [CMC] (1): that the Committee on Propagation of the National Defense Research Committee be requested to act as a coordinating agency for all meteorological information associated with short-wave propagation, (2) that the Committee on Propagation be requested to forward periodically to the CMC a list of all reports and papers dealing with the meteorological aspects on short wave propagation which have been received or transmitted by that Committee. Later the Combined Meteorological Committee in its thirty-seventh meeting on Tuesday, February 22, 1944, agreed that the NDRC Committee on Propagation be recognized as the supervising committee on all basic research being done in the United States on the related prob- lems of radar propagation and weather, in addition it shall be the recognized channel whereby international exchange of papers of the two related sciences will be effected. The Joint Communications Board [JCB] therefore approved the following policy, which was concurred in by NDRC and by the Joint Meteorological Committee: 1. The NDRC Propagation Committee and its associated working groups will initiate and exercise technical supervision over such tests and investigations as they deem necessary to ascertain the nature of the above mentioned propagation anomalies in the VHF, UHF, and SHF bands, to devise the most practicable methods to determine the occurrence and characteristics of these anomalies from appropriate meteorological forecasts, with a view to improving the interim solutions offered by the Joint Wave Propagation Committee of the JCB. 2. The Army and Navy will furnish by direct coordination between them the basic staff guidance for such tests and investigations. They will accomplish this by determining (a) the specific forms in which basic prediction data shall be presented, and (b) the method of use required for operational forecast of propagation anomalies in the VHF, UHF, and SHF bands. 3. When the NDRC requires the cooperation of the operating units of the Army and Navy in conducting such tests and investigations as it deems necessary and this cooperation is of such an extent and nature that it cannot be furnished by informal coordination, it will be requested through the Joint Wave Propagation Committee of the JCB. Such requests will be initiated by the NDRC representative on the Wave Propagation Committee and recom- mended to the JCB by the Joint Wave Propagation Committee for consideration. 4. The Joint Wave Propagation Committee will be responsible for devising and furnish- ing immediately interim operational forecasting guides based upon information already available. On April 3, the Coordinator of Research and Development requested that the Army Project SOS-9 be made a joint Army-Navy project. Project No. AN-16 was assigned to this. On May 23, 1944, the Chief Signal Officer requested that under Project AN-16 the following work be inaugurated: Project AC 230.04, ‘Wave Propagation Study of Line-of-Sight Communication and Navigation.” 201 INDEX The subject indexes of all STR volumes are combined in a master index printed in a separate volume. For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. A scope, calibration, 161-162 Aircraft targets, radar cross section measurement, 185-186 Antenna, general characteristics beam width, 22 characteristics in transmission, 6 diameter, 29-31 effective length, 13 function, 22 horns, 44 impedance of nearby conductors, 24 pattern factors in ground reflection, 23 radiation patterns, 22-23 radiation resistance, 24 Antenna arrays, 33-39 binomial, 38-39 broadside, 34-38 colinear, 34-38 dipole, basic types, 34 multidimensional, 38 principle, 33-34 ring, 39 two-dipole side-by-side, 34-35 unidirectional, 38 Antenna gain calculation, 16-17 definition, 16 description, 22 Jungle locations, effect on gain, 195- 196 propagation factor in interference region, 69 wood location, effect on gain, 195- 196 Antenna types, 22-33, 39-43 directive antennas, 22-23 multiple half-wave, 27-28 parabolic reflector antennas, 42-43 parasitic antennas, 39-42 resonant antennas, 23 standing-wave antennas, 23, 25-32 traveling-wave antennas, 23-24, 32- 33 Atmospheric ducts effect on set performance, 163 formation, 4 types, 4 Atmospheric stratification, effect on refraction, 50-52 Attenuation, definition, 5 Attenuation factors, radio gain caleu- lation, 61 Beacons, performance characteristics, 169 Beam width, antennas, 22 Binomial arrays, antenna, 38-39 Brewster angle of reflection, defined, 54 Broadside arrays, antenna, 34-38 one-dimensional array, 35-38 side-by-side array, 34 unidirectional array, 38 Calibration, A scope, 161-162 Clarendon Laboratory at Oxford, con- ductivity of sea water, 55 Coastal diffraction, transmission, 178— 181 cliff site, 181 field strength, 179-181 level site near coast, 178-179 Colinear arrays, antenna, 34-38 one-dimensional array, 38 two half-wave dipole array, 35 unidirectional array, 38 Columbia University Wave Propaga- tion Group, | Communication equipment, perform- ance characteristics, 169 Conductivity, soil, 56-57 Cophased dipole antenna, 28-29 Corner-reflector antenna, 42 Cornu’s spiral, diffraction theory, 172- 173, 175 Coverage diagrams, generalized coordi- nates, 144-159 basic parameters, 144-145 normalized free-space distance, 145- 147 path-difference parameter, 145, 147 use of charts, 147-159 Coverage diagrams, methods of con- struction, 132-144 lobe-angle method, 138-144 P-Q method, 132-135 U-V method, 135-138 Coverage diagrams, plane earth, 129- 131 field strength, 129 horizontal polarization, 129-130 vertical polarization, 130-131 Coverage diagrams, spherical earth, 131-132 Curvature of earth calculation of radio gain below inter- ference region, 92 geometrical relationships, 62 Curvature of radio waves curvature relationships, 47-48 diffraction by earth’s curvature, 58 rays in standard atmosphere, 3-4 Definition of propagation, 1 Dielectric constant, soil, 56-57 Dielectric constant, water, 54-55 Dielectric earth graphical calculations, radio gain, 95-108 radiation field characteristics, 65-67 sea water as dielectric earth, 66 Diffraction by terrain, 170-181 coastal diffraction, 178-181 Cornu’s spiral, 172-173, 175 definition, 58 earth’s curvature diffraction, 58 Fresnel diffraction theory, 170-175 hill diffraction, 175-178 raindrop diffraction, 59 reflecting ground, 177-178 slot diffraction, 175 summary, 58 target diffraction, 58-59 transmission factor, 7 Diffraction regions, radio gain caleula- tion, 62-63 Dimensionless coordinates, radio gain calculations, 77-78 Dimensionless parameters, radio gain calculation, 75-77 Dipole arrays, basic types, 34 Directive antennas, 22-23 Divergence factor ground reflection, 57 lobe length determination, 139-140 propagation in interference region, 68-69 spherical earth radio gain calcula- tions, 75 transmission characteristic, 7 Doublet antennas, radio gain, 5 Ducts, atmospheric effect on set performance, 163 formation, 4 types, 4 Earth conductivity, 61 Echo constant, radar coverage measure- ment, 182 Ieffective length, antenna, 13 Electric doublet antenna in free space, 12-15 radiation, 12-13 received power, 13-14 scattered power, 13-14 transmission, 14-15 End-fire arrays, antenna, 34 203 204 INDEX Equipment performance, propagation aspects, 160-169 A scope calibration, 161-162 communications and radar equip- ment data, 169 free space high-angle coverage, 162- 163 low-angle and surface coverage, 163- 169 performance figure, measurement of efficiency, 160 reflection effect, 160-161 signal-to-noise ratio, 161 Equivalent height, radio gain calcula- tion, 71 Field strength, definition, 5 Free space distance, normalized, 145— 147 Free space field, definition, 5 Free space gain factor below interference region, 91 definition, 5 equation, 60 Free space radio gain, definition and formula, 15 Fresnel diffraction theory, 170-175 Fresnel integrals, 172-173 Irresnel zones, 174-175 mechanism of diffraction, 171 polarization, 174 slot diffraction, 175 straight edge formula, 171-172 Fresnel formulas, reflection, 54 Fresnel-Kirchoff optical theory see Fresnel diffraction theory Generalized reflection coefficient, 69-70 Grazing angle corresponding to lobe maxima, 129-130 Grazing angle in radio gain calcula- tions, 78 Ground reflection, 52-58 analysis of problem, 52-53 conductivity of soil, 56-57 dielectric constant of soil, 56-57 dielectric constant of water, 54 divergence factor, 57 Fresnel formulas, 54 irregularity of ground, 57-58 overland transmission, 56 RQ plane reflecting surface, 53 Half-wave antennas, 25 Half-wave dipole, antenna, 25-29 comparison of alternate and cophased half-wave dipoles, 28 cophased dipoles, 28-29 folded dipole, 27 gain, 26 impedance, 26 quarter-wave dipole, modification, 26-27 radiation field, 25-26 radiation resistance, 26 Hill diffraction, 175-178 criterion, roughness of surface, 176 field near the line of sight, 177 reflecting ground diffraction, 177-178 straight ridge diffraction, 176-177 Horizontal polarization angles of lobe maxima, 129-130 radio gain curves, 8-10 Horns, antenna use, 44 Huyghens principle, radio wave diffrac- tion, 7 Impedance, half-wave dipole antenna, 26 Index of refraction, definition, 3 Induction field, definition, 12 Integral half-wave antennas, 27-28 Interference, propagation factor in radio gain calculation, 67 Isotropic radiator, hypothetical an- tenna, 16 Jungles, obstacles to radio wave propa- gation, 195-196 Line of sight, definition, 62 Linear antennas, diameter lengths, 29— 31 Linear antennas, types, 24-25 Lobe maxima and minima, definition, 129 Lobe-angle method, coverage diagram construction, 138-144 basic equation, 138 correction for low angles, 141-143 lobe angles with horizontal, 139 lobe construction, 140-141 modified divergence factor, 139-140 reflection point curves, 138-139 vertical polarization, 143-144 Maximum range, 162-166 low-angle aircraft coverage, antenna heights and target heights, 163— 164 low-angle aircraft coverage, height curves versus maximum range, 164 low-angle and surface range coverage, 163 one-way communication, 162 radar, 162 Microwave beacons, ring arrays, 39 Moist standard atmosphere, propaga- tion, 3-4 Multidimensional arrays, antenna, 38 Multiple half-wave long antennas, 27- 28 Noise figure of radio receiver, 17-19 definition, 18 measurement, 18-19 Nonstandard atmosphere, tion, 4 propaga- Operator loss, radar reception, 19 Optical region, radiation field charac- teristics, 66 Optical region, radio gain calculation, 62-63 Parabolic reflectors for antennas, 42-43 Parasitic antennas, 39-42 corner reflector, 42 half-wave dipole and parasite, 39-41 multiple parasites, Yagi antenna, 41 reflecting screens, 41-42 Path difference loci construction, coverage diagrams, 134-135 parameters, 145, 147 path difference variable equation, 80-81 plane earth, 70 spherical earth, 74-75 Path gain factor, definition, 5 Performance figure, measurement of set efficiency, 162-168 Permanent echoes, prediction, 193-195 profile method, 193-195 radar test at site, 193 RPD (radar planning device), 193 supersonic method, 193 Permanent echoes, site selection fac- tors, 191-195 permanent echo diagrams, 191-192 shielding, echo control, 193 Plane earth calculation of radio gain below inter- ference region, 91-92 coverage diagrams, 129-131 path difference, 70 Polarization angles of lobe maxima, horizontal polarization, 129-130 angles of lobe maxima, vertical polar- ization, 130-131 Fresnel diffraction theory, 174 horizontal versus vertical, optical region, 66 horizontal versus vertical, radio gain calculations, 109 radio gain, vertical effect, 83 radio gain curves, horizontal polar- ization, 8-10 radio wave diffraction, 174 polarization INDEX radio waves, 52-53 transmission factor, 6 Power transmission, 15-17 antenna gain and polarization, 16-17 radio gain, 15-16 reciprocity principle, 17 P-Q method of coverage diagram con- struction, 132-135 path-difference — loci 134-1385 range loci construction, 1338-134 | Profile method, echo determination, 193-195 Propagation, assumption of standard conditions, 61-62 construction, Propagation, atmospheric considera- tions, 2-4 moist standard atmosphere, 3-4 nonstandard atmosphere, 4 standard atmosphere, 3 Propagation, basic relationships, 12-21 electric doublet in free space, 12-15 power transmission, 15-17 radar cross section, 19-20 radar gain, 20-21 receiver sensitivity, 17-19 Propagation, definition, 1 Propagation, general characteristics, 1-11 atmospheric considerations, 2-4 basic problems, 2 radiation field characteristics, 7-8 radio gain, 4-5, 8-10 transmission factors, 6-7 units and symbols used in propaga- tion study, 11 Propagation aspects of low-angle and surface coverage performance, 163-169 ducts, effect on set performance, 163 low heights, effect on ranges, 163-164 maximum range, 163 maximum range versus height curves 164-166 performance check before operation, 166 radar cross section of surface craft, 164 ship size estimation, 166 Propagation below interference region, radio gain calculation, 91-128 curved earth calculations, 92 general problem analysis, 91-95 graphical solution, dielectric earth calculations, 95-108 plane earth calculations, 91-92 radio gain near line of sight, 115 sample calculations for general solu- tion, 116-128 sea water calculations, 108-115 Propagation in interference region, radio gain calculation, 66-91 antenna gain and directivity, 69 divergence, 68-69 factors affecting radio gain, 67-69 general solution, 69-70 imperfect reflection, 67-68 interference, 67 plane earth calculations, 70-71 sample calculations, 79-91 spherical earth calculations, 71-78 spreading effect, 67 Quarter-wave dipole antenna, 26-27 Radar cross section aircraft target, 185-186 characteristic length L, 20 maximum range calculations, 163 scattering cross section, 19-20 scattering parameters, radar cover- age measurement, 182 surface craft, 164 Radar cross section, simple forms, 183- 185 circular plate, 183-185 corner reflector, 185 cylinders, 183 flat plates, 183 rectangular plate, 185 spheres, 183 Radar gain, 5, 20-21 Radar planning device for echo deter- mination (RPD), 193 Radar receivers, sensitivity, 19 operator loss, 19 performance characteristics, 169 scanning loss, 19 sweep-speed loss, 19 Radar siting see Siting, terrain selection and util- ization Radar targets, 182-186 aircraft, cross section measurement, 185-186 radar cross section of simple forms, 183-185 scattering parameters, coverage measurement, 182-183 Radiation antenna radiation patterns, 22-23 antenna radiation resistance, 24 electric doublet antenna, 12-13 induction field, 12 reciprocity principle with reception, 17 resistance, 13 resistance, half-wave dipole, 26 Radiation field electric doublet antenna, 12-13 general nature, 7-8 half-wave dipole antenna, 25-26 Radiation field in standard atmosphere, radio gain calculations, 63-67 field variation, 63-65 high antenna calculations, 65 low antenna calculations, 65 ultra short waves in diffraction re- gion, 65-67 Radio gain basic equation, 15 calculation, 60-128 defined, 4-5, 60-61 doublet antennas in free space, 5 radio gain curves, 8-10 Radio gain calculations, below inter- ference region, 91-128 analysis of first mode, 91—94 effect of linear variation of refractive index of atmosphere, 94-95 general solution for dipole over a smooth sphere, 116-122 graphical aids for sea water, v-h-f, vertical polarization, 108-115 graphs for the case of the dielectric earth, 95-108 radio gain near the line, 115 sample calculation for very dry soil, 123-128 Radio gain calculations, general con- siderations, 60-63 attenuation factors, 61 curved-earth geometrical relation- ships, 62 definition, 60-61 optical and diffraction regions, 62-63 standard propagation conditions as- sumed, 61-62 Radio gain calculations, in interference region, 67-71 general solution, 69-70 plane earth, 70-71 propagation factors, 67-69 Radio gain calculations, in standard atmosphere, 63-67 Radio gain calculations, optical-inter- ference region, 79-91 coverage problem, 87-89 for fixed heights and distance, 82-84 maximum range vs. receiver height, 89-91 radio gain vs. distance for given an- tenna heights, 85-87 radio gain vs. receiver height for given distance, 84-85 Radio gain calculations, earth, 71-78 angle determination, 71 dimensionless coordinates, 77-78 dimensionless parameters, 75-77 distance measurement, 71 divergence factor, 75 equivalent height, 71 spherical 206 INDEX grazing angle, 78 path difference, 74-75 reflection point determination, 71-74 Raindrop diffraction, 59 Range loci construction, 133-134 Rayleigh criterion, radiation theory application, 176 Receiver sensitivity, 17-19 definition, 18 noise figure, 17-19 radar receivers, 19 thermal noise, 17 Reciprocity principle, reception and radiation, 17 Reflection Brewster angle, 54 effect on equipment performance, 160-161 ground reflection, 52-58 imperfect reflection, interference re- gion, 67-68 reflection coefficient, 53, 69-70 transmission factors, 6-7 Reflection point curves, coverage dia- gram construction, 138-139 Reflection point determination, 71-74 Reflection point variable, 80 Refraction, 45-52 atmospheric stratification, 50-52 curvature relationships, 47-48 definition, 3 graphical representation, 46-47 Snell’s law, 45-46 standard refraction, 45 transmission factor, 6 Refractive index, 48-52 computation, 48-50 function of temperature and height, 49-50 function of temperature and relative humidity, 51 modified refractive index, 46 standard atmosphere, 61 Resonant antennas, 23 Rhombie antenna, traveling wave an- tenna, 32-33 Ring arrays, antenna, 39 RPD (radar planning device) for echo determination, 193 Scanning loss, radar reception, 19 Scattered power, radio wave reception, 13-14 Scattering parameters, radar coverage measurement, 182-183 echo constant, 182 equivalent plate area, 183 radar cross section, 182 scattering coefficient, radar length of target, 183 target gain, 182 Sea water dielectric earth behavior, 66-67 influence on radio wave transmission, 54-55 radio gain calculation, LO8-115 Sectoral horn, antenna, +4 Shadow zone, 7 Shielding, permanent echo control de- vice, 193 Ship roll, effect on radar coverage, 160- 161 Ship size estimated by strength of re- turned radar signal, 166 Side-by-side array, two-dipole antenna, 34-35 Signal-to-noise ratio, radar equipment performance, 161 Siting, terrain selection and utilization, 187-196 geometrical limits of visibility, 188— 191 permanent echoes, 191-195 radar, overland transmission, 56 requirements of siting, 187 trees and jungles, 195-196 Siting topography, 187-188 maps, 187 orientation, 187-188 profiles, 187 Slot diffraction, 175 Snell’s law of refraction, 45-46 Soil as radio wave conductor, 56-57 Solar azimuths, calculation methods, 187-188 Spherical earth coverage diagrams, 131-132 path difference, 74-75 radio gain calculation; see Radio gain calculations, spherical earth Spreading effect, propagation factor in interference region, 67 Standard atmosphere, refractive index, 61 Standard dry atmosphere, properties, 3 Standard refraction, definition, 45 Standing-wave antennas, 24-32 half-wave dipole, 25-26, 28-29 linear antennas, 24-25, 29-31 multiple half-wave long antennas, 27-28 V antennas, 31-32 Supersonic method of echo determina- tion, 193 Surface craft, estimation of size by strength of returned radar sig- nal, 166 Surface craft, radar cross section, 164 Sweep-speed loss, radar reception, 19 Symbols adopted for frequency ranges, 11 Target diffraction, 58-59 Terrain diffraction, 170-181 Terrain selection see Siting, terrain selection and util- ization Thermal noise in radio receivers, 17 Time computations necessary for radar site selection, 188 Topographic maps and profile prediction of echoes, 193-19 -siting selection use, 187 Transmission, 6-7, 45-59 antenna characteristics, 6 diffraction, 7, 58-59 divergence, 7 ground reflection, 52-58 polarization, 6 reflection, 6 refraction, 6, 45-52 Traveling-wave antennas, 23-24, 32-33 Trees as obstacles for high frequency radio wave, 195-196 Troposphere, propagation role, 2 Ultra-short waves in diffraction region 65-67 Unidirectional arrays, antenna, 38 Units used in propagation study, 11 U-V method of coverage diagram con- struction, 185-188 V antennas, 31-32 Vertical polarization angles of lobe maxima, 130-131 effect on radio gain, 83 Visibility, geometrical limits, 188-191 degree of shielding, 190-191 horizon distance of transmitter, 188-189 obstacle height, 189-190 Yagi antenna, multiple parasite, 41