Extremely Low Frequency (ELF) Propagation Formulas For .
NUWC-NPT Technical Report 11,36914 May 2002Extremely Low Frequency (ELF) PropagationFormulas for Dipole Sources Radiating in aSpherical Earth-Ionosphere WaveguideJohn P. CaseySubmarine Electromagnetic Systems DepartmentIMAVSEAUndersea Warfare Center DivisionNaval Undersea Warfare Center DivisionNewport, Rhode IslandApproved for public release; distribution is unlimited.20020905 060
PREFACEThis work was conducted in support of the "ELF CommunicationsProgram," program manager Robert J. Aiksnoras (Code 3491). Thesponsoring activity is the Space and Naval Warfare Systems Command,Acquisition Program Manager for Strategic Systems, Paul Singer (PMW173-2).The technical reviewers for this report were Thomas A. Wettergren(Code 2002) and Peter R. Bannister of Sebastian, FL. The author isgrateful to these reviewers for their helpful comments and suggestions.Reviewed and Approved: 14 May 2002Peter M. TraskHead, Submarine Electromagnetic Systems DepartmentJ
Form ApprovedOMB No. 0704-0188REPORT DOCUMENTATION PAGEPublic reporting for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection ofinformation, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway,Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.1. AGENCY USE ONLY (Leave blanK)2. REPORT DATE14 May 20023. REPORT TYPE AND DATES COVERED4. TITLE AND SUBTITLE5. FUNDING NUMBERSExtremely Low Frequency (ELF) Propagation Formulas for Dipole Sources Radiatingin a Spherical Earth-Ionosphere Waveguide6. AUTHOR(S)John P. Casey8. PERFORMING ORGANIZATIONREPORT NUMBER7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)Naval Undersea Warfare Center Division1176 Howell StreetNewport, Rl 02841-1708TR 11,3699. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)10. SPONSORING/MONITORINGAGENCY REPORT NUMBERSpace and Naval Warfare Systems Command (PMW 173)4301 Pacific Highway (Bldg. OT-4)San Diego, CA 92110-312711. SUPPLEMENTARY NOTES12a. DISTRIBUTION/AVAILABILITY STATEMENT12b. DISTRIBUTION CODEApproved for public release; distribution is unlimited.13. ABSTRACT (Maximum 200 words)Extremely low frequency (ELF) propagation formulas are derived for dipole sources radiating in a spherical earth-ionospherewaveguide. In these formulas, the earth and ionosphere boundaries are modeled as scalar surface impedances. The sphericalwaveguide formulas are applied to predict the electromagnetic fields produced by vertical and horizontal electric dipoles (locatedon the surface of the earth) at antipodal ranges for several frequencies and propagation conditions. These results are used toestablish the maximum ranges of validity of ELF propagation formulas that are based on the earth-flattening approximation.Numerous derivations are given in the appendices.14. SUBJECT TERMSElectromagnetics15. NUMBER OF PAGESExtremely Low Frequency17. SECURITY CLASSIFICATIONOF REPORTUnclassifiedNSN 7540-01-280-5500Earth-Ionosphere Waveguide18. SECURITY CLASSIFICATIONOF THIS PAGEUnclassified19. SECURITY CLASSIFICATIONOF ABSTRACTUnclassified18616. PRICE CODE20. LIMITATION OF ABSTRACTSARStandard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102
TABLE OF CONTENTSSectionPageLIST OF ILLUSTRATIONSiiiLIST OF TABLESvii1INTRODUCTION12APPLICATION OF PARALLEL-PLATE WAVEGUIDE MODEL FORDETERMINATION OF APPROXIMATE MODECUTOFF FREQUENCIESTEM and TM ModesTE Modes2.12.2ELF PROPAGATION FORMULAS BASED ON THE EARTHFLATTENING APPROXIMATION3.1 Earth-Flattening Approximation to the Legendre Function of the First Kind3.2 Bannister's Formulas3.2.1 Vertical Electric Dipole3.2.2 Horizontal Electric Dipole3.3 Formulas for a HED above an Anisotropie .4.24.5131314151820SPHERICAL WAVEGUIDE PROPAGATION FORMULASPropagation Parameters for TM and TE Modes at ELFQuasi-TEM FieldsVertical Electric Dipole,Horizontal Electric DipoleAntipode-Centered FormulasVertical Electric DipoleHorizontal Electric DipoleReduction to Bannister's Formulas via the Earth-Flattening ApproximationVertical Electric DipoleHorizontal Electric DipoleModification of HED Spherical Waveguide Formulas to Accountfor an Anisotropie Ground25253030323333343636384055.15.2COMPARISONS OF PROPAGATION FORMULASResults at 76 HzResults at Other Frequencies4142626SUMMARY AND CONCLUSIONS87
TABLE OF CONTENTS (Cont'd)SectionPageAPPENDIX A—ELECTROMAGNETIC FIELDS IN TERMS OFPOTENTIALS IN SPHERICAL COORDINATESA-lAPPENDIX B—SOLUTION OF THE HELMHOLTZ EQUATION INSPHERICAL COORDINATESB-lAPPENDLX C—TRANSVERSE MAGNETIC (TM) ANDTRANSVERSE ELECTRIC (TE) MODAL EXPANSIONS IN ASPHERICAL WAVEGUIDEC-lAPPENDIX D—PROOF OF ORTHOGONALITY OF RADIAL FUNCTIONS.D-lAPPENDLX E—EXCITATION COEFFICIENTS FOR TRANSVERSEMAGNETIC (TM) AND TRANSVERSE ELECTRIC (TE) MODESE-lAPPENDIX F—DERIVATION OF VERTICAL ELECTRIC DIPOLE (VED)ELECTROMAGNETIC (EM) FIELDSF-lAPPENDIX G—DERIVATION OF VERTICAL MAGNETIC DIPOLE (VMD)ELECTROMAGNETIC (EM) FIELDSG-lAPPENDLX H—DERIVATION OF HORIZONTAL ELECTRIC DIPOLE (HED)ELECTROMAGNETIC (EM) FIELDSH-lAPPENDIX I—THIN-SHELL APPROXIMATION TO THE RADIALDEPENDENCE OF THE FIELDS1-1APPENDIX J—LEGENDRE FUNCTION Pv(-cos 0) ANDITS DERIVATIVESJ-lAPPENDIX K— EARTH-FLATTENING APPROXIMATIONK-lREFERENCESR-l
LIST OF ILLUSTRATIONS (Cont'd)Figure5-9a5-9bPageComparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Producedby a VED Under Typical Daytime Propagation Conditions at 30 Hz66Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Phase of the Azimuthal Magnetic Field Produced by aVED Under Typical Daytime Propagation Conditions at 30 Hz675-10a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Vertical Electric Field Produced by aHED Under Typical Daytime Propagation Conditions at 30 Hz685-1 Ob Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Phase of the Vertical Electric Field Produced by aHED Under Typical Daytime Propagation Conditions at 30 Hz695-11 a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Radial Magnetic Field Produced by aHED Under Typical Daytime Propagation Conditions at 30 Hz705-1 lb Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Phase of the Radial Magnetic Field Produced by aHED Under Typical Daytime Propagation Conditions at 30 Hz715-12a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Producedby a HED Under Typical Daytime Propagation Conditions at 30 Hz725-12b Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Phase of the Azimuthal Magnetic Field Produced by aHED Under Typical Daytime Propagation Conditions at 3 0 Hz735-13a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Vertical Electric Field Produced by aHED Under Typical Nighttime Propagation Conditions at 30 Hz745-13b Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Radial Magnetic Field Produced by aHED Under Typical Nighttime Propagation Conditions at 30 Hz755-13c Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Producedby a HED Under Typical Nighttime Propagation Conditions at 30 Hz76
LIST OF ILLUSTRATIONS (Cont'd)FigurePage5-14a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Vertical Electric Field Produced by aVED Under Typical Daytime Propagation Conditions at 300 Hz775-14b Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Produced by aVED Under Typic al Daytime Propagation Conditions at 3 00 Hz785-15a Comparison of Spherical Waveguide and Bannister' s ELF PropagationFormulas for the Magnitude of the Vertical Electric Field Produced by aHED Under Typical Daytime Propagation Conditions at 3 00 Hz795-15b Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Radial Magnetic Field Produced by aHED Under Typical Daytime Propagation Conditions at 300 Hz805-15c Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Producedby a HED Under Typical Daytime Propagation Conditions at300Hz815-16a Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Vertical Electric Field Produced by aHED Under Typical Nighttime Propagation Conditions at 300 Hz845-16b Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Radial Magnetic Field Produced by aHED Under Typical Nighttime Propagation Conditions at 300 Hz855-16c Comparison of Spherical Waveguide and Bannister's ELF PropagationFormulas for the Magnitude of the Azimuthal Magnetic Field Produced by aHED Under Typical Nighttime Propagation Conditions at 3 00 Hz86H-lH-2H-3VICoordinate Systems Defined in the Application of the Reciprocity Theoremfor Determination of the HED FieldsH-2Electric Field Component E g* Radiated by a VED (Source 2) Projectedonto the jc-Axis at the HED (Source 1) LocationH-3Electric Field Component E . Radiated by a VMD (Source 2) Projectedonto the x-Axis at the HED (Source 1) LocationH-5
LIST OF ILLUSTRATIONS (Cont'd)FigureJ-lJ-2J-3J-4J-5J-6PageComparison of the Infinite Series (Exact) and Approximate FormulasforPv(-cos 0) for V 4.75 -j 0.440; (a) Magnitude; (b) PhaseJ-10Comparison of the Infinite Series (Exact) and Approximate Formulasfor dPv(- cos G)/d9 for v - 4.75 -j 0.440; (a) Magnitude; (b) PhaseJ-l 1Comparison of the Infinite Series (Exact) and Approximate Formulasfor d 2PV(- cos 0)/dO2 for v 4.75 -j 0.440; (a) Magnitude; (b) PhaseJ-12Comparison of the Infinite Series (Exact) and Approximate Formulasfor Pv(- cos 0) for v 15.9 -j 1.32; (a) Magnitude; (b) PhaseJ-13Comparison of the Infinite Series (Exact) and Approximate Formulasfor dPv(- cos 6)/dd for v 15.9 -j 1.32; (a) Magnitude; (b) PhaseJ-14Comparison of the Infinite Series (Exact) and Approximate Formulasfor d2Pv(- cos 6)/d02 for v 15.9 -j 1.32; (a) Magnitude; (b) PhaseJ-15LIST OF TABLESTable2-13-14-la4-lb4-2aPageCutoff Frequencies for TEM, TM1? and TM2 Modes for a Parallel-PlateWaveguide with Perfect Electrically Conducting Boundaries forSeveral Waveguide Heights9Conversion Table Relating Bannister's HED Propagation Formulaswith Those of Wolkoff and Kraimer23Phase Velocity Ratio civ for the Dominant Modes in a SphericalEarth-Ionosphere Waveguide at ELF Frequencies Under DaytimeConditions with h 50 km, jg 10"3 S/m, and at 10"5 S/m27Attenuation a in dB/Mm for the Dominant Modes in a SphericalEarth-Ionosphere Waveguide at ELF Frequencies Under DaytimeConditions with h 50 km, ag 10"3 S/m, and at 10"5 S/m28Phase Velocity Ratio civ for the Dominant Modes in a SphericalEarth-Ionosphere Waveguide at ELF Frequencies Under NighttimeConditions with h 75 km, ag 10"3 S/m, and at 10"5 S/m29Vll
LIST OF TABLES (Cont'd)Table4-2bPageAttenuation a in dB/Mm for the Dominant Modes in a SphericalEarth-Ionosphere Waveguide at ELF Frequencies Under NighttimeConditions with h 75 km, ag 10-3 S/m, and o; 10'5 S/m30Magnitude and Phase of Difference Factor C in HED Antipode-CenteredELF Propagation Formulas at Several Frequencies Under Daytime Conditions35Propagation Parameters for Typical Daytime and Nighttime Conditionsat Several Frequencies as Obtained from Bannister (Reference 16)42Ranges Over Which Bannister's Direct Great-Circle Path Field Formulas Agreeto Within 1 dB in Magnitude of the Spherical Waveguide Formulas at 76 Hz56Ranges Over Which Bannister's Total Field Formulas Agree to Within 1 dB inMagnitude of the Spherical Waveguide Formulas at 76 Hz56Ranges Over Which Bannister's Total Field Formulas Agree to Within 1 dB inMagnitude of the Spherical Waveguide Formulas at 30 Hz63Ranges Over Which Bannister's Total Field Formulas Agree to Within 1 dB inMagnitude of the Spherical Waveguide Formulas at 300 Hz82Average Normalized Ranges Over Which Bannister's Total Field FormulasAgree to Within 1 dB in Magnitude of the Spherical Waveguide Formulas(Averages Taken Over 30, 76, and 300 Hz)83J-lPhase Adjustment Terms for the Indirect Great-Circle Path FieldsJ-3J-2Number of Iterations Required for the Infinite Series Formulas for Pv(- cos 0)and Its First Two Derivatives to Converge to 15 DecimalPlaces with v 11.7 j 0.880J-84-35-15-2a5-2b5-35-45-5Vlll
EXTREMELY LOW FREQUENCY (ELF) PROPAGATION FORMULASFOR DIPOLE SOURCES RADIATING IN ASPHERICAL EARTH-IONOSPHERE WAVEGUIDE1. INTRODUCTIONExtremely low frequency (ELF) propagation formulas for dipole sources radiating in aspherical earth-ionosphere waveguide (figure 1-1) have been derived by various authors.Developments of such formulas are provided in the texts Written by J. R. Wait (reference 1) andJ. Galejs (reference 2). These spherical waveguide formulas have been derived for a uniform earthand ionosphere and, depending on the field component, have a range dependence characterized byeither the Legendre function of the first kind of complex degree and order zero or one of its firsttwo derivatives. In the ELF band (defined here as 30 to 300 Hz), because the ionosphericreflection height is less than one-half of a free-space wavelength, the only propagating mode in theearth-ionosphere waveguide is the zeroth-order transverse magnetic (TM) mode, which iscommonly referred to as the quasi-transverse electromagnetic (quasi-TEM) mode.IONOSPHEREFigure 1-1. Spherical Coordinate System Description of the Earth-Ionosphere Waveguide
To predict the fields more accurately at ranges closer to the source or to account for morecomplicated boundary conditions, various investigators have incorporated the earth-flatteningapproximation into their ELF propagation formulas. In this approximation, the Legendrefunction range dependence is approximated by the product of a Hankel function and a curvaturecorrection term (spherical earth spreading factor). As a result, the fields are derived from aplanar earth-ionosphere waveguide model and then multiplied by the curvature correction term.For example, Bannister (reference 3) has derived ELF propagation formulas based on the earthflattening approximation that extend the results of Wait and Galejs to closer ranges from thesource. Whereas the spherical waveguide formulas given by Wait and Galejs are valid for rangesgreater than approximately three ionospheric reflection heights from the source, Bannister'sformulas are valid in the quasi-nearfield range, which is defined as the range where themeasurement distance is greater than an earth wavelength, but much less than a free-spacewavelength. However, Bannister's approximate formulas are not valid at field points close to theantipode where the simple spherical focusing factor fails.To account for the anisotropic surface impedance in the vicinity of a horizontal electricdipole (HED) at ELF, Wolkoff and Kraimer (references 4 and 5) have listed propagationformulas that are modifications of Bannister's HED formulas. Wolkoff and Kraimer's formulasaccount for the anisotropic ground through the use of two complex-valued antenna patternfactors. These antenna pattern factors are unique for a given HED and must be determined fromnear-field measurements of the antenna. Wolkoff and Kraimer have determined the antennapattern factors for each of the U.S. Navy's ELF transmitting antennas (reference 4). Wolkoffand Kraimer's propagation formulas have been formally derived from Bannister's formulas byCasey (reference 6) through use of the reciprocity theorem.For prediction of the ELF fields from dipole sources at antipodal ranges, propagationformulas that are valid out to approximately 20 Mm from the source must be applied. In a recentreport (reference 7), approximate formulas for a HED source that contain the proper rangedependence at antipodal ranges in a spherical earth-ionosphere waveguide, referred to as"antipode-centered propagation formulas," were derived. These HED formulas are based onBurrow's simple parallel-plate waveguide approximation of the earth-ionosphere waveguide(reference 8) and include a curvature correction factor. In reference 7, the antipode-centeredpropagation formulas were compared with Bannister's HED formulas (direct and indirect great-
circle path fields were combined) under various propagation conditions, where both the sourceand field points are located on the earth's surface. The results showed that Bannister's verticalelectric field and radial magnetic field formulas (magnitude only) agree to within 1 dB of thecorresponding antipode-centered formulas for ranges greater than 0.97 Mm to 1.13 Mm from theantipode, depending on the propagation conditions. In addition, Bannister's azimuthal magneticfield formula agrees to within 1 dB of the corresponding antipode-centered formula for rangesgreater than 3.17 Mm to 3.72 Mm from the antipode, depending on the propagation conditions.In this report, ELF propagation formulas for dipole sources radiating in a spherical earthionosphere waveguide are derived from first principles. These derivations are presented becausethe developments given by previous authors were found to be difficult to follow. The formulasderived here are based on the assumptions of a homogeneous, isotropic earth and ahomogeneous, isotropic ionosphere of constant reflection height. As a result, the earth andionosphere boundaries are modeled as scalar surface impedances. The spherical waveguideformulas are expressed in terms of series expansions of TM and transverse electric (TE) modes.However, at ELF, the only mode of practical interest is the quasi-TEM mode. The computedresults for the spherical waveguide formulas presented in this report are based on an exacthypergeometric series representation of the Legendre function of the first kind. The propagationformulas for a horizontal magnetic dipole (HMD) can be derived from the HED formulasthrough application of the duality principle (reference 2).In addition, in this report, through appropriate approximations of the range dependence, itwill be shown how the spherical waveguide formulas (for both vertical electric dipole (VED) andHED sources) can reduce to Bannister's formulas or to the antipode-centered formulas. It willalso be shown how the quasi-TEM spherical waveguide formulas derived for a HED located atthe surface of the earth can be modified to account for the anisotropic surface impedance in thevicinity of the antenna. These modified spherical waveguide formulas will include Wolkoff andKraimer's antenna pattern factors and will be useful for prediction of the electromagnetic (EM)fields radiated by the U. S. Navy's four transmitting antennas at antipodal ranges. Detailedderivations of approximate formulas for the Legendre function range dependence are given in theappendices. Comparisons of the spherical waveguide propagation formulas with bothBannister's direct great-circle path and total field (direct plus indirect great-circle paths)formulas are presented for the surface magnetic field and vertical electric field components undervarious propagation conditions. The field comparisons are presented at several frequencies for
both VED and HED sources at ranges that extend from 1 Mm from the
3.2.2 Horizontal Electric Dipole 18 3.3 Formulas for a HED above an Anisotropie Ground 20 4 SPHERICAL WAVEGUIDE PROPAGATION FORMULAS 25 4.1 Propagation Parameters for TM and TE Modes at ELF 25 4.2 Quasi-TEM Fields 30 4.2.1 Vertical Electric Dipole , 30 4.2.2 Horizontal Electric Dipole 32 4.3 Antipode-Centered Formulas 33 4.3.1 Vertical Electric .
There is ongoing interest in applications of pulsed electromagnetic field (PEMF) radiation as an alternative therapy for different medical conditions [1]. Studies have demonstrated that extremely low frequency (ELF) PEMF radiation facilitates the process of wound repair [2,3]. ELF PEMF is a sub-class of electromagnetic field (EMF)
5 Sh elf 6 Sh elf 7 Sh elf 8 Sh elf E ndKit C apa city ** P er Sh elf D imen sio ns (in che s) B a sic U nit C at. N o. B a sic U nit C at. N o. B a sic U nit C at. N o. B a sic U nit C at. N o. E ndKit C at. N o. P ounds W D H 1 H8015 1 H8025 1 H8035 1 H8016 1 H8026 1 H8036 1 H8017 1 H8
The ELF file format, like any other file format, is an array of bits and bytes interconnected through data structures. When interpreted by an ELF parser, an ELF file makes sense, depending upon the parsing context: runtime (execution view) or static (linking view). In 1999, ELF was chosen as the standard binary file format for *NIX systems, and .
Build The COSMAC "ELF" A Low-Cost Experimenter's Microcomputer Part 1 PE Tested Simple-to-build computer trainer can be expanded for advanced applications. BY JOSEPH WEISBECKER There are basically two ways in which you can get involved with microcomputers on the nonprofessional level. You can buy one of several reasonably priced hobbyFile Size: 722KBPage Count: 56Explore furtherCOSMAC ELF Manualwww.cosmacelf.com/publications/books/ CDP1802A, CDP1802AC, CDP1802BCwww.cosmacelf.com/publications/data-s RCA 1802 BASIC level 3 ver. 1.1 User Manual - Sunrise EVwww.sunrise-ev.com/MembershipCard/B Recommended to you b
pulsed electromagnetic fields (PEMF) for the treatment of delayed union or nonunion fractures, failed joint fusions, and congenital pseudarthroses 1, 2. For therapeutic purposes, PEMF is typically applied at extremely low frequencies between 5 and 300 Hz – Ex-tremely Low Frequency Pulsed Electromagnetic Magnetic Field (ELF-PEMF).
on radio propagation. This handbook also provides basic information about the entire telecommunications environment on and around Mars for propagation researchers, system . 1.2 Radio Wave Propagation Parameters. 4 2. Martian Ionosphere and Its Effects on Propagation (Plasma and Magnetic Field). 7
1 How Plant Propagation Evolved in Human Society 2 2 Biology of Plant Propagation 14 3 The Propagation Environment 49. part two. Seed Propagation. 4 Seed Development 110 5 Principles and Practices of Seed Selection 140 6 Techniques of Seed Production and Handling 162 7 Principles of Propagati
Unit 5: American Revolution . 2 A m e r i c a n R e v o l u t i o n Political and Economic Relationships between Great Britain and the Colonies England became Great Britain in the early 1700s, and it was throughout this century that the British colonies in America grew and prospered. The growth of the colonies made it more and more difficult for Great Britain to remain in control. King .
