The Incremental Far Field and Degrees of Freedom of the Sinusoidal Current Filament E. K. Miller Los Alamos National Laboratory (retired) 597 Rustic Ranch Lane, Lincoln, CA 95648 USA Tel: +1(916) 408-0916; E-mail:
[email protected]
Abstract From where radiation originates for a specified current source or PEG boundary seems not to be as well-understood or agreed-upon as many other aspects of electromagnetics. Two propositions are explored in this article that relate to this issue. One is that if some incremental portion of a source distribution increases (changes) the total far field at any observation angle, then by definition radiation originates from that portion of the source distribution. The other is that the greater the number of degrees of freedom in a radiation pattern, the greater will be the proportion of the source distribution that actually radiates. These issues are explored here in terms of the radiation from a sinusoidal current filament. Keywords: Electromagnetic fields; electromagnetic radiation; antenna radiation patterns; antenna theory; wire antennas; degrees of freedom; sinusoidal current filament; singular values; far field
1. Introduction
Radiation
is perhaps the most important phenomenon in elecJitromnagnetic theory. Yet, some uncertainty continues about how radiation occurs, and whether its quantitative distribution over a specified source -such as the sinusoidal current filament (SCF), or a PEG (perfect electrically conducting) object -can be determined, even though its fundamental cause -charge acceleration is well known. Even for such simple objects as PEG straight wires or simple variations thereof, the places where radiation originates are subject to debate. One viewpoint is that radiation from a linear current filament or PEC wire comes only from the ends and feedpoint [1-4], while another is that radiation comes from along the length of the current [5-10], the latter being the viewpoint that the author holds [11, 12].
- Pulse-excited straight wire: "Radiation is produced when the spatially integrated value of the product of the charge density and its acceleration is nonzero. This is supported by the results presented in this article. Between the feed point and tip (or subsequently between the tip and its image), the charge-density pulse travels along the monopole at the speed of light, with no dispersion. During these times, the integrated value of charge density times acceleration is zero," i.e., there is no radiation along the monopole [4]. Examples of the latter viewpoint, which have the common theme of current decay as an indicator of radiation for PEC wires, include:
Some examples of the former viewpoint include:
- Finite dipole antenna: "If damping due to the emission of radiation is neglected... the current along the antenna can be taken as sinusoidal .... [5].
- Sinusoidal current filament: "Radiates from only its center and ends" [1].
. Finite dipole antenna: "An outgoing traveling wave ...decays slowly ...as a result of radiation." [6].
* Hertzian dipole: "The aim of this present paper is to show how to determine where static and reactive energy are converted into radiation in the fields around an antenna." [2].
- Infinite cylindrical antenna: "The current wave on the infinite rod decreases slowly with distance and it is damped by radiation losses." [7].
- Pulse-excited straight wire: "In this case, after the positive pulse due to the initial charge acceleration, the current propagates along the antenna without radiating until it reaches the end of the wire, where the charges are decelerated and the radiation of the negative pulse takes place." [3]. IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
- Conical spiral antenna: "Most of the radiation takes place in the exponential decay (of the current) region." [8]. - Infinite cylindrical antenna: "The current wave decays slowly with distance along the rod...gradually radiating energy." [9]. ISSN 1045-9243/2007/$25 @2007 IEEE
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. Pulse-excited traveling-wave and standing-wave (dipole) current elements: "The results presented above show that there is a net amount of energy leaving every unit length of both filamentary current distributions: the element and the standing-wave traveling-wave dipole.... .For the traveling-wave element, some of this energy is radiated away from the element, and the rest is stored in the field of the point charges that remain at the two ends of the element. For the standing-wave dipole, this energy is all radiated away." [ 10]. It can be concluded from the above quotations that there is no unanimity of opinion about where radiation takes place, even for simple geometries. While having a definitive answer to where radiation originates is obviously not necessary to solving realworld problems, radiation is a central phenomenon of electromagnetics, the better understanding of which would add to our knowledge base. That is the motivation of the discussion that follows. The purpose of this brief note is to explore radiation of the sinusoidal current filament from two different perspectives. One is to examine the fundamental mathematical operation of obtaining a far field from integrating over a source distribution, in this case the sinusoidal current filament. This simple and straightforward approach shows explicitly that interior regions of the sinusoidal current filament, i.e., away from the ends and feedpoint, do indeed radiate. The other is to explore the number of degrees of freedom (DoF), or rank, of the sinusoidal current filament's radiation pattern. This is motivated from a synthesis viewpoint, that of determining the minimum number of discrete sources needed to radiate a given pattern to some prescribed accuracy [131. If the sinusoidal current filament really does only radiate from three isotropic point sources, then its radiation-pattern rank should be expected to be no greater than three. But radiation from along its length would result in more than three such sources being needed to produce its pattern, and the rank of its radiated field would be greater than three. Observe that although what follows is restricted to the frequency domain, its implications apply to time-domain results as well, since they're related via a Fourier transform.
Z
4;r
r
+
1 k
2
1
-k)j'
()
where r=x2
2
2?
is the field-observation point. Also observe that the directivity of the current is sin (9), with the observation angle, 9, measured from the positive z axis. The l/r, or first, term in the brackets of Equation (1) is the radiation E field. Since a current filament of arbitrary spatial variation can be approximated by a piecewise 14
1(Z')=10sin[k(L-jZ11)1, _L _
Z
I o:
-;7---
05 001
Case, a, 3 point sources
~
Case b, 33point sourcestbimesSin( o
7
14
NUMBER OF SINGULAR VALUE
Figure 4. The singular-value spectra for the three patterns computed from Equation (4b), with the open-circle plot being that of three isotropic point sources.
20-
0-
w LU~
-20
-1
~01
COSINE OF OBSERVATION ANGLE Figure 5a. The power pattern for Case (a) associated with Equation (4b). For the pattern fit obtained using 19 and 21 point sources, the sampling interval was from -0.99 to +0.99 in cos(O). For Case (a) and three point sources, the sampling interval was -0.1 to +0.1. The curves of the actual pattern and using 21 point sources are essentially graphically indistinguishable, as is that for Case (a) with three point sources. IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
117
d
0-
LU
H-20-
0~0
COSINE OF OBSERVATION ANGLE Figure 5b. The power pattern for Case (b) associated with Equation (4b). The other comments in the caption of Figure 5a apply.
20-
49
0 LU
LU L.
-60
-
FIT USING 19 POINT SOURCES
-
-
-0-
-80
-1
ACTUAL PATTERN FIT USING 21 SOURCES 0
COSINE OF THE OBSERVATION ANGLE Figure 5c. The power pattern for Case (c) associated with Equation (4b). The other comments in the caption of Figure 5n apply. Is
IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
The synthesized patterns shown in Figure 5 are graphically indistinguishable from the actual patterns when using 21 point sources, with the rms difference between them being on the order of 10-
7. Current and Charge Relationship to Radiation It's well known that source radiation is due to accelerated charge, as shown by the Lienard-Wiechert potentials [27]. However, attention here has so far been directed only to the current to identify' from where radiation originates. Are these two apparently different concepts mutually consistent? It's been discussed elsewhere by the author [28] that for standing-wave currents such as the sinusoidal current filament, current maxima occur where the maxima of oppositely propagating charge waves of opposite sign meet. In addition, these current maxima are locations where the local tangential electric field is a maximum. These two phenomena in turn result in a maxima in the induced EMF (IEMF or input) power there, through the acceleration this field imposes on the charge. Thus, a current maximum should be expected to produce a spatial maximum in the distributed radiation along the standing-wave current filament, consistent with the discussion above concerning the current contribution to the far field. These results are demonstrated by FARS (Farfield Analysis of Radiation Sources) [29], and are consistent with various other observations. However, as shown by FARS, the distribution of radiated power along the sinusoidal current filament is not equally proportional to the current magnitude everywhere, being largest at the ends and at the center feedpoint when the filament length is an even number of half wavelengths. A somewhat similar situation holds for the current on a PEC wire. However, in that case the local or IEME input power is zero away from the feedpoint. The charge acceleration in this case comes from a partial reflection of the counter-propagating charge waves due to the spatially varying wave impedance of the wire. Again, radiation maxima occur at a current maxima where opposite-signed maxima in the charge waves meet and are reflected in opposite directions to produce a radiated E field of the same sign. Thus, although charge acceleration is the fundamental physical cause of far-field radiation, the current distribution that the charge comprises provides a simple and direct connection to the radiated field. Of course, this is consistent with the far field coming from an integral over the current. So, a qualitative estimate of where radiation comes from does not require explicit consideration of accelerated charge, but can, instead, be identified from the current distribution. Thus, the simplest, although a somewhat superficial, answer from a physics' viewpoint to the question, "Where does radiation originate from a source distribution?" is, after all, "It's the current," the answer most often given by undergraduate EM students. Finally, note that radiation in the time domain can be evaluated by Fourier transforming frequency-domain results, or from direct time-domain solutions. It follows from frequency-domain results like those presented above that time-domain radiation also occurs from along the length of a PEG wire or specified current, a conclusion that is consistent with time-domain solutions [10, 29], the former for an assumed Gaussian current/charge pulse and the latter for a PEC wire. IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
8. Concluding Remarks The results and observations presented above are consistent with various others in supporting the conclusion that radiation originates from along the length of a linear current distribution. This is true whether that distribution satisfies the boundary condition on a PEG wire or one that has no boundary condition, as is the case of the sinusoidal current filament. To summarize: 1. For the most part, the far field at the maximum of the traveling-wave lobe is increased by current all along the sinusoidal current filament. 2. The singular-value spectrum for the sinusoidal current filament extends well beyond the three singular values that apply to the spectrum of three point sources. 3. The total power radiated by a sinusoidal current filament of increasing length grows in proportion to log(kL), showing a length dependence inconsistent with radiation from three isotropic point sources located at the ends and feedpoint. The power from three point sources as in Equation (4b) instead varies between two fixed limits with increasing length [28]. 4. FARS results exhibit a lobed, distributed radiated power along the entire length of the sinusoidal current filament [28]. 5. The induced EMF distributed input power is essentially identical to FARS for the sinusoidal current filament, in accord with the fact that the time-averaged power flow along a standing wave is zero, requiring the input power and radiated power to be equal as a function of position [28]. 6. The Schellcunoff-Feldman (S-F) distributed radiation
resistance, when multiplied by [,,-(Zn )]2/2, yields results essentially identical to FARS and the induced-EMF distributed power [28].
7. The radial power flow a few thousandths of a wavelength from the sinusoidal current filament is within a few percent of the FARS, induced EMF, and Schelkunoff-Feldman powers, except at the center feed point, the effect of which is included explicitly in the radial power flow, whereas it is not in the others [28]. 8. In the time domain, Gaussian-pulse-excited current/charge pulses propagating on a straight PEG wire decrease monotonically in amplitude with increasing distance from the feedpoint, as do their associated energies measured by spatial integrals of 12 and (QC) 2 [31].
9. References 1. R. C. Hansen, "Duality in Antennas," North America Radio Science Meeting, The Queen Elizabeth Hotel, Montreal, Canada, July 1997.
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2. Hans Gregory Schantz, "Electromagnetic Energy Around Hertzian Dipoles," IEEE International Symposium on Antennas and Propagation, Renaissance Orlando Resort, Orlando, FL, June 2 1-26, June 1999.
18. Roger F. Harrington and Joseph R. Mautz, "Computation of Characteristic Modes for Conducting Bodies," IEEE Transactions on Antennas and Propagation,AP-19, September 1971, pp. 629639.
3. Rafael G6mez Martin, Amelia Rubio Bretones, Salvador Gonzilez Garcia, "Some Thoughts About Transient Radiation by Straight Thin Wires," IEEE Antennas and PropagationMagazine, 41, June 1999, pp. 24-33.
19. N. Inagaki and R. J. Garbacz, "Eigenfunctions of Composite Hermitian Operators with Application to Discrete and Continuous Radiating Systems," IEEE Transactions on Antennas and Propagation, A-P-30, July 1982, pp. 57 1-575.
4. Colin C. Bantin, "Radiation from a Pulse-Excited Thin Wire Monopole," IEEE Antennas and PropagationMagazine, 43, June 2001, pp. 64-69.
20. David M. Pozar, "Antenna Synthesis and Optimization Using Weighted Inagaki Modes," IEEE Transactions on Antennas and Propagation,AP-32, February 1984, pp. 159-165.
5. J. D. Jackson, Classical Electrodynamics, New York, John Wiley & Sons, 1962, p. 401.
21. D. Liu, R. J. Garbacz, and D. M. Pozar, "Antenna Synthesis and Optimization Using Generalized Characteristic Modes," IEEE Transactionson Antennas and Propagation,AP-38, June 1990, pp. 862-868.
6. Liang-Chi Shen, Tai Tsun Wu, and Ronold W. P. King, "A Simple Formula of Current in Dipole Antennas," IEEE Transactions on Antennas and Propagation,AP-16, September 1968, pp. 542-547. 7. J. Bach Anderson, "Admittance of Infnite and Finite Cylindrical Metallic Antenna," Radio Science, 3, 6, 1968, pp. 607-62 1. 8. Y. S. Yeh and K. K. Mei, "Theory of Conical Equiangular-Spiral Antennas Part II -Current Distributions and input Impedances, IEEE Transactions on Antennas and Propagation,AlP-1 6, 1, 1968, pp. 14-21. 9. D. S. Jones, Methods in Electromagnetic Wave Propagation, Gloucestershire, UK, Clarendon Press, 1994, p. 295. 10. Glenn S. Smith, "On the Interpretation for Radiation from Simple Current Distributions," IEEE Antennas and Propagation Magazine, 40, August 1998, pp. 39-44. 11. E. K. Miller and G. J. Burke, "Time-Domain Far-Field Analysis of Radiation Sources," IEEE International Symposium on Antennas and Propagation Digest, Salt Lake City, UT, 2000, pp. 2058-2061. 12. E. K. Miller and G. J. Burke, "A Multi-Perspective Examination of the Physics of Electromagnetic Radiation," Applied ComnputationalElectromagneticsSociety Journal, 16, 3, 2001, pp. 190201. 13. E. B. Ojeba and C. H. Walter, "On the Cylindrical and Spherical Wave Spectral Content of Radiated Electromagnetic Fields," IEEE Transactions on Antennas and Propagation, AP-27, September 1979, pp. 634-639. 14. C. Balanis, Antenna Theory, Analysis and Design, New York, Harper & Row, 1982.
22. E. K. Miller, "Computing Radiation and Scattering Patterns Using Model Based Parameter Estimation," IEEE International Symposium on Antennas and Propagation Digest, Renaissance Waverly Hotel, Atlanta, GA, June 2 1-26, 1998, pp. 66-69. 23. E. K. Miller, "Using Windowed, Adaptive Sampling to Minimize the Number of Field Values Needed to Estimate Radiation and Scattering Patterns," Proceedings of 14th Annual Review of Progress in Applied ComputationalElectromagnetics, Naval Postgraduate School, Monterey, CA, 1998, pp. 958-963. 24. E. K. Miller, "Model-Based Parameter Estimation in Electromagnetics: Part I. Background and Theoretical Development," IEEE Antennas and PropagationMagazine, 40, 1, 1998, pp. 42-52. 25. E. K. Miller, "Model-Based Parameter Estimation in Electromagnetics: Part HI.Applications to EM Observables," IEEE Antennas and PropagationMagazine, 40, 2, 1998, pp. 51-64. 26. E. K. Miller, "Model-Based Parameter Estimation in Electromagnetics: Part Ill. Applications to EM Integral Equations," IEEE Antennas and PropagationMagazine, 40, 3, 1998, pp. 49-66. 27. W. K. H. Panofsky and M. Phillips, ClassicalElectricity and Magnetism, Addison-Wesley Publishing Co., Inc., Cambridge, MA, 1955. 28. E. K. Miller, "Comparison of the Radiation Properties of a Sinusoidal Current Filament and a PEC Dipole of Near-Zero Radius," IEEE Antennas and PropagationMagazine, 48, 4, August 2006, pp. 37-47. 29. E. K. Miller, "Some Further Results from FARS: Far-Field Analysis of Radiation Sources," Proceedings of 16th Annual Review of Progress in Applied Computational Electromagnetics, Naval Postgraduate School, Monterey, CA, 2000, pp. 278-285.
15. G. J. Burke, private communication, 1980. 16. R. J. Garbacz and R. H. Turpin, "A Generalized Expansion for Radiated and Scattered Fields," IEEE Transactions on Antennas and Propagation,AP-19, May 1971, pp. 348-358. 17. Roger F. Harrington and Joseph R. Mautz, "Theory of Characteristic Modes for Conducting Bodies," IEEE Transactions on Antennas and Propagation,AP-19, September 1971, pp. 622-628.
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30. E. K. Miller, "Further Investigation Using Far-Field Analysis of Radiation Sources (FARS)," IEEE International Symposium on Antennas and Propagation Digest, Salt Lake City, UT, Hilton Hotel, July 16-21, 2000, pp. 2058-61. 31. E. K. Miller, "Exploring Electromagnetic Physics Using ThinWire Time-Domain (TWTD) Modeling," Proceedings of 14th Annual Review of Progressin Applied ComputationalElectroinagnetics, Naval Postgraduate School, Monterey, CA, 1998, pp. 583588. IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
Introducing the Feature Article Author
Since earning his PhD in Electrical Engineering at the University of Michigan, E. Y.. Miller has held a variety of government, academic, and industrial positions. These include 15 years at Lawrence Livermore National Laboratory, where he spent seven years as a Division Leader, and over four years at Los Alamos National Laboratory, from which he retired as a Group Leader in 1993. His academic experience includes holding a position as Regents Distinguished Professor at Kansas University and as Stocker Visiting Professor at Ohio University. Dr. Miller has served as an AP-S Distinguished Lecturer, and wrote the column "PCs for AP and Other EM Reflections" from 1984 to 2000. He received (with others) a Certificate of Achievement from the IEEE Electromagnetic Compatibility Society for Contributions to Development of NEC (Numerical Electromagnetics Code) and was a recipient (with others) in 1989 of the best paper award given by the Education Society for "Computer Movies for Education."~ Dr. Miller served as Editor or Associate Editor of IEEE Potentials Magazine from 1985 to 2005, for which he wrote
a regular column "On the Job." hii connection with this, he was a I
member of the IEEE Technical Activities Advisory Committee of the Education Activities Board, and a member of the IIEEE Student Activities Committee. As a member of the Technical Program Committee for the MTT Symposium in Albuquerque, NM, he was Guest Editor of the Special Symposium Issue of the IEEE Transactions on Microwave Theory and Techniques for that meeting. He was involved in the beginning of the IEEE Computing in Science and Engineering Magazine (originally called Computational Science and Engineering), for which he has served as Area Editor or Editor-at-Large. Dr. Miller has lectured at numerous short courses in various venues, such as ACES, AP-S, MTT-S, and local IEEE Chapter/Section meetings, and at NATO Lecture Series and Advanced Study Institutes. Dr. Miller edited the book Time-Domain Measurements in Electromagnetics, (Van Nostrand Reinhold, 1986), and was coeditor of the IEEE Press book Computational Elecfromagnetics: Frequency-Domain Moment Methods (1991). He was organizer and first President of the Applied Computational Electromagnetics Society, for which he also served two terms on the Board of Directors. He served a term as Chair of Commission A of USNCIURSI, and is or has been a member of USNC/URSI Commissions B, C, and F; was on the Technical Program Committee for the URSI Electromagnetic Theory Symposia in 1992 and 2001; and was elected as a member of the US delegation to several UTRSI General Assemblies. He is a Life Fellow of the IEEE, from which he received the IEEE Third Millennium Medal in 2000. His research interests include scientific visualization, model-based parameter estimation, the physics of electromagnetic radiation, validation of computational software, and numerical modeling. He is listed in Who's Who in the West, Who's Who in Technology, American Men and Women of Science, and Who 's Who in America. I
Online Access to AP-S Symposia When Renewing Membership The IEEE Antennas and Propagation Society has arranged for its members to have online access to the current and past proceedings of the AP-S annual Symposia (and the proceedings of a few other conferences that AP-S has sponsored financially) as a member benefit for 2008, included in the AP-S annual membership renewal price. However, due to a quirk in the IEEE membership renewal system, it will be necessary for AP-S members to choose what is identified as the Society Member Digital Library (SMDL) option when renewing their membership in order to receive this benefit. Note that this is not the same as the AP-S Digital Archive, a set of CDs or DVDs containing all prior AP-S publications. The AP-S Digital Archive is available separately for purchase by AP-S members: see the announcement elsewhere in this issue of the Magazine. IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007
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