S.A. PODOSENOV, A.A. POTAPOV, J. FOUKZON, E.R. MEN’KOVA
Fields, Fractals, Singularities. Classical and Quantum Control
2015
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УДК 621.396 Fields, Fractals, Singularities and Quantum Control. Podosenov S.A., Potapov A.A., Foukzon J., Men’kova E. R., 2015, 383 pp. The book is devoted to the questions of radiation and detection of Ultra-Wide band electromagnetic pulses (EMP). The new analytical method of field calculation in time domain permitting to determine electromagnetic fields from complex structures both neighbouring and far zones is considered. Developed body of mathematics is applied to field calculation in EMP simulators and to calculate the pulse radiation of wire and horn antennas, arrays and reflector antennas. The metod permitting to determine the polyrizability tensors of arbitrary shape conducting bodies is represented. The theory of wave scattering by anisotropic statistically rough surfaces, which is an important part of statistical radiophysics, is considered. A new analytic method is developed and generalized for solving problems of radar imaging. The method involves analytic determination of the functionals of stochastic backscattered fields and can be applied to solve a wide class of physical problems with allowance for the finite width of an antenna’s pattern. The unified approach based on this method is used to analyze the generalized frequency response of a scattering radio channel, a generalized correlator of scattered fields, spatial correlation functions of stochastic backscattered fields, frequency coherence functions of stochastic backscattered fields, the coherence band of a spatial–temporal scattering radar channel, the kernel of the generalized uncertainty function, and the measure of noise immunity characterizing radar probing of the Earth’s surface or extended targets. The introduced frequency coherence functions are applied for thorough and consistent study of techniques for measuring the characteristics of a rough surface, aircraft altitude, and distortions observed when radar signals are scattered by statistically rough, including fractal, surfaces. To exemplify urgent applications, radiophysical synthesis of detailed digital reference radar terrain maps and microwave radar images that was proposed earlier is considered and improved with the use of the theory of fractals The book presents, a new large deviations principles (SLDP) of non-Freidlin-Wentzell type, corresponding to the solutions Colombeau-Ito’s SDE. Exact quasi-classical asymptotic beyond WKB-theory and beyond Maslov canonical operator to the Colombeau solutions of the -dimensional Schrodinger equation is presented. Quantum jumps nature is considered successfully. An explanation of quantum jumps can be found to result from Colombeau solutions of the Schrödinger equation alone without additional postulates. The book is written to engineers and research workers dealing with problems of nonstationary electromagnetic radiation and electromagnetic compatibility. Ill. 101. Bibl. 371 pp. 383.
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CONTENTS Preface…………………………………………………………………………… 7 Chapter 1. Pulse radiation theory by field-forming systems…………………. 10 1.1. General problem formulation of electromagnetic field coupling to a conducting bodies ……………………………………………………………….. 1.2. New analytical method of field calculation by travelling current waves…… 1.2.1. Equations for antenna currents……………………………………………. 1.2.2. Field tensor calculation…………………………………………………… 1.2.3. Radiation from a V – antenna……………………………………………... 1.2.4. Radiation from a linear antenna. Discussion……………………………… 1.2.5. Application of the results to calculation of complex wire structures……... 1.2.6. On pseudoparadoxes in the classic theory of radiation from subsystems……………………………………………………………………….. 1.2.7. Calculation of radiation field by openings in a free space by means of modified Huygens-Kirchhoff method in time domain………………….……….. 1.2.8. Calculation method of pulse radiation by TEM-horn array in a free space…………………………………………………………………………....... 1.2.9. New method for calculating of pulse radiation from an antenna with a reflector………………………………………………………………................... 1.2.10. New calculation method of pulse electromagnetic fields from traveling current waves in complex wire structures……………………………...................
10 15 16 18 28 34 42 49 59 74 78 85
Chapter 2. Linear two-wire transmission line coupling to an external electromagnetic field. Theory…………………………………………………... 105 2.1. Introduction…………………………………………………………………. 2.2. Derivation of transmission line equations…………………………………... 2.3. Analytical calculation of a plane wave coupling to a two-wire line………... 2.4. Analysis of the results……………………………………………………….
105 105 111 119
Chapter 3. Tensors of electric and magnetic polarizability of arbitrary shaped conducting bodies……………………………………..……………….. 123 3.1. Method for determining the polarizability tensor of arbitrary shaped conducting bodies………………………………….…………………………….. 123 3.2. Relation between the polarizability tensor component for conducting bodies of revolution……………………………………………………………………… 132
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3.3. Analytical calculation of polarizability tensor of bodies with shape like a cubic syngony……………………………………………………………………. 136 3.4. Some general polarizability properties of arbitrary shaped conducting bodies…………………………………………………………............................. 147 3.5. Experimental results………………………………………………………… 153 Chapter 4. The theory of functionals of stochastic backscattered fields……………………………………………………………………………… 158 4.1. Introduction………………………………………………………………….. 4.2. The angular spectrum of wave fields…………………………………........... 4.3. The angular spectrum of modulated waves…………………………………. 4.4. Simulation of the spatial-temporal structure of the field scattered by a statistically rough anisotropic surface: a mathematical model taking into accout the effect of an antenna…………………………………………………………... 4.5. Generalized frequency response of a scattering radio channel………............ 4.6. Generalized correlator of fields scattered by a statistically rough anisotropic surface…………………………………………………………............................. 4.7. Generalized correlator of anisotropically scattered fields for the Gaussian pattern of the antenna…………………………………………..………………… 4.8. Normalized generalized correlator of fields scattered by a statistically rough anisotropic surface…………………………………………………...................... 4.9. Spatial correlations of stochastic backscattered fields……………….……… 4.10. Frequency coherence functions of stochastic backscattered fields………… 4.11. Calculation and analysis of the frequency coherence functions of millimeter-wave stochastic backscattered fields…………….…………………… 4.12. Frequency coherence band of a spatial-temporal microwave probing radar channel…………………………..……………………………………………….. 4.13. Radar measurements of the characteristics of a rough surface and of a flight’s altitude according to the frequency coherence function……………………………………………………….….......................... 4.14. The kernel of the generalized ambiguity function and a measure of noise immunity involved in radar probing of a rough surface and extended targets……………………………………………………………………………. 4.15. Formalism of high-order correlators and bispectra in the wave theory and multidimensional-signal precessing………………………..…............................. 4.16. The effect of irregularities of a scattering surface on the structure of reflected signals………………………………………..………………................ 4.17. The effect of a fractal scattering surface on the structure of reflected signals………………………………………………………………………......... 4.18. The effect of hydrometeors on radar imaging……………………………....
158 159 164
166 170 171 175 179 181 184 187 191
196
200 205 206 208 212
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4.19. Radiophysical model of formation of reference detailed digital radar maps of terrain in the MMW band………………………………………..……………. 4.20. A radiophysical method of DDRM synthesis based on the theory of fractals……………………………………………………………………............. 4.21. Indicatrices of millimeter-wave scattering by a fractal surface……............. 4.22. Conclusions…………………………………………………………………
215 222 225 227
Chapter 5. Strong large deviations principles of non-Freidlin-Wentzell type. Optimal control problem with imperfect information. Jumps phenomena in financial markets………………………………………………………………... 230 5.1. Introduction………………………………………………………………….. 5.2. Proposed approach…………………………………………………………... 5.3. Homing missile guidance with imperfect measurements capable to defeat in conditions of hostile active radio-electronic jamming……….…………………... 5. 4. Jumps in financial markets..………………………………………………… 5. 5. Comparison of the quasi classical stochastic dynamics obtained by using Saddle-point approximation with a non perturbative quasi classical stochastic dynamics obtained by using SLDP………………………………………………. 5.6. Strong large deviations principles of Non-Freidlin-Wentzell Type.…............ 5.7. Conclusions…………………………………………………………..............
230 231 264 273
284 287 328
Chapter 6. Exact quasi-classical asymptotic beyond Maslov canonical operator and quantum jumps nature………………………………………….. 330 6.1. Introduction………………………………………………………………….. 6.2. Colombeau solutions of the Schrödinger equation and corresponding path integral representation……………………………………………………………. 6.3. Exact quasi-classical asymptotic beyond Maslov canonical operator………. 6.4. Quantum anharmonic oscillator with a cubic potential supplemented by additive sinusoidal driving……………………………………………………….. 6.5. Comparison exact quasi-classical asymptotic with stationary-point approximation…………………………………………………………………….
330
Appendix A………………………………………………………………............
348
Appendix B………………………………………………………………............
355
References………………………………………………………………………..
369
331 335 340
342 6.6. Conclusions………………………………………………………………...... 347
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PREFACE The problem of generation and measurement of electromagnetic pulse parameters (EMP) having a front on the order of 10 8 s was arisen mainly in connection with creation of nuclear explosion EMP simulators. This problem was successfully solved in practice simultaneously both in USA [1, 2] and in Russia [3, 4]. However last years the leading firms and research laboratories of the world have concentrated their efforts to solve the same problems for Short Ultra-Wide Band (UWB) EMP with duration of the order of 1010 s . It is proposed to apply these pulses for effective energy and information transmission. In accordance with publications at present the scientific-technical revolution in utilization of electromagnetic waves is realized on this basis. This permits, firstly, qualitatively improving the technical, economic and ecological characteristics of radioelectronic devices and, secondly, to solve a number of new problems inaccessible to classic radiotechnics which were arisen in contemporary society. Excellent perspectives of UWB - technology were convincingly presented on "The International Ultra-Wideband Conference Washington, D. C. September 29, 1999" and following conferences. Information transmission by means of ultrashort electromagnetic pulses is energetically profitable in comparison with existing methods of radiowave modulation, since the energy is radiated only during very short time intervales. Diminishing of average power of radiocommunication means permits to decrease of their mass-size characteristics, cost and to attenuate the harmful influence of electromagnetic waves to persons and environment. On the basis of UWB EMP application the radiotelephones of reticent radio communication, the systems of non-wire communication between the computers were developed, and the experiments on transmission of a TV signal are carried out. If reflected radiopulse of ordinary radar permits to determine the distance and angular target coordinates and reflected short electromagnetic pulse contains besides information about sizes, aspect and structure of the target. This permits rapidly and exactly target information. More over by means of modern computer engeneering it is possible to restore the object image. When pulse duration is 1010 s the resolution of a few centimeters is achieved. Also, it is determined that similar radar allows effective detection of targets located under forest cover and under soil surface. The developments of radars to detect and identify the plastic mines are successfully carried out. Developed diminutive UWB radars permits to control the pulsations of heart, vessels, bronchi and vocal cords on a distance and without contact essentially increasing diagnostic possibilities of medicine. The above mentioned small part of UWB EMP application fields explains the great interest to this problem. The hundreds of works devoted to theoretical and experimental investigations in this field were published. However the quantity of books on pulse electromagnetics is insufficiently. Proposed work is written to fill in the gap between abundance of journal articles and deficiency of books in the investigation field of transient elecntromagnetic fields. In considerable part the book is the generalization of before published works by authors in Russia and in abroad. The efficiency of radar, radio-wave imaging, remote sensing, and many other systems has been enhanced by the intense application of MMWs. Therefore, it is necessary to study the processes and characteristic features of MMW scattering by terrain. This study has adequately described the situation in this field, which, today, is again attracting the attention of numerous radiophysicists and radio engineers. The mathematical approach that I have been continually developing involves functionals of
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stochastic backscattered fields and allows qualitative estimation of the spatial–temporal and spatial– frequency characteristics of a scattered signal in the case of combined or spaced radio systems and an adequate description of the formation of observed radiophysical fields. The expressions obtained take into account the effects of irregularities’ slopes and the dimensions of the antenna’s aperture more accurately. Fluctuation effects occurring during random and chaotic nonstationary reflection of waves by a medium’s boundary have been investigated in the case when the boundary has an arbitrary (integer or fractal) topological dimension. The approach proposed makes it possible to study spatial and frequency functionals of stochastic fields scattered by rough anisotropic surfaces with allowance for the mutual statistical relationship between irregularities’ slopes and to specify the limits of applicability of the approximations employed. The developed generalizations and new solutions substantially extend the scope of problems of the statistical theory of wave diffraction. The application of the FCF absolute value and phase characteristic for precision estimation of an aircraft’s flight altitude and of the height of large irregularities have been investigated more accurately and justified quantitatively for various cross-correlation factors of irregularities’ slopes. The measurement errors of the aforementioned quantities increase for large probing angles and wide antenna patterns. It has been shown that measurement techniques using the absolute values and phase characteristics of FCFs are competitive with standard shortpulse methods for measuring flight altitudes and the characteristics of large-height irregularities of terrain. Analytic expressions for the kernel of the generalized AF have been derived with allowance for the antenna’s characteristics and angular orientation and the characteristics of an isotropic or anisotropic scattering surface. These expressions are valid over the entire range of the correlation factor of large-height irregularities’ slopes. It has been shown that the kernel of the generalized AF can be expressed through elementary functions. The results obtained can be used to choose the antenna’s parameters, the type of modulation of the probing signal, the detection characteristics, and the measure of noise immunity for specified ranges of the statistical characteristics of a scattering surface. More general solutions that fit known results can be obtained with the use of FCFs. The investigations have shown that the generalized complex radiophysical model and the method developed by A. A. Potapov for formation of MMW reference DDRMs of an inhomogeneous terrain and for DDRN synthesis, respectively, are promising and highly effective. Combination of the developed methods and fractal description of wave scattering will undoubtedly result in the discovery of new physical laws in the wave theory. We are sure that, combined with the formalism of fractal operators, the theory of fractals and deterministic chaos applied to the problems considered above will make it possible to synthesize more adequate radiophysical and radar models that will substantially reduce discrepancies between theoretical predictions and experimental results. This study rather comprehensively covers the variety of modern problems of wave scattering that arise in theoretical fields and applications of radiophysics and radar and, generally, involve the theory of integer and fractal measures. Thus, the use of the formalism of dynamics of dissipative systems (the fractal character, fractal operators, a non-Gaussian statistic, distributions with heavy tails, the mode of deterministic chaos, the existence of strange attractors in the phase space of reflected signals, their topology, etc.) makes it possible to expect that the classical problem of wave scattering by random media will remain an area of fruitful future investigations. The results obtained can be widely applied for designing various modern radio systems in the microwave, optical, and acoustic bands.
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The book presents, a new large deviations principles (SLDP) of non-Freidlin-Wentzell type, corresponding to the solutions Colombeau-Ito’s SDE. Using SLDP we present a new approach to construct the Bellman function , and optimal control , directly by way of using strong large deviations principle for the solutions Colombeau-Ito’s SDE. As important application such SLDP, the generic imperfect dynamic models of air-to-surface missiles are given in addition to the related simple guidance law. A four, examples have been illustrated proposed approach and corresponding numerical simulations have been illustrated and analyzed. Using SLDP approach, Jumps phenomena, in financial markets, also is considered. Jumps phenomena, in financial markets is explained from the first principles, without any reference to Poisson jump process. In contrast with a phenomenological approach we explain such jumps phenomena from the first principles, without any reference to Poisson jump process. In this book exact quasi-classical asymptotic beyond WKB-theory and beyond Maslov canonical operator to the Colombeau solutions of the -dimensional Schrodinger equation is presented. Quantum jumps nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrödinger equation alone without additional postulates. The experimental investigations described in this book were carried out by K. Yu. Sakharov, V. A. Turkin, and O. V. Mikheev. The authors would like to thank Dr. of Science (techn.) V. N. Krutikov Director of All-Russian Research Institute for Optical and Physical Measurements for his encouragement and support. Potapov Alexander A. Kotel’nikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, ul. Mokhovaya 11/7, Moscow, 125009 Russia E-mail:
[email protected] Podosenov Stanislav A. All-Russian Research Institute for Optical and Physical Measurements, ul. Ozernaya 46, Moscow, 119361 Russia E-mail:
[email protected] Men’kova Elena R. All-Russian Research Institute for Optical and Physical Measurements, ul. Ozernaya 46, Moscow, 119361 Russia E-mail:
[email protected] Foukzon Jaykov Israel Institute of Technologies, Technion City, Haifa, 32000 Israel E-mail:
[email protected] S.A. Podosenov, A. A. Potapoov, J. Foukzon, E. R. Menkova 2015.
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Chapter 1 PULSE RADIATION THEORY BY FIELD-FORMING SYSTEMS This chapter contains: 1. The system of integro-differential equations for charge densities and currents of charged metallic bodies and wires located at external heterogeneous transient electromagnetic field was obtained. 2.New simple calculation method of pulse radiation in time domain is suggested. This one is based on direct determination of electromagnetic field tensor by travelling current wave arbitrary shaped without preliminary calculation of delayed potentials. 3. On the basis of proposed method different field-forming systems are calculated and analysis of existing works in this field have been carried out. 1.1. General problem formulation of electromagnetic field coupling to a conducting bodies Let us consider a system of N arbitrary shaped conducting bodies located in vacuum. We shall also consider that these bodies are at some prescribed external transient heterogeneous electromagnetic field and contain field sources themselves: charges and currents. It is known that Maxwell's equations are applied for change of a system consisting of fields and charges. However in a number of practical problems a densities of charges and currents are not known. The purpose of present part is the close system derivation of integro-differential equations for charges and currents. The solution of these ones will permit in principle to determine the electromagnetic fields outside the body system. To solve this problem let us use a standard four-dimensional Maxwell's equations [5]
F =
4 j .(1.1.1) c
e F = 0.(1.1.2) where e is the absolutely antisymmetric single 4-tensor, F is the field tensor, j is the four-dimensional current vector.
F = 2[ A ] = 2[ A ].(1.1.3) A is the 4-potential satisfying with Lorentz condition.
A = 0.(1.1.4) Obviously, that when presentating the field tensor as (1.1.3), equation (1.1.2) becomes identical, and one (1.1.1) has the form
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1 2 4 .(1.1.5) j ,□ □A c 2 t 2 c
The indentical fulfilment of continuity equation follows from asymmetry of field tensor F j = 0.(1.1.6)
Solution of system (1.1.5) is expressed by means of delayed potentials [5]
1 j ( r, t R/c) A = dV , R = r r, dV = dxdydz, (1.1.7) c R
where
r ( x, y, z ) = ( x1 , y1 , z1 ), r ( x1 , x2 , x3 ). R is the distance from volume element d V to observation point. Three space components of 4-vector A form three-dimentional vector A or vector field potential, time component A0 = is the scalar potential, id est
A ( , A).(1.1.8) Four-dimentional current vector is connected with three-dimentional current density j and charge density by the relationship
j (c , j ).(1.1.9) Using formulae (1.1.7) and (1.1.3) we shall calculate the field tensor F in arbitrary point outside of conductors. F =
2 1 1 j[ 0 1 j[ 1 { ] ]k R k 2 j[ ]k R k }dV , c R c cR R
R k = x k xk ,
=t
R , (1.1.10) c
that forms in components
F0k =
1 R k 1 j k Rk { c }dV , c R 2 cR R3
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Fpr =
2 R[ p j r ] R[ p j r ] { }dV , (1.1.11) c cR 2 R3
where alternation is realized in accordance with indexes in a square brackets. On the other hand the field tensor may be expressed by components of intensity vectors of electric E and magnetic H fields. F0 k = E k = ( E x , E y , E z ) = ( E 1 , E 2 , E 3 ), (1.1.12)
1 H k = eklm Flm = ( H x , H y , H z ) = ( H 1 , H 2 , H 3 ), (1.1.13) 2 where eklm is the absolutely antisymmetric unit pseudotensor of 3 rank. Formulae (1.1.11)-(1.1.13) permit to calculate electromagnetic field in accordance with charge and current j densities at the conductor. If N conductors are present, then according to linearity of electrodynamics equations the complete field is a result of superposition of all ones. In this case the complete field is expressed as N
= F( k ) F* , (1.1.14) k =1
where k in a parenthesis brackets is a number of conducting body, F* is the tensor of external electromagnetic field. As it is known inside ideal conductors the fields E and H are equal to zero and the equalities [6] are valid on their surfaces: c 1 g= n H , (a ) = n E , (b), 4 4 n E = 0, ( c )
n H = 0, ( d ).(1.1.15)
In formula (1.1.15) n is the unit normal vector to conductor surface, g is the surface current density, is the surface charge density. As charges and currents at ideal conductors are located at the surface then the formal substitution is possible in the relationships (1.1.10) and (1.1.11): j k dV = g k dS , dV = dS , (1.1.16) where dS is the conductor surface element. In accordance with (1.1.14) the complete electric field in a system at arbitrary point outsite conductors N
0i = E~i = E i ( k ) E *i , (1.1.17) k =1
where E~ i is the summary electric field, E *i is the external electric field. Choosing on conductor surface with number s an arbitrary point and taking into account that field in the vicinity of this
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point created by all the rest of the charges except one at this point is equal to half of a field from surface and using condition (1.1.15) (b), we shall obtain 2
(s)
* ( s ) 1 N ( k ) R ( k ) n ( s ) = E n [ ( k ) c k =1 R ( k )2
(k ) (s) (k ) 1 n ( s ) g (k ) R (n ( k ) ) ( k ) c ]dS ( k ) .(1.1.18) ( k )3 cR R In formula (1.1.18) s has a values from 1 to N , R ( k ) connects surface element dS ( k ) of conductor with number k and point on conductor surface with number s , n ( s ) is the unit vector normal vector to surface with number s . In a case of electrostatics
= 0,
g =0
and set of equations (1.1.18) goes over into set of integral equations obtained by Grinberg [7]. Complete magnetic field H~ outside the conductors from expressions (1.1.13), (1.1.14) has the form ~ t = 1 etlm .(1.1.19) H lm 2 Vector product n ( s ) H~ in components at the axis has the form
~ t = 1 e prt etlmn ( s ) r = e prt n ( s ) r H lm 2 1 = ( pl rm pm rl )n ( s ) r lm = n ( s ) r rp .(1.1.20) 2 From relations (1.1.11), (1.1.14), (1.1.15), (1.1.16), (1.1.20) we obtain N 2 ( s ) p g = Frp* n ( s ) r Frp( k )n ( s ) r = c k =1
= Erp* n ( s ) r
2 N R ( k )[ p g ( k ) r ] ( s ) r R ( k )[ p g ( k ) r ] ( s ) r n n }dS ( k ) .(1.1.21) { c k =1 cR ( k )2 ( k ) R ( k )3
Relationship (1.1.21) may be rewrite in a vector form 2 ( s ) ( s ) * 1 N R ( k ) g ( k ) ( s ) g = n H { ( k )2 ( ( k ) n ) c c k =1 cR
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( R( k ) n ( s ) ) g ( k ) R( k ) ( k ) ( s ) g (k ) (k ) ( s) ( k )3 ( g n ) ( k )3 ( R n )}dS ( k ) , (1.1.22) ( k )2 (k ) cR R R where H * is the intensity of external field. The set of 4 N integro-differential equations connects 4 N unknown functions: N values and 3 N vector components g . As space components of the tensors were increased and decreased by means of kl , we did not distinguish the covariant and contrvariant tensor components in spite of the standard signature was used ( ) . Values g (s ) and ( s ) are connected by continuity equations (22.6), which when transiting to surfaces will represent to the form: 1 ( s ) ( ( s ) g k ( s ) ) = 0, ( k = 1,2) k (s) (s) t u
s det ik s , dl s 2 ik s du s i du s k .(1.1.22) ik( s ) is the metric surface tensor with number s , and u k ( s ) are the curvature coordinates at the surfaces. A body of mathematics developed here is applied not only to ideal conductors but real ones too at strong skin effect. However a number of essential factors connected with thermal losses are not considered in this case. Proposed method may be used, for example, for analytical computations or computer calculations in tasks of electromagnetic field coupling to a conducting bodies. It should be noted that proposed system (1.1.18), (1.1.22) is a closed one. The boundary conditions (1.1.15)(a) and (1.1.15)(b), used to obtain the system are directly followed from field equations (1.1.1). The equation coinciding with (1.1.22) at N = 1 was obtained in work [8] for the single conductor. However in [8] the condition (1.1.15)(c) was selected instead of the (1.1.15)(b) one as a second boundary condition. Therefore the system obtained in [8] contains 6 equations for four unknown functions. This circumstance do not contradicts to boundary conditions (1.1.15). For the ideal conductors the original singularity takes place when electromagnetic field inside of conductor is equal to zero and four unknown functions have to satisfy to eight equations (1.1.15). For numerical calculations only four independent equations are required, these ones are determined in present work. Other equations for model under consideration are determined from founded ones. For example we shall obtain the integral equations for current calculations in a thin wires. Let quantity of wires is N and their form in a general case is different. The complete electric field outside of the wires is determined in accordance with formula (1.1.17). At the surface of each wire the ~ is perpendicular to wire surface. Therefore for the wire with number s we have complete field E
E~ l ( s ) = 0, ( s = 1,2,...N ),
where l ( s ) is the unit vector at the surface of wire s directed along corresponding wire. From relationships (1.1.11) taking into account the equations
dV K dl ,
j k dV J k dl ,
where K is the linear charge density, J is the current in wire and also formulae (1.1.17) and
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(1.1.25) we shall obtain * ( s) 1 N K ( a ) R ( a ) l ( s ) E l = [ ( a ) c a =1 R ( a )2
(a) (s) ( s ) J ( a ) 1 l (a ) R (l ( a ) ) ( a ) cK ]dl ( a ) .(1.1.26) ( a )3 cR R From the continuity equation for one-dimensional case it is follows ( a ) J ( a ) (l ( a ) , t ) K ( a ) J ( a ) (a ) (a) (a) = , K ( l , ) = l( a ) dt, (1.1.27) ( a ) l ( a )
that in turn results in relationships * ( s ) 1 N J ( a ) R ( a ) l ( s ) E l = [ ( a ) c a =1 l R ( a )2
( s ) J ( a ) 1 R( a ) l ( s ) (l ( a ) ) ( a ) c cR R( a )3
(a )
J ( a ) (l ( a ) , t) dt]dl ( a ) .(1.1.28) l ( a )
Formula (1.1.28) at N = 1 (converted to International System) coincides with expression obtained in [8] and [9]. Body of mathematics developed in this part is considered in work [10]. 1.2. New analytical method of field calculation by travelling current waves
In this part a simple analytic method is suggested for calculation radiation fields from travelling current waves propagating along curvilinear wires. The solutions obtained are valid in both the near and the far zones. The results are used to calculate the radiation of linear and V-antennas, as well as those composed of rectilinear segments. The main results of this part are considered in works [11], [12]. In connection with the problem of screening electronic equipment from transient electromagnetic waves, there are some prblems related to investigation of transient radiation. Therewith the fields in the near zone must be calculated from non-sinusoidal current waves travelling along thin curvilinear wires. Analytic solutions have been found by now only for the simplest cases. Paper [13] investigates in detail the transient radiation problem for a linear antenna. The results are, however, applicable only in the far zone. In works [14-16] the same problems were solved for both the far and the near zones. The results obtained in [14] and [16] for the magnetic field coincide in the whole domain, whereas those for the electric field differ from each other considerably, although in the far zone they go over to the result obtained in [13]. In our opinion, the authors of work [15] commited an error while deriving formula for the magnetic field. Its correction leads to the coincidence of the results of [15] and [16]. In work [16] the results obtained in [14] are sharply criticized, which, in our opinion, is not justified, as we will show later on. Work [17] contains a calculation of the electric radiation field of an element of a rectilinear
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wire of finite lenth, outside which there is no current by definition. The derivation starts from considering the exitation of a wire element by a -shaped current pulse which propagates along the wire with a velocity not necessarily equal to that of light in vacuum. Then, employing a standard method (the convolution theorem), one finds the field of an arbitrary-shaped current. However, in our view, this solution method is not quite correct, since the vector potential A is found by solving the inhomogeneous D'Alembert equation, whereas the scalar potential is computed from the Lorentz gauge condition using the previously found vector potential. The scalar potential calculated in this way will not in general satisfy the inhomogeneous D'Alembert equation and the electric field strength E will be different from that calculated conventionally. (Further this problem will be discussed in more detail). Paper [18] offers an efficient numerical method for studying electromagnetic fields in wire EMP simulators. The results of numerical estimations have been compared in [18] with experimental data. However, the results of [18], based on the theory of [17], require an additional analysis. Paper [19] has developed, for calculating antenna pulse characteristics, the so-called charge model which had been actually used earlier [20] to explain the processes and experimental results of pulse excitation. As is well-known, the classical radiation theory rests on calculations of delayed potentials followed by those of currents. The delayed potentials are calculated by integration over the domain occupied by charges and currents. It is clear that for currents of arbitrary functional form the integral is, as a rule, incalculable analytically. However, for practical purposes, of importance are not the potentials themselves but the electric and magnetic field strengths related to the potentials through differentiation in the coordinates and time at the observation point. The objective of the present work is a direct field tensor determination for a travelling current wave of arbitrary shape, unpreceded by a calculation of delayed potentials. In particular, in the case of a fractured linear wire we find an exact analytic expression for the electromagnetic field with the radiation from the break taken into account. This is evidently impossible for the potentials. Mathematically it is connected with dependence of travelling wave current from argument that permits to express the observation point coordinate derivatives by means of integration field point coordinate derivateves and to calculate the integrals in explicit form. The present work is a further development of papers [21, 22] published earlier. 1.2.1. Equations for antenna currents
To calculate the radiation fields, it is necessary to know the current distribution in the antenna subject to a given electromotive force. The electrodynamic theory of vibrators was built in the frequency representation by Hallen [81, 23], Leontovich and Levin [24]. Nowadays many modifications of their equations exist. The basic results and numerous references are given in the papers [9, 25, 26]. In numerical methods one frequently uses the space-time integro-differential equation for thin wires in the form [9] (EFIE) which coincides with equation (1.1.28) at N = 1 . s Ei
0 s n s R = [ J ( s , ) c J ( s, ) 4 R R 2 s s R c J ( s, )d ]ds, (1.2.1) 3 R s 2
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where n is a unit vector along the wire axis, s is a unit vector on the surface of the wire at the observation point, R is the vector connecting the points of the integration variable s and the observation points on the wire surface, = t R/c , t is time, c is the velocity of light in vacuum, E i is the external electric field, J is the current to be found and 0 is the vacuum permeability. We consider, along with the rigorous equation (1.2.1), a simple, physically clear model, the so-called one-dimensional model in terms of the transmission line theory [27]. An equation for the current in this model, taking into account the external field, can be obtained in a very simple manner, just as it has been done in the well-known paper [28] in the absence of an external field. In common with the theory of multiple-wire line excitations [29], we present the total field t E as a sum of the incident (external) field E i and the scattered one (due to the wire current), E s . Consider a very thin, perfectly conducting wire coinciding with a curve s , requiring that the s -component of the total electric field Est vanish along the wire and that the continuity equations be satisfied, i.e. Est =
u As Esi = 0, s t
J = 0.(1.2.2) s t
In the relations (1.2.2) u is the scalar potential near the wire surface, As is the projection of the vector potential onto the wire, is the linear charge density. Since u and As are specified near the surface of a very thin wire, having a radius of curvature much greater than its thickness, u and As may be found from the formulae for an infinitely long straight wire, as is done in [28]:
u=
r ln , 2 0 r1
As =
0 J r ln , (1.2.3) 2 r2
where r1 and r2 are constants depending on the shape of the wire and 0 is the vacuum permittivity. Unlike [28], we do not assume at the outset that the constants r1 and r2 , involved in the scalar and vector potentials, are equal to each other and to the unit of length. A more rigorous reasoning of (1.2.3) may be found, for instance, in Ref. [30] where it is proved that the delayed potentials of a wire of finite length contain singular terms tending to infinity at r 0 and having the form (1.2.3), while the other terms remain finite when the wire radius r tends to zero. Let us introduce, by definition, using (1.2.3), a wire capacitance per unit length C~ by the formula
2 0 ( s, t ) = .(1.2.4) C~ = u( s, t ) ln( r1/r ) For instance, a thin cylindrical wire of length l and radius r can be approximated by a strongly elongated rotation ellipsoid whose capacitance is determined by a well-known formula [6], whence
18
4 0 l 2 /4 r 2 2 0 .(1.2.5) C~ = 1 ln (l/r ) cosh [l/(2r )] The comparison of (1.2.5) and (1.2.4) yields r1 = l . The relations (1.2.2), (1.2.3) and (1.2.4) imply
r J u 0 0 J 0 ~ ln| 1 | = Esi ( s, t ); s C t 2 r2 t U J C~ = 0.(1.2.6) t s The relations (1.2.6) for a single wire become similar in appearance to the telegraph equations for a two-wire transmission line excited by an external electromagnetic field [29] if the coefficient before J/t is replaced by an induction per unit length L . From (1.2.4) it follows that for curvilinear wires with a constant cross-section C~ = const, according to (1.2.6), one obtains:
1 2 J 2 J ~ Esi =C , v 2 t 2 s 2 t
c=
1
0 0
2u 1 2u Esi = , s 2 v 2 t 2 s
v = c(1
C~ 2 0
ln| r1/r2 |)1/2 .(1.2.7)
To find the constant r2 in (1.2.7), let us make use of the results of [31] and [32] where it is shown that, in the perfect conductor approximation, the field propagation velocity along the conductors is entirely determined by the properties of the environment. In this approximation the velocity v depends neither on the configuration of the conductors, nor on their size [31, 32]. For the case under consideration the environment is air, so that v = c and therefore r1 = r2 and the logarithmic term in (1.2.6) and (1.2.7) falls out. Eqs. (1.2.7) can be used for both receiving and transmitting antennas where the external field acts only at the input. In the latter case Eqs. (1.2.7) lead to wave equations whose solutions are travelling waves. It will be shown in what follows that in some cases the rigorous theory based on Eq. (1.2.1) leads to the same result as the approximate one. We will show that for infinite rectilinear and V antennas Eqs. (1.2.1) and (1.2.7) are equivalent, while for curved wires Eqs. (1.2.7) possess a logarithmic accuracy [6], i.e., the wire length to radius ratio is assumed to be so large that its logarithm is large. This condition is valid, in particular, for wire EMP simulators. 1.2.2. Field tensor calculation
We proceed to calculate the radiation field of a symmetric wire antenna of arbitrary shape (Fig. 1).
19
Fig. 1. Problem geometry Assume, for simplicity, that there is no ohmic resistance and the antenna arms are semi-infinite. The wire length is reckoned along each arm from the excitation point. Let the antenna be excited by a voltage pulse u (t ) determined by the equation
0, t 0 (1.2.8) u t u0 f t t , u0 const , t 1, t 0. The solution to the set of equations (1.2.7) outside the sources is determined by D'Alembert's well-known formula:
u( s, t ) = u0 f (t s/c) (t s/c), u0 = J 0W , J ( s, t ) = J 0 f (t s/c) (t s/c), W = 1/(cC~ ).(1.2.9) It turned out to be easier to calculate the fields in the tensor form. We will always assume that Greek indices range from 1 to 4, Latin ones from 1 to 3 and summation over repeated indices is carried out. We perform our calculation in the Cartesian coordinates x1 = x , x2 = y , x3 = z , x4 = ict , i = 1 . The four-vector potential A from thin wires may be presented in the form
A =
0 4
I (t s/c R/c ) ds, I k = Jnk , I 4 = ic , (1.2.10) (l ) R
where R is the distance from a current element of length ds and nk is a unit vector in the direction of the current J in the wire. The electromagnetic field tensor F is defined conventionally [5, 33] by the formula
20
F =
A A .(1.2.11) x x
The coordinates of the observation points xk enter into the integrand of (1.2.10) through the distance R = [ kl ( xk xk ( s))( xl xl( s))]1/2 , where kl is the Kronecker symbol and the wire equation is specified in the parametric form
xk = xk ( s),
dxk = nk , nk nk = 1, (1.2.12) ds
where the wire length is chosen as a parameter. Eqs. (1.2.10--1.2.12) imply
Fkl =
0 m[ k nl ] J/s J ds, (1.2.13) 4 R 1 R/s R 0 I 4 I 4 /s [ mk 2 4 R R(1 R/s)
F4 k =
nk J/s ]ds.(1.2.14) iR (1 R/s)
The following relations were used in the derivation of (1.2.13) and (1.2.14):
R x xk ( s) R = mk k , xk xk R nk mk
m R R k = 0, s s
R = mk nk , s (1.2.15)
where mk is the unit vector directed from a current element towards the observation point. Confining of indices in square brackets denotes alternation without division by two, i.e. m[ k nl ] mk nl ml nk . To further simplify the field tensor, let us introduce the expressions M kl
Gk
J m[ k nl ] ; s R (1 R/s)
J (nk mk ) .(1.2.16) s R(1 R/s)
Making use of Eqs. (1.2.15), the expression
21
2 R R 1 = K ( N k mk ) [1 ( )2 ], (1.2.17) 2 s s R Frenet's formula known from differential geometry [34]
dnk = KN k (1.2.18) ds and the continuity equation along the wire
I = 0, (1.2.19) x we obtain I 4 = i J ; (1.2.20)
Fkl =
Ykl [(l )] [ (l )
F4 k =
0 [ M kl ]ds Ykl [(l )]; 4 (l )
JK (m p N p ) JKm[ k N l ] ] ds ; (1.2.21) m[ k nl ] 2 R(1 R/s) R(1 R/s)
JKN p m p (nk mk ) 0i JKN k [Gk ] ds, (1.2.22) 2 4 ( l ) R(1 R/s) R(1 R/s)
where K is the curvature of the line and N k is the unit vector in the direction of the first curvature of the line. As follows from Eqs. (1.2.16), (1.2.21), (1.2.22), the integrals (1.2.21) and (1.2.22) can be computed explicitly for rectilinear wire segments, for which K = 0 . To simplify the calculation of radiation from curvilinear segments, let us approximate the smooth curve by a broken, piecewise rectilinear line. An intersection of two segments is equivalent to their joining by an arc of infinitesimal curvature radius. That does not, however, result in vanishing of the integrals in (1.2.21) and (1.2.22), since K as ds 0 , while the quantity K ds = (d/ds) ds where is a finite wire fracture angle. Fig. 2 depicts symmetric segments of the antenna with a fracture formed by an intersection of two rectilinear segments. The beginning of the rectilinear segment, its end and the fracture point are situated at distances s1 , s2 and s0 , respectively, from the antenna excitation point. The beginning of the rectilinear segment is on a distance s1 , the end is on a distance s2 and fracture is on a distance s0 from antenna excitation point.
22
Fig. 2. Geometry of the symmetric segments with a fracture Let us find out the contribution of the antenna element, shown in Fig. 3a, to the electromagnetic field.
Two rectilinear antenna segments 1 A and B 2 are connected by the circular arc AB of radius r . The integral (1.2.21) on the curve (L) is
Fkl =
0 J (t s2 /c R2 /c)m2[ k n2 l ] [ R2 (1 m2 n2 ) 2
J (t s1/c R1/c)m1[ k n1l ] Ykl [ AB ]]. R1 (1 m1 n1 )
Eq. (1.2.15) implies
mk r(mk m p n p nk ) = . R Evidently, if r/R = 1 , then the vector m on the arc practically preserves its orientation. Therefore it is possible to use a certain easily verifiable relation. Indeed, r/R 0 implies m/ 0 , whence Km[ k nl ] Km[ k nl ] ( m N ) m[ k nl ] ( . ) = s 1 m n 1 m n (1 m n ) 2
The limit r 0 corresponds to intersecting straight lines. We obtain:
lim Ykl [ AB ] = r 0
m0[ k n1l ] J (t s0 /c R0 /c ) m0[ k n2 l ] [ ]. R0 1 m0 n2 1 m0 n1
23
Fig. 3. Connection of two antenna elements: (a) - connection by the circular arc, (b) - connection by the intersecting straight lines Thus the contribution of the two intersecting rectilinear fragments (Fig. 3b) to the field is
Fkl =
0 J (t s2 /c R2 /c) m2[ k n2 l ] { 4 R2 1 m2 n2
J (t s1/c R1/c ) m1[ k n1l ] R1 1 m1 n1
m0[ k n2 l ] J (t s0 /c R0 /c ) m0[ k n1l ] [ ]}, (1.2.21a ) R0 1 m0 n1 1 m0 n2
where m0 is the unit vector with the direction from the fracture to the observation point, R0 is the distance between the fracture and the observation point, s1 and s2 are the distances from the antenna excitation point to the points 1 and 2 along the wire, s is the distance from antenna excitation point to fracture along the wire, R1 , R2 are the distances from points 1 and 2 to observation point. Let us find the field of the symmetric part of the antenna (the lower part in Fig. 2). The current direction in this part (unlike the upper one) is opposite to that of excitation wave propagation.
24
Therefore in Eq.(1.2.12) for the lower part of the antenna dxk /ds = nk where nk coincides, by definition, with the current direction, while s increases from point 1 to point 2 . Integration in s for the lower part is carried out in the current direction, i.e., from point 2 to point 1 . Hence it follows that the contribution to the field from the lower part of the antenna can be obtained from (1.2.21a) by the formal substitution n1 n1, n2 n2 , J J , m0 m0 , R0 R0 , m1 m1, R1 R1, m2 m2 , (1.2.21b) R2 R2 , s1 = s1, s0 = s0 , s2 = s2 .
To find the electric field F4 k , we integrate (1.2.22) in a similar way using an easily verified relation which follows from the conditions r/R 0, m/ 0 . These imply
n m KN K (n m)( N m) . = s 1 m n 1 m n (1 m n )2 The vector n , unlike m , varies along this arc from n1 to n2 . The contribution from the lower element of the antenna (Fig. 2) is obtained from that of the upper one with the aid of the above formal substitution. Using Eqs. (1.2.16), (1.2.21), (1.2.22) and (1.2.21a), we obtain as a result the fields Fkl and F4 k explicitly, with the radiation from the fractures taken into account: Fkl =
m0[ k n2 l ] 0 J (t s0 /c R0 /c) m0[ k n1l ] { ( ) 4 R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c) m2[ k n2 l ] R2 1 m2 n2 m n J (t s0 /c R0 /c) m0[ k n1l ] ( 0[k 2 l] ) R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c) m2[ k n2 l ] J (t s1/c R1/c) m1[ k n1l ] R2 1 m2 n2 R1 1 m1 n1
J (t s1/c R1/c ) m1[ k n1l ] }, (1.2.23) R1 1 m1 n1
25
F4 k =
0i J (t s0 /c R0 /c ) n1k m0 k n2 k m0 k { ( ) 4 R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c ) n2 k m2 k R2 1 m2 n2 J (t s0 /c R0 /c ) m0 k n1k m0 k n2 k ( ) R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c ) n2 k m2 k J (t s1/c R1/c ) n1k m1k R2 1 m2 n2 R1 1 m1 n1
J (t s1/c R1/c) n1k m1k }.(1.2.24) R1 1 m1 n1
The radiation from fractures corresponds to the first terms in (1.2.23) and (1.2.24). It can be shown that the radiation field from the angles can be described as spherical TEM waves centered at the fractures. The energy density depends on the angular coordinates of the vectors m0 and m0 . Thus the fractures are actually secondary radiation sources influenced on the antenna current. This influence on principle may be taken into account introducing to (1.2.7) as a source the angle radiation field. Formulae (1.2.23) and (1.2.24) are key ones for complex antenna design. The transition from the field tensor F to the electric field strength Ek and the magnetic induction Bk is carried out conventionally [5, 33]:
F4 k =
i Ek , F12 = B3 = Bz , F13 = B2 = B y , F23 = B1 = Bx .(1.2.25) c
Let us introduce for our analytic studies the vector Bt , dual to the spatial components Fkl of the field tensor. Bt coincides with the magnetic induction vector and is determined by formula [5]
Bt = 12etkl Fkl (1.2.26) where etkl is the unit antisymmetric tensor. In view of the importance of the relations (1.2.23) and (1.2.24), let us rewrite them in the more popular vector form. Using (1.2.25) and (1.2.26) and the definition of the vector product of two arbitrary vectors, P and M , known from vector algebra,
K = P M , K p = e pmn Pm M n ,
we obtain for E and H = B/0 the expressions
26 1 J (t s0 /c R0 /c ) m0 n1 m0 n2 { ( H= ) 4 1 m0 n1 1 m0 n2 R0 J (t s2 /c R2 /c ) m2 n2 J (t s1/c R1/c ) m1 n1 1 m2 n2 1 m1 n1 R2 R1 J (t s0 /c R0 /c ) m n m n ( 0 1 0 2 ) R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c) m2 n2 J (t s1/c R1/c) m1 n1 }, (1.2.23a ) R2 R1 1 m2 n2 1 m1 n1 J (t s0 /c R0 /c ) n1 m n2 m0 0 ( E= 0{ ) 4 1 m0 n1 1 m0 n2 R0 J (t s2 /c R2 /c ) n2 m2 J (t s1/c R1/c ) n1 m1 1 m2 n2 1 m1 n1 R2 R1 J (t s0 /c R0 /c ) m0 n1 m0 n2 ( ) R0 1 m0 n1 1 m0 n2
J (t s2 /c R2 /c) n2 m2 J (t s1/c R1/c) n1 m1 }, (1.2.24a ) R2 R1 1 m2 n2 1 m1 n1
where 0 = ( 0 / 0 )1/2 is the free-space intrinsic impedance. Let us analyze the relations (1.2.23a) and (1.2.24a). These relations describe the contribution to the electromagnetic field from the symmetric antenna elements which form a part of some complex wire travelling-wave antenna. The field from the whole antenna can be obtained by summing the contributions from all its parts. The solutions obtained are exact under the assumption that the current is determined by the relation (1.2.9) following from (1.2.7). The question arises how the solution (1.2.9) is connected with the solution to EFIE (1.2.1)? To find an answer, let us find a solution to (1.2.1) outside the source in the form of a travelling wave. Evidently outside the source, when external fields are absent, E i = 0 . For a travelling wave J ( s, t ) = J (t s/c R/c) where R is the distance between the observation point, located on the antenna surface, and the integration point s ; = t R/c . For travelling waves the following evident relation is valid:
J/s |t =const 1 J J ( s, ) | =const = = |s=const . c 1 R/s s Therefore we have in (1.2.1): s R J ( s, ) s R J s R c d = c 3 d = c 3 J . R 3 s R R 2
27 In our notation R = m R . Taking into account the aforesaid, Eq. (1.2.1) outside the source, in the absence of an external electric field, takes the form
s (m n ) J/s |t =const s m J 0 = [ ]ds. R 1 R/s R2
Since the unit vector s at the observation point does not depend on s , we can factor it outside the integral, sign which, in tensor notation, results in
sk [(
J n J/s J/s )mk k ]ds = 0. 2 R R(1 R/s) R 1 R/s
Using the equalities (1.2.14), (1.2.20) and (1.2.25), one can assure that the last equality is equivalent to the relation
sk Ek = s E = 0, which means the absence of an electric field tangential component on the wire surface. The expression for F4 k in (1.2.14) is valid both inside the conductor and on its surface. In particular, Eq. (1.2.14) may be rewritten in the form (1.2.22). If we manage to find the field E from (1.2.22) at an arbitrary point outside the wire and, in particular, find that the tangential component Et = 0 on the wire surface, we will thus prove that Eq. (1.2.1) is satisfied by the travelling-wave solution. Simultaneously the equivalence of (1.2.1) and (1.2.7) is proved. It is not the case in general, but we will demonstrate this equivalence in some special cases cons below. The basic reason for a non-equivalence of (1.2.1) and (1.2.7) is connected with the fact that a travelling-wave solution to (1.2.7) implies that a current in a curvilinear wire at a distance s along the wire from the oscillator cannot appear earlier than in the time t = s/c from the oscillator switch-on instant. What actually happens is that the radiation field, propagating rectilinearly, reaches the same point s in a time t < s /c and causes a current at that point at an earlier instant. Fig. 4 depicts a rhombic travelling-wave antenna loaded by an active resistance equal to the wave one. For the antenna under consideration, from (1.2.24a) we obtain:
s1 = 0, s2 = 2l , s0 = l , R1 = r = R1, m1 = m = m1 , where r is the radius vector from the origin to the observation point. If the observation point coincides with point A in Fig. 4, then the terms of the form
n1 m J (t r/c) n1 m E1 = 0 ( )(1.2.26a ) r 4 1 m n1 1 m n1 represent a wave propagating along r with the vector E1 perpendicular to r = m r . This wave evidently reaches the point A earlier than the wave travelling along the wire, and excites an additional current.
28
Fig. 4. Rhombic antenna The existence of additional currents affects the accuracy of the suggested method, therefore the set (1.2.7) has a so-called logarithmic accuracy [6] when the logarithm of the wire length to its cross-section radius ( b ) ratio is big as compared to unity. In what follows we will show that such an accuracy is quite satisfactory in real wire EMP simulator design. Let us consider concrete special cases. 1.2.3. Radiation from a V -antenna Let us consider the radiation from a V-antenna formed by two thin cones originating from a single point, with an angle 0 between the axes (Fig. 5). Two cases will be studied: (a) an infinite antenna and (b) a finite antenna. (a) An infinite antenna. From Eqs. (1.2.23-1.2.25), with
R0 , R0 , R2 , R2 , R1 = R1 = r , n1 = n , n1 = n , m1 = m1 = m,
we have Fkl =
m[ k nl] 0 J (t r/c ) m[ k nl ] ( ), (1.2.27) 4 r 1 m n 1 m n
Ek =
0 J (t r/c) nk mk nk mk ( ).(1.2.28) r 4 1 m n 1 m n
or, with (1.2.26), in the vector notation,
1 J (t r/c) m n m n H = ( ), (1.2.29) 4 r 1 m n 1 m n
29
J (t r/c) n m n m E= 0 ( ).(1.2.30) r 4 1 m n 1 m n Asfollows from (1.2.29) and (1.2.30), an infinite V antenna emits a spherical TEM wave. Indeed, ( B m ) = ( E m ) = 0, ( B E ) = 0 , while the calculation of ( B B ) and ( E E ) leads to the relation
E = 0 H =
1 n n 0 J 1/2 [ ] , (1.2.31) 2 2r (1 m n )(1 m n )
which proves the above statement. It follows from the present consideration that the electric field strength vector on the antenna surface is perpendicular to it. This, as shown earlier, means that the travelling-wave current is a solution to (1.2.1) and, at the same time, a solution to (1.2.7). Therefore for this particular problem the EFIE (1.2.1) and (1.2.7) are equivalent. Let us calculate the wave impedance of a V-antenna. Since such an antenna radiates a spherical TEM wave, it is sufficient to find the potential difference between the points a1 and a2 (Fig. 5) and divide it by the value of the current at these points. Let the curve a1a 2 be an arc of a circle coinciding with one of the meridians of the spherical coordinate frame whose z axis coincides with the axis of the upper cone. We find for the only nonzero electric field component E from (1.2.31):
Fig. 5. V-antenna
E =
1 cos 2 0 J 0 J cos [ ]1/2 = .(1.2.32) 2r(sin sin ) 2 2r (1 cos( ))(1 cos( ))
30
The potential difference U is obtained by the formula ( )
U =
E rd =
0 J cos( /2) ln .(1.2.33) sin (/2)
~ The wave impedance Z is
cos( / 2) Z 0 ln . (1.2.34) sin( / 2) With = 0 a V-antenna becomes a bicone one and formula (1.2.34) coincides with the one known from Ref. [35]. At small , Z (1/ )( 0 / 0 )1/2 ln(d / a), (Fig. 5), which corresponds to the wave impedance of hyperparallel wires [31], while at close to /2 corresponds to the wave impedance of a two-wire transmission line. Let us study in more detail the expressions for the fields of an infinite V-antenna. The angle is reckoned from the z axis, while from the y axis to the x axis. The arms of the antenna are situated in the YZ plane. To construct the directional pattern, we make use of formula (1.2.31). As follows from the aforesaid,
n = {0, sin , cos }; n = {0, sin , cos }; m = {sin sin , sin cos , cos }.(1.2.34a ) It is more convenient to introduce, instead of , the angle = /2 reckoned from the Y axis. As follows from (1.2.31), the E field is the strongest at the bicone surface. It is of practical interest to know the field in the XY plane, which corresponds to = 90 . We have in this case:
E ( r, /2, ) =
0 J (t r/c) sin .(1.2.34b) 2r(1 cos cos )
The value E max evidently corresponds to = 0 . Introducing F ( ) = E/Emax , we obtain:
F ( ) =
1 cos (1.2.34c) 1 cos cos
where the angle is included as a parameter. Let us analyze (1.2.34b) for = 0 :
E ( r, /2, 0) =
0 J cot .(1.2.34d ) 2r 2
31
It is of interest to consider the case 0 and small , which is close to a two-wire transmission line. In this case (1.2.34d) yields:
E ( r, /2, 0) =
0 J
0 J , 2r sin(/2) d1
where d1 is the perpendicular dropped from the observation point to one of the antenna arms. The last formula coincides with that for the field E in the middle of a two-wire transmission line with the spacing 2 d1 between the wire axes, under the condition that the wire radii are much smaller than their spacing [36]. For 0, 0 Eq. (34b) gives E 0 , which means the absence of radiation sideway a two-wire transmission line. At = 0 we have a bicone antenna for which (1.2.31) and (1.2.32) give
E = 0 H =
0 J , (1.2.34e) 2r sin
which coincides with the well-known result [35]. Fig. 6 shows the directional pattern F ( ) with the parameter . At = 90 the V-antenna becomes a linear one, with a circular directional pattern. As decreases, the pattern stretches. For instance, at = 60 , = 180 , F (180 ) = 0.33 , i.e., the backward radiation is three times weaker than the forward one. From (1.2.32) (Fig. 5) we find the field value along the meridian in the aperture plane:
E =
0 J sin . 2r(sin cos )
At = 90 we obtain (1.2.34d) and at = = /2 we find the field E 1 on the upper cone surface:
E 1 =
0 J sin . 2r[cos( ) cos ]
Case (b). The radiation of a V-antenna with a finite arm length l may be treated from two equivalent viewpoints: either considering the current in such an antenna as that in a long line with an open end, or assuming that the current propagates along a line with a 180 break at its end and returns to the oscillator. We will adhere to the second viewpoint and assume for simplicity that the antenna is matched to the oscillator output impedance. The field will be calculated by formulae (1.2.23a) and (1.2.24a), where we put
32
n2 = n1 , n2 = n1, m2 = m2 = m, s1 = 0, s0 = l , s2 = 2l , R2 = R2 = r = R1 = R1, m1 = m1 = m.
Fig. 6. Directional pattern of infinite V-antenna The result is J (t r/c ) n1 m n m E= 0 { ( 1 ) r 4 1 n1 m 1 n1 m J (t 2l/c r/c ) n1 m n m ( 1 ) r 1 n1 m 1 n1 m J (t l/c R0 /c ) n1 m0 n1 m0 ( ) R0 1 n1 m0 1 n1 m0
J (t l/c R0 /c) n1 m0 n1 m0 ( )}, (1.2.35) R0 1 n1 m0 1 n1 m0 m n1 1 J (t r/c ) m n1 { ( H = ) r 4 1 m n1 1 m n1 J (t 2l/c r/c ) m n1 m n1 ( ) r 1 m n1 1 m n1 m0 n1 2 J (t l/c R0 /c ) ( ) R0 (1 n1 m0 )(1 n1 m0 )
33
m0 n1 2 J (t l/c R0 /c) ( )}.(1.2.36) R0 (1 n1 m0 )(1 n1 m0 ) It follows from the solutions (1.2.35) and (1.2.36) that the radiation contains a sum of four spherical waves two of which emerge from the center, one of them shifted by the time 2 l/c , the current propagation time to the antenna ends and back. Two other waves form at the ends of the antenna at the instant when the signal from the excitation point is received. If 0 0, 0 , then the antenna turns into a two-wire transmission line for which n1 = n1 , m0 = m0 and R0 = R0 . In this case the fields E and B vanish outside the line. Let us analyze formulae (1.2.35) and (1.2.36). Assume that a step signal J (t ) = J 0 (t ) is fed to the input of a V-antenna. Then in t = 2l/c the current in the antenna disappears and after the instant t = 2 l/c r/c the magnetic field at the observation point must vanish. Let us prove that Fig. 5 gives the following relations:
n1l m0 R0 = mr, r m0 n1 = m n1 , R0
n1l mr = mR0 , r m0 n1 = m n1 . R0
By the theorem of sines,
R0 = r[
1 (n1 m)2 1/2 ] ; 1 (n1 m0 )2
R0 = r[
1 (n1 m)2 1/2 ] . 1 (n1 m0 )2
As t > 2l/c r/c , Eq. (1.2.36) may be rewritten in the form H
J 0 m n1 m n1 m n1 [ 4r 1 m n1 1 m n1 1 m n1 2m n1 2m n1 m n1 ] 0. 1 m n1 1 ( m n1 ) 2 1 ( m n1 ) 2
=
The disappearance of the magnetic field is connected with that of the current. The electric field at the observation point does not disappear when the current stops since the antenna arms acquire electric charges. We compared the presently obtained results with those of Ref. [19] where, on the basis of the charge model, the radiation field from a linear antenna bended over a perfectly conducting plane is found, when the antenna is excited by a -shaped current. The comparison showed that the formulae from Ref. [19] for the electric field on the Y axis (Fig. 5), follow from (1.2.35) with J (t ) = q (t ) , where q is the ``charge'' and (t ) is the Dirac function, under the condition that in the former one takes into account a delay for the signal propagation from the antenna base to the observation point. The charge model of wire antennas is in a certain sense equivalent to describing radiation by the Lienard-Wiechert potentials, while in the conventional model the delayed potentials are used.
34
1.2.4. Radiation from a linear antenna. Discussion A linear antenna is obtained from a V-antenna when 0,0 = (Fig. 5). Let us first consider the case when the antenna is infinite but we will take into account the contribution to the field only from the arms of length l (Fig. 7), ignoring that from the remaining part. From general formula (1.2.23a) for the case under consideration we obtain an expression for B in cylindrical coordinates in the form
B =
0 2 J (t r/c) cot(1/2) { J (t R0 /c l/c) 4 R0 tan( 2 /2)
J (t R0 /c l/c)}.(1.2.37) R0 An expression for the field E is obtained more conveniently in spherical coordinates, which by (1.2.24a) yields the following expressions for the nonzero components E and Er :
Fig. 7. Geometry of the linear antenna
0 2 J ( t r /c ) { 4 [cot (1/2) cos 1 sin 1 ]
E =
R0
J (t R0 /c l/c )
[ tan(2 /2) cos 2 sin 2 ] J (t R0 /c l/c)}(1.2.38) R0
Er =
0 [sin 1 cot(1/2) cos1 ] { J (t R0 /c l/c) 4 R0
35
[sin 2 tan(2 /2) cos2 ] J (t R0 /c l/c)}.(1.2.39) R0
Let us now calculate the fields of a linear antenna of a finite length 2l taking into account the reflections from its ends. For that purpose we will use formulae (1.2.35) and (1.2.36) where we put 0 = . The result is
B =
o {J (t r/c) J (t 2l/c r/c) 2
J (t l/c R0 /c ) J (t l/c R0 /c )}, (1.2.40)
E =
0 {J (t r/c) J (t 2l/c r/c) 2
cos 1 J (t l/c R0 /c ) cos 2 J (t l/c R0 /c)}, (1.2.41)
Er =
0 {sin 1J (t l/c R0 /c) 2
sin 2 J (t l/c R0 /c)}.(1.2.42) Let us compare our results for nonstationary radiation of a linear vibrator with those from other papers [13--17]. All the formulae obtained, (1.2.37)--(1.2.42), pass into the known formulae [13] in the far zone. Eq. (1.2.37) coincides with the results of Refs. [14, 16]. In [15] there is an error in deriving this formula, which resulted in an inaccurate expression for the electric field. This inaccuracy having been corrected, the results of [15] and [16] coincide. Our formulae (1.2.38) and (1.2.39) for the electric field do not coincide with the corresponding formulae from [16] in the near zone, although formulae (1.2.40-1.2.42) coincide identically. Let us compare our formulae (1.2.38) and (1.2.39) with the results of the paper [14] which was sharply criticized by the authors of Ref. [16]. Changing the oblique-angled coordinate frame of Ref. [14] to the spherical one, it is easy to verify the coincidence between (1.2.38) and (1.2.39) with the similar formulae from [14]. Thus our results for the electric field do coincide with those from [14] and do not with those from [16]. The discrepancy is connected with different methods of calculating the electric field created by a finite current segment selected from an infinite straight line along which the current is propagating. If one considers the radiation from the whole domain to which the current has propagated (as is the case, e.g., with our treatment of a finite-length vibrator matched to the oscillator), then the delayed potential method and that of [15] and [16] lead to coinciding results (1.2.40--1.2.42). If, on the other hand, one considers the field of a bounded segment, mentally distinguished on an unbounded straight line, then the delayed potential method makes it possible to calculate the field from the current on this segment at a given time instant (taking into account the delay due to signal propagation from the current element to the observation point). In Refs. [16] and [17] the magnetic field was also sought from the delayed vector potentials. Unlike that, the electric field was calculated by time integration of the Maxwell equation
36
E 0 = H, t using the already found expression for the magnetic field H . So in such a way of calculation the obtained electric field includes the information on the pre-history of the current behavior on the segment ( l , l ) from the oscillator switch-on instant till the instant under consideration. For comparison with (1.2.38) we show a similar formula from [16] in our notation:
2 J (t r/c ) cos 1 cot (1/2) E ( r , t ) = 0 { J (t R0 /c l/c ) 4 R0
tan ( 2 /2) cos 2 J (t R0 /c l/c )} R0 1 4 0
{
sin 1 sin 2 Q (t R0 /c l/c ) 2 Q ( t R0 /c l/c )}, 2 R0 R0 t
Q (t ) = J (t )dt .(1.2.43)
A comparison of (1.2.38) and (1.2.43) shows that the expressions differ in the coefficients before sin 1 and sin 2 . For example, it follows from (1.2.43) that when the antenna is excited by a step voltage, the field E due to a current which is constant along the segment, linearly grows with time, which is lacking in physical content. Fig. 8 presents a comparison of dimensionless fields
E 4 E /( 0 / 0 J 0 ] calculated by the formulae (1.2.38) and (1.2.43) under antenna excitation in the plane z = 0 (Fig. 7) as a function of the dimensionless time T = ct/ , б = l , 1 = 2 = /4 , = /2, 2 = /4, 1 = 3/4 .
Fig. 8. Comparison of the fields
37
As seen from Fig. 8, after the arrival of a signal from the end of the segment l to the observation point, the field E , calculated by (1.2.43) [16], infinitely grows in time, while the field E calculated by (1.2.38) remains finite. Thus the criticism of the work [14] contained in Ref. [16] is, in our view, not quite correct. Let us compare our results and those of [17]. Ref. [17] calculates the electric radiation field from a travelling current wave propagating along a linear segment. By definition, there is no current outside the linear segment of length (b a ) , while the wave propagation velocity v is in general unequal to the light propagation velocity in vacuum, c . For the case of interest for us, that of uncoated wires, so that v = c , Eq. (16) from [17] with a = 0 and b = l in our notation (Fig. 7), in spherical coordinates has the form
E =
sin cos1 sin 1 0 [ J (t r/c) J (t l/c R0 /c) ] r (1 cos ) R0 (1 cos1 ) 4
0c sin 1 Q(t R0 /c l/c); (1.2.44) 4 R02
Er =
0 sin 1 sin 1 J (t l/c R0 /c) 4 R0 (1 cos1 )
1 0c 1 [ 2 Q(t r/c) 2 cos1Q(t l/c R0 /c)].(1.2.45) 4 r R0
If the fields (1.2.44) and (1.2.45) are supplemented by those due to the current existing in the lower part of the antenna (Fig. 7), then, using the substitution (1.2.21b), one obtains: E =
sin 0 J (t r/c ) sin [ ( ) 4 1 cos 1 cos r
J (t l/c R0 /c ) cos 1 cot (1/2) R0 J (t l/c R0 /c ) cos 2 tan( 2 /2)] R0
Q (t l/c R0 /c ) 0c [sin 1 R02 4
sin 2
Q(t l/c R0 /c) ]; (1.2.46) 2 R0
38
Er =
0 J (t l/c R0 /c ) [ sin 1 cot (1/2) 4 R0
J (t l/c R0 /c ) sin 2 tan ( 2 /2)] R0
0c cos 1 [ Q (t l/c R0 /c ) 4 R02
cos 2 2 Q (t l/c R0 /c )].(1.2.47) R0
Thus, Eqs. (1.2.46) and (1.2.47) are identical to (5b) and (5c) from [16] and differ from [14] and our formulae (1.2.38) and (1.2.39). So the results of Refs. [16] and [17] have turned out to be equivalent, although they were obtained by different methods. Ref. [17] refers to [14] with no comments and contains no reference to the close papers [15] and [16]. We see a paradoxical situation. Solving the simplest problem, that of finding the electric field of a linear antenna in the near zone, different authors arrive at different results. Let us find out the reason for this “paradox”. As is well-known from the electromagnetic field theory [5, 33], the representation of a solution in the form of the delayed potentials (1.2.10) requires that the following two conditions be satisfied: 1. The continuity equation along the wires, (1.2.19). 2. The Lorentz condition,
A A4 i = A = 0 0 A = 0, A4 = .(1.2.48) x x4 t c In Ref. [33] a theorem has been proved for the general case that, provided the charge conservation law is valid, the validity of the Lorentz condition is connected with the existence of such a closed surface S bounding the volume V that the current at this surface is zero. In view of the importance of this theorem for what follows, let us reproduce the result of [33] in our notation: 0 0 A = 0 t 4
1 j 0 V r [( j )t*=const t* ]dv 4 V ( r )dv, (1.2.49)
where, unlike [33], the unprimed coordinates correspond to the observation point, whereas the integration variable is primed and t * = t r/c . Evidently if the system where currents are specified is confined to a finite domain, then, recalling the Gauss theorem, one can choose the surface S so big that the current density on this surface vanish, so that
j n V ( j /r)dv = S r dS = 0.(1.2.50)
39
In this case, as follows from (1.2.49), the validity of the continuity equation
/t * j | * t
=const
= 0(1.2.51)
implies the validity of the Lorentz condition. Let us consider our formulae (1.2.38) and (1.2.39) in the light of the theorem from [33]. We have been considering an infinite antenna but took into account the contribution to the field from only a finite segment of length 2l , outside which there is no current, so that the continuity equation (1.2.51) or (1.2.19) is valid both inside and outside it. However, when integrating the last term in (1.2.49), we have to draw the surface S , by definition, thorough the points z with the coordinates z = l and z = l . If we drew it through points on the z axis outside the segment l < z < l , we would have taken into account the currents outside the domain boundary. On the contrary, if S were drawn through a point inside this interval, it would mean that not the whole current on the segment is included. In view of all that, the relation (1.2.49) (Fig. 7) may be rewritten in the form
0 0
J (t l/c R0 /c) J (t l/c R0 /c) A = 0 ( ).(1.2.52) R0 R0 t 4
Thus the Lorentz condition for a segment mentally selected on a straight line, outside which there is a current, is invalid. However, for real finite antennas, where the currents are reflected from the ends, the surface S can always be selected outside the antenna, so that the Lorentz condition will be satisfied. Let us use the theorem from Ref. [33] for a more thorough analysis of the E field calculation in [15] and [16], where it was found from the Maxwell equation
E H = 0 j , (1.2.53) t
1 and the field H = A is determined from the solution in the form of the delayed vector
0
potential A . As E = A/t , Eq.(1.2.53) is equivalent to
2 A ) = A 0 0 2 j .(1.2.54) ( A 0 0 t t
Since the right-hand side of (1.2.54) is zero, the method of finding E from known H [15,16] is equivalent to solving the D'Alembert equation for the vector potential A and satisfying the Lorentz condition, from which the scalar potential is found and after that E . However, by (1.2.52) the Lorentz condition for a segment is not satisfied, therefore the suggested method of finding E is not quite correct. In [17] ([17], Eq.(1)) the current density in our notation has the form
40 j = n1 ( x ) ( y ) J ( t z /c )[ ( z a ) ( z b )].(1.2.55 )
The authors of [17] assume that the current appears at the point z = a and is entirely absorbed at the point z = b after traversing the wire. Evidently, with such problem setting one can point out such a surface S outside the segment [a, b] for (1.2.49) that the condition (1.2.50) will be satisfied. A substitution of (1.2.56) into (1.2.49), with the obvious equality /t * = /t gives:
A = t 0 1 J = {( )[ ( z a ) ( z b)] 4 r t z
0 0
J (t * z/c)[ ( z a ) ( z b)]}dz =
=
1 0 1 { J (t Ra /c a/c) J (t Rb /c b/c)}, (1.2.56) 4 Ra Rb
J = 0 due to (1.2.2), (1.2.10) and (1.2.20). t z We conclude that the representation of j in the form (1.2.55) does not lead to vanishing right-hand side of (1.2.49) due to continuity violation at the ends of the segment [a, b] . In Ref. [17], to find the radiation field from a current on a segment [a, b] , the D'Alembert equation for the vector potential A was solved and a solution was found in a standard way in the form [17], Eq. (2)]. However, instead of solving the D'Alembert equation for the scalar potential , Ref. [17] employed the Lorentz condition, which, as we showed in (1.2.56) is not valid. Thus we have revealed the cause of the ``paradox'' when the authors of [16] and [17] obtained the same result for the E radiation field of a linear antenna, while the result of [14] and ours coincided with each other but differed from those of [16] and [17]. It is curious to notice that, although the Lorentz condition violations in [16] and [17] had different causes (in [16] it is connected with the nonzero value of the last integral in the right-hand side of (1.2.49) and in [17] with that of the first integral), formulae (1.2.52) and (1.2.56) turned out to be equivalent. This has led to the equivalence of the results of [16] and [17]. To continue the discussion of the topic, we present a solution to the problem mentioned in [17] and presented in Fig. 1 of Ref. [17]. Let, for simplicity, J (t ) = e (t ) where e is a certain ``charge'' and v = c . We are solving the problem by two methods: 1. The standard one. 2. Ours. 1. From the solution [17, Eq. (11)] we find:
where in (1.2.56)
Ak = Azh =
0ce k 3 4 ct x3
( (t a/c Ra /c) (t b/c Rb /c )).(1.2.57)
41
Similarly for the scalar potential
A4 =
i i0ce 1 = c 4 ct x3
( (t a/c Ra /c) (t b/c Rb /c )).(1.2.58) Calculating in a standard way Ek = Ak /t /xk , we find
Ek = 0e[
ct z
(t a/c Ra /c)
= 0e[
(t b/c Rb /c)
ct z
( k 3
( k 3
Rb ) xk
Ra )] = xk
(t b/c Rb /c) R ( k 3 b ) Rb (1 cosb ) xk
(t a/c Ra /c) R ( k 3 a )].(1.2.59) Ra (1 cosa ) xk
2. From Eq. (1.2.22) of the present work, taking into account (1.2.15), (1.2.16) and K = 0 along with J = e (t ) , we find
c (t b/c Rb /c) Ek = F4 k = 0e[ ( k 3 mkb ) i Rb (1 cosb )
(t a/c Ra /c) ( k 3 mka )].(1.2.60) Ra (1 cos a )
Formulae (1.2.59) and (1.2.60) are identical. Let us calculate A /x using (1.2.57) and (1.2.58):
A /x =
0e (t a/c Ra /c) (t b/c Rb /c) ( ).(1.2.61) Ra Rb 4
Comparing (1.2.61) and (1.2.56) for J (t ) = e (t ) , we assure their identity. Ref. [17], instead of (1.2.58), used for finding the Lorentz condition, which is invalid due to (1.2.61). That is what has led to the erroneous, in our view, formula [17, Eq.(16)]. Concluding the discussion, we would like to note:
42
1.As follows from the field superposition principle, which is in turn a caused by the linearity of the Maxwell equations, the electromagnetic field of the whole system is a sum of electromagnetic fields of each of its parts taken separately. 2.The Lorentz condition, being valid for the system as a whole, is not necessarily valid for each of its parts taken separately. That is just the cause of the coincidence of Eqs. (1.2.40-1.2.42) of the present paper with Eqs. (8a-c) from [16] and the non-coincidence of (1.2.38) and (1.2.39) with the similar formulae (5b) and (5c) from [16]. 3.The scalar ( ) and vector ( A ) potentials are auxiliary quantities of no direct physical meaning. Evidently, in a gauge transformation A = A /x , which leaves the field tensor F (1.2.11) unchanged, the function can be chosen in such a way that the Lorentz condition for A be satisfied. 4.Dealing with radiation fields from closed structures, one can use the solutions for delayed or from the vector potentials A and find the scalar potential either as a scalar delayed potential, Lorentz condition. There is no such arbitrariness for unclosed structures. Both A and must be found from the D'Alembert equation. 5.As finite structures are always considered in the practice, the result of field summing for the whole system does not depend on the calculation method, although for separate structure elements it does.
1.2.5. Application of the results to calculation of complex wire structures Obtained formulae (1.2.23a) and (1.2.24a) can be used for wire EMP simulator design. A typical element of an EMP simulator is a curvilinear wire placed over a perfectly conducting plane. One of the wire ends is connected with a pulse generator, the other one, through a load, with the conducting plane. It is convenient to use for analytic and numerical calculations the equivalent system depicted in Fig. 9 where the conducting plane is replaced by a wire reflected in it.
Fig. 9. Geometry of the EMP simulator typical element The current in such a system can be found by solving Eqs. (1.2.6) for r1 = r2 , Esi = 0 , using, for instance, the Laplace transformation.
43
We use the following relations valid at the terminals of the generator and the impedance R1 :
u0 (t ) = u0 f (t ) (t ); u0 (t ) = R0 J (t ,0) u(t ,0); u(t , l ) = J (t , l ) R1 , (1.2.62) where l is the length of the upper part of the wire, from the generator to the load, R0 is the generator resistance and R1 is the load resistance. Omitting standard intermediate computations, we can write down the result as follows: J ( s, t ) =
u0 s 2ln s 2ln ( 1 0 ) n [ f ( t ) (t ) W R0 n =0 c c
1 f ( t
1 =
2l ( n 1) s 2l ( n 1) s ) (t )], c c
R1 W R W 1 , 0 = 0 , W = ~ .(1.2.63) R1 W R0 W cC
Evidently, for t , f (t ) = (t ) Eq. (1.2.63) yields: J ( s, ) =
u0 u0 ( 1 0 ) n (1 1 ) = , (1.2.64) W R0 n =0 R1 R0
which coincides with the full-circuit Ohm law, so that W is dropped from the formula. The travelling-wave regime is determined by the condition 1 = 0 , i.e., R1 = W , ( , 0 )0 = 1 . In this case
J ( s, t ) =
u0 f (t s/c) (t s/c).(1.2.65) W R0
The contribution of the wires from Fig. 9 to the electromagnetic field can be obtained as a sum of the fields from the wires ABC, ABC , CDF, CDF using Eqs. (1.2.23a) and (1.2.24a). For practical purposes it is the electromagnetic field value on the conducting surface that is of interest. Therefore let us consider a point on the x2 axis, for which we find: H1 =
1 1 { [ J (t r/c ) J (t l1/c R1/c )] 2 h1
1 [ J (t l1/c R1/c ) J (t (l1 l2 )/c R2 /c )] h2
1 [ J (t (l1 l2 )/c R2 /c ) J (t (l1 l2 l3 )/c R3/c )] h3
1 [ J (t r/c ) cos J (t l1/c R1/c ) cos 1 ] h1
1 [ J (t l1/c R1/c ) cos 1 J (t l1/c l2 /c R2 /c ) cos 2 ] h2
1 [ J (t l1/c l2 /c R2 /c ) cos 3 h3
44
J (t l1/c l2 /c l3/c R3/c) cos ]}, (1.2.66) where h1 , h2 , h3 are the ``impact'' spacings. If, in particular, J ( s, t ) = J 0 (t s/c ) , then for t > (l1 l2 l3 R)/c , i.e., for a stationary regime, Eq. (1.2.66) corresponds to the magnetic field H calculated according to the Biot-Savart law. We would note that the wire geometry in Fig. 9, with different relations between the angles and the arm lengths l1 , l2 , l3 allows one to embrace the geometries of many existing coupled-wave simulators, including the radiating ones, which include linear and V-antennas as elements. The E field will have only one nonzero component E3 at a point on the x2 axis. We have: cos 1 [ J (t r/c ) J (t l1/c R1/c )] E3 = 0 { 2 h1 cos [ J (t l1/c R1/c ) J (t l1/c l2 /c R2 /c )] h2 1 cos [ J (t (l1 l2 )/c R2 /c ) h3 1 J (t l1/c l2 /c l3/c R3/c )] | h3 | 1 [ J (t r/c ) cos( 1 ) J (t l1/c R1/c )] h1 1 [cos(1 ) J (t l1/c R1/c ) h2 cos( 2 ) J (t l1/c l2 /c R2 /c )] 1 cos( 3 ) J (t l1/c l2 /c R2 /c ) h3
1 J (t l1/c l2 /c l3/c R3/c)}.(1.2.67) | h3 |
For instance, for = = we have l1 l2 l3 = l , h1 = h2 , which corresponds to a rhombic antenna (Fig. 4). A rhombic antenna is a constituent of an EMP simulator considered in Ref. [18] ([18], Fig. 6). The obtained formulae can be used to calculate the wire EMP simulators. For example, EMP simulator [18] consists of many curvilinear wires placed over a perfectly conducting plane. One of the wire ends is connected with a pulse generator, the other one, through a load, with the conducting plane (Fig. 10). The theory [17] is used in ref.[18] as a basis formula to calculations. The electric field of one of the simulator wires is obtained from (1.2.24a) at s1 = 0, s2 = 2l , s0 = l . The basic formula used for one of the wires used in [18] is [18, Eq.(16)], which in our notation has the form
J (t r/c) n m ( m n n m( m n [ E1 = 0 ].(1.2.68) 4 1 n m 1 n m r
45
Fig. 10. Geometry of the rhombic EMP simulator The contributions to the total field from the upper and lower wires in (1.2.68) differ from those of similar wires in (1.2.30) but the summed fields in (1.2.,30) and (1.2.68) coincide identically, as follows from item 5 of the discussion at the end of the previous section V. To calculate the EMP simulator [18] we shall use the formulae (1.2.24a) taking into account the wire fractures. The calculation was carried out in the travelling-wave regime using the formulae obtained from (1.2.24a) by the substitutions: s0 l0i , R0 R) i , n1 n1i , m0 m0i , n2 n2i , s2 2li , R2 R2 , m2 m2 , s1 = 0 , and similarly for the primed quantities. Later on, summing over all wires, we find: J E (r , t ) = 0 0 4
N
{ i =1
f (t l0i /c R0i /c ) n1i m0i n2i m0i ( ) 1 m0i n1i 1 m0i n2i R0i
f (t l0i /c R2 /c) n2i m2 n2 i m2 ( ) R2 1 m2 n2i 1 m2 n2 i f (t r/c) n1i m n1i m ( ) r 1 m n1i 1 m n1i
f (t l0i /c R0i /c) n1i m0 i n2 i m0 i ( )} R0i 1 m0 i n1i 1 m0 i n2 i
i /2 i
/2
F ( ,
N a )d , (1.2.69) N 1
where, in accordance with [18], we have introduced
a =
2 N 1 2 a 0 ; = = 0 ; i = (i 1) a = N N 1 N
46
0
=
N
(2i 1 N ),
F ( , 0 ) =
1
02 2
.(1.2.70)
Calculation of the integral in (1.2.69) gives:
i /2 i
/2
F ( , 0 )d =
1
[arcsin
2i N 2i 2 N arcsin ].(1.2.71) N N
From Fig. 10 we find for the i -th wire and its mirror image the following relations:
n1i = cosi cos0 e1 sin i e2 cosi sin 0 e3 , n1i = cosi cos0 e1 sin i e2 cosi sin 0 e3 , n2i = cosi cos0 e1 sin i e2 cosi sin 0 e3 ,
n2 i = cosi cos0 e1 sin i e2 cosi sin 0 e3 , l0i = l0 / cosi = L/(2 cos0 cosi ).(1.2.72) The remaining values of the vectors and other quantities in (1.2.69) depend on the coordinates of the observation point. To determine the constant J 0 , it is necessary to know the input impedance Z . To find it, we will use the result [18] or a similar expression obtained in [37]. In particular, for the EMP simulator ([18], Fig. 6), consisting of 76 wires, we find from [18]: Z = 78.8 Ohm. Hence it follows: J 0 = u0 /Z , u(t ) = u0 f (t ), u0 being a known voltage at the simulator input. On the basis of (1.2.69) we made up a computer program for calculating the EMP simulator fields. The computational results are presented in Figures. 11a--11d. Fig. 11a presents the E z component of the electric field on the conducting surface at the center of the simulator, where the fields of the wire fractures and their mirror images cancel each other. Therewith the E z pulse shape repeats that of the input voltage. At other points inside the EMP simulator the radiation fields from the fractures and their mirror images do not cancel each other. Fig. 11b presents the E field components at the point ( x = 142.5 m, y = 0 , z = 40 m). It follows from the symmetry that E y = 0 . The computation results of Fig. 11b may be easily explained on the basis of simple geometric considerations. As follows from [18, Fig.6], the point under consideration is at a distance of 148 m from the generator, therefore the field at this point emerges in 493.3 ns. The length of the central wire up to the fracture is 155.5 m and the spacing between the central wire fracture and the observation point is 22.3 m. Accordingly, the signal from the fracture reaches the observation point at the time instant 592.7 ns.
47
Fig. 11. The calculation results of rhombic EMP simulator The lengths of the extreme wires up to their fractures is 172.6 m and the observation point is spaced from these fractures by 78.1 m, therefore the field from the extreme wire fractures comes to the observation point at the instant 835,7 ns. Thus, between 593 and 836 ns the E x component of the TEM wave is superposed by the field form the fractures. The mirror image of the central wire fracture is spaced from the observation point by 102.3 m and the length of the image of the central wire is 155.5 m. Thus the signal from the image of the central wire fracture comes to the observation point in 859.3 ns. The E x field components from the fracture and its image have different signs. This leads to an increase of the E x component between 859.3 ns and 998 ns. The instant 998 ns corresponds to the arrival of signals from the images of the extreme wire fractures at the observation point. The calculation presented in Fig. 11b is in perfect agreement with these elementary considerations. The results presented in Figs. 11c and 11d can be explained in a similar way. Fig. 12e presents the 377 H y component of the magnetic field and the E z component of the electric field at the center of the simulator on the conducting surface. At the simulator center, E y = 0 due to the symmetry of the system and E x = 0 because the radiation fields from the wire fractures and their images mutually cancel each other. The magnetic fields of the fractures and their images have the same sign at the simulator center, leading to a violation of the equality E z /377 = H y valid for a TEM wave.
48
Fig. 12. The calculation results of rhombic EMP simulator, continuation It is obvious from geometric considerations that in 726 ns signals from the central wire fracture and its mirror image arrive and those from the extreme wire fractures and their images arrive in 900 ns. Their summing results in the fields depicted in Fig. 12e. Fig. 11a coincides with its counterpart [18]. Other figures coincide qualitatively but there is no coincidence on the time scale. The latter circumstance seems rather strange since the time coordinates of the fracture-point signals are easily obtained from the above simplest geometric considerations. Fig. 11f presents the fields at the point x = 100 m, y = 0 , z = 0 and Fig. 11g those at the point x = 185 m, y = 0 , z = 0 . As is easily understandable from physical considerations, knowing the Pointing vector direction and the direction of the magnetic field due to a fracture at the observation point, the jump of the derivative of E z at x = 100 m is opposite in sign to that of the derivative of H y , while at x = 180 m the changes of the derivatives of E z and H y are equally directed. Fig. 12g for E z is
analogous to [38]. So far we have been leaving aside the problem of accuracy of the method suggested. The results making possible the corresponding estimation to be done, are given in Ref. [39]. First, it is shown there by a numerical solution of the EFIE set that the current distribution in the input wire TEM horn of the EMP simulator corresponds to the analytically calculated current distribution in a solid TEM horn. This enables one to specify the current distribution in the wires of the input TEM horn. Second, it has been shown that the current magnitudes along the wires in the real EMP simulator geometry are constant within 6 %, despite the flexures and the interaction of currents.
49
Taking into account the possible additional error connected with the variations in the EMP simulator geometries, the upper bound of the error of the method can be estimated as 10 %. 1.2.6. On pseudoparadoxes in the classic theory of radiation from subsystems
The present part is devoted to an analysis of some vaguenesses which are hitherto found in scientific publications, while considering the problems of radiation from subsystems [105]. Different authors, solving one and the same problem, arrive at different results. And vice versa, solving different problems, they obtain identical results. Among such problems are, e. g., a calculation of transient radiation of travelling waves by wire antennas. As known, while solving the problems of radiation in the classical theory used are the d'Alembert equations both for a vector potential A and a scalar one . With the aid of these potentials an electric field E and a magnetic one H are calculated by the formulae A , E = t
B = 0 H = A.(1.2.73)
If one imposes the Lorentz conditions on the potentials
0 0
A = 0, (1.2.74) t
then from Maxwell's equations the d'Alembert ones are obtained
1 2 A 1 2 A 2 2 = 0 j , 2 2 = .(1.2.75) 0 c t c t The current density j and the charge one are connected with the continuity equation expressing the charge conservation law.
j = 0.(1.2.76) t The above formulae are contained in any course on field theory but they are necessary for references. The solution to the d'Alembert equations (1.2.75) both for the vector potential A and the scalar one generally comprises three terms
A = A1 A2 A0 , (1.2.77) where A1 is the retarded potential, A2 is the advanced one, A0 is the general solution to the homogeneous equation (1.2.75). Similarly for the scalar potential = 1 2 0 .(1.2.78)
50
To solve the problems of radiation, solutions in the form of retarded potentials are usually used. Advanced potentials are omitted as contradicting the causality principle, and the solutions A0 and 0 are identified with an external field acting on the system [5]. Having represented the solutions to the d'Alembert equations in the form of retarded potentials, Stratton [33] proved that the Lorentz conditions were satisfied. The latter appeared to be satisfied in a closed system, provided the charge conservation law is valid. Hence, in all real systems, where charges and currents are set in a finite space domain, the solutions to Maxwell's equations and those to the d'Alembert ones in the form of retarded potentials are completely equivalent. There arises another situation, while considering the subsystems being part of the closed system. In real physical problems, while considering fields from the whole system, one has to divide the system into a finite number of subsystems, which is exemplified by any numerical calculations. In [1.2.(49)] on the basis of [33] analyzed is the expression 0 0 A = 0 4 t
1 0 j V r [( j )t*=const t * ]dv 4 V ( r )dv.(1.2.79)
For closed systems the first integral in (1.2.79) vanished due to satisfying the continuity equation. The second integral vanished due to the Gauss theorem and the choice of a surface bounding the volume of a closed system at such far distances that the flux j/r through the surface S should vanish. If one considers part of the radiating system (a subsystem), then the second integral in (1.2.79) will not vanish since the currents through the surface bounding the subsystem volume are nonzero. Hence, while the continuity equation is satisfied, relation (1.2.79) reduces to the form 0 j n 0 0 A = ( )dS 0.(1.2.80) t 4 S r
Thus for a subsystem the Lorentz condition are not satisfied. Prove the statement: For radiating susbsystems the solutions to the d'Alembert equations in the form of retarded potentials are allowed not to satisfy Maxwell's equations. For radiating subsystems the Maxwell and d'Alembert equations may prove to be inconsistent. It is the d'Alembert equations but not Maxwell's that should be considered " primary". Proof. While deriving a particular solution to equations (1.2.75), all space is divided into infinitesimal sections, and defined is a field being created by a charge in one of such volume elements [5]. Due to linearity of the equations the total field from the whole closed system is equal to a sum of fields being created by all such elements. The field from a subsystem incorporates part of system particles. If a solution in the form of retarded potentials is applicable for each radiating "physical" particle of the system, then it is evidently applicable for each of the particles of the subsystem as part of the system. From this it follows that the validity of the d'Alembert equations for the system involves their validity for a subsystem. But for the subsystem from (1.2.79) it follows that the Lorentz conditions are violated. Since the d'Alembert equations (1.2.75) are obtained from Maxwell's using the Lorentz conditions (1.2.74) and formulae (1.2.73), and conditions (1.2.74) have been broken, then from the validity of the d'Alembert equations it follows that Maxwell's are not valid.
51
E H 0 j .(1.2.81) t The solution concerning the radiation of linear antenna, we have considered, drew a sharp criticism in the scientific press, in our opinion, without solid grounds. In the recent comment on our paper [41] derived is the statement, implicitly containing in (1.2.53), (1.2.54), of the Lorentz conditions not being satisfied results in Maxwell's equations not being satisfied as well, if the d'Alembert equations for the vector potential A are valid. This is really a plight, if formulae [41, (1)-(4)] are referred to the system as a whole but not to a subsystem as we do. So the criticism based on [41, (1)-(4)] misses. For example, relations (1.2.37)-(1.2.39) for a subsystem do not satisfy Maxwell's equations and the Lorentz conditions (1.2.74), but satisfy relations (1.2.73) and the d'Alembert equations (1.2.75) for A and in the form of retarded potentials. On the other hand, to the whole system, i. e. For antenna with finite length taking into account the current reflection from the antenna ends (1.2.40) -(1.2.42) the Maxwell's equations, the d'Alembert ones and the Lorentz conditions are satisfied. Hence, however, formulae (1.2.40)-(1.2.42) are correct, and formulae (1.2.37)-(1.2.39) are incorrect. Both relations were obtained by one and the same method, using a solution in the form of retarded potentials. We consider both solutions to be true. Therefore the paradox that formulae (1.2.37)-(1.2.39) satisfy (1.2.81) is really a pseudoparadox. This should be the case for radiating subsystems. While analyzing papers [13-17], one has to encounter one more "paradox" when solution to one and the same problem leads to different results (obtained by different authors), and vice versa, when the solution to different problems leads to identical results. These questions are partly investigated here, however, the appearance of paper [41] indicates that there is no clarity in these questions. In papers [13-16] an antenna of length 2L is considered, and the current in the antenna is assumed to vary following the law
I ( z, t ) = I 0 (t | z | /c ), | z | L, (1.2.82)
i. e. the current is a travelling wave. The solutions for the fields E and H found in [17] are interpreted as solutions for which the current wave is completely absorbed at the antenna ends. The formulae for the magnetic field obtained in papers [14] and [16] exactly coincided with formula (1.2.37). The same result for the magnetic field also follows from formula in Ref. [15] after some corrections. The most curious in this situation is the fact that in here solved was a problem other than that in [14-16]. Formula (1.2.37) expresses a contribution, to the field, of the current on the segment [ L, L] of infinite antenna, where the current is a continuous function of coordinates at the segment ends. The current also exists outside the segment, and there is no absorption at the segment ends. If one is based on the Liénart-Wiechert solution [5] or the charge model [20], in some sense, equivalent to it, then a coincidence of the expressions for the magnetic field in essentially different problems seems to be very strange. On the basis of the charge model, absorption of the wave current at the antenna ends should result in emergence of "charge" acceleration, and hence, radiation. In (1.2.37) there is no "charge" acceleration. The last two terms in (1.2.37) are not due to a "charge" acceleration at the segment ends. The end point of the segment is an interface between the "flow" and the "source" lying on both sides from the boundary point. The total contribution to the field from the "flow" and the "source" is zero for each of the arms. But since, by definition, we take account of a contribution to the field only from the "flow" of one of the arms and from the "source" of the other arm, there appear terms mentioned above. Consider how the general formulae (1.2.21),(1.2.22) alter for a piece of curvilinear wire, if in accordance with [17] the current in the wire is taken in the form
52
J = J ( t s/c ) ( s, s1 , s2 ),
( s, s1 , s2 ) [ ( s s1 ) ( s s2 )].(1.2.83 )
Here ( s, s1 , s2 ) is a pulse of unit height and width s2 s1 . We shall consider the wire length to vary from 0 to , and we shall calculate a contribution, to the field, of a piece of length s2 s1 . The substitution of (1.2.83) into formula (1.2.82), with regard to (1.2.20, gives A =
0 4
0
I (t s/c R/c ) ( s, s1 , s2 ) ds, R
I k = Jnk , I 4 = ic , 0 < s1 < s2 < .(1.2.84) Using the same calculation procedure, we obtain expressions for the electromagnetic field tensor
JK ( m p N p ) Fkl = 0 {M kl [ m[ k nl ] R (1 R/s) 2 4 0
JKm[ k N l ] ]}( s, s1 , s2 )ds.(1.2.85) R(1 R/s)
Using the obvious equalities
( s, s1, s2 ) M kl =
J( s, s1, s2 ) m[ k nl ] ] [ s R(1 R/s)
J ( s s1 ) m[ k nl ] R(1 R/s)
J ( s s2 ) m[ k nl ] , (1.2.86) R(1 R/s)
taking into account that (0, s1 , s2 ) = ( , s1 , s2 ) = 0 , we find Fkl =
0 J (t s2 /c R2 /c ) m2[ k n2 l ] { 4 R2 1 m2 n2
J (t s1/c R1/c ) m1[ k n1l ] R1 1 m1 n1
53
s2
[ s1
JK ( m p N p ) m[ k nl ] R (1 R/s) 2
JKm[ k N l ] ]ds}.(1.2.87) R(1 R/s)
In the same way from (1.2.84) we find the expression
F4 k =
0i J (t s2 /c R2 /c) n2 k m2 k { 4 R2 1 m2 n2
J (t s1/c R1/c) n1k m1k R1 1 m1 n1 s2
[ s1
JK ( m p N p ) ( nk m k ) R (1 R/s) 2
JKNk ]ds}.(1.2.88) R(1 R/s)
Expressions (1.2.87) and (1.2.88) coincide exactly with expressions (1.2.21), (1.2.22). It is evident that the break of the function J in (1.2.83) on the boundaries s1 and s2 is due to multiplication of the function J (t s/c) , which is considered to be continuous for a positive argument, by a discontinuous function ( s, s1 , s2 ) . Relation (1.2.49) for the function (1.2.83) for a semi-infinite curvilinear thin wire may be represented in the form
0 0
A = 0 t 4
1
r ( t ( ( s, s , s )) 1
2
0
( J ( s, s1 , s2 )))d s s
0 ( J ( s, s1 , s2 )) d s = 4 0 s
1 J = 0 ( ) ( s, s1 , s2 )d s 4 0 r t s
54
=
0 4
0
J (t * s/c ) [ ( s s1 ) ( s s2 )]d s = r
1 0 1 { J (t R1/c s1/c) J (t R2 /c s2 /c)}.(1.2.89) 4 R1 R2
While deriving (1.2.89), we took into account that (0, s1 , s2 ) = ( , s1 , s2 ) = 0.
In [17] the formula for a current similar to (1.2.83) was based on the assumption that "the current arises at the point z = a and is completely absorbed at z = b after traversing the wire". From an identical coincidence of the expressions for the electromagnetic field (1.2.87), (1.2.88) and formula (1.2.89) with the corresponding expressions (1.2.21), (1.2.22), (1.2.52), (1.2.56) one may arrive at the conclusion that two different models actually coincide. Thus, from our viewpoint, for the current law in the form (1.2.82) or (1.2.83) in [14-17], another problem, but not that was formulated by the author, has been solved. It is not the problem of radiation of a linear antenna of finite length with a complete absorption of the current at the ends that was solved in [14-16] but the problem of radiation from part of length 2L of infinite antenna. In [17], from our viewpoint, the current does not arise at the point z = a and is not absorbed at the point z = b at all but there is only an explicit taking into account of a contribution of the current on the segment of length b a to the retarded potential. From our viewpoint, the current is a continuous function at the points a and b and exists outside the segment. The multiplication of a continuous current by a discontinuous function ( z, a, b in [17] actually contributes to the field from the current at a finite segment set on an infinite straight line. The assumption that the current arises at the point a and is completely absorbed at the point b contradicts the charge conservation law. The authors of [13] treat the formula for the current as a travelling wave with a complete absorption at the ends of the antenna of length 2L . We treat the solution for the electric field as a contribution of the current on the segment 2L of infinite antenna taking no account of the remaining part (1.2.38), (1.2.39). Formulae [13], with regard to a complete reflection of the current wave from the antenna ends, cast no doubt and are adequate to formulae (1.2.41), (1.2.42). In [14], in our opinion, the problem of finding the electric and magnetic field on the segment 2L of infinite length is solved correctly, but not the problem for the current of the antenna of finite length 2L , as the author contends. As known, the potentials A and are auxiliary quantities which have no immediate physical sense. It is the field intensities E and H , or what is the same, the electromagnetic field tensor F (1.2.11). that have physical sense. For the fields E and H one may also select other potentials A and which leave the fields E and H , i. e. the field tensor F , unchanged. The invariance of the fields E and H with respect to various types of gauges is called gauge or gradient invariance [5]. It is evident that transforming the 4-potential A in accordance with the rule
55
A A
, (1.2.90) x
where (r , t ) is an arbitrary function, does not change the antisymmetric field tensor, and hence the vectors E and H as well. The gauge invariance allows one to make use of some arbitrariness of the potentials and choose them so that the field theory equations should reduce to the simplest form. In particular, one may always choose the function such that the Lorentz conditions (1.2.74) should be satisfied for the new 4-potential A ' . It is clear that for a subsystem, where the Lorentz condition (1.2.80) is not satisfied, one may, in principle, using (1.2.90), select the function so that the Lorentz condition should be satisfied. But since the fields in so doing do not change, we shall not take it up. It is much more interesting to clarify the question of how the potentials should be gauged to link, e. g., the solutions [16] with the solutions (1.2.37)-(1.2.39)]. Since for the magnetic field expressions [16] and (1.2.37) coincide identically, we transform only the scalar potential to find this correspondence. First consider the general case of the transformation for the subsystem being a piece of curvilinear wire. Introduce a new scalar potential for which the Lorentz conditions (1.2.74) will be satisfied. Let the new scalar potential be related to the old one by the formula
= .(1.2.91)
The substitution of (1.2.91) into (1.2.74), with regard to (1.2.89), results in the equation for
0 0
1 1 = 0 { J (t R1/c s1/c) J (t R2 /c s2 /c)}.(1.2.92) t 4 R1 R2
Integrating (1.2.92), we obtain
=
1 4 0
{
1 1 Q(t R2 /c s2 /c) Q(t R1/c s1/c)}, R2 R1 t
Q (t ) = J (u )d u.(1.2.93)
From the requirement that should satisfy the d'Alembert equation (1.2.75), we find
1 2 = .(1.2.94) 2 2 0 c t
From (1.2.94), (1.2.95) and (1.2.91) we have
56
1 2 = 0.(1.2.95) c 2 t 2
Since the potential transformation does not belong to the type of (1.2.90), it should lead to changing the electromagnetic field tensor. This transformation in the four-dimensional form reads
i A = A 4 , (1.2.96) c where A is defined in (1.2.84). The calculation of the electromagnetic field tensor F , based on (1.2.96) and (1.2.11), gives
F4k =
i0 cmk cm [ 2 Q(t R2 /c s2 /c) 2k Q(t R1/c s1/c) 4 R2 R1
mk m I (t R2 /c s2 /c) k I (t R1/c s1/c)] F4 k , (1.2.97) R2 R1
where F4 k is calculated by formula (1.2.88). The spatial components of the field tensor Fkl = Fkl i.e. remain unchanged and are determined from (1.2.87). Analyze the expressions obtained. Relation (1.2.97) applied, in particular, to a segment of linear wire K = 0 after simple transformations reduces to formula [17] or equivivalent formulae (1.2.44), (1.2.45). In (1.2.46), (1.2.47) it is shown that from [17] easily derivable are relations [16] which were obtained by integrating Maxwell's equations
E H = 0 j (1.2.98) t
over the found vector field H that was obtained from formula (1.2.73) where A satisfied (1.2.75) whose solution corresponded to the retarded vector potential. Based on the calculations performed, we can obtain the result [16], using (1.2.91), having substituted by = , where is calculated in accordance with (1.2.93) and satisfies a d'Alembert homogeneous equation (1.2.95). It is evident that from (1.2.93) is really a solution to (1.2.95) and corresponds to the value 0 in formula (1.2.78). Thus, to obtain the electric field [16] in terms of d'Alembert equation solutions, the scalar potential should be a sum of the retarded potential and the external field = 0 . The vector potential A is retarded and contains no external field. A violent (disconnected with field gauge invariance) imposing of the Lorentz condition (1.2.91)--(1.2.97) results in changing the electric field with leaving the magnetic one unchanged. While deriving (1.2.87), (1.2.88) or (1.2.21), (1.2.22), the current J on the curve (straight line) was considered to be a continuous function existing outside the segment. The natural assumption that the transformation (1.2.96) means a transition to considering a
57
piece of wire with absorption (emergence) of a current at the wire ends is wrong. The transformation (1.2.96) leaves the magnetic field unchanged, and hence formula (1.2.97) may not be treated as the field from a piece of curvilinear (rectilinear) wire of finite length with absorption (emergence) of a current at the wire ends. One and the same current cannot simultaneously be absent (a complete absorption hypothesis) and be present (invariance of the magnetic field) outside the segment of a curve (straight line). The emergence of an additional external electric field and a simultaneous insensitivity of the magnetic field to the "charge" acceleration in the "absorption" contradicts the foundations of electrodynamics. This indicates that the method suggested in [16] for finding the electric field is faulty. As proved in (1.2.53), (1.2.54)] papers [16] and [17] are equivalent. For this reason, formulae [41, (5), (6)] for subsystems are wrong. Instead of making use of the solutions to the d'Alembert equations both for the vector A and scalar potentials and calculating the fields in accordance with (1.2.73), the authors [41], [15], [16] integrate (1.2.98), which provided the conditions (1.2.74) are not satisfied results in a conflict with the solution for the retarded potentials from (1.2.75). Apart from the retarded potentials, the integration of (1.2.98) led to emergence of fictitious sources being a general solution to the homogeneous equation for the scalar potential from (1.2.75) in the form 0 from (1.2.78). It is curious to note that, recognizing the validity of the d'Alembert equations for the vector potential, the authors [41], [15], [16] ignore the d'Alembert equation for the scalar potential. The latter seems to be very strange, since, while deriving the d'Alembert equations from Maxwell's for a closed system, using (1.2.74), equations (1.2.75) are always obtained in a pair. And the validity of a vector equation implies the validity of a scalar one. In our opinion, the problem with a complete absorption cannot be consistently solved in the framework of the travelling wave model, which itself is rather approximate. In [42] the authors on the basis of formalism [11] calculated the electromagnetic field by travelling wave current propagated in thin curvalinear wire with free ends. In our opinion the solution obtained which is exactly equivalent to (1.2.97) do not corresponds to proposed model. In [42] the substitution one problem on another was occured. Actually the other problem was solved but not one formulated by authors. Instead of radiation problem of finite length curvalinear antenna with free ends the problem of radiation by part of infinite antenna wire and two external point sources arisen when forced demanding of satisfying a Lorentz condition for subsystems where this condition does not operate was solved. It should be noted that the criticism of paper [11] is groundless in many points. The accusation that the theory [11] is inapplicable to antennas of finite length is erroneous. An impression is forming that, having noticed formulae [11] (1.2.37 - 1.2.39) the authors of the article being commented did not pay attention to formulae [11], (1.2.40-1.2.42) where the problem of radiation of an antenna of finite length is just considered. The phrase in the in introduction "In [11], the authors did not consider the accelerated or decelerated effect of the charges at the ends of the finite linear antenna, therefore their results were wrong..." gives rise to perplexity. If the formilae [11] (1.2.37 - 1.2.39) are meant, then it is clear that there are no accelerations of the charges at the segment and they could not be in principle! Formulae [11] (1.2.37 - 1.2.39) contribute to the field from a segment mentally isolated on the stright line, where at the ends of the segment the current J is the continuous function of its argument. On the other hand in formulae [11] (1.2.40 - 1.2.42) when reflecting from the antenna end the current vector changes its direction to the opposite one, which is equivalent to emergence of an acceleration. In [11] this procedure had been described in detail, however, the authors did not notice it. Based on the aforesaid, we consider that the method to find the electric field proposed in [42] is erroneous. The appearance of an additional external field and the simultaneous nonsensitivity of the magnetic field to the acceleration of a "charge" in "absorption"
58
contradicts the fundamentals of electrodynamics. The basic result obtained in the present paper is a proof of the statement that for radiating subsystems the main equations are not Maxwell's equations but the d'Alembert ones for the scalar and vector potentials. A substitution of the solution to the d'Alembert scalar equation by integrating Maxwell's equations (1.2.98) over a given magnetic field H results in emergence of fictitious external sources making no reasonable physical sense. Hence the chief objection of the opponents that the field solutions do not satisfy (1.2.98) for subsystems is correct but ineffective. Maxwell's equations (1.2.98) for subsystems should not be satisfied at all! Therefore, the formulae in [41] suggested for calculating the electric field from subsystems, while not satisfying the Lorentz condition, are wrong.
1 E(r , t ) = [ A J ]dt,
A 1 E (r , t ) = Adt t It is these formulae that are a source of "paradoxes". However, falliability of these formulae cannot be established experimentally, since the fields being measured by instruments react on fields of the system as a whole but not on fields from mentally isolated subsystems. For radiating subsystems as a whole, which the experimentalist deals with, Maxwell's equations and the d'Alembert ones are equivalent. In numerical electrodynamical calculations of radiation fields for arbitrary field-forming systems, the system is divided into a finite number of subsystems. The calculation of the electric field for each of the subsystems by formulae [37] are erroneous, however adds up to a correct result. The disadvantages of the method lies in superfluous calculations of integrals for each of the subsystems adding up to zero. As an example, one may refer to paper [18] whose author has obtained results correct in toto, having used the theory [17] for calculations. It is follows from the process investigation of excitement and radiation of thin wire antennas with arbitrary sharp in time domain when arbitrary sharp of excitement and radiating signal: 1. The system of differential equations to excitement of single wire by external electromagnetic field or concentrated voltage source which is similar to telegraph equations for two-wire line [27] has been obtained as a approximation of infinite thin wire. 2. The simple direct calculation method of elecyromagnetic field tensor from travelling current wave without preliminary calculation of delay potentials has been proposed. This enables to find exact analytical expressions for electromagnetic field caused by arbitrary current on a wire segment. Obviously for potentials this is impossible. 3. The diagram of antenna radiation field calculation composed of linear segments taking into account angle radiation has been proposed. For example, the radiation field of V - antenna both infinite and finite length and linear one has been determined. It is shown that infinite V - antenna radiates a TEM wave and approximated current system gives the same result as EFIE. The characteristic impedance of such antenna was obtained. 4. A comparison with the results of other works has been carried out and the reason of "paradox" arisen when considering of current radiation on a linear antenna segments is determined [14-18], [41], [42]. The rise of the "paradox" is connected with the circumstance that authors of works
59
instead of d'Alembert equation solution for scalar potential used the Lorentz condition. We proved that the Lorentz condition which is valid to whole system is not always fulfilled to each separate part. Just on this reason the solution of the most simple problem of linear antenna element radiation by travelling wave current result in differences in [14] and [15-18], [41], [42]. 5. As in practice always the finite wire structures are considered then the Lorentz condition are filfilled. Therefore determination of the scalar potential from the d'Alembert equation for whole system is equivalent to determination of this one from the Lorentz condition taking into account the known vector delay potential A . Thus, the result of field summation for whole system is invariant and for structure elements this one is noninvariant. 6. In proposed method the terms containing the current-time integrals are absent that considerably simplifies the field summation in digital calculating of complex wire structures. The proposed method of complex structures field calculation requires considerably less time then the method of moments [40]. 7. The formulae obtained are convenient to engineering calculations on usual personal computers. Calculating time is a few seconds. 1.2.7. Calculation of radiation field by openings in a free space by means of modified Huygens-Kirchhoff method in time domain
Theory described in previous part is applied in particular for radiation field calculation of horn antennas. The two methods for calculation of radiation field by horn (or horn system) are applied. 1. Each horn is broken to a system of V - antennas. In each one the current is calculated in accordance with exact Yang, Lee model used in paper [18]. The field of each V - antenna is calculated analytically to the theory proposed above and in paper [11]. A program was written for calculation of the fields from the horn system in arbitrary space with an arbitrary excitement voltage. The fields from each horn antenna and horn system are computed using superposition. 2. The modified Huygens-Kirchhoff method for field calculation in far zone to determine the directional pattern was suggested. It allows one to determine the fields directly in time domain for excitement voltage of arbitrary form and permits a rapid estimation of fields expected over a field in array aperture. The great attention is called to radiation by horn antennas both in theoretical and experimental investigations. Many works are devoted to investigation of radiaiton by horns in frequency domain on the basis of well-known Huygens-Kirchhoff principle [33] widely applied in optics. However this principle may be used for electrodynamics too. Generally the electric E and magnetic H intensities are determined at a some field of space T taking into account that field sources outside this field are unknown but the fields E s and H s at restricting surface S are known. Two variants of calculation are possible: the task may be both inside and outside one. For outside task knowledge of surface fields is quite enough for single determination field in T domain. Fourier method is used for solution of this task in time domain for pulse fields. This method demands the numerical calculations and has a bad convergence. That is why the different numerical methods instead of analytical ones are used for pulse fields radiated by horn antenna. It is clear that if we have a horn system (not one horn) then direct numerical calculations are very complicated and their practical application becomes difficult. Hence there arise a problem of simultaneous analytical and numerical calculation of pulse fields radiation. The field calculation model considered in previous part allows one determine the exact field distribution at any space point. However this model is not clearly evident because of the awkwardness of the formulae. That is why the theory of transient radiation on the basis of Huygens method was
60
developed. In principle it is possible to determine the radiation field created by a horn if the electrical current on its surface is known (see previous part). Instead of current it is possible to consider the aperture plane field replaced by a source distribution of Huygens element type. Harmonic oscillations correspond to steady-state electromagnetic processes. It is known that the equivalence theorem is widely applied for calculation of SHF antennas. The essence of this theorem is as follows. Instead of real but unknown field sources inside the antenna the equivalent sources on the antenna aperture are introduced and field vectors are determined. The equivalence theorem is the generalization of known Kirchhoff theorem for scalar fields to vector ones. Let us apply this theorem for the description of pulse TEM-horn radiation with aperture area S. We shall consider on this surface S, that the distribution of the electrical field E and magnetic field H is specified. In the general case it depends on surface coordinates and time. Let us also consider at the external metallic surface of the TEM-horn shell, the tangential components E and H of fields are absent, and on the aperture surface (at the rise of electromagnetic wave) the normal field components E and H are equal to zero. We shall determine the fields at an arbitrary space point by means of Maxwell equations in time domain in accordance with the fields over the aperture surface. As in scientific literature in general the proposed method is applied for frequency domain it is interesting to consider in more detail the derivations of the main relations directly for time domain. Our derivation is based on the modernization of method [43] from frequency domain to time domain. We shall write the Maxwell equations in the form
D B H = e , E = m , t t D = E = 0, B = H = 0,
B = H ,
D = E .(1.2.99)
In (1.2.99) on the basis of reciprocity theorem the fictitious equivalent electric and magnetic current densities are introduced as a sources. e Ё m . At the opening plane the surface densities e and m are connected with electromagnetic fields on the surface by the formulae
e = n0 H s =
1 Es , m = n0 Es = WH s , (1.2.100) W
where W has the sense of surface resistance. Like the case of steady state processes [43] the fields to be determined we shall represent as E = E E ,
H = H H , (1.2.101)
where fields E and H are excited only by electric sources and E and H fields are excited by magnetic ones. If one introduces the potentials A , A , , in such manner that
1 1 A H = A, E = A E = , (1.2.102) t then from the equality to zero of the electromagnetic field normal components in a horn aperture it is follows
61 A H = .(1.2.103) t
Using Lorentz condition to potentials (one consider that current aperture is a closed system)
A = 0, A = 0, (1.2.104) t t we shall obtain the following d'Alembert equations for potentials
2 A 2 A 2 A = e , 2 A = m .(1.2.105) t t
These potentials are connected with fields to be determined E and H by means of next relations
1 1 A H = A ( A)dt , t 1 1 A E = A ( A)dt .(1.2.106) t It is follows from the relations (1.2.105) and (1.2.106) that when substituting , e m , A A we have for fields E H . That is in accordance with Pistolkors principle of permutable duality. Solution of d'Alembert equations is expressed by means of delayed potentials [5].
e ( r, t R/c) A = dS , 4 R
m ( r, t R/c) A = dS , (1.2.107) 4 R
where volume densities of "electric" e and "magnetic" m charges are changed by surface densities e and m using the equality dS = dV . In relation (1.2.107) and m are determined from (1.2.100), R is the distance between surface element of horn aperture dS and view point, r is the radius-vector of "charge" element at aperture plane. To simplify first let us consider the ideal case when aperture field only depends on retarded time but not on r in the aperture. This case corresponds to one when the waves of highest types are absent like in TEM-horn. Of interest is the radiation field at great distances from horn aperture. Let horn radiation surface is located at XOY plane, horn has a rectangular form with size a1 along x axis and b1 along y one with oriigin of coordinates at the centre of horn aperture at O point. At the great distances from aperture (as compared with the horn sizes) the next relation is valid
R = R0 n r , (1.2.108)
62
where unit vector n is directed from aperture centre to observation point and R0 is the distance from aperture centre to observation point. It is known [5], at the great distances from radiating system at denominator of formula (1.2.107) the value n r is negligible as compared to R0 , at the argument t R /c this is not always possible. The possibility of such negligibility is not determined by relative small value ( r n )/R0 as compared to unit. The possibility is determined by rapidity of change e and m with time ( r n )/c . Taking into account these remarks we have the expressions for retarded potentials A = e (t R0 /c (n r )/c)dxdy, 4R0 A = m (t R0 /c (n r )/c)dxdy.(1.2.109) 4R0 Let us choose the field direction at the horn aperture as follows: we shall direct the vector Es of electric field at the aperture plane along Y axis and vector of magnetic field at the aperture plane H s at the reverse side of X axis. Pounting's vector at the aperture will be directed along Z axis. Carrying out the calculations for radiation fields at a wave zone including the terms of equation proportional to 1/R0 and neglecting the terms contained this multiplier at a higher degrees using (1.2.100) and (1.2.109) we obtained j H s j H s t A = dS , dS = 4R0 t t 4R0 t t A i E s = dS , 4R0 t t A A H s t A = i y k y = (i dS z 0 x0 4R0 t z0
H t H H k s dS ); (i cos s dS k cos s dS ), t t x0 4R0c t E E A; ( j cos s dS k cos s dS ), t 4R0c t cos H s cos Es dS , A; dS , A; 4R0c t 4R0c t
63
cos n 2 H s dS , ( A); 4R0c 2 t 2
cos n H s ( A ) dt ; dS , 4R0c 2 t cos n 2 Es dS , ( A); 4R0c 2 t 2
cosn Es ( A ) dt ; dS.(1.2.110) 4R0c 2 t To derivate relations (1.2.110) the next formulae were used
t x y cos 2 = cos cos , sin x0 c cR0 cR0 y x t cos 2 = cos cos , sin y0 c cR0 cR0 x y cos t = cos cos cos cos , z0 c cR0 cR0 cos =
x0 y z , cos = 0 , cos = 0 .(1.2.111) R0 R0 R0
Using (1.2.100), (1.2.106), (1.2.110) and (1.2.111) we find the expression for electric intensity E
E=
1 Es [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ]dS.(1.2.112) 4R0c t
For magnetic field at the observation point we have
H=
1 H s [i ((W/W0 ) cos ) (W/W0 )n cos k cos ]dS.(1.2.113) 4R0c t
In (1.2.112) and (1.2.113) the integration inperformed over the horn aperture plane. R0 is the distance from the aperture center to the observbation point, Es and H s are the values of electric and magnetic intensities in the aperture, i , j , k are the unit vectors along the coordinate axes x , y
64
Ё z accordingly, , , are the direction cosines along the coordinate axes, t is the retardation time calculated in accordance with the formula
t = t R0 /c x cos /c y cos /c, (1.2.114) n is the unit vector directed from the aperture center to the observation point. Vector Es is directed along y axis, and vector H s is directed in the negative x direction. It easy to check that the calculated fields, E and H , are diametrical that is they are orthogonal with each other, and likewise, orthogonal to vector n in the far field. It is important to point out that formulae were obtained at time domain for arbitrary time and coordinate dependence of the aperture plane field. To compare our method with frequency one let us consider the field in the aperture of a rectangular TEM-horn changing sinusoidally with frequency . The real part of this one is represented in the form Es = ( E0eit ) = E0 cos t .(1.2.115) Substituting in (1.2.112) and integrating we shall obtain
E=
1 [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] 4R0c
{E0i exp(i (t
=
R0 x cos y cos ))dxdy} = c c c
E0a1b1 [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] 4R0c
sin(a1 cos/(2c)) sin(b1 cos /(2c)) sin[(t R0 /c)].(1.2.116) a1 cos/(2c) b1 cos /(2c)
Converting in (1.2.116) from Cartesian coordinares to spherical ones, and regarding that W = W0 one can be directly convinced that the result determined identically coincides with [44] which was calculated using Huygens-Kirchhoff method in the frequency domain for radiation from rectangular aperture. Thus the necessary validity condition of the proposed method is fulfilled. For practical purposes of interest was in a step function signal rather than a harmonic one. To simplify the calculation we shall consider that field in the aperture practically independent of the coordinates of the aperture points. We shall choose the law of field changing in a horn aperture as E
s
(t ) = E0 (1 exp( t / ), H s (t ) =
1 E0 (1 exp( t / ), (1.2.117) W
where is the duration of a signal front, E0 is the mean value of electric intensity in aperture
65
obtained below. To integrating over the aperture plane we shall introduce the Cartesian coordinates in which we have E=
a1/2 b1/2 W E0 exp( t/ R0 /c ) dx [ j ( 0 cos ) 4R0c W a /2 b /2 1
1
W0 x cos y cos n cos k cos ] exp( )dy (1.2.118) W c c
Calculating the integral we shall obtain
E a b sinh p sinh k t R E= 0 1 1 exp[( 0 )] k c 4R0c p t R [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] ( 0 p k ) = c
=
E0a1b1 1 exp(2 p) 1 exp(2k ) t R exp[( 0 p k )] c 4R0c 2p 2k
t R [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] ( 0 p k ), (1.2.119) c where the next designations are introduced
p=
b cos a1 cos , k= 1 , (1.2.120) 2 c 2 c
(t ) is the Heaviside's step function which takes into account the delay of signal radiated by different points of the horn aperture when pulse comes to ovservation point. If to consider the field at the horn axis perpendicular to aperture plane we shall obtain cos = 1 , cos = cos = 0 ,
sinh p sinh k = = 1, p k that gives
E ab t R W t R E = 0 1 1 exp[( 0 )] j ( 0 1) ( 0 ).(1.2.121) c W c 4R0c Approximation function proposed in (1.2.117) frequently is used at scientific literature when considering the pulse field radiation, however this function has essential lack - the derivative in zero is differed from zero. This results in step front of radiated pulse and contradicts to experimental
66
investigations. Therefore other approximation function is proposed which is converted to zero in zero and has the zero derivative in zero. Such function frequently is realized on the output of pulse generators. We shall choose the field Es (t ) in the horn aperture as
Es (t ) =
t 1 E0 [(1 cos( )) ( (t / ) ((t / ) 1)) 2 ((t / ) 1)].(1.2.122) 2 t Es (t ) E0 = sin( ) ( (t / ) ((t / ) 1)), (1.2.123) t 2
where is the time of function growth from zero to maximum. For following time values t > function value Es (t ) is remained constant and equal to E0 . Calculation of field E is connected with the calculation of integral
E E = 0 [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] 8R0c
t
sin(
)) ( (t / ) ((t / ) 1))dxdy, (1.2.124)
Which may be represented in the form E E = 0 [ j ((W0 /W ) cos ) (W0 /W )n cos k cos ] 8R0c
exp(
it
)) ( (t / ) ((t / ) 1))dxdy.(1.2.125)
The most simple this integral is calculated at points of far zone located at the horn axis. For this case we have E ab W E = 0 1 1 j [ 0 1] W 8R0c
sin(
(t R0 /c) ) ( ((t R0 /c )/ ) ((t R0 /c)/ 1)).(1.2.126)
To calculate the directional pattern at XOZ plane perpendicular to vector Es we shall determined the field E at = /2 , = /2 . The result may be represented in the form Eb W a /2 (t R0 /c x sin /c ) E = 0 1 j [ 0 cos ] 1 sin ( ) a /2 W 8R0c 1
( ((t R0 /c x sin /c)/ ) ((t R0 /c x sin /c)/ 1))dx.(1.2.127)
67
To calculate the integral we shall substitute
u = (t R0 /c x sin /c)/ , T = (t R0 /c)/ , T0 =
a1 sin , 2 c
u1 = (t R0 /c (a1/2) sin /c )/ , u2 = (t R0 /c (a1/2) sin /c)/ , (1.2.128) then calculation of integral is converted to the form E=
W u E0b1 j [ 0 cos ] 2 sin (u )( (u ) (u 1)) du.(1.2.129) u 8 sin R0 W 1
Evident analytical calculation of integral depends on difference u2 u1 . Let the general condition is fulfilled
u2 u1 = 2T0 =
a1 sin < 1.(1.2.130) c
Integration results in next expressions differed from zero: 1. Let lower limit of integration u1 is not located in the field of variable u , arranged over the range from zero to unit, and upper limit u2 is located in this field. Analytically this case corresponds to inequalities
0 < u2 < 1, u1 < 0, T > T0 , T < 1 T0 , T < T0 , T0 < 1/2, solution of these ones is equivalent to relations
T0 < T < T0 ,
R0 a1 sin R a sin T0 , T < 1 T0 , T < T0 , T0 > 1/2, solution of these ones has the form
T0 < T < 1 T0 ,
R0 a1 sin R a sin < t 1, T0 > 1/2, solution of these ones has the form
1 T0 < T < T0 ,
R0 a1 sin R a sin 1, T0 > 1/2, solution of these ones is equivalent to relations
71
R0 a1 sin R a sin < t T1 , T < 1 T1 , T < T1 , T1 < 1/2,
solution of these ones is equivalent to relations
T1 < T < T1 ,
R0 b1 sin R b sin T1 , T < 1 T1 , T < T1 , T1 > 1/2,
solution of these ones has the form
T1 < T < 1 T1 ,
R0 b1 sin R b sin < t 1, T1 > 1/2,
solution of these ones has the form
1 T1 < T < T1 ,
R0 b1 sin R b sin 1, T1 > 1/2,
solution of these ones is equivalent to relations
T1 < T < 1 T1 ,
R0 b1 sin R b sin < t i1 , then i2 = i j , i1 = j i , (2.14)
where i is out-phase, and j is in-phase currents. Using (2.4, 2.5, 2.11-2.14) and considering that the dielectric plate and environment have i = e = 1 , we find, allowing for equality
i j h s B n dx = L3 L4 (2.15) t 0 t t the expression
u s i j j ( r2 r1 ) i ( r1 r2 ) ( L1 L2 L3 ) ( L2 L1 L4 ) z t t h 11 ( E(ie )1 z E(ie )2 z ) = (1 11 ) B(ie ) ( x, z , t ) n dx , (2.16) 0
where L3 is an external per-unit-length inductance for the out-phase current, L4 is the same for in-phase one. If the conductors in the line are identical, then r2 = r1 , L2 = L1 L3 =
1 0 h b1 1 h b1 , dx = 0 ln 2 b1 b1 x h x
0 h b1 1 1 dx = 0, (2.17) b 2 1 x hx where b1 is wire radius, b1 = h , 0 is the vacuum permeability. L4 =
Since for the in-phase currents the charges flowing through an arbitrary section z = const are equal in sign and magnitude, then the potential difference between the wires, being due to these charges, vanishes. Indeed, the field outside the wires at the cross-section passing through their axes is
110
Er
1 1 , (2.18) 2 i 0 x h x q
V =
h b1
b1
E r dx = 0, (2.19)
where q is per-unit-length charge of the wire in the case of an in-phase current, 0 is the vacuum permittivity. From (2.19) it follows that the value of the scattering voltage u s in formula (2.16), which is, generally speaking, equal to a sum of the scattering voltage of the out-phase and in-phase currents, is equal to the scattering voltage of the out-phase current. From (2.16), (2.17), (2.19) we obtain finally
u s i h iR L = 11 ( E(ie )2 z E(ie )1z ) (1 11 ) B(ie ) ( x, z, t ) n dx, z t t 0 R = r1 r2 = 2r1 , L = L1 L2 L3 = 2 L1 L3.(2.20) Formula (2.20) as compared with a similar formula obtained in paper [29], gives at i = e the same result to within the sign choice for the voltage and current (the sign of voltage and currents we choose is opposite to that from paper [29]). Thus, if the two-wire line is in a homogeneous medium, then in equation (2.20) i = e and a magnetic field in the source vanishes. The derivation of the second equation of the transmission line for our case, based on the second Maxwell equation and the charge conservation law followed from it, does not give a radically new result as compared to the case of homogeneous medium. Therefore, using the results of papers [29] or [36], for zero values of the environmental conductivity and that of a dielectric plate we have the expression
i u s C = 0, (2.21) z t where C is a per-unit-length capacitance. Relations (2.20) and (2.21) form a system of equations of the transmission line wherein the values sought are the current i and scattering voltage u s . The absence of a source of the external magnetic field of the wave in transmission line equation (2.20) in homogeneous medium is related to an equality of the electromotive force in the line, being due to a change of the magnetic flux of the external wave, and the voltage of the external electric field of the wave. And since these voltages have different signs, their total vanishes. The situation will be different, if one places, into a gap between the wires (Fig. 18), a dielectric plate, which attenuates the external electric field component normal to the interface, remaining unchanged the magnetic field value. This decompensates the electric and magnetic tensions and gives rise to an additional magnetic source distributed along the length of the transmission line. The experimental investigations confirm the theory.
111
2.3. Analytical calculation of a plane wave coupling to a two-wire line The theory being developed in the present paper is illustrated by a calculation of a plane wave coupling to the transmission line. Since Poynting's vector P , the vector E (ie ) and vector B(ie ) are mutually orthogonal, each of them will be aligned with corresponding unit vectors e'k ( k=1,2,3) whose set forms a triad. Let P = P e' 3 , ,
E(ie ) = E(ie ) e'1 ,
P = E(ie ) H (ie ) ,
B i ( e ) = B(ie ) e'2 ,
B i ( e ) = 0 H (ie ) .(2.22)
Since the line under consideration is flat, while passing an external electromagnetic wave through it, the orientation of each triad vector e'k is the same as with reference to the local triad vectors ek parallel to x, y, z axes. It is evident that the triads e'k and ek are related by a rotation transform
e'k = gkl el .(2.23) Since
e'k e'l = ek el = kl ,
the matrix gik is orthogonal to
gki gli = kl = gki gil .(2.24) From (24) it follows that for orthogonal matrices a transposed matrix coincides with a reciprocal one. (According to Einstein's rule, with summing in (2.23) and later on over the repeated indices). The rotation matrix g ki depends on three parameters. Fig. 24 presents so-called Eulerian ~ angles , , [72], being an important type of parameters in the rotational group. If (L~ ) is a 1
2
straight line of intersection of planes xC2 y and x C 2 y , then 1 is an angle between C 2 x and ~ (L~ ) , 2 is an angle between (L~ ) and the axis C 2 x , is an angle between positive directions of C 2 z and C 2 z .
Fig. 24. Eulerian angles
112
To orient the triad e'k arbitrarily with reference to the triad ek , the rotational matrix gik , with i being the line number and k being the column one, can be represented in the form gik = e'i ek , (2.25) or in terms of the Eulerian angles 1 , ~ cos 1 cos 2 cos ~ sin 1 sin 2 gik = sin 1 cos 2 cos ~ sin 2 cos 1 sin 2 sin ~ cos 1 sin 2 cos ~ sin 1 cos 2 sin 1 sin 2 cos ~ cos 1 cos 2 cos 2 sin ~
sin 1 sin ~ cos 1 sin ~ , (2.26) cos ~
~ where the angles 1 and 2 may change from 0 to 2 , and the angle from 0 to . Using the above-mentioned geometrical notes, one can represent system of equations (2.20), (2.21) of the transmission line in the form
u s i iR L = 2 g13 ( E(ie )2 E(ie )1 ) z t 1
h g22 B(ie ) ( x, z, t )dx, 0 t
1 = 1 2 , 2 = 11 ,
i u s C = 0.(2.27) z t
With the wave propagating along the vector e'3 , system (27) can be simplified g z e' r E(ie )1 = E(ie )0 f t 3 1 = E(ie )0 f t 33 , v0 v0 ( e' ( e z e1h )) e' r = E(ie )2 = E(ie )0 f t 3 2 = E(ie )0 f t 3 3 v0 v0 g z g 31h .(2.28) = E(ie )0 f t 33 v0
113
( e'3 ( e3 z e1 x )) h h i i dx = g 22 B( e ) ( x, z, t )dx = g 22 B( e )0 g 22 f t v t 0 t 0 0 = g 22 B(ie )0
g z g 31 x h dx = g 22 f t 33 v0 t 0
t B(ie )0v0 g 22 = g31 t
g33 z g31h v0 t
B(ie )0v0 g22 g33 z = f t g31 v0
z
f (u )du =
g33
v0
g z g31h .(2.29) f t 33 v0
where f is the function determining a pulse waveform, v0 is the velocity of light in the environment surrounding the line. Since in an electromagnetic wave B(ie )0 v0 = E(ie )0 system (2.27), allowing for (2.28) and (2.29), is representable in the form g u s i iR L = E(ie )0 1 22 2 g13 z t g31
g z f t 33 v0
g z g31h , f t 33 v 0
u s i = 0.(2.30) C t z System (2.30) will be solved using the Laplace transform and considering the initial current and voltage to be zero. Using the known shift theorem, for the first equation of system (2.30) we obtain dU s ~ ( z, p) F~( p), ( Lp R ) I = dz p ( g33z g31h ) pg33z v0 ~ ( z, p ) = E i ( g /g g ) e v0 e , (2.31) ( e )0 1 22 31 2 13
where U s , I , F~ ( p ) Laplace transforms for the voltage, current and function respectively
114
depending on z and a parameter p . For the second equation of system (2.30) we obtain
dI pCU s = 0.(2.32) dz Eliminating the operator voltage from (2.32), we obtain
d 2I ~ F~, (2.33) 2 I = pC dz 2 where
= [Cp( Lp R)]1/2 (2.34) is a wave propagation coefficient. For the operator voltage U s we obtain
~ F~ ~ d 2U s ~ d .(2.35) 2 s = U g p F = 33 dz 2 v0 dz The solution to equation (2.33), (2.35) creates no difficulties. To facilitate the comparison with the results of other papers on two-wire lines, we write down this solution in terms close to those adopted in [56], where this problem is solved in the frequency representation. However, the coefficients in the solution will, generally speaking, differ from [56]
I ( z, p) = [ K1 M ( z )]ez [ K2 Q( z )]ez (2.36) U s ( z, p) = W {[ K1 M ( z )]ez [ K2 Q( z )]ez }.(2.37) Here the designations are introduced as follows: 1/2
Lp R 1 W = , M ( z) = 2W pC
Q( z ) =
1 2W
z
e 0
z
e 0
~ ( , p ) F~ ( p )d ,
~ ( , p ) F~( p )d .(2.38)
W is a characteristic resistance of the line, K1 and K 2 are constants ( depending on p ) to be found from the boundary conditions. Let the boundary conditions at line ends be set in the form [29] u t ( z1 , t ) = u s ( z1, t ) ui ( z1, t ) = i ( z1, t ) Z1,
115
ut ( z2 , t ) = u s ( z2 , t ) ui ( z2 , t ) = i ( z2 , t ) Z 2 .(2.39) We consider the line to be loaded by resistances Z1 and Z 2 . For our case z1 = 0 , z 2 = l . To make it convenient to compare the results with those of other papers, we retain the form of (2.39). System (2.39) in Laplace transforms with using (2.36) and (2.37) reduces to the form
K1 1K 2e
2 K1 K 2e
2z1
2z2
= 1Q ( z1 )e
2z1
= 2 M ( z2 ) e
e
2z2
z1
U i ( z1 ) , Z1 W
U i ( z2 ) , (2.40) Z2 W
solving the system we obtain K1 =
1 2 e
2z1
M ( z2 ) 1Q ( z1 )e 2 z 2 z e 2 1 2 e 1
2 ( z1
z2 )
e e
( z1
2 z 2
z2 )
1 2 e
2z1
U i ( z1 )ez2 1ez1U i ( z2 ) , (2.41) Z W Z W 1 2 K2 =
1 2 e
2z1
Q ( z1 ) 2 M ( z2 ) 1 2 z 2z1 2 z e 1 2 e e 2 1 2 e 1 2 z 2
U i ( z1 )ez1 ez2U i ( z2 ) 2 .(2.42) Z W Z W 1 2 In (2.40-2.42) 1 and 2 are reflection coefficients at the line ends
1 =
Z1 W Z W , 2 = 2 .(2.43) Z1 W Z2 W
If one assumes U i = 0 in (2.41) and (2.42), the solution for K1 , and K 2 , presented in [56] in Fourier's representation, but not in Laplace's. However, the latter is unimportant, since a formal transition from Fourier's transform to Laplace's is performed by substituting i p , where is the frequency. Comparing the basic formulae of papers [56] and [29], one may be convinced of their difference, which is due to the scattering voltage u s of paper [29] being substituted by u t in paper [56], and the equations of the transmission line in terms of u t [56] being the same as those of paper [29] in terms of u s .
116
Expressions (2.38), allowing for (2.31), may be integrated excplicitly. As a result we obtain E(ie )0 g 22 g hp 2 g13 1 exp 31 M ( z, p) 1 2W g31 v0 1 pg33 pg33 exp z 1 F ( p ) .(2.44) v0 v0 E(ie )0 g 22 g hp Q( z, p) 2 g13 1 exp 31 1 v0 2W g 31 1 pg 33 pg33 pg33 exp z exp l F ( p) .(2.45) v0 v0 v0
The external field voltage drop across the loads Z1 and Z 2 will be determined from (2.6), (2.10), (2.25). h g x u i (0, t ) 2 g11 E(ie )0 f t 31 dx, 0 v0 h g x g33l u i (l , t ) 2 g11 E(ie )0 f t 31 dx.(2.46) 0 v0
Using the shift theorem, we obtain an expression for the operator voltage h
U i ( z1 ) 2 g11 E(ie )0 F ( p) e 0
U ( z2 ) i
pg31 x v0
dx
2 g11 E(ie )0 F ( p )v0 g31 p
pg31h 1 exp , v0
2 g11 E(ie )0 F ( p )v0
pg 31h pg33l 1 exp exp , g31 p v0 v0 pg l U i ( z2 ) U i ( z1 ) exp 33 .(2.47) v0
Find the currents in the loads induced by the wave. At the beginning of the line
I (0, p) K1 K 2 Q(0, p).(2.48) At the end of the line
I (l , p) K1 M (l , p) e l K 2e l .(2.49)
117
In further calculations we shall consider that per-unit-length ohmic resistance R is essentially smaller then wave resistance. Using formulae (2.41-2.48), we find
I (0, p)
N ( p)
{(
2 l 1 1)[ 2 e
e l (1 g33 0 ) 1 g33 0
1 e l (1 g33 0 ) 2 g11e l (1 g33 0 ) ] 1 g33 0 0 g31 ( Z 2 W ) 2 g11 2 e 2 l 1}. 0 g31 ( Z1 W )
N 1 12e2 l , 0 (e / i )1/2 v1 / v0 , p / v1 p( LC )1/2 , E(ie )0 F ( p) 1 1 e g31h 0 , ( 1 g 22 / g31 2 g13 ).(2.50) 2W v1 vi is the electromagnetic wave velocity in a dielectric layer. At the line end we obtain: I (l , p) 1
N ( p)
l (1 g33 0 )
{(1 )e [ e l
2
1
1 g33 0
1 e l (1 g33 0 ) 2 g11 ] 1 g33 0 0 g31 ( Z1 W )
2 g11e lg
33
0 g31 ( Z1 W )
1 e }.(2.51) 2 l
1
Allowing for 1 i 1 , (i 1, 2)(2.52) Zi W 2W
relations (2.50), (2.51) be of the form
I (0, p)
( 1 1)
{ [
2 g11 ] 0
2 N ( p)Wg31 1 g33 0 g [e l (1 g33 0 ) e2 l ] [ 2 11 ][1 e l (1 g33 0 ) ]}(2.53) 1 g33 0 0 (1 2 ) {[ 2 g11 ] I (l , p) 2 N ( p)Wg31 1 g33 0 0 g [e lg33 0 e 2 l ] 1[ 2 11 ][e l e l (2 g33 0 ) ]}, (2.54) 1 g33 0 0 2
118
where
1 g22 2 g13 g31.(2.55) Using the evident equality 1 ( 1 2 ) k e 2 lk , (2.56) N ( p ) k 0
where ( 1 2 )0 = 1 for 1 or 2 equal to zero as well, and the known multiplication theorem, we find for original functions (2.53) and (2.54) the expressions:
i (0, t )
E(ie )0 ( 1 1)v0 2Wg31
( ) k
k 0
1
2
0
f ( )
{2 A[ (t2,k ; t1,k ) (t3,k ; t4,k )]
B[ ((2lk / v0 ) ; t2,k ) ((2lk / v0 ) ( g31h / v0 ) ; t3,k )]}d , (2.57) where the designations are introduced as follows
A
0 1 g33
2 g11 , B
0 1 g33
2 g11 ,
2l l (k 1), t2, k (2k 1 g33 0 ), v1 v1 g h g h t2,k 31 0 , t4,k t1, k 31 0 , v1 v1
t1,k t3, k
(a; b) (t a) (t b)(2.58) and (t ) is the Heaviside step function.
i(l , t )
E(ie )0 (1 1 )v0 2Wg31
( ) k 0
k
1
2
0
f ( )
{A[ (t5,k ; t6,k ) (t7,k ; t8,k )]
1B[ (t6,k ; t9,k ) (t10,k ; t11,k )]}d , (2.59) where t5, k
l l (2k g33 0 ), t6, k (2k 1), v1 v1
119
t7,k t5,k t9,k
g31h 0 g h , t8,k t6,k 31 0 , v1 v1
l (2k 2 g33 0 ), t10,k t8,k , v1 g h t11,k t9,k 31 0 .(2.60) v1
Formulae (2.57) and (2.59) do not assume that h l . If the condition as follows is satisfied
cos(v0 z ) h , (2.61) l cos(v0 x )
g31h lg33 ,
then, using the evident formula, according to which at t / a 1
(a; b) (a t; b t ) [ (t a) (t b)]t, (2.62) where is Dirac's delta function, we find
i (0, t )
E(ie )0 h( 1 1)
( ) { A[ f (t t ) f (t t )] 2W B[ f (t 2lk / v ) f (t t )]}(2.63) k
k 0
1
2
0
i(l , t )
i ( e )0
E
h(1 2 )
2, k
2
1, k
2, k
( ) {A[ f (t t ) f (t t )] 2W B[ f (t t ) f (t t )]}.(2.64) k 0
1
k
1
6, k
5, k
2
6, k
9, k
2.4. Analysis of the results Formulae (2.57), (2.59), (2.63), (2.64) found in the paper allow one to calculate the currents in the loads of the two-wire transmission lines being excited by electromagnetic pulses of arbitrary form at any orientation of Poynting's vector and different polarization angles. Compare our results with the ones of other papers on this subject, e.g., [59], [60]. We express the rotational matrix components gik for convenience to be compared in terms of the parameters , , , following paper [60], where is an azimuth angle, is a polarization angle, is an elevation angle (Fig.25) but not through the Eulerian angles.
Fig. 25. Polarization angles
120
Fig.25 presents a straight line mk lying in the incidence plane and perpendicular to P , a straight line ab lying in the y,z plane and perpendicular to the incidence plane dotted in the figure. The vector E(ie ) lying in the plane kb, may be resolved into components aligned with the axes ab and mk. If E(ie ) is parallel to mk, i. e. / 2 , then it lies in the incidence plane (vertical polarization), if E(ie ) is parallel to ab, i.e. 0 then the polarization is horizontal. From Fig. 25
and formulae (2.22) and (2.25) we find the required rotational matrix components
g11 e1' e1 sin sin , g31 e3' e1 cos , g33 e3' e3 sin cos , g13 e1' e3 cos sin sin cos cos , g 22 e2' e2 e2 e3' e1' e3 e1 e3' e1' g33 g11 g31 g13 . (2.65) Using formulae (2.55), (2.58) and (2.65), we obtained the expressions for coefficients A and B : A
B
1 g 22
g 33 1 0
1 g 22
g 33 1 0
2W , (2.66) 2W , (2.67)
where, to compare with paper [60], we have introduced a function
W
cos cos sin sin (cos 01 sin ) , (2.68) 01 sin cos
which for the case of homogeneous medium 0 1 reduces identically to the angular function [60]. If one assumes 1 0 , 2 1 in formulae (2.66) and (2.67), their substitution into expression (2.63) after some algebraic transformations results (to within the sign) in the relationship for the current of paper [60], which, in our view, is valid provided (2.61). Thus, the results of papers [60] are in agreement with the theory, developed in [52], [29], i with equation (2.30) being free of magnetic field B( e ) , which is naturally valid for transmission lines in homogeneous media. In paper [59] a problem of an electromagnetic pulse coupling to the wire parallel to a conducting plane has been solved on the basis of the theory [29]. The ends of this wire are connected with the plane through resistances R1 and R2 . Two cases of polarization considered are: vertical (parallel) and perpendicular (horizontal).
121
In particular, for (h / l ) 1 in the case of parallel polarization and matched loads the load current (within factor 2 due to reflection from the conducting surface), with a misprint in formula (14c) [59] being corrected (where the factor sin preceding Z0 in the third line of the formula should be corrected for sin ), coincides with [60], and hence with ours calculated by formula (2.63) with 1 0 , 2 1 , 0 1 . It should be noted that formula (2.63) is a corollary to more general formula (2.57) provided (2.61). Choose a step function (2.58) and vertical polarization of the incident wave as a function f in formula (2.57) for convenience to be compared with the results of papers [55], [56]. Calculate a short-circuit current ( 1) , considering the line to be semi-infinite, and the environment to be air ( 1 0, 2 1, 0 1) . Since for the semi-infinite line there exists no signal reflected from its end, without loss in generality, we may consider that 2 0 , i.e. Z 2 W . The calculation by formula (2.57), taking account of (2.58), results in
i(0, t )
BE(ie )0 v0 Wg31
t
BE(ie )0 v0
0
Wg31
( ) ( ; g31h / v0 )d
F (0;
g31h ), (2.69) v0
where the symbol F (a; b) introduces the function being defined by the equality
0, t a F (a; b) t a, a t b (70) b a, t b The quantity B calculated by formulae (2.67), (2.68) for 2 0 1 and / 2 (vertical polarization) is of the form
B W
cos sin (2.71) 1 sin cos
If the incidence plane coincides with the x , z plane (Fig.25), then 0 and B W 1 for any angle . Fig. 26 presents a plot of the quantity i(0, t ) i1 (t ) , depending on time t , for B 1 .
Fig. 26. Comparison of exact and approximate formulae for the current
122
i On the same figure the dependence i2 (t ) ( E( e )0 h (t )) / W is represented.
When using formula (2.63) instead of (2.57) to solve the problem considered, we obtain the i current i2 (t ) ( E( e )0 h (t )) / W , i.e. approximate formula (2.63) conserves the incident pulse front slope, and (2.57) expands this front (Fig.21). In papers [55], [56] the formula similar to (2.69) was compared with the exact solution, revealing good agreement of the latter with the solution based on the theory of long lines in homogeneous medium. In further consideration we apply the results obtained for calculation of strip lines with dielectric filling and compare the theoretical results with experimental investigations. The equations of a transmission line consisting of two linear wires separated by a dielectric layer are obtained on the basis of Maxwell's equations. An arbitrary nonstationary electromagnetic field is used as the source of excitation. If the line is in homogeneous medium (a dielectric layer is absent), the equations obtained coincide with the equations of papers [50], [51] and [29]. The presence of a layer between the wires leads to an additional source, distributed along the magnetic source, in the transmission line equation, which (as will be shown in the next chapter of the book) changes radically the pattern of voltage-and-current distribution in the transmission line. The equations obtained in the paper are solved analytically in the case of the line excitation source being an electromagnetic pulse of arbitrary form with arbitrary orientation of Poynting's vector and arbitrary polarization. The solutions obtained in a particular case of homogeneous medium without dielectric layer change into the known solutions of other authors [55], [56], [59], [60].
123
Chapter 3 TENSORS OF ELECTRIC AND MAGNETIC POLARIZABILITY OF ARBITRARY SHAPED CONDUCTING BODIES 3.1. Method for determining the polarizability tensor of arbitrary shaped conducting bodies
When considering the problems of electromagnetic wave diffraction on small bodies and when estimating of quasi-stationary coupling of bodies polarized by electric and magnetic field each other and with field-forming system (FS) the determination necessity of the polarizability tensor components of these bodies [6, 68, 69] is arisen. In particular the polarizability tensor of the electric and magnetic field transducer determine the error component of its calibration in FS connected with coupling the transducer and FS. A numerical calculation of the polarizability tensor for arbitrary conducting bodies is possible, in principle, but it is rather cumbersome and requires a great deal of effort. In the present paper a solid basis is given for the experimental method of determining the electric (magnetic) polarizability tensor of arbitrarily shaped conducting bodies. The idea of the method is based on the assumption as follows. If a body introduced into the field-forming system distorts the electric (magnetic) field structure due to the induced dipole moment, then there must exist a relation between the body polarizability and the additional capacitance (inductance) introduced by the body into the system. Link the capacitance increment C of a flat capacitor with the electric polarizability tensor ik , considering that the charged capacitor of capacitance C is not connected with the source. From the theorem on the energy of an uncharged conductor [33], it follows that introduction of the uncharged conductor into the field of charged ones diminishes the total field energy U by the value
U =
0
2 | E | dv 0 V 2 0 2
V1
| E E |2 dv, (7.1)
where E and E are electric field intensities before and after introducing the conductor into the system, V0 is a volume of the body being introduced, V1 is a volume of the surrounding space minus the volume of the body being introduced and that of capacitor plates. The second integral in formula (7.1) may be reduced to four surface integrals bounding the volume
0
| E E |2 dv = 0 V1 2
2
=
0 2
[( )( E E )]dv = V1
[ ( )( E E ) n dS ( )( E E ) n1 dS S
S1
( )( E E ) n2 dS ( )( E E ) n0 dS ], (7.2) S2
S0
where S1 and S2 are surfaces bounding the capacitor plates; S0 is a surface bounding the body
124
being introduced; S is an infinitely distant surface; and are potentials on the surfaces before and after introducing the body, n0 , n1 , n2 , n --- are unit vectors of the normals to the corresponding surfaces (Fig. 27).
Fig. 27. Integration domain Since the charges on the capacitor plates before and after introducing the body are invariant, and the potentials and are constant, the integrals over S1 and S2 vanish. Hence in formula (7.2) there remains only integrals over the surface S , S0 . 2 lim | ( )( E E ) r |= 0, r ( )( E E ) n dS = 0 S
where r is a distance from the origin of coordinates to a surface at infinity. Since the conductor being introduced is unchanged, the integral of
E n dS = 0. S0
0
vanishes. Due to E = 0 the integral of E becomes zero inside S0 .
E n dS = (E )dv = 0
S0
V0
= E dv = | E |2 dv. V0
V0
Substituting the expressions obtained into (7.1), we find
125
U =
0
1
E n dS = dS , (7.3) 2 2 0
S0
S0
where is a surface density of the conductor being introduced. The last-mentioned formula is exact. On the other hand, while introducing an uncharged conductor into the field of two arbitrary charged conductors with charges q and q , we have a relation U =
q2 q2 q q , C= , C = , 2C 2C | 1 2 | | 1 2 |
where C is a capacitance of the system of two conductors with charges q , C is a capacitance of this system after introducing the charged conductor. Thus 1 q2 1 1 dS = ( ).(7.4) 2 S0 2 C C
In particular, if an uncharged body is introduced into the homogeneous domain of a flat capacitor, which is not connected with the source, then in formula (7.4)
= E r0 , P = r0 dS , q = CEd , S0
where d is a distance between the plates, r0 --- is a radius vector from the origin of coordinates to the point of surface S0 , P --- is a dipole moment (Fig. 27). Hence E 2d 2CC .(7.5) PE = C C Since in the field E , which was homogeneous before introducing the body, an equality for the dipole moment Pi is valid
Pi = ik Ek , where ik is a polarizability tensor of the body. Then from (7.5) it follows that a formula for the capacitor capacitance decrement C after introducting a test body into the capacitor is C =
ik ni nk C , (7.6) Cd pm n p nm 2
where ni = Ei / | E | is a unit vector in the direction of E . Formula (7.5) allows the dipole moment of
126 the bodies, for which its direction coincides with that of the field E , to be calculated at once. Among such bodies are, e.g., the arbitrary bodies of revolution with an axis of rotation parallel to the field as well as the bodies having a symmetry plane and arranged so that the latter should be perpendicular to the field. For these bodies
d 2 CE .(7.7) P= 1 C/C Formulae (7.6) and (7.7) are valid for any relations C/C providing inhomogeneity of an initial (unpertubed) field E in the body location zone. We link rigidly the arbitrarily shaped conductor with a unit orthonormal triad ei (i=1,2,3) forming the Cartesian coordinate system. If one conducts six independent measurements of the capacitor capacitance at different orientations of a triad "frozen" into the conductor with respect to the field direction, then we obtain a system of six equations to find six independent components of the polarizability tensor. To have a unique solution to this system, it is necessary for the conductor to be oriented in the field so that the sixth-order determinant should not vanish, each of its lines contains a combination of values
n12 , n22 , n32 , 2n1n2 , 2n1n3 , 2n2n3 (7.8) calculated for six different orientations. In particular, a system of unit vectors satisfies this condition ni = i1 , ni = i 2 , ni = i 3 , ni =
ni =
i1 i 3 2
, ni =
i1 i 2
i2 i3 2
2
,
, (7.9)
where ik is a Kronecker symbol. A successive substitution of values (7.9) results in a system of six equations whose solution is of the form
ii d 2 Ai i 1,2,3, no summation over i, 1 2
ik = d 2 [ Aik ( Ai Ak )], (i k ), Ai =
Ci Cik , Aik = , (7.10) 1 Ci /C 1 Cik /C
where Ci is a capacitor capacitance change at the field vector E directed along the vector ei and Cik is that for the field vector parallel to the plane passing through ei and ek , and forming the angle = /4 . In principle, the angle is arbitrary. But at the value of = /4 the solution to the system takes the simplest form. Formula (7.10) links the components of the electric polarizability
127
tensor ik and the capacitor capacitance change C . It is evident that such a method may be developed to determine the magnetic polarizability tensor of arbitrarily conducting bodies as well. As this takes place, the inductances Li being introduced by a body into the system forming a homogeneous magnetic field will appear to be experimentally measurable parameters at different orientations of the body with respect to the field vector [90]. If the skin-layer is less than typical dimensions of the body, then the results obtained will be applicable to all frequencies satisfying this condition. It should also be noted that this method can de extended to dielectric and magnetic arbitrary bodies as well. When the body has the symmetry planes or axises the number of measurements to determine the polarizability tensor may be diminished. Consider some particular cases. 1. Introduce a conducting ball of radius R between the capacitor plates. It is evedent from symmetry considerations that
A1 = A2 = A3 = A12 = A13 = A23 =
C . C 1 C
From formula (7.10) we find
ik =
Cd 2 ik .(7.11) C 1 C
As known [6], for the ball in a homogeneous field ik = 4 0 R3 ik . Hence
C =
4 0 R 3 (7.12) 4R 3 2 ) d (1 Sd
where S is an area of the capacitor plates. If R3 = Sd then C = 4 0 R3/d 2 , which coincides with the solution obtained in paper [6]. 2. Let a body of revolution be introduced into the capacitor and the vector e3 be directed along the axis of rotation. From the symmetry it follows that A1 = A2 = A12 = A, A13 = A23 = B, A3 = F and Ek = E 3k , and we obtain an expression Pi = E i 3 for the dipole moment. On the other hand, from the symmetry it follows that in this case PPE , hence 13 = 23 = 0 . From formula (7.10), taking account of the condition A1 = A2 = A12 we have 12 = 0 . Thus, if the unit vector e3 coincides with the axis of the body of revolution, then any orientation of the
128 dyad e1 and e2 the tensor ik of the body of revolution is diagonal. Thus, the above-mentioned triad ei sets a direction of the principal axes of an arbitrary body of revolution. Hence, to find the polarizability tensor of an arbitray body of revolution, it will suffice to conduct two measurements of the capacitor capacitance with a body introduced into the capacitor: with the axis parallel and perpendicular to the field E . Using formula (7.10), we find an expression for the polarizability tensor of an arbitrary body of revolution in the pricipal axes:
C d 2 C 1 C
ik =
0
0
0 C d 2 C 1 C 0
0
0
(7.13)
CPd 2 , CP 1 C
where C and CP are changes of the capacitor capacitance, when the axis of body symmetry is perpendicular or parallel to the field E respectively. From formulae (7.6), (7.13) we calculate the value of C () , when the axis e3 forms the angle with the field (Fig. 28):
C sin2 CP cos2 1 C /C 1 CP /C C ( ) = (7.14) 1 C sin2 CP cos2 1 [ ] C 1 C /C 1 CP /C
Fig. 28. Body of revolution in the electric field
129
3. Calculate the components of the polarizability tensor of a parallelepiped with sides a1 , a2 , a3 by bringing the triad ei into coincidence with the corresponding edges ai going out of one of the vertices. If one directs the field E along each of the unit vectors e1 , e2 , e3 , then based on the symmetry, Pi ei may be expected in these cases. Hence, for the parallelepiped the tensor ik is diagonal, i. e. ei are principal axes of the parallelepiped. If the parallelepiped base is a square, i.e. a1 = a2 , then 11 = 22 . From linear algebra we know that in this case the principal axes e1 , and e2 , remaining orthogonal to each other and to the axis e3 , may be arbitrarily oriented in the plane of the base. Physically, this means that the field E directed perpendicular to the axis e3 results in an induced dipole moment P being parallel to E and the moment value remaining constant independently of orientation of the vector E with respect to the dyad e and e . In other words, a 1
2
bar of square cross-section polarized by the field parallel to the cross-section behaves as a body of revolution. If one considers a cube instead of a parallelepiped, then 11 = 22 = 33 . In this case the principal axes ei may be arbitrarily oriented in space. Physically, this means that a polarized cube is "similar" to a sphere, i.e. by orienting the cube arbitrarily in a homogeneous field E , we obtain one and the same dipole moment, parallel to the field and invariable in magnitude, independently of orientation. Thus, the cube as well as the sphere is isotropic with respect to polarizability. In terms of the group theory the cube and sphere belong to the same syngony, i.e. have to do with a system of the same symmetry group. One may show that among the same syngony are the tetrahedron and the remaing regular polyhedrons [88]. For the bodies of cubic syngony each orthogonal unit vector constructed in the body coincides with the direction of the principal axes, and the polarizability tensor is globular. The bodies of trigonal, tetragonal and hexagonal syngonies, as known [89], have one axis each of the third, fourth and sixth order respectively (an axis of symmetry of the order n is called the straight line, a turn about which at the angle 2/n brings the body into coincidence with its initial position). The tensor ik , in the principial axes, has two equal eigenvalues. Thus these bodies are "similar" to bodies of revolution with respect to polarizability. The above-mentioned parallelepiped with edges a1 a2 a3 in the principal axes has three different eigenvalues of the polarizability tensor. It has three axes of the second order, which are orthogonal to one another. Therefore it may be referred to the rhombic syngony. The monocline system bodies have one axis of the second order, which may be chosen as a principal axis. And, finally, the triclinic system bodies have no axes of symmetry. 4. Consider the polarization of arbitrarily shaped flat thin plates. Let two vectors of the triad and , be rigidly connected with the body, lie in the plate plane, and the vector e be e1 e2 3 perpendicular to the plate. It is evident that introducing such a plate into a flat capacitor to the capacitor plate,will not change the capacitor capacitance. From formula (7.6) at C = 0 and ni = i 3 we find 33 = 0 . If one transforms the plate polarizability tensor to the principal axes, then from the problem symmetry one of the principal axes (e. g., e3 ) must coincide with the vector e3 of the initial "frozen" triad e . The directions of the other two vectors e and e lying in the plate i
1
2
plane may be calculated. In the principal axes the polarizability tensor components ik are of the form
130
1
0
ik = 0
2
0
0
0 0. 0
The initial tensor of polarizability of a flat arbitrarily shaped thin plate is generally representable in the form
11 12 0 ik = 12 22 0 .(7.15) 0
0
0
Using the well-known method of linear algebra, one can find the directions of the principal axes of the polarizability tensor that are set by an orthogonal normalized unit vector:
1
ei = 1
(i 11 )
2
12
122
i 11 ( ) 2 1 i 2 11 12
0
(i = 1,2), e3 = (0, 0, 1),
1 =
2 =
11 22 ( 11 22 ) 2 4 122 2
11 22 ( 11 22 ) 2 4 122 2
,
.(7.16)
From the aforesaid it follows that the polarization properties of arbitrarily shaped flat plates can be determined with the aid of three measurements, e.g., A1 , A2 and A12 by formula (7.10). For example, we shall consider the flat body having an axis of symmetry of the order n , (n 3) . Let the two main axes e1 , e2 are located in the plate plane and their origin coincides with its geometrical centre. Unit vector e3 is orthogonal to the plate plane. The polarizability tensor ik in the main axes has the form
ik = A1i1k B 2i 2 k .(7.17) If one turns the plate around the axis e3 on angle = 2/n then the body properties relatively "non-frozen" triad conservating space orientation will be unchanged. Rotation transform converts tensor ik to one ik .
131
ik = gil g kn ln = Agi1 g k 1 Bgi 2 gk 2 , (7.18) where cos gil = sin 0
sin 0 cos 0 (7.19) 0 1.
We shall represent the polarizability tensor components ik and ik as a matrix form
A cos2 B sin2 ik = ( A B ) sin cos 0
( A B ) sin cos A sin2 B cos2
0 0
0
0.
A 0 0 ik = 0 B 0 (7.20) 0 0 0.
2 in accordance with symmetry requirement does n not change the initial tensor ik . From the equality ik = ik we have A = B. Thus, polarizability tensor of the body under consideration in the main axes has two identical own meanings 11 = 22 = A , 33 = 0 . Physically, this means that the plate plane being parallel to the field E results in an induced dipole moment P being unchanged in magnitude and direction at any turn of the plate in this plane. Thus, when polarizing this body is similar to the disk. In particular, all regular n -gons possess this "similarity" property. If a given body is turned in the capacitor field so that the axis e3 normal to its surface should form an angle with the field direction, then the capacitor change C () may be calculated by formula (7.14) taking account of CP = 0 , and C , from the aforesaid, does not change at any orientation of the body in this plane. For an arbitrary flat figure the capacitance change in the rotation is determined by two angles , , where is an angle between the field vector projection onto the plane ( e1 , e2 ) and the vector e1 . Thereby e1 , e2 , e3 set the direction of the principal axes of the figure (Fig. 29). On the other hand the turn on angle =
A=
C (/2,0) cos2 C (/2,/2) sin2 , 1 C (/2,0)/C 1 C (/2,/2)/C C ( , ) =
A sin 2 .(7.21) A 2 1 sin C
132
Fig. 29. Arbitrary flat figure in the electric field
3.2. Relation between the electric and magnetic polarizability tensor components for conducting bodies of revolution While measuring the magnetic polarizability tensor components of a conducting body in practice, there may arise difficulties related to providing the requirements of skin-layer smallness as compared with the body sizes. Under these conditions it would be appropriate to find the relation between the components of the electric and magnetic polarizability tensors, if only for bodies of revolution - the most abundant form of transducers, permitting the magnetic measurements to be substituted by the electric ones. To calculate the fields E and H , we choose an elliptic system of coordinates ( , ) in the meridional planes ( r, x ) (Fig. 30) related to ( r, x ) by formulae
x = c cosh cos , 0 , r = c sinh sin, 0 2 , (7.22) where c is a focal length of the family of coordinate lines the confocal ellipses and hyperbolas.
Fig. 30. Problem geometry
133
If one assumes
cosh = , cos = , 1 ,1 1, then
x = c , r = c 2 1 1 2 .(7.23) It is evident that the problem of finding the magnetic field potential satisfying the Laplace equation outside the body of revolution in a strong skin-effect is equivalent to the hydrodynamic problem of a longitudinal liquid flow about the bodies of revolution [91]. Making use of the result [91], we obtain in our designations
= cH 0 [AnQn ( ) Pn ( ) ], (7.24) n =1
where Pn ( ) and Qn ( ) are Legendre's functions of 1st and 2nd kind respectively. The coefficients An may be determined, if one takes into account that the normal component of the magnetic field H n = 0 on the conductor surface, which is equivalent to the equality
2 An
n(n 1) n =1
dQn dPn 1 = 0, (7.25) d d
determining a zero surface of the current in the longitudinal flow about the bodies of revolution. It should be added by a given equation of a body-of-revolution profile
= f ( ).(7.26) Of course, from the calculational viewpoint finding the coefficients An from system (7.25), (7.26) is rather a complicated problem (simply solved, e.g., for the ellipsoids with = 0 = const ). To find the electric field potential outside the body, when the field direction E (Fig. 59) is perpendicular to the axis of rotation, we use the well-known general solution to the Laplace equation in the elliptic coordinates [91]
( , , ) = Qnk ( ) Pnk ( ) n =0 k =0
[ Ank cos k Bnk sin k ] E0 y , (7.27) where
134
Pnk ( ) = (1 2 ) k/2
k d k Pn ( ) k k/2 d Qn ( ) 2 ( ) = ( 1) (7.28) Q n d k dk
are Legendre's associated functions,
y = r cos = c 2 1 1 2 . Assuming that in the plane y = 0 ( = /2) , we find Bnk = 0 . From the symmetry it follows that the dependence of the potential on contains only cos . Hence, using relations(7.27),(7.28), we have
( , , ) = cE0 2 1 1 2
dQn dPn Cn 1].(7.29) n =1 d d
cos [
To find the coefficients Cn , we take advantage of the potential being constant on the conductor surface and, in particular, equal to zero. From (7.29) it follows that
C n =1
n
dQn dPn 1 = 0(7.30) d d
at = f ( ) . Comparing (7.25), (7.26) and (7.30), we find Cn =
2 An .(7.31) n(n 1)
Formula (7.31) links solutions (7.24) and (7.29). Consider formulae (7.24) and (7.29) in the dipole approximation at the distances
R = x 2 r 2 much larger then the body sizes. It is evident that the greater R corresponds to the asymmetrically greater , both being of the same order. To conduct a further analysis, we expand the function Qn ( ) in a power series of 1/ [92] Qn ( ) =
F(
2
n 1
( n 1) ( n 3/2) n 1
n 1 n 2 2n 3 1 ; ; ; 2 ), (7.32) 2 2 2
where ( z ) is the gamma-function, F (a; b; c; z ) is the hypergeometric function. The dipole approximation allows one term of the expansion only to be retained in formulae (7.29) and
135
(7.24). From (7.32) it follows that Q1 ( ) 1/(32 ) 1/(3R 2 ) ,
= H 0 x (1
2C A1 ) = E0 y (1 13 ).(7.33) 3 3R 3R
From formula (7.31) it follows that C1 = A1 . As seen from Fig. 30, formula (7.33) is representable in the form
M R A1H 0 = H0 R ,M = , 3 R3 P R 2 A1E0 = E0 R 3 , P = , (7.34) 3 R where M and P are induced magnetic and electric dipole moments respectively. On the other hand, in the principal axes of the electric polarizability ik and magnetic one
ik we have by definition M x H 0 , Py = E0 , where 11 , = 22 = 33 , and the orientation of the principal axes of the tensors is: x1 = x , x2 = y , x3 = z . Comparing with (7.34), we obtain
1 . (7.35) 2 Formula (7.35) admits a simple analytic verification in some particular cases. For example, the problems on an ellipsoid of revolution in an electrostatic field and on a conducting ellipsoid in a variable magnetic field are solved in [6], [93]. For the conducting ellipsoid of revolution whose axis is perpendicular to E0 we have
= abc/(3n ) , where a, b = c are lengths of the ellipsoid semiaxes, n is a depolarization coefficient. In the case of a strong skin-effect we have 11 = abc/(3[1 nP ]) for the same ellipsoid whose axis is parallel to the variable magnetic field H 0 . Taking into account that n 2n 1 , we obtain / 1 / 2 . In particular, for a ball we have
1 a3 n n , , a3. 3 2
136
3.3. Analytical calculation of polarizability tensor of bodies with shape like a cubic syngony
When determining polarizability tensor of bodies with shape like a sphere, cube, tetrahedron etc. (that is to cubic syngony bodies) by means of method above mentioned the sensitivity of measuring device may be small to detect the tensor distinction from sphere. Let us consider analytically the polarization of such bodies. Work [94] contains the method to obtain the formula of dipole moment P when the body surface in spherical coordinates was specified by the equation
r0 ( , ) = R(1 F ( , )), (7.36) where F ( , ) is the arbitrary smooth function satisfying to the condition 0 | F ( , ) | 1 , is the dimensionless small parameter so that value 2 F 2 = 1 is neglected remaining only the 1-order infinitesimal . R is the "effective" sphere radius of the cubic syngony body which coincides with the sphere radius for body with a sharp like a sphere. For other cubic syngony bodies the "effective" radius is known either from handbook numerical calculations or it is measured experimentally by means of the method above mentioned. The purpose of the present part is both analytical derivation of polarizability tensor and determination of the general mechanisms permitting to estimate the upper error of the transducers distorting the initial electric field when coupling to this field. Let we locate the quasi-cubic syngony transducer to homogeneous electric field with intensity E parallel to axis z . Let at the surface of the body the constant potential Psi0 is maintained. Potential ( r, , ) at r R is sought as a sum
Bn Y ( , ), (7.37) n 1 n n =0 r
= Er cos
where spherical harmonics Yn ( , ) are expressedby the linear combination 2 n 1 of spherical functions Yn(m) Ё Yn( m ) . n
Yn ( , ) = ( AnmYn( m ) BnmYn( m ) ), (7.38) m =0
and spherical functions are connected with Legendre's functions Pn(m) by means of known relations
Yn( m ) = Pn( m ) sin(m ), Yn( m ) = Pn( m ) cos(m ).(7.39) It is known that relation (7.37) satisfies to Laplace equation in spherical coordinate system for external problem. Taking into account that the conductor surface potential is constant and expansing the factor Bn in formula (7.37) into a power series , namely
137
Bn = Bn( p ) p , (7.40) p =0
we obtain the set of equations connected the factors to be sought Bn( p )
Bn(0) Y ( , ), (7.41) n 1 n n =0 R
0 ER cos =
1 ( Bn(1) Bn(0) (n 1) F )Yn ( , ), (7.42) n 1 n =0 R
EFR cos =
1 (n 1)(n 2) 2 ( Bn(0) F Bn(1) (n 1) F Bn(2) )Yn ( , ), (7.43) n 1 2 n =0 R
0=
............................................... It is known that any twice differentiated function f ( , ) is expanded in uniformly and absolutely converging terms of spherical functions [30]. Expanding the left part of equality (7.41) in terms of spherical functions one can to be convinced that only following factors will be different from zero
A00 B0(0) R 1 = 0 , A10 B1(0) R 2 = ER.(7.44) Using (7.44), (7.40), (7.38) and (7.37) we obtain in zero-order approximation the known expression for potential (0) = 0
R R3 Er (1 3 )(7.45) r r
of conducting sphere located in external homogeneous electric field [6]. Taking into account equations (7.44) the equation (7.42) has the form
1 (1) B Yn ( , ), (7.46) n 1 n n =0 R
3EFRP1(0) 0 F =
To further calculations we shall expand the function F ( , ) , determining the body surface in terms of spherical functions
F ( , ) = (CklYk( l ) DklYk( l ) ), (7.47) k = 0 l =0
then relation (7.46) has the form
138
3ER (Cnm cos(m ) Dnm sin(m )) Pn( m ) P1(0) = n =0 m =0
= ( nm cos(m ) nm sin(m )) Pn( m ) , (7.48) n = 0 m =0
where
nm = R ( n1) Bn(1) Anm Cnm 0 , nm = R ( n1) Bn(1) Bnm Dnm 0 .(7.49) The factors to be sought nm and nm one can to calculate in accordance with the general formulae [30]
nm =
3ER k =0 l =0
2
(C Y
0
( l ) kl k
0
DklYk( l ) )Yn( m )Y1( 0) sin dd
|| Yn( m ) ||2
, (7.50)
where || Yn( m ) ||2 = 2
1 0 m ( n | m |)! .(7.51) 2n 1 ( n | m |)!
The factors nm are calculated similarly with substitution in formula (7.50) Yn( m ) Yn( m ) . To calculate the integrals we shall use the known formula [95] Yn( m )Y1( 0) =
( n m 1)Yn(1m ) ( n m )Yn(1m ) .(7.52) 2n 1
Taking into account the orthogonality of spherical functions with different indexes we have
nm = 3ER || Y
(m) n
2
||
[ k =0 l = 0
= 3ER( Similarly we obtain that
n m 1 Ckl ml n 1,k || Yn(1m ) ||2 2n 1
nm Ckl ml n 1,k || Yn(1m ) ||2 ] = 2n 1
n m 1 nm Cn 1,m Cn 1,m ).(7.53) 2n 3 2n 1
139
nm = 3ER(
nm n m 1 Dn 1,m ).(7.54) Dn 1,m 2n 3 2n 1
Omitting the interval calculations we write the final expression for potential distribution outside of the body. ( r, , ) = 0
R R3 Er cos (1 3 ) r r
R n 1 (CnmYn( m ) DnmYn( m ) ) n 1 n =0 m =0 r
n
0
R n 1 n m 1 nm {[ Cn 1,m Cn 1,m ]Yn( m ) n 1 2n 3 2n 1 n =1 m = 0 r
n
3ER
[
n m 1 nm Dn 1,m Dn 1,m ]Yn( m ) }, (7.55) 2n 3 2n 1
where factors Ckl are determined by function F ( , ) in accordance with formulae Ckl =
(2k 1) l (k l )! 2 d F ( , )Yk( l ) sin d , 0 0 4 (k l )!
0 = 1, l = 2 (l = 1,2...), (7.56) and factors Dkl are calculated similarly substituting under integral Yk( l ) Yk( l ) . Let us analyze the obtained relation to compare with other works. If one supposes that in (7.55) E = 0 then the obtained result coincides with problem solution proposed in [96]. To uncharged body (transducer) with potential 0 = 0 located in homogeneous electric field with intensity E the additional contribution to dipole potential (1) owing to departure of body surface shape from spherical has the form (1) =
3ER 3 2 3 3 {[C00 C20 ]Y1( 0) C21Y1( 1) D21Y1(1) }, 2 5 r 5 5
Y1( 0) =
z y x = cos , Y1( 1) = , Y1(1) = .(7.57) r r r
This spherical functions are the directed cosines of the unit vector n = r/r. It is follows from the aforesaid that if the uncharged metallic quasi-cubic syngony body with "effective" radius R is located in homogeneous electric field directed along axis z then this one acquires the dipole moment calculated in accordance with the formula
140
9 2 9 P = R 3 E{ D21e1 C21e2 [1 3 ( C20 C00 )]e3}(7.58) 5 5 5 Here the unit vectors e1 , e2 , e3 are choosen along axes x , y , z accordingly and factors Ckl and Dkl are determined from the formula (7.56) and its analogue. Let in the body under consideration the orthobench mark is frozen-in hi which at first coincides with one ei . Formulae (7.36), (7.56) and (7.58) are valid for both bench marks. In the bench mark ei the electric field in tensor designations has the form
Ei = E i 3 , (7.59) and relations (7.58) have the form
Pi = ik Ek = Ei 3 , 13 =
9 3 9 R D21 , 23 = R 3C21, 5 5
2 5
33 = R 3[1 3 ( C20 C00 )].(7.60) Formula (7.60) determines 3 - rd column of polarizability matrix in bench marks ei and ei . To determine two other columns we shall turn the body around axis e1 (or h1 ) so that h3 will be coincided with ( e2 ), and h2 with e3 . Physically it is equivalent to the situation when the field E "illuminates" the body on axis h2 while relatively the initial system ei the field is directed along axis e3 . For new body position formula (7.60) is valid if one substitutes
~ , ik ~ik , Pi P~i , D21 D 21
C21 C~21 , C20 C~20 , C00 C~00 . It is possible to establish the connection between C~kl and Ckl since the surface equation (7.36) and formula (7.56) are known. Let xk are the coordinates of radius-vector mapping some body surface point specified by equation (7.36). Rotation operator g converts this vector to one xk that means
xk = g kl xl (7.61) Rotation matrix around axis e1 on angle has the form
141
1 0 g kl = 0 cos 0 sin
0 sin .(7.62) cos
As a result of rotation we shall obtain x1 xk = x2 cos x3 sin .(7.63) x2 sin x3 cos
In particular if xk are the components of unit radius-vector in a spherical coordinate system and angle = /2 then from formula (7.63) we shall obtain sin ~ cos ~ = sin cos , sin ~ sin ~ = cos , cos ~ = sin sin , (7.64)
where ~ , ~ are the polar coordinates of vector xk in the basis ei , , are the polar coordinates of vector xk at the same basis. Solution of the system (7.64) gives
~ ~ sin = 1 sin2 sin2 , sin =
cos ~ .(7.65) 1 sin2 ~ sin2 ~
If one substitutes , obtained from (7.65) to equation (7.36) we shall obtain the equation of body rotation surface in initial basis ei
~ r0 (~, ~) = R(1 (~, ~ )).(7.66) One can to produce the factors C~kl from formula (7.56) substituting function F ( , ) on ~ . It should be noted that this operation is not substitution in definite one (~, ~ ) . Similarly and D kl integral (7.56) the variables , on ones ~ , ~ at which the integral value is not changed. If in bench mark hi "frosen-in" to the body the equation of surface in spherical coordinates of this one is invariant then relatively initial bench mark ei the surface of rotated body is described by other equation. It is clear that from equation (using (7.65)) F ( , ) = F ( (~ , ~ ), (~ , ~ )) = (~ , ~ )
in general case it is not followed that F ( , ) = ( , ) , so C~kl Ckl . Physically this means that induced dipole moment of arbitrary body depends on orientation of the conductor in the field.
142
~ Let mn is the body polarizability tensor in the bench mark ei after rotation. Then the polarizability tensor ik of the body in the bench mark hi will be determined by the formula
ik = gim gkn ~mn = gim ~mn gnkT , (7.67) T where gnk is the matrix obtained by means of transposition of matrix g kn . From (7.67) in matrix designations using (7.62) at = /2 we shall obtain
11 12 21 22 31 32
13 1 0 1 ~11 ~12 23 = 0 0 1 ~21 ~22 33 0 1 0 ~31 ~32
~13 1 0 1 ~23 0 0 1 = ~33 0 1 0
~11 ~13 ~12 = ~31 ~33 ~32 .(7.68) ~21 ~23 ~22 Formula (7.68) permits to determine the factors
12 = ~13 , 22 = ~33 , (7.69) ~ ~ where 13 and 33 was obtained by means of the rule above mentioned. One component is unknown 11 . To determine this one we shall rotate the body around axis e2 (or h2 ). Rotation matrix around axis e on angle has the form 2
cos gik ( ) = 0 sin
0 sin 1 0 .(7.70) 0 cos
Using (7.61) and (7.70) we shall obtain for arbitrary vector the action of rotation operator x1 cos x3 sin xk = x2 .(7.71) x1 sin x3 cos
If xk are the components of radius-vector in spherical coordinate system and = /2 then from (7.71) it is follows
sin ˆ cos ˆ = cos , sin ˆ sin ˆ = sin sin , cosˆ = sin cos , (7.72)
143
Solution of the system (7.72) has the form
cos = sin ˆ cos ˆ,
cos =
cosˆ .(7.73) 1 sin2 ˆ cos2 ˆ
Surface equation (7.36) in the new coordinates
rˆ0 (ˆ, ˆ) = R(1 (ˆ, ˆ)).(7.74) We shall obtain the factors Cˆ kl from formula (7.56) substituting F ( , ) by function
(ˆ, ˆ) . Let bˆmn is the body polarizability tensor in bench mark ei after rotation (7.70) at = /2 . The polarizability tensor ik of this body in bench mark hi is determined by formula similar (7.67):
ik = gim gkn ~mn = gim ~mn gnkT , (7.75) or in matrix designations taking into account (7.70) at = /2 we shall obtain
11 12 13 0 0 1 ˆ11 ˆ12 ˆ13 0 0 1 21 22 23 = 0 1 0 ˆ21 ˆ22 ˆ23 0 1 0 = 31 32 33 1 0 0 ˆ31 ˆ32 ˆ33 1 0 0 ˆ33 ˆ32 ˆ31 ˆ21 .(7.76) = ˆ23 ˆ22 ˆ13 ˆ12 ~11 From formulae (7.76) and (7.68) it is follows that
ˆ33 ~13 13 ik = ~13 ~33 23 .(7.77) 13 23 33 Relation (7.77) gives the components of polarizability matrix in bench mark ei for arbitrary body with shape like sphere or any body of quasi-cubic syngony. For example, we shall calculate the polarizability tensor of lengthened ellipsoid of revolution which has small difference from sphere. Let the revolution axis of ellipsoid coincides with direction e3 . Then ellipsoid surface equation may be represented in the form
144
r0 ( ) =
c
c (1
1 e 2 cos2
1 2 e cos2 ), a > b = c, (7.78) 2
where a , b, c are the lengths of ellipsoid semi-axes and e is the eccentricity. Comparing (7.78) and (7.36) we obtain
R = c, =
1 2 e , F = cos2 .(7.79) 2
From formula (7.56) we determine the necessary factors C21 , D21 , C20 , C00
C21 =
5 12
2
0
d cos2 Y2( 1) sin d , 0
where
3 Y2( 1) = P2(1) (cos ) cos = sin 2 cos. 2 It is follows from the last relations that C 21 = 0 and similarly D21 = 0 . Similarly we obtain that
C20 =
5 4
C00 =
2
0
1 4
2 d cos2 P2 (cos )d = , 0 3
2
0
1 d cos2 sin d = . 0 3
From formula (7.58) taking into account the obtained factors we obtain that
9 P = R 3 E[1 e2 ]e3.(7.80) 10 It is follows from formula (7.60) that
33 = R3 (1
9 2 e ), 13 = 23 = 0.(7.81) 10
For ellipsoid rotated around e1 on angle = /2 , the surface equation will be calculated in accordance with formula (7.66) where taking into account (7.65)
~ ~ ~ ~ ( , ) = cos2 = sin2 sin2 .(7.82)
145
From formula (7.56) taking into account the substitution F ( , ) (~ , ~ ) we determine ~ , C~ : ~ , C the necessary factors C~21 , D 20 00 21
5 C~21 = 12
~ ~ ~3 ~ ~ ~ d sin2 sin2 sin 2 cos d = 0, 0 2
2
2
0
~ = 5 D 21 12
0
~ ~ ~3 ~ ~ ~ d sin2 sin2 sin 2 cos d = 0. 0 2
Similarly we obtain that
5 C~20 = 4
2
0
1 ~ ~ ~ ~ ~ ~ d sin2 sin2 P2 (cos ) sin d = , 0 3
1 C~00 = 4
2
0
~ ~ ~ ~ ~ 1 d sin2 sin2 sin d = . 0 3
From formula (7.60) we find
~33 = R3 (1
3 2 ~ ~ e ), 13 = 23 = 0.(7.83) 10
For ellipsoid rotated around axis e2 on angle /2 , the surface is specified by formula (7.74) which taking into account (7.73), (7.79) result in relation
(ˆ, ˆ) = sin2 ˆ cos2 ˆ.(7.84) For factors Cˆ 21 , Dˆ 21 , Cˆ 20 , Cˆ 00 we obtain the values
5 Cˆ 21 = 12 5 Dˆ 21 = 12
2
0
3 dˆ sin2 ˆ sin2 ˆ sin 2ˆ cos ˆdˆ = 0, 0 2
2
0
3 dˆ sin2 ˆ sin2 ˆ sin 2ˆ cos ˆdˆ = 0. 0 2
Similarly we obtain that
5 Cˆ 20 = 4
2
0
1 1 dˆ sin2 ˆ sin2 ˆ (3 cos2 ˆ 1) sin ˆdˆ = , 0 2 3
1 Cˆ 00 = 4
2
0
1 dˆ sin2 ˆ sin2 ˆ sin ˆdˆ = . 0 3
146
From formula (7.60) we find
ˆ33 = R3 (1
3 2 e ), ˆ13 = ˆ23 = 0.(7.85) 10
From formulae (7.77), (7.81), (7.83), (7.85) we find the expression for polarizability tensor
1
ik = R 3
3 2 e 10 0 0
1
0
0
3 2 e 10
0
0
1
.(7.86)
9 2 e 10
The polarizability tensor is diagonalized since symmetry axis coincides with one of the coordinate axises e3 , 11 = 22 that is valid for all revolution bodies. Let us compare the formula (7.86) with exact solution obtained for ellipsoid of revolution in work [6]. It is known from this work that the main values of revolution ellipsoid polarizability tensor with axis directed along e3 have the form
(1) =
abc abc abc , (2) = (2) , (3) = (3) , (1) 3n 3n 3n
n (1) = n (2) =
1 e2 1 e 1 n (3) , n (3) = [ln 2e], 2 2e 3 1 e
e = 1
b2 , b = c < a.(7.87) a2
If the ellipsoid shape is like a sphere (namely formula (7.86) is intended for this case) then from (7.87) taking into account (7.78) and (7.79) we obtain
n (3)
1 2 2 1 1 c 1 e , n (1) = n (2) e 2 , a = R(1 e 2 ), 2 3 15 3 15 2 1 e
(1)
1 R 3 (1 e2 ) 3 2 R 3 (1 e2 ) = (2) , 1 10 1 e2 5
147
(3)
1 R 3 (1 e2 ) 9 2 R 3 (1 e2 ). 2 10 1 e2 5
So, relation (7.86) coincides with approximation of strict theory that is the confirmation of proposed method validity. 3.4. Some general polarizability properties of arbitrary shaped conducting bodies
Calculated estimation of the arbitrary construction transducer coupling to a measured field in general case is irrational because of computational complexity. Therefore in some cases it is sufficient to point out the some upper estimation taking into account the transducer coupling to a field. For example, it is possible to show that induced by the field dipole moment of smooth conducting body is smaller than one of conducting sphere in which the body may be inscribed. Since for technological reasons in many cases the transducer represents the body of revolution at first we shall show that for arbitrary smooth revolution bodies with axis parallel to field E the dipole moment | P | is not greater than one of the sphere in which the bodies may be inscribed. Let us break the arbitrary revolution body to rings with planes which are perpendicular to body revolution axis coinciding with axis z of Cartesian coordinate system 0, x, y, z . Let origin of coordinates 0 is located on the revolution axis and surface equation of revolution body has the form
r0 = r0 ( ), (7.88) where r0 is the radius-vector directed from origin of coordinates 0 to arbitrary surface point. is the angle between the radius-vector and axis z . Potential d1 from elementary charge dq is determined by known formula [97] d1 =
dq r0 n 1 ( ) Pn (cos ) Pn (cos ), (7.89) r0 n =0 r
where r (r, ) is the radius-vector from origin of coordinates to observation point located outside the body. is the angle between radius-vector and axis z . Elementary charge value on elementary ring with area dS is determined by the relation dq = dS = 2r0 r02 (
dr0 2 ) sin d , (7.90) d
where is the surface charge density. Potential 1 ( r , ) , where = cos , = cos provided that r r0 max has the form
1
1 ( r , ) = 2 ( ) r02 ( n =0
1
dr0 2 r ) (1 2 ) ( 0 ) n 1 Pn ( ) Pn ( )d.(7.91) d r
148
Solution for potential around conducting uncharged body in external homogeneous electric field E we find in the form = Er 1.(7.92)
From relations (7.92) and (7.91) we identically have
r 2 Er 3 = P Pn ( ) 1 n 1 dr r ( ) r02 ( 0 ) 2 (1 2 ) Pn ( )d.(7.93) n 1 1 0 d n =2 r
2
We took into account that the full body charge is equal to zero and dipole moment value P is generally calculated in accordance with the formula 1
P = 2 ( ) r02 ( 1
dr0 2 ) (1 2 ) r02 ( )d.(7.94) d
Let multiply relation (7.93) by and integrate between 1 and 1 at fixed value r = R , where R = r0 max corresponds to maximum radius of revolution body. Integrating we obtain
P = ER 3
3 2 1 R ( R, )d.(7.95) 2 1
It is known [6] that potential distribution to any electrostatic field satisfies to following theorem: "Function may to achieve maximum and minimum values only at the field boundaries." It is follows from formula (7.93) that at great distances
Er 2
2P r2
< 0.
At the conductor surface r = r0 ( ) we choose ( r0 ) = 0 . Since at great distances for any fixed 0 < 0 , and at conductor surface = 0 then it is follows from above mentioned theorem that is negative at any points if 0 . Considering that the currents in which > 0 are present we obtain the extremum presence for potentials outside the body boundaries that contradicts to above mentioned theorem. It is follows from aforesaid that in formula (7.95) integrand 0 and equality sign corresponds to the point belonging to conductor surface. Thus we obtain the inequality
P ER3 , (7.96) and equality sign corresponds to the spherical conductor case. For all known calculated conducting
149
bodies in homogeneous field, for example, such as thin cylinder, ellipsoid induced dipole moment satisfies to inequality (7.96) [6]. Our purpose is the extension of the bodies class inscribed in sphere for which induced electric dipole moment is smaller than value calculated for described sphere. Let the metallic body has a form like a sphere and its surface is specified by equation (7.36). Induced dipole moment of the body is determined by formula (7.58) and may be represented in the form
P = R 3 E P, (7.97) where
9 2 9 P = R 3 E{ D21e1 C21e2 [3 ( C20 C00 )]e3}.(7.98) 5 5 5 Evidently, additional dipole moment in general case is not parallel to vector of external field E . In this case the variation of modulus P is smaller then when P is parallel to E . Full dipole potential (1) is
(1)
P r R3 = 3 = 3 E r (1) , (7.99) r r
where (1) is determined from (7.57). If using revolution transformation one chooses a new coordinate system so that axis ~z of the new system coincides with direction of additional dipole moment P then additional dipole potential (1) will have axial symmetry relatively new axis ~ z. We shall construct the proof of this statement. If the surface of conducting body is described by equation (7.36) then the distortion of initial field E because of body influence in dipole approximation is smaller than one because of influence of sphere with radius r0 = R(1 ) . To prove the statement at first we shall calculate additional (in comparison with radius R sphere) dipole potential of sphere with radius r0 = R(1 ) . From general formula of function decomposition F ( , ) from (7.36) of spherical functions
F ( , ) = (CklYk( l ) DklYk( l ) )(7.47) k =0 l =0
it is follows that for the sphere the decomposition factors have the form
Ckl = k 0 l 0 , Dkl = 0.(7.100) Substitution (7.100) in third term of formula (7.55) at 0 = 0 gives
150
(1) =
3ER 3 ( 0) 3ER 3 Y1 = cos .(7.101) r2 r2
In new coordinate system where axis ~z is directed along additional dipole moment P (7.98), additional dipole potential of sphere with radius R(1 ) is
~ (1) =
3ER 3 ~ 2 [C00 C~20 ]Y~1( 0) .(7.102) 2 5 r
In this system surface F~ (~, ~ ) is specified in accordance with formula (7.47) with ~ and functions Y~( l ) (~, ~) , Y~( l ) (~, ~) , the connection between transformated factors C~ and D kl
k
kl
k
these ones is determined in accordance with formula (7.56). Under such conditions function F~ (~ , ~ ) 1 in all range of variable modification. From relation (7.56) we have
2 1 ~ [ 1 Y~ ( 0) ]F~(~, ~).(7.103) C~00 C~20 = d 2 5 2 2 Removing the brackets in expression under the integral sign we are convinced that this is not negative. 1 ~ ( 0) 3 ~ z2 3 2 ~ = cos 0.(7.104) Y2 = 2 2~ r2 2 Therefore it is possible to apply the known mean-value theorem that gives
2 1 ~ ~ ~ ~ [ 1 Y~ ( 0) ] = F~(~ , ~ ) 1.(7.105) C~00 C~20 = F (0 , 0 ) d 2 0 0 2 5 2 Comparing (7.101) and (7.102) we have ~ (1) |=| 3ER F~ (~ , ~ ) cos ~ || (1) |, | 0 0 r2 3
that proves the statement. Let us consider the revolution body around axis Z . For such bodies
Ckl = ck l 0 , Dkl = 0.(7.106) From formula (7.55) at 0 = 0 we find ( r, , ) = Er cos (1
R3 ) r3
151
3ER n =1
R n 1 n 1 nC n 1 [ Cn 1 ]Pn (cos ), (7.107) n 1 r 2n 3 2n 1
where Pn (cos ) are the Legendre's polynomials. For dipole potential of such body we obtain (1) =
ER 3 2 [1 3 (C0 C2 )] cos , (7.108) 2 r 5
and dipole moment P coincides with field direction E and axis z . Similarly it is possible to prove that electric field of uncharged revolution body with shape like sphere located in homogeneous electric field parallel to symmetric axis is smaller than field value of uncharged metallic sphere diameter of which is equal to maximum size of revolution body. When derivating both statements it is necessary to expand functions F ( , ) for the first case and F (cos ) for the second one in terms of spherical functions and Legendre's polynomials that is possible for any twice differentiable functions [30]. In conclusion we shall show that single body with shape (7.36) possesses smaller electric capacitance in comparison with described sphere. This result may be useful to upper distortion estimation of initial electric field E by transducers which have been acquired additinal potential V and additional charge q = CV distorted the field, where C is the transducer capacitance. Assuming in (7.55) E = 0 , to calculate C we shall use the formula
C=
1 dS , (7.109) 0
where surface charge density is calculated in accordance with the formula
= 0
0 0 |r = R (1F ) = n R(1 F ) 2
n 1 (CnmYn( m ) DnmYn( m ) )].(7.110) n (1 F ) n =0 m =0
[1
Taking into account that surface element dS may be represented in the form
dS = E1G F12 dd , where E1 = (
X 2 Y 2 Z 2 X Y Z ) ( ) ( ) , G1 = ( ) 2 ( ) 2 ( ) 2 ,
152
F1 =
X X Y Y Z Z ,
X = R(1 F ) sin cos , Y = R(1 F ) sin sin , Z = R(1 F ) cos , F = F ( , ), retaining the first order of smallness on we find that
C = 4 0 R[1
4
2
0
d sin d 0
n
| (n 1)(CnmYn( m ) DnmYn( m ) |].(7.111) n = 0 m =0
Taking into account the orthogonality of spherical functions it is follows from the last formula
C = 4 0 R[1
00
], (7.112)
where C00 may be calculated in accordance with the formula
C00 =
1 1 F ( , )Y0( 0)d = 4 4
2
0
d F ( , ) sin d .(7.113) 0
Let the surface specified by equation (7.36) possesses the property
0 F ( , ) 1, that is the surface is located in a spherical layer
R r0 R(1 ). Using the mean-value theorem from formula (7.113) we have
0 C00 = F (1 , 1 ) 1. For capacitance C we find
4 0 R C = 4 0 R(1 F ( , )) 4 0 R(1 ).(7.114) Brief summary. 1. The disturbances introduced in homogeneous electric field by charged and uncharged arbitrarily shaped bodies slightly differed from spherical one are determined in general form.
153
2. It is proved that in dipole approximation arbitrarily shaped bodies with surface (7.36) where F ( , ) is the twice differentiable function distort the field no more than metallic sphere in which these bodies may be inscribed. 3. If the body surface may be enclosed between two spheres with radiuses R and R(1 ) then electric capacitance of such single body is no more than one of sphere with radius R(1 ) and is no less than one of sphere with radius R . 4. The measuring error of electric intensity connected with induced body dipole moment is not greater than corresponding error for sphere with radius R(1 ) if the body surface is enclosed between radiuses R and R(1 ) . 3.5. Experimental results
While conducting the measurements, the problems posed were as follows: 1. To give experimental evidence for the validity of the method proposed for determining the polarizability tensor. It is evident that the method is verifiable on the bodies that can be calculated (ball, ellipsoid, disk) as well as on others of different degree of symmetry. 2. To expand experimentally the validity of the conclusion presented in [94] on the inscribed body polarizability being less than that of a circumscribed one. In paper [94] this conclusion is proved theoretically only for bodies inscribed in the sphere. The proof of this assertion for other shapes of the circumscribed body allows its universality (i.e. correctness for any shape of the circumscribed and inscribed body) to be spoken about almost plausibly. There is no doubt of this statement being of practical importance, which permits the upper estimate of the polarizability of any complex body inscribed in a geometrically simpler body of the known polarizability. 3. To determine quantitatively values of the polarizability tensor components for simplest geometry bodies not calculated yet (cube, cylinder etc.). This is of importance for estimating the polarizability of objects close in shape to the given bodies. 4. To provide an experimental evidence in favor of the validity of the relation between the electric and magnetic polarizadility of the conducting bodies of revolution derived above. Determining the electric polarizability tensor components was carried out in a flat capacitor, and measuring the magnetic polarizability tensor components - in a three-coil field-forming system (FS). Functional apparatus diagram and overall dimensions of FS are presented in [68]. While analyzing and representing the results, it is most convenient to operate on the electric and magnetic polarizability tensor components normalized to the calculated measure polarizabilities ( e.g., for the ball or disk) 0 and 0 respectively ( but not their absolute values ik and ik ). Under the condition C/C = 1 , L/L = 1 , from relation (7.10) for the normalized values of the polarizability components it follows that the expressions
ii Ci ii Li = , = , (7.115) 0 C0 0 L0 are valid, where C0 and L0 are a capacitance and an inductance respectively introduced into the system by a measure placed at the same point of the working zone of the field-forming system. An aluminium ball of diameter 2 R = 120 0.2 mm was used as a measure of the electric and magnetic polarizability. As known, the quantity of the electric polarizibility of the ball is = 4 0 R 3 . Hence
154
for the chosen value of R we have calc = 2.403 1014 F .m2 , with the relative error of determining
calc 3R/R = 0.5% . Since the condition C/C = 1 and meas = Cd 2 was satisfied under the measurements, then the total relative error of determining calc / meas may be estimated as follows
= [
( C ) C
]2 [2
d d
]2 [3
R R
]2 .(7.116)
The experiment has shown that at d=290 mm the value of the capacitance introduced by the ball amounted to 0.29 pF. In this case the quantities meas , calc differs by 1.5 % , which lies within the estimate = 3.6% corresponding to the value of (C ) = 0.01 pF, d = 1 mm. Thus, the method proposed for determining the polarizability tensor components is efficient. Consider the results obtained. Fig. 31 depicts dependence of the components , and
, of the electric and magnetic polarizability tensors on the geometry of the conducting circular cylinder inscribed in the sphere. The angle between the generatrix and the diagonal of the axial cross-section of the cylinder is chosen as a parameter characterizing the cylinder shape. From the figure it is seen that the polarizability of any cylinder inscribed in the ball is less than the ball polarizability. Coincidence within the accuracy of measuring the curves / 0 and / 0 indicates that for any cylinder there takes place the same relation between and
as for the
ball, i.e. formula (7.35) obtained in the second part of this chapter is valid for the cylinder of a circular cross-section.
Fig. 31. Polarizability of conducting cylinder Of utmost interest, from the viewpoint of proving the statement on the polarizability of an inscribed body being less than that of a circumscribed one, are the objects consisting of elements with a small radius of curvature. Fig. 32 presents results of measuring the polarizability of a complex body of revolution composed of identical conducting disks and a disk circumscribed in the cylinder with
155
the parameter = 45 . The values of and
in the figure are normalized to the polarizability
of the ball circumscribed about the cylinder. For the cylinder with the angle = 45 , from Fig. 31 / 0 / 0 0.55 . The dependences obtained confirm once again that the polarizability of an inscribed body does not exceed that of a circumscribed one. Fig. 33 presents results of measuring the polarizability of the cylinder whose lateral is made of the increasing number of parallel wires. From the plot it follows that the polarizability of such a body is less than that of the cylinder. The polarizability tensor components of two bodies of revolution inscribed in the sphere were measured: a circular cone of angle 60 and a body having a regular octagonal cross-section in the axial plane. The results presented in [68] support the validity of formula (7.35), within the accuracy of the measurements, and the relation between the polarizabilities of inscribed and circumscribed bodies. Measurements of the electric and magnetic polarizabilities of the cube and tetrahedron, inscribed in the ball at differend orientations of these bodies with respect to the field vector, confirm the globular structure of the cubic syngony body polarizability tensors. The dependences of introduced capacitance C on the angle between the body rotation axis and the field vector were measured for four bodies of revolution inscribed in the sphere of diameter 120 mm. The experimental results presented in Fig. 34 give evidence for the validity of calculated relation (7.14) (the continuous lines in Fig. 34).
Fig. 32. Polarizability of a complex body of revolution
Fig. 33. Polarizability of a cylindre composed of n parallel wires
156
Of particular interest is a comparison of the measured components of the polarizability tensor with their calculated values. The measurements have been performed of the components of the electric polarizability tensor for an ellipsoid of revolution and for a thin disc (Fig. 63, curves 1,4 ), whose exact analytic expressions are known [6]. The ellipsoid of revolution was chosen to have axes a = 60 mm b = c = 20 mm . The capacitance of a flat capacitor with a gap d = 206 mm amounted to C = 57.3 pF . The coupled capacitance of the ellipsoid oriented parallel and perpendicular to the electric field vector amounted to C 0.21 pF , C = 0.05 pF (Fig. 34 15 Fm2 . = 0 , = /2 ). From (7.13) we find 11 = 22 = 2.1 1015 Fm2 , 33 = 8.9 10
Fig. 34. Polarizability of bodies of revolution From [6] we find
11calc = 22calc =
=
4 0abc 4 0abc , 33calc = , n (1) = n (2) (1) 3n 3n (3)
b2 1 1 e2 1 e n e e (1 n (3) ), n (3) = ( l 2 ), = 1 .(7.116) a2 2 2e 3 1 e
From (7.116) we have 11calc = 1.995 1015 Fm2 , 33calc = 8.177 1015 Fm2 . The related deviations of the measured and calculated values amounted to 6.3% for 11 and 9% for 33 . Similar comparisons have been carried out for the thin disc as well (Fig. 34, curve 4). The coincidence of the experimental results with theoretical ones [6] for the ellipsoid of rotation and the thin disc indicates that the method proposed is effective. The present chapter substantiates the method allowing one to determine experimentally the
157
electric (magnetic) polarizability tensor of arbitrary conducting bodies by measuring an additinal capacitance (inductance) being introduced by a body into the field-forming system at different orientations of the body with respect to the field vector. The method is extendible to the bodies made of different matetials (magnetic, dielectric) and having complex structure ( e.g., containing electron components). The polarizability tensor defines the body dipole moment induced by the acting field and an additional one acting on the body due to a mirror image of the dipole being a reflection from the conducting surfaces located in the neighborhood. The polarizability of simplest geometry bodies has been measured. The polarizability of an inscribed body is shown to be less than that of a circumscribed one. A relation between the tensors of the electric and magnetic polarizability is found for conducting bodies of revolution. The paper results may be used to estimate errors of measuring the electric and magnetic field intensities in the neiborhood of the conducting surfaces for any field transducers. The results of even greater importance have been obtained to estimate the additional field acting on the object being tested in strip simulators of the electromagnetic field. Since the polarizability of such bodies is proportional to their volume, then in this case the polarizability tensor may be determined in the laboratory using the object scale model. The paper results may be also useful to solve the problem of diffraction on small arbitrarily shaped bodies.
158
Chapter 4 THE THEORY OF FUNCTIONALS OF STOCHASTIC BACKSCATTERED FIELDS
The theory of wave scattering by anisotropic statistically rough surfaces, which is an important part of statistical radiophysics, is considered. A new analytic method is developed and generalized for solving problems of radar imaging. The method involves analytic determination of the functionals of stochastic backscattered fields and can be applied to solve a wide class of physical problems with allowance for the finite width of an antenna’s pattern. The unified approach based on this method is used to analyze the generalized frequency response of a scattering radio channel, a generalized correlator of scattered fields, spatial correlation functions of stochastic backscattered fields, frequency coherence functions of stochastic backscattered fields, the coherence band of a spatial–temporal scattering radar channel, the kernel of the generalized uncertainty function, and the measure of noise immunity characterizing radar probing of the Earth’s surface or extended targets. The introduced frequency coherence functions are applied for thorough and consistent study of techniques for measuring the characteristics of a rough surface, aircraft altitude, and distortions observed when radar signals are scattered by statistically rough, including fractal, surfaces. To exemplify urgent applications, radiophysical synthesis of detailed digital reference radar terrain maps and microwave radar images that was proposed earlier is considered and improved with the use of the theory of fractals. 4.1. Introduction
At present, interest in processing of fields (spatial–temporal signals) scattered by statistically rough surfaces has grown substantially because of the following factors. In contrast to the past situation, today, the possibilities of solving various applied problems with the use of millimeter waves (MMWs) have again attracted the attention of researchers [1–7]. Lying between two classical (centimeter and optical) bands, MMWs often are more suitable for specific applications. This circumstance results from the progress of the MMW circuit technology and from the characteristic features of this band, in particular, the higher immunity of MMW radars to electromagnetic countermeasures [6, 7]. Digital processing of spatial–temporal signals and digital control of an antenna’s aperture (when an antenna is considered as a dynamic spatial–temporal filter) allows formation and reception of radar images (RIs) in almost real time. Thus, attempts to effectively use the spatial–temporal structure of the electromagnetic field for maximizing the amount of information obtained from received signals have stimulated the development of systems forming millimeter- and centimeter-wave RIs. The main problem of radar—detection and discrimination of targets in the presence of reflections from terrain and of intrinsic radar noise—has been solved simultaneously. An RI can generally be interpreted as the map (matrix) of specific radar cross sections (RCSs) * of a probed object or the signature (portrait) of a probed object in the case of a high angular resolution. When a probing beam is wide, a real RI is associated with an RCS map with smeared contours. Enhancement of RI resolution necessitates complex probing signals, such as chirp, nonlinearly frequency-modulated, or phasecode-shift keyed signals [1–9]. Processing of current images often yields specific detailed digital radar maps (DDRMs) of terrains or reference maps [1, 2, 7]. At present, rapid progress in the development of methods of DDRM synthesis is taking place [7, 10].
159
Spatial–temporal signal processing requires the use of array antennas. However, theoretical investigations of problems of radar imaging and detailed digital radar mapping often deal with a continuous antenna aperture. This assumption simplifies solution of the aforementioned problems and allows determination of potential radar characteristics exhibited in the case when the entire space assigned for observations is employed [4, 7, 11, 12]. When an RI is formed, the structure and parameters of the wave field induced by a distant statistically rough surface in the region where this field is analyzed depend on the reception point and surface characteristics. If the above factors are taken into account, it is fundamentally important to obtain a complete mathematical description of the scattered field in the space–time continuum [7]. Therefore, in the late 1970s, I formulated the problem of theoretical simulation of a spatial–temporal MMW signal with allowance for a linear radio channel consisting of an antenna aperture, the atmosphere, and a chaotic vegetationless surface and the problem of forming new classes of radar signatures [7, 13]. This study is the first to systemically consider a unified approach to mathematical description of spatial–temporal scattering radio channels. The approach proposed allows investigations of wide classes of various problems in the statistical theory of diffraction. In the study, methods of investigation and analysis are developed that provide for explicit formulas and noticeable results in specific physical problems. Some of these techniques are new [7] and have not been reported in the literature. The method developed in the study is used for solution of certain physical problems. Thus, the purpose of the study is to present numerous results consistently and as comprehensively as possible and to demonstrate application of these results to solution of practical problems that often arise in radiophysics, radar, acoustics, and optics. The presented results were obtained at the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences before the 1990s, and some of the results have been employed at a number of organizations. Most of the results are reported in my doctoral thesis [7] and summarized in monographs [1, 2, 5, 6]. Later, my scientific interests concentrated exclusively on investigations, development, and application of fractal methods of radiophysical data processing based on the theory of fractal measures, fractal operators, and scaling relationships. The effectiveness of radiophysical investigations can be enhanced substantially if the fractal character of wave phenomena that occur during all the stages of wave radiation, scattering, and propagation in various media is taken into account. In addition to pure theoretical significance, fractal methods are important for solution of practical radar and telecommunication problems and problems of medium monitoring on various space–time scales [1–3]. 4.2. The angular spectrum of wave fields
Since the microwave band is located close to the optical band, it is suitable to solve problems of diffraction of microwave fields with the use of integral Fourier transforms, which is an optical technique, and spatial (angular) and temporal spectra [7, 14–19]. Assume that a monochromatic wave propagates in free space along the z axis in the region where z > 0. In a homogeneous medium without currents or charges, complex amplitude E≡E(x,y,z) of a monochromatic wave satisfies the Helmholtz equation 2E k 2E 0
A partial solution to Eq. (1) has the form
(1)
160 E ( x, y , z ) E 0 ( x, y ,0) exp(ik r ) E 0 ( x, y,0) exp[i ( k x x k y y k z z )]
(2)
and describes a plane wave. In (2), E0 ( x, y,0) is the complex wave amplitude in the plane z 0 , is the wave vector with
the rojections {k x , k y , k z k 2 k x2 k y2 } , and r x0 x y0 y z 0 z is the radius vector of an observation point. If, inCartesian coordinates (x, y, z), the irection of wave propagation is specified by angles α, β, and γ, we have cos 2 k x / k , cos 2 k y / k , cos2 k z / k , cos2 cos2 cos2 1 .
(3)
The 2D Fourier transform of function E ( x, y, z ) has the form
E 0 ( x, y,0)
1 4 2
F0 ( x , y ) exp[i( x x y y )]d x d y ,
(4)
where F0 ( x , y )
E 0 ( x, y ) exp[ i ( x x y y )]dxdy .
(5)
Comparing expressions (2) and (4) and taking into account (3), we see that the integrand in (4) can be regarded as a plane wave with the direction cosines
cos f x , cos f y , cos 1 (f x ) 2 (f y ) 2 .
(6)
The independent variables
fx
y k y cos x k x cos , fy 2 2 2 2
(7)
are linear spatial frequencies of the dimension that is the reciprocal of the unit length. For any fixed two frequencies { f x , f y } , the phase of the elementary function exp[i 2( f x x f y y ) is zero along the line described by the equation
y
fx n x , fy fy
where n is an integer. These parallel straight lines form a set with the spatial period
(8)
161
L ( f x2 f y2 ) 1 / 2 ,
(9)
and their slope with respect to the x axis is specified by the angle =arctg f y / f x .
(10)
The complex amplitude of a plane wave from (4) equals F0 ( x , y ) d x d y . Therefore, function (5), which can be represented as
F0 (
cos cos cos cos , ) E 0 ( x, y,0) exp[i( x y )]dxdy .
(11)
is the angular (spatial) spectrum of perturbation E0 ( x, y,0) . In order to find a general solution, let us define the angular spectrum of the perturbation in a plane that is parallel to the XOY plane and has arbitrary coordinate z in the following form: F0 ( x , y , z )
E 0 ( x, y, z ) exp[ i ( x x y y )]dxdy .
(12)
Then, perturbation E(x, y, z) can be represented as
E 0 ( x, y , z )
1 4 2
F ( x , y , z ) exp[i( x x y y )]d x d y .
(13)
The substitution of (13) into Helmholtz equation (1) yields the solution
F ( x , y , z) F0 ( x , y ) exp(iz k 2 2x 2y ) .
(14)
for the traveling-wave mode. Relationship (14) implies that, as the distance of point z from the origin grows, the angular spectrum changes. When k 2 2x 2y 0
(15)
this change manifests itself in changes of relative phases of different components of the angular spectrum. These phase shifts result from the fact that plane waves propagating at different angles to the z axis cover different distances by the moment when they reach a point considered. When
162
k 2 2x 2y 0
(16)
expression (14) describes plane nonuniform waves whose amplitudes decrease exponentially as the distance from the point z = 0 grows. The limiting case k 2 2x 2y 0
(17)
corresponds to plane waves that propagate perpendicularly to the z axis and do not transfer energy in the z direction. A perturbation occurring at arbitrary point (x, y, z) can be expressed through the angular spectrum as
E ( x, y , z )
1 4 2
F ( x , y ) exp(iz k 2 2x 2y ) exp[i( x x y y )]d x d y . (18)
0
Despite the formal resemblance, representation (18) and the Fourier representation should not be confused. In contrast to representation (18), which involves the angular spectrum, the Fourier representation of a function of three real arguments contains triple integrals rather than double integrals. The assumption that field E(x, y, z) is known only in a finite region rather than in the entire space implies that the Fourier decomposition (in contrast to representation (18)) is not unique and does not yield a representation in terms of wave-field modes. In radiophysics, integral (18) is known as the Rayleigh representation. Spatial frequencies must satisfy the single condition 2x 2y 2z
2 . c2
(19)
The function under consideration satisfies the wave equation only when condition (19) is fulfilled. It is seen from relationship (14) that a spatial layer of thickness z performs signal transformation as a linear dispersion filter with the transfer function
K ( x , y ) exp(iz k 2 2x 2y ) . Amplitude–frequency and phase–frequency characteristics K ( x , y )
(20) and ( x , y )
have the form 2 2 2 K ( x , y ) 1 , ( x , y ) z k x y ,
in frequency range (15) and
(21)
163
K ( x , y ) exp( z 2x 2y k 2 ) , ( x , y ) 0 .
(22)
in frequency range (16). The passband of such a filter is limited by the cutoff frequency
f x2 f y2 1 / .
(23)
Hence, the passband of a free-space region of length z decreases as wavelength λ grows (at fixed z) or as length z grows (at fixed λ). The well-known Weyl decomposition [20] can be regarded as a representation in the form of the angular spectrum of the wave field produced in free space by a point source located at the origin. The Hankel transformation and formula (18) yield the plane-wave decomposition of the field of an outgoing spherical wave, 2 2 2 exp(ikr ) i exp[i ( x x y y ) iz k x y ] d x d y . 2 r k 2 2x 2y
(24)
Interestingly [19], the well-known formula
sin(kr ) 1 exp(ik r )d . kr 4 4
(25)
can be obtained with the use of the Weyl decomposition for outgoing and incoming spherical waves. Therefore, from the physical viewpoint, the function sin( kr ) /( kr ) is a superposition of uniform plane waves that propagate omnidirectionally and have the amplitudes 1 / 4 . Dirichlet and Neumann boundary value problems for the Helmholtz equation in a half-space are solved in monograph [19] with the help of the Weyl transformation and the concept of an angular spectrum. A solution to the Dirichlet problem or the Rayleigh diffraction formula of the first kind for the half-space z 0 has the form E ( x, y , z )
e ikR 1 E ( x ' , y ' , 0 ) z R 2 0
dx' dy ' ,
(26)
where boundary value E0 ( x, y,0) is specified on the plane z 0 . A solution to the Neumann problem or the Rayleigh diffraction formula of the second kind for the halfspace z 0 has the form E ( x, y , z )
1 E ( x' , y ' , z ) e ikR R 2 z z 0
dx' dy '
(27)
164
where the boundary value of the derivative E ( x, y , z ) / z is specified on the plane z 0 . Note once again the dualism of variables x and y [16]. In formulas (4) and (5), variables
x and y have the meaning of spatial frequencies. At the same time, in formula (18), variables x and y determine the direction of propagation of plane waves. This direction is specified by angles (3) and (6). The spatial spectrum of complex amplitudes distributed over a plane and the angular spectrum of radiation may differ substantially. 4.3. The angular spectrum of modulated waves
The above solutions have been obtained under the assumption that the field under consideration is monochromatic. In a linear approximation, a solution can be constructed for the general case of nonmonochromatic radiation [7, 15, 16]. Let arbitrary field E ( x, y.t ) be specified on the plane z 0 . By representing this field as a sum of harmonic components, we can consider each of these as a monochromatic field satisfying relationship (18). Then, a general solution to the wave problem is obtained via the Fourier transform where integration is performed over all frequency components: 2 iz 2 2 F ( , ) exp x y x y 8 3 c2 exp[i ( x x y y t )]d x d y d ,
E ( x, y , z , t )
1
F ( x , y )
E ( x, y ,0) exp[ i ( x x y y )]dxdy ,
(28)
(29)
Here, E ( x, y,0) is the spectral density of the complex amplitude. If the functions of time and coordinates involved in the above relationships are factorable, E ( x, y , t ) E ( x, y ) E (t ) ,
(30)
then solution (28) can be simplified to take the form
E ( x, y , z , t )
1 F () E ( x, y, z ) exp(it )d , 2
(31)
where field E ( x, y, z) , which is a function of as well, can be found from integral relationship (18). Furthermore, this description is valid when a function is not factorable but can be represented as a sum of factorable functions [16]:
E ( x, y, t ) E n ( x, y )E n (t ) . n
(32)
165
In (32), value En ( x, y) is the distribution of an individual source producing oscillation
E n (t ) . For each component, it is necessary to use expression (31) and, then, sum the results obtained. In various radio systems, information on an object observed is extracted from the field reradiated or radiated by the object. It is always a relatively small region with this field that is incident on an antenna’s aperture and examined. The pattern (i.e., the main characteristic) of an antenna is the Fourier transform of amplitude–phase distribution A(s ) over aperture d : F (u ) A( s )e ius ds .
(33)
In (33), integration is performed over the plane of the antenna’s aperture with respect to coordinates such that the antenna’s aperture coincides with a coordinate plane. The physical meaning of an antenna’s pattern is the frequency response of a spatial filter connected in series with free space [16]. Scanning performed by means of antenna rotation is equivalent to tuning the corresponding spatial filter to another frequency without changing the filter’s shape. Neither the frequency response of an antenna regarded as a spatial filter nor the antenna’s pattern can be specified arbitrarily. This function has a bounded spatial spectrum and belongs to a certain class of functions [21]. In practice, the angular spatial frequencies
x
x fx y fy , y , c c
(34)
are often used [11]. These frequencies characterize the rate of variation of a plane harmonic wave with frequency f along the directions specified by direction cosines that are measured relative to the x and y axes of the aperture, respectively:
(cos ) 2 x cos 2
y cos x cos , (cos ) 2 y cos 2 ,
(35)
For the linear spatial frequencies specified by formulas (6) and (7), we have
( x) 2f x x 2
x cos y cos , ( y ) 2f y y 2 .
(36)
Since the right-hand sides of relationships (35) and (36) coincide, it is possible to introduce frequencies x, y or f x , y the choice depends on specific circumstances [7, 11]. Both of the approaches can be applied to analyze spectra of narrowband signals, because the form and width of the spatial-frequency spectrum are determined by an antenna. For example, the width of the spatial-frequency spectrum is determined by the antenna’s pattern for linear spatial frequencies f x , y and by the aperture function for angular spatial frequencies x, y . These characteristics of the antenna are related through the Fourier transforms; therefore, under the aforementioned conditions, both of the approaches are equivalent.
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When angular spatial frequencies x, y are used, the analysis of wideband signals is impeded by the ambiguous correspondence between the generalized angular coordinate and spatial frequency. For different components of the frequency spectrum of a signal, one value of the angular coordinate is associated with different values of the spatial frequency. In the case of spherical waves, the analysis is impeded more substantially when frequencies x, y are used, because each x, y component depends on all coordinates of the antenna’s aperture and on all coordinates of a target. The spatial and temporal characteristics of a radio system cannot be considered independently when signals with wider spectra and ultrawideband (UWB) signals or ultrashort electromagnetic pulses are analyzed [5]. For example, one can speak about the pattern formed by a given instant. Thus, when UWB signals are used, the dimension of the space of radar characteristics and features, i.e., radar signatures, abruptly increases owing to the dynamics of the processes developing in a system [1–4, 7, 22].
4.4. Simulation of the spatial-temporal structure of the field scattered by a statistically rough anisotropic surface: a mathematical model taking into accout the effect of an antenna Let us apply the spectral approach presented above and take into account an antenna’s pattern,
G , to obtain, for a wide range of incidence angles , a general analytic solution in the 3D case, R
R3, for the field of a monochromatic wave scattered by a statistically rough surface [7, 13]. In a real situation when RIs of terrains with objects or DDRMs are formed, probing angles often range in the interval 30 0...40 0 . Let us analytically solve the 2D scattering problem in the Kirchhoff approximation or with the method of a tangent plane (MTP) [5–7, 23, 24]. The applicability conditions for this method can be generalized as follows: the height of surface irregularities is much greater than unity, l / 1 ; the smoothness of a surface is specified by the inequality a / 1 ; and the flatness of a statistically rough surface (X', Y', Z') that is flat on the average is determined by the inequality 2
