Electromagnetic Waves Propagation in Complex Matter Part 13 docx

High Frequency Techniques: the Physical Optics Approximation and the Modified Equivalent Current Approximation (MECA) 227 the observation points analytically, the incident wave is supposed to be a plane wave which impinges on the surface and generates a current density distribution with constant amplitude and linear phase variation. Assuming a flat triangular facet, the radiation integral can be solved by parts. The good behaviour was proven in the validation examples, where the results from the frequency sweep in the high frequency region agreed the theoretical values. Likewise, an excellent overlapping was obtained for different angles of incidence when dealing with a non-PEC electrically large surface. Because one of the constraints to employ PO and MECA is the determination of line of sight between the source and the observation points, some algorithms to solve the visibility problem were described. The classic methods are complemented by acceleration techniques and then, they are translated into the GPU programming languages. The Pyramid method was explained as an example of fast algorithm which was specifically developed for evaluating the occlusion by flat facets. Undoubtedly, this can be employed in joint with the MECA formulation, but the Pyramid method can also be helpful in other disciplines of engineering. Throughout the section “ Application examples”, the way MECA becomes a powerful and efficient method to tackle different scattering problems for electrically large scenarios was satisfactorily demonstrated by means of the example consisting in the evaluation of the radio electric coverage in a rural environment. In addition to this, other fields of application were suggested from the RCS computation to imaging techniques, covering a wide range of electromagnetic problems. 7. References Arias, A.M., Rubiños, J.O., Cuiñas, I., & Pino, A.G. (2000). Electromagnetic Scattering of Reflector Antennas by Fast Physical Optics Algorithms, Recent Res. Devel. Magnetics, Vol. 1, No. 1, pp. 43-63 Adana, F.S. de, Lozano, P., Gisbert, F., Sudupe, I., Pérez, J., & Cátedra, M.F. (2000) Application of the PO to the Computation of the Monostatic RCS of Arbitrary Bodies Modeled by Plane Facets of Dielectric and Magnetic Material, Proceedings of 2000 USCN/URSI National Radio Science Meeting , Salt Lake City, Utah, USA, 2000 Balanis, C. A. (1989). Advanced Engineering Electromagnetics, John Wiley & Sons, ISBN 978- 0471621942, New York, USA Bittner, J., & Wonka, P. (2003). Visibility in Computer Graphics, Environment and Planning B: Planning and Design , Vol. 30, No. 5, Boag, A., & Letrou, C. (2003). Fast Radiation Pattern Evaluation for Lens and Reflection Antennas. IEEE Transactions on Antennas and Propagation, Vol. 51, No. 5, pp. 1063- 1068 Burkholder, R.J. & Lundin, T. (2005). Forward-Backward Iterative Physical Optics Algorithm for Computing the RCS of Open-Ended Cavities. IEEE Transactions on Antennas and Propagation , Vol.53, No.2, pp. 793- 799 Cátedra, M. F., Pérez, J., Sáez de Adana, F., & Gutiérrez O., (1998). Efficient Ray-Tracing Techniques for Three Dimensional Analyses of Propagation in Mobile Communications: Application to Picocell and Microcell Scenarios. IEEE Antennas and Propagation Magazine , Vol. 40, No. 2, pp. 15-28 Electromagnetic Waves Propagation in Complex Matter 228 Cátedra, M.F., & Arriaga, J.P. (1999). Cell Planning for Wireless Communications, Artech House, ISBN 978-0890066010, Boston, USA Dewey, B.R. (1988). Computer graphics for Engineers, Harpercollins College Div., ISBN 978- 0060416706, USA Engheta, N., Murphy, W.D., Rokhlin, V., & Vassiliou, M.S. (1992). The Fast Multipole Method (FMM) for Electromagnetic Scattering, IEEE Transactions on Antennas and Propagation , Vol. 40, No. 6, pp. 634-641 Foley, J.D. (1992). Computer Graphics: Principles and Practice in C (2 nd edition), ISBN 0201848406, Addison-Wesley, USA Fuch, H., Kedem, Z.M., & Naylor, B.F. (1980). On Visible Surface Generation by Priori Tree Structures. ACM SIGGRAPH Computer Graphics, Vol. 14, No. 3, pp. 124-133 Glassner, A.S. (1989). An Introduction to Ray Tracing, Academic Press, ISBN 978-0-12-286160- 4, San Diego, USA Gordon, D., & Chen, S. (1991). Front-to-Back Display of BSP Trees. IEEE Computer Graphics and Applications , Vol. 11, No. 5, pp. 79-85 Griesser, T., & Balanis, C. (1987). Backscatter Analysis of Dihedral Corner Reflectors Using Physcal Optics and the Physical Theory of Diffraction, IEEE Antennas and Propagation Magazine , Vol. 35, No. 10, pp. 1137-1147 Harrington, R.F. (2001). Time-Harmonic Electromagnetic Fields (2 nd edition), McGraw Hill, ISBN 978-0471208068, USA Hodges, R.E., & Rahmat-Samii, Y. (1993). Evaluation of Dielectric Physical Optics in Electromagnetic Scattering, Proceedings 1993 Antennas and Propagation Society InternationalSymposium , USA, 1993 Keller, J.B. (1962). Geometrical Theory of Diffraction. Journal of the Optical Society of America, Vol. 52, No. 3, pp. 116-130 Kempel, L.C., Chatterjee, A., & Volakis, J.L. (1998). Finite Element Method Electromagnetics (1 st edition), IEEE, USA Lorenzo, J. A. M., Pino, A. G., Vega, I., Arias, M., & Rubiños, O. (2005). ICARA: Induced- Current Analysis of Reflector Antennas. IEEE Antennas and Propagation Magazine, Vol.47, No.2, pp. 92-100 Meana, J. G., Las-Heras, F., & Martínez-Lorenzo, J. Á. (2009). A Comparison Among Fast Visibility Algorithms Applied to Computational Electromagnetics. Applied Computational Electromagnetics Society Journal , Vol.24, No.3, pp. 268-280 Meana, J. G., Martínez-Lorenzo, J. Á., Las-Heras, F., & Rappaport, C. (2010). Wave Scattering by Dielectric and Lossy Materials Using the Modified Equivalent Current Approximation (MECA). IEEE Transactions on Antennas and Propagation, Vol. 58, No. 11, pp. 3757-3761 Medgyesi-Mitschang, L.N., Putnam, J.M., & Gedera, M.B. (1994). Generalized Method of Moments for Three-Dimensional Penetrable Scatterers, Journal of the Optical Society of America A , Vol. 11, No. 4, pp. 1383-1398 Pathak, P.H., & Kouyoumjian, R.G. (1974). A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface. Proceedings of the IEEE, Vol. 62, No. 11, pp. 1448-1461 High Frequency Techniques: the Physical Optics Approximation and the Modified Equivalent Current Approximation (MECA) 229 Papkelis, E.G., Psarros, I., Ouranos, I.C., Moschovitis, C.G., Karakatselos, K.T., Vagenas, E., Anastassiu, H.T., & Frangos, P.V. (2007). A Radio-Coverage Prediction Model in Wireless Communication Systems Based on Physical Optics and the Physical Theory of Diffraction. IEEE Antennas and Propagation Magazine, Vol. 49, No. 2, pp. 156-165 Rappaport, C.M., & McCartin, B.J. (1991). FDFD Analysis of Electromagnetic Scattering in Anisotropic Media Using Unconstrained Triangular Meshes. IEEE Transactions on Antennas and Propagation , Vol. 39, No. 3, pp. 345-349 Rengarajan, S.R., & Gillespie, E.S. (1988). Asymptotic Approximations in Radome Analysis. IEEE Transactions on Antennas and Propagation, Vol. 36, No. 3, pp. 405-414 Ricks, T., & Kuhlen, T. (2010). Accelerating Radio Wave Propagation Algorithms by Implementation on Graphics Hardware, In: Wave Propagation in Materials for Modern Applications , Andrey Petrin, pp. 103-122, Intech, Vienna, Austria Rokhlin, V. (1985). Rapid Solution of Integral Equations of Classical Potential Theory. Journal of Computational Physics , Vol. 60, No. 9, pp. 187-207 Rokhlin, V. (1990). Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions. Journal of Computational Physics, Vol. 86, No. 2, pp. 414-439 Ross, R.A. (1966). Radar Cross Section of Rectangular Flat Plates as a Function of Aspect Angle. IEEE Transactions on Antennas and Propagation, Vol. 14, No. 8, pp. 329-335 Rossi, J.P., & Gabillet, Y. (2002). A Mixed Launching/Tracing Method for Full 3-D UHF Propagation Modeling and Comparison with Wide-Band Measurements. IEEE Transactions on Antennas and Propagation , Vol. 50, No. 4, pp. 517-523 Saeedfar, A., & Barkeshli, K. (2006). Shape Reconstruction of Three-Dimensional Conducting Curved Plates Using Physical Optics, NURBS Modeling, and Genetic Algorithm. IEEE Transactions on Antennas and Propagation, Vol.54, No.9, pp. 2497-2507 Sáez de Adana, F., González, I., Gutiérrez, O., Lozano, P., & Cátedra, M.F. (2004) Method Based on Physical Optics for the Computation of the Radar Cross Section Including Diffraction and Double Effects of Metallic and Absorbing Bodies Modeled with Parametric Surfaces. IEEE Transactions on Antennas and Propagation, Vol. 52, No. 12, pp. 3295-3303 Shreiner, D. (2004). OpenGL Reference Manual: the Official Reference Document to OpenGL,Version 1.4 , Addison-Wesley, ISBN 978-0321173836, London, UK Staelin, D.H., Morgenthaler, A.W., & Kong, J.A. (1993). Electromagnetic Waves, Prentice Hall, ISBN 978-0132258715, USA Taflove, A., & Umashankar, K.R. (1987). The Finite Difference Time Domain FD-TD Method for Electromagnetic Scattering and Interaction Problems. Journal of Electromagnetic Waves and Applications , Vol. 1, No. 4, pp. 363-387 Ufimtsev, P.Y. (1962). Method of Edge Waves in the Physical Theory of Diffraction, Izd-vo Sov. Radio , pp. 1-243 (translated by U.S. Air Force Foreign Technology Division, Wright-Patterson AFB, OH) Uluisik, C., Cakis, G., Cakis, M., & Sevgi, L. (2008). Radar Cross Section (RCS) Modeling and Simulation, Part 1: A Tutorial Review of Definitions, Strategies, and Canonical Examples. IEEE Antennas and Propagation Magazine, Vol. 50, No. 1, pp. 115-126 Electromagnetic Waves Propagation in Complex Matter 230 Van-Bladel, J. (2007). Electromagnetic Fields (2 nd edition), IEEE Press,Wiley-Interscience, ISBN 978-0471263883, USA Part 4 Propagation in Guided Media 0 Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode Hiroshi Toki and Kenji Sato Research Center for Nuclear Physics (RCNP), Osaka University and National Institute of Radiological Sciences (NIRS) Japan 1. Introduction In the modern life we depend completely on the electricity as the most useful form of energy. The technology on the use of electricity has been developed in all directions and also in very sophisticated manner. All the electric devices have to use electric power (energy) and they use both direct current (DC) and alternating current (AC). Today a powerful technology of manipulation of frequency and power becomes available due to the development of chopping devices as IGBT and other methods. This technology of manipulating electric current and voltage, however, unavoidably produces electromagnetic noise with high frequency. We are now filled with electromagnetic noise in our circumstance. This situation seems to be caused by the fact that we do not have a theory to describe the electromagnetic noise and to take into account the effect of the circumstance in the design of electric circuit. We have worked out such a theory in one of our papers as "Three-conductor transmission-line theory and origin of electromagnetic radiation and noise" (Toki & Sato (2009)). In addition to the standard two-conductor transmission-line system, we ought to introduce one more transmission object to treat the circumstance. As the most simple object, we introduce one more line to take care of the effect of the circumstance. This third transmission-line is the place where the electromagnetic noise (electromagnetic wave) goes through and influences the performance of the two major transmission-lines. If we are able to work out the three-conductor transmission-line theory by taking care of unwanted electromagnetic wave going through the third line, we understand how we produce and receive electromagnetic noise and how to avoid its influence. To this end, we had to introduce the coefficient of potential instead of the coefficient of capacity, which is used in all the standard multi-conductor transmission line theories (Paul (2008)). We are then able to introduce the normal mode voltage and current, which are usually considered in ordinary calculations, and at the same time the common mode voltage and current, which are not considered at all so far and are the sources of the electromagnetic noise (Sato & Toki (2007)). We are then able to provide the fundamental coupled differential equations for the TEM mode of the three-conductor transmission-line theory and solve the coupled equations analytically. As the most important consequence we obtain that the main two transmission-lines should have the same qualities and same geometrical shapes and their distances to the third line should be the same in order to decouple the normal mode from the common mode. The symmetrization is the key word to minimize the influence of the circumstance and hence the electromagnetic noise to the electric circuit. The symmetrization makes the normal mode decouple from the common mode and hence we are able to avoid the 9 2 Will-be-set-by-IN-TECH influence of the common mode noise in the use of the normal mode (Toki & Sato (2009)). The symmetrization has been carried out at HIMAC (Heavy Ion Medical Accelerator in Chiba) (Kumada (1994)) one and half decade ago and at Main Ring of J-PARC recently (Kobayashi (2009)). Both synchrotrons are working well at very low noise level. As the next step, we went on to develop a theory to couple the electric circuit theory with the antenna theory (Toki & Sato (2011)). This work is motivated by the fact that when the electromagnetic noise is present in an electric circuit, we observe electromagnetic radiation in the circumstance. In order to complete the noise problem we ought to couple the performance of electric circuit with the emission and absorption of electromagnetic radiation in the circuit. To this end, we introduce the Ohm’s law as one of the properties of the charge and current under the influence of the electromagnetic fields outside of a thick wire. As a consequence of the new multi-conductor transmission-line theory with the antenna mode, we again find that the symmetrization is the key technology to decouple the performance of the normal mode from the common and antenna modes (Toki & Sato (2011)). The Ohm’s law is considered as the terminal solution of the equation of motion of massive amount of electrons in a transmission-line of a thick wire with resistance, where the collisions of electrons with other electrons and nuclei take place. This consideration is able to put the electrodynamics of electromagnetic fields and dynamics of electrons in the field theory. We are also able to discuss the skin effect of the TEM mode in transmission-lines on the same footing. In this paper, we would like to formulate the multi-conductor transmission-line theory on the basis of electrodynamics, which includes naturally the Maxwell equations and the Lorentz force. This paper is arranged as follows. In Sect.2, we introduce the field theory on electrodynamics and derive the Maxwell equation and the Lorentz force. In Sect.3, we develop the multiconductor transmission-line (MTL) equations for the TEM mode. We naturally include the antenna mode by taking the retardation potentials. In Sect.4, we provide a solution of one antenna system for emission and absorption of radiation. In Sect.5, we discuss a three-conductor transmission-line system and show the symmetrization for the decoupling of the normal mode from the common and antenna modes. In Sect.6, we introduce a recommended electric circuit with symmetric arrangement of power supply and electric load for good performance of the electric circuit. Sect.7 is devoted to the conclusion of the present study. 2. Electrodynamics We would like to work out the multiconductor transmission-line (MTL) equation with electromagnetic emission and absorption. To this end, we should work out fundamental equations for a multiconductor transmission-line system by using the Maxwell equation and the properties of transmission-lines. We shall work out electromagnetic fields outside of multi-conductor transmission-lines produced by the charges and currents in the transmission-lines. In this way, we are able to describe electromagnetic fields far outside of the transmission-line system so that we can include the emission and absorption of electromagnetic wave. For this purpose, we take the electrodynamics field theory, since a multiconductor transmission-line system is a coupled system of charged particles and electromagnetic fields. In this way, we are motivated to treat the scalar potential in the same way as the vector potential and find it natural to use the coefficients of potential instead of the coefficients of capacity as the case of the coefficients of inductance. We discuss here the dynamics of charged particles with electromagnetic fields in terms of the modern electrodynamics field theory. For those who are not familiar to this theory, you can skip this paragraph and start with the equations (6) and (7). In the electrodynamics, we 234 Electromagnetic Waves Propagation in Complex Matter Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode 3 have the gauge theory Lagrangian, where the interaction of charge and current of a Fermion (electron) field ψ with the electromagnetic field A μ is determined by the following Lagrangian, L = 1 4 F μν (x)F μν (x)+ ¯ ψ (iγ μ D μ −m)ψ . (1) with D μ = ∂ μ − ieA μ , where A μ is the electromagnetic potential. Here, F μν (x)=∂ μ A ν (x) − ∂ ν A μ (x) is the anti-symmetric tensor with the four-derivative defined as ∂ μ = ∂ ∂x μ =( ∂ c∂t , ∇) and the four-coordinate as x μ =(ct, x). Here, electrons are expressed by the Dirac field ψ, which possesses spin as the source of the permanent magnet and therefore we do not have to introduce the notion of the perfect conductor anymore (Maxwell (1876)). The vector current is written by using the charged field as j μ = ¯ ψγ μ ψ. The variation of the above Lagrangian with respect to A μ provides the Maxwell equation with a source term expressed in the covariant form (Maxwell (1876)). ∂ μ F μν (x)=ej ν (x) (2) They are Maxwell equations, which become clear by writing explicitly the anti-symmetric tensor in terms of the electric field E and magnetic field B. F μν = ⎛ ⎜ ⎜ ⎝ 0 1 c E x 1 c E y 1 c E z − 1 c E x 0 −B z B y − 1 c E y B z 0 −B x − 1 c E z −B y B x 0 ⎞ ⎟ ⎟ ⎠ (3) Here, E = −∇V − ∂A ∂t and B = ∇×A. The two more equations are explicitly written as ∇·E = 1 ε q and ∇×B − 1 c 2 ∂E ∂t = μj by using the above equation of motion (2). It is convenient to write the Maxwell equation in the covariant form for the symmetry of the relevant quantities without worrying about the factors as c, μ and ε. The four-vector potential is written by the scalar and vector potentials as A μ (x)=(V(x)/c, A(x)) and the four-current, which is a source term of the potentials, is given as ej μ = μ(cq, j ). Here, the charge q and current j are both charge and current densities. The contra-variant four vector x μ is related with the co-variant four vector x μ as x μ = g μν x ν . Here, the metric is g μν = 1 for μ = ν = 0 and g μν = −1 for μ = ν = 1, 2, 3 and zero otherwise (Bjorken (1970)). The Maxwell equation (2) gives the following differential equation (Maxwell (1876)). ∂ μ ∂ μ A ν (x) −∂ μ ∂ ν A μ (x)=ej ν (x) (4) In order to simplify the differential equation and also to keep the symmetry among the scalar and vector potentials, we take the Lorenz gauge ∂ μ A μ (x)=0 (Lorenz (1867); Jackson (1998)). In this case, we get a simple covariant equation for the potential with the source current. ∂ μ ∂ μ A ν (x)=ej ν (x) (5) This expression based on the field theory shows the fact that the dynamics of the four-vector potential A ν is purely given by the corresponding source current j ν . This fact should be contrasted with the standard notion that the time-dependent electric and magnetic fields are the sources from each other through the Ampere-Maxwell’s law and the Faraday’s law in the Maxwell equation. When there is no source term j ν = 0 in the space outside of the conductors, the four-vector potential satisfies the wave equation with the light velocity. In 235 Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode 4 Will-be-set-by-IN-TECH the electrodynamics, the propagation of electromagnetic wave with the velocity of light is the property of a vector particle with zero mass. We express now the four-vectors in the standard three-vector form. The scalarpotential V (x, t) and the vector potential A(x, t) should satisfy the following equations with sources in the Lorenz gauge.  ∂ 2 c 2 ∂t 2 −∇ 2  V (x, t)= 1 ε q (x, t) (6)  ∂ 2 c 2 ∂t 2 −∇ 2  A (x, t)=μj(x, t) (7) These two second-order differential equations (6) and (7) clearly show that the charge and current are the sources of electromagnetic fields. For the propagation of electromagnetic power through a MTL system, we are interested in the electromagnetic fields outside of thick electric wires with resistance. In this case, we are able to solve the differential equations by using retardation charge and current (Lorenz (1867); Rieman (1867); Jackson (1998)). V (x, t)= 1 4πε  dx  q(x  , t − |x−x  | c ) |x −x  | (8) A (x, t)= μ 4π  dx  j(x  , t − |x−x  | c ) |x −x  | (9) These expressions are valid for the scalar and vector potentials outside of the transmission-lines. The presence of the retardation effect in the time coordinate in the integrand is important for the production of electromagnetic radiation. The retardation terms generate a finite Poynting vector going out of a surface surrounding the MTL system not only at a far distance but also at a boundary. This part is related with the derivation of the Lorentz force from the field theory. You may skip this part and directly move to the next section. It is important to derive the current conservation equation of the field theory, which is related with the behavior of charged particles. The current conservation is derived by writing an equation of motion for ψ using the above Lagrangian as (iγ μ ∂ μ + eγ μ A μ −m)ψ(x)=0 . (10) Using this Dirac equation together with the complex-conjugate Dirac equation, we obtain ∂ μ j μ (x)=0 , (11) which is the charge conservation law of the field theory. The electromagnetic potential for a charged particle is given from the above equation as ej μ A μ . From this expression, we are able to derive an electromagnetic force exerted on a charged particle. To write it explicitly, we ought to use a Lagrangian of a point particle with the electromagnetic potential ej μ A μ , where j μ =(c, v). L = 1 2 m ( dx dt ) 2 −eV(x)+ev ·A(x) (12) 236 Electromagnetic Waves Propagation in Complex Matter [...]... a line antenna Here, the current is obtained by solving the TEM mode wave equation with the boundary condition at the center and its ends of a line antenna We try to understand the meaning of the input impedance by setting M = 0 In this case, the input impedance is written as 246 14 Electromagnetic Waves Propagation in Complex Matter Will-be-set-by -IN- TECH Zs = 2 =2 Zkc e− jkl + e jkl ω e jkl − e− jkl... one-line antenna in contrast to the standard understanding that the TEM mode is associated with at least two conductors At the same time, the input impedance has a resonance structure around k R l = nπ with n being an integer due to the sine-function in the denominator The real part has a peak structure at this point, while the imaginary part changes sign and the small additional term makes the imaginary... current Ii from the integral by taking their arguments at x This fact indicates that the electric field has the perpendicular component to the transmission-line and the magnetic field has the axial component produced by the current at the same coordinate Hence, the 240 8 Electromagnetic Waves Propagation in Complex Matter Will-be-set-by -IN- TECH TEM mode propagates through the transmission-lines We shall call... described in a book of Ohta (Ohta (2005)) We shall see that these two retardation terms provide naturally the emission and absorption of electromagnetic waves through the multiconductor transmission-line system We emphasize here that electromagnetic waves go through a multiconductor transmission-line system in the TEM mode while making electromagnetic radiation 4 TEM mode of one line antenna Since we... the initial input energy and is written in terms of Z, R, M and l for a given ω, which are the properties of the transmission-line It is the first time to obtain the input impedance of one resistive conductor antenna This expression should be contrasted with the EMF method for the input impedance, which is obtained by assuming the expression for the current in a line antenna Here, the current is obtained... conservation equation (11) of the field theory, which indicates the conservation of charge ∂q/∂t + ∇j = 0 and the continuity equation of the standard electromagnetism We introduce i-th current and i-th charge by integrating j and q over the cross section of each transmission-line at a space-time position x, t taking into account the skin effect in the transmission-line, Ii ( x, t) = dsjix ( x, y, z, t) and Qi... conductor in the direction of the current through the resistance Ri with the current Ii ( x, t) Ri Ii ( x, t) = Eix ( x, t) (18) Here, the superscript x denotes the x component of the electric field of the i-th transmission-line We note that the resistance Ri should depend on the wave-length of the 238 6 Electromagnetic Waves Propagation in Complex Matter Will-be-set-by -IN- TECH electromagnetic wave going... k I l Z (k R + jk I )c k I l + jsin(k R l )cos(k R l )(1 − k I l ) ω sin2 (k R l ) + cos2 (k R l )(k I l )2 1 cos(k R l ) + j2Z ∼ Rl 2 sin(k R l ) sin (k R l ) Zs = 2 1 and (58) In the last step, we take the dominant terms for the case that sin(k R l ) is not close to 0 The above expression indicates that the real part corresponds to the resistance and the imaginary part corresponds to the characteristic... dx c Zc2 (44) 244 12 Electromagnetic Waves Propagation in Complex Matter Will-be-set-by -IN- TECH We note that this equation is a second order linear differential equation with a constant, if we consider the last term is known In this case, we can write a general solution as I ( x ) = ie jkx + i e− jkx − j M I I (l ) Rc 1 + j Zω Z (45) Rc Here k = k R + jk I = ω 1 + j Zω By inserting this solution to... the MTL equation including radiation, we would like to discuss an isolated system of one-conductor transmission-line so that we write explicitly how the electromagnetic energy is converted into Joule energy and radiation energy In principle, we may have to consider the in uence of the circumstance even for one-line antenna However, for simplicity and also for the sake of understanding the antenna mode, . (51) 244 Electromagnetic Waves Propagation in Complex Matter Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode 13 Substituting this expression to Eq. (49), we get I(0) in. Lagrangian of a point particle with the electromagnetic potential ej μ A μ , where j μ =(c, v). L = 1 2 m ( dx dt ) 2 −eV(x)+ev ·A(x) (12) 236 Electromagnetic Waves Propagation in Complex Matter Electrodynamics. Waves Propagation in Complex Matter 230 Van-Bladel, J. (2007). Electromagnetic Fields (2 nd edition), IEEE Press,Wiley-Interscience, ISBN 978-0471263883, USA Part 4 Propagation in Guided