Electromagnetic Waves Propagation in Complex Matter Part 13 docx

Electromagnetic Scattering of Reflector Antennas by Fast Physical Optics Algorithms, Recent Res.. The Fast Multipole Method FMM for Electromagnetic Scattering, IEEE Transactions on Anten

High Frequency Techniques: the Physical Optics Approximation and

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

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Part 4

Propagation in Guided Media

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) fieldψ with the electromagnetic field A μis determined by the following Lagrangian,

L= 1

4F 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 fieldψ,

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 1c E x 1c E y 1c 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

⎞

⎟

Here, E = −∇ V − ∂A

∂t and B = ∇ ×A The two more equations are explicitly written as

∇ ·E= 1

ε q and ∇ ×B− 1

c2∂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 forμ=ν=0

and g μν = −1 forμ=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 scalar potential V(x, t)

and the vector potential A(x, t) should satisfy the following equations with sources in the Lorenz gauge



∂2

c2∂t2 − ∇2

V(x, t) = 1ε q(x, t) (6)



∂2

c2∂t2− ∇2

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  |)

A(x, t) = μ

4π dx 

j(x , t − |x−x c  |)

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 forψ using

the above Lagrangian as

Using this Dirac equation together with the complex-conjugate Dirac equation, we obtain

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

2m(dx

dt)2− eV(x) +ev ·A(x) (12)

236 Electromagnetic Waves Propagation in Complex Matter