Advanced Theory of Electromagnetic Fields

北京理工大学研究生课程高等电磁场理论教材

Advanced Theory of Electromagnetic

Fields

XU Xiao-Wen

(Beijing Institute of Technology)

北京理工大学研究生课程高等电磁场理论教材

Contexts

Chapter 1: Some Basic Principles

Chapter 2: Modal Expansion Method—Plane Waves

Chapter 3: Modal Expansion Method—Cylindrical Waves

Chapter 4: Modal Expansion Method—Spherical Waves

Chapter 5: Hertzian Dipoles in Stratified Media

Chapter 6: Integral equations and Method of Moments

Chapter 7: Geometrical Theory of Diffraction

Chapter 8: Dyadic Green’s Functions in Electromagnetics

Appendix

1 10 18 34 45 62 80 104 124

北京理工大学研究生课程高等电磁场理论教材

Advanced Theory of Electromagnetic Fields Some Basic Principles 1

Chapter One Some Basic Principles

1.1 Electromagnetic Duality

If the time factor is taken as ejωt, the frequency domain Maxwell’s equations in the linear space can be written as follows, i.e., ×= jωμ , (1.1a) =ρε, (1.1b)

= jωρ, (1.1c) ×=jωε+, (1.1d) =mμ, (1.1e) = jωm, (1.1f)

where and are respectively the electric and magnetic field in the space,

respectively the electric and magnetic current density, ρ and α and

respectively the electric and magnetic charge density, ε and μ respectively the permittivity and permeability, and ω represents the angular frequency.

According to Eq.(1.1), we can obtain the following electromagnetic duality relationships, i.e., in the case of replacing the electric quantities with the magnetic ones, they should be → → ρ→m ε μ η→ k→k; (1.2)

in the case of replacing the magnetic quantities with the electric ones, then they should be → → m→ ρ μ ε η→ k→k. (1.3)

The electromagnetic duality is very helpful to the memory of formulas and the simplification of analysis. For example, suppose that when there is a certain uniform

J0, in the XOY (z=0) plane, the electric field in electric current distribution, =x

J0 jkz

eη (z>0)2the space is obtained as Ex= . Then, according to the duality J0jkz(z<0) ηe

2

K0, in the XOY principle, for a certain uniform magnetic current distribution, =x

(z=0) plane, one can directly write out the magnetic field in the space as

北京理工大学研究生课程高等电磁场理论教材

Advanced Theory of Electromagnetic Fields Some Basic Principles 2

K0 jkz e (z>0) 2ηHx= .

K0jkz(z<0) e 2η

1.2 Theorem of Uniqueness

It is very important to know whether or not a problem has a unique solution. The reasons for this can be summarized as follows.

1) The theorem of uniqueness points out the essential conditions to obtain such a solution. 2) The theorem of uniqueness allows several different methods to be used in the evaluation, so as to increase the efficiency.

3) The theorem of uniqueness establishes the field-to-source one-by-one corresponding conditions, so as to be able to evaluate the source (or sources) from the fields, or vice versa. The theorem of uniqueness can be obtained as follows.

Consider the following closed surface S in a linear medium space, within which a group of electric and magnetic sources and are enclosed, as shown in Fig.1.1.

Fig.1.1 A closed surface S enclosing the sources and in the linear space

Suppose that in the space there are two groups of possible field solutions (a,a) and (b,b). Substitute them respectively in the curl equations in Eq.(1.1) and subtract the corresponding two results, and then we have ×δ=jωμδ, (1.4a) ×δ=jωεδ, (1.4b)

where δ=a b and δ=a b stand for the electric and magnetic difference fields, respectively. By applying the complex Poynting theorem to the difference fields, we can obtain

22 ~~(δ×δ) d+(zH+yE)dv=0, (1.5) ∫∫∫

where ~z=jωμ, ~y=jωε. If on the surface S

北京理工大学研究生课程高等电磁场理论教材

(δ×δ) d=0, (1.6)

then we have

22~~Re(z)δH+Re(y)δEdv=0, (1.7a) ∫∫∫

[~[Im(∫∫∫z)δH

2

2

Im(~y)δE

]

]dv=0. (1.7b)

For lossy media, Re(~z) and Re(~y) are always positive. Therefore, as long as there is some loss in the space (no matter how small), it is required that δ=δ=0 everywhere inside S.

Consequently, we can finally obtain the theorem of uniqueness. This theorem states that, in the lossy region, the electromagnetic field solution will be uniquely specified in any one of the following three situations. The first situation is when the sources within that region and the tangential components of electric field over the boundary surface S are given. The second one is when the sources within that region and the tangential components of magnetic field over the boundary surface S are obtained. The last one is when the sources within that region and the tangential components of electric field over part of the boundary surface S and the tangential components of magnetic field over the rest of it are presented.

According to the concept of limit, it is easily shown that the above theorem also applies to all of th …… 此处隐藏：19513字，全部文档内容请下载后查看。喜欢就下载吧 ……