Notes Electromagnetic Fields

Lecture Notes

on

Electromagnetic Field Theory

PRELIMINARY VERSION

November 2012

DEPARTMENT OF ELECTRONICS

AND

TELECOMMUNICATIONS

POLITECNICO DI TORINO

Contents 1 1 Fundamental concepts 3 1.1 Maxwell’s Equations . . . 3 1.2 Phasors . . . 5 1.3 Constitutive relations . . . 9 1.4 Boundary conditions . . . 11

2 Waves in homogeneous media 14 2.1 Plane waves . . . 14

2.2 Cylindrical waves . . . 20

2.3 Spherical waves . . . 21

2.4 Waves in non homogeneous media . . . 22

2.5 Propagation in good conductors . . . 23

3 Radiation in free space 28 3.1 Green’s functions . . . 28

3.2 Elementary dipoles . . . 34

3.3 Radiation of generic sources . . . 41

4 Antennas 49 4.1 Antenna parameters . . . 49

4.1.1 Input impedance . . . 50

4.1.2 Radiation pattern, Directivity and Gain . . . 51

4.1.3 Effective area, effective height . . . 54

4.2 Friis transmission formula . . . 57

4.3 Examples of simple antennas . . . 59

4.3.1 Wire antennas . . . 60

4.3.2 Aperture antennas . . . 67 1

5 Waveguides 76

5.1 Waveguide modes . . . 77

5.2 Equivalent transmission lines . . . 80

5.3 Rectangular waveguide . . . 86

5.3.1 Design of a single mode rectangular waveguide . . . 92

5.3.2 Tunneling effects . . . 94

5.3.3 Irises and waveguide discontinuities . . . 100

A Mathematical Basics 1 A.1 Coordinate systems and algebra of vector fields . . . 1

A.2 Calculus of vector fields . . . 10

A.3 Multidimensional Dirac delta functions . . . 17

B Solved Exercises 20 B.1 Polarization and Phasors . . . 20

B.2 Plane Waves . . . 23

B.3 Antennas . . . 28

B.4 Waveguides . . . 36

Bibliography 45

Fundamental concepts

1.1

Maxwell’s Equations

All electromagnetic phenomena of interest in this course can be modeled by means of Maxwell’s

equations _       ∇ × E(r,t) = −∂ ∂tB(r,t) − M(r,t) ∇ × H(r,t) = ∂ ∂tD(r,t) + J (r,t) (1.1)

Let us review the meaning of the symbols and the relevant measurement units.

E(r,t) electric field V/m

H(r,t) magnetic field A/m

D(r,t) electric induction C/m2

B(r,t) magnetic induction Wb/m2

J (r,t) electric current density (source) A/m2

M(r,t) magnetic current density (source) [V/m2_]

Customarily, only electric currents are introduced; it is in particular stated that magnetic charges and currents do not exist. However, it will be seen in later chapters, that the introduction of fictitious magnetic currents has some advantages:

• The radiation of some antennas, such as loops or horns, is easily obtained • Maxwell’s equations are more symmetric

• (surface) magnetic currents are necessary for the formulation of the equivalence theorem, a fundamental tool for the rigorous modelling of antennas

In circuit theory, one has two types of ideal generators, i.e. current and voltage ones: likewise, in electromagnetism one introduces two types of sources.

Concerning the symmetry of Maxwell’s equations, we cite the principle of duality: performing the exchanges

E ↔ H B ↔ −D J ↔ −M

Maxwell’s equations transform into each others.

Experiments show that the electric charge is a conserved quantity. This implies that electric current density and electric charge (volume density are related by a continuity equation

∇ · J (r,t) +∂ρ(r,t)

∂t = 0 (1.2)

By analogy, we assume that also magnetic charges are conserved, so that a similar continuity equation must be satisfied:

∇ · M(r,t) +∂ρm(r,t)

∂t = 0 (1.3)

It can be proved that eqs.(1.1), (1.2) (1.3) imply the well known divergence equations

∇ · B(r,t) = ρm(r,t) ∇ · D(r,t) = ρ(r,t) (1.4)

Some authors prefer to assume eqs.(1.1), 1.4) as fundamental equations and obtain the conservation of charge (1.2) (1.3) as a consequence.

Maxwell’s equations can be written in differential form as above, so that they are assumed to hold in every point of a domain, or in integral form, so that they refer to a finite volume. The integral form can be obtained by integrating eq.(1.1) over an open surface Σo with boundary Γ

and applying Stokes theorem: I Γ E · ˆτ ds = −d dt Z Σo B · ˆν dΣo− Z Σo M · ˆν dΣo I Γ H · ˆτ ds = d dt Z Σo D · ˆν dΣo+ Z Σo J · ˆν dΣo (1.5)

In words, the first equation says that the line integral of the electric field, i.e. the sum of all voltage drops along a closed loop, equals the time rate of change of the magnetic induction flux plus the total magnetic current. The second equation says that the line integral of the magnetic field along a closed loop equals the time rate of change of the electric induction flux plus the total electric current.

Then we integrate eq.(1.4) over a volume V with surface Σ and apply the divergence theorem: I Σ B · ˆν dΣ = Z V ρmdV I Σ D · ˆν dΣ = Z V ρ dV (1.6)

Figure 1.1. Open surface Σo for the application of Stokes theorem. Notice that the

orien-tations of ˆν and ˆτ are related by the right-hand-rule: if the thumb points in the direction

of ˆν, the fingers point in that of ˆτ .

Figure 1.2. Closed surface Σ for the application of the divergence theorem.

which is the usual formulation of Gauss theorem. The same procedure on eq.(1.2) produces:

I Σ J · ˆν dΣ + d dt Z V ρ dV = 0 (1.7)

This says that the total current out of a volume equals the time rate of decrease of the internal charge.

1.2

Phasors

Field variables can have any time dependence but a particularly important one is the so called time-harmonic regime. Consider a general time-harmonic electric field in a particular point:

The three cartesian components are sinusoidal functions of time with different amplitudes and phases, but the same frequency. This equation can be transformed in the following way.

E(t) = RnEx0ej(ω0t+ϕx) o ˆ x + RnEy0ej(ω0t+ϕy) o ˆ y + RnEz0ej(ω0t+ϕz) o ˆ z = R©¡Ex0ejϕxˆx + Ey0ejϕyy + Eˆ z0ejϕzˆz ¢ ejω0tª = R©E ejω0tª (1.8)

The complex vector E is called the phasor of the time-harmonic vector E(t) and is measured in the same units. It can be decomposed into real and imaginary parts:

E = E0_{+ jE}00

with

E0 _{= E}

x0cos ϕxˆx + Ey0cos ϕyy + Eˆ z0cos ϕzˆz

and

E00_{= E}

x0sin ϕxˆx + Ey0sin ϕyy + Eˆ z0sin ϕzˆz

Consider again eq.(1.8):

E(t) = R©E ejω0tª = R©(E0_{+ jE}00_{) e}jω0tª = R {(E0_{+ jE}00_{) (cos ω} 0t + j sin ω0t)} = E0_{cos ω} 0t − E00sin ω0t

We have obtained a representation of the time-harmonic electric field as the sum of two vectors, with arbitrary directions, in time quadrature one with respect to the other: in other words both these vectors are sinusoidal functions of time, with the same frequency but with a relative delay of a quarter of a period. This representation is useful to study the polarization of the time-harmonic vector, i.e. the form of the curve traced by the vector E(t) as a function of time. It can be shown

t

E( )

E"

E'

Figure 1.3. Elliptically polarized field

that, in general, this curve is an ellipse inscribed in a parallelogram that has E0 _{and E}00 _{as half}

• for t = 0, E = E0

• for t = T /4, E = −E00

• for t = T /2, E = −E0

• for t = 3T /4, E = E00

where T = 2π/ω is the period. Hence the sense of rotation is from E0 _{to −E}00_{. In this case the}

field is said to be elliptically polarized.

There are particular cases. When E0 _{and E}00 _{are parallel or one of the two is zero, the}

paral-lelogram degenerates into a line and the polarization is linear. The two cases can be condensed in the single condition (cross product, i.e. vector product):

E0× E00= 0 The other particular case is that in which

|E0_{| = |E}00_{| and E}0_{· E}00_{= 0}

The parallelogram becomes a square and the ellipse a circle: the field is circularly polarized. The plane defined by the two vectors E0 _{and E}00 _{is called polarization plane. Suppose we}

introduce a cartesian reference in this plane with the z axis orthogonal to it. The phasor E has only x and y components,

E = Exˆx + Eyyˆ

where Ex and Ey are complex numbers. This means that the original time-harmonic field is

represented as the sum of two sinusoidally varying orthogonal vectors, with arbitrary magnitudes and phases. On this basis, the type of polarization is ascertained with the following rules:

• if the phase difference between the two components δ = arg Ey − arg Ex is 0 or π the

polarization is linear, along a line that forms an angle ψ = arctan(|Ey|/|Ex|) (if δ = 0) or

ψ = − arctan(|Ey|/|Ex|) (if δ = π)

• if δ = ±π/2 and |Ey| = |Ex| the polarization is circular, clockwise if δ = π/2,

counterclock-wise if δ = −π/2

• in the other cases, the polarization is elliptical To illustrate these concepts consider the following example.

Example

The phasor of a magnetic field is H = (1 + j)ˆx + (1 − 3j)ˆy. Determine the type of polarization, write the expression of the time varying field H(t) and draw the polarization curve defined by this vector.

Solution Find real and imaginary part of the phasor

Compute

H0_{× H}00_{= (ˆx + ˆy) × (ˆx − 3ˆy) = (−3 − 1)ˆz 6= 0}

H0_{· H}00_{= (ˆx + ˆy) · (ˆx − 3ˆy) = 1 − 3 = −2 6= 0}

The polarization is neither linear nor circular, hence it is elliptical counterclockwise (H(t) goes from H0 _{to −H}00_).

The time varying field is

H(t) = (ˆx + ˆy) cos ω0t − (ˆx − 3ˆy) sin ω0t

The plot is shown in Fig. 1.4

−4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 y x

Figure 1.4. Polarization curve defined by H(t) above

¥ The time-harmonic regime is important because of the property

d dtE(r,t) = d dtR © E(r) ejω0tª_{= R}©_jω 0E(r) ejω0t ª (1.9) so that time derivatives become algebraic operations. If we substitute the representation (1.8) into (1.1) we obtain, after canceling common factors exp(jω0t):

∇ × E(r) = −jω0B(r) − M(r)

∇ × H(r) = jω0D(r) + J(r)

(1.10)

If the time dependence is of general type, eq. (1.8) is generalized by the spectral representation (inverse Fourier transform)

E(r,t) = 1

2π Z ∞

−∞

In words, a generic time varying electric field is represented as a sum of an infinite number of harmonic components of all frequencies and amplitude E(r,ω)dω/(2π), where

E(r,ω) = Z ∞

−∞

E(r,t) e−jωt_{dt (1.12)}

(direct Fourier transform). E(r,ω) is a spectral density of electric field, hence it is measured in V/(m Hz). Due to the fact that E(r,t) is real, E(r, − ω) = E∗_{(r,ω), so that the previous equation}

can also be written

E(r,t) = 2R ½ 1 2π Z _∞ 0 E(r,ω) ejωtdω ¾

in terms of positive frequencies only.

The importance of the spectral representation is related to the property

d dtE(r,t) = d dt 1 2π Z ∞ −∞ E(r,ω) ejωt_{dω =} 1 2π Z ∞ −∞ jωE(r,ω) ejωt_dω

If we take the Fourier transform of (1.1), we get

∇ × E(r,ω) = −jωB(r,ω) − M(r,ω)

∇ × H(r,ω) = jωD(r,ω) + J(r,ω)

(1.13)

While ω0 is a specific frequency value, (1.13) must be satisfied for all frequencies. We refer to

these system as Maxwell’s equations in the frequency domain. The variables will be interpreted as phasors or Fourier transforms, depending on the application.

1.3

Constitutive relations

Clearly Maxwell’s equations as written in the previous section form an underdetermined system: indeed there are only two equations but four unknowns, the two fields and the two inductions. It is necessary to introduce the constitutive relations, i.e. the equations linking the inductions to the fields. From a general point of view, matter becomes polarized when it is introduced into a region in which there is an electromagnetic field, that is, the electric charges at molecular and atomic level are set in motion by the field and produce an additional field that is summed to the original one. The inductions describe the electromagnetic behaviour of matter.

The simplest case is that of free space in which

B(r,t) = µ0H(r,t)

D(r,t) = ²0E(r,t)

(1.14) where ²0, dielectric permittivity, and µ0 magnetic permeability, have the values

µ0 = 4π · 10−7 H/m ²0 = 1 µ0c2 ≈ 1 36π · 10 −9 _F/m

and the speed of light in free space c has the value

c = 2.99792458 · 108 m/s.

In the case of linear, isotropic, dielectrics, the constitutive relations (1.14) are substituted by B(r,ω) = µ(ω) H(r,ω) D(r,ω) = ε(ω) E(r,ω) (1.15) where µ(ω) = µ0µr(ω) ²(ω) = ²0²r(ω)

and µr(ω), ²r(ω) (pure numbers) are the relative permittivity and permeabilities. All non

fer-romagnetic materials have values of µr very close to 1. Notice that since molecular and atomic

charges have some inertia, they cannot respond instantaneously to the applied field, so that the response depends on the time rate of change of the excitation. The description of such a mechanism is best performed in the frequency domain, where ε(ω) and µ(ω) play the role of transfer functions. The fact that they depend on frequency is called dispersion: hence free space is non dispersive. In general ε(ω) and µ(ω) are complex:

ε = ε0_{− jε}00 _{µ = µ}0_{− jµ}00

It can be shown that the volume density of dissipated power in a medium is related to their imaginary part Pdiss= 1 2ε 00_|E|2₊1 2µ 00_|H|2

Notice that ε00_{, µ}00 _{are positive in a passive medium because of the phasor time convention}

exp(jω0t). Some authors use the convention exp(−jω0t): in this case passive media have

neg-ative ε00_{, µ}00_{. Clearly with the time convention we adopt, negative ε}00_{, µ}00 _{characterize active}

media, such as laser media.

When the dielectric contains free charges, the presence of an electric field E(r,ω) gives rise to a conduction current density Jc(r,ω):

Jc(r,ω) = γ(ω) E(r,ω)

where γ(ω) is the conductivity of the dielectric, measured in S/m. The previous equation is the microscopic form of Ohm’s law of circuit theory.

The conduction current enters into the second Maxwell’s equation (1.13), which becomes

∇ × H = jωεE + γE + J

where the term J is the (independent) source. It is customary to incorporate the conduction current into the polarization current by means of an equivalent permittivity. Indeed we can write

jωεE + γE = jω ³ ε − jγ ω ´ E = jωεeqE

with εeq = ε0− j(ε00+ γ/ω). In practice the subscript eq is always dropped: the imaginary part of

ε takes into account all loss mechanisms, both those due to molecular and atomic transitions and

In the case of low loss dielectrics one often introduces the loss tangent tan δ = ε

00

ε0

so that we can write

ε = ε0_{(1 − j tan δ)}

Values of tan δ ' 10−3_{: 10}−4 _{characterize low loss dielectrics.}

It is interesting to note that for fundamental physical reasons, there is relationship between the real and the imaginary part of the dielectric permittivity and of the magnetic permeability. Indeed, in the case of the permittivity, just as a consequence of causality, ε0_{(ω) − ε}

0and ε00(ω) are

Hilbert transforms of each other, that is

ε0(ω) − ε0= 1 πP Z _∞ −∞ ε00_(α) α − ω dα ε00_{(ω) =} 1 πP Z _∞ −∞ ε0_{(α) − ε} 0 α − ω dα

These equations are called Kramers-Kr¨onig relations. The symbol P denotes the Cauchy principal value of the integral, that is, for f (0) 6= 0

P Z ∞ −∞ f (x) x dx = lima→0 ½Z _−a −∞ f (x) x dx + Z ∞ a f (x) x dx ¾

The constitutive equations (1.15) imply that the inductions are parallel to the applied fields, which is true in the case of isotropic media but not in the case of crystals. These media are said to be anisotropic and are characterized by a regular periodical arrangement of their atoms. In this case the permittivity constitutive equations must be written in matrix form:

  D_Dx_y Dz   =   ε_εxx_yx ε_εxy_yy ε_εxz_yz εzx εzy εzz     E_Ex_y Ez  

In the case of an isotropic dielectric, the matrix is a multiple of the identity: ε = εI.

1.4

Boundary conditions

Maxwell’s equations (1.13) are partial differential equations (PDE), valid in every point of a given domain, which can be finite or infinite. If it is finite, we must supply information about the nature of the material that forms the boundary. In mathematical terms, we must specify the boundary conditions, i.e. the values of the state variables on the boundary.

Often the boundary is a perfect electric conductor (PEC), that is a material with infinite conductivity. Note that copper is such a good conductor that up to microwave frequencies it can be modeled as a PEC. If the conductivity is infinite, the electric field must vanish everywhere in the volume of a PEC, otherwise the conduction current would be infinite. The first Maxwell’s equation shows that also the magnetic field is identically zero, provided the frequency is not zero. Since we are essentially interested in time-varying fields, we conclude that in a PEC both fields vanish identically. At the surface, since the conduction current cannot have a normal component,

only the tangential component of the electric field must be zero. If ˆν is the unit normal at the

boundary, this condition can be written ˆ

ν × E = 0 on the boundary (1.16) Indeed, ˆν × E is tangential to the boundary, as shown in Fig. 1.5.

ƚĂŶŐĞŶƚ ƉůĂŶĞ

ˆ

ν

E

W

ˆ

ν

Figure 1.5. Boundary condition at the surface of a perfect electric conductor. Relationship between the tangential component of the electric field Etg and ˆν × E

Sometimes the permittivity or the permeability change abruptly crossing a surface in the do-main. By applying some integral theorems to Maxwell’s equations, it can be shown that the following continuity conditions hold

ˆ

ν × (H(rΣ+) − H(rΣ−)) = 0 ν × (E(rˆ Σ+) − E(rΣ−)) = 0 (1.17)

ˆ

ν · (B(rΣ+) − B(rΣ−)) = 0 ν · (D(rˆ Σ+) − D(rΣ−)) = 0 (1.18)

where ˆν is the normal to the surface and rΣ±are infinitely close points, lying on opposite sides of

the surface, as shown in Fig. 1.6. It can also be proved that if a surface current Js or Ms exist,

then the fields are discontinuous

ˆ

ν

r

r

Σ

ε µ

ε µ

Figure 1.6. Boundary conditions at the surface of separation between different dielectric media

ˆ

ν × (H(rΣ+) − H(rΣ−)) = Js ν × (E(rˆ Σ+) − E(rΣ−)) = Ms

Since it can be proved that when a PEC is present in a magnetic field, the induced current flows only on the surface of it and the magnetic field is identically zero in the PEC, we can write

ˆ

Notice that this is an equation that does not force a condition on H but allows the computation of Js. Hence the boundary condition to be enforced at the surface of a PEC is only (1.16). Notice also

that the units of a surface electric current are A/m and those of a surface magnetic current V/m. This is necessary for the validity of the previous equations, but it is also obvious for geometrical reasons, as Fig. 1.7 shows.

s J γ W ˆ ν

Figure 1.7. Surface electric current induced on a perfect electric conductor (PEC). γ is a curve lying on the PEC surface. Js is a current density per unit length measured along γ (A/m)

Sometimes an approximate boundary condition of impedance type is used: it is a linear relation between the tangential electric and magnetic fields, called also a Leontovich boundary condition, that can be written

ˆ

ν × E(rΣ+) = Zs(ˆν × H(rΣ+)) × ˆν (1.20)

in terms of a suitable surface impedance. This condition is typically applied when the boundary is a real metal and one desires a more accurate model than just a PEC. The double vector product on the right hand side makes the tangential electric and magnetic field orthogonal on the surface. If the surface is not smooth but has a sub-wavelength wire structure, the surface impedance is not a scalar but a tensor (matrix).

If the domain is infinite and sources are all at finite distance from the origin, the only necessary boundary condition is that the field is only outgoing at large distance from the origin.

Sometimes the geometry of dielectric and metal bodies possesses sharp edges or sharp vertices (e.g. plates, cubes, cones), as shown in Fig. 1.8. In this case some field components can become infinite at the geometric singularity: however the fields must always be locally square integrable. In physical terms this condition means that the electromagnetic energy contained in a finite neigh-borhood of the singularity must always be finite. Apart from these cases of true singularities of the geometry, fields are always regular, i.e. differentiable. This is to be noted in particular when the geometry singularity is only apparent because it is related to the particular coordinate system. For example if we use cylindrical coordinates in a homogeneous medium the fields must be regular in the origin even if the coordinates have a singularity there.

,ĂůĨƉůĂŶĞ tĞĚŐĞ ŽŶĞ

Waves in homogeneous media

At this point we have finished the preliminaries. We have decided to use the electric and magnetic field as state variables and we have the relevant equations:

       ∇ × E = −jωµH − M ∇ × H = jωεE + J + boundary conditions (2.1)

where for brevity we have dropped the dependence of all variables on r, ω, but it is understood. The line “boundary conditions” contains the form of the domain where we want to compute the fields created by the given sources J,M and information on the nature of the material that forms the boundary. In general permittivity and permeability are functions of r and provide information on the shape of the bodies in the domain and on their nature. In this way the problem is well posed and it can be shown it has a unique solution, provided there is at least a region in the domain where energy is dissipated. Needless to say the problem (2.1) can be very complicated and can be solved only by approximate numerical techniques. For this reason it is useful to proceed by small steps, by analyzing first a very idealized problem that is so simple to allow an analytical solution.

2.1

Plane waves

Let us start by assuming that the domain of interest is infinite and the medium is homogeneous and lossless, so that ε, µ are real constants. The only boundary condition to enforce is that the field is regular everywhere, in particular at infinity.

Moreover we assume that sources are identically zero. This is, at first sight, a strange assump-tion since it would seem to imply that also the fields must be identically zero! However, if there are no sources, Maxwell’s equations become a homogeneous system of differential equations: it is well known that homogeneous differential equations have nontrivial solutions. Let us review some examples:

•

d

dxf (x) = 0, x ∈ R ⇒ f (x) = const

• Harmonic oscillator µ d2 dt2+ ω 2 0 ¶ f (t) = 0, t ∈ R ⇒ f (t) = A cos ω0t + B sin ω0t • Transmission line      −d dzV = jω L I −d dzI = jω C V , z ∈ R ⇒    V (z) = V₀+e−jkz_{+ V}− 0 e+jkz I(z) = Y∞V0+e−jkz− Y∞V0−e+jkz

These are actually generalizations of the simple case of a linear system of algebraic equations A x = 0

which has nontrivial solutions if the matrix A is non invertible. So the problem we want to solve is

(

∇ × E = −jωµH

∇ × H = jωεE (2.2)

Equations (2.2) are written in a coordinate-free language. However, in order to solve them it is necessary to select a coordinate system. Several choices are at our disposal, the more common being cartesian, cylindrical and spherical coordinates. The corresponding solutions of (2.2) will be plane, cylindrical and spherical waves, respectively. The simplest case is the first and we start with that.

Recalling the expression of the ∇ operator in cartesian coordinates

∇ = ˆx ∂ ∂x+ ˆy ∂ ∂y + ˆz ∂ ∂z

and that the medium is homogeneous, it is clear that (2.2) is a linear system of constant coefficient equations. On the basis of the experience with ordinary differential equations, we can expect that the solution is of exponential type, hence we assume tentatively

E(r) = E0exp(−jkxx) exp(−jkyy) exp(−jkzz) (2.3)

and likewise for the magnetic field, where E0and kx,ky,kzare constants to be found. The constants

kx,ky,kz have dimensions rad/m and are wavenumbers along the three coordinate axes. It is

convenient to work with a vector formalism, even if the coordinate system is fixed. By recalling that r = xˆx + yˆy + zˆz and defining the wavevector k = kxˆx + kyy + kˆ zˆz, the assumed form of the

solution is

E(r) = E0exp(−jk · r) H(r) = H0exp(−jk · r) (2.4)

Before substituting it into (2.2) it is useful to compute

∇ exp(−jk · r) = µ ˆ x ∂ ∂x+ ˆy ∂ ∂y+ ˆz ∂ ∂z ¶

exp(−jkxx) exp(−jkyy) exp(−jkzz)

= (−jkxˆx − jkyy − jkˆ zˆz) exp(−jkxx) exp(−jkyy) exp(−jkzz)

Moreover we recall the identity of vector calculus

∇ × (A(r)φ(r)) = φ(r)∇ × A(r) + ∇φ(r) × A(r)

so that the substitution of (2.4) into (2.2) yields

−jk × E0exp(−jk · r) = −jωµH0exp(−jk · r)

−jk × H0exp(−jk · r) = jωεE0exp(−jk · r)

Canceling common factors

−jk × E0 = −jωµH0

−jk × H0 = jωεE0

Note that the fact that these equations do not contain the variable r any longer confirms the correctness of the assumption (2.4).

Recalling the properties of vector products we learn that E0, H0, k form a righthanded triple

of mutually orthogonal vectors. Next, to proceed, we eliminate H0between the two equations. To

this end, we solve the first equation with respect to H0:

H0= k × E0

ωµ (2.5)

and substitute into the second one

k × (k × E0) + ω2µεE0= 0

The double vector product can be expanded

k (k · E0) − (k · k) E0+ ω2µεE0= 0

The first term is zero because of the orthogonality of k and E0, hence

¡

k · k − ω2µε¢E0= 0 (2.6)

We are interested in nontrivial solutions of this equation, so that the following condition must hold

k · k = ω2_{µε (2.7)}

A relationship between frequency and wavenumbers is called in general a dispersion relation. We can read it from right to left or vice versa: in the first instance it tells us what the frequency must be so that the field distribution (2.4) with a specific wavevector k is a solution of Maxwell’s equations. From this point of view, the value of ω can be considered to be a resonance frequency of the structure. Notice that the requirement that the solution be regular everywhere (in particular at infinity) forces the wavector to be real. Apart from this condition there is no constrain on the possible values of the wavenumbers, hence the resonance frequencies of the system are infinite in number and even distributed continuously. We start seeing a property that characterizes all field problems. Whereas lumped element circuits have a finite number of resonances, distributed systems always have an infinite number of them. Moreover, if the structure has finite size its resonances are denumerably infinite: this means that they can be labeled with integers ω1, . . . ,ωn, . . .. If the

variable. To simplify the notation, we omit this labeling variable and remember that ω can take every real value.

The dispersion equation can also be read from left to right: in this case the frequency is considered fixed and we look for the wavevectors that satisfy eq.(2.7). It is convenient to introduce a unit vector ˆs, called direction of propagation, directed along k, so that k = kˆs; then the dispersion relation becomes

k = ω√εµ (2.8)

Clearly the direction ˆs can be whatever, only the wavenumber k is specified. In other words for any given frequency there are an infinite number of waves with arbitrary directions of propagation. Considering again eq.(2.6), we see that if the dispersion relation is satisfied, the vector E0

can be arbitrarily chosen, provided it is orthogonal to ˆs. For a given ˆs, there are two linearly independent waves, but they are degenerate, because they have the same value of the wavenumber

k. The corresponding magnetic field is obtained by the impedance relation (2.5):

H0= ω√εµˆs × E0 ωµ = r ε µˆs × E0= Y ˆs × E0 (2.9)

where the wave admittance has been introduced. Its inverse is the wave impedance Z = 1/Y . In free space it has the value

Z0=

r

µ0

ε0 ≈ 120π ≈ 377 Ω

In conclusion a solution of the problem (2.2) is

Eˆs(r) = E0exp(−jkˆs · r)

Hˆs(r) = H0exp(−jkˆs · r)

(2.10) where ˆs is the direction of propagation, k = ω√εµ, H0 = Y ˆs × E0 and E0, H0, ˆs are mutually

orthogonal: electromagnetic waves are transverse. Since the problem is linear, the general solution of (2.2) can be written as a linear combination of waves of the type (2.10) with all possible directions ˆs.

Wavefronts are defined to be the surfaces on which the phase Φ(r) of the wave is constant. In

this case

Φ(r) = −kˆs · r = constant

is the equation of a family of planes perpendicular to ˆs: hence the fields (2.10) are called plane waves because the wavefronts are planes.

Let us now study the polarization of plane waves. This requires going back to time domain via eq.(1.8)

E(r,t) = R {E0exp(−jkˆs · r) exp(jω0t)}

= E0

0cos (ω0t − kˆs · r) − E000sin (ω0t − kˆs · r)

(2.11) We see clearly from this equation that the type of polarization in every point of space is specified by E0, which is the electric field in the origin. What changes from point to point is the time

evolution, due to the propagation delay. The constant phase surfaces of the time varying field are

ω0t − kˆs · r = const, from which we find

ˆs · r = ω0

W

E

H

r

ˆs

Figure 2.1. Wavefronts of a plane wave with direction of propagation ˆs. They move at the phase velocity along ˆs. The vector ˆu denotes an arbitrary direction

This means that the wavefronts are not fixed but are moving. Indeed, consider a specific wavefront, say the zero-phase one, i.e. the one for which const=0, as shown in Fig.2.1; the vector r that denotes its points is such that its projection on ˆs increases linearly with time. In other words, the plane moves as a whole with speed

vph=ω0 k = ω0 ω0√εµ = 1 √ εµ = c √ εrµr (2.12)

This velocity is called phase velocity of the wave, because it has been defined by means of the constant phase surfaces. Now consider an arbitrary straight line with direction ˆu, an let P be its intersection with the zero-phase wavefront. The velocity of P is

vph(ˆu) = 1 ˆ u · ˆs c √ εrµr

Clearly, for all directions ˆu 6= ˆs this velocity is larger than c/√εrµr. Notice, however, that even

when this velocity is larger than the speed of light in empty space, the theory of relativity is not violated. Indeed no matter or energy is moving in the direction of ˆu, but only a mathematical point, e.g. a maximum or a node of the oscillation. This concept is at the basis of the fact that the phase velocity in a waveguide is always greater that the speed of light in vacuum.

Fig.2.2 shows the fields of a linearly polarized plane wave propagating in the ˆs = ˆz direction: remember that the electric and magnetic field are orthogonal. For clarity, the field vectors have been drawn only for a number of points on the z axis, even if they are defined in every point of space.

Wave propagation is always associated to energy flow. The Poynting vector has the meaning of power density (per unit surface) associated to the wave. Let us compute the Poynting vector S in the case of a plane wave:

S = 1 2(E × H ∗_{) =} 1 2(E0exp(−jkˆs · r) × H0 ∗_{exp(jkˆs · r))} = 1 2(E0× H0 ∗_{) =} 1 2(E0× (ˆs × E0 ∗_)Y∗₎ = 1 2 |E0|2 Z ˆs (2.13)

where we used the impedance relation (2.9) and the property

nj

dž

LJ

,

z t

E( )

,

z t

H( )

Figure 2.2. Snapshot of a linearly polarized plane wave propagating in the ˆs = ˆz direction

because of the orthogonality between E0 and ˆs. Note that the Poynting vector is real, hence

no reactive power is associated to plane waves in a lossless medium. We see that the active power density (magnitude of S) associated to a plane wave is constant: this implies that the total power, obtained by integration over the whole wavefront, is infinite. Hence, a single plane wave is not a physically realizable field. This property, however, does not destroy the usefulness of the concept. Indeed, since Maxwell’s equations are linear, the superposition principle holds and linear combinations of plane waves are also solutions. It turns out that a continuous sum (integral) of plane waves not necessarily has infinite power: indeed all physically realizable fields can always be represented as integrals of plane waves.

We recall that to each plane wave not only a power flow is associated, but also a flow of linear momentum and of angular momentum. In particular the linear momentum flow, which has direction ˆs, is responsible for the radiation pressure, that explains, for instance, the shape of comet tails and has been considered as a possible “engine” for interplanetary travels.

The properties of plane waves do not change much if the dielectric is lossy. In this case the permittivity is complex and the dispersion relation (2.8) becomes

k = ωp(ε0_{− jε}00_{)µ = β − jα, α ≥ 0}

where β is the true phase constant, measured in rad/m and α is the attenuation constant, measured in Nepers/m. The electric field, for instance, obeys the propagation law

E(r) = E0exp(−jβˆs · r) exp(−αˆs · r) (2.15)

Clearly the magnitude of the field is no longer constant in space and the wavefronts are also surfaces of constant field magnitude. Obviously, it must be remarked that the value of the phase velocity cannot be computed by (2.12), but it is given by

vph =ω

β

and the wavelength is

λ = 2π β =

vph

Indeed, both the phase velocity and the wavelength are defined on the basis of the phase of the wave and β is exactly the phase rate-of-change, measured, as said above, in rad/m.

Let us compute the power flow. S = 1 2(E × H ∗_{) =} 1 2(E0exp(−jkˆs · r) × H0 ∗_exp(jk∗_{s · r))ˆ} = 1 2(E0× (ˆs × E0 ∗_)Y∗_{exp(−2αˆs · r)) =} 1 2Y ∗_|E 0|2exp(−2αˆs · r)ˆs (2.16)

where we have used again the property (2.14). In this case the Poynting vector is complex. The active power density of the wave decreases during propagation because part of it is transferred to the dielectric in the form of heat.

All the plane waves considered up to now are called uniform because their propagation direc-tion ˆs is real. If we go through all the steps of the derivation, we realize that even if ˆs is complex

(whatever this means!) the expression (2.10) is a valid solution of Maxwell’s equations, although only in a halfspace. Such a generalization leading to non uniform plane waves is required when solving a scattering problem where a plane wave is incident on the interface separating two di-electrics. It is to be remarked that also the plane wave (2.15) is a valid solution only in a halfspace. Indeed, if ˆs · r → −∞, the field diverges, which is not physically acceptable.

2.2

Cylindrical waves

To solve eqs.(2.2) it is also possible to use a cylindrical coordinate system instead of a cartesian one. The mathematics is considerably more complicated in this case. The reason is that the unit vectors of the cylindrical coordinate system are not constant but change from point to point. As a consequence, the expression of the differential operators is no longer with constant coefficients and the solutions are no longer of exponential type, but are expressed in terms of Bessel functions. These special functions of mathematical physics were actually introduced, along with many others, in order to solve the wave equation.

If a cylindrical or spherical coordinate system is used, Maxwell’s equations (2.2) are not attacked directly but are first transformed into a single second order equation. We write them again for

convenience (

∇ × E = −jωµH

∇ × H = jωεE

Since it is a system of equations, we can eliminate one of the two unknowns. We solve the first equation with respect to H and substitute in the second

       H = ∇ × E −jωµ ∇ × (∇ × E) = ω2_εµE

As expected, the second equation contains only the electric field: the price to pay for it is that it is second order in the space derivatives; it is called the curl-curl equation. However its form can be simplified recalling the identity

where we have used the fact that (2.2) do not have sources, hence also the charge density ρ(r) is zero and the electric field has zero divergence, ∇ · E = ρ = 0. We obtain in this way the vector

Helmholtz equation

∇2E + ω2εµE = 0 (2.17)

Even if it is written in coordinate-free language, its meaning is easily understood in cartesian coordinates only, where

E(r) = Ex(r)ˆx + Ey(r)ˆy + Ez(r)

Since the unit vectors are not function of r, each cartesian component of the electric field satisfies Helmholtz equation, which then becomes scalar:

∇2_{ψ + ω}2_{εµψ = 0}

where ψ(r) denotes any component of E. It is interesting to note that even if we are using cartesian

components to represent the electric field, we are not forced to use necessarily cartesian coordinates

to specify the observation point, i.e. the components of r. By using the classical method of the separation of variables in cylindrical coordinates, we can find

ψ(ρ,ϕ,z) = ψ0Hm(2)(kρρ)e−jkzze−jmϕ (2.18)

where m = 0, ± 1, ± 2, . . ., kρ ∈ [0,∞) and kz identify the various outgoing cylindrical waves. These

three parameters play the role of kx, ky, kz in the case of plane waves. The function Hm(2)(kρρ) is

a Hankel function of second kind and order m. Its asymptotic expansion is

Hm(2)(kρρ) ∼ s 2 πkρρexp h −j(kρρ − mπ 2 − π 4) i

The dispersion relation is

kz=

q

ω2_{εµ − k}2 ρ

Notice that all three components have this form, but the values of ψ0for each must be interrelated

so that the resulting vectors E and H satisfy Maxwell’s equations.

We are not going to describe in detail the properties of these waves. To explain the name, it is enough to say that the wavefronts are cylinders having ˆz as axis, at least in the case m = 0 and kρ= ω√εµ.

2.3

Spherical waves

The case of spherical waves is similar, from a certain point view, to that of cylindrical waves. Again the mathematics is fairly complicated and new special functions are introduced. In this case the scalar Helmholtz equation is solved in spherical coordinates and the result is

ψ(r,ϑ,ϕ) = ψ0h(2)l (kr)Plm(cos ϑ)e−jmϕ

where l = 0,1,2, . . . and −l ≤ m ≤ l identify the various outgoing spherical waves. The functions

h(2)_l (kρ) are spherical Hankel functions of second kind and order l, whereas Pm

l (cos ϑ) are associated

1 1

ε

µ

ε

µ

ε

µ

ε

µ

t

i

Figure 2.3. Scattering from a stratified dielectric: i, incident wave; r reflected wave; t, transmitted wave. For clarity, the couple of plane waves existing in each of the internal layers has not been indicated

components must be related so that the resulting E and H satisfy Maxwell’s equations. The asymptotic expansion of the spherical Hankel functions is

h(2)_l (kr) ∼ 1 krexp h −j(kr − mπ 2 − π 4) i

Hence the wavefronts are spheres with center in the origin and this justifies the name.

2.4

Waves in non homogeneous media

The case that has been considered, namely a homogeneous medium filling the whole space is highly idealized. In a realistic situation, ε(r), µ(r) are not constant and obviously the plane waves (2.10) are not solution of Maxwell’s equations (2.2). In order to consider a simple case, let us assume that the medium is piecewise homogeneous and that the interfaces between the different materials are planar: the structure is called a stratified dielectric. In the left half space an incident plane wave is assumed. In each layer, plane waves are solutions of (2.2), but the continuity conditions (1.17) must be obeyed. It can be proved that in each one of the internal layers two plane waves are present, one forward (incident on the following interface) the other backward (reflected from the following interface); in the right half space only one, because the medium extends to infinity and there is no other interface. All these plane waves have the same transverse (to z) component of the wavevector and their amplitudes can be easily determined so that the continuity conditions are satisfied. The single wave in the fourth medium is the transmitted field, the second one in the first medium is the reflected field, as sketched in Fig. 2.3.

If the interfaces are not planar, the problem becomes much more difficult. Consider, for ex-ample, the case of Fig. 2.4, where a plane wave is incident on a cylinder with parameters ε2, µ2,

embedded in a homogeneous medium with parameters ε1, µ1. It can be shown that the continuity

conditions require that an infinite number of plane waves are excited, each one with the right am-plitude. Collectively. these are called scattered waves. Hence the difficulty of the problem stems from the necessity to solve a linear system with an infinite number of unknowns.

If the medium is not even piecewise homogeneous but arbitrarily inhomogeneous, no analytical solution is at our disposal. It is, however, to be mentioned that when the variations of ε(r), µ(r) are small on the wavelength scale, a well known approximate method can be used, i.e. Geometrical Optics. Whereas the plane waves discussed up to now can be defined global plane waves since each one is defined over the whole space, the elementary geometrical optics field is a local plane wave. For instance, a spherical wave in free space can be approximated by a collection of local plane

i

s

s

s

s

ε

µ

ε

µ

Figure 2.4. Scattering from a non planar interface: i, incident wave; s, scattered waves. For clarity, the plane waves existing inside the cylinder have not been indicated

waves because its wavefront (a sphere) can be approximated locally by the relevant tangent plane. The k vectors of these local plane waves define a vector field, whose field lines are the geometrical

optics rays. It turns out that rays are also the field lines of the Poynting vector field: hence a plot

of the rays provides information about the power flow.

Geometrical optics is a very powerful technique, but sometimes yields definitely wrong results. This happens when rays cross in a point or along a line, because in this case it predicts a field of infinite intensity. These singularities are called caustics and the focus of a converging lens is an example: in such a point the electromagnetic field can be very large but is certainly finite. Hence geometrical optics can be safely used only away from caustics.

2.5

Propagation in good conductors

Apart from the case of optical fibers, guided wave propagation is possible in structures containing metal conductors. Examples are coaxial cables, parallel wire transmission lines, microstrip lines, waveguides with any cross section. Since the metals used in the applications (such as copper) are characterized by a very large conductivity, in a first approximation they can be considered to be perfect conductors (PEC), an assumption that greatly simplifies the study. However, in order to build more accurate numerical models of real devices, it is necessary to take into account the finite conductivity of real metals. In this section we consider the propagation of plane waves in good conductors, in order to draw some conclusions pertaining to transmission systems.

Metals are characterized by so a large conductivity that the displacement currents can be safely neglected with respect to the conduction currents, so that some simplifications in the general formulas of Section 2.1 are possible. Starting with the wavenumber,

km= ω p (ε − jγ/ω)µ ≈ ωp(−jγ/ω)µ if γ ωεÀ 1 (good conductor) Recalling that p −j = ±1 − j√ 2

and that Imk ≤ 0 for a passive medium, we find km= 1 − j√ 2 √ ωγµ = 1 − j δ (2.19)

where we have introduced the so called skin depth

δ =

r 2

ωµγ (2.20)

which, of course, should not be confused with the loss angle, introduced in Section 1.3, indicated with the same symbol. This relation can also be written

p

f δ =

r 1

πµγ = const

where f is the frequency and the constant depends only on the material. For instance, in the case of copper, γ = 5.8 · 107_{S/m and µ = µ}

0= 4π · 10−7 H/m, hence

p

f δ = 0.0661√Hzm (2.21) The reason for the name will be explained below.

The wave impedance is computed with the same approximation:

Zm= r µ ε − jγ/ω ≈ r µ −jγ/ω = s jµω γ = 1 + j √ 2 r ωµ γ that is Zm= Rs(1 + j) (2.22)

where we have introduced the surface resistance

Rs= r ωµ 2γ = 1 γδ

for which we can write

Rs √ f = r πµ γ = const i E r E t E x z ŵĞƚĂů ĨƌĞĞƐƉĂĐĞ t H r H i H

Figure 2.5. Good conductor in a plane wave field. In the free space region both an incident and a reflected wave exist, in the metal only the transmitted one. The wavevector of the transmitted wave is drawn dashed, to indicate that it is complex.

Again, in the case of copper,

Rs

√

f = 2.6090 · 10

−7_Ω/√_Hz

Consider now a (highly idealized) conductor in the form of a half space, which faces free space, with a linearly plane wave incident normally on it, as shown in Fig. 2.5. The tangential electric and magnetic fields are continuous at the interface, then the ratio of their magnitudes is the same in z = 0− _{and in z = 0}+_{. But in z = 0}+ _{this ratio is Z}

mby definition, so we can easily understand

that the expressions of the electric fields are

Ei= E0e−jk0zˆx

Er_{= ΓE}

0ejk0zˆx

Et_{= (1 + Γ)E}

0e−jkmzxˆ

where the reflection coefficient is

Γ = Zm− Z0

Zm+ Z0

Since |Zm| ¿ Z0, Γ is close to −1. Indeed,

1 + Γ = 2Zm Zm+ Z0 ≈ 2Zm Z0 = 2(1 + j) s πε0f γ from which Γ ≈ −1 + 2(1 + j) s πε0f γ

In the case of copper,

Γ ≈ −1 + 2(1 + j) · 6.9252 · 10−10p_f

(frequency in Hz). We can also say that the metal enforces an impedance type boundary condition (see (1.20)) with Zmas surface impedance.

The magnetic field is

Hi= Y0E0e−jk0zˆy

Hr= −Y0ΓE0ejk0zˆy

Ht_{= Y}

0(1 − Γ)E0e−jkmzˆx ≈ 2Y0E0e−jkmzyˆ

(2.23)

Note that the total magnetic field at the interface is approximately twice the incident one because Γ is very close to −1.

The electric field in the metal produces a conduction current in the ˆx direction Jc= γEt= 2(1 + j)

p

πε0f γE0e−jkmzˆx

In the case of copper, this becomes

Jc = 2(1 + j) · 0.402

p

f E0e−jz/δe−z/δˆx

(frequency in Hz) where we have used (2.19). The magnitude of this current density is maximum at the interface and then decays exponentially in the metal. At a depth z = δ, it has reduced by a factor 1/e = 0.368. Eq.(2.21) allows a simple computation of δ for various frequencies, reported in

Table 2.1. Skin depth for copper at various frequencies

Frequency Skin depth 50 Hz 9.3 mm 1 kHz 2.1 mm 1 MHz 66.1 µm 1 GHz 2.1 µm

Table 2.1. We see clearly that as the frequency increases, the current density remains appreciable only in a very thin layer close to the metal surface, which can be considered as its “skin”. Even if this analysis strictly refers to a metal half space, we can use it to draw qualitative conclusions in the case of finite thickness conductors or even round conductors, provided the thickness is much larger than the skin depth. At the power frequency of 50Hz, the skin depth is so large that the current has a uniform distribution in ordinary wires. At the frequency of 1MHz, instead, most of the conductor copper is not used. At microwave frequencies, a few microns of copper deposited on an insulator perform as an excellent conductor.

The consequence of the skin depth change with frequency is that the resistance of a conductor is an increasing function of frequency: indeed, the “effective” cross-section of the conductor decreases as the frequency increases. This phenomenon is generally called skin effect.

Let us compute the impedance of the structure of Fig. 2.5, viewed as a current carrying con-ductor. Since the fields and the current density does not depend on y, we consider a strip of unit length in this direction. We compute first the total current I, flowing in the ˆx direction, per unit length along y: I = Z ∞ 0 Jc(x,z) · ˆxdz = Z ∞ 0 Jc0e−j(1−j)z/δdz = Jc0δ 1 + j (2.24)

Notice that the dimensions of I are correctly A/m, since Jc0is a surface current density with value

Jc0= 2(1 + j)

p

πε0f γE0 (2.25)

Next, consider a unit length in the ˆx direction of this conductor and compute the potential differ-ence along this length by integrating the electric field Ex along the x axis (y = 0, z = 0). Note

that Exdoes not depend on x, hence Ex itself coincides numerically with this potential difference.

Finally, the impedance per unit width in the y direction and unit length in the x direction is given by Zpul= Ex(0,0) I = Jc0/γ Jc0δ/(1 + j) =1 + j γδ = Zm

where we used (2.22). In conclusion we have this remarkable result: the impedance seen by a current flowing through a square of unit sides coincides with the wave impedance in the metal. Notice that, apart from the similarity in the symbols,

Zm= Ex

Hy

hence it is a completely different concept. Moreover, since the conductor we are considering has unit width in the y direction, unit length in the x direction (and infinite thickness in the z direction)

the previous analysis shows why often the value of the surface resistance Rs is expressed in Ω/¤

(read Ohm per square).

As a final remark, we note that the material becomes a perfect conductor when γ → ∞. In these conditions, the skin depth vanishes and the value of the current density at the interface tends to infinity, according to (2.25). Nevertheless, we see from (2.24) that the total current is finite and its value, independent of γ is

I = Jc0δ

1 + j = 2 r

ε0

µ0E0

where (2.20) and (2.25) have been used. This means that in a perfect conductor the current density can be written

Jc(x,z) = 2

r

ε0

µ0E0δ(z) = Jσδ(z)

On the other hand, from (2.23) we see that in the limit γ → ∞ the magnitude of the total magnetic field at the z = 0− _{interface coincides with J}

σ. Taking the directions of the vectors into account,

we conclude that if a perfect conductor is immersed in an electromagnetic field, on its surface a current density Jσ (A/m) appears, such that

Jσ= ˆν × H

where ˆν is the normal to the PEC surface, pointing toward free space. In practice, this is the proof

of Eq.(1.19).

Another example that we consider now is that of sea water: because of the salt contained in it, the conductivity is γ = 5 S/m, whereas the relative permittivity, up to the microwave region, does not change very much and will be taken to be εr= 80. We compute the complex wavenumber and

the attenuation constant by the general equation

km= ω

p

(ε − jγ/ω)µ The results are the following:

• At f = 100Hz, γ/(2πf ε0εr) = 1.1234 · 107, so sea water behaves as a good conductor;

k = (4.4429 − j4.4429) · 10−2_m−1 _{α = 0.3859dB/m}

• At f = 10kHz, γ/(2πf ε0εr) = 1.1234 · 105, so sea water behaves as a good conductor;

k = (0.4429 − j0.4429)m−1 _{α = 3.8590dB/m}

• At f = 1GHz, γ/(2πf ε0εr) = 1.1234, so the displacement currents cannot be neglected;

k = (209.7536 − j94.1066)m−1 _{α = 817.3998dB/m}

• At f = 10GHz, γ/(2πf ε0εr) = 0.1123, so the displacement currents cannot be neglected;

k = (1877.5266 − j105.1341)m−1 _{α = 913.1833dB/m}

Obviously, at microwave frequency, the attenuation of sea water precludes the possibility of commu-nicating with submarines during subsurface navigation. This becomes possible at low frequencies, where, however, the available bandwidth is very narrow.

Radiation in free space

The fundamental problem in electromagnetics is computing the fields created by a specified set of sources in a given region of space. This means that the functions ε(r), µ(r) are assigned, as well as the form of the region boundary and the material of which it consists. Then the sources are specified in terms of electric and magnetic current densities J(r), M(r).

In order to understand the basic mechanism of radiation, it is convenient to consider first a highly idealized problem, wherein the sources radiate in an infinite homogeneous medium. Later we will see how to apply the results of this chapter to the real antenna problem.

3.1

Green’s functions

The radiation problem is mathematically formulated as (

∇ × E = −jωµ0H − M

∇ × H = jωε0E + J

(3.1) in an infinite homogeneous domain that we assume to be free space. These equations are linear with constant coefficients and the independent variable is r. We can interpret them as the equations of a Linear Space Invariant system (LSI), where the source currents play the role of input and the radiated fields that of output, see Fig. 3.1. The box represents a system with two inputs and two outputs. E(r) H(r) J(r) M(r) EJ G HJ G HM G EM G

Figure 3.1. Linear system view of the radiation phenomenon

LSI systems are clearly a multidimensional generalization of Linear Time Invariant (LTI) sys-tems. Let us review briefly the properties of the latter. LTI systems, as shown in Fig. 3.2 are completely characterized in time domain by their impulse response h(t), that is the output that is obtained when the input is a Dirac delta function δ(t). An arbitrary (continuous) input x(t) can be represented as a linear combination of pulses thanks to the sifting property of the delta function

x(t) =

Z ∞ −∞

x(t0_{)δ(t − t}0_)dt0

Because of linearity, the response y(t) to x(t) can be found by convolution

y(t) = h(t) ∗ x(t) =

Z ∞ −∞

h(t − t0_)x(t0_)dt0

Alternatively, an LTI system can be characterized by its transfer function: when the input is x(t) = exp(jωt), the output is proportional to it and the constant of proportionality is, by definition, H(ω), so that y(t) = H(ω) exp(jωt). It can be proved that the impulse response and the transfer function of a system are related by a Fourier transform

H(ω) = Z _∞ −∞ h(t) exp(−jωt)dt

( )

x t

h t

( )

y t

( )

X

( )

ω

H

( )

ω

Y

( )

ω

Figure 3.2. Time domain and frequency domain description of an LTI system

As a preparation for (3.1), let us consider the simpler case of an infinite, uniform transmission line excited by a distribution of voltage and current generators, vs(z) and is(z), as shown in Fig. 3.3.

Since these generators are distributed continuously, vs(z) and is(z) are densities per unit length

of generators described, as usual, in terms of their open circuit voltage (V/m) and short circuit

current (A/m), respectively. The differential equations of the system are      −dV dz = jωLI + vs −dI dz = jωCV + is (3.2)

+

+

v z

( )

+

+

( )

i z

( )

V z

I z

( )

( )

v z

( )

i z

G

G

G

G

Figure 3.4. Linear system view of the transmission line with distributed generators.

Generally, transmission lines are excited at one end by a generator that acts as a transmitter. The model shown in Fig. 3.3 refers to a situation of electromagnetic compatibility, where a line is excited by an electromagnetic wave that couples to the line along a certain segment of it. It is easy to recognize that this is the one dimensional analogue of Maxwell’s equations (3.1). The problem is again LSI and can be schematized as in Fig. 3.4. Hence, as suggested by this picture, the solution can be expressed as

V (z) = Z∞ Z _∞ −∞ GV is(z − z 0_)i s(z0)dz0+ Z _∞ −∞ GV vs(z − z 0_)v s(z0)dz0 I(z) = Z _∞ −∞ GIis(z − z 0_)i s(z0)dz0+ Y∞ Z _∞ −∞ GIvs(z − z 0_)v s(z0)dz0

The system here has two inputs and two outputs: each output depends on both inputs, so that in practice there are four Green’s functions, each one a pure number. They can be obtained by applying the spatial Fourier transform to the system equations (3.2). However, by the very definition of Green’s function

• GV is(z) is the voltage wave V (z)/Z∞created by a unit amplitude current generator located in z = 0

• GV vs(z) is the voltage wave V (z) created by a unit amplitude voltage generator located in

z = 0

• GIis(z) is the current wave I(z) created by a unit amplitude current generator located in

z = 0

• GIvs(z) is the current wave I(z)Y∞created by a unit amplitude voltage generator located in

z = 0

so that they can be found by simple circuit theory, just recalling that the input impedance of an infinitely long line is Z∞. The resulting expressions are

GV is(z) = GIvs(z) = − 1 2e −jk|z| GV vs(z) = GIis(z) = − 1 2sgn(z)e −jk|z|

where sgn(z) is the sign function

sgn(z) = ½

1 if z > 0

As another preparatory example before tackling (3.1), let us consider the case of sound waves. It can be shown that the excess pressure p(r) with respect to the background pressure satisfies the scalar Helmholtz equation _µ

∇2+ω 2 V2 s ¶ p(r) = −S(r) (3.3)

where Vsis the sound velocity and S(r) is a source term. This equation corresponds to the picture

of Fig. 3.5, where S(r) is the input and p(r) the output. In this case the system has only one input and one output but it is multidimensional, since both depend on the three independent variables x, y, z. In perfect analogy with LTI systems, LSI systems are completely characterized in space domain by their “impulse response” G(r), which is traditionally called Green’s function. This is the output of the system when the input is a point source located at the origin of the coordinate system, which can be represented mathematically by a three-dimensional Dirac delta function S(r) = δ(r) = δ(x)δ(y)δ(z). The fundamental property of this multidimensional Dirac δ

function is _Z δ(r)dr = Z ∞ −∞ Z ∞ −∞ Z ∞ −∞ δ(x)δ(y)δ(z)dx dy dz = 1

When the input is an arbitrary function, the output is found by (three-dimensional) convolution

p(r) =

Z

G(r − r0_)S(r0_)dr0

Alternatively, an LSI system can be characterized in the spectral domain. When the input is a harmonic function of x, y, z, that is S(r) = exp(−j(kxx + kyy + kzz)) = exp(−jk · r), the output

is proportional to it and the coefficient of proportionality is, by definition, the transfer function

H(k). Again, transfer function and Green’s function of the same system are related by a Fourier

transform: however, in this case, it is triple, since it operates on the three variables x, y, z. The couple of inverse and direct 3-D Fourier transform is given by

G(r) = 1 (2π)3 Z H(k) exp(−jk · r)dk H(k) = Z G(r) exp(jk · r)dr (3.4)

where dk = dkxdkydkz. It can be shown that in the case of free space, the transfer function is

H(k) = 1 k2_{− ω}2_/V2

s

(3.5) and the corresponding Green’s function is

G(r) = exp(−jk0r)

4πr (3.6)

( )

S r

G r

( )

p r

( )

S k

( )

H k

( )

p k

( )

with k0 = ω/Vs denoting the wavenumber. This expression describes a diverging spherical wave.

Indeed, the constant phase surfaces are obviously r = const, a series of concentric spheres with center in the origin. Moreover, assuming that the source is harmonic with frequency ω0, the

expression of the Green’s function in time domain is

g(r,t) = R ½ exp(−jk0r) 4πr exp(jω0t) ¾ = cos(ω0t − k0r) 4πr from which it is evident that the phase velocity is

Vph= ω0/k0= Vs> 0

As another well known example of LSI system, let the frequency ω go to zero in (3.3), so that the Helmholtz equation becomes Poisson equation. This, for example, relates the electric potential

V (r) to a charge distribution ρ(r), which acts as its source: ∇2V (r) = −ρ(r)

ε

The transfer function associated to this equation is (from (3.5))

H(k) = 1 εk2

and the corresponding Green’s function (from (3.6))

G(r) = 1

4πεr

We recognize immediately this expression as the potential generated by a point charge q = 1 C in a dielectric with permittivity ε.

We are ready now for Maxwell’s equations (3.1), which are still more complicated because in addition to being multidimensional and multiple input/output, they are vector equations: this means that the output is a vector that is not necessarily parallel to the input. This implies that each of the four Green’s function is not a scalar but a linear operator (a tensor ), which, in a basis, is represented by a 3 × 3 matrix. This means that, differently from the case of sound waves, the Green’s function is not directly the field radiated by a point source. The source is really a point but is also a vector, which can have all possible orientations. From a certain point of view, we can say that the Green’s tensor yields the field radiated in a given point by a point source in the origin with all possible orientations. This concept will be better clarified in Section 3.2.

In coordinate-free language E(r) = −jωµ0 Z GEJ(r − r0) · J(r0)dr0− Z GEM(r − r0) · M(r0)dr0 H(r) = Z GHJ(r − r0) · J(r0)dr0− jωε0 Z GHM(r − r0) · M(r0)dr0 (3.7)

To check the dimensions of the various Green’s functions, it is useful to note that

ωµ0= k0Z0 ωε0= k0Y0= k0/Z0

Hence we recognize that GEJ and GHM are measured in m−1, GEM and GHJ in m−2. The