As discussed previously, the Green’s function is the basic tool for the computation of the field radiated by any source by means of eq. (3.7). However, it is convenient to start with the simplest one, i.e. a point source, and this will help in understanding the properties of the Green’s functions.
Consider first an elementary source of electric type located at the origin of the coordinate system, modeled by the current distribution
J(r) = Meδ(r)
The vector Me is called the electric dipole moment of the current distribution and is measured in Am (recall that the dimensions of the three dimensional δ function are m−3). An arbitrary current distribution can be characterized by its moments. This concept is used also in the theory of probability: if ξ is a random variable with density function Wξ(x), moments of all orders can be defined by
mn= E{ξn} = Z ∞
−∞
xnWξ(x)dx
where E{ } denotes the expectation value. In the case of the current distribution, the role of Wξ(x) is played by J(r), but the situation is more complicated because its vector nature implies that the moments beyond the first are tensors. The first moment (dipole moment) is a vector, defined by
Me= Z
J(r)dr (3.10)
In the case of the point source introduced above, thanks to the properties of the delta function, the previous equation becomes an identity and we understand the reason for the name of the coefficient.
From a practical point of view, we can imagine to obtain this source by a limiting process, starting from a rectilinear current I, whose length l is progressively reduced without changing the aspect ratio (diameter/length of the wire), while, at the same time, the current is increased, so that the value of the integral (the dipole moment Il) remains constant.
Introducing the dipole current into (3.7), we find that the radiated fields are given by E(r) = −jωµ0
Z
GEJ(r − r0) · Meδ(r0)dr0= −jωµ0GEJ(r) · Me
H(r) = Z
GHJ(r − r0) · Meδ(r0)dr0= GHJ(r) · Me
(3.11)
Since we know the expressions of the matrices representing the Green’s functions in the spherical basis, it is necessary to express the vector Me in the same basis. Let us assume that the polar axis of the coordinate system is in the direction of Me, i.e. assume Me = Meˆz. Note that this step is allowed because the Green’s function does not depend on the angular variables, as a
consequence of the isotropy of free space. It is to be remarked, as a general rule, that the Green’s function depends only on the structure and, hence, shares its symmetries. This choice guarantees the simplest description of the radiated field. Since the radiated field must have the direction of Meas a symmetry axis, orienting the polar axis of the coordinate system in this direction allows the expressions to be independent on ϕ.
Me= (Me· ˆr)ˆr + (Me· ˆϑ) ˆϑ + (Me· ˆϕ) ˆϕ
= Me(ˆz · ˆr)ˆr + Me(ˆz · ˆϑ) ˆϑ + Me(ˆz · ˆϕ) ˆϕ
= Mecos ϑˆr − Mesin ϑ ˆϑ
(3.12)
where we have exploited (A.12). Now recalling the expression of the Green’s function (3.8), we obtain
where use has been made of
ωµ0= k0Z0=2π λZ0
and Z0 is the wave impedance. Concerning the meaning of (3.12), it is to be remarked that the matrix (3.8) represents the Green’s function in the spherical basis consisting of the unit vectors ˆr, ˆϑ, ˆϕ defined in the observation point r. Hence, even if the source is located in the origin, its components are evaluated in the basis associated to the point r.
We can proceed similarly for the magnetic field:
H(r) = −jk0
In conclusion, the electromagnetic field radiated by an electric dipole has the following expression E(r) = −jZ0Meexp(−jk0r)
This wave has two components of electric field and only one of magnetic field. Imagine a geo-graphical system of coordinates such that the direction of the dipole moment defines the direction of the earth axis. The angle ϑ is the colatitude (= 90◦−latitude), the angle ϕ is the longitude.
Then the electric field is contained in the meridian planes and the magnetic field is tangent to the parallels. This type of wave is called TM (Transverse Magnetic) since the magnetic field has no radial component. We recognize also that the radial component of the electric field is dominant close to the source, but negligible with respect to the others at large distance. Here the wave is essentially TEM, since neither field has a (significant) radial component.
Let us compute the energy budget by means of the Poynting theorem. The Poynting vector is
Substituting in the previous equation we get S = Z0Me2
According to Poynting theorem, the surface density of active power flow is dP
dΣ = 1
2R{S · ˆν} (3.16)
where ˆν is the normal to the surface element. In order to compute the total radiated active power, we have to evaluate the flux of the Poynting vector across a closed surface surrounding the source.
For maximum simplicity we choose a sphere of radius r:
Prad= 1
Here we have used the following facts
• the normal to the spherical surface is ˆν = ˆr
• the area element in spherical coordinates is dΣ = r2sin ϑ dθdϕ
• the integrand does not depend on ϕ, so the ϕ integration yields the 2π factor
• the ϑ integration yields Z π
0
sin3ϑ dθ = 4 3
The factor 1/2 has been left explicit to make it clear that Me is a peak value, that is the time domain dipole moment is Me(t) = Mecos(ω0t). If, on the contrary Me is an effective value, the factor 1/2 has to be dropped.
We notice that the total radiated power does not depend on the radius of the sphere chosen to compute it. Algebraically this is the result of the cancellation between the r2 factor in the denominator of the Poynting vector and the one in the area element dΣ. To get a more physical explanation, consider the fluxes through two concentric spheres of different radii: if they were different, power would be lost or generated in the shell, which is impossible by conservation of energy in a lossless medium.
In (3.17) we took the real part of the integral. The imaginary part is the reactive power Q = 1
It is reasonable that Q depends on r: the reactive power is the energy that twice per period is exchanged between generator and load, hence crosses the spherical surface of radius r. Moreover, it may be remarked that in the ϑ direction, the structure is closed as a (virtual) cavity. This explains why the ϑ component of the Poynting vector is pure imaginary. Indeed, a real part of Sϑ
would imply a steady energy flow in that direction; however, this is impossible, since the angular domain is finite (0 ≤ ϑ ≤ π/2) and the dielectric is lossless.
It is useful to explicitly indicate the dominant components close to the source and far from it.
In the far field region, r À λ
We see that the fields tend to be linearly polarized, orthogonal and proportional to each other and also orthogonal to the radial direction. These properties are summarized in the impedance relation
H(r) ∼ 1
Z0ˆr × E(r) E(r) ∼ Z0H(r) × ˆr
The radiated field is a spherical wave (because of the factor exp(−jk0r)), diverging from the origin with phase velocity c = ω/k0= 1/√
ε0µ0. Indeed, the time varying electric field is given by E(r,t) = R{E(r)ejωt}
∼ −Z0Me
2rλ sin ϑ sin(ω0t − k0r) ˆϑ
Note that there is no contradiction between the statement that the radiated field is a spherical wave and the fact that this field has the sin ϑ dependence: the former refers to the constant phase surfaces, the latter to a magnitude factor.
The field amplitudes decay as 1/r. This behavior is strictly connected with the principle of conservation of energy. Indeed, it is easy to see that the total radiated active power computed in
(3.17) is due only to these 1/r components.
The active power density per unit surface dP/dΣ, at any distance, (dΣ orthogonal to ˆr) is given, according to (3.16), by
dP dΣ =1
2 Z0Me2
4r2λ2 sin2ϑ (3.19)
Thus, in the far field,
dP dΣ ∼ 1
2
|E|2 Z0
exactly as in the case of a plane wave, see (2.13). This fact should not be surprising, since a spherical wave can be locally approximated by a plane wave with direction of propagation coincident with the radial direction. In the near field region r ¿ λ
E(r) ∼ −jZ0Me
which may be reduced to
E(r) ∼ −jZ0Meλ
We notice that the exponential has been dropped, since its value, for r ¿ λ is essentially 1. Clearly, very close to the source, the propagation delay is negligible. We can recognize that the electric field coincides with that of a static dipole, whose moment, however, is a sinusoidal function of time. Likewise, the magnetic field coincides with that created, in accordance with the Biot-Savart law, by an infinitesimal current element, where the current is a sinusoidal function of time. This regime is called quasi-static.
Fig. 3.6 shows a sketch of the electric field lines in a plane ϕ = const; close to the source the field line behavior is similar to that of a static dipole, further away, entering in the radiation region, they are completely different. Note that they are closed: we are accustomed to magnetic field line being closed because ∇ · B = 0, i.e. B is solenoidal. Actually in this case also the electric field is (almost) everywhere solenoidal. Recall the divergence equation
∇ · E = ρ
ε = 0 for r 6= 0 because the source is point like and located in the origin.
Note that the image shown in the figure refers to a specific time; as time passes the contours move outward radially. Fig. 3.7 shows the traditional way to plot the ϑ component of the electric field in the far field region, in a plane ϕ =const. It is a polar plot and the curves have equation ρ = sin ϑ (with ρ denoting the length of OP ), hence they are circles.
The other elementary source that we describe now is the dual of the electric dipole, i.e. a magnetic dipole. We consider then formally a point source of magnetic current,
M(r) = Mmδ(r)
Figure 3.6. Dipole electric field lines
x z
ϑ P
O 1
Figure 3.7. Dipole far field. Normalized polar plot of Eϑand Hϕat large distance. The maximum field value is reached on the equator ϑ = π/2 and is normalized to 1. For any ϑ the length of the segment OP is proportional to the value of Eϑ
The vector Mm is called the magnetic dipole moment of the current distribution and is defined as Mm=
Z
M(r)dr
Its units are Vm. It is well known that magnetic currents do not exist, as flow of magnetic charges, but can be introduced formally to describe electric currents circulating in closed loops. Indeed, this elementary source can be imagined to be obtained with a limiting procedure from a loop, enclosing an area S, on which the current I is circulating. The “equivalent” magnetic current has a direction perpendicular the plane of the loop as indicated by the thumb of the right hand when the fingers are aligned along the electric current flow. The point source is obtained by shrinking the loop and increasing the current so that the magnetic dipole moment remains constant. It can
be shown that the magnetic dipole moment of a small loop is Mm= jωµSI
Notice that since ωµ = k0Z0, this quantity is measured in Ω/m and Mmturns out with the right dimensions. In order to compute the fields radiated by such a source, we substitute its expression in (3.7) and obtain
E(r) = −
As in the case of the electric dipole, we choose a spherical coordinate system with the polar axis aligned along Mm, which then takes the form Mm= Mmˆz. Recalling (3.8), (3.12) and computing the scalar products we obtain
E(r) = −jMmexp(−jk0r)
If we compare these expressions with those of the fields radiated by an electric dipole (3.15), we see that they (obviously!) satisfy the principle of duality, in the sense that the expressions of electric and magnetic fields are exchanged in the two cases. This wave is of T E type since the electric field has no radial component and tends to become T EM in the far field region.
The Poynting vector associated to these fields is S = E × H∗= Y0Mm2
4r2λ2 h
B∗C sin2ϑˆr + A∗C sin ϑ cos ϑ ˆϑi
which is just the complex conjugate of the corresponding expression for the electric dipole. Hence the result is written down by inspection
S = Y0Mm2
The total radiated active power is given by the real part of the flux of S through a sphere concentric with the source and is
Prad=1
For completeness we report the expression of the reactive power Q =1
which has the opposite sign with respect to the case of the electric dipole, since this is a magnetic source. This must not be surprising: also in circuit theory the reactive power in inductors and capacitors has opposite signs.
It is useful to explicitly indicate the dominant components close to the source and far from it.
In the far field region, r À λ
E(r) ∼ −jMmexp(−jk0r) 2rλ sin ϑ ˆϕ
H(r) ∼ jY0Mmexp(−jk0r) 2rλ sin ϑ ˆϑ
(3.25)
The impedance relation is the same as the one for the electric dipole fields:
H(r) ∼ 1 Z0
ˆr × E(r) E(r) ∼ Z0H(r) × ˆr In the near field region r ¿ λ
E(r) ∼ −Mm
2rλ 1
k0rsin ϑ ˆϕ
H(r) ∼ −jY0Mm
2rλ
· 2
(k0r)2cos ϑˆr + 1
(k0r)2sin ϑ ˆϑ
¸ (3.26)
which may be reduced to
E(r) ∼ −Mm
4πr2sin ϑ ˆϕ
H(r) ∼ −jY0Mmλ 8π2r3
h
2 cos ϑˆr + sin ϑ ˆϑ
i (3.27)
Obviously, Figs. 3.6 and Fig. 3.7 can be used also for magnetic dipoles.