Complex Poynting's Theorem and Radiation Resistance
To the generator supplying its terminal current, a radiating
antenna appears as a load with an impedance having a resistive part.
This is true even
if the antenna is made from perfectly conducting material and
therefore incapable of converting electrical power to heat. The power
radiated away from the antenna must be supplied through its terminals,
much as if it were dissipated in a resistor. Indeed, if there is no
electrical dissipation in the antenna, the power supplied
at the terminals is that radiated away. This statement of power
conservation makes it
possible to determine the equivalent resistance of the antenna simply
by using the far fields that were the theme of Sec. 12.4.
Complex Poynting's Theorem
For systems in the sinusoidal steady state, a useful alternative
to the form of Poynting's theorem introduced in Secs. 11.1 and 11.2
results from writing Maxwell's equations in terms of complex
amplitudes before they are combined to provide the desired
theorem. That is, we assume at the outset that fields and sources
take the form
Suppose that the region of interest is composed either of free
space or of perfect conductors. Then, substitution of complex
amplitudes into the laws of Ampère and Faraday, (12.0.8) and
The manipulations that are now used to obtain the desired "complex
Poynting's theorem" parallel those used to derive the real,
time-dependent form of Poynting's theorem in Sec. 11.2. We dot
E with the complex conjugate of (2) and subtract the dot product
of the complex conjugate of H with (3). It follows
The object of this manipulation was to obtain the "perfect" divergence
on the left, because this expression can then be integrated over a
volume V and Gauss' theorem used to convert the volume integral on the
left to an integral over the enclosing surface S.
(A x B) = B x A - A x B
This expression has been multiplied by , so that its real
part represents the time average flow of power, familiar from Sec.
11.5. Note that the real part of the first term on the right is
zero. The real part of (5) equates the time average of the
Poynting vector flux into the volume with the time average of the power
imparted to the current density of unpaired charge, Ju, by the
electric field. This information is equivalent to the time average
of the (real form of) Poynting's theorem. The imaginary part of (5)
relates the difference between the time average magnetic and electric
energies in the volume V to the imaginary part of the complex
Poynting flux into the volume. The imaginary part of the complex
Poynting theorem conveys additional information.
Consider the perfectly conducting antenna system surrounded by the
spherical surface, S, shown in Fig. 12.5.1. To exclude sources from
the enclosed volume, this surface is composed of an outer surface,
Sa, that is far enough from the antenna so that only the radiation
field makes a contribution, a surface Sb that surrounds the
source(s), and a surface Sc that can be envisioned as the wall of a
system of thin tubes connecting Sa to Sb in such a way that
Sa + Sb + Sc is indeed the surface enclosing V. By making the
connecting tubes very thin, contributions to the integral on the left
in (5) from the surface Sc are negligible. We now write, and
then explain, the terms in (5) as they describe this radiation
Figure 12.5.1 Surface S encloses the antenna
but excludes the source. Spherical part of S is at "infinity."
The first term is the contribution from integrating the radiation
Poynting flux over Sa, where (12.4.2) serves to eliminate H.
The second term comes from the surface integral in (5) of the
Poynting flux over the surface, Sb, enclosing the sources
(generators). Think of the generators as enclosed by perfectly
conducting boxes powered by terminal pairs (coaxial cables) to which
the antennae are
attached. We have shown in Sec. 11.3 (11.3.29) that the integral
of the Poynting flux over Sb is equivalent to the sum of
voltage-current products expressing power flow from the circuit point
of view. The first term on the right is the same as the first on the
right in (5). Finally, the last term in (5) makes no contribution,
because the only regions where J exists within V are those modeled
here as perfectly conducting and hence where E = 0.
Consider a single antenna with one input terminal pair.
The antenna is a linear system, so the complex voltage must be
proportional to the complex terminal current.
Here, Zant is the impedance of the antenna. In terms of this
impedance, the time average power can be written as
It follows from the real part of (6) that the radiation
resistance, Rrad, is
The imaginary part of Zout describes the reactive power
supplied to the antenna.
The radiation field contributions to this integral cancel out. If the
antenna elements are short compared to a wavelength, contributions to
(10) are dominated by the quasistatic fields. Thus, the electric
dipole contributions are dominated by the electric field (and the
reactance is capacitive), while those for the magnetic dipole are
inductive. By making the antenna on the order of a wavelength, the
magnetic and electric contributions to (10) are often made to
essentially cancel. An example is a half-wavelength version of the
wire antenna in Example 12.4.1. The equivalent circuit for such
resonant antennae is then solely the radiation resistance.
Example 12.5.1. Equivalent Circuit of an Electric Dipole
An "end-loaded" electric dipole is composed of a pair of
perfectly conducting metal spheres, each of radius R, as shown in Fig.
12.5.2. These spheres have a spacing, d, that is short compared to a
wavelength but large compared to the radius, R, of the spheres.
Figure 12.5.2 End-loaded dipole and equivalent
The equivalent circuit is also shown in Fig. 12.5.2. The statement that
the sum of the voltage drops around the circuit is zero requires that
A statement of power flow is obtained by multiplying this expression
by the complex conjugate of the complex amplitude of the current.
Here the dipole charge, q, is defined such that i = dq/dt. The real
part of this expression takes the same form as the statement of
complex power flow for the antenna, (6). Thus, with provided by (12.2.23) and (12.2.24), we can solve for the
Note that because k /c, this radiation resistance is
proportional to the square of the frequency.
The imaginary part of the impedance is given by the right-hand side
of (10). The radiation field contributions to this integral cancel
out. In integrating over the near field, the electric energy storage
dominates and becomes essentially that associated with the quasistatic
capacitance of the pair of spheres. We assume that the spheres are
connected by wires that are extremely thin, so that their effect
can be ignored. Then, the capacitance is the series capacitance
of two isolated spheres, each having a capacitance of 4 R.
Radiation fields are solutions to the full Maxwell equations. In
contrast, EQS fields were analyzed ignoring the magnetic flux linkage
in Faraday's law. The approximation is justified if the size of the
system is small compared with a wavelength. The following example
treats the scattering of particles that are small compared with the
wavelength. The fields around the particles are EQS, and the currents
induced in the particles are deduced from the EQS approximation.
These currents drive radiation fields, resulting in Rayleigh
scattering. The theory of Rayleigh scattering explains why the sky
is blue in color, as the following example shows.
Example 12.5.2. Rayleigh Scattering
Consider a spatial distribution of particles in the field of an
infinite parallel plane wave. The particles are assumed to be small
as compared to the wavelength of the plane wave. They get polarized in
the presence of an electric field Ea, acquiring a dipole moment
where is the polarizability. These particles could be
atoms or molecules, such as the molecules of nitrogen and oxygen of
air exposed to visible light. They could also be conducting
spheres of radius R. In the latter case, the dipole moment
produced by an applied electric field Ea is given by (6.6.5) and
If the frequency of the polarizing wave is and its
propagation constant k = /c, the far field radiated by the
particle, expressed in a spherical coordinate system with its
= 0 axis aligned with the electric field of the wave, is, from
where d = j is used in the above
expression. The power radiated by each dipole, i.e., the power
scattered by a dipole, is
The scattered power increases with the fourth power of frequency when
is not a function of frequency. The polarizability of
N2 and O2 is roughly frequency independent. Of the visible
radiation, the blue (high) frequencies scatter much more than the red
(low) frequencies. This is the reason for the blue color of the sky.
The same phenomenon accounts for the polarization of the scattered
radiation. Along a line L at a large angle from the line from the
observer O to the sun S, only the electric field perpendicular to the
plane LOS produces radiation visible at the observer position (note
the sin2 dependence of the radiation). Thus, the scattered
radiation observed at O has an electric field perpendicular to LOS.
The present analysis has made two approximations. First, of course,
we assumed that the particle is small compared with a wavelength.
Second, we computed the induced polarization from the unperturbed
field E of the incident plane wave. This assumes that the
particle perturbs the wave negligibly, that the scattered power is
very small compared to the power in the wave. Of course, the
incident wave decreases in intensity as it proceeds through the
distribution of scatterers, but this macroscopic change can be
treated as a simple attenuation proportional to the density of