Electrodynamic Fields in the Presence of PerfectConductors
The superposition integral approach is directly applicable to the
determination of electrodynamic fields from sources specified
throughout all space. In the presence of materials, sources are induced as
well as imposed. These sources cannot be specified in advance. For
example, if a perfect conductor is introduced, surface currents and
charges are induced on its surface in just such a way as to insure
that there is neither a tangential electric field at its surface nor a
magnetic flux density normal to its surface.
We have already seen how the superposition integral approach can
be used to find the fields in the vicinity of perfect conductors, for
EQS systems in Chap. 4 and for MQS systems in Chap. 8. Fictitious
sources are located in regions outside that of interest so that they
add to those from the actual sources in such a way as to satisfy the
boundary conditions. The approach is usually used to provide simple
analytical descriptions of fields, in which case its application is a
bit of an art- but it can also be the basis for practical numerical
analyses involving complex systems.
We begin with a reminder of the boundary conditions that
represent the influence of the sources induced on the surface of a
perfect conductor. Such a conductor is defined as one in which E
0 because .
Because the tangential electric
field must be continuous across the boundary, it follows from
Faraday's continuity condition that just outside the surface of the
perfect conductor (having the unit normal n)
In Sec. 8.4, and again in Sec. 12.1, it was argued that (1)
implies that the normal magnetic flux density just outside a perfectly
conducting surface must be constant.
The physical origins and limitations of this boundary condition were
one of the subjects of Chap. 10.
Method of Images
The symmetry considerations used to satisfy boundary conditions
in Secs. 4.7 and 8.6 on certain planes of symmetry are equally
applicable here, even though the fields now suffer time delays under
transient conditions and phase delays in the sinusoidal steady state.
We shall illustrate the method of images for an incremental dipole.
It follows by superposition that the same method can be used with
arbitrary source distributions.
Suppose that we wished to determine the fields associated with an
electric dipole over a perfectly conducting ground plane. This
dipole is the upper one of the two shown in Fig. 12.7.1. The
associated electric and magnetic fields were determined in Sec. 12.2,
and will be called Ep and Hp, respectively. To satisfy
the condition that there be no tangential electric field on the
perfectly conducting plane, that plane is made one of symmetry in an
equivalent configuration in which a second "image" dipole is mounted,
having a direction and intensity such that at any instant, its
charges are the negatives of those of the first dipole. That is, the
+ charge of the upper dipole is imaged by a negative charge of
equal magnitude with the plane of symmetry perpendicular to and
bisecting a line joining the two. The second dipole has been arranged
so that at each instant in time, it produces a tangential E =
Eh that just cancels that of the first at each location on the
symmetry plane. With
we have made E satisfy (1) and hence (2) on the ground plane.
There are two ways of conceptualizing the "method of images."
The one given here is consistent with the superposition integral
point of view that is the theme of this chapter. The second takes
the boundary value point of view of the next chapter. These
alternative points of view are familiar from Chaps. 4 and 5 for EQS
systems and from the first and second halves of Chap. 8 for MQS
systems. From the boundary value point of view, in the upper
half-space, Ep and
Hp are particular solutions, satisfying the inhomogeneous wave
equation everywhere in the volume of interest. In this region, the
fields Eh and Hh due to the image dipole are then
solutions to the homogeneous wave equation. Physically, they
represent fields induced by sources on the perfectly conducting
To emphasize that the symmetry arguments apply regardless of the
temporal details of the excitations, the fields shown in Fig. 12.7.1
are those of the electric dipole during the turn-on transient
discussed in Example 12.2.1. At an arbitrary point on the
ground plane, the "real" dipole produces fields that are not
necessarily in the plane of the paper or perpendicular to it. Yet
symmetry requires that the tangential E due to the sum of the
fields is zero on the ground plane, and Faraday's law requires that
the normal H is zero as well.
Figure 12.7.1 Dipoles over a ground plane together
with their images: (a) electric dipole; and (b) magnetic dipole.
In the case of the magnetic dipole over a ground plane shown in
Fig. 12.7.1b, finding the image dipole is easiest by nulling the
magnetic flux density normal to the ground plane, rather than the
electric field tangential to the ground plane. The fields shown are
the dual [(12.2.33)-(12.2.34)] of those for the electric dipole turn-on
transient of Example 12.2.1. If we visualize the dipole as due to
magnetic charge, the image charge is now of the same sign, rather
than opposite sign, as the source.
Image methods are commonly used in extending the superposition
integral techniques to antenna field patterns in order to treat
the effects of a ground plane and of reflectors.
Example 12.7.1. Ground Planes and Reflectors
Quarter-Wave Antenna above a Ground Plane
The center-fed wire antenna of Example
12.4.1, shown in Fig. 12.7.2a, has a plane of symmetry, =
/2, on which there is no tangential electric field. Thus,
provided the terminal current remains the same, the field in the upper
half-space remains unaltered if a perfectly conducting ground plane is
placed in this plane. The radiation electric field is therefore given
by (12.4.2), (12.4.5), and (12.4.8). Note that the lower half of the
wire antenna serves as an image for the top half. Whether used for
AM broadcasting or as a microwave mobile antenna (on the roof of an
automobile), the height is usually a quarter-wavelength. In
this case, kl = , and these relations give
where the radiation intensity pattern is
Figure 12.7.2 Equivalent image systems for three
Although the radiation pattern for the quarter-wave ground plane is
the same as that for the half-wave center-fed wire antenna, the
radiation resistance is half as much. This follows from the fact that
the surface of integration in (12.5.9) is now a hemisphere rather
than a sphere.
The integral can be converted to a sine integral, which is tabulated.
9 It is perhaps easiest to carry out the integral
numerically, as can be done with a programmable calculator. Note that
the integrand is zero at = 0. In free space, this radiation
resistance is 37 .
Two-Element Array over Ground Plane
The radiation pattern from an array of elements vertical to a ground
plane can be deduced using the same image arguments. The pair of
center-fed half-wave elements shown in Fig. 12.7.2c have lower
elements that serve as images for the quarter-wave vertical elements
over a ground plane shown in Fig. 12.7.2d.
If we consider elements with a half-wave spacing that are driven
180 degrees out of phase, the array factor is given by (12.4.15) with
ka = and 1 - o = . Thus, with o from (5),
the electric radiation field is
The radiation pattern is proportional to the square of this function
and is sketched in Fig. 12.7.2d. The field initiated by one element
arrives in the far field at = 0 and = with a phase
that reinforces that from the second element. The fields produced
from the elements arrive out of phase in the "broadside" directions,
and so the pattern nulls in those directions ( = /2).
Phased arrays of two or more verticals are often used by AM stations
to provide directed broadcasting, with the ground plane preferably
wet land, often with buried "radial" conductors to make the
ground plane more nearly like a perfect conductor.
Ground-Plane with Reflector
The radiation pattern for the pair
of vertical elements has no electric field tangential to a vertical
plane located midway between the elements. Thus, the effect of one of
the elements is equivalent to that of a reflector having a distance of
a quarter-wavelength from the vertical element. This is the
configuration shown in Fig. 12.7.2f.
The radiation resistance of the vertical quarter wave element
with a reflector follows from (12.5.9), evaluated using (7). Now
the integration is over the quarter-sphere which, together with the
ground plane and the reflector plane, encloses the element at a radius
of many wavelengths.
Demonstration 12.7.1. Ground-Planes, Phased Arrays, and Reflectors
The experiment shown in Fig. 12.7.3 demonstrates the effect of the
phase shift on the radiation pattern of the array considered in
Example 12.7.1. The spacing and length of the vertical elements are
7.9 cm and 3.9 cm, respectively, which corresponds to /2 and
/4 respectively at a frequency of 1.9 GHz. The ground
plane consists of an aluminum sheet, with the
array mounted on a section of the sheet that can be rotated. Thus, the
radiation pattern in the plane = /2 can be measured by
rotating the array, keeping the receiving antenna, which is many
wavelengths away, fixed.
An audible tone can be used to indicate the amplitude of the received
signal. To this end, the 1.9 GHz source is modulated at the desired
audio frequency and detected at the receiver, amplified, and made
audible through a loud speaker.
The 180 degree phase shift between the drives for the two driven
elements is obtained by inserting a "line stretcher" in series with
the coaxial line feeding one of the elements. By effectively
lengthening the transmission line, the delay in the transmission line
wave results in the desired phase delay. (Chapter 14 is devoted to
the dynamics of signals propagating on such transmission lines.) The
desired 180 degree phase shift is produced by rotating the array to a
broadside position (the elements equidistant from the receiving
antenna) and tuning the line stretcher so that the signals are nulled.
With a further 90 degree rotation so that the elements are in the
end-fire array position (in line with the receiving antenna), the
detected signal should peak.
One vertical element can be regarded as the image for the other in
a physical situation in which one element is backed at a
quarter-wavelength by a reflector. This quarter-wave ground plane
with a reflector is demonstrated by introducing a sheet of aluminum
halfway between the original elements, as shown in Fig. 12.7.3.
With the introduction of the sheet, the "image" element is shielded
from the receiving antenna. Nevertheless, the detected signal should
be essentially unaltered.
Figure 12.7.3 Demonstration of phase shift on
The experiment suggests many other interesting and practical
configurations. For example, if the line stretcher is used to null
the signal with the elements in end-fire array position, the elements are
presumably driven in phase. Then, the signal should peak if the array
is rotated 90 degrees so that it is broadside to the receiver.
Boundaries at the Nodes of Standing Waves
The TM fields found in Example 12.6.1 were those produced by
a surface charge density taking the form of a standing wave in the y
= 0 plane. Examination of the analytical expressions for E,
(12.6.27)-(12.6.28), and of their graphical portrayal, Fig. 12.6.3,
shows that at every instant in time, E was normal to the planes
where kx x = n (n any integer), whether the waves were
evanescent or propagating in the y directions. That is, the fields
have nodal planes (of no tangential E) parallel to the y - z
plane. These fields would therefore remain unaltered by the
introduction of thin, perfectly conducting sheets in these planes.
Example 12.7.2. TM Fields between Parallel Perfect Conductors
To be specific, suppose that the fields found in Example 12.6.1
are to "fit" within a region bounded by perfectly conducting
surfaces in the planes x = 0 and x = a. The configuration is
shown in Fig. 12.7.4. We adjust kx so that
where n indicates the number of half-wavelengths in the x
direction of the fields shown in Fig. 12.6.3 that have been made to
fit between the perfect conductors. To make the fields satisfy the
wave equation, ky must be given by (12.6.14) and (12.6.13). Thus,
from this expression and (10), we see that for the n-th mode
of the TM fields between the plates, the wave number in the y
direction is related to the frequency
Figure 12.7.4 The n = 1 TM fields between
parallel plates (a) evanescent in y direction and (b) propagating
in y direction.
We shall encounter these modes and this dispersion equation again
in Chap. 13, where waves propagating between parallel plates will be
considered from the boundary value point of view. There we shall
superimpose these modes and, if need be, comparable TM field modes, to
satisfy arbitrary source conditions in the plane y = 0. The sources in
the plane y = 0 will then represent an antenna driving a parallel plate
The standing-wave fields of Example 12.6.1 are the superposition of
two traveling waves that exactly cancel at the nodal planes to form
the standing wave in the x direction. To see this, observe that a
standing wave, such as that for the surface charge distribution given
by (12.6.18), can be written as the sum of two traveling
10 sin u = (exp(ju) - exp (-ju))/2j
By superposition, the field responses therefore must take this same
form. For example, Ey as given by (12.6.28) can be written
where the upper and lower signs again refer to the regions above and
below the sheet of charge density. The first term represents the
response to the component of the surface current density that travels
to the right while the second is the response from the component
traveling to the
left. The planes of constant phase for the component waves
traveling to the right, as well as their respective directions of
propagation, are as for the TE fields of Fig. 12.6.4. Because the
traveling wave components of the standing wave have phases that
advance in the y direction with the same velocity, have the same
wavelength in the x direction and the same frequency, their
electric fields in the y-direction exactly cancel in the planes x
= 0 and x = a at each instant in time. With this recognition, we
may construct TE modes of the parallel plate conductor
structure of Fig. 12.7.4 by superposition of two countertraveling
waves, one of which was studied in Example 12.6.2.