This chapter looks at various methods of determining the interaction of an antenna and a structure and discusses diffraction theory in detail.
The ways of determining antenna performance when installed on a structure include the following
Each method has particular advantages and disadvantages which are discussed below.
The Integral Equation method [ 2-1 ] is often referred to loosely as `Method of Moments' and models a structure in terms of a set of conducting wires or flat patches which can be various shapes (triangular, quadrilateral) depending on the method of software implementation. It is usually a surface model, that is, the structure only has to have its surface modelled and not the whole volume. In some cases where the software implementation allows the presence of a dielectric, a volume model may be necessary.
The method determines the current on every segment (wire or patch) in the model and thus the dimensions of each segment must be less than 0.25 wavelength; in practice 0.1 lambda is taken as a maximum dimension. Complex portions of the structure, particularly near the excitation point, may be modelled using segments down to 0.02 lambda in dimensions. The method is extremely good at determining far-field and nearfield radiation patterns [ 2-2 ] and also the impedance. The correspondence between prediction and measurement is good. However, there are serious limitations caused by the amount of computer memory storage required and the long runtimes.
The procedure is as follows
Clearly if the maximum segment length is 0.1 lambda , the number of entries in the matrix grows rapidly with the dimensions of the structure and matrix inversion becomes difficult and time-consuming as the number of segments increases. For example, a wire-grid model of a P-3/CP-140 aircraft had 327 segments for 2-30 MHz [ 2-3 ] . This corresponds to a segment length of 0.2 lambda . A model for a BAC-111 aircraft at HF had 7736 segments and took 30,000 seconds to run on a CRAY-2 computer [ 2-4 ] .
A typical pc (PENTIUM 200 MHz) with 128 MBytes of memory can handle up to 4000 segments with a runtime of 1.5 hours. A typical wiregrid model which uses 128 MBytes is shown in here .
Figure 1. Wire Grid Model of a Cylinder with a radius of 1 lambda and a length of 5.7 lambda . The antenna is a quarterwave monopole
The problem with such codes is that the radiation patterns are more accurate than the impedance due to the difficulty of modelling a `real' excitation. The choice of detail in the model around the excitation point is critical to the accuracy and of course there are severe limitations in the size of problem which can be handled.
This is a volume method [ 2-5 ] and the volume under consideration has to be subdivided into cells whose shape is determined by the software implementation used. In general, the cells are cuboid. The advantage of FDTD is that each cell can be described as having any RF properties so that imperfect conductors and lossy dielectrics can be modelled. There is no matrix inversion and the method works by advancing time steps. However runtimes can be long and there is still a memory problem. For example, a volume of 120 by 120 by 120 cells can be handled in a memory of 128 MBytes but the runtime may be several hours. Again farfield, nearfield and impedance can be computed. The accuracy depends on the boundary conditions set to the volume investigated.
The geometrical Theory of Diffraction (GTD) is based on Geometric Optics (GO) and Diffraction Theory [ 2-6 ] . It assumes that all waves are `well-formed' and are locally plane waves. This enables ray tracing to be used. The implications of these assumptions are
An immediate problem is raised by Item 2 above when an antenna is mounted on a surface of the structure. However this has been solved by the introduction [ 2-8 ] of UTD (=Unified Theory of Diffraction).
In Integral Equation and FDTD methods, the geometry is set up and the same method is applied to the entire structure. In diffraction theory, this is not possible because there are different diffraction coefficients for different types of geometric objects. These diffraction coefficients are all derived from theory
The advantages of GTD/UTD are the following
The rest of this chapter discusses the implications of GTD/UTD and should assist the user in determining the best method or structure model to use.
The assumption is made here that the structure is a perfect conductor. Diffraction coefficients for imperfect conductors and dielectrics are poorly determined at present. Available diffraction coefficients [ 2-7 ] , [ 2-8 ] include
Reflection at a surface follows the standard GO laws, see Figure 2 and Figure 3. One point to note is that the cone of rays from a flat reflector is not changed in semi-angle and therefore the power density remains unchanged. For a curved surface, the power density does change on reflection and this change depends on the geometry, distances et cetera. Usually, the power is spread over a larger angle and is therefore lower but, of course on concave surface, focusing could occur.
Figure 2. Reflection in a flat conducting sheet showing normal, n, at the point of reflection and the incident and reflected polarisation vectors, E i and E r
Figure 3. Reflection in a curved conducting sheet showing normal, n, at the point of reflection and the incident and reflected polarisation vectors, E i and E r . The directions of the principal tangents (principal radii of curvature) at the reflection point are also shown
A ray incident on an edge is diffracted into a cone of rays (see Figure 4). Since the angle of diffracted energy is increased, the diffracted rays are lower in signal level than the incident ray. The amount of power diffracted depends on
Figure 4. Diffraction at a straight edge, showing the cone of output rays (semi-angle, b) produced by an input ray from S to Q. One output ray goes in the direction Q to P
Figure 5 to Figure 8 show the computed radiation patterns of a quarterwave monopole on the upper side of a square plate with sides whose length vary from 2.5 to 200 wavelengths. The patterns show edge diffraction which decreases in amplitude as the plate becomes larger. When the plate becomes infinite, there would be no radiation at all below the plate and the upper pattern would be quite smooth. As the size decreases, there is more radiation below the plate and there is also fringing. That below the plate is due to diffracted rays from opposite edges going in and out of phase. That above the plate is due to the two edge diffracted rays adding and subtracting from the direct radiation. Because the distances are smaller for smaller plates, the angular spacing between the nulls increases while the depth of the fringes increases.
 Figure 5 |
 Figure 6 |
 Figure 7 |
 Figure 8 |
B) Radiation Pattern
Figure 9. Structure of radiation pattern of a dipole 3.47 wavelengths above a square plate of side 4.75 wavelengths
Figure 9 shows the radiation of a horizontal halfwave dipole at a distance of 3.47 lambda above a conducting square ground plane 4.75 lambda on a side. The dipole is aligned parallel to the y-axis and the radiation pattern is in the x-z plane and is a cut at right angles to the dipole length. Measured results are from Pathak [ 2-8 ] .
Again the interference fringes when diffracted rays, reflected rays and the direct ray are present together are very clear. Figure 10 shows the radiation pattern when the geometry is unchanged but the frequency is increased by a factor of 4. The fringes are closer together and smaller from peak to trough, the cut-off at the shadow boundary is sharper.
Figure 10. Radiation pattern of a dipole 13.88 wavelengths above a square plate of side 19.0 wavelengths
A ray incident on a corner diffracts into a sphere; thus the contribution from a corner is even lower than from an edge. Corner diffraction is not currently included in ALDAS Version 2.40 but it is intended to include this contribution shortly.
An area of diffraction which is different from GO is that of `creeping rays'. A ray incident tangential to a curved surface, sets off a creeping ray which moves down a geodesic of the surface (Figure 11). The postulate is that, when a ray strikes a curved surface, the ray is restrained to run along the local geodesic (by Fermat's principle). The ray/geodesic sheds power as it travels over the surface. The requirements are that
The ray and geodesic geometry is described in terms of the local triad, that is, the normal to the surface, n, the tangent along the geodesic at the same point, t, and the binormal, b given by t x n.
This situation has been generalised to the case when the source is on the curved surface. Geometrical Theory of Diffraction is not an accurate formulation for this type of ray and the Uniform Theory of Diffraction (UTD) has been developed for curved surfaces, in both the lit and shadow regions. It gives good radiation pattern match across shadow boundaries, deals with sources on the curved surface and can be used on smaller objects, down to a radius of 1 wavelength.
The form of the equations for the diffracted field is different for the source on or off the surface but the diffraction coefficients are dependent upon
Finding geodesics on complex curved surfaces is difficult and time-consuming particularly when the antenna is not on the curved surface [ 2-9 ] . However, many structures can be modelled using elements which are developable surfaces (cylinders, cones) or which are simple enough to be treated using standard search methods (ellipsoids).
Figure 11. Creeping rays on a curved surface showing the shedding of output energy all the way along the geodesic
Two examples of radiation patterns containing geodesics are shown in Figure 12 and Figure 13.
The first (Figure 12) shows the crosspolarised radiation pattern from a monopole mounted centrally at the top of an ellipsoid which has dimensions small in terms of wavelengths. The radiation pattern is one half of a complete cut in Azimuth (the X-Y plane), that is, in the horizontal plane and shows the high level of crosspolarisation which comes from diagonal cut radiation patterns on a curved surface.
The second example (Figure 13) shows a radiation pattern from a dipole mounted above an ellipsoid and parallel to the X-axis. The radiation pattern is at right angles to the dipole and the X-axis. The comparison radiation pattern in Figure 13 comes from computations by ASL using a Method of Moments code, NEC [ 2-2 ] .
Again the signal level in the shadow region is dependent upon the frequency and as the frequency increases, the signal level at angles between 150 and 210 degrees in Figure 13 will decrease while the fringes in the lit region will become closer together and the peak-to-peak ripple will decrease.
Figure 12. Monopole on the Z-axis of an ellipsoid with semiaxes 4, 2, 2, 4 l .
A) The radiation pattern in the horizontal X-Y plane. The measured results are from Pathak [ 2-8 ] .
B) Geometry
Figure 13. Horizontal dipole on the Z-axis of an ellipsoid with semiaxes 0.54, 0.3, 0.3, 0.54 l .
A) Radiation pattern in the Y-Z plane
B) Geometry
It is possible to have multiple interactions in one ray, that is, a reflection followed by an edge diffraction and then another reflection. In most cases, inclusion of two interactions is sufficient for good accuracy. Examples are illustrated in Figure 14. The general levels of interactions are typically as given here .
| Interaction | Level in dB wrt source |
| Reflection at flat plate | 0.0 |
| Reflection at curved surface | <-6.0 |
| Diffraction at straight edge | -10.0 |
| Diffraction at curved edge | <-16.0 |
| Diffraction at vertex | <-16.0 |
| Diffraction round cylinder | <-20.0 |
| Diffraction round cone | <-15.0 |
| Diffraction round curved surface | <-20.0 |
| Multiple interactions | <-20.0 |
Table 1.
Figure 14. Schematic aircraft illuminated by a source at S, showing the rays with multiple diffractions which must be summed in the output direction
The following interactions are included in ALDAS Version 2.40 [ 2-10 ]
Please note that the less important higher order interactions are being added and your current version of ALDAS may contain extra interactions.
The following canonical shapes are available
1) The `fuselage' structure chosen from
2) Up to 99 circular cylinders of any orientation
3) Up to 99 plates each having between 3 and 15 corners
4) Up to 99 obstacles which are flat plates with the number of corners fixed at 4.
The difficulty in computing an installed radiation pattern using diffraction is due to the geometry, both in the determination of the best mixture of basic elements to use and in the computation of the correct ray paths. This is particularly time consuming when structures curved in three dimensions are incorporated. For most structures, allowing one complex three dimensional element plus a large number of simpler elements is a compromise solution which allows several different types of structure to be modelled ( Table 2). The simpler elements are flat plates which have reflections and edge diffractions and a relatively easy geometry and cylinders and cones which require the computation of geodesics but are developable surfaces and therefore easy to compute.
The errors in the resulting radiation patterns stem from
| Type of structure | Structure elements | Multiple diffractions important? |
|---|---|---|
| Civil airliner | Fuselage as two half-ellipsoids of differing length, engines as cylinders, engine pylons as flat plates, tails, tailerons, wings as collections of abutting flat plates | No |
| Military aircraft | Fuselage as elliptical cylinder with nose and tail cone, engines as abutting flat plates, tail, tailerons, wings as abutting flat plates | Sometimes |
| Helicopter | Body as ellipsoid with tail boom of cylinders and flat plates, skids, blades as flat plates | Sometimes |
| Spacecraft | Collection of flat plates and cylinders for body plus cylinders, cones for other parts | Yes |
| Ships | Flat plates and cylinders | Yes |
| Ground vehicles | Flat plates. | Sometimes |
| Lattice masts | Flat plates and cylinders | Yes |
Table 2. Structure Geometries
Diffraction theory can be used to compute the radiation patterns of low gain antennas installed on conducting structures provided the structures are larger than several wavelengths. The larger the structure, the more accurate the results will be. It is sometimes important to include rays which suffer multiple interactions with the structure. Chapters 3 and 5 cover examples of such structures.
Where the structure immediately adjacent to the antenna is too small for diffraction theory but has an effect on the antenna radiation pattern, other steps should be taken. It is better to measure the antenna with such structure present or to model using, say, a Method of Moments or FDTD code and to incorporate the results into ALDAS. What you are doing is to treat the antenna plus its immediate neighbourhood as a single antenna unit. There are problems in deciding where the geometrical boundary between one method and the other should lie. Without care, the junction may introduce errors. The problems of integrating such data into ALDAS is covered in Chapter 6.
The choice of geometry is very important and is covered in detail in the tutorials of Chapters 4 and 5 along with the constraints which are set by diffraction theory on the relative sizes of parts of the structure.