The basic devices for modeling structures with the NEC code are short, straight segments for modeling wires and flat patches for modeling surfaces. An antenna and any other conducting objects in its vicinity that affect its performance must be modeled with strings of segments following the paths of wires and with patches covering surfaces. Proper choice of the segments and patches for a model is the most critical step to obtaining accurate results. The number of segments and patches should be the minimum required for accuracy, however, since the program running time increases rapidly a this number increases. Guidelines for choosing segments and patches are given below and should be followed carefully by anyone using the NEC code. Experience gained by using the code will also aid the user in developing models.
A wire segment is defined by the coordinates of its two end points and its radius. Modeling a wire structure with segments involves both geometrical and electrical factors. Geometrically, the segments should follow the paths of conductors as closely as possible, using a piece-wise linear fit on curves.
The main electrical consideration is segment length Delta relative to the wavelength Lambda. Generally, Delta should be less than about 0.l Lambda at the desired frequency. Somewhat longer segments may be acceptable on long wires with no abrupt changes while shorter segments, 0.05 Lambda or less, may be needed in modeling critical regions of an antenna. The size of the segments determines the resolution in solving for the current on the model since the current is computed at the center of each segment. Extremely short segments, less than about 10-3 Lambda, should also be avoided since the similarity of the constant and cosine components of the current expansion leads to numerical inaccuracy.
The wire radius, a, relative to Lambda is limited by the approximations used in the kernel of the electric field integral equation. Two approximation options are available in NEC: the thin-wire kernel and the extended thin-wire kernel. These are discussed in reference 1. In the thin-wire kernel, the current on the surface of a segment is reduced to a filament of current on the segment axis. In the extended thin-wire kernel, a current uniformly distributed around the segment surface is assumed. The field of the current is approximated by the first two terms in a series expansion of the exact field in powers of a2. The first term in the series, which is independent of a, is identical to the thin-wire kernel while the second term extends the accuracy for larger values of a. Higher order approximation are not used because they would require excessive computation time.
In either of these approximations, only currents in the axial direction on a segment are considered, and there is no allowance for variation of the current around the wire circumference. The acceptability of these approximations depends on both the value of a/Lambda and the tendency of the excitation to produce circumferential current or current variation. Unless 2Pi a/Lambda is much less than 1, the validity of these approximations should be considered.
The accuracy of the numerical solution for the dominant axial current is also dependent on Delta/a. Small values of Delta/a may result in extraneous oscillations in the computed current near free wire ends, voltage sources, or lumped loads. Use of the extended thin-wire kernel will extend the limit on Delta/a to smaller values than are permissible with the normal thin-wire kernel. Studies of the computed field on a segment due to its own current have shown that with the thin-wire kernel, Delta/a must be greater than about 8 for errors of less than 1%. With the extended thin-wire kernel, Delta/a may be as small as 2 for the same accuracy (ref. 3). In the current solution with either of these kernels, the error tends to be less than for a single field evaluation. Reasonable current solutions have been obtained with the thin-wire kernel for Delta/a down to about 2 and with the extended thin-wire kernel for Delta/a down to 0.5. When a model includes segments with Delta/a less than about 2, the extended thin-wire kernel option should be used by inclusion of an EK card in the data deck.
When the extended thin-wire kernel option is selected, it is used at free wire ends and between parallel, connected segments. The normal thin-wire kernel is always used at bends in wires, however. Hence, segments with small Delta/a should be avoided at bends. Use of a small Delta/a at a bend, which results in the center of one segment falling within the radius of the other segment, generally leads to severe error.
The current expansion used in NEC enforces conditions on the current and charge density along wires, at unctions, and at wire ends. For these conditions to be applied properly, segments that are electrically connected must have coincident end points. If segments intersect other than at their ends, the NEC code will not allow current to flow from one segment to the other. Segments will be treated as connected if the separation of their ends is less than about 10-3 times the length of the shortest segment. When possible, however, identical coordinates should be used for connected segment ends.
The angle of the intersection of wire segments in NEC is not restricted in any manner. In fact, the acute angle may be so small as to place the observation point on one wire segment within the volume of another wire segment. Numerical studies have shown that such overlapping leads to meaningless results; thus, as a minimum, one must ensure that the angle is large enough to prevent overlaps. Even with such care, the details of the current distribution near the intersection may not be reliable even though the results for the current may be accurate at distances from this region.
NEC includes a patch option for modeling surfaces using the magnetic-field integral equation. This formulation is restricted to closed surfaces with nonvanishing enclosed volume. For example, it is not theoretically applicable to a conducting plate of zero thickness and, actually, the numerical algorithm is not practical for thin bodies (such as solar panels). The latter difficulty is due to the possibility of poor conditioning of the matrix equation.
Wire-grid modeling of conducting surfaces has been used with varying success. The earliest applications to the computation of radar cross sections and radiation patterns provided reasonably accurate results. Even computations for the input impedance of antennas driven against grid models of surfaces have oftentimes exhibited good agreement with experiments. However, broad and generalized guidelines for near-field quantities have not been developed, and the use of wire-grid modeling for near-field parameters should be approached with caution. A single wire grid, however, may represent both surfaces of a thin conducting plate. The current on the grid will be the sum of the currents that would flow on opposite sites of the plate. While information on the currents on the individual surfaces is lost the grid will yield the correct radiated fields.
A conducting surface is modeled by means of multiple, small flat surface patches corresponding to the segments used to model wires. The patches are chosen to cover completely the surface to be modeled, conforming as closely as possible to curved surfaces. The parameters defining a surface patch are the Cartesian coordinates of the patch center, the components of the outward-directed, unit normal vector and the patch area. These are illustrated in figure 1 where r0=x0 ^x + y0 ^y + z0 ^z is the position of the segment center; ^n=nx ^x + ny ^y + nz ^z is the unit normal vector and A is the patch area.
Figure 1. Patch Position and Orientation
Although the shape (square, rectangular, etc.) may be used to define a patch on input it does not affect the solution since there is no integration over the patch unless a wire is connected to the patch center. The program computes the surface current on each patch along the orthogonal unit vectors ^t1 and ^t2, which are tangent to the surface. The vector ^t1 is parallel to a side of the triangular, rectangular, or quadrilateral patch. For a patch of arbitrary shape, it is chosen by the following rules:
For a horizontal patch,
^t1=^x .
For a non horizontal patch,
^t1=( ^z X ^n ) / I ^z X ^n I ,
^t2 is then chosen as ^t2=^n X ^t1. When a structure having plane symmetry is formed by reflection in a coordinate plane using a GX input card, the vectors ^t1 ^t2 and ^n are also reflected so that :he new patches will have ^t2=-^n X ^t1.
When a wire is connected to a surface, the wire must end at the center of a patch with identical coordinates used for the wire end and the patch center. The program then divides the patch into four equal patches about the wire end as shown in figure 2, where a wire has been connected to the second of three previously identical patches. The connection patch is divided along lines defined by the vectors ^t1 and t2 for that patch, with a square patch assumed. The four new patches are ordinary patches like those input by the user, except when the interactions between the patches and the lowest segment on the connected wire are computed. In this case an interpolation function is applied to the four patches to represent the current from the wire onto the surface, and the function is numerically integrated over the patches. Thus, the shape of the patch is significant in this case. The user should try to choose patches so that those with wires connected are approximately square with sides parallel to ^t1 and t2. The connected wire is not required to be normal to the patch but cannot lie in the plane of the patch. Only a single wire may connect to a given patch and a segment may have a patch connection on only one of its ends. Also, wire may never connect to a patch formed by subdividing another patch for a previous connection.
Figure 2. Connection of a Wire to a Surface Patch.
As with wire modeling, patch size measured in wavelengths is very important for accuracy of the results. A minimum of about 25 patches should be used per square wavelength of surface area, with the maximum size for an individual patch about 0.04 square wavelengths. Large patches may be used on large smooth surfaces while smaller patches are needed in areas of small radius of curvature, both for geometrical modeling accuracy and for accuracy of the integral equation solution. In the case of an edge, a precise local representation cannot be included; however, smaller patches in the vicinity of the edge can lead to more accurate results since the current magnitude may vary rapidly in this region. Since connection of a wire to a patch causes the patch to be divided into four smaller patches, a larger patch may be input in anticipation of the subdivision.
While patch shape is not input to the program, very long narrow patches should be avoided when subdividing the surface. This is illustrated by the two methods of modeling a sphere shown in figure 3. The first uses uniform division in azimuth and equal cuts along the vertical axis. This results in all patches having equal areas but with long narrow patches near the poles. In the second method, the number of divisions in azimuth is increased toward the equator so that the patch length and width are kept more nearly equal. The areas are again kept approximately equal. The results of the two segmentations are shown in figure 4 for scattering by a sphere of ka (2Pi . radius/wavelength) equal to 5.3. The uniform segmentation used 14 increments in azimuth and 14 equal bands along the vertical axis. The variable segmentation used 13 equal increments in arc length along the vertical axis, with each band from top to bottom divided into the following number of patches in azimuth: 4, 8, 12, 16, 20, 24, 24, 24, 20, 16, 12, 8, 4. Much better agreement with experiment is obtained with the variable segmentation.
Figure 3. Patch Models for a Sphere.
Figure 4. Bistatic RCS of a Sphere with ka=5.3.
In general, the use of surface patches is restricted to modeling voluminous bodies. The surface modeled must be closed since the patches only model the side of the surface from which their normals are directed outward. If a somewhat thin body, such as a box with one narrow dimension, is modeled with patches the narrow sites (edges) must be modeled a well as the broad surfaces. Furthermore, the parallel surface on opposite sides cannot be too close together or severe numerical error will occur.
When modeling complex structures with features not previously encountered, accuracy may be checked by comparison with reliable experimental data if available. Alternatively, it may be possible to develop an idealized model for which the correct results can be estimated while retaining the critical features of the desired model. The optimum model for a class of structures can be estimated by varying the segment and patch density and observing the effect on the results. Some dependence of results on segmentation will always be found. A large dependence, however, would indicate that the solution has not converged and more segments or patches should be used. A model will generally be useable over a band of frequencies. For frequencies beyond the upper limit of a particular model, a new set of geometry cards must be input with a finer segmentation.
Several options are available in NEC for modeling an antenna over a ground plane. For a perfectly conducting ground, the code generates an image of the structure reflected in the ground surface. The image is exactly equivalent to a perfectly conducting ground and results in solution accuracy comparable to hat for a free-space model. Structures may be close to the ground or contacting it in this case. However, for a horizontal wire with radius a, and height h, to the wire axis, [h2 + a2]1/2 should be greater than about 10-6 wavelengths. Furthermore, the height should be at least several times the radius for the thin-wire approximation to be valid. This method doubles the time to fill the interaction matrix. A finitely conducting ground may be modeled by an image modified by the Fresnel plane-wave reflection coefficients. This method is fast but of limited accuracy and should not be used for structures close to the ground. The reflection coefficient approximation for the near fields can yield reasonable accuracy if the structure is a least several tenths of a wavelength above the ground. It should not be used for structures having a large horizontal extent over the ground such as some traveling-wave antennas. An alternate method (Sommerfeld/Norton), available for wires only, uses the exact solution for the fields in the presence of ground and is accurate close to the ground. For a horizontal wire the height restriction is the same as for a perfect ground. When this method is used NEC requires an input file (TAPE21) containing field values for the specific ground parameters and frequency. This interpolation table must be generated by running a separate program, SOMNEC, prior to the NEC run. The present NEC code uses the Sommerfeld/Norton method only for wire-to-wire interactions. If Sommerfeld/ Norton is requested for a structure that includes surfaces, the reflection coefficient approximation will be used for surface-to-surface and surface-to-wire interactions. Computation of wire-to-wire interactions by the Sommerfeld/ Norton method take about four times longer than for free space. In addition, computation of the interpolation table requires about 15 s on a CDC 7600 computer. However, the file of interpolation tables may be saved and reused for problems having the same ground parameters and frequency. The Sommerfeld/ Norton method is not available in the earlier code NEC-l.
A wire ground screen may be modeled with the Sommerfeld/Norton method if it is raised slightly above the ground surface. A ground stake cannot be modeled in NEC since there is presently no provision to compute interactions across the interface. Wires may end on a ground plane with a condition that the charge density (i.e., derivative of current) be zero at the base of the wire, but this is accurate only for a perfectly conducting ground. A wire may end on a finitely conducting ground with the charge set to zero at the connection, but this will not accurately model a ground stake. If a wire is driven against a finitely conducting ground in this way, the input impedance will typically be dependent on length of the source segment.
NEC also includes options for a radial-wire ground-screen approximation and two-medium ground approximation (cliff) based on modified reflection coefficients. These methods are implemented only for wires and not for patches, however. For the radial-wire ground-screen approximation, an approximate surface impedance - based on the wire density and the ground parameters - is computed at specular reflection points. Since the formula for surface impedance yields zero at the center of the screen, the current on a vertical monopole will be the same as over a perfect ground. The ground screen approximation is used in computing both near-field interactions and the radiated field. It should be noted that defraction from the edge of the screen is not included. When limited accuracy can be accepted, the ground screen approximation provides a large time saving over explicit modeling with the Sommerfeld/Norton method since the ground screen does not increase the number of unknowns in the matrix equation.
The two-medium ground approximation permits the user to define a linear or circular cliff with different ground parameters and ground height on opposite sides. This aprroximation is not used for the near-field interactions affecting the currents but is used in computing the radiated field. The reflection coefficient is based on the ground parameters and height at the specular-reflection point for each ray. This option may also be used to compute the current over a perfect ground and then compute radiated fields for a finitely conducting ground.
