1. Introduction

SuperNEC (SNEC) is an object-oriented version of the FORTRAN program NEC-2. This technical manual is therefore very similar to the NEC-2 manual. In fact, many sections have been copied verbatim from the NEC-2 manual. The reason this document has been produced (as opposed to referring the reader to any readily available NEC-2 manual) is because new theory will be added to SNEC and some features that exist in NEC-2 will not be implemented in SNEC. The easiest way of keeping an up to date reference of the theory implemented in SNEC is to start with the an electronic version of the NEC-2 manual and modify the document as the SNEC code evolves. Many extensions have been made to SNEC in prototype form. These features include hybridisation with UTD, fast iterative solvers, MBPE amongst other features. When these extensions are formally incorporated into the SNEC program, then the theory behind the extension will be added to this manual.

2. Computations specific to cylinders

This sections details some of the computations that apply to cylinders.

2.1. Radius of curvature in the principal planes

The principal planes of the cylinder were chosen such that is tangent to the cylinder in the XY plane and is in the direction of the z-axis of the cylinder. The radius of curvature in the XY plane for an elliptical cylinder is given by

(1)

2.2. Radius of curvature in plane of incidence

The plane of incidence is defined as the plane containing .In general the radius of curvature in a plane at an angle from the principal surface axes is given by Euler’s theorem as :

(2)

For the cylinder, is as given in section 2.1, whilst . The angle is computed as :

(3)

2.3. Finding the length of a geodesic path

This details how to find the distance along the surface of the cylinder from elliptical angle to in a given direction. The arc length is given by :

(4)

where

is the elevation angle of the path traversed between and .

3. Computations specific to diffraction

3.1. The diffraction transition function

The transition function used in both the UTD edge and curved surface diffraction is defined by [ 1]:

(5)

For large argument

(6)

For small argument

(7)

For argument , the interpolation scheme used for numerical computation is :

(8)

where the values for the iterative scheme are obtained from :


	Real	Imaginary	Real	Imaginary
0.3	0.0	0.0	0.5729	0.2677
0.5	0.5195	0.0025	0.6768	0.2682
0.7	0.3355	-0.0665	0.7439	0.2549
1.0	0.2187	-0.0757	-0.8095	0.2322
1.5	0.1270	-0.680	0.8730	0.1982
2.3	0.0638	-0.0506	0.9240	0.1577
4.0	0.0246	-0.0296	0.9658	0.1073
5.5	0.0093	-0.0163	0.9797	0.0828

4. UTD diffraction off a wedge

4.1. The UTD equation

The field diffracted from a wedge is given by :

(9)

where

is the field incident on the edge at point
is the dyadic diffraction coefficient.
A(s) is the spreading factor.

Written in matrix form in the edge coordinate system :

(10)

where

is the diffracted field in the b direction (parallel to the plane of diffraction)
is the diffracted field in the f direction (perpendicular to the plane of diffraction)
is the soft / parallel diffraction coefficient
is the hard / perpendicular diffraction coefficient
is the incident field in the b direction (parallel to the plane of incidence)
is the incident field in the f direction (perpendicular to the plane of incidence)

4.1.1. The UTD edge diffraction coefficients

The soft and hard diffraction coefficients are given by :

(11)

for all angles other than grazing incidence where

(12)

Where :

are the distance parameters for the incident shadow boundary and the reflection shadow boundary for the o-face and the n-face respectively.
is the diffraction angle with respect to the o-face
is the incident angle with respect to the o-face
is the angle between the incident ray and the edge
is the wedge angle number.
are the soft and hard reflection coefficients of the surfaces of the wedge at the edge.
are the components of the diffraction coefficients.

The diffraction components for 3-D diffraction are given by :

(13)

(14)

(15)

(16)

where

is the transition function.
is the wave number
is defined in section 4.1.2

4.1.2. The a function

The function used in the diffraction coefficient calculation is defined as :

(17)

where

(18)

and are the integers that most nearly satisfy the equations :

(19)

The values of are computed as :

(20)

NOTE : (,) are associated with the n-face and (,) are associated with the o-face.

4.1.3. The 3D distance parameters

The distance parameters are defined here [ 3]:

(21)

where

is the radius of curvature of the incident wavefront at in the plane of incidence i.e., the plane containing .
is the radius of curvature of the incident wavefront at transverse to the in the plane of incidence.
is the radius of curvature of the incident wavefront at in the edge-fixed plane of incidence i.e., the plane containing .

(22)

where

are the principal radii of curvature of the reflected wavefront at from the o-face and n-face respectively.
are the radii of curvature of the reflected wavefront at in the plane containing the reflected ray and the edge i.e., the plane containing .

For an edge formed by the junction of two flat surfaces, the radii of curvature of the reflected wavefront are equal to the radii of curvature of the incident wavefronts. Thus

(23)

4.1.4. The spreading factor

The spreading factor for the edge diffracted field is given by :

(24)

where

is the edge caustic distance.

For flat plates the edge caustic is equal to , i.e., the radius of curvature of the incident wavefront at in the plane containing .

4.1.5. Computing the diffracted wavefront radii

The principal directions of the diffracted wavefront are :

1. The plane containing the edge and the reflected ray ()

2. The plane transverse to plane 1 ().

The two unit vectors are computed as :

(25)

where

is the unit normal of the o-face
is the surface tangent of the o-face
is the angle of the reflected plane measured from the o-face.

The radii of curvature of the reflected wavefront at the point of observation are :

(26)

5. UTD slope diffraction off a wedge

This section details the aspects of slope diffraction that differ from edge diffraction.

5.1. The UTD equation

The slope diffracted field from a wedge is given by :

(27)

where

is the field incident on the edge at point
is the normal to the edge
A(s) is the spreading factor.

Written in matrix form in the edge coordinate system :

(28)

where

is the slope diffracted field in the b direction (parallel to the plane of diffraction)
is the slope diffracted field in the f direction (perpendicular to the plane of diffraction)
is the soft / parallel slope diffraction coefficient
is the hard / perpendicular slope diffraction coefficient
is the derivative of the incident field in the b direction (parallel to the plane of incidence) with respect to .
is the derivative of the incident field in the f direction (parallel to the plane of incidence) with respect to .

5.1.1. The UTD slope diffraction coefficients

The soft and hard diffraction coefficients are given by :

(29)

for all angles other than grazing incidence where

(30)

Where :

The slope diffraction components for 3-D diffraction are given by :

(31)

(32)

(33)

(34)

where

is given in terms of the transition function .
is the wave number
is defined in section 4.1.2

5.1.2. Common functions

See the appropriate subsections of section 4 for

· The function

· Computing the 3D distance parameters.

· Computing the 3D spreading factor.

· Computing the diffracted wavefront radii.

6. UTD corner diffraction

The corner diffracted wave for spherically incident plane waves is described in this section.

6.1. The UTD equation

The UTD equation for corner diffracted fields is given by :

(35)

where

are the b and f components of the corner diffracted field.
are the b and f components of the field incident at .
The angles and are illustrated in Figure 1.
is the distance parameter associated with the corner.
is the transition function.
are the corner diffraction coefficients.
is the spreading factor.

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Figure 1: The Geometry for corner diffracted fields [ 1].

6.1.1. The UTD corner diffraction coefficients

The soft and hard diffraction coefficients are given by :

(36)

or for grazing incidence

(37)

where

are the soft and hard reflection coefficients of the surfaces of the wedge.
are the components of the diffraction coefficients as defined in equations.(13) to (16).

and

(38)

6.1.2. The spreading factor

The spreading factor for the corner diffracted field is given by :

(39)

6.1.3. The corner distance parameter for spherical incidence

The distance parameter is given by :

(40)

where

is the distance from the source to the corner
is the distance from the corner to the observation point

7. UTD reflection off a cylinder

The SNEC MOM/ITD hybrid program uses the UTD theory for computing the electric field reflected off the curved surface of a cylinder. The implemented equations are outlined in this section.

7.1. The UTD equation

The field reflected off a cylinder from an electric field source is given by [ 1] as

(41)

where

is the E-field incident on the cylinder at
are the soft and hard UTD reflection coefficients
is the 3-D spreading factor.

7.1.1. The UTD reflection coefficients

The UTD soft and hard reflection coefficients [ 2] are given by :

(42)

where

and are the soft and hard Fock scattering functions.
is the diffraction transition function.
is the transition function argument.

and

where is the unit normal at the point of incidence and is the reflected ray.
is the wave number
is the curvature parameter with the radius of curvature of the surface in the plane of incidence .
is the 3-D distance parameter.

The reflection coefficients are computed in the following form :

(43)

7.1.2. The 3-D distance parameters

The distance parameter [ 2] for the computation of the transition function argument is given by :

(44)

where

is the radius of curvature of the incident wave in the plane of incidence.
is the radius of curvature of the incident wave transverse to the plane of incidence.
is the radius of curvature of the reflected wave in the plane of reflection.
is the radius of curvature of the reflection wave transverse to the plane of reflection.
the distance from point of incidence to the observation point.

7.1.3. The 3D spreading factor

The spreading factor for the reflected wave is given by :

(45)

where

is the radius of curvature of the reflected wave in the plane of reflection.
is the radius of curvature of the reflection wave transverse to the plane of reflection.
the distance from point of incidence to the observation point.

7.1.4. Computing the reflected wavefront radii

The principal radii of curvature of the reflected wavefront , and the principal directions (axes) of the wavefront are given in this section. The plane of incidence may be different from the principal planes of the reflecting surface, so that the principal directions of the incident wavefront are quite distinct from those of the reflecting surface. , are given as [ 3] :

(46)

and

(47)

where

(48)

The matrix is defined in terms of the principal planes of the incident wavefront and the principal directions of the curved surface .The principal directions of the cylinder have been chosen such that : is tangent to the cylinder in the XY plane and is in the direction of the z-axis of the cylinder. Note : All computations are done in the cylinder coordinate system.

(49)

The determinant of is represented as .

The principal directions of the reflected wavefront are given in terms of the vectors , which are unit vectors perpendicular to the reflected ray. The are determined by reflecting the unit vectors in the plane tangent to the surface at i.e.,

(50)

The principal directions of the reflected wavefront are then given by :

(51)

and

(52)

Where

(53)

(54)

(55)

NOTE : The principal directions of the wavefront are distinct from the principal directions associated with the reflected wave directions i.e., and

8. UTD diffraction off a cylinder

The SNEC MOM/UTD hybrid program uses the UTD theory for computing the electric field diffracted from the curved surface of a cylinder. The implemented equations are outlined in this section.

8.1. The UTD equation

The field diffracted off a cylinder from an electric field source is given by [ 1] as

(56)

where

is the E-field incident on the cylinder at .
are the soft and hard UTD diffraction coefficients.
The square root term is the conservation of energy term as wave creeps around cylinder.
is the 3-D spreading factor.

8.1.1. The UTD diffraction coefficients

The UTD soft and hard diffraction coefficients [ 2] are given by :

(57)

where

and are the soft and hard Fock scattering functions.
is the diffraction transition function.
is the transition function argument.
where

and

The reflection coefficients are computed in the following form :

(58)

Where the argument for the Fock scattering functions is computed as :

where

Thus

is the radius of curvature is the direction of incidence.

8.1.2. The conservation of energy term

The conservation of the energy flux in the surface-ray strip from to is given by :

(59)

This is computed (for spherically incident waves) as :

(60)

where is the creep distance along the surface geodesic between and .

8.1.3. The 3-D distance parameters

The distance parameter [ 2] for the computation of the transition function argument is given by :

(61)

where

8.1.4. The 3-D spreading factor

The 3D spreading factor for diffraction is given by :

(62)

where

is the radius of curvature of the reflection wave transverse to the plane of reflection.
the distance from point of incidence to the observation point.

8.1.5. Computing the diffracted wavefront radii

The two principal radii of curvature of the diffracted wave front at the point of observation is computed as follows :

(63)

where is the distance the wave crept around the cylinder and

(64)

i.e., is the radius of curvature of the incident wavefront in the plane tangent to the surface. Plane containing . is the vector tangent to the surface in the plane of incidence, is the vector perpendicular to on the surface and is the normal to the surface. All vectors are computed at .

(65)

The principal directions and are computed as :

(66)

and

(67)

8.1.6. Evaluating the transition function

See section 3.1.

8.2. The UTD equation for grazing incidence

The field diffracted off a cylinder at grazing incidence from an electric field source is given by the sum of half the incident field and the diffracted field modified for angles of grazing incidence at the observation point.

(68)

where

is the E-field incident on the cylinder at , the point of grazing incidence
are the soft and hard UTD diffraction coefficients modified for the point of grazing
Sign is positive if and negative otherwise
are the radii of curvature of the direct field at in the plane of grazing for 1 (tangential to surface) and in the plane transverse to grazing for 2 (direction of normal to surface). It is noted that
are the radii of curvature of the diffracted field in the same planes as above. In this case if the point of reference is taken to be and .
The first square root term is the 3-D spreading factor for the incident field
The second square root term is the spreading factor for the diffracted field

The field is computed as :

(69)

where the term in square brackets is grouped to form a modified diffraction coefficient.

8.2.1. The UTD diffraction coefficients for grazing

The UTD soft and hard diffraction coefficients for grazing are given by :

(70)

where

and are the soft and hard Fock scattering functions.

is the curvature parameter with the radius of curvature of the surface in the plane of incidence .

8.2.2. Computing the grazed wavefront radii

The grazing wavefront radii and planes are the same as the diffracted values, except the creeping distance is zero. See section 8.1.4.

9. References

[ 1]	McNamara D.A., Pistorius C.W.I, Malherbe J.A.G “Introduction to the uniform geometrical theory of diffraction”, Artech House, 1990.
[ 2]	Pathak P.H, Burnside W.D, Marhefka R.J, “A Uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface”, OEEE Trans. Antennas Propagat., Vol. AP-28, pp. 631-642, Sept. 1980.
[ 3]	Kouyoumjian R.G, Pathak P.H, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface”, Proc IEEE, Vol. 62, pp. 1448-1461, Nov. 1974