Diffraction at Rounded Edges
Gokhan Apaydin
Levent Sevgi
Department of Electronics Engineering
Uskudar University
Istanbul, Turkey
gokhan.apaydin@uskudar.edu.tr
Department of Electrical and Electronics Engineering
Okan University
Akfirat, Istanbul, Turkey
levent.sevgi@okan.edu.tr
Abstract—This study aims to visualize the diffraction effects
of objects with rounded edges using method of moments (MoM).
The comparison of scattering effects of wedges and trilateral
cylinders are shown with and without rounded edges using fringe
integral equations with physical theory of diffraction (PTD).
Keywords—diffraction; method of moments; MoM; physical
theory of diffraction; PTD; wedge.
I. INTRODUCTION
This study shows an extension of PTD for objects with
rounded edges and compares the wedge and trilateral cylinders
with and without rounded edges [1,2]. It is important to
understand the behavior of electromagnetic scattering in
complex environment. The word scattering includes reflection,
refraction, and diffraction, etc. produced from the interaction of
electromagnetic waves with objects. One reason for diffraction
is a sharp boundary discontinuity as an edge and/or a tip [3-11].
Rounded edges instead of sharp edges have also scattering
effects by considering physical theory of diffraction (PTD)
when the wavelength is small compared with the interacted
object size [1,2]. According to PTD, source-induced surface
currents can be divided into uniform (physical optics-PO)
currents caused by planar boundaries and non-uniform (fringePTD) currents caused by edges and/or tips [5]. The PTD
currents generate fringe waves and have been modeled using
method of moments (MoM) [8-11].
combination of three rounded wedges. L01, L02 , L03 are parts
of the circular cylinders, which are smoothly conjugated, with
the faces of the tangential wedges [2].
For the computation of fields around the wedge and
trilateral cylinder, the receivers are located on the observation
circle around the origin. The total field is the addition of the
incident and scattered fields. The subtraction of the incident
and reflected fields from the total field yields the diffracted
field. The fringe field is the part of the diffracted field
generated by the source-induced fringe currents. Here, the
fringe waves for the rounded structures are compared with
those for the sharp structures without rounded edges (when
a=0).
Fig. 1. The 2D PEC wedge with a rounded edge.
In this paper, the effects of rounded edges for wedges and
trilateral cylinders are shown using PTD and MoM [1,2]. Note
that, understanding and reducing diffracted fields is also critical
in EMC engineering.
II. MODELING WITH ROUNDED EDGE
A wedge with a rounded edge in Fig. 1 is constructed as a
combination of a circular cylinder with radius a. The surface
L0 is smoothly conjugated with two half-planes L1 and L2.
These are the faces of the tangential wedge. The origin
coincides with the apex of the sharp wedge. Double side
illumination is considered by a plane incident wave
propagating along the x-axis (see [1] for details).
A trilateral cylinder with rounded edges is constructed in
Fig. 2. The trilateral cylinder can be considered as a
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Fig. 2. The 2D PEC trilateral cylinder with rounded edges.
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III. EXAMPLES AND COMPARISONS
First, the fringe fields of rounded wedge are directly
computed using (11) of [1]. The PO contributions are added to
obtain the total scattered fields. Finally, the addition of the
incident field to the scattered field yields the total field.
The frequency of all simulations is 30 MHz (i.e., λ=10 m).
The observation circle with r=2 is chosen. The lengths of L1
and L2 parts of the wedge are taken 50-long which is tested
to be enough and the number of segments in one wavelength is
chosen as 20 for MoM calculations. The number of segments
of the rounded part (L0 part) is at least 20 to satisfy the
rounded curvature [1].
Figures 3 and 4 shows the total and fringe fields of soft
wedges with different wedge half angles (β=5°, 15°, 30°, 45°).
The three field curves belong to a=0 (sharp wedge), a=λ/10,
and a= λ/5 cases. The total fields of the 90 wedge for all three
cases are almost identical. The differences in the total fields
around the wedge become significant, as the wedge interior
angle gets smaller. This is because the locations of the
receivers shift significantly for narrow wedges. The same
observation also holds for the fringe field variations in Fig. 4
(See [1] for details).
For the sharp wedge diffracted field occurs only
backwards (i.e., towards to the angle of incoming plane wave).
On the other hand, there is a strong backward reflection for the
rounded wedge.
Fig. 4. Fringe fields around soft wedges with different wedge angles
(β=5°,15°,30°,45°) and radius (a=0 for sharp wedge, a=λ/10,λ/5 for
rounded wedges) [1].
Fig. 5. Fringe fields generated by PTD currents induced on a soft rounded
trilateral cylinder (a=λ/5, D=3λ, dl=λ/200) [2].
Fig. 3. Total fields around soft wedges with different wedge angles
(β=5°,15°,30°,45°) and radius (a=0 for sharp wedge, a=λ/10,λ/5 for rounded
wedges) [1].
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Note that, as the radius increases, the distance between the
receiver on x-axis and backward specular reflection point
increases and the amplitude of the scattered field along this
direction decreases in Fig. 4.
Second, the bistatic scattering at a soft rounded trilateral
cylinder is shown with PTD and direct MoM solutions. Figure
5 demonstrates the field generated by fringe currents. It is seen
that the main maxima relate to shadow radiation and specular
reflection. According to Fig. 5, the multiple diffracted fringe
waves actually do not reveal themselves. Therefore, the firstorder PTD is totally applicable for objects about three
wavelengths long, and certainly for longer objects.
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IV. CONCLUSIONS
This study shows an extension of PTD for objects with
rounded edges and compares the wedge and trilateral cylinders
with and without rounded edges. Comparison of the results
agrees well and the effects of round edges are clearly observed.
REFERENCES
[1]
Fig. 6. Comparison of scattering from a soft trilateral cylinder with sharp and
rounded edges (a=λ/5, D=3λ, dl=λ/200) [2].
Figure 6 compares the scattering from a soft trilateral
cylinder with sharp and rounded edges. The data for the sharp
cylinder are based on the solution where the curvature radius
is set to zero. Note that, here the MoM and PTD lines coincide
within the graphical accuracy. For the sharp cylinder, they also
coincide with the PTD curve in Fig. 5.11 of [5]. This figure
also admits clear physical interpretation. Shadow radiation and
specular reflection for both objects are nearly the same due to
their close dimensions. However, in the vicinity of
backscattering (120º≤φ≤180º), the rounded object creates
higher scattering due to the specular reflection from the front
rounded edge in Fig. 2.
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G. Apaydin, L. Sevgi, and P. Ya. Ufimtsev, “Diffraction at rounded
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