Transcript
I
Sensor
and Simulation Note 147
Notes
-’
(Radiation
Laboratory
Report
March
MODIFIED T. B.A.
1972
BICONICAL
Senior
No. O1O748-1-T)
and G.A.
. Desjardins
The University of Michigan Radiation Laboratory Department of Electrical Engineering Ann Arbor, Michigan 48105
Abstract The early time behavior of the fields radiated by two types of EMP antennas is investigated. The first type is a bicone-cylinder in which each cone is mated directly to a semi-infinite cylinder of constant radius, thereby creating a ring discontinuity in slope at the junction. The second assumes a continuation of the bicone which matches the slcpe at the junction with the cone, producing there a ring discontinuity in curvature. Using geometrical diffraction theory techniques and knowing the diffraction matrices for the hvo forms of surtice singularity, a high frequency asymptotic development of the radiated field is obtained for each autenna. Application of an inverse Fourier transform then yields a time domain solution valid for sufficiently early times. The solutions are computed and their validity investigated. Examples are given showing the reduction in the perturbation of the early time response associated with the smoother geometry.
,.
. .-
-. —..
—.-
.-.
.-
. ..
. . . . . -,.
—
s.--..—.
—
.._
. _
. .
,
.
.. .._
.—
—----
Q..4?/ikllqt~(
.. ●
b
1.
Introduction
● .
From ideal device
an electromagnetic for radiating
infinitesimal
spherical
which is identical because
each half of the bicone
whilst
By choosing
acterized
by a slope illuminated,
pulse radiated
the multiple
interactions
of these
the net field will be affected, to the diffraction cylindrical
structure
in the shadowed yet further infinite
wave excited face,
however,
by the join.
the cylinder
illuminated
relevance
would not extend
the bicone, limit
circular
and an infinity
the transverse
at a selected
of the antenna, being attributable
effect
in nature,
and can also distort
back reflection. a traveling
further
affect
or surface of the sur -
on the field in the of an infinite
and we must emphasize regions
continuations
are possible antenna.
that the
of space.
of other
1
On the semi-
loading
is only one type of continuation
.-O
in the
on the field with-
cylinder
of the resulting
be the
with the rest
would an exact analysis
to the deep shadowed
dimension
will still
by progressive
antenna,
which is
at the joins and
field is primarily
to a realistic
region
Any modification
without
cylinder,
( 1971) and char-
the spatial
through
If this were attenuated
distance
circular
time perturbations
Only on such a premise
be relevant
The right
.
the surface
is to mate
of diffraction
diffractive
could then be terminated
region.
bicone-cylinder
pulses
region
finite
contribution
y components.
the field in the illuminated
cylinder,
Within
but because
which is entirely
is unrealistic
and Varvatsis
beyond the join will have a dominant
region
pulse
to the axis of
structure
by Sancer
with the short
of the high frequent
a radiating
to be a semi-infinite
diffracted
a pure frequency-
dimensions
that the dominant
by the gap,
the
some of the electromagnetic
geometrical
at the join.
it is expected
produces
is an
across
such an antenna
the transverse
studied
discontinuity
antenna
applied
pulse will create
to retain
making
voltage
and perpendicular
this structure
at the bicone-cylinder
undistorted
parallel
to some other
we arrive
directly
a voltage
which seeks
biconical
of the bicone
Unfortunately,
dimensions
of the bicone
from the gap.
Since a constant
in all respects.
A compromise
advantages
the infinite
the two halves
TEM wave,
of its infinite
symmetry.
*
an EMP.
gap separating
independent
viewpoint
structure
which also
One class
for
serve
of these
to
produces
., b
*
a smoother
transition
between
leaving
the curvature
slope whilst matching
the bicone
could be entertained.
the high frequency (in the first derivative
diffraction,
derivative
pulse for at least a high frequent continuity
sible to quantify
Such smoothing
of a surface
singularity
that a change
to a curvature
will decrease
expansion
has recently
this effect.
We here examine the radiated
and still higher
bolic continuation
time.
been obtained
a cylindrical as to remove
expansion
minated
for each of the two geometries.
ier transformation. is valid cannot
is to calculate
of the diffracted
applied
across
Although
be stated
with any real certainty,
analysis.
surprisingly,
time
at every
rather
ting this effect and some general ent
is sufficient
to outweigh
time behavior
of the bicone
(first
times
point in spac,e there
than slope.
conclusions
order)
term
in the
point in the illusolution
for a
by an inverse
Four-
for which this so~ution
in the frequency is some range pulse is smaller
Data are presented
are drawn,
the disadvantages
by a hyper -
an upper bound is available
of the undistorted
of
in slope at the junction.
which are ignored
for which the perturbation
the discontinuity y is in curvature
the early
The time domain
of elapsed
for a line dis -
Note.
field at any arbitrary
the range
of those ray interactions
elapsed
of the present
the leading
in
1971), it is now pos-
the gap is then obtained
a consideration Not
(Senior,
the discontinuity
high frequency
voltage
(in the second
coefficient
continuation
that is adopted
step function
discontinuity
as regards
The procedure
region
from a slope discontinuity
of the diffraction
the consequences
‘so chosen
reduces
of the direct radiated ,< Since the precise leading term
This is the purpose
field of replacing
order
the perturbation
some span of elapsed
in curvature
the
at the join,
implying
y asymptotic
by matching
discontinuous
of the profile)
of the profiie)
and the continuation
but whether
of this new geometry
from domain
of when
illustrathe improvemis a question
beyond the scope of this Note. These produce
disadvantages
the smooth
For any smooth ceeds sis,
the radius
junction
are two-fold: and an increase
and monotonic of”the join,
the discontinuity
continuation,
increased
of curvature
2
of construction
in the overaH
diameter
the maximum
radius
and for the hyperbolic
in radius
difficulty
profile
is proportional
assumed
to
of the antenna. necessarily y exin the analy-
to the fractional
. 6
*
increase
in maximum
discontinuity).
Indeed,
the final presentation tain the advantages portion
radius
on changing
this fractional
of the continuation
strength
1966) . Because
tinuation,
and attenuation
will be less
with a smaller
increase.
to the analysis
field here is basically
are known (Weston, the requirements
stringent
(slope
it should be possible
which is essential
The surface
of this attenuation,
mate termination
geometry
continuation
is one of the key parameters
but in practice
structure
which is just beyond the join. wave whose
increase
of our results, of the smoother
from a cylindrical
in to ob-
The only is that
a creeping
1965; Hong and Weston,
on loading
to permit
than was the case for a cylindrical
ulti con-
acceptable
and at some point of the curved surface beyond the join it would be ‘., to change to another surface profile more nearly approximating that
of a right
circular
a transition
region
the second
surface
have no effect early
behveen
The hyperbolic the bicone
discontinuity
geometry
and the structure
was well within
on the field within the illuminated
time behavior.
irrelevant
cylinder.
However,
to the main thrust
these
further
the shadow, region,
are practical
of our analysis.
would then apply only to out, and provided
its presence
certainly
considerations
should
as regards which are
the
.!
2.
Preliminary 2.1
Considerations
The Infinite
The antennas infinite
biconical
field radiated
Biconical
Antenna
to be expiored
antenna
are all geometrical
and it is therefore
by such an antenna
modifications
appropriate
when excited
of the
to consider
by a point generator
first
the
at the apex
of the cones, The geometry coordinates
referred
monic voltage
is now as shown in Fig. to an orgin at the apex,
generator
of circular
1.
the field radiated
frequency
A(w) eik(r - et) ~=&—— 9 r sin (3
If (r, O, @)are sphericai
l!=
polar
by a time har-
u = kc is
? *
(1)
eit<(r-et)
in which A(w) = :
where
Z = l/Y
surface
current
is the intrinsic entering
cone to the value
impedance
of free space
and 1(u, O) is the total
the upper cone from the generator
Knowing the structure line r = constant,
1(u, O)
of the field,
$1= constant i3 = r -60
we can integrate
at r = O. E.
along the field
from the va~ue @ = 00 appropriate
to the upper
at the lower cone to obtain V(W,r) = I(w, r) Zc
where
is the characteristic
impedance
of the antenna.
v(u) = r(u, o) z where
V(u) = V(u, O) is the strength
In particular,
c
of the voltage
4
at r = O
source,
and if we now write
z
I
\
\
I
\
M’
I
/
/
/
point genera
/ /
FIG.
●
1: INFINITE
I
I
BICONICAL
\
ANTENNA.
.
(2) a we have A(u) = V(u)fo Tbe radiation component
field(l)
of the electric
is independent
field,
it is a function
frequency
dependence
excitation
is the voltage
amplitude
scaling
Heaviside)
of frequency
step function
pulse
by the 6
(4)
.
only through voltage.
v(t),
the phase factor
It therefore radiate
it will
and the
foUows that if the
undistorted
and in particular,
fo/(r sin(3), U(t),
specified
viz.
of the excitation
factor
of @ andentirely
‘(w)fO ik(r-et) —e r sin 19
‘6 = Moreover,
.
except
for the
if v(t) is the unit (or
then P
‘0
‘e We shalt henceforth 2.2
Fourier
restrict
of the radiated In the illuminated
(4) appropriate attributable this expansion
to diffraction
discussed
geometries first
field for a simple region
the leading
biconical
pulse.
these
terms
in Section
a high frequency
harmonic term
antenna.
at the surface
can be found,
(5)
Relations
antenna
to an infinite
:) ~
to such a voltage
that we shall follow is to develop
expansion cones.
U(t -
— r sine
ourselves
Transform
For the modified cedure
=
source
asymptotic at the aWx of the
in the expansion The higher
singularities,
2.3 the pro-
order
is the field terms
and to the accuracy
are that
are of the form .
1
‘n- ~eik(t
V(w)w
6
-et)
*
for
n = O, 1, where
1 is some distance,
field is then obtained term
by appiying
an inverse
with V(w) as the spectrum To permit
imaginary
part
c,
m,
Fourier
time behavior transform
of the unit step source
the rigorous
not tend to zero as t +
The early
Fourier
transformation
it is necessary
With this extended
to assume
definition,
of the radiated
to each individual
voltage, of functions
f{t) which do
that w has a small
the Fourier
transform
positive of f(t) is
m J {f(t)]
(6)
e ‘Wt f(t) dt
= F(w) =
\
J-a)
and the inverse
transform
is
(7)
J-m+ i~ If f(t) is tie unit step function
U(t),
direct
integration
shows that
F(u) = i/w . Moreover,
3-1 {F(w)}
can now be evaluated
u(t) =
&
)
unambiguously,
‘e, w
-iwt
implying
(8)
dw.
-m + i~
Consider w = O is assumed integration upper
where
by the addition
half u plane and the transform
integrals
the cut originating
to lie along the negative
can be closed
of a semi-circle
.-
~ -’{Q-’/21
in the lower
along both sides
real
If t <0,
of a semi-circle
of large
shown equal to zero.
half plane reduces
of the cut,
w axis.
at the branch
and these
● 7
the path of radius
If t >0,
the transform can be evaluated
point
in the
the addition
to the sum of trivially.
Hence
Integration
by parts
now yields
the two results
which are needed
in the sequel,
viz. (9)
and
3-1{iu which are special
-5/2 ~ikl
=-
1
cases
+ei”’4(’-:Y’2@
of the general
~
“0)
formula
(11) valid for positive 2.3
integer
Specific
In the region half-cone
angle
of the cone. ordinates
Geometry adjacent
to the gap the antenna
190, extending
out to a radius
At this point the profile
with origin
the requirements function
n.
on the new profile
.)
configuration
to be a bicone of d is the slant length
and if p, z are cylindrical
co-
with the axis of symmetry,
p = P(z) are that it be an analytic,
monotonic
m and
7
go =tane
at z = dcos80
.
~
is symmetrical
about the plane
Z = O, it is suffi -
only the upper cone and its continuation.
The most simple we therefore
is changed,
finite limit as z +
P = dsin60
cient to consider
d sin 13~ where
at the gap and z axis coincident
of z with p +
Since the entire
is assumed
function
choose’ the hyperbolic
having the above properties profile
8
is a hyperbola,
and
●. .
*
p-dsin
OO=Btan60
1-
~ ~co~~
~B
0
{-
for z~dcoseo, with the upper (inward)
where cone,
curvature
B is some constant
p = d sin 6 0 and dp/dz in any plane containing
al=B This is the quantity junction
of the bicone
to the problem.
p +dsin
OO+Btan~O.
to express
“B in terms
For this purpose,
Iim 2+0 e is the fractional cylindrical
At the junction
as required,
and the
0“
the strength
Let us therefore
,
the z axis is
and its continuation.
natural
from a circular
= taneo,
k sin 260COS6
which determines
It only remains
where
yet to be chosen.
1
of the field diffracted
‘. of a convenient
at the
geometrical
quantity
we note that as z -+m, write
p = d(l+e)sin@o
increase
in the overail
continuation
(13)
radius
of the bicone
of the antenna
on changing
to a slope matching
one.
Then B = edcose
o
and A sin 20 o al = cd with c = O implying
.
..*
.
a slope discontinuity
at the junction.
(14)
.
3.
Ray Techniques The procedure
from the antenna the field radiated time variation metrical
is to first
of diffraction
is basically
diffracted
rays
produced
the formulae
optics,
it is convenient
niques
as are appropriate
by surface
optics
singularities, still
to begin with a generaI to a problem
radiation
expansion
for
(1957,
1962).
to include
The
the concept
of
and since we shall re -phrase more the similarity survey
to geometrical
of such aspects
of ray tech-
such as this.
An Overview
according
optics
therefore,
the optical
The variation
the propagation
to Fermat’s
and P must be stationary
by energy
asymptotic
by Keller
of geometrical
of GTI) to emphasize
In geometrical P occurs
a high frequency
(GTD) originated
an extension
slightIy
3.1
develop
the transient
when the voltage across the gap is a simpLe harmonic one with -iut e . The only effective method for doing so is to use the geo -
theory
theory
that we shall follow in analyzing
with respect
rays
of intensity
conservation:
principle
between
that the optical
to small
in a homogeneous of the geometrica~
the energy
of energy
distance
variations
isotropic optics
two points between
in path.
medium
Q and Q
In particular,
are straight
a
lines,
field along a ray is dictated
flux in a tube of rays
must be the same at all
points along the tube. Let us consider With reference
where
dS
Q
to Fig.
the wavefront.
Thus,
and specifically
sections
and are inversely if we denote
P from a fixed origin
the astigmatic
case,
the electric
.
field
~.
2 we have that
and dSp are the cross
and P respectively,
point
the vector
of the elementary
proportional
tube of rays
to the Gaussian
by s the oriented
distance
curvature
Q,
10
then since
of
of the observation
Q, and by pl = AQ and p2 = BQ the dis~nces
line-s A and B from
at Q
the polarization
of
is unchanged
Q
.-
dS
Q
/ dSP
FIG.
2: ASTIGMATIC
- ,. .
,. ●
,,1
.
TUBE OF RAYS.
--
along the ray, ik(s -et)
&(P) = EJ(Q) ~e where
r= /-is the so-called elementary
divergence
tube of rays
Equation B where failure
either
factor
from
( 15) yields
by Kay and Keller r
a positive
However,
of the
and provided
A and
This is a universal
for obtaining
a finite,
of a caustic
frequency-
has been discussed
a ray through
we interpret
r -1
on passing
through
a caustic
as -i,
and
Eq. (16)
a line caustic
in
direction.
medium,
of a body or,
the direction
discontinuously, cases
depending
on the principal
high frequency
limit,
ray strikes.
the incident
is in accordance
and polarization.
There
radii
of the surface
of curvature
If both radii
situation
the surface
by the Fresnel
reflection
A general
a single
by Fock ( 1965), and can be written
most compactly
E_r(Q) = ii* ~i(Q)
12
reflected
in the
ray whose
of the refiected
field at
for plane wave
for — Er(Q) as
at the
at a pLane interface.
coefficients
expression
are now two
are non zero then,
and the strength
in a
with any ray change
with Snell~s laws of reflection
optics
on a plane interface.
of discontinuity
associated
ray will produce
This is the geometrical is specified
at any surface
curvature
as do the field strength
the incident
orientation
indeed,
and wavefront
point Q where
incidence
r = m.
on following
the known phase delay of ~/2
At the surface
distinct
implying
for the field in the vicinity
( 1954).
of the spreading
value for the fieid at the caustics
and a procedure
is again finite,
does predict
measure
Q to P.
s = -91 or s = -P2,
expression
beyond,
which isa
an infinite
of any ray technique
dependent
(16)
has been derived
e
where
the tensor
reflection
and the electromagnetic
coefficient
~ is a function
of the angle of incidence
of the surface. For a perfectly conducting r =fi”ql, where ?? is a unit vector ~ is such that fi~ ~r = -a ~ J and $sIJ
body,
parameters
normal. To calculate the surface radii
the field on the reflected
we can again use Eq. ( i5) where
of curvature
general,
these
of the reflected
depend
the local normal, at Q,
in and perpendicular
the corresponding
radii,
( 1965) and Senior
( 1972),
Observe
of curvature,
rl
at the point of reflection,
1 —=
_+2seca 1
‘1
‘1
LY, measured
s ~ and S2, to the plane
and r2,
P which is away from
p ~ and p2 are now the principal
on the angle of incidence,
the radii
measured
wavefront
ray at a point
of the surface
with respect
of the incident
of incidence at Q.
Q.
In to
wavefront
respectively,
As shown by Fock
_=~+2cosa 1
,
P2
‘1
and
%2
ra
(18)
“
that
s ~, S2 and r , r are not necessarily principal radii of curvature. 12 if s ~ and s; are the principal radii for the incident wave front at Q and ~
Indeed,
is the angle between
the plane of s f and the plane of incidence, 1 (19)
and a similar
pair of formulae
The second principal
radii
point (vertex) applicable, produces
hold for the surface
case to be considered
of curvature singularity
geometrical an infinity
in the first
derivative
derivative,
the earlier
is that in which one or both of the
are zero at the point Q, respectively.
diffraction
of diffracted
only the line singularity
curvatures.
Although theory
rays.
is important. of the surface
corresponding
geometrical
now takes
over,
For the purposes
optics
ones being continuous
13
is no longer
and each incident of the present
This may be produced (wedge-like
to a line or
singularity)
at Q, but in either
ray
problem,
by a discontinuity or in some higher case
the diffracted
.-
.
rays are confined
to the surface
of the Keller
of Q the field on any one diffracted
a local
a canonical
geometry
in question.
In principal
reflected
field strength
at a point close
similar
In the immediate
ray is entirely
face at Q, and can be found by soiving
an equation
cone.
at least,
property
problem
this enables
of the sur -
displaying
us to relate
to Q to the incident
to (17) in which a tensor
vicinity
diffraction
the the
field strength
coefficient
a“
takes
via the
place of F. Unfortunately diffracted
field at a remote
s insularity
is a caustic
the singularity, from
we now run into a conceptual
Q.
immediately
the diffracted
field there
along the singularity along the edge, towards diffracted
ray,
customary
point
tensor
Since the surface
(16) to proceed
to
outwards
(following
Keller,
for points
close
P distance
=\
r
ei7r/4
‘1
s (>> k) from
1957) to Q, and Q, writing
~1+5
radius
m
.
of curvature
the incident
3 is the principal of curvature)
(20)
‘ll’(S ‘Ct) ‘“Ei(Q) of the diffracted
unit vector
or “edge’~r the local radius
the center
to find the
as
pl is the transverse
If ~ is the angle between
factor
of a diffraction
to a “remote”
Ed(P)
where
it has become
statement
ray.
in trying
ray tube in a plane perpendicular
use the divergence
For this reason,
to proceed
P along a diffracted
for the diffracted
we cannot
to omit any explicit
point
difficulty
unit vector
~ and a tangent of,which
normal
ray tube at Q. unit vector
is r ~, 7 is arclength
to the edge (i.e.
and ~ is a unit vector
?
in the direction
pointing of the
then
(21)
For a wedge-type
singularity 3=
where oriented
. -cosec
~T
~ is given by Senior and Uslenghi base vectors. 14
(22) (i971)
in terms
of a set of surface-
0 --
Even if we grant justifiable
that the above procedure
..one, we can still enhance
tion processes a diffracted
by regarding
the similarity
~ as a diffraction
ray at a distance
is the most rigorously
s 21
from
of the diffraction
tensor
which relates
Q to the incident
point is denoted
analogous
to (15) and ( 17), viz.
the field on
field at ‘Q.
by Q’, Eq. (20) can be decomposed
displaced
and reilec -
If this
into two equations
(23)
~d(Q’) = ~ o~i(Q) ,
gd( P) = lJd(Q’) where
factor
the divergence
r
~e
ik(s -et)
has the standard
(24) form ( 16) with (25)
P2=~
and the proviso
c
i7r/4
i=e
that
.
Thus
(26)
and though for
Eqs.
(23) , (24) and (26)
ks >> 1 (the condition
under
are no longer
equivalent
which the diffraction
to (20) for ks & 1,
tensor
~ was originally
deduced),
(27)
and the equivalence
is restored.
In a problem
where
advantage
of the present
similarly
and proceed
for
interpretation stepwise
i’ . But the advantages
pay is that
both diffraction
and reflection
is that we can treat
along all rays
the wave number,
15
the main
the two processes
using the same
are not overwhelming,
~ can now involve
occur,
general
and one penalty k.
formula
that we do
.
3, 2 Diffraction The diffraction singularity)
Tensors
was originally
what more
general
in curvature
by Keller
and Uslenghi
was obtained
Consider
by Senior
The analogous
a wedge-like
singularity
remarks,
it is sufficient
for a discontinuity
shown in Fig.
resuR
for
( 1971), and to present,
of one pair of parameters
half angle of the wedge be f2 and choose A N
( 1971).
in slope since the result
only in the replacement
in slope (wedge-like
( L957) and was put into a some-
in the form impkied by our preceding
take “only a discontinuity differs
deveioped
form by Senior
a line discontinuity the tensors
for a line discontinuity
tensor
to
in curvature
by another. 3.
Let the interior
a base set of unit vectors
f?’,I?,;
with
A
normal
to the edge and pointing
into the shadowed
half space,
Then if the incident
out,
B binormal
to the edge and pointing
and ; = R A~ to make the system
ray direction
right-handed.
is (28)
with
O < P < ~ and the diffracted
-~/2+ f2 lJS
of a valid estimate
the implications
of ray techniques
for the field radiated
as regards
by the antenna.
the
Although
we
-1
,
.“
shall obtain the time domain high frequency
response,
solution
by inverse
quite different
Fourier
factors
transformation
influence
.
of the
the accuracies
in the
m
two domains. In the frequent
to the field is determined tude of a direct
ray contribution
Ak-’/2{l+0()}’)}
that the reflection the other
ha~d,
ampIitude
diffraction
problem
(or re -diffraction)
as the geometry
ray contribution at least
a ray diffracted
at a surface
though it should be noted
obvious
order,
will reduce
that the nature
will depend on the impedance
On .
the
If, for example,
of a re-radiated
of this re as well
this process
is quite
necessary
and the consequences
bat!< to the
of the source
which quantifies
It is therefore
viewpoint.
at a surface
reflected
is
for a ray to be diffracted
It is intuitively
from our analysis,
of k, the contribution
at a slope discontinuity
of k only to the leading
it is possible
k dependence.
from a rigorous
If the ampli -
A is independent
at a slope discontinuity
of the gap, but the factor
unimown even in its
is A, where
contributions
.
and then re-radiated.
radiation
amplitudes.
is unaffected,
is independent
to Ak-l {1 + O(k-l)]
of individual
ray is subsequently
the k dependence
a secondary
In the present source
singly diffracted
coefficient
of their
from the source
. If a diffracted
are large,
importance
by the k dependence
of k, that for a contribution
whose radii
the relative
y domain,
to exclude
of this are unfortunate.
the factor
were independent
ray would be of the same order
singularity y, regardless
any such
of its nature,
as that for
and this would
.invalidate to direct
all terms
in the high frequency
expansion
beyond the first,
corres
radiation
proportional
from the source; it is not inconceivable that the factor 1/2 to k , which would invalidate even the leading term !
The validity field now rests tion factor. time behavior
of even a single
on an assumption
On the premise
about the frequency
field comparable
would have its gap configuration factor
in the high frequency
that any EMP antenna
of the radiated
that the re-radiation
term
changed
is proportional
20
dependence
expansion
could be
of the
of the re -radia-
which did not produce
to that of the excitatiori
(perhaps
an early
voltage
until it did, we shall henceforth to some negative
pending
assume
fractional)
a
power of k, thereby that of a single consequence, direct
reducing
diffracted
ray,
tinuities,
either
directly
reflection
off the sides
of any such ray contribution
but not below that of a double diffracted
the only contributions
ray from the source
domain
the importxmce
which can be entertained
and the rays
singly
diffracted
to the point of observation, of the bicone.
below
ray, ‘f’ In
are those
of the
at the surface
disc on -
or via an intermediate
The resulting
expansion
in the frequency
is then A+Bk ‘1/2{
for a slope discontinuity,
l+ O(k-e)}
or A+ Bk-3/2{J+0(k+)}
for a curvature factors
discontinuity,
produced
where
by the different
c >0 and we have again omitted
ray paths. From
Let us now turn to the time domain. the field we can deduce verse
Fourier
Whereas
the early
transformation,
the lengths
time behavior but the validity
of the individual
paths
field point are no direct
concern
time domain
are reflected
since
these
path whose contribution
is omitted
the bound is provided
one, whichever not entirely
certain.
the frequency diffraction
is shorter,
domain,
coefficient
Take,
a high frequency
of the transient of the result
response
is not without
domain,
now provides response
an upper
is patently
The shortest
invalid.
the doubly diffracted
only the leading was included
a singly term
diffracted
in the asymptotic
in estimating
by inquestion. reach
ray
In the present
ray or the re -radiated of the solution
ray contribution. expansion
the contribution,
is
In
of the
yet all terms
‘f For a surface discontinuity in curvature it is even less justifiable to include doubly diffracted rays since their contribut”o is 0(k-3) whereas the correction a singly diffracted ray contribution is o(k -ifl, .
21
the
bound on the elapsed
but even within this bound the accuracy for example,
of
they are vital in the
in time separations.
by either
estimate
by which the ray contributions
in the frequency
time beyond which the time domain problem
the phase
to
would yield some contributior~ impossible
to estimate
and on a strictly transformation
in the time domain.
the error
mathematical of an asymptotic
resulting basis
there
expansion
Unfortunately,
from the omission are difficulties in the first
.
it is almost
of these
in justifying
terms,
e
the Fourier
place.
O..22
.
4.
Analysis 4.1
Frequency
Domain
We shall now use the geometrical frequency
expansion
of the field radiated
which will be included larities
are those
and which then reach
the sides
the previous two orders
section,
the resulting
The contributions
diffracted
contribution
asymptotic
at the surface
directly
of the source expansion
singu-
or by reflection
from the source.
properties
a high
off
Subject
referred
is accurate
to
to in
to the first
in k.
The polar
coordinates
of the field point
($0 < e < 7/2)
is measured
For simplicity
it will be assumed
portion,
singly
the field point either
about the re-radiation
to develop
of diffraction
by the antenna.
for rays
plus the direct
of the cone,
the assumption
theory
which allows
with respect
(r, 6) where
to the symmetry
that
us to treat
PO are
8
(z) axis of the antenna.
PO is in the far field of the biconical
as parallel
all rays
reaching
PO.
The procedure to be followed is the same whether the surface singularities I and since the only difference between are discontinuities in slope or curvature, the formulae
is in the expression
cient to describe
the analysis
is now as shown in Fig. Observe
that because
P are confined
for the diffraction
in terms
4, where
we have included
angles
associated
symmetry
plane through
Let us begin by establishing with the rays
and
~P ~OPO = @-0.,
The geometry
the ray paths to be considered. of the problem,
all rays
shown in Fig.
4.
diffracted
important
path lengths
The direct--ray rays
are
and
is OPO, and
OP ~PO and 0P2P0.
(38)
;(7-6.)
the diffraction
reaching
the z axis.
some of the more
The two simple OPO = r by definition. [1 Since the interior half wedge angle ~ is 0’
D, it is suffi-
of slope discontinuities.
of the azimuthal
to a single
coefficient
angle
y (see Fig.
3) for the upper
path is
en
Q“.
(39)
23
..
1---1
FIG.
4: THE GEOMETRY, RAYS.
SHOWING THE MORE IMPORTANT
24
Also,
.
●
[PIPO]= .- [OP~COS~PIOPo where
d is the slant
responding
results
length of the cone. can be obtained
former, exist,
L 0P4PL
= r -@o-6
it is necessary
we + d
either
the
are
and hence
OP ~ P o,
the cor-
6 by ~ -0. OP i P4P0
and 0P2P3P0.
~OPl P4 = 30.+0
-r.
For the
If this path is to
o<3eo+e-w30
excluded
ource ‘7
rays
path
that
! where
For the lower
by replacing
The two diffracted-reflected
(40)
= .-d.os(@-Oo)
the extremes
O or the surface
for which the reflection singularity
P2,
point coincides
and thus
T-360 <8 ~+2c0.
eo-c0.(e-eo)
34
,
restrictive,
*
making .
doubie diffraction
has no solution
the basis
for our criterion.
P2 and then reflection
produced
by diffraction
off the upper
is t~, and it can be verified
surface
g
and at the upper
f<:
{
singularity
the associated
boundary
time delay
of the vertically
}
shaded
< :
is rather
off the lower
of the horizontally
T is determined
points of the region therefore,
of the region,
t3 ~ 2d/c. in Fig.
lying above the dotted
It is obvious
with equaiity 5,
by the doubly diffracted
this reflected
it is legitimate
for the contribution
of the cone.
region
, to include
solution.
different
surface
shaded
1+2 cosf30-cos(6-80)]
the interior
only for 19= 901 and
purposes,
At the lower
l+2coseo-cos(e-eJ
in our time domain
The situation
ever,
{
t4 < T throughout
this contribution
equality
of the cone,
surface
boundary
‘4 =
reflection
at the lower
5,
‘4=C
and since
Eq, (74)
that
for all 6 for which the path exists. in Fig.
t10 >36.9°,
for 6 ~ 90°.
For the contribution
region
For
line in Fig.
ray contribution
domain.
35
that
from
t3 ~ t4 with
on the lower
Throughout ray,
which results
boundary
most of this region,
and 5.
how-
t3 < T only at those For most practical
is of no interest
in the time
., ,,
5.
.
Data and Coriclusions. Expressions
for the normalized
transverse
(rj G) in the far zone of the biconical of the eiapsed
time
of a cylindrical
t in Eqs.
continuation
and the second
portion
Both are vaiid for only a short
vature. sufficient)
condition
required
for validity
E at a point
aregiven
as functions
of these applies
a slope discontinuity
to the case
at the junction, is in cur-
for which the discontinuity
range
of times,
and a necessary
is that given in Eq. (72).
(but not
In particular,
it is
that ad
t<”=
where
The first
which produces continuation
field
of theantenna
(69) and (70).
to a hyperbolic
electric
t is measured
in nanoseconds
Each of the expressions of a direct produced entirely
contribution
physically
respectively,
subtract
contributions
AH of the contributions
are negative
from the direct
With increasing
7r/2 and consists
and up to four secondary
Ds(tl) and ~c(6)
contributions
required. 1/2 as (t -ti) or (t-t
for E(t) is valid for 00 f 6<
and diffraction-reflection.
real and since
secondary
and d in meters.
from the source
by diffraction
(75)
for 60<6
one.
These
t, each secondary
~ r/2,
are the
properties
contribution
e
are increases
)3/2 according as the discontinuity is in siope or curvature i t = ti is the time of onset. Of necessity, there must now
exist a time
where ::: t = ti at which the secondary
geometries,
all other
parameters
contributions
being equal.
For
are the same for the two a discontinuity .,,,. for t > ti a slope dis-
Ok L
Hence,
5C( T/ 2)
(57), (71) and (14),
from Eqs.
.,. t
(76)
-1
(c.st?,-sin
2X sin AT
J-g
4’-,1
(77)
COSA,+C?C)S+
showing
an increase
proportional
can ensure
that the curvature
throughout
the entire
range
value of e for which function
For values
discontinuity
of 6 other
to calculate
meters.
The program
analyzing
the data,
parameters
than
~ affects
the perturbation
changes
Eqs.
and needs
d affects
reach
early
has
of paraand in
by the number
of
with time is a fum tion
discontinuity, parameters
of ~ as well. is rather
For-
trivial.
Thus,
by a curvature
discontinuity
only as a scaling
2 decreases
the perturbation
by the same
proportionally
the field point
of all perturbations
of the computer
show the normalized
as a
program
no comment,
is that produced
the times
(see Eqs. through
of E(t) on 6 and 60 is much more
The form
and the
values
(69) and (70) for any combination
of some of these
produced
length
the magnitudes
is valid,
A computer
of the field strength
in e by a factor
contributions
dependence
E(t) from
the effect
The slant
we
perturbation
to turn to computed
about the performance.
in the case of a curvature
however,
secondary
7r/2 it is necessary
The variation
tunately,
amount.
the smaller
enough,
from Eq. (77) and is plotted
is quite straightforward
involved.
an increase
e large
t < T for which our analysis
the only complication
of 0, 60 and d and,
factor:
produces
)
6.
of E(t) to draw any conclusions been written
By choosing
t’x = T has been computed
of 00 in Fig.
‘o sin ~
sin 26 0 l+cosoo (
to e and d.
of’ times
$)3
E(t),
(68)) and,
a factor
d-1/2,
in Tables
as a function
“e= O implies a slope discontinuity and, hence, Eq. (69), corresponds to a cul*vature discontinuity, i. e. Eq. (70).
37
in addition, but the
involved.
output is illustrated
time response,
ti at which the
i and 2 which of t for’f @@90 0 ,
whereas
c # O
,
. .-.
145
.
1.0
e .
0.5
0
10
20
30
40
50
$O(deg.) FIG. 6: THE .!. FRACTIONAL INCREASE, t~ = T WHEN 19= 90°.
e, IN RADIUS FOR WHICH
. .
,,
●✌ 38
TABLE
1.
Individual d=
fSd E 0.0 0.20 0.43 O.bl 0.81 1.01 1.21 1.41 1.62 1.8A 2.02 2.22 2.42 2. bJ 2.83 3.03 3.23 3.43 4.64 3.84 4.04 4.24 *.k4 4.65 4.85 5.95 5.25 5.45 %66 5. U* b. Li13
6.26 6.u6 b.67 h87 7.07 7.27 7.47 1. b~ 1.88 *.08 &2d d.4tl i3.69 8.$9 S.09 9. .?9 9.49 9.70 9.90 10.10 10.30 10.51 10.71 10.91 11.11 11.31 11. 5,2 11.72 11.94 12.12 12.32 12.53 12.73 12.93 t3. 13 13.33 13.54 13.74 13.94 14.14 94. J4 14.5s 14.75 14.95 15.15 :5.3s 15.56 ls.7b 15. 9b
l,5m
to E(t) for e = 90°,
Contributions andc’O.
iu’L4k l.dd 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.93 0.90 0.67 O.do 0.7s 0.71 0.b8 0.b5 O.bd 0.60 0.513 0.55 0.5s 0.51 6.49 0.47 0.4b 0.4U 0.4,? 0.41 0.39 0.J7 0.3b 0.34 0.J3 0.J2 9.30 0..29 0..28 0.20 0.25 oei4 0..22 0.21 0.20 0.19 a. 18 0.16 0.13 0.11 0.03 0.07 0.05 LI.03 U.04 -J. (JO -0.02 -0.03 -0.05 -d.06 -0.07 -0.05 -d. 10 -0.11 -0. lJ -o. 14 -0.15 -0.17 -0. lu -0.19 -0.20 -0.21 -O*JJ -0..44 -0.25 -0.20 -Q*L7 -o.2a
e. = 30°,
Lud Ld 0.0
0.0
0.0
0.0
0.0 0.4 0.0 0,0 0.0 0.0 O.u U.o 0.0 0.0 O.u 0.0 -0. Ob -0. oa -0.11 -0. lA -0.1* -0.15 ‘0. lb -0.18 -0.19 -0.20 -0.21 -0..42 -0.23 -0.23 -0..lti -0.25 -0..26 -0..)7 -0.27 -0.28 -0.29 -0.3.3 -0..I2 -0.31 -0.31 -0..),? -0.33 -O*3J -0. J4 -0.35 -0. JJ -0. Jb -0.36 -9,37 -0. J7 -0.38 -0.3d -0.39 -0.39 -C.40 -0.40 -0.41 -0.41 -0.+2 -0.$2 -0.43 ‘o. QJ -0.44 ‘0. Q14 -d. *5 -d.45 -0.43 -0.46 ‘(?.40 -0.47 -O. *7 -o.4d ‘Oe4d -0.48 -0.+9 -o. @’J -U.50 -0.50 -0.53 -0.51
,
0.0 0.0 0.0 0.0 0,0 0.0 (J. O 0.0 0.0 0.0 -0.07 -0.10 -o, 13 -0.15 -0.17 -0.18 -0.20 -0. J.1 -0.22 -0.24 -0.25 -o. Je -0.47 -o. &u -0.29 -0. 30 -0.31 -0.32 -0.33 -0.33 -0.34 -0.35 -0. Jb -0, 37 -0.37 -0.38 -0.39 -0.40 -0.40 -0.41 -0.42 -0.42 -0,43 -0.44 -0.44 -0.45 -0.40 -0.46 -0.+7 -o. U7 -0.48 -0.49 -0.49 -0.50 -0.50 -0.51 -0.51 -0.5.2 -0.53 -0.53 -0.54 -0.54 -0,55 -0.55 ‘0.5b -o.5b -0.57 -0.57 -0.58 -0.58 -0.59 -0.59 ‘0,60 ‘0. bo -0.61 -0.61 -0. bl -0.02
39
.42P 0.0 d.o 0.0 9.0 0.0 J.O ).0 0.0 0.0 3.0 0.0 2.0 O*O 0.0 0.0
U.o 0.0 0.0 0.0 0.0 2.0 0.0 0.0 U*O 3.0 3.0 0.0 3.0 ‘).0 0.0 5s0 0.0 Jao 0.0 0.0 0.0 2.0 0.0 O.d 0.0 0.0 0.0 0.0 0.0 0.0 9.0 0.0 0.0 0.0 0.0 -3.92 -0.03 -0.04 -Oaos -J.06 -0.07 -J .07 -9.0$3 -0.08 -!3.09 -J .09 -0.10 -9.10 -o. 10 -3.11 -0.11 -0.12 -0.12 -0. 12 -0.13 -0.13 -0.13 -Oa 13 -0.14 -0.14 -O*14 -0. 15 -0.15 -0+15 -0.15
UPPiH
REP 0.0 0.0 0.0 0.0 0.0 0.0 0.0
M M 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0, 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 .0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
SUB TOTAL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 !).0 0.0 0.0 0.0 -0.07 -0.10 -0.13 -0.20 -0.25 -0.29 -0.32 -0.35 -0.38 -0.40 -0. s2 -0.45 -0.47 -0.49 -0.51 -0.53 -0.56 -0.56 -0..58 -0.59 -0.61 -o.133 -0.64 -0.66 -0.67 -0.68 -0.70 -0.71 -0.72 -0.74 -0.75 -0.76 -0.78 -0.79 -0.80 -0.81 -0.82 -0.84 -0.87 -0.89 -0.91 -0.93 -0.95 -0.97 -0.98 -1.00 -1.0/! -1.03 -1.05 -1.06 -1.07 -1.09 -1.10 -1.11 -1.13 -1.14 -1.15 -1.17 -1.18 -1.19 -1.20 -1.21 -1.43 -1.24 -1.25 -1.26 -1..f7 -1.28
TABLE
2: Individual d=l.5rn
LuIAL
1.00 1.00 l.o~ 1..23 0.99 a.97 0.94 0.91 IJ.87 0.63 0.7* a.74 0.69 0.04 0.59 0.5J 9.47 0.41 a.~ii @.z#
0.11 0. 1+ 6.07 -0.0 1 -0. cal -a. w -0.24 -0.3A -0.41 .-0.50 -a.59 -0.68 -43.77 -13. d7 -a. sb 7.07 7.27 1.47 7.6* 7.UU a.oti 8.2# 8.48 8.6’9 *. ki9 9.9$+ 9.29 9.49 9.7a 9.9a la IO*JO 10.51 toll k?. 91
to.
12.3J 12.53 12.73 12.*3
-1.l)b
10 -1.,it -t. J7 -i.4J -1.%1 -1.69 -1. MO -1.91 -i. a3 -2. 1* -2. 2* -4. W -.t.50 -L.6L -J. 74 -,z. d6 -i.9j -3.11 -3.24 -i. il -J .50 -3.6J -3.77 -3.90 -*.04 -4.11 -+.31 -4.45. -4.5’4 -.).71 -4. dd -3.04 -5. 17 -j. Jl ->.4* -5.01 -5.70 -> .91 -U. co -G. AI -G.37 -~.5.2 -%.
‘o.
b@
-0.
tl.l
Contributions = 1, and
Ludi!i
LtLt d.o O.J C.a 0.!3 0.0 0.0 0.0 a.o 0.0 o.a
a.~ 0.0 0.0 0.0 O.J 0.0 0.0 0.3 0.0 0.0 0.0 0.0 O.J 0.0 0.0 o.a 0.9 -0.00 -0. do -a.s~ -0.0 t -0.01 -a.o~ -a. a~ -0.0.? -0.od -b.03 -0.04 -0.04 -0.05 -0.05 ‘0. Ob -0.07 -0. J7 -0. od -o. a9 -0.09 -6.10 -0.11 -0.12 -0. 1,+ -O.l J -0. tu -0.15 -0. lb -0.1? -0.17 -0.16 -0.19 -o. &fJ -0../ I -Q.22 -0..2J -0.24 -a..25 ‘C.
db
-0..?7 -0..?5 -0.29 -o .40 -3.31 -0.4.4 -0. JJ -0.34 -0. db -0. J7 -Old -0. JY -0.40 -0.41
to E(t) for6
LU.? a.a 0.0 0.0 -0.00 -a. ol -0.03 -0.06 -o. a~ -0.13 -0.17 -0.21 -0..26 -0.31 -o. Jb -0.41 -a.47 -0. >3 -0.59 -0. b6 -0.72 -0.79 -0.46 -o. !33 -1:01 -t. oti -1.16 -1.24 -1.32 -1. ut -1.49 -1.51J -1.67 -1.76 -I .*5 -1.94 -2.03 -2. 13 -2.23 -2.33 -2.43 -A. 53 -2.63 -.2.73 -2.44 -2.95 -3.05 -3.16 -3. .?7 -3.39 -3.50 ‘3. bl -3.73 -3.$5 -3. Y7 -4. a9 -4.21 -4*A3 -4.45 -6.57 -4. 70 -4.83 -u’. !)5 -5. Oa -5.2! -5.44 -5,47 -5.61 -5. /u -5*67 -b. al -e. li -b, JY -0.42 -6. >b -b. 71 -6. M5
ukti~d
-b.
YY
-7.13 -7. dli -7. UJ
40
LUSU
4JF J.o 1.0 3.0 J.o .?.0 d.a ;:: J.o 0.9 N J.a 0.0 J.a ‘3.0 9.a 0.0 il. o 3.0 0.0 0.0 0.0 4.0 0.0 a. o J.11 0.0 0.0 o.a 0.0 o.a 0.0 0.0 0.0 M 0.0 0.0 M M 0.0 0.0 9.0 0.3 0.0 0.0 0.0 0.0 N’ ::! 0.0 ~.a J.J 3.0 9,0 O.d J.o 0.0 0.0 J.a 1.0 3.0 3.0
O.u 3.0 3.0 0,0 J.Ll ].a 0.0 .3.0 J.cl J*O d.o 0.0
=90°,
,60 ‘30°,
UI?PZU
Ml! 0.0 0.0 0.0 o.a 0.0 0.0 0.9 0.0 o.a 0.0 0.0 0.0 0.0 0.0 a.o 0.0 o.a 0.0 0.0 0.0 0.0 a.o 0.0 0.0 0.0 o.a a.o 0.0 o.a 0.0 0.0 0.0 0.0 0.0 0.0 0.0 o.a 0.0 0.0 0.6 0.0 0.0 0.0 0.0 0.0 o.a 0.0 0.0 0.0 0.0 (7.0 0.0 0.0 0.0 0.0 a.o 0.0 0.0 0.0 0.0 0.0 o.a 0.0 ::: o.a 0.0
::; 0.0 G.o 0.0 0.0 0.0 0.0 G.o G.o o.a 0.0 a.o
SUB
TuTAL a.o
0.0 0.0 -9.00 -oat -0.03 -0.06 -0.09 -0.13 -0.17 -0.21 -a.26 -0.31 -0.36 -0.41 -0.47 -0.53 -0.59 -0.46 -0.72 -0.79 -0. i56 -0.93 -).01 -i. os -1.16 -1.2U -1.32 -1.41 -1.50 -*.59 -1.68 -1.77 -1.87 -1.96 -2.06 -2. 16 -2.26 -.4.37 -2.67 -2.5E -2.69 -2* aa -2.9 1 -3.03 -3.14 -3.26 -3.38 -3.50 -3. b2 -3. ’lb -3.86 -J. 99 -4.11 -4.24 -4.37 -$.50 -4.63 -4.77 -4.90 -5. a4 -5. 17 -5.31 -5.&5 -5.59 -5.73 -5.8a -b.02 -0. 1? -6.31
-6.46 -b.6 1 -6.76 -6.91 -7.06 -7.21 -7.37 -7.52 -7. ba -7. h4
●
.
e = 1, the cdkr
~ = O and f3~60°,
~ 300).
The total field is broken
‘o last column
(labelled
“sub-total”)
i. e. the net perturbation. cluded
data for times
As noted earlier in just before
the limit
of allowable
le~s than
= O in Eqs.
e = 1.
The crossing
time
of the field perturbation, the slope discontinuity d enhances
upper
and lower
con-
due to reflection, in addition,
include
discontinuity
00 = 29°,
which is not far short
is
@= 90° and of the limit
As judged by the magnitude
are admissible.
time span,
and to increase
and the curves
and discontinuities This is illustrated
6 = 90°, the contributions come in at the same in Figs.
8 snd 9.
are for a slope discontinuity
e = 1. time is
contributions
In Fig.
8, @= 75°
T = 18.7 nsec. are clearly
only
from 90°.
d = 1.5 m,
When
pur -
i. e. put
over most of the admissible
d = 3.0 m
the two diffracted
For values
is superior
0 # 90°.
allowable
mag-
discontinuity
bicones
having
a significant
for most practical
from the slope to a curvature
this superiority.
-co,
0 = 90° comes
curvature
for
maximum
Thus,
markedly
t* is 8.3 nsec,
neg-
holds for the diffracted
2).
any contribution
this particular
is not true
discontinuity
(see Table
for which the solutions
e or
field for
that can be entertained.
conclusion
7 for the case in which
we have in-
as t -+ co, E(t)+
and only achieves
when e differs
of changing
in Fig.
T = 10 nsec
times
columns,
time approximation.
(69) and (70) for all 6 and @o, and,
contribution
The effect illustrated
to ignore
Indeed,
1, the reflected
the same
90°,
and the
the increasingly
of the initial
than those
discontinuity
it is sufficient
one diffracted
0“
in Table
of e markedly
of the lower
to demonstrate
of the failure
and evident
components
for both Tables,
as t increases.
which are greater
‘1=V2
T ~ 10 nsec
up to 16 nanoseconds
nitude at times
poses,
down into its various
Although
but this is just a consequence
b;ing the same (d = 1.5 m,
is just the sum of the four previous
ative value of the field amplitude
tribution
parameters
and
to
from the
time,
bat this
In both cases
(e ‘ O) and a curvature 00 = 35°
The breakpoints differentiated
for which the associated
with
and even the contribution
due to reflection off the lower bicone enters within the allowable time span, as . predicted by Fi&. 5. The curve for the curvature discontinuity has these same
●
breakpoints,
though they are not so immediately 41
apparent.
0,5
E(t)
o
u
3
6
4 t
8
r’”
10
(nsec)
FIG. 7: NORMALIZED ELECTRIC FIELD E(t) FOR SLOPE (1) AND CURVATURE (II: e = 1) DISCONTINUITIES WITH 6 = 90°, 00 = 29° ANDd=l.5m.
f
.. .
,,
,“””
42
1.0
0.5
I
i
I&
E (t)
0
GJ
I
I
i
I I
I
I
I I
I
-0.5
I
I I -1
Ot
1
I I
i I
I I I I 1
I I I 1“
I
I I
I
-1.0
\
I
1
5
t
‘2
1
I
10
t
t
:
1
15
\
I ,
u
‘4
a
20
\
-i
25
(nsec)
FIG. 8: NORMALIZED ELECTRIC FIELD E(t) FOR SLOPE (I) AND CURVATURE (II: 6 = 1) DISCONTIN(JITIES WITH 0 = 750, e. = 35° AND d = 3. Om.
1.0
0,5
I@
I&
E (t)
o
-O* 5
I I I
I I
I
I
I
I
I I I
I I
!
I
I I I
-1.0
#
‘1
1
5
I
10 t2
t*
t(nsec) FIG.
9: NORMALIZED ELECTRIC FIELD E(t) FOR SLOPE (I) AND CURVATURE (II: e = 1) DISCON INUITIES WITH 6 = 57°, (30 = 29° AND d = 3. Orn.
b
,
● /b
*
The effect of decreasing illustrated
in Fig,
Observe
9.
19 and O“ h 57° and 29° respectively that the crossing
time
is
t ‘;: is now 7.5 nsec
which is less than half of the allowable time span T ‘ 18.7 nsec, whereas >~ By and large, t’;f varies little with 190but decreases in Fig. 8, t = 13.9 nsec. uniformly times
t3. This is evident
with decreasing
are listed
does not differ apparent
for d = L 5 m, c =‘1 and a variety drastically
from
that for 6<66°
50 percent
because
times;
discontinuity
@ and
is better
of the slope of the time domain
in size and shape,
6..
crossing Since
8.,
the performance but which differ
TABLE
curves
of antennas
for less than it is much,
(see Figs.
\
.. .
AS A FUNCTION
i3
6(
15
OF
6 AND
in the form of the continuation
3:
FOR d = 1. 5m AND ~ = 1
30
25
20
e.
T
40 b
8.05
8.02
8.08
8,28
7.62
7.60
7.68
7.88
7.27
7.19
7.18
7.27
7.49
6.95
6.84
6.77
6.77
6.86
7.09
78
6.52
6.41
6.35
6.35
6.46
6.70
75
6.09
5.99
5.93
5.94
6.06
6.32
72
5.66
5.57
5.52
5.52
5.67
5.95
66
4.84
4.75
4.72
4.76
4.91
5.25
60
4.05
3.98
3.96
4.02
4.21
4.64
54
3.32
3.25
3.25
3.34
3.61
4.21
90
8.27
8.13
87
7.83
7.70
84
7.39
81
~
45
7-9).
whose bicones
;::
t
T
it is
and we note that when it is worse,
So far we have compared are identical
3 where
of 6 and
10 nsec for all of these
the curvature
of the allowable
much worse,
from Table
“
4
c
*
This is not the only possible
structure. to a cylinder
of constant
the final radius
vature,
necessarily
particular
the continuation
radius. is the maximum
radius,
practical
considerations
limit
legitimate
comparison
between
radius
rather
superiority
of the hyperbolic
d
6
1.0
90°
E — o
.5
90°
1
continuation
of the antenna,
.33
90°
radius
of. the antenna,
of antenna
in Table
1.0
90°
0
1.0
90°
1
90°
2
3.0
.90°
0
1.5
1.5
90°
1
1.5
1.0
90°
2
3.0
60°
1.5 1.0
When this is done,
the
,
having
a half-cone
10
6
—
-.04
.22 -.69
,-2.7
I .81
1.0
1.0
on this
) 4
1 I .63 I
and that a
should be based
4 for antennas
that
4
.5
2.0
(12), the! final
with a S1OPSdiscontinuity, three different v~ues
Final Radius
.5
profile
in cur-
and in the
is far from overwhelming.
is illustrated
I
a discontinuity
●
i.e.
( i + e}d sin (lO. It can now be argued
the two types
TABLE
60
radius
the maximum
For antennas
190= 29°.
.66
leaving
has the hyperbolic
namely
at the junction,
of the ring discontinuity,
than the radius of the ring discontinuity.
Such a comparison angle
the radius
the maximum
When the cone is mated
discontinuity
to match the slopes,
increases
case where
a slow
is simply
On the other hand,
d sin 6..
creating
radius,
of the antenna
comparison.
I
.52
.57 .06
[ -0.9
-5.2
-.23
-8.1
I .43I
I -,69 -.88
.31
-1.62
I
1-1.82 .52
.81
.63
.90
.63
.27
-.16
1.5
.82
.53
.16
-..31
0
1.5
.55
.41
.29
.20
60°
1
1.5
●
60°
2
1.5
.59
1.0
22
= .51 rad. 46
-.56
.34
-1.54
.09
-2.68 -.16
.
of d are considered
and the response
is compared
antenna
noted at t ‘ 4, S, 8 and 10 nsec,
with two having curvature
discontinuities
d/ (l+e) which are such as to yield the same overall the response purposes)
differs
from
the poorer
in Table
change
slope-discontinuity
d changes
when d= 1.0 2.22 nsec
the limits
to most
the validity
arrived.
at ail,
inferior, A more
since
purpose
criticism
but yield different
practice,
values
it is not even necessary and having
the upper
limit
followed
antennas
has been made at times
antennas.
times
in Table
The same
of 1.0,
antennas
of the comparison continuation
would have been
is possible.
can be found which achieve radius
that the continuation
for some distance
function
than is implied 47
substantially
a much smaller
analytic
beyond
can be introduced.
in slope without require
In
be a single
of a discontinuity
All such geometries
the same
of the antenna.
profile
profile
profile
We have
is only one of many serving
for the maximum
of the antenna
no
would not yet have
could even take the form
in the radius
had
to
showing
lies well within the shadow
increase
criticism
such as 1, 2, 3 nsec
which is created
time behavior.
4, the
On the other hand,
vided the new singularity
the early
and
an amount.
Others
a different
T =3.33
three
contribution
(say) a hyperbolic
of the cone,
and thus,
and 0.33, lines
To
the first
the curvature-discontinuity
of the bicone.
listed
infe rior
counts.
is invalid,
had a response
diffracted
that the hyperbolic
the slope
at the times
it would have been found that in almost
antenna
the first
fundamental
and (for our
fair on several
in the Table.
at sampled
aIbeit by not so large
remarked
to match
for these
of the comparison,
Once again,
already
as regards
comparisons
the slope-discontinuity
~rturbation
is not entirely
of applicability
on the responses
The more
is substantially
T beyond which the analysis
Hence,
of the other
we concentrated
all cases,
antennas
with the curvature-discontinuity
far beyond
curve,
the field perturbation
T = 6.66 nsec, but when d= 0.5
respectively.
comparison
applies
and e=l,
antenna radius..
one.
however,
the time
at slant distances
As judged by the responses
the antenna.
This conclusion,
judged
the greater
4, each of the curvature-discontinuity
to the equivalent
ensure
1.0,
E aoh
Pro-
region,
it
affecting (fractional)
by (14) for the same discon-
..
. A
tinuity antenna
in curvature,
although
increase
of high frequency
a
diffraction
being equal,
of the antenna. but whether and difficulty
?
beyond that of the slope-discontinuity
is inevitable, By its very nature,
factors
some
a discontinuity
the improvement of construction
produces
a smaller
than a discontinuity y in S1Ope and hence,
the former
The analysis
in curvature
is preferable
as regards
we have given enables is sufficient of the antenna,
the early
the effects
to compensate
amount
all other time behavior
to be determined,
for the increased
radius
is a topic beyond the perview
of this
report.
48
*’/’
..
.’ .
J
References
9
*
diffraction
and propagation
Fock,
V. A. ( 1965), Electromagnetic Pergamon Press, New York.
Hong,
S. and V.H. Weston (1966), A modified Fock function of currents in the penumbra region with discontinuity Sci. ~, 1045-1053.
Kay, I. and J. B, Keller (1954), Asymptotic J. Appl. Phys. &, 876-883.
evaluation
by an aperture,
problems,
for the distribution in curvature, Radio
of the field at a caustic,
Keller,
J. B, (1957),
Diffraction
Keller,
J. B. (1962), 116-130.
Geometrical
Sancer,
M. I. and A. D. Varvatsis ( 1971), Geometrical diffraction solution for the high frequency - early time behavior of the field radiated by an infinite cylindrical antenna with a biconical feed, Sensor and Simulation Notes No. 129.
Senior,
T, B.A. ( 1971), The diffraction matrix Sensor and Simulation Notes No. 132.
Senior,
T. B. A. ( 1972), Divergence factors, University Laboratory Memorandum No. O10748 -5 O4-M.
Senior,
T. B. A. and P. L. E. Uslenghi ( 1971), a finite cone, Radio *i. 6_, 393-406.
Weston,
V. H. ( 1965), Extension region, Radio Sci. 69D,
theory
of diffraction,
of Fock theory 1257-1270.
.
49
J. Appl.
Phys.
2&, 426-444.
J. Opt. Soc. Amer.
for a discontinuity of Michigan
High-frequency for currents
~,
in curvature, Radiation
backscattering in the penumbra
from
