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Oct. 16, 1962
R. H. DU HAMEL ETAL
3,059,234
LOGARITHMICALLY PERIODIC ANTENNA ARRAY
Filed Sept. 21, 1959
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Filed Sept. 21, 1959
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United States Patent 0 ’ "ice
1
3,059,234
Patented Oct. 16, 1962
2
point on any given tooth in a speci?c radial member bears
3,059,234
a constant ratio 1- to the radial distance from the vertex
LOGARITHMICALLY PERIODIC ANTENNA
ARRAY
Raymond H. Du Hamel and David G. Berry, Cedar
to the corresponding point on the next adjacent tooth
which is farther removed from the vertex than said given
tooth. In the most general case where each antenna
element employs two radial members lying in the same
plane, the teeth of one radial member are positioned
Rapids, Iowa, assignors to CoHins Radio Company,
Cedar Rapids, Iowa, a corporation of Iowa
Filed Sept. 21, 1959, Ser. No. 841,391
22 Claims. (Cl. 343-795)
This invention relates generally to antenna arrays and
more speci?cally to frequency independent end ?re an
opposite the gaps of the other radial member.
An object of the present invention is to provide an
antenna array capable of providing a radiation pattern
which will remain constant over large changes in fre
tenna arrays employing logarithmically periodic antennas
quency.
as described in patent application Serial No. 721,408,
Another object of the invention is an antenna array
whose impedance remains substantially constant over
?led March 14, 1958, by Raymond H. Du Hamel and
Fred R. Ore entitled “Logarithmically Periodic Antenna” 15 large changes of frequency.
and patent application Serial No. 804,357, now Patent
A third object of the invention is to provide a reliable
No. 2,989,749, ?led April 6, 1959, by Raymond H. Du
Hamel and David G. Berry and entitled “Unidirectional
and relatively simple antenna array employing logarith
mically periodic antenna elements in which the impedance
Frequency-Independent Coplanar Antenna,” both of
and the radiation pattern remain substantially constant
which are incorporated by reference as a part of this 20 over large changes of frequency.
speci?cation.
Another object of the invention is the improvement of
antenna arrays generally.
In accordance with the invention there is provided
some even number of logarithmically periodic antenna
arrangement of the rungs in a ladder to form an array. 25 elements arranged either in a coplanar or nonplanar rela
tionship or some combination of coplanar and nonplanar
The pattern for any given frequency is determined by
relationship with respect to each other. In order to main
the spacing between the various individual antennas, said
spacing being measured in wavelengths. It is well known
tain substantially constant pattern and impedance over
that as the frequency changes the wavelength spacing
large frequency changes it is necessary that the locations
of the elements with respect to each other be de?ned by
between the various antennas will change, thus causing
angles rather than distances. By de?nition the foregoing
a change in the radiation pattern and also producing a
change in antenna impedance. The aforementioned
sentence means that all of the elements of the array will
have their vertices (or feed points) at a common point.
characteristics, i.e., change of impedance and radiation
pattern with frequency, exist in all known end ?re an 35 The particular radiation pattern desired can be obtained
by selecting the proper values for a, 5, 3b, and g, and 7-,
tenna array systems. Now, it is possible, by employing
as de?ned in the aforementioned copending application,
a large number of antennas, and ‘by switching from one
group of antennas to another group of antennas as the
Serial Number 721,408.
frequency shifts over wide ranges, to maintain a some
Due to the fact that the vertices of the individual ele
In the prior art there are known many forms of end
?re antenna arrays. For example, a group of dipoles
can be arranged parallel to each other similar to the
what constant pattern and a somewhat constant imped 40 ments of the array meet at a common point, the phase
ance. The degree of constancy in impedance and pattern
centers of the individual elements of the array will lie
will be dependent upon the complexity and number of
in an are about said common vertex and will not lie in
antennas employed. The higher the degree of constancy
a straight line, thus preventing the formation of a plane
‘wave front. The foregoing sentence assumes that all of
desired, the greater the complexity will be. Such a pro
cedure for obtaining a wide frequency bandwidth an—
the antenna elements have the same construction and are
tenna system is quite expensive, quite space consuming,
fed by signals having the same phase.
and in many cases is not practical. It is evident that it
It is to be noted that arrays can be formed to produce
would be quite desirable to have an antenna array ca~
horizontally polarized ?elds, vertically polarized ?elds,
pable of producing a highly directive beam pattern which
or elliptically polarized ?elds, or the special case of cir
would remain constant over large frequency changes and 50 cularly polarized ?elds. In the case of the elliptically
having an impedance which would also remain substan
or circularly polarized ?elds the antenna arrays can be
tially constant over large frequency changes.
composed of a plurality of antenna elements each of
Recently a new-type antenna such as described in the
which has more than two radial elements connected to
above-identi?ed patent applications 721,408 and 804,357
a single central conductive member. Alternatively, el
liptically or circularly polarized ?elds may be generated
has been developed. This new-type antenna known gen
erally as a logarithmically periodic antenna is comprised
of individual antenna elements, each element being gen
erally triangular in shape and having a vertex and side
elements de?ned by an angle a. More speci?cally, each
element is comprised of at least two radial sections 60
de?ned on one side by the center line of the antenna
element and on the other side by a radial line extending
from the vertex at \an angle
0.
2
Each of these radial members is comprised of a plurality
of teeth which are all similar to each other in shape but
by a plurality of groups of two pairs of individual an
tenna elements having a common vertex and arranged
in quadrature.
'
An important feature of the invention is the fact that
all of the linear dimensions (i.e. radial distances meas
ured from the vertex) of an antenna element may be
either shrunk or stretched by a constant factor to pro
duce a change in the phase relationship between the signal
at the feed point and a signal‘ at the phase center of
65 the element without changing the distance of the phase
center of the element from the vertex. Thus, by proper
ly stretching or shrinking an antenna element or antenna
group the phase of the radiated signal therefrom can be
altered to compensate for the difference in phase with
sively farther apart as the distance from the vertex in 70 respect to the radiated signal of another element or group
creases. The above relationship may be expressed by
so as to produce a common phase at a given reference
stating that the radial distance from the vertex to any
plane representative of the electric ?eld in space.
which become progressively larger and spaced progres
3,059,234.
4
The above-mentioned and other features and objects
vidu‘al element, disregarding the interaction with adjacent
of the invention will be more fully understood from the
elements. In the particular embodiment of the invention
shown in FIG. 1, if a plane 116 is drawn perpendicular
to the bisector of the angle between elements 110 and
115, the phase centers of the elements 110, 111, 112, 113,
114 and 115 would, in the absence of stretching or shrink
ing of certain of the elements, be at different normal dis
tances from the line 116. More speci?cally, the phase
detailed description thereof when read in conjunction
with the drawings, in which:
FIG. 1 shows a perspective view of a nonplanar array
of six logarithmically periodic antenna elements;
FIG. 2 shows a schematic diagram of logarithmically
periodic antenna elements arranged in an array with
vertices of the various elements meeting at a substantially
common point;
FIGS. 3a
terns of six
FIG. 4 is
beam width
centers of elements 112 and 113 would be farther from
10 the line 116 than would be the phase centers of the ele
through 3]‘ show predicted vs. measured pat
elements;
a chart showing the relationship radiated of
vs. T, both for the H plane and the E plane
ments 111) and 115 or 111 and 114. Under certain cir
cumstances this ‘would be an undesirable situation. For
centers of single antenna elements as 1 varies for different
elements in a manner to be described in detail later herein.
example, if it is desired to generate a plane radiation pat
tern in the H ?eld (a plane formed by the center lines
where 1 is the ratio of the radial distance from the vertex 15 of the elements 110 through 115) the phase of the signal
radiated by each element should be the same as it reaches
as de?ned in the above-mentioned copending applica
the plane 16. Such a phased array can be accomplished
tion, Serial Number 721,408;
by shrinking or stretching various ones of the antenna
FIG. 5 is a chart showing the variations of the phase
20 For the present the characteristics and de?nitions of an
antenna con?gurations;
individual logarithmically periodic antenna will be dis
FIGS. 6a, 6b, ‘and 6c, show a plane view of three of
the six antenna elements of FIG. 1 and illustrate the re
cussed brie?y.
lationship of the dimensions of the three elements;
In FIG. 1, RN is the distance from the vertex t0 the
wire element 119 of element 110, rN is the distance from
center of an antenna element vs. changes of frequency; 25 the vertex to the wire element 120 of element 110, RNH
FIG. 8 is an elementary diagram of an antenna element
is the distance from the vertex to wire element 121 and
showing one of the dimensions multiplied by a stretching
rN+1 is the distance from the vertex to the wire element
FIG. 7 shows a chart of the relationship of the phase
factor k;
122. The ‘following ratios exist:
FIG. 9 is a chart showing the relationship between
the phase delay of an antenna element as it varies with 30
R11
different stretching factors;
FIGS. 10a through 10d show the predicted and the
measured H and E plane radiation ?elds for a six-element
11.
and
n.
phased array at two dilferent frequencies;
Rn
FIG. 11 show a pair of coplanar elements which may 35
The
angle
a
de?nes
the
side
elements such as sides 123
be utilized with other pairs of antenna elements to form
an array;
of the wire forming the shaped con?guration. To obtain
structural symmetry of the element, cr=\/F:
FIGS. 12a through 12d show radiation patterns of a
The antenna element such as antenna element 110 (FIG.
two-element coplanar array such as shown in FIG. 11;
FIG. 13 is a chart showing half-power beam width vs. 40 1) when fed against another similar antenna element such
as element 111 will have a natural tendency to produce
the angle of separation of the center lines of two-element
a radiation pattern off the end of the ‘antenna element.
coplanar arrays;
Such radiation ?eld or pattern is ordinarily de?ned as
FIG. 14 is a schematic sketch of a four-element array
having an H plane and an E plane. The H ?eld lies in
of identical elements;
FIGS. 15a through 15d show E and H plane radiation 45 the plane perpendicular to the individual transverse ele
ments such vas elements 122 and 121 of antenna element
patterns of the ‘four-element array of FIG. 14 for two
110 and passing through the central conductive member
different frequencies;
such as member 136. The E ?eld is the ?eld lying in the
FIG. 16 shows a six-element array of identical ele
plane parallel to transverse elements such as elements
ments;
FIGS. 17a through 17d show the radiation patterns for 50 122 and 121 and passing through the bisector of angle
formed by the end antenna elements 110 and 115. An
the H plane only of the array of FIG. 16 for four differ
other term frequently used in connection with logarith
ent frequencies;
FIG. 18 shows a ten-element array which can produce
an electrically steerable radiation pattern;
FIGS. 19a through 19d show radiation patterns in the
H plane for the steerable array of FIG. 18;
FIGS. 20, 21, 22, and 23 show various types of an
tenna elements for logarithmically periodic antenna arrays
which may be employed in the various arrays described
mically periodic antenna elements are “image elements”
and “non-image elements.” Image elements are de?ned
as a pair of elements which are positioned so as to be
substantially mirror images of each other. Non-image
elements exist when one element of a pair of image ele
ments is rotated about its center line 180”. In the struc
ture of FIG. 1 antenna element 110 is ‘a non-image of
60 element 111.
herein;
Throughout much of this discussion of antenna arrays
FIGS. 24, 25, 26, and 27 shows antenna element con
the particular type antenna element employed as an ex
?gurations which can be employed to generate elliptically
or circularly polarized radiation patterns; and
ample will be the type shown in FIG. 1, i.e., the type hav
ing rectangularly shaped teeth formed of a wire or a rod.
FIG. 28 shows the means for feeding a logarithmic
65 It is to be understood, however, that other type antenna
periodic antenna ‘array with a coaxial cable.
elements such as the antenna elements shown in FIGS. 20
Referring now to FIG. 1 there is shown a six element
through 27, and others, may be substituted freely for the
type depicted in FIG. 1.
114, and 115. Each of these elements feeds against the
In the following paragraphs the theory and element
adjacent elements. More speci?cally, element 110 feeds
against element 111, element 111 feeds against elements 70 characteristics of an array will be discussed in some de
tail. Subsequently, the specific method for construction
110 and 112, 'and element 112 feeds against elements 111
of an array to produce a desired pattern will be discussed.
and 113, etc. Although there is some interaction between
Considering now the general theory of an array using
adjacent elements due to the fact that they are not image
logarithmically periodic antennas, reference is made to
elements, the resultant array can be calculated with a
high degree of accuracy on the contribution of each indi 75 FIG. 2 which shows an array of end ?re elements. All
coplanar array comprised of elements 110, 111, 112, 113,
3,059,234
6
5
of these elements have their vertex (or apex) at a com
mon point 100. It is to be noted speci?cally that the '
antennas are not electrically connected at this common
point 100. In actual practice the antennas are terminated
short of their vertex so that actual connections between
the various elements are not thereby effected. The odd
numbered antenna elements are supplied from one wire of
tion characteristics of a single element.
It has been
found that this di?iculty can be circumvented by feeding
a logarithmically periodic element vagainst a vertical wire
which is perpendicular to the teeth of the logarithmically
periodic element and which is connected to the center
conductor of a coaxial cable, which cable forms the cen
ter line of the logarithmically periodic antenna element.
Although the input impedance of the element is no
a transmission line and the even-numbered elements are
longer frequency independent, the patterns are very simi
supplied from the other wire of a transmission line. In a
general sense it will be observed in FIG. 2 that the result 10 lar to the patterns of the element when placed in an ar
ray. Since the vertical wire radiates vertical polarization
ant radiation pattern is controlled by three principal fac
in the E ?eld, it is possible to measure the principal plane
tors. The ?rst of these factors is the radiation pattern of
horizontal polarization patterns (in the E ?eld) of a peri
each individual element. The means for determining the
odic element alone. This technique can also be em
radiation pattern of each individual element will be dis
cussed later. The second factor is the fact that each of 15 ployed to determine the phase center of a single element.
Sample patterns for various values of the parameters
the elements radiates in a different direction from every
a and 'r are shown in FIGS. 3a through 3f. These are
other element. More speci?cally, each element radiates
relative ?eld intensity patterns. The main beam in each
in a direction determined by its angle 6 which results in
case represents the end ?re characteristic of the single
the phase centers of the various elements to beat different
distances from a plane parallel to the direction of radia 20 element. The graph of FIG. 4 summarizes the pattern
data taken on the various types of elements. It is to be
tion. The phase center is de?ned as the apparent point
noted that the E plane beam widths are relatively insensi
from which radiation is originating. ‘For example, as
tive to change in 1' but that the H beam widths generally
suming the Z coordinate to be perpendicular to the plane
decrease with increasing 1'. The graph of FIG. 7 will be
of the drawing of FIG. 2, the phase centers of elements
1 and N will be closer to the XZ plane than the phase 25 employed in a manner to be described later in selecting
the constants to be used in constructing an array having
centers of elements 2 and N-l. It is apparent that if it
is desired to create an array that will radiate a directive
a desired radiation pattern.
beam symmetrically about the Y axis, for example, ‘it
would be desirable to have the phases of all of the ele
The phase centers of the elements can be determined
by mounting the elements on a vertical rotating mast and
measuring the phase of the received signal at a distant
antenna. The center of rotation of the element is ad
justed so that the phase variation over a 60° sector in the
ments the same as they pass through any plane perpendic
pular to the direction of radiation such as, for example,
X-—Z plane. By a procedure to be explained in detail
direction of radiation was minimum. It was found that
later the phases of ‘signals radiated by elements 2 and
the distance d, that is the wavelengths to the vertex from
N-l can be caused to lead the phases of the signals radi
ated by elements 1 and N by an amount equal to the dis 35 the phase center, was essentially independent of 1- but quite
dependent upon a. The results of the immediately afore
tance I measured in wavelengths. The distance I, as
mentioned tests are shown by the curves of FIG. 5.
measured in wavelengths, remains constant with a change
,The phase center position of various logarithmically
in frequency due to the inherent characteristics of log
periodic elements was measured over a period of fre
arithmically periodic antennas. More speci?cally, as the
quency. A typical result is shown in FIG. 7. Since d is
phase centers move toward the vertex with signals of
proportional to the wavelengths within the accuracy of
higher frequency, the distance I will decrease in actual
the measuring equipment, it can be implied from the
distance but will remain the same as measured by wave
curve that the phase center does not shift when a logarith
lengths which become shorter as the frequency becomes
mically periodic element is expanded or contracted, by
The general expression for the radiation pattern of the 45 some constant K. Worded in another way it can be
stated that the phase center is substantially independent
array shown in FIG. 2 is given ‘by
higher.
of the number of teeth between said phase center and the
vertex (within certain limitations) as long as 'r and a re
main constant.
1
l
l
where f(¢—6n) represents the radiation pattern con?gura 50 The concept of shrinking or stretching an antenna ele
ment is a rather important one in the consideration of
tion of each ‘element and where the portion at cos (qt-5n)
antenna arrays employing logarithmically periodic an
of the exponent represents the phase advance of the phase
center relative to the origin 100, the values d, <15, and 6
tenna elements.
To clarify the concept the following
analogy might be useful. Assume the antenna element is
being indicated in FIG. 2. The value of the feed-point
voltage for the nth element is given by An. The parame 55 composed of a spring wire formed in the shape of an ele
ment shown in FIG. 1 and fastened securely to some ?xed
ter 7,, is the relative phase of the ?eld radiated from the
point at the vertex. Now, if the end of the antenna op
n‘:11 element. More speci?cally, 7,, is the change in phase
posite the vertex is moved away from the ?xed vertex, the
introduced into any given element by stretching or ‘shrink
antenna will be stretched, i.e. every point in the antenna
ing that element so as to produce a desired phase relation
ship between the ?elds radiated by the various elements. 60 will move out from the apex radially by a constant factor
K. Conversely, if the antenna is compressed back toward
-As indicated hereinbefore, shrinking or stretching an ele
the apex it is saidto be shrunk and every point in the
ment will result in a phase shift of the radiated signal with
antenna is moved back toward the apex by a constant
respect to the phase of the input (or feed) signal.
factor K. In order for the foregoing analogy to be valid
The assumptions made in Equation 1 are that the element
patterns and input impedances are identical. Although 65 two assumptions must be made. The ?rst assumption is
that the bending point of the spring must not be exceeded.
mutual effects can introduce some error into these assump
The second assumption is that if a spring were in fact
tions, good correlation between theory and experiment
streched as described above, the angle determined by the
has been obtained. As indicated above, it is necessary, in
outer edges of the spring would vary, decreasing with
the construction of an array, to determine the radiation
pattern of each individual element used therein. The 70 stretching and increasing with compression. In the case
of stretching or compressing an antenna this angle a must
radiation pattern of a single element will depend primar
ily upon the design parameters a and '1'. Since it is neces
remain the same. Consequently, it can be seen that the
analogy of the spring is not a completely accurate anal
sary to feed two logarithmically periodic elements against
ogy. An ‘additional point to be noted is that a stretched
each other in order to obtain frequency independent op
eration, it would appear di?icult to determine the radia 75 or shrunk antenna should have the same over-all length
3,059,234
8
as another antenna which is stretched or shrunk in a dif
?re directivity of the logarithmically periodic antenna ele
ferent degree.
ments tends to enhance the effective aperture. The dis
tance between the phase centers of the two outer elements
(see outer elements 1 and N of FIG. 2) must be approx
imately D. Experimental tests indicate that a reasonable
However, even though the phase center of an element
does not vary when stretched or shrunk it has been found
experimentally that the phase of the radiated signal at the
phase center will vary with respect to the phase of the
input signal fed to the vertex. This characteristic of
logarithmically periodic antennas is de?ned as the phase
rotation phenomenon. It has been veri?ed experimen
maximum spacing between the phase centers of adjacent
element is 0.7 wavelengths. Thus the number of ele
ments may be determined approximately from
tally that when an element is shrunk or stretched through 10
one complete period, the phase of the signal would be
advanced or delayed 360°.
In FIG. 8 the distance to an element is given by KRN.
Assume the desired bandwidth BW is 11.4° the number of
elements N is then equal to 6, and D is equal to 3.5 wave
The expansion through a period is accomplished by let
lengths.
ting K increase from one to 1/ 7'. During this expansion
One of the limitations of the design is that the maxi
mum value of the angle (Bn—61) (see FIG. 2) be less
all lengths involved in the structure are multiplied by K.
In FIG. 9 the phase delay in radians is plotted versus the
logarithm of K to the base 7‘. The ideal phase variation
is given by the solid straight line. Measurements have
indicated that the actual phase variation is somewhat like
the dashed line. The measurements made to date indi
cate that the deviation of the dashed line from the
straight line is not more than 20°.
It is to be noted that the phase of a signal will be ad
vanced 360° when the structure is shrunk through a com 25
than the half power beam width of an individual element.
This can be understood more clearly when it is realized
that the various elements radiate toward the common ver
tex at diiferent angles with respect to the plane normal to
the direction of radiation. If the angle ( (Sn-61) is greater
than the element beam width, then energy radiated in the
desired direction from the end elements will be less than
half of its maximum radiated energy, and will contribute
relatively very little to the desired pattern of radiation.
plete period. Somewhat different expressions are used to
It has been found experimentally when the radiation pat
de?ne shrinkage and stretching. For shrinkage the ex
tern de?ned by the half power limits lays entirely outside
pression for K in term of y is:
the central direction of the desired main ?eld, that the
1‘
30 contribution of said element is primarily in the produc
K: T 21r
tion of side lobes.
Further consideration will now be given to determine
where 7 equals the amount of phase delay in radians.
the factors controlling the angle 6n—61. It will be ap
For stretching an element the relationship between K
parent that the larger 04 is made, the shorter the antenna
and 'y is as follows:
35 element will have to be to cover a desired bandwidth.
However, certain limitations on the size of a exist as
follows. For any given frequency the wavelength 7\ will
be measured in some distance such as centimeters or
where 7 equals the phase advancement in radians. The
inches. In FIG. 2 the aperture D is represented by the
phase center and the radiation patterns are substantially
independent of the expansion or contraction of a logarith 40 letter D and the difference between phase centers is indi
mically periodic element provided that on and 1- remain
cated as 0.7%. It will also be noted that A bears a de?nite
unchanged.
relationship to the transverse dimension of an individual
element at the phase center. Such transverse dimension
is not equal to
The information that has been supplied above is suffi
cient for predicting the pattern of an array of similar end
?re elements with the only difference between individual
elements being the scale factor K. The method for pre
dicting a pattern of an array could be generalized to in
clude arrays of elements with different a’s and possible
different r’s. However, if different 1-’s are used, it would
be necessary that the logarithm of any 1- to the base of
any other 7- be a i integer, i.e., 11:11“. Also, if differ
ent a’S are used it is necessary to take into account the
relative phase of the radiated ?eld compared to the phase
of the feed point current of the various logarithmically
periodic elements.
but is equal to
where K is usually less than unity as indicated in the
curves of FIG. 5. It has been found experimentally that
the phase center does not lie at the half wavelength point
of the antenna element. However, even though k varies
55 somewhat as a varies, the variation of a is greatly pre
Design Procedure
In designing an array of antennas employing’ logarith
mically periodic antennas the designer should ?rst deter
mine the radiation pattern which is desired. Then, a
judicious choice of the parameters and u, 1-, and 8N will
have to be made so as to achieve a minimum amount of
space and material to produce the desired array. It is to
be noted that although the design method to be described
infra is a cut and try method, a fair approximation may
be obtained thereby. The procedure is the same for ar
rays in the E plane and in the H plane.
Given a desired beam width the equivalent aperture D
may be calculated from the expression
2__4O_
x _BW
where BW is the half power beam width in degrees. The
number 40 is employed instead of the number 50 (as
dominating. For example, if or is increased, the phase
center will move closer to the vertex (k will change only
a little), thus necessitating an increase in the angle
(an-a1) if the number of elements used and the wave
length spacing therebetween is to remain constant. Thus
it can be concluded that as on is increased, the end ele
ments of the array, i.e., elements 1 and elements N will
tend to contribute less and less to the main ?eld. How
ever, by referring to FIG. 4 it can be seen that if 7- is de
creased, the half power beam width of the element is
increased, thus tending to compensate for the increased
angle (§n—51) necessitated by an increase in the angle a.
Further, if 7' is decreased, the amount of material required
70 for an antenna element will be decreased. However, 1
cannot be decreased to too small a value or the radiation
pattern of each element will tend to break up. From the
foregoing discussion it can be seen that it is desirable to
construct the antenna elements with as large an a and. as
is normally used in this design formula) because the end 75 small a 'r as permitted by their limitations.
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