Log-Period Dipole Array Optimization
You Chung Chung
University of Nevada, Reno
Electrical Engineering
Reno, NV 89557
775-784-6927
youchung @unr.edu
Randy Haupt
Utah State University
Electrical and Computer Engineering
4 120 Old Main Hill
Logan, UT 84322-4120
435-797-2840
haupt (@
ieeo.org
Abstract-The log periodic dipole array is a commonly used
broadband antenna. Design equations exist for calculating
the size and spacings of the wires that make up the antenna.
These equations fail to take into account the electromagnetic
interactions between the wires, though. The Numerical
Electromagnetics Code is an excellent wire modeling code
for analyzing the full electromagnetic interactions of a wire
antenna. In this paper we use the Nelder-Mead downhill
simplex algorithm and a genetic algorithm to optimize the
design of a log periodic dipole array. Modeling results show
that a hybrid genetic a1gorith"elder-Mead
downhill
simplex algorithm produce superior antenna designs to
traditional analytical approaches or using the genetic
algorithm or Nelder-Mead algorithm alone.
TABLE OF CONTENTS
1.
2.
3.
4.
5.
INTRODUCTTON
COMPUTERMODEL OF LPDA
NUMERICAL
OPTIMIZATION AF'PROACH
Rl5SULTS
CONCLUSIONS
I
Figure 1. LPDA geometry.
The LPDA was introduced by Isbell [I], and improved using
techniques shown in references [2-51. The performance of a
LPDA is a function of number of elements as well as
element length, spacing, and diameter. Antenna element
lengths and spacings have proportionality factors given by a
scale factor
1.INTRODUCTION
The log-periodic antenna finds use in many broadband
communications applications. Antenna components are
proportional in size to each other and to the wavelength. Its
geometry forces the antenna impedance and radiation
properties to repeat periodically as the logarithm of
frequency. These antennas come in many different shapes,
including a wire dipole construction called the log-periodic
dipole array (LPDA) shown in Figure 1. Increasing the
bandwidth of the antenna requires adding more antenna
elements.
I t
2=- Ll,+1
L"
and spacing factor
where, the L,, is the length of nth element, and d, is the
element spacing between nth and (n+l)*. The longest
element is approximately a half wavelength at the lowest
frequency. Similarly, the shortest element is approximately a
half wavelength at the highest frequency. This antenna is fed
with a voltage source at the high frequency end. Figure 1
shows a uniformly spaced LPDA with an angle, a,which
0-7803-5846-5/00/$10.00 0 2000 IEEE
449
bounds the dipole lengths. The angle a is calculated from z
and G by
cx=2tan -1 I - r
(z)
in order to model a variety of antenna structures. The
antenna model can have nonradiating networks and
transmission lines connecting parts of the structure, perfect
or imperfect conductors, and lumped element loading.
At low frequencies, the larger antenna elements are active.
As the frequency increases. the active region moves to the
shorter elements. When an element is approximately one
half wavelength long, it is resonant.
Reference [6] has a plot of the gain of a well-designed
uniformly spaced LPDA as a function of z and G. The
voltage standing wave ratio is given by,
vswR =
(1 + IPI)
(4)
Figure 2. Wire segments for NEC.
where, p is the reflection coefficient given by
An integral equation formulation works best for antennas
Z, = antenna impedance
2, = transmission line impedance
Most antenna books give equations for the design of logperiodic antennas. Unfortunately, these design equations do
not include the complex interactions between the dipoles
that compose the antenna. Consequently, a numerical model
and numerical optimization of that model are important for
developing realistic designs.
2. COMPUTER MODELOF LPDA
Wire antennas, like the log-periodic dipole array (LPDA),
are accurately modeled with the Numerical Electromagnetics
Code (NEC). This FORTRAN code is widely used and
forms the basis for commercial and noncommercial wire
modeling codes. The geometry of the log-periodic antenna is
specified in terms of wire segments. A method of moments
(MOM) solution [7] is used to find the antenna impedance
as a function of frequency and the antenna gain as a function
of angle and frequency. Each wire is broken into segments
as shown in Figure 2. The locations of the segments are
specified in an input file that the program reads. Using the
geometry and induced voltage source, the currents are
calculated for each of the wire segments. Once the currents
are known, the program calculates the antenna impedance
and radiation characteristics.
NEC numerically solves integral equations for the currents
induced on antennas by voltage sources. This code combines
an integral equation for smooth surfaces with one for wires
450
with dimensions of up to several wavelengths. Beyond that
size, the MOM requires a rather large matrix equation, since
the matrix order increases relative to a wavelength. Hence,
modeling very large structures may require more computer
time and file storage than is practical on a particular
machine. In such cases high-frequency codes based on
approximations such as geometrical optics, physical optics,
or geometrical theory of difhction are more appropriate.
The NEC program uses an electric-field integral equation
(EFIE) and a magnetic-field integral equation (MFIE). The
EFIE works best for thin-wire structures of small conductor
volume while the MFIE (doesn’t work for the thin-wire
case) works best for voluminous structures with large
smooth surfaces. The EFIE can be used to model very thin
surfaces. Although the EFIE is specialized to thin wires in
NEC, it is frequently used to represent surfaces may be
modeled by wire grids with reasonable success for far-field
quantities but with variable accuracy for surface fields.
Each wire model of the LPDA is defined by the (x, y, z)
coordinates of its two end points and its radius. A wire
segment is defined by the coordinates of the element
containing the segment and the number of segments in the
element. A segment should be less than O.lh and larger than
0.001h [SI. The center of the last element is excited. In
addition, the 50 ohm crossed line transmission line is
connected between the center segment of each element, so
there is always an odd number of segments.
3. NUMERICAL
OFTIMEATION
APPROACH
Our numerical
routines are written in
a Nelder-Mead local
MATLAB. A GA andor
algorithm couples the optimization routines to a compiled
version of the NEC2 code. The algorithms vary the element
length, spacing and radius in order to optimize the gain and
VSWR. A Nelder-Mead downhill simplex local optimization
method and a GA are well-known optimization tools, and
they are described well in the references [9] and [lo]. The
Nelder-Mead method is executed after the GA finds a
reasonably good design for the LPDA. The flow chart of the
antenna design with a genetic algorithm & Nelder-Mead is
shown in Figure 3.
The initial population is evaluated by NEC. Chromosomes
in the population are ranked from the best fit to the worst fit
based on their costs. The bottom 50% of the initial
population is discarded, and the remaining top half is called
the parents. Two partners are randomly selected from the
parents, and they mate. A random number, p, and a random
crossover point are selected. The variables at the right side
of the random crossover point are swapped, and the
variables at the crossover point are calculated with p. The
sample of a chromosome and the parents having variables
for an LPDA, and their mating and offspring are shown as
follows in Figure 4.
mom1 = [L,,,I ... L, dmlIdm2i
dm3... d,.l RmI... R],
dad 1 = [Ldl ... Ld, ddi ddi dd3 ... ddn-l&l ... &,,I
JN
Figure 4. Continuous chromosome mating with a random
crossover point in a GA.
GA optimization
The total number of new offspring is the same as the number
of discarded population members. A small percentage of
variables in the new offspring and their parents are randomly
switched. This process is called mutation, and a high
percentage of mutation gives more freedom to search new
LPDA designs.
There is no mutation of the best
chromosome on each generation. The new population is
evaluated, and these processes repeat until the GA satisfies
the results or reaches a given maximum generation.
I Local optimization I
I
Great design
Figure 3. Flowchart of the computer model.
The GA begins by creating an initial random population of
continuous parameters for element size, spacing and radius.
This set of variables for a LPDA is placed in a vector called
a chromosome. The values of the variables are limited to
those that are physically feasible for a LPDA. A
chromosome is shown in equation (6), where LI and RI are
the length and radius of the lstelement, respectively, and dl
is the spacing between the lstelement and the 2ndelement.
chromosome=[Ll L2 . . . Ln dl d2 . . . dn-1 Ri R2 . . . Rn1 (6)
The cost function of the adaptive genetic algorithm linked
with NEC evaluates the gain and VSWR corresponding to
the element size, spacing and radius settings. The cost
function includes the mean and standard deviation of the
gain and the VSWR of the LPDA, and it is shown in
equation (7), where a, b, c and d are constants determined by
the user.
Cost = -axMEAN( Gain(dB)) + bxSTD( Gain(dB)) +
cxMEAN{VSWR) + dxSTD{VSWR)
(7)
For the hybrid GA and Nelder-Mead downhill simplex
method, the parameters optimized by a GA are transferred to
a Nelder-Mead local optimization method, and single or
multiple runs of the Nelder-Mead algorithm refine the
solution.
4.RESULTS
The GA, Nelder-Mead, and hybrid GA & Nelder-Mead
methods linked with NEC are applied to three LPDA
designs. The first design is a five element LPDA (LPDAl)
with a frequency range of 300 to 400 MHz. The second
design is a 7 element LPDA (LPDA2) with a frequency
range of 800 to 1600 MHz. The third design is a 20 element
LPDA (LPDA3) with a frequency range of 200 to 1300
MHz. The mean and standard deviation of gain and VSWR
are included in the cost function in equation (7). The
constants in equation (7) are different for each LPDA
design. The results of the 3 designs-lement
lengths,
spacings, radii, gain and VSWR-are
shown in the
following subsections.
LPDAl: 5 elements over 300 to 400 MHz
45 1
The 5 element lengths and radii and 4 spacings (14
variables) for LPDAl are optimized with the 3 methods. The
constants of equation (7) are a=1.5, b=l, c=1.5, and d=l for
the CA.
The parameters of the GA are initial
population=48, discard rate=0.5 and mutation rate =5%. The
maximum number of function evaluations is 5000 for all 3
methods.
The 7 element lengths and radii and 6 spacings (20
variables) for LPDAl are optimized with the 3 methods. The
constants of equation (7) are a=l, b=l, c=2, and d=l for the
GA. The parameters of GA are initial population=48,
discard rate=0.5 and mutation rate =5%. The maximum
number of evaluations is 6000 for all 3 methods.
11,
The gain and VSWR are sampled every IOMHz. The
element lengths, spacings, radii, mean and standard
deviation of the gain and VSWR over the frequency range
are shown in Table 1. The average gain over the frequency
range improved from 7.4561dB to 8.691dB (Nelder-Mead),
8.4285 (GA), and 8.8126 (hybrid). In addition, the average
VSWR of LPDAl improved from 1.90 to 1.1585 (NelderMead), 1.3336 (GA), and 1.1278 (hybrid). The boom length
of initial the design is 0.5937m, and the boom length of the
optimized LPDAs by Nelder-Mead, GA and hybrid are
11.I%, 39% and 11.4% larger than the initial boom length.
The initial and optimized gain and VSWR vs. frequency are
shown in Figures 5 and 6. The highest VSWR over the
frequency range for Nelder-Mead, CA and CA & NelderMead are 1.24, 1.7 and 1.23, in that order.
Table 1. The 5 element LPDAl for 300 to 400MHz:
Element lengths, spacings and radii, and gain & VSWR.
105
.
*
10
I
I'
Initial gain
Gain by GA
Gain by Nelder
Gain by GA & Nelder
0
+
95
m
U
;
+
9,;
*
*
8
851.
**
*
.
*
i
8-
_,
*
&
.
*
4
320
340
360
380
400
Frequency in MHz
1;
I
Figure 5. Gain of 5 element LPDAl for 300 to 400 MHz.
2.0-
-
2.6
0
Initial VSWR
VSWR by GA
VSWR by Nelder
VSWR by GA & Nelder
242.21
0
0
Nelder
R1
L2
R2
L3
R3
0.00041
0.4860
0.00032
0.4374
0.00032
L4
0.3937
R4
0.00032
0.3543
L5
0.00032
R5
0.1728
SI
s2
0.1555
s3
0.1397
s4
I 0.1257
Gain
7.4561
STD(gain), ,I 0.7610
VSWR
1 1.900
STD(VSWR) I 0.4129
I
\"
I
I
I
0.0058
0.3794
0.0040
0.3424
0.0072
0.3826
0.0009
0.3774
0.0001
0.090
0.1750
0.1997
0.1953
8.6891
0.2983
1.1585
0.0899
I 0.0050
I 0.2168
0.0050
0.1998
0.0050
0.1914
0.0050
0.1846
0.0048
0.2500
0.2266
0.1521
1 0.2008
I 8.4285
I 0.3974
1.3336
0.1821
0.0054
0.1929
0.0053
0.1778
0.0021
0.1687
0.0054
0.1827
0.0014
0.1290
0.1382
0.1579
I 0.2363
I 8.8126
I 0.3966
1.1278
0.0851
do0
452
340
360
380
400
Frequency in MHz
Figure 6. VSWR of 5 element LPDAl for 300 to 400 MHz.
I
The GA & Nelder method gives best gain, VSWR, standard
deviation of gain and VSWR compared to the results of
other methods, while Nelder-Mead method gives better
results than the CA for 5 element LPDA optimization.
LPDAZ: 7 elements over 8300 to 1600 MHz
320
The gain and VSWR are sampled every 80MHz from
800MHz to 1600MHz. The element lengths, spacings, radii,
mean and standard deviation of gain and VSWR over the
frequency range are shown in Table 2. The average gain
over the frequency range improved from 8.0673dB to
8.7606dB by the Nelder-Mead method, and to 8.0781dB by
a CA. In addition, the gain is improved to 8.9040dB with
low variation all over the frequency range by the hybrid
method. The VSWR of LPDA2 improved from 1.7974 to
1.1051 (Nelder-Mead), 1.0864 (GA) and 1.0114 (hybrid).
All are very closed to 1. The boom length of initial design is
0.2997m, and the boom length of optimized LPDAs by
Nelder-Mead, GA, and hybrid are 1.1%, 1.8% and 1.2%
larger than initial boom length. The initial and optimized
gain and VSWR vs. frequency are shown in Figures 7 and 8.
In Figure 7, the highest VSWR over the frequency range for
1
Nelder-Mead, GA and GA & Nelder-Mead are 1.23, 1.14
and 1.013, in order.
Table 2. The 7 element LPDA2 for 300 to 400MHz:
Element lengths, spacings and radii, and gain & VSWR.
Initial 7 element LPDA design. z=0.9 and o=O.17
I
I
I
Ontimized
281
~
26
VSWR by Nelder
VSWR by GA & Nelder
24
~
$
2 2c
I
1
~
~
~
0
*
0
*
*
:
320
*
340
*
*
*
360
*
380
,
400
Frequency in MHz
Figure 8. VSWR of 7 element LPDA2 for 800 to 1600
MHz.
1I
1
I
R6
L7
R7
s1
s2
s3
s4
~
I
I
~.
I
I
I
R5
L6
1
I
1
I
1
s5
S6
Gain
STD(gain)
VSWR
STD(VSWR)
I
I
1
I
I
I
0.004 I 0.0039 I 0.0054 I 0.0039
0.1107 0.0924
0.0950 1 0.0924
0.003 10.0027 10.0045 10.0027
0.0996 I 0.0934 I 0.0836 1 0.0934
0.003
0.0001 0.0017
O.OOO1
0.0638 I 0.0731 I 0.0607 1 0.0731
0.0574 I 0.0404 1 0.0571 I 0.0404
0.0516 0.0607 I 0.0516 I 0.0607
0.0465 1 0.0413 I 0.0542 0.0413
0.0428 I 0.0399 0.0346 1 0.0399
0.0376 0.0481 I 0.0471 I 0.0481
8.0673 1 8.7606 I 8.0781 I 8.9040
1.0326 I 0.3206 I 0.3363 1 0.2816
1.8232 I 1.1051 I 1.0864 I 1.0114
0.6935 10.0611 10.0415 IO.0110
I
I
I
1
I
I
Initial gain
Gain by GA
Gain by Nelder
Gain by GA & Nelder
*
+
P
*
:
*
.
O
.
I
I
0
*
I
+
*
1I
I
1
The hybrid method gives the best gain, VSWR, standard
deviation of gain and VSWR compared to the results of the
other methods, while the Nelder-Mead method gives better
results than the GA for 7 element LPDA2 optimization.
Table 3. The 20 element LPDA3 for 200 to 1300MHz:
Initial element lengths, spacings and radii, and optimized
gain & VSWR.
Initial 20 element LPDA desi n, -0.9 and 0=0.16
4
]
I
+
*
*
O
O
*
0.75000
0.67500
0.60750
4
0.54675
0.49207
5
0.44286
6
7
0.39858
I 0.35872
8
0.32285
9
10
I 0.29056
11
0.26151
I 0.23536
12
13
I 0.21182
14
0. I9064
0.17 157
15
0.15442
16
0.13897
17
I 0.12508
18
I 0.11257
19
20
1 0.1013 1
Gain / STD(Gain)
VSWR I STD(VSWR)
I
I
7.L-l
6.5
800
320
340
360
380
400
1
Frequency in MHz
Figure 7. Gain of 7 element LPDA2 for 800 to 1600 MHz.
I
I
I
I
I
1
I
1
0.24000
0.004
0.21600
0.004
0.19440
0.004
0.17496
0.004
0.15746
0.004
0.14171
0.004
0.12754
0.004
I 0.11479
I 0.10331
0.004
0.004
I 0.09288
0.003
1 0.08368
0.003
1 0.07531
0.003
I 0.06778
0.003
0.06100
0.003
0.05490
0.003
0.04941
0.003
0.04447
0.003
I 0.04002
0.002
I 0.03602
0.002
8.5017 dB 10.9707
1.4158 10.5462
~
I
J
LPDA3: 20 eleinents over 200 to 1300 MHz
The 20 element lengths and radii and 19 spacings (59
variables) for LPDA3 are optimized with the 3 methods. The
constants of equation (7) are a=l, b=l, c=2, and d=l for the
GA. The parameters of the GA are initial population=24,
453
~
discard rate=0.5 and mutation rate =lo%. The maximum
number of evaluation is 2000 for the GA, and 3300 for the
Nelder-Mead method. The Nelder-Mead requires more
number of iteration due to the large number of variables to
optimize. The gain and VSWR are sampled every 5OMHz
from 200MHz to 1300MHz. The gain, VSWR and element
lengths, spacings and radii of initial 20 element LPDA are
shown in Table 3. The optimized gain, VSWR, standard
deviation of gain and VSWR are compared in Table 4.
Initial
Design
Performance
Gain
STD(gain)
VSWR
STD(VSWR)
8.5017
I
I
I
0.9707
1.4158
0.5462
Optimized
Nelder-Mead I
GA
90118
I
0.5472
1.1495
0.1676
I
~~
1
I
9 9798
1
0.6646
1.2324
~ _
0.1462
I
I
~
_
The gain over the frequency range improved from 8.5017 to
9.01 18dB by the Nelder-Mead method, and to 9.9798dB by
a GA. The VSWR of the 20 element LPDA3 improved from
1.4158 to 1.1495 and 1.2411 by a Nelder-Mead and a GA,
respectively. The initial and optimized gain and VSWR vs.
frequency are shown in Figures 9 and 10. In Figure 9, most
of optimized gains over the frequency range by a GA are
larger than the initial gain, and the gain optimized by a
Nelder-Mead method. In Figure 10, the highest VSWR over
the frequency range for Nelder-Mead, and GA are 1.498 and
1.58, respectively. The optimized VSWR by both methods
are low compared to the initial VSWR. Figure 11 compares
the convergence speed of both methods. The convergence
speed of the GA outperforms that of Nelder-Mead. The GA
improves the gain of 20 element LPDA with reasonably low
VSWR quickly, compared to the Nelder-Mead method.
1 1/1
'7I
11
10 5
I
f
1
I
element, and seven element LPDAs are optimized with the
three methods, and the performance compared. The average
gain of the five element LPDA improved from 7.4561dB to
8.8126dB an average VSWR of 1.1278 over the frequency
range 300 to 400MHz. The seven element LPDA average
gain improved from 8.0673dB to 8.9040dB with an average
VSWR of 1.0114 over the frequency ranges 800 to
1600MHz. The GA & Nelder methods yield improved
designs over the analytical methods. The Nelder-Mead
method performs better than a GA for the five and seven
element LPDAs. The total number of variables for five
element LPDA and seven element LPDA are 14 and 20,
respectively, and the range of element lengths, spacings and
radii are limited for having a feasible size of antenna.
Therefore, the local optimization method, Nelder-Mead
downhill simplex method performs better than a GA for a
limited number of variables.
26
VSWR by GA
VSWR by Nelde
24
Frequency in MHz
Figure 10. VSWR of 20 element LPDA3 for 200 to1300
MHz.
I
Initial gain
Gain by GA
Gain by Nelde
-7
-8
Frequency in MHz
Figure 9. Gain of 20 element LPDA3 for 300 to 1300 MHz.
5. CONCLUSIONS
This paper presented results for optimizing a complex
antenna system with a GA, Nelder-Mead local optimization
method, and hybrid GA & Nelder-Mead method. The five
454
500
1000
1500
2000
Number of Evaluation
2500
3000
Figure 11. Cost vs. number of evaluation for 20 element
LPDA3 by a GA and a Nelder-Mead.
The twenty element LPDA is optimized with a GA and the
Nelder-Mead method to compare the performance of the
methods with a large number of variables. The total number
of yariables are 59, including 20 element lengths and radii,
and 19 spacings. The gain of twenty element initial LPDA
design improved from 8.5017dB to 9.01 18dB by NelderMead and to 9.9798dB by a GA. In addition the VSWR is
improved from 1.415 to 1.1495 and to 1.2324 with low
standard deviation 0.1676 and 0.1462, respectively. The GA
outperforms Nelder-Mead optimization method due to the
large number of variables.
In all cases, the hybrid GA & Nelder-Mead algorithm
worked better than either the GA or Nelder-Mead algorithm
alone.
REFERENCES
[l] D. E. Isbell, “Log periodic dipole arrays,” IRE Trans.
Anteiirza Propagat., AP-8, pp. 260-267, May., 1960.
[2] R. L. Carrel, “Analysis and design of the log-periodic
dipole antenna,” Ph.D. Dissertation, Elec. Eng. Dept.,
University of Illinois, 1961, University microfilms, Inc, Ann
Arbor, MI.
[3] G. DeVito and G. B. Stracca, “Comments on the design
of log-periodic dipole antennas,” IEEE Trans. Antenna
Propagat., AP-21, pp. 303-308, May., 1973.
141 G . DeVito and G. B. Stracca, “Further comments on the
design of log-periodic dipole antennas,” IEEE Trans.
Arztenna Propagat., AP-22, pp. 714-718, Sep., 1974.
[SI P. C. Butson and G. T. Thomson, “A note on the
calculation of the gain of log-periodic dipole antennas,”
IEEE Trans. Antenna Propagat., AP-14, pp. 105-106. Jan.,
1976.
[6] W. L. Stutzman and G. A. Thiele, Antenna Theory and
Design, John Wiley & Sons, New York, NY, 1981.
[7] R.F. Harrington, Field Computation by Moment
Methods, Robert E. Krieger Publishing Company, Malabar,
FL, 1985.
[SI G. J. Burke and A. J. Poggio, Numerical
Electromagnetics Code (NEC2) User’s Manual, Lawrence
Livermore Lab., Jan., 1981.
[9] J. A. Nelder and R. Mead, “A simplex method for
function minimization,” Comput. Journal, vol. 7, pp. 308313, 1965.
Randy Haupt is Department
Head and Professor of Electrical
and Computer Engineering at
Utah State University. He was
Dept. Chair and Professor of
Electrical
Engineering
at
University of Nevada Reno,
of
Electrical
Professor
Engineering at the USAF
Academy, research engineer at RADC, and project engineer
for the OTH-B Radar Program. He has a PhD from the
University of Michigan, MS from Northeastern University,
MS from Western New England College, and BS from the
USAF Academy. He is co-author of the book Practical
Genetic Algorithms, has 8 antenna patents, is an IEEE
Fellow, and is recipient of the 1993 Federal Engineer of the
Year Award.
You Chung Chung received his
BS in electrical engineering from
Jnha University, Inchon, Korea
in 1990, and MSEE degree from
University of Nevada, Reno
(UNR) in 1994. He recently
completed all requirements for
the Ph.D. degree in electrical
engineering at UNR and is working as a Post Doctoral
Research Engineer at Utah State University. Since 1995, he
has been employed as a teaching and research assistant in
electrical engineering at UNR. His research interests
include computational electromagnetics, optimized antenna
and array design, conformal and fractal antennas, adaptive
array processing, optimization techniques, and genetic
algorithms. In 1996, he received an Outstanding Teaching
He also received an
Assistant Award from UNR.
Outstanding Graduate Student Award in 1999. The NSF
sponsored his 1999 IEEE AP-S paper presentation.
[lo] R. L. Haupt and S. E. Haupt, Practical Genetic
Algorithms, John Wiley & Sons Inc., New York, NY, 1998.
455