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     In fact, [6] tells how to extend this program with very
little modification to include arrays of log-periodic dipole
arrays.

     The excitation matrix, I = [1 0 0 ... 0]T can be used in
other useful ways.  For instance, changing the excitation matrix
to I = [ ... 0 0 0 1]T allows the user to investigate problems
arising from reflections at the termination.  I = [1 1 ... 1]T
allows the user to investigate arrays of independently driven
dipoles.  Of course, the transmission line matrix, YT, must be
removed (or set to 0) if the elements are not connected.

     The input impedances can be confusing.  One could see
50 Ohms of input impedance as measured at the source (ZinAS)
while the input impedance seen at the shortest element of the
antenna is something completely different.  Therefore, one must
ensure that ALL impedance match for the highest efficiency.
That is, ideally Rs = 0, ZinAS = ZinA = Rin.

     The optimal value of the termination impedance will change
with frequency.  If the antenna is radiating from the shortest
elements, Zout should equal ZO.  If the antenna is radiating from
the longest elements, then Zout approximately should equal Rin.
Since these values are close together, picking Zout = Rin should
be sufficient.

     The gain of this antenna can be increased beyond that
predicted by Figure 11-13 and [11].  The classic LPDA antenna uses
a metal boom as a transmission line.  If the antenna transmission
line were allowed to meander slightly between elements, the phase of
the current could be forced to change precisely (not approximately)
by 180 degrees between elements.


5.  OUTPUT PARAMETERS

5.1.  DESCRIPTION

     Many of the output parameters have already been described. 
All input parameters (and quantities easily derivable, such as
Alpha from Sigma and Tau) are also output, so they will not be
discussed here.  The output depends on what analysis options
were chosen.  What follows is a list of all available outputs.

     Voltages and currents at all elements, the source, and the
        termination.
     Gain pattern (in dBi) of the E-plane, H-plane, and a custom 
        plane for a single frequency.
     Swept frequency analysis of the gain, front-to-back ratio,
        front-to-sidelobe ratio, input impedance, and VSWR.
     Single frequency analysis of the gain, front-to-back ratio, 
        front-to-sidelobe ratio, input impedance, and VSWR.
     Characteristic impedance of the antenna transmission line.


5.2.  OUTPUT FILES

     Log-Perd.FOR outputs several files depending on the options
chosen during the input portion of the program.  These files are

     Log-Perd.INI An input file which is automatically updated
                  every time the program is executed.  If the file
                  does not exist in the current directory, it is
                  created using default values.

     LP_DES.OUT   File containing a design summary.

     LP_CPL.OUT   File containing a design summary and parameters
                  analyzed at a particular frequency and measured
                  in the custom plane.

     LP_EPL.OUT   File containing a design summary and parameters
                  analyzed at a particular frequency and measured
                  in the E-plane.

     LP_HPL.OUT   File containing a design summary and parameters
                  analyzed at a particular frequency and measured
                  in the H-plane.

     LP_SWEPT.DAT File containing the result of a swept frequency
                  analysis and measured in the E-plane.

     LP_GAIN.DAT  File containing two columns: frequency and gain
                  on boresight (dBi).  Suitable for use with the
                  plotting program.

     LP_FTBR.DAT  File containing two columns: frequency and front
                  to back ratio (dB).  Suitable for use with the
                  plotting program.

     LP_SRC.DAT   File containing two columns: frequency and 
                  magnitude of the input impedance measured at the
                  source looking toward the antenna.
                  Suitable for use with the plotting program.

     LP_Z_ANT.DAT File containing two columns: frequency and 
                  magnitude of the input impedance measured at the
                  input terminals of the antenna (not at the source).
                  Suitable for use with the plotting program.

     LP_VSWR.DAT  File containing two columns: frequency and VSWR
                  measured in the input line.  Suitable for use with
                  the plotting program.

     LP_EPAT.DAT  File containing the E-plane gain (dBi) as a function
                  of angle.  Suitable for use with the plotting program.

     LP_HPAT.DAT  File containing the H-plane gain (dBi)as a function
                  of angle.  Suitable for use with the plotting program.

     LP_CPAT.DAT  File containing the C-plane gain (dBi) as a function
                  of angle.  Suitable for use with the plotting program.


6.  VERIFICATION AND VALIDATION SUMMARY

     This program has undergone extensive, though not
exhaustive, testing to ensure that each module and the total
program work as they were intended.  However, no rigorous
testing has been performed which ensures that the models are
accurate (validation).  Instead, the outputs from this program
were casually compared to previously published results, such as
earlier editions of the book [9] and other sources already cited. 
Furthermore, some hand waving, such as occurs in the description
of the program in the book on page 565, allows us to feel good about
the accuracy of the program without rigorously proving it.

Assumptions include

     All conductors are lossless.
     The medium surrounding the antenna has unity relative 
        permittivity and permeability.
     All parts are perfectly manufactured and connected. In
        particular, the feed is perfectly balanced.
     The current distribution on each element is sinusoidal.
     The antenna transmission line can be spaced precisely and 
        uniformly along its length.
     The computer has no round-off error, especially with regard 
        to matrix inversion.

     At the subroutine level, each module was checked with
simple test cases to verify its accuracy.

     FINDST  Tested with sample inputs and compared to Figure 
             11.13 in the book.  Note that the book makes a 1 dB 
             correction to the results originally published[7].  
             It is known that there is a mistake in this 
             reference, and the amount of error varies from
             about 0.5 to 2 dB.  Therefore, the optimum design as  
             calculated by this program will be slightly in 
             error.  This program could be used to find a 
             corrected version of Figure 11.13, to the accuracy 
             of the other error sources and assumptions.  This
             exercise has been done, although with a different
             program, and the results are published[11].
     FINDZ   Tested by comparison of the results to identical 
             cases calculated in MathCAD which used the integral 
             equations of 7-29 and 7-39a.  As described in Part 4,
             SUBTLETIES AND ASSUMPTIONS, an approximation is made
             with regard to removing all elements except two:  the 
             excited one and the one under observation.  
     INPUT   Tested by direct observation of the results.
     LUSOLV  Tested by sample test matrices[12].
     LUDEC   Tested by sample test matrices[12].
     OUTPUT1 Tested by direct observation of the results.
     OUTPUT2 Tested by direct observation of the results.
     PATTERN Tested indirectly by comparison of the analysis 
             results to published results[].
     R2POL   Tested with sample inputs with the ouput compared to
             hand calculations.
     SICI    Tested with sample inputs and compared to [13]
     SLL     Tested by direct observation of the results.

     At the program level, the results were checked against
previously published results, against previously written
code in MathCAD, and against sanity checks.


7.  FORTRAN Compilation

     The code contained in Log-Perd.FOR is written for the FORTRAN
77 language standard.  It was developed on a PC using Microsoft
FORTRAN Power Station for Windows and Windows 95 and has also been
successfully compiled on a Sun workstation.


8.  CREDITS

     This program and all its subroutines were created by
Mr. Chris Bishop with the exception of the matrix inversion
routines, LUSOLV and LUDEC, and the sine and cosine integrals,
SI and CI.  The matrix inversion routines were created by
Dr. James T. Aberle, and the sine and cosine integrals were
created by Mr. Anastasis Polycarpou.  In both cases the routines
were made for the Telecommunications Research Center at Arizona
State University.


Biography

MR. CHRIS BISHOP received his bachelor's and master's degrees in
electrical engineering from the Georgia Institute of Technology
in 1990 and 1991, respectively.  His research focused precisely
on modeling log-periodic dipole arrays.  From 1992 through 1995
he worked for Phase IV Systems, Inc. testing radar seekers in 
Hardware-in-the-Loop environments for US Army Missile Command.
His chief tasks there included analysis of existing electronics,
microwave devices, and radiating elements, as well as 
specification of a compact antenna range.  Currently, Mr. Bishop
attends Arizona State University where he pursues the Ph.D. in
electrical engineering.


Acknowledgements

Mr. Bishop would like to thank Dr. Ed B. Joy of the Georgia
Institute of Technology for inspiring him to pursue antennas as
an area of research, Dr. Michael D. Fahey for making him realize
engineering is less of a job than it is a way of life, Dr. James
T. Aberle for allowing him time from his regular duties to write
the code for this application, and Dr. Constantine A. Balanis
for agreeing to include this code in his book.


9.  REFERENCES

[1]  V.H. Rumsey, "Frequency Independent Antennas."  1957 IRE
     National Convention Record, pt 1, pp 114 - 118.

[2]  D.K. Cheng, FIELD AND WAVE ELECTROMAGNETICS, 2ed, Addison-
     Wesley Publishing Co., Reading, Massachusetts, pp. 449-455,
     1989.
  
[3]  S. Ramo, J.R. Whinnery, and T. van Duzer,  FIELDS AND WAVES
     IN COMMUNICATION ELECTRONICS, 2ed, John Wiley & Sons, 
     New York, p 252, 1984.

[4]  G. De Vito and G.B. Stracca, "Further Comments on the Design
     of Log-Periodic Dipole Antennas," IEEE Trans. Antennas
     Propag., vol. AP-22, No. 5, pp. 714-718, September 1974.

[5]  R.C. Jasik and H. Jasik, HANDBOOK OF ANTENNA ENGINEERING,
     McGraw-Hill, New York, 1984.

[6]  R.H. Kyle, "Mutual Coupling Between Log-Periodic Dipole
     Antennas," General Electric Tech. Info. Series, Report No.
     R69ELS-3, Chapter 2, December 1968.

[7]  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,
     Michigan.

[8]  M.T. Ma, THEORY AND APPLICATION OF ANTENNA ARRAYS, John Wiley
     & Sons, New York, Chapter 5, 1974.

[9]  C.A. Balanis, ANTENNA THEORY ANALYSIS AND DESIGN, John Wiley
     & Sons, New York, 1982.

[10] H.E. King, "Mutual Impedance of Unequal Length Antennas in
     Echelon," IRE Trans. on Antennas and Propag., AP-5, pp 306-313,
     July 1957.

[11] Y.T. Lo and S.W. Lee,  ANTENNA HANDBOOK THEORY, APPLICATIONS,
     AND DESIGN, Van Nostrand Reinhold Company, New York, pp 9-23,
     1988.

[12] J.R. Westlake, A HANDBOOK OF NUMERICAL MATRIX INVERSION AND
     SOLUTION OF LINEAR EQUATIONS, John Wiley & Sons, New York,
     Appendix C, 1968.

[13] M. Abramowitz and I. Stegun, editors, HANDBOOK OF MATHEMATICAL
     FUNCTIONS, ninth printing, Dover Publications, New York, 1970.