INSTRUCTIONS FOR USING THE LOG-PERIODIC

                  DIPOLE ARRAY DESIGN PROGRAM



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     This documentation file contains information for the correct
use of the log-periodic dipole array (LPDA) design program
included with the book, ANTENNA THEORY:  ANALYSIS AND DESIGN, by 
Constantine A. Balanis.  When the design program is first run, 
the many input variables can be confusing, especially since many
of them represent impedances - characteristic, load, source, and
desired input impedances.  Furthermore, the program utilizes 
some approximations which need to be fully understood by the
user.  Only with this kind of detailed knowledge can the user
fully appreciate the difficulty of accurately modeling antennas.  
Finally, many of the sources for the equations come directly 
from this book, but others come from elsewhere.  For instance, 
the transmission line theory from introductory electromagnetics 
forms a key part of the analysis.  Therefore, the references for 
this work are summarized at the end of the document.

     This file contains ten parts as follows:

          1.     Geometry definitions
            1.1.  Antenna Geometry
            1.2.  Coordinate Geometry
          2.     Discussion of input parameters
            2.1.  Design Parameters
            2.2.  Analysis Parameters
          3.     Algorithm development
            3.1.  Self and Mutual Impedances
            3.2.  Transmission Line Admittance Matrix
            3.3.  Combining the Matrices
            3.4.  Finding the Input and Termination Currents
            3.5.  Finding the Critical Parameters
          4.     Subtleties and assumptions
          5.     Output parameters
          6.     Verification and validation summary
          7.     FORTRAN Compilation
          8.     Credits
          9.     References


1.  GEOMETRY DEFINITIONS


1.1.  ANTENNA GEOMETRY

     The geometry utilized for the analysis largely corresponds
to the geometry of Figure 11.9(a) in the book.  FIGURE 1 shows
this geometry redrawn.

               Element Number:
               1  2   3   ...  N
                                |
                       |        |
                  |    |        |
               |  |    |        |
           x
         ->| R1|<-|    |        |
           |<-R2->|    |        |
           |<--- R3 -->|        |
           |<------  RN  ------>|
           |
         Apex

         FIGURE 1.  Array Geometry

     Several important additions are not shown in this figure. 
First, something must energize the antenna.  This source is
generally a voltage source with an internal resistance, Rs. 
This voltage source is connected to the shortest element,
element 1, by means of a source transmission line which is
different from the transmission line (often referred to as the
"boom") which connects the antenna elements.  Often this source 
transmission line is a coaxial cable.  The center conductor is 
connected to one side of element 1, and the shield is connected 
to the other side of element 1.  There are many subtleties 
associated with this connection, but this analysis ignores them.

     The effect of including the source resistance, Rs, is a
reduction in the antenna efficiency from 100% to something less.
For instance, let us assume the antenna has an input impedance
of 50 Ohms as measured at the input terminals of element 1.
Further assume the source transmission line characteristic
impedance is 50 Ohms so that it is matched to the antenna.  If
there were no source resistance, the antenna efficiency would be
100%.  Now assume that a 5 Ohm source resistance (internal to the
voltage generator) were present.  The resulting circuit can be
analyzed as shown in FIGURE 2.

          |--- 5 Ohms --------------|
          |   (source               |
          |    resistance)          |
      1V  O                       50 Ohms (antenna)
      ac  |                         |
          |                         |
          |-------------------------|

      FIGURE 2.  Equivalent Circuit Considering the
                 Antenna as a Load

The proportion of voltage transferred to the load (50 Ohms) is

                50
         %V = ------ 100% = 91%
              50 + 5

The fraction of power received by the load is equal to (%V)**2 or

         %P = (0.91)(0.91) 100% = 83%

83% is the antenna efficiency neglecting other sources of
inefficiency.  This efficiency factor results in a decrease in
gain from the published values, such as in Figure 11.13.  The moral
of this calculation is that design equations tend to yield the
best possible answer and that other mechanisms can degrade the
real-world results.  Only careful modeling and attention to
detail can prevent substandard performance.

     The other side of the antenna, that is, the side with the
longest element, also has an additional feature:  a termination
impedance is added across the terminals of element N (the
longest element).  To understand why this is necessary, consider
first an antenna without a termination impedance.  Instead, the
transmission line is left open.  What happens if some energy,
injected at element 1, manages to continue down the antenna
transmission line past the active region to strike the open
circuit.  Transmission line theory says it reflects from the
open and travels back toward the source.  While this means more
energy could be radiated (the reflected energy has a second
chance to radiate), it also means that interference effects will
occur.  Experimentation with this program will show that this
interference will result in a design whose VSWR versus frequency
contains many spikes at particular frequencies.  In contrast, if
a termination impedance is added which has the same value as the
effective antenna transmission line characteristic impedance,
all energy which travels past the active region will be absorbed
by the load.  The resulting VSWR is much smoother than the
previous case.

     Why would a designer choose one reflection elimination
technique over another?  Using a matched load is the best
solution in terms of performance, but often it is difficult,
undesirable, or not cost effective to solder a resistor across
the longest element's terminals.  The quarter-wave transformer
is cheaper and easier to construct and provides an improvement
over not doing anything.  However, it also makes the physical
size of the antenna longer, a possible disadvantage.

     FIGURE 3 shows the resulting antenna geometry.

                                |
                       |        |
     <- LLin ->   |    |        |<- LLout ->
     __________|  |    |        |___________
   Rs__________                  ___________Zout
         ->| R1|<-|    |        |
           |<-R2->|    |        |
           |<--- R3 -->|        |
           |<------  R4  ------>|
           |
         Apex

   FIGURE 3.  Geometry Showing the Source and
              Termination Transmission Lines
              and Impedances

     In addition to the placements of the elements, sources, and
terminations, one must also consider the construction of the
antenna transmission line.  Because this transmission line
generally, though not always, provides a structure upon which to
mount the antenna elements, it is also called a "boom."  The
boom often consists of a twin lead transmission line made from
two copper tubes as shown in detail in Figure 11.9(d) in the
book.  This construction actually represents a departure from a
truly log-periodic design.  The geometry of truly frequency
independent antennas is a function of (apex) angle only[1].  For
instance, the truncation of the antenna at elements 1 and N is
not a function of the apex angle, Alpha, but rather of the
distances R1 and RN (see Figure 11.9(a) in the book).  The
result is an antenna which operates over a frequency band
(albeit a large one), not over all frequencies.  Therefore, the
spacing and diameters of the two tubes which form the boom
should increase linearly with distance from the apex.  At the
apex they should be spaced by zero and have zero diameter.  It
is also for this reason that the diameters of the elements
increase with distance from the apex.

     Although maintaining constant element diameters can have a
noticeable effect on the pattern, maintaining constant boom
spacing and boom tube diameters has a very minor effect on the
pattern.  For this reason and for convenience, the boom
center-to-center spacing is fixed at SB, and the boom tube
diameters is fixed at DB.  These two parameters are sufficient
to calculate the characteristic impedance of the boom
transmission line (without the elements attached)[2][3].


1.2.  COORDINATE GEOMETRY

     While the antenna geometry given in Section 1.1 is enough to
design the antenna, a coordinate system is needed for the
analysis.  FIGURE 4 shows the coordinate system for analysis.

                     Z axis
                       ^
                       |
              ________ | ________
                       |
                       |
                ______ | ______
                       |
                   ___ | ___
                       |
                     _ | _
                       |
            Apex ----->O-------------->  y axis
                     X axis
                 (out of page)

            FIGURE 4.  Coordinate System

Furthermore, the angle Phi is measured from the x axis toward
the y axis in the x-y plane.  The angle Theta is measured from
the z axis toward the x-y plane.

     The E-plane is defined as the plane which contains the
electric field vectors and also the major axis (here, the z
axis) of the antenna.  Since the E-field develops where there is
a voltage drop (V = Integral(-E * dl) and since there is a
voltage drop from one side of each element to the other, the
E-plane is the y-z plane.  The H-plane also contains the z axis 
and is perpendicular to the E-plane.  Therefore, the H-plane is 
the x-z plane.  In terms of Phi and Theta, the E-plane has Phi 
fixed at 90 or 270 degrees, and Theta is allowed to vary from
0 to 180 degrees on both sides of the z axis.  The H-plane has
Phi at 0 or 180 degrees and Theta is allowed to vary from 0 to
180 degrees on both sides of the z axis.  One important
consequence of these definitions is that the boresight of the
antenna is at Theta equal to 180 degrees, not Theta equal to
0 degrees.  (Here, "boresight" refers to the direction of the
mechanical axis of the antenna.  In other contexts, "boresight"
is the direction where the pattern is maximum.) 


2.  DISCUSSION OF INPUT PARAMETERS

     Now that we have defined the geometry, it is important to
precisely understand each term and its representation in the
program.  Some parameters can be calculated from others.


2.1.  DESIGN PARAMETERS

   INPUT VARIABLES *
     TITLE  = Design title
     D0     = Desired gain
     Fhigh  = Upper design frequency
     Flow   = Lower design frequency
     Rs     = Source impedance internal to the voltage generator
     ZCin   = Characteristic impedance of the input transmission
              line
     Rin    = DESIRED input impedance, measured at the terminals
              of element 1 (the shortest element)
     LLin   = Line length of the input transmission line
     Zout   = Termination impedance