INSTRUCTIONS FOR USING THE LOG-PERIODIC                  DIPOLE ARRAY DESIGN PROGRAMNote:  This document should be viewed from an editor which doesnot use proportional fonts such as the DOS EDIT editor or theWindows NOTEPAD editor.  Alternatively, it can be printed orviewed with a non-proportional font such as Courier.  In thismanner, the various text figures can be fully appreciated.     This documentation file contains information for the correctuse of the log-periodic dipole array (LPDA) design programincluded 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 manyof them represent impedances - characteristic, load, source, anddesired input impedances.  Furthermore, the program utilizes some approximations which need to be fully understood by theuser.  Only with this kind of detailed knowledge can the userfully 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.     References1.  GEOMETRY DEFINITIONS1.1.  ANTENNA GEOMETRY     The geometry utilized for the analysis largely correspondsto the geometry of Figure 11.9(a) in the book.  FIGURE 1 showsthis 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 isgenerally 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 isdifferent 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 areduction in the antenna efficiency from 100% to something less.For instance, let us assume the antenna has an input impedanceof 50 Ohms as measured at the input terminals of element 1.Further assume the source transmission line characteristicimpedance is 50 Ohms so that it is matched to the antenna.  Ifthere were no source resistance, the antenna efficiency would be100%.  Now assume that a 5 Ohm source resistance (internal to thevoltage generator) were present.  The resulting circuit can beanalyzed as shown in FIGURE 2.          |--- 5 Ohms --------------|          |   (source               |          |    resistance)          |      1V  O                       50 Ohms (antenna)      ac  |                         |          |                         |          |-------------------------|      FIGURE 2.  Equivalent Circuit Considering the                 Antenna as a LoadThe proportion of voltage transferred to the load (50 Ohms) is                50         %V = ------ 100% = 91%              50 + 5The 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 ofinefficiency.  This efficiency factor results in a decrease ingain from the published values, such as in Figure 11.13.  The moralof this calculation is that design equations tend to yield thebest possible answer and that other mechanisms can degrade thereal-world results.  Only careful modeling and attention todetail can prevent substandard performance.     The other side of the antenna, that is, the side with thelongest element, also has an additional feature:  a terminationimpedance is added across the terminals of element N (thelongest element).  To understand why this is necessary, considerfirst an antenna without a termination impedance.  Instead, thetransmission line is left open.  What happens if some energy,injected at element 1, manages to continue down the antennatransmission line past the active region to strike the opencircuit.  Transmission line theory says it reflects from theopen and travels back toward the source.  While this means moreenergy could be radiated (the reflected energy has a secondchance to radiate), it also means that interference effects willoccur.  Experimentation with this program will show that thisinterference will result in a design whose VSWR versus frequencycontains many spikes at particular frequencies.  In contrast, ifa termination impedance is added which has the same value as theeffective antenna transmission line characteristic impedance,all energy which travels past the active region will be absorbedby the load.  The resulting VSWR is much smoother than theprevious case.     Why would a designer choose one reflection eliminationtechnique over another?  Using a matched load is the bestsolution in terms of performance, but often it is difficult,undesirable, or not cost effective to solder a resistor acrossthe longest element's terminals.  The quarter-wave transformeris cheaper and easier to construct and provides an improvementover not doing anything.  However, it also makes the physicalsize 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, andterminations, one must also consider the construction of theantenna transmission line.  Because this transmission linegenerally, though not always, provides a structure upon which tomount the antenna elements, it is also called a "boom."  Theboom often consists of a twin lead transmission line made fromtwo copper tubes as shown in detail in Figure 11.9(d) in thebook.  This construction actually represents a departure from atruly log-periodic design.  The geometry of truly frequencyindependent antennas is a function of (apex) angle only[1].  Forinstance, the truncation of the antenna at elements 1 and N isnot a function of the apex angle, Alpha, but rather of thedistances R1 and RN (see Figure 11.9(a) in the book).  Theresult is an antenna which operates over a frequency band(albeit a large one), not over all frequencies.  Therefore, thespacing and diameters of the two tubes which form the boomshould increase linearly with distance from the apex.  At theapex they should be spaced by zero and have zero diameter.  Itis also for this reason that the diameters of the elementsincrease with distance from the apex.     Although maintaining constant element diameters can have anoticeable effect on the pattern, maintaining constant boomspacing and boom tube diameters has a very minor effect on thepattern.  For this reason and for convenience, the boomcenter-to-center spacing is fixed at SB, and the boom tubediameters is fixed at DB.  These two parameters are sufficientto calculate the characteristic impedance of the boomtransmission line (without the elements attached)[2][3].1.2.  COORDINATE GEOMETRY     While the antenna geometry given in Section 1.1 is enough todesign the antenna, a coordinate system is needed for theanalysis.  FIGURE 4 shows the coordinate system for analysis.                     Z axis                       ^                       |              ________ | ________                       |                       |                ______ | ______                       |                   ___ | ___                       |                     _ | _                       |            Apex ----->O-------------->  y axis                     X axis                 (out of page)            FIGURE 4.  Coordinate SystemFurthermore, the angle Phi is measured from the x axis towardthe y axis in the x-y plane.  The angle Theta is measured fromthe z axis toward the x-y plane.     The E-plane is defined as the plane which contains theelectric field vectors and also the major axis (here, the zaxis) of the antenna.  Since the E-field develops where there isa voltage drop (V = Integral(-E * dl) and since there is avoltage drop from one side of each element to the other, theE-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 from0 to 180 degrees on both sides of the z axis.  The H-plane hasPhi at 0 or 180 degrees and Theta is allowed to vary from 0 to180 degrees on both sides of the z axis.  One importantconsequence of these definitions is that the boresight of theantenna is at Theta equal to 180 degrees, not Theta equal to0 degrees.  (Here, "boresight" refers to the direction of themechanical 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 toprecisely understand each term and its representation in theprogram.  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     LLout  = Line length of termination transmission line     LD     = Length to diameter ratio of antenna elements     Navail = Number of available wires or tubes for the elements     Davail = Diameters of available tubes for the elements     SB     = Spacing of boom tubes or wires     DB     = Diameter of boom tubes or wires     Tau    = Geometric ratio (see Section 11.4.2)     Sigma  = Spacing factor (see Section 11.4.2)   DESIGN VARIABLES **     L(n)   = Total length of element n     D(n)   = Diameter of element n     ZL(n)  = Location along the z axis of element n     ZO     = Characteristic impedance of antenna transmission              line     ZinA   = ACTUAL input impedance, measured at the terminals              of element 1 (the shortest element)                 *  Note that some of the input variables listed in the book      are listed in Section 2.2., ANALYSIS PARAMETERS.   ** Note that this list is somewhat abbreviated and that some      variables, such as Tau and Sigma, can be considered as      belonging to more than one category.          Now that each design parameter is defined, how does one goabout choosing values?  For starters, some parameters areavailable for more precise modeling of a particular applicationor design.  For this reason, the source impedance (Rs), inputline length (LLin), and the output line length (LLout) all canbe set to zero.  Furthermore, setting the option to not quantizethe element diameters forces the program to ignore Navail andDavail.     Certain parameters must be known by the designer before thedesign.  These include the characteristic impedance of thesource transmission line (ZCin), the frequency range (Fhigh andFlow), and the desired gain (D0).  If the characteristicimpedance of the source transmission line (ZCin) is not known,guess:  for most coaxial cable in the UHF band is 50 or 75 Ohms. Since we want to match the antenna to this cable, the desiredinput impedance (Rin) should be equal to the source transmissionline characteristic impedance (ZCin).  Selecting the desired gainsets Tau and Sigma for an optimum design.  Alternatively,selecting Tau and Sigma allows independent control for specialapplications.  All other parameters will be calculated by theprogram.     Since the program only estimates the transmission linecharacteristic impedance (ZO) for the antenna to achieve thedesired input impedance (Rin), the actual input impedance (ZinA)may not be correct.  For instance, assume that the actual inputimpedance (ZinA) comes out to be 60 to 65 Ohms for a 50 Ohmdesired input impedance (Rin).  In this case, lower the desiredinput impedance (Rin) so that the actual input impedance (ZinA)comes out nearly correct (that is, approximately equal to ZCin).More accurate estimates for ZO exist.[4][5]     Now that the design is complete and satisfactory, quantizethe element diameters to the available wire or tube diameters. This quantization rounds each calculated element diameter to thenearest available size.  The variable, Navail, tells how manysizes are available.  Davail contains the available diameters. Next, perturb the design by adding a source impedance of about 5 Ohms.  This should decrease the gain by about 1 dB for anantenna with an input impedance (ZinA) 50 Ohms matched to a50 Ohm source transmission line.     Finally, add a matched load (Zout) to suppress reflectionsfrom the open-circuit termination.  Notice the resultingdecrease in VSWR at many frequencies.2.2.  ANALYSIS PARAMETERS     In addition to the design parameters, the input screensalso ask for certain analysis parameters.  The user can selectsingle frequency E- and H-plane analyses, single frequencycustom plane analysis, and/or swept frequency analysis.  Theinput parameters are as follows.     AFSEH  = Frequency for single frequency analysis of E- and              H- planes     AFSC   = Frequency for single frequency analysis of custom              plane     AFhigh = Upper analysis frequency for swept frequency              analysis     AFlow  = Lower analysis frequency for swept frequency              analysis     Phi    = Angle of custom plane (90 degrees equals E-plane,              0 degrees equals H-plane)     AFpowr = Number of frequency steps per octave