INSTRUCTIONS FOR USING THE LOGPERIODIC
DIPOLE ARRAY DESIGN PROGRAM
Note: This document should be viewed from an editor which does
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This documentation file contains information for the correct
use of the logperiodic 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
realworld 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 quarterwave 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 logperiodic 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
centertocenter 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 xy plane. The angle Theta is measured from
the z axis toward the xy plane.
The Eplane 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 Efield 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
Eplane is the yz plane. The Hplane also contains the z axis
and is perpendicular to the Eplane. Therefore, the Hplane is
the xz plane. In terms of Phi and Theta, the Eplane 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 Hplane 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
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 go
about choosing values? For starters, some parameters are
available for more precise modeling of a particular application
or design. For this reason, the source impedance (Rs), input
line length (LLin), and the output line length (LLout) all can
be set to zero. Furthermore, setting the option to not quantize
the element diameters forces the program to ignore Navail and
Davail.
Certain parameters must be known by the designer before the
design. These include the characteristic impedance of the
source transmission line (ZCin), the frequency range (Fhigh and
Flow), and the desired gain (D0). If the characteristic
impedance 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 desired
input impedance (Rin) should be equal to the source transmission
line characteristic impedance (ZCin). Selecting the desired gain
sets Tau and Sigma for an optimum design. Alternatively,
selecting Tau and Sigma allows independent control for special
applications. All other parameters will be calculated by the
program.
Since the program only estimates the transmission line
characteristic impedance (ZO) for the antenna to achieve the
desired input impedance (Rin), the actual input impedance (ZinA)
may not be correct. For instance, assume that the actual input
impedance (ZinA) comes out to be 60 to 65 Ohms for a 50 Ohm
desired input impedance (Rin). In this case, lower the desired
input 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, quantize
the element diameters to the available wire or tube diameters.
This quantization rounds each calculated element diameter to the
nearest available size. The variable, Navail, tells how many
sizes 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 an
antenna with an input impedance (ZinA) 50 Ohms matched to a
50 Ohm source transmission line.
Finally, add a matched load (Zout) to suppress reflections
from the opencircuit termination. Notice the resulting
decrease in VSWR at many frequencies.
2.2. ANALYSIS PARAMETERS
In addition to the design parameters, the input screens
also ask for certain analysis parameters. The user can select
single frequency E and Hplane analyses, single frequency
custom plane analysis, and/or swept frequency analysis. The
input 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 Eplane,
0 degrees equals Hplane)
AFpowr = Number of frequency steps per octave
All parameters are selfexplanatory except the last. For
swept frequency analysis, the program starts at the lowest
analysis frequency, AFlow, and after each iteration increases
the analysis frequency by a certain amount until it exceeds the
upper analysis frequency. The step size, AFstep, is defined by
the following relation.
AFstep = FMHz * 10 ** (1/AFpowr)
where FMHz is the current analysis frequency. This relation has
the advantage of providing equally spaced points when frequency
is plotted on a logarithmic scale. Increasing AFpowr increases
the resolution.
3. ALGORITHM DEVELOPMENT
Some of the algorithms have already been explained, but for
the sake of clarity, all will be explained here. There is
nothing fantastically difficult about any particular step of the
algorithm, but taken together there are so many topics that it
is easy to become confused.
Since the design of the antenna is covered in the book, it
will be omitted here. Let us begin the discussion knowing the
topology of the design, as if we had an antenna in hand, so that
we are ready to analyze it.
There are several steps to the analysis of the antenna.
First, we must calculate the self and mutual impedances of each
antenna element. These calculations tell us how each excited
element interacts with each other element and how well each
antenna radiates. Next, we need to calculate the
characteristics of the antenna transmission line. This
calculation accounts for the termination impedance (Zout) as
well as the source impedance (Rs). It tells us how energy
propagates down the transmission lines. Third, we need to solve
for the currents on each antenna element as well as the input
current (Iin) and voltage (Vin) and the termination current
(Iout) and voltage (Vout). Finally, we calculate the gain,
VSWR, and other important parameters.
3.1. SELF AND MUTUAL IMPEDANCES
The self and mutual impedances of radiating elements is a
subject of much research. To understand the basic idea[8],
consider several radiating elements located in space. Now
remove all elements except one, for instance element 1, and
excite it with a current. The input impedance can be measured
as the ratio of the input voltage over the input current. This
is the self impedance denoted Z11. Now replace one of the other
elements, say element 2, and short its terminals together.
Recalculate the input impedance of element 1. This is the
mutual impedance denoted Z12 (impedance of element 1 due to
element 2). Repeat the process for every pair of antennas.
(The measurements can be roughly halved, because reciprocity
assures us that Z12 = Z21. This means the resulting matrix is
symmetric.)
To obtain very accurate results, finite difference methods
and projection methods, which include moment methods and
finite element methods often are used. Finite difference
methods approximate the governing differential equations by
mathematically dividing the surface into very small segments, then
approximating a derivative by the equation
f(x2)  f(x1)
df = 
x2  x1
For projection methods, a set of basis functions which satisfy
the boundary conditions are weighted and summed to form an
approximate answer. For moment methods, the basis functions are
valid for the entire antenna surface. For finite element
methods, the basis functions are valid over a small part of the
antenna surface.
An alternative approach, and the one used here, assumes
that the current distribution along each antenna varies
sinusoidally. This assumption is valid for infinitesimally thin
dipoles sitting alone in space. While this assumption is not
totally valid, the approximation is still a good one.
With this assumption, the self and mutual impedances can be
written as integral equations. These equations can be
recast in terms of the sine and cosine integrals[9][10]. With
these equations, the symmetric matrix, called ZA in the program,
can be found.
3.2. TRANSMISSION LINE ADMITTANCE MATRIX
Next, we must compute the characteristics of the antenna
transmission line. Good documentation can be found in the
literature [6][7][8], although these references can
sometimes be hard to find. Therefore, a summary will be
presented here. We wish to create the same kind of data for the
transmission line as we did for the self and mutual impedances.
That is, we will excite the transmission line with a unit
voltage everywhere an antenna element attaches, one place at a
time. The remaining places will be shorted. This procedure is
derived directly from Nport admittance matrix theory. The task
is to find the resulting current at each port (place of antenna
attachment).
FIGURE 5 illustrates the concept for an N element antenna
excited at port 4. Only a few of the ports are shown. The
distance between ports is given by the variable DZL.
PORT: 2 3 4 5

  + 
  
  
  
o o o   V  o o o
  
  
  
   

<DZL2><DZL3><DZL4>
FIGURE 5. Transmission Line Equivalent Circuit
For the following discussion, please refer to any standard
electromagnetics text. For example, see [2][3]. Also see
[4][7][8].
It is obvious by inspection that given an excitation at
element 4, a current will flow only through the shorts at ports
3, 4, and 5. Current will not flow at the other ports. The
current at port 4 is the current through the source, V. The
transmission line equation tells us that the impedance seen by
port 4 in the direction of port 3 is equal to
Z43 = j ZO tan(Beta * DZL3)
where ZO is the characteristic impedance of the line, and Beta
is the wavenumber. The admittance (Y) is one over the impedance
so,
Y43 = j YO cot(Beta * DZL3)
where YO = 1/ZO. Similarly, the admittance seen by port 4 in
the direction of port 5 is
Y45 = j YO cot(Beta * DZL4)
Therefore, the current flowing through port 4 (excited by a unit
voltage) is
Y43 + Y45 = I4 / 1
To find the current in port 3, the following general
transmission line equation is used.
V(DZL3) = V3 * cos(Beta*ZL3) + j * I3'' * ZO * sin(Beta*ZL3)
where V3 and I3'' are the voltage and current at port 3,
respectively. (Note: the primes do not represent
differentiation.) Since port 3 is shorted, V3 = 0.
Furthermore, V(DZL3) is just the excitation voltage,
V4 = 1 Volt. Therefore,
1 = j * I3'' * ZO * sin(Beta*ZL3)
or
I3'' / 1 = j YO * csc(Beta*L)
However, this definition of I3'' is reversed from that of the
standard Nport definitions; it flows out of the port instead
of in to it. Therefore, we remove one prime ('') and change the
sign.
I3' / 1 = j YO * csc(Beta*L)
One step remains. Since we arranged for a phase reversal by
crossing the wires in the transmission line [], we must account
for this by changing the sign again. The final form for the
current is then
I3 / 1 = j YO * csc(Beta*L)
The admittance seen in port 3 due to an excitation at port 4 is
then
Y34 = j YO * csc(Beta*L)
Notice that the resulting matrix will be tridiagonal.
That is, it will have nonzero elements only along its major
diagonal and the diagonals on either side of it.
The termination impedance is accounted when the admittance
seen by element N is calculated. The transmission line equation
tells the equivalent impedance seen by element N in the direction
of the load.
The program does not use the admittance matrix to account for
the source transmission line and source resistance. Instead,
it accounts for the effect of these items later after the solution
for the antenna is obtained.
The resulting tridiagonal admittance matrix is called YT in
the program.
3.3. COMBINING THE MATRICES
At this point we have one matrix, ZA, which describes how
the elements interact with each other. Another matrix, YT,
describes how the transmission line propagates energy. We wish
to combine the two. Following [7], the connection of the
Nport antenna elements network (ZA) and the Nport transmission
line network (YT) amounts to connecting the two Nports in
parallel. Therefore we can write that
Iel = YA Vel
IT = YT VT
where Iel is the current at the input to the antenna elements
(A column vector)
YA is the inverse of ZA
(A square matrix)
Vel is the voltage at the input to the antenna elements
(A column vector)
IT is the current at the ports of the transmission line
(A column vector)
VT is the voltage at the ports of the transmission line
(A column vector)
Since the connection occurs in parallel, the total current is
then
I = Iel + IT
Here, I represents the excitation current. For a logperiodic
dipole array, we excite the shortest element, element 1, only.
Therefore, I = [1 0 0 ... 0]T (T denotes transpose). When
we make the connection, Vel = VT. Therefore,
I = (YA + YT) Vel
(Here this text departs from [7].) We want the voltages on each
antenna element measured at the port. Therefore, we rewrite
this equation as
(ZAZT) = inverse of (YA + YT)
Vel = (ZAZT) I
From this,
Iel = YA Vel
The references [4][6][7] recast the equations slightly to make
them faster to evaluate. For this program, the straightforward
approach is fast enough.
3.4. FINDING THE INPUT AND TERMINATION CURRENTS
Now at last we have the antenna currents and voltages.
Before we calculate the critical parameters, let us find the
source and termination currents and voltages. For this theory,
we again consult [2][3], and write
I(z') = IL * cos(Beta*z') + j * VL / ZO * sin(Beta*z')
V(z') = VL cos(Beta*z') + j * IL * ZO * sin(Beta*z')
Here, z' is a distance measured from the load to the source
VL is the voltage at the load
IL is the current at the load
ZO is the characteristic impedance of the transmission line
To find the source current and voltage, we reduce the antenna to
the source generator, Vin, and an equivalent impedance which
replaces everything but the source and the input transmission
line (ZinA). The result is a diagram similar to FIGURE 2 where
the antenna becomes the load. Therefore, we set
z' = LLin (Length of source transmission line)
V(z') = Vin'
I(z') = Iin
VL = Vel1
IL = 1
ZO = ZCin
Note that
ZinA = Vel1 / 1 Amp = Vel1
This is the input impedance measured at the shortest element of
the dipole array. The result of the above substitutions is
Iin = cos(Beta*LLin) + j * Vel1 / ZO * sin(Beta*LLin)
Vin' = Vel1 * cos(Beta*LLin) + j * ZO * sin(Beta*LLin)
These are explicit expressions for Iin and Vin'. Vin' is the
voltage, not at the source, but after the current drop across
the source resistance is accounted as shown in Figure 6.
Iin>
 Rs  + Vin'
 (source 
 resistance) (ZinAS  antenna impedance
Vin O as seen by the
 voltage source)
 
 
FIGURE 6. Vin, Iin, and Vin'
Rs + ZinAS
Therefore, Vin =  Vin'
ZinAS
ZinAS can be determined from the transmission line equation
or by
ZinAS = Vin'/Iin
It is the antenna input impedance as seen by the source.
The power accepted by the antenna is
1
Pin =  (Iin) (Iin*) [real(ZinAS) + Rs]
2
where * indicates complex conjugate.
Note that confusion might exist about the definition of
the input impedance. There are two definitions: ZinA is the
input impedance measured at the shortest element of the
antenna array, and ZinAS is the input impedance as seen by the
voltage source. Note that the magnitude of ZinAS will vary
with the length of the input transmission line. The reason
for this is that for an antenna which is not precisely matched
to the characteristic impedance of the input transmission line,
a standing wave exists on this line. For this reason, a design
engineer might be interested in both measures od input impedance.
Finding the termination current and voltage is similar.
This time the antenna is the source and the termination is the
load. The above equations must be turned around to yield
Vout = VelN * cos(Beta*LLout)  j*IN' * ZO*sin(Beta*LLout)
Iout = IN' * cos(Beta*LLout)  j*VelN / ZO*sin(Beta*LLout)
where
VelN is the voltage present at the terminals of the last
(longest) element
LLout is the length of the termination transmission line
ZO is the termination transmission line characteristic
impedance (equal to the antenna transmission line
characteristic impedance)
and where
IN' = VelN / ZR
Zout + j * ZO * tan(Beta * LLout)
ZR = ZO * 
ZO + j * Zout * tan(Beta * LLout)
Zout is the termination impedance.
3.5. FINDING THE CRITICAL PARAMETERS
From the currents and voltages just calculated, we can find
all the critical parameters of the design. Before we do, it is
desireable to take care of one small point. It is convenient to
normalize all the currents and voltages to 1 Watt of input
power. This normalization allows us to compare the directivity
of this antenna with that of an isotropic antenna driven by an
ideal source which has the same input power. That is, for the
isotropic source there will be no source resistance. To calculate
the scale factor, we first need to know the amount of power accepted
by the antenna. This value was calculated in Section 3.4 as
1
POWERIN =  (Iin) (Iin*) [real(ZinAS) + Rs]
2
The scale factor is
SCALE = sqrt(WATTS / POWERIN)
where WATTS = 1 Watt of input power
sqrt() is the square root function.
Now we multiply all currents and voltages by the value of SCALE.
The gain of an isotropic antenna is given in dB as
Gain_Iso = 10.0 * log10(WATTS / (4 * PI * RADIUS**2))
where RADIUS is a distance sufficiently far from the antenna to
ensure the measurement is in the farfield.
Next, the pattern of the antenna is calculated as given in
the book (old eq. 652) using the appropriate direction cosines
for our geometry. The fronttoback ratio is found by examination
of the pattern on boresight and 180 degrees from boresight. The
fronttosidelobe level is found by searching for the next
largest local maximum besides the main beam. It is possible
that the main beam has split, and the fronttosidelobe level is
negative (in dB). The program then converts the currents and
voltages to dB at some phase angle in degrees (this is for
output purposes). Finally, it finds the VSWR by computing
the reflection coefficient, ABS_GAM as
 ZinA  ZCin 
ABS_GAM =   
 ZinA + ZCin 
where z represents the magnitude of the complex argument, z.
The VSWR is then
1 + ABS_GAM
VSWR = 
1  ABS_GAM
This is the VSWR in the source transmission line relative to the
characteristic impedance of that line.
4. SUBTLETIES AND ASSUMPTIONS
This section contains a list of assumptions and subtle
points associated with this design and analysis program. For
further comments, please see Section 6., VERIFICATION AND
VALIDATION SUMMARY.
The technique used to find the antenna impedance matrix is
actually an approximation which [6] calls "significant."
Excitation of one element has an effect on another element
which in turn has an effect on a third, and so forth. The method
used disregards the secondary effects on the third and
subsequent interactions.
By applying a unit voltage at element 1, we have normalized
the source voltage, Vin, located away from element 1 down the
source transmission line, to a value for which we later solve.
That is, for the purpose of calculation, we merely assume an
input current of 1 Amp and take care of the scaling later.
Notice that this routine can analyze other antenna arrays
as well. For instance, proper modification of the transmission
line matrix, YT, and the antenna elements matrix, ZA, allow
us to analyze an array of dipoles located arbitrarily in space
whether or not they are connected. Additional modifications to
the routine which finds the gain allows analysis of elements
other than dipoles.
In fact, [6] tells how to extend this program with very
little modification to include arrays of logperiodic 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 1113 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 Eplane, Hplane, and a custom
plane for a single frequency.
Swept frequency analysis of the gain, fronttoback ratio,
fronttosidelobe ratio, input impedance, and VSWR.
Single frequency analysis of the gain, fronttoback ratio,
fronttosidelobe ratio, input impedance, and VSWR.
Characteristic impedance of the antenna transmission line.
5.2. OUTPUT FILES
LogPerd.FOR outputs several files depending on the options
chosen during the input portion of the program. These files are
LogPerd.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 Eplane.
LP_HPL.OUT File containing a design summary and parameters
analyzed at a particular frequency and measured
in the Hplane.
LP_SWEPT.DAT File containing the result of a swept frequency
analysis and measured in the Eplane.
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 Eplane gain (dBi) as a function
of angle. Suitable for use with the plotting program.
LP_HPAT.DAT File containing the Hplane gain (dBi)as a function
of angle. Suitable for use with the plotting program.
LP_CPAT.DAT File containing the Cplane 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 roundoff 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 729 and 739a. 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 LogPerd.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 logperiodic dipole arrays. From 1992 through 1995
he worked for Phase IV Systems, Inc. testing radar seekers in
HardwareintheLoop 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. 449455,
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 LogPeriodic Dipole Antennas," IEEE Trans. Antennas
Propag., vol. AP22, No. 5, pp. 714718, September 1974.
[5] R.C. Jasik and H. Jasik, HANDBOOK OF ANTENNA ENGINEERING,
McGrawHill, New York, 1984.
[6] R.H. Kyle, "Mutual Coupling Between LogPeriodic Dipole
Antennas," General Electric Tech. Info. Series, Report No.
R69ELS3, Chapter 2, December 1968.
[7] R.L. Carrel, "Analysis and Design of the LogPeriodic 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., AP5, pp 306313,
July 1957.
[11] Y.T. Lo and S.W. Lee, ANTENNA HANDBOOK THEORY, APPLICATIONS,
AND DESIGN, Van Nostrand Reinhold Company, New York, pp 923,
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.