log-perd.doc

天线方向图程序

     All parameters are self-explanatory 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 N-port 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 N-port 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 tri-diagonal. 
That is, it will have non-zero 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 tri-diagonal 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
N-port antenna elements network (ZA) and the N-port transmission
line network (YT) amounts to connecting the two N-ports 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 log-periodic
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)