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fortran语言的程序

     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)

     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 far-field.

     Next, the pattern of the antenna is calculated as given in
the book (old eq. 6-52) using the appropriate direction cosines
for our geometry.  The front-to-back ratio is found by examination
of the pattern on boresight and 180 degrees from boresight.  The
front-to-sidelobe 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 front-to-sidelobe 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.