NEC-LIST: re: modeling short dipoles (revised)

It looks as though I had a typo in the efficiency equations in my last
post. I hope that this is right, finally.

Several people have written suggesting that the Z0 source resistance
that I like to put in series with the feed points adds no value to the
analysis. This is true for single-feed networks. The easiest thing
is to just multiply the NEC efficiency by 1-(abs(rho))^2, rho =
(Zin-Z0)/(Zin+Z0). I usually work with arrays, and it is important to
add source resistances to all of the feed points in order to get
correct embedded-element patterns. I've just gotten into the habit of
using them all of the time. I also am assuming that no matching
network is being used.

In simple MM programs that I've written, I feed the elements with
voltage sources in series with pure 50-Ohm resistors. The voltage
sources are sized such that the total power dissipated by the antenna
would be equal to one watt if the antenna consisted of pure 50-Ohm
loads. For a single source, the voltage would be sqrt(2*z0) =
14.14... V. With this voltage set, the reported E-Field will be
correct for any input impedance, assuming a 1-watt source with a
50-Ohm characteristic impedance. Absolute gain, including mismatch
effects, falls right out. NEC, however, does not compute absolute
gain this way. It computes efficiency as the total radiated power
divided by the total input power, and multiplies all computed gain
numbers by this efficiency value. For a practical system, however,
efficiency is better represented by the total radiated power divided
by the total power that WOULD HAVE been input had the impedance been
perfectly matched.

  I = abs(feed current), Z0 = feed characteristic impedance,
  RL = Real(Zant) = Real(Zin-Z0), Vs = Abs(source voltage)

For a single element:
  NEC efficiency = radiated power / total power = (I^2 * RL) / (I^2 * Z0
+ I^2 * RL)
  practical efficiency = [I^2 * RL] / [(Vs^2 / 4 * Z0^2) * Z0] = [I^2 *
RL] / [Vs^2 / (4 * Z0)]
  (pract. effic.) / (NEC effic.) = {4*Z0*(Z0+RL)} / {[abs(Zin)]^2} =
{4*Z0*Zin} / {[abs(Zin)]^2}

For multiple elements:
  NEC efficiency = sum[I^2*RL] / sum[I^2*(Z0+RL)]
  practical efficiency = sum [I^2*RL] / sum[Vs^2 / (4*Z0)]
  (pract. effic.) / (NEC effic.) = {sum[I^2*(Z0+RL)]} / {sum[Vs^2 /
(4*Z0)]}
  
For example, I ran a tenth-wavelength dipole. With no series resistor,
NEC gave an input impedance of 2.4-j1234, a gain of 1.7 dBi (1.5 in
linear units), and an efficiency of 100%. With a 50-Ohm resistor the
gain showed up as -11.6 dBi and the efficiency was given as 0.046. This
is too high, since the efficiency penalty due to mismatch loss alone is:

  efficiency = 1-rho^2 = 1 - [(Zl-50) / (Zl+50)]^2 = 0.00315

which leads to a practical gain of approximately -23 dBi.

The equation above will give the ratio of these two values, which can be
used as a correction factor for NEC gain patterns when it is desired to
account for mismatch loss.

Comments would be appreciated.

                                Dave
Received on Wed Dec 10 1997 - 09:39:45 EST