It's possible to get exact self and mutual impedances from EZNEC. I'll
explain the method for two identical elements.

Excite the two elements with equal, in-phase currents. Record the
feedpoint impedance of either element (they should be the same) as Z0
(= R0 + jX0). Then change the phase of one of the currents to 180
degrees, so the elements are fed exactly out of phase. Record the
feedpoint impedances with this excitation as Z180 (= R180 + jX180).

The mutual impedance Zm = (Z0 - Z180) / 2

The self impedance can also be found as Zs = (Z0 + Z180) / 2

For example, use the Cardioid.EZ EZNEC example file. Change the phase of
the second source to zero, click Src Dat, and note the element impedance
Z0 = 56.11 - j14.22. Change the phase of the second source to 180, click
Src Dat again, and note the impedance Z180 = 16.54 + j16.37. The mutual
Z is then (56.11 - 16.54) - j(-14.22 - 16.37) = 19.8 - j15.3. The self Z
is 36.3 + j1.1. These values can be used in SIMPFEED program Lewall1.

As it turns out, you can also calculate the exact self and mutual
impedances from the feedpoint impedances of two elements fed 90 degrees
out of phase. For identical elements fed with equal magnitude 90 degree
phased currents, where Z1 is the feedpoint impedance of the leading
element (that is, element 2 is fed at -90 degrees relative to element 1)
and Z2 is the feedpoint impedance of the lagging element,

Rm = (X2 - X1) / 2
Xm = (R1 - R2) /2

and

Rs = (R1 + R2) / 2
Xs = (X1 + X2) / 2

Caution: Don't think that because the self impedance is the average of
the two feedpoint impedances in the above two special cases, that it's
always true. It isn't.

Going back to the Cardioid model as it comes with EZNEC, note that
Z1 = 21.03 - j18.71 and Z2 = 51.61 + j20.86 when the elements are fed at
90 degrees. So

Rm = (20.86 - -18.71) / 2 = 19.8
Xm = (21.03 - 51.61) / 2 = -15.3
Rs = (21.03 + 51.61) / 2 = 36.3
Xs = (-18.71 + 20.86) / 2 = 1.1

exactly the values calculated before. Note that the values of mutual
impedance are very close to the values from the graph in Chapter 8 of
the ARRL Antenna Book.

The equations for these special cases are derived from the more general
equations which can be found in Chapter 8 of the ARRL Antenna Book, and
numerous other references. In the 20th Edition of the Antenna Book,
they're Eq 20 and 21 on p. 8-19. Equations can easily be derived for two
dissimilar elements from feedpoint impedances with in-phase and
out-of-phase excitation with equal currents. And although it's possible
to derive equations for self and mutual Z from the feedpoint impedances
of more complex arrays, it requires more "measurements" in order to have
enough equations for the increased number of unknowns.

Roy Lewallen, W7EL

Cecil Moore wrote:
acepilot wrote:

Cecil, I think you were implying that the dipoles you modeled were
parallel to each other, correct? Our ELF antennas were dipoles that
were perpendicular to each other. In theory, there should be minimal
interaction between them because of the nulls off of each end of the
antennas, correct? Somebody else mentioned that the antennas, when
driven, feed power into each other. Placing them at 90 degrees to
each other should minimize interaction, would it not?


Yep, that's true, and a turnstile is an example. But for a phased beam,
one needs maximum interaction. The dipoles in my example are 1/4WL
apart, parallel, and in the same horizontal plane.

Incidentally, one of the disadvantages of Roy's SIMPFEED program is
that one needs to know the mutual coupling impedance between the
elements. For a two-element system, with identical elements, there
is a way to use EZNEC to calculate (estimate) the mutual coupling
impedance, Rm +/- jXm.

For two identical (resonant) elements, the feedpoint impedances reported
by EZNEC will be of the form, (Rs +/- Xm) +/- jRm, where Rs is the
resonant resistance of a single element alone (second element
open-circuited).

For instance, in my earlier example of two 33 ft dipoles, 33 ft apart
at a height of 66 ft, fed 90 degrees apart - the feedpoint impedances
a

109+j34 and 29-j34

That makes Rm = 34 ohms and makes Rs (109+29)/2 = 69 ohms, which
makes Xm = -j39 ohms. Those Rm and Xm values can then be plugged
into Roy's SIMPFEED program to obtain the length of the feedlines.

Note that two phased 20m dipoles work just fine as a beam on 17m.
All it takes is different phasing of the feedlines.