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Re: [TenTec] NEC, ground, grounds, and radials.

To: tentec@contesting.com
Subject: Re: [TenTec] NEC, ground, grounds, and radials.
From: Dave Kelley <dkelley@bucknell.edu>
Reply-to: Discussion of Ten-Tec Equipment <tentec@contesting.com>
Date: Thu, 06 Jan 2011 16:02:51 -0500
List-post: <tentec@contesting.com">mailto:tentec@contesting.com>
Jerry makes some good points about taking care when using NEC. I have a 
few comments to add, interspersed below.

On 1/6/2011 3:00 PM, tentec-request@contesting.com wrote:
> Message: 1
> Date: Thu, 06 Jan 2011 11:07:39 -0600
> From: "Dr. Gerald N. Johnson"<geraldj@weather.net>
> Subject: Re: [TenTec] NEC, ground, grounds, and radials.
> To: tentec@contesting.com
> Message-ID:<4D25F6DB.7040305@weather.net>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
> [snip]
> I've been using NEC and MININEC since the days of the Apple II+. Since
> the program for the Apple II+ came in source code, my copy drops the
> gain of an antenna over a ground plane by 3 dB.
> Segmentation. The docs with MININEC and NEC suggest you start with a few
> segments and keep increasing them until small changes in the number of
> segments results in practically no change in results. Maybe I got it
> into round off error problems from the larger matrix, but on the Apple
> II+ with MININEC, I found that phenomena and then if I kept increasing
> segments there came a number where the solutions got wilder and wilder
> with more segments. That didn't give me super confidence in that concept
> for selecting segments. Segmenting the wires and emulation shapes other
> than wires with wire meshes are some of the arts required to get useful
> results from NEC, often limited by the number of total segments allowed
> by EZNEC or the computer available memory and acceptable computation time.
The improvement in accuracy followed by degradation as the number of 
segments increases that you have noted is well known. What happens is 
that when too many segments are used to model a wire, the matrix formed 
by NEC becomes ill conditioned. A matrix is a collection of numerical 
values arranged in a two-dimensional array. In NEC's case, the number of 
rows and columns in the matrix is always equal, and its size can be over 
100 by 100, 1000 by 1000, or even 10000 by 10000 or more, depending on 
the type of problem. Ill conditioning is a mathematical phenomenon 
familiar to experts in linear algebra. It takes a few years of 
college-level math to truly understand, but a simplified way to view it 
is that as the matrix becomes larger the numbers stored in the matrix 
become closer in value to their neighbors. Since computers can only 
store a few significant digits of any given number (for example. 
3.14159265359 might be stored as 3.141593), as the numbers get closer 
together in value, some of them might be stored as identical values. If 
sets of identical values stored in the matrix become numerous enough, 
then the solution calculated by NEC starts to diverge from the correct 
answer. Again, this is something of an oversimplification, but it is 
relatively close to what happens.

The usual advice for using NEC is to increase the number of segments as 
long as the solution seems to be converging, then back off once the 
solution begins to diverge again. This same phenomenon is observed with 
other software based on the method of moments and other kinds of 
numerical analysis methods used in antenna design.

> As for the effects with ground planes and my claim of error. I base it
> on this: Model a quarter wave vertical on a perfect ground plane. It
> will show 3 dB more gain than a half wave dipole in free space. Yet the
> theory of images in the ground plane insists that the quarter wave
> vertical on the ground plane has a image of the other half making it the
> exact equivalent of a half wave dipole. I claim that while the program
> in free space is comparing the signal intensity from the antenna to that
> of a perfect isotropic radiator located at the 0,0,0 origin of the axes,
> that when the ground plane is present it cuts that isotropic radiator in
> half, shielding half of its radiated power and so the reference to a
> full isotropic radiator is 3 dB in error. 3 dB too much gain.
Yes, this is basically what is happening. I agree with your disagreement 
[ :) ] with the way NEC calculates gain. Gain (or directivity, more 
precisely) is defined as the power density (measured in watts per square 
meter, or W/m^2) of the radiation leaving the antenna in the direction 
of maximum radiation divided by the power density radiated by an 
isotropic source fed with the same input power. Mathematically, this is 
stated as:

G = S_max/S_iso,

where S_max and S_iso are the power densities of the antenna (maximum) 
and the isotropic radiator, respectively. The power density from an 
isotropic radiator is simply the input power divided by the surface area 
of the sphere surrounding the antenna at some chosen distance:

S_iso = Pin/(4*pi*r^2)    for a whole sphere.

The distance r is either chosen to be some convenient value, or it is 
normalized out of S_max and S_iso. At any rate, this formula is for a 
whole sphere. If you now analyze an antenna (a monopole, for instance) 
over a ground plane, there is no radiation into the half-space below 
ground. As Jerry says, the radiation pattern of the antenna and its 
image is the same as that of a full antenna in free space, so the power 
radiated by the antenna is distributed throughout space the same way. 
However, because no power is radiated below ground, that power is added 
to the radiation in the upper half-space. Thus, twice as much power 
density is found in a given direction above ground (including the 
maximum direction) than when the antenna radiates in free space. For 
example, if 100 W is fed to a half-wave dipole in free space, 50 W is 
distributed throughout the upper half-space and 50 W throughout the 
lower half-space. (The power is distributed unevenly because of the 
doughnut-shaped radiation pattern of the dipole; however, the power is 
split evenly between the upper and lower half-spaces.) Now, if 100 W is 
feed to a quarter-wave monopole mounted on a perfectly conducting 
ground, then all 100 W is radiated into the upper half-space.

To obtain a "fair" comparison, the power density from the isotropic 
radiator should be doubled as well when ground is present, because the 
isotropic radiator only radiates into the upper half-space. Another way 
to view it is that the formula given above for S_iso should be modified to

S_iso = Pin/(2*pi*r^2)  [for a half sphere, which has an area of 2*pi*r^2]

when a ground plane is present. But NEC doesn't do that. It compares 
S_max (which when a ground plane is present is double the value of the 
free space case) to S_iso for a free space isotropic radiator. Because 
S_max is doubled in the ground plane case and S_iso is not, the antenna 
appears to have twice the gain (i.e., 3 dB greater) over what it would 
have in free space. If a "fair" comparison were made between S_max and 
S_iso for an isotropic radiator over a ground plane, then the gain 
figure would be the same because both S_max and S_iso would be doubled 
and the factors of two would cancel each other. That is, a monopole 
would exhibit 2.15 dBi of gain toward the horizon if the ground were a 
perfect conductor, but now the "i" in "dBi" would be understood to mean 
an isotropic radiator radiating into a half-space.

Nevertheless, frequent users of NEC know about this idiosyncracy, and it 
is a simple matter to subtract 3 dB to get a more realistic picture of 
the performance of an antenna over ground. You just have to be aware of 
the issue.

73 and Happy New Year,
Dave ND3K

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