I was doing some simulations of fractal antennas recently and a friend said, "I'm talking about the postage sized antenna for 160 meters."
The statement intrigued me for several reasons:
1) Perhaps fractal antennas scale differently because of fractal dimension?
2) How small would could an antenna for 160 meters (1.9 Mhz) REALLY be?
3) How does fractal dimension affect the performance of the antenna?
Analysis Tools
The questions were answered via extensive analysis. Analysis was performed using 4NEC2 an excellent adaption by Arie Voors of the Numerical Electromagnetics Code by J, Burke and A. Poggio. These links will open in a new window if you would like to check them out. Arie Voors has made his tool freely available to the technical community. The Minkowski Antenna configuration used in this study is described here by Camps-Raga and Islam at the University of Missouri, Columbia. The 'C' program minkowski.c, the bash script, doit.bash, and the parameter file, tail.txt used to generate the antenna examples are included at the bottom of this technical note.
Simulation Results
1) A fractal antenna of a given configuration and dimension scales
linearly in wavelength just like all antennas do.
2) A Minkowski fractal antenna for 160 meters would be 36 feet wide.
It would have a gain of -19.3 dB relative to an isotropic radiator.
3) Increasing fractal dimension decreases antenna size with
gain dropping faster than size.
Discussion
1) A fourfold reduction in antenna size is accompanied by a gain reduction of 21.3 dB so there appears to be is a severe penalty for size reduction. My current hypothesis is that as fractal dimension increases, the antenna starts to look more like a mirror than an absorber of electromagnetic energy.
2) A level 0 Minkowski fractal is a square loop antenna! The makes a nice basis for comparison.
Analysis Details
'C' Code Used to Generate the Minkowski Fractal Antennas
The 'C' code was written entirely by the author using the principle of segmenting a single line and offsetting the middle segment by a distance h. The code was compiled using the cygwin cc compiler from the fantastic and free cygwin running on top of Windows XP on a dual core Intel processor and ASUS motherboard from newegg.com. If you have never had the pleasure of building your own computer from off the shelf components I urge you to do so. You will save a ton of money and get a computer more suited to your needs. Anyway, here is the 'C' code:
------- cut here -------
// Minkowski fractal antenna generator by van at wdv dot com
#include
#define SOUTH 1
#define EAST 2
#define NORTH 3
#define WEST 4
#define MAX(a,b) ((a)>(b)?a:b)
int maxLevel = 1;
void edge(double xA, double yA, double xF, double yF, int type, int level)
{
double W = 1.0 / 3.0;
double H = 1.0 / 3.2;
double L, w, h;
double xB, yB;
double xC, yC;
double xD, yD;
double xE, yE;
if(level >= maxLevel)
{
printf("1\t0\t%9.4f\t%9.4f\t0\t%9.4f\t%9.4f\t1.e-4\n",
xA, yA + 1, xF, yF + 1);
}
else
{
if(type == SOUTH)
{
L = xF - xA; w = L * W; h = L * H;
xB = xA + w; yB = yA + 0;
xC = xA + w; yC = yA + h;
xD = xF - w; yD = yA + h;
xE = xF - w; yE = yA + 0;
edge(xA, yA, xB, yB, SOUTH, level + 1);
edge(xB, yB, xC, yC, WEST, level + 1);
edge(xC, yC, xD, yD, SOUTH, level + 1);
edge(xD, yD, xE, yE, EAST, level + 1);
edge(xE, yE, xF, yF, SOUTH, level + 1);
}
else if(type == EAST)
{
L = yF - yA; w = L * W; h = L * H;
xB = xA + 0; yB = yA + w;
xC = xA - h; yC = yA + w;
xD = xA - h; yD = yF - w;
xE = xA + 0; yE = yF - w;
edge(xA, yA, xB, yB, EAST, level + 1);
edge(xB, yB, xC, yC, NORTH, level + 1);
edge(xC, yC, xD, yD, EAST, level + 1);
edge(xD, yD, xE, yE, SOUTH, level + 1);
edge(xE, yE, xF, yF, EAST, level + 1);
}
else if(type == NORTH)
{
L = xA - xF; w = L * W; h = L * H;
xB = xA - w; yB = yA + 0;
xC = xA - w; yC = yA - h;
xD = xF + w; yD = yA - h;
xE = xF + w; yE = yA + 0;
edge(xA, yA, xB, yB, NORTH, level + 1);
edge(xB, yB, xC, yC, EAST, level + 1);
edge(xC, yC, xD, yD, NORTH, level + 1);
edge(xD, yD, xE, yE, WEST, level + 1);
edge(xE, yE, xF, yF, NORTH, level + 1);
}
else if(type == WEST)
{
L = yA - yF; w = L * W; h = L * H;
xB = xA - 0; yB = yA - w;
xC = xA + h; yC = yA - w;
xD = xA + h; yD = yF + w;
xE = xA - 0; yE = yF + w;
edge(xA, yA, xB, yB, WEST, level + 1);
edge(xB, yB, xC, yC, SOUTH, level + 1);
edge(xC, yC, xD, yD, WEST, level + 1);
edge(xD, yD, xE, yE, NORTH, level + 1);
edge(xE, yE, xF, yF, WEST, level + 1);
}
}
}
main(int argc, char **argv)
{
if(argc > 1) maxLevel = atoi(argv[1]);
edge( 0, 0, 16, 0, SOUTH, 0);
edge(16, 0, 16, 16, EAST, 0);
edge(16, 16, 0, 16, NORTH, 0);
edge( 0, 16, 0, 0, WEST, 0);
}
Bash Script Used to Generate the Minkowski Fractal Antennas
The 'C' program above, though somewhat minimal, produces extra copies of the segments during the recursion. The output lines are only part of the "card entry" fragments necessary for 4NEC2. To get the output file into a form usable by 4NEC2 the following script was run:
$doit.bash 2
which produces a level 2 Minkowski fractal antenna.
------- cut here --------------
#!/bin/bash
minkowski.exe $1 | sort -u | nl | sed -e 's/ //g' | sed -e 's/^/GW\t/' > head.txt
cat head.txt | wc -l > /tmp/tmp.txt
lineCount=`cat /tmp/tmp.txt`
echo $lineCount
sedCommand="s/NSEG/$lineCount/"
echo $sedCommand
cat tail.txt | sed -e "$sedCommand" > foo.txt
cat head.txt foo.txt > MI3ALVW.nec
u2d MI3ALVW.nec
~
The Last File
The last file "tail.txt" is used by the bash script to provide analysis parameters for 4NEC2. It looks like this:
------- cut here ----------
GE 0
EK
LD 5 0 0 0 5.74713e7 0
EX 6 NSEG 1 0 1 0
GN 2 0 0 0 13 0.005
FR 0 1 0 0 1.8 0
Addendum:
"Chip" W1YW owns US patents 6452553 and 7256751 on fractal antennas. He has measured the performance of a configuration similar to the one pictured at 175 MHz and does not report the dip in gain that appears in the 1.9 MHz version above. He states and I quote:"For example,the MI2 (second iteration) modeled in NEC4, reveals a 97% efficiency and dipole equivalent gain. Recently an open house with members of the Radio Club of America showed this to: K1VR; WB2MGP; K1BG; WA6RNU; W0BIW; KD0FAA; and several others. In addition, the visitors were treated to actual chamber measurements of a wire-bent MI2 at 175 MHz, Its gain was indisitinguishable (within 0.2dB) from an ETS calibrated dipole of 1.7 dBi. The fractal antenna is thus unity gain to a dipole."
Thus W1YW does not observe the same drop in gain that occur in my simulations. Work is proceeding to rectify this discrepancy.
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