Driving Point Impedance - CEARLcearl.ee.psu.edu/Projects/Assets/Project2/Project2_2_5/Driving... ·...
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An Efficient Method for An Efficient Method for Calculating the Driving Point Calculating the Driving Point Impedance of Fractal ArraysImpedance of Fractal Arrays
D.H. Werner, D. Baldacci, and P.L. WernerThe Pennsylvania State University
Department of Electrical Engineering
Driving Point Impedance of Fractal Driving Point Impedance of Fractal ArraysArrays
• The selfThe self--similarity property of fractal arrays may be similarity property of fractal arrays may be exploited to develop efficient recursive algorithms for exploited to develop efficient recursive algorithms for calculating driving point impedance.calculating driving point impedance.
•• The algorithms have been shown to be particularly useful The algorithms have been shown to be particularly useful for fractal arrays containing a large number of elements.for fractal arrays containing a large number of elements.
•• Two specific fractal antenna configurations have been Two specific fractal antenna configurations have been considered: considered:
Cantor linear arraysCantor linear arrays
Sierpinski carpet planar arraysSierpinski carpet planar arrays
The First Four Stages of the Triadic The First Four Stages of the Triadic Cantor Linear ArrayCantor Linear Array
Cantor Array Cantor Array (continued)(continued)
Driving Point ImpedanceDriving Point Impedance
+
=
−+
+=
− ∑P
mND
N
Nn
Pnm
PmDP
mD
P
P
P
Z
ZZZ
1
12/,
1
wherewhere1
2,11
1,11
21
1 ZZZZ DD +==
For m = 1, 2, … , NFor m = 1, 2, … , NPP / 2 and P / 2 and P >> 2 2
For m = NFor m = NPP / 2 + 1, N/ 2 + 1, NPP / 2 + 2, … , N/ 2 + 2, … , NPP and P and P >> 22
Cantor ArrayCantor ArrayTable of Driving Point Impedances Table of Driving Point Impedances
Element Element ImpedanceImpedanceStage 1:Stage 1: 1, 21, 2 60.5989+j12.616260.5989+j12.6162
Stage 2:Stage 2: 1, 41, 4 59.7959+j9.6762359.7959+j9.676232, 32, 3 62.7372+j18.054162.7372+j18.0541
Stage 3:Stage 3: 1, 81, 8 59.7723+j9.4996659.7723+j9.499662, 72, 7 62.7573+j18.290362.7573+j18.29033, 63, 6 62.8102+j18.523162.8102+j18.52314, 54, 5 59.6418+j8.9664659.6418+j8.96646
Stage 4:Stage 4: 1, 161, 16 59.7716+j9.4944259.7716+j9.494422, 152, 15 62.7573+j18.296262.7573+j18.29623, 143, 14 62.8103+j18.530962.8103+j18.53094, 134, 13 59.6408+j8.9575659.6408+j8.957565, 125, 12 59.6429+j8.9859059.6429+j8.985906, 116, 11 62.8078+j18.499962.8078+j18.49997, 107, 10 62.7537+j18.256362.7537+j18.25638, 98, 9 59.7761+j9.5415559.7761+j9.54155
Cantor ArrayCantor ArrayTable of Driving Point Impedances Table of Driving Point Impedances –– continuedcontinued
Stage 5:Stage 5: Element Element ImpedanceImpedance1, 321, 32 59.7708+j9.4943559.7708+j9.494352, 312, 31 62.7565+j18.296362.7565+j18.29633, 303, 30 62.8095+j18.530962.8095+j18.53094, 294, 29 59.6400+j8.9574859.6400+j8.957485, 285, 28 59.6422+j8.9860159.6422+j8.986016, 276, 27 62.8070+j18.499862.8070+j18.49987, 267, 26 62.7529+j18.256162.7529+j18.25618, 258, 25 59.7753+j9.5417059.7753+j9.541709, 249, 24 59.7753+j9.5419259.7753+j9.5419210, 2310, 23 62.7529+j18.255962.7529+j18.255911, 2211, 22 62.8070+j18.499462.8070+j18.499412, 2112, 21 59.6422+j8.9863959.6422+j8.9863913, 2013, 20 59.6400+j8.9568259.6400+j8.9568214, 1914, 19 62.8095+j18.531762.8095+j18.531715, 1815, 18 62.7565+j18.297262.7565+j18.297216, 1716, 17 59.7707+j9.4933759.7707+j9.49337
The First Four Stages of the The First Four Stages of the Sierpinski Carpet ArraySierpinski Carpet Array
P3P2P0 P1
Active Element
Sierpinski Carpet Dipole Array
Stage 1
8 elements
Sierpinski Carpet Dipole Array
Stage 2
64 elements
Sierpinski Carpet Dipole Array
Stage 3
512 elements
The First 3 Stages for the Impedance The First 3 Stages for the Impedance Matrix of the Sierpinski Carpet Planar Matrix of the Sierpinski Carpet Planar
Fractal ArrayFractal Array
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