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

0.6

0.65

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0.95

1

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7

80.3

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100 200 300 400 500

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