Fractal antennas and fractal resonators
View Patent ↗An antenna includes at least one element whose physical shape is at least partially defined as a second or higher iteration deterministic fractal. The resultant fractal antenna does not rely upon an opening angle for performance, and may be fabricated as a dipole, a vertical, or a quad, among other configurations. The number of resonant frequencies for the fractal antenna increases with iteration number N and more such frequencies are present than in a prior art Euclidean antenna. Further, the resonant frequencies can include non-harmonically related frequencies. At the high frequencies associated with wireless and cellular telephone communications, a second or third iteration, preferably Minkowski fractal antenna is implemented on a printed circuit board that is small enough to fit within the telephone housing. A fractal antenna according to the present invention is substantially smaller than its Euclidean counterpart, yet exhibits at least similar gain, efficiency, SWR, and provides a 50Ω termination impedance without requiring impedance matching.
1. An apparatus comprising:
an antenna undefined by an opening angle and having a first element whose physical shape is defined substantially as a deterministic fractal of iteration N≧2 for at least a portion of said first element, wherein said deterministic fractal is defined as a superposition over at least N=2 iterations of a fractal generator motif, an iteration being placement of said fractal generator motif upon a base figure through at least one positioning selected from the group consisting of (i) rotation, (ii) stretching, and (iii) translation, wherein said antenna has a perimeter compression (PC) parameter defined by:
PC
=
full
-
sized
antenna
element
length
fractal
-
reduced
antenna
element
length
where:
PC=A log [ N ( D+C )]
wherein A and C are constant coefficients for a given said fractal generator motif, N is an iteration number, and D is a fractal dimension given by log(L)/log(r), where L and r are one-dimensional antenna element lengths before and after fractalization, respectively; and
wherein iteration N=2, PC is approximately 1.9, termination impedance is substantially 50Ω, and gain is substantially at least within 1 dB of unity.
2. An apparatus comprising:
an antenna undefined by an opening angle and having a first element whose physical shape is defined substantially as a deterministic fractal of iteration N≧2 for at least a portion of said first element, wherein said deterministic fractal is defined as a superposition over at least N=2 iterations of a fractal generator motif, an iteration being placement of said fractal generator motif upon a base figure through at least one positioning selected from the group consisting of(i) rotation. (ii) stretching, and (iii) translation, wherein said antenna has a perimeter compression (PC) parameter defined by:
PC
=
full
-
sized
antenna
element
length
fractal
-
reduced
antenna
element
length
where:
PC=A log [ N ( D+C )]
wherein A and C are constant coefficients for a given said fractal generator motif, N is an iteration number, and D is a fractal dimension given by log(L)/log(r), wherein L and r are one-dimensional antenna element lengths before and after fractalization, respectively; and
wherein iteration N=3, PC is approximately 2.4, termination impedance is substantially 50Ω, and gain is substantially at least within 1 dB of unity.
3. The antenna of claim 1 or 2 , wherein said fractal generator motif has x-axis, y-axis coordinates for a next iteration N+1 defined by X N+1 =f(x N , y N ) and Y N+1 =, (x N , y N ) where (x N , y N ) are coordinates for iteration N, and where f(x,y) and g(x,y) are functions defining said fractal generator motif and behavior.
4. The antenna of claim 1 or 2 , wherein said first element is fabricated in a mariner selected from the group consisting of(i) shaping conductive wire into said fractal, (ii) forming upon an insulator substrate a conductive layer defining traces shaped to form said fractal, (iii) forming upon a flexible insulator substrate conductive traces shaped to form said fractal; and (iv) forming upon a semiconductor substrate a layer of conductive material shaped to form said fractal.
5. The antenna of claim 1 or 2 , wherein said antenna is selected from the group consisting of (i) a fractal quad, (ii) an at least third iteration fractal quad, (iii) a Minkowski fractal quad, (iv) a dipole, and (vi) a vertical.
6. A fractal resonating system, comprising:
an inductor including an element portion whose physical shape is defined substantially as a deterministic fractal of iteration N≧2 for at least a portion of said element, wherein said deterministic fractal is defined as a superposition over at least N=2 iterations of a fractal generator motif, an iteration being placement of said fractal generator motif upon a base figure through at least one positioning selected from the group consisting of (i) rotation, (ii) stretching, and (iii) translation; and
a capacitive element coupled with said inductor to define at least one resonant frequency for said system, including frequencies non-harmonically related to each other.
7. An apparatus comprising:
an antenna undefined by an opening angle and having an element whose physical shape is defined substantially as a deterministic fractal of iteration N≧2 for at least a portion of the element; and
wherein said antenna has a perimeter compression (PC) parameter defined by:
PC
=
full
-
sized
antenna
element
length
fractal
-
reduced
antenna
element
length
where:
PC=A log [ N ( D+C )]
wherein A and C are constant coefficients for said fractal generator motif, N is an iteration number, and D is a fractal dimension given by Iog(L)/log(r), where L and r are one-dimensional antenna element lengths before and after fractalization, respectively; and
wherein iteration N=2, PC is approximately 1.9, termination impedance is substantially 50Ω, and gain is substantially at least within 1 dB of unity.
8. An apparatus comprising:
an antenna undefined by an opening angle and having an element whose physical shape is defined substantially as a deterministic fractal of iteration N≧2 for at least a portion of the element; and
wherein said antenna has a perimeter compression (PC) parameter defined by:
PC
=
full
-
sized
antenna
element
length
fractal
-
reduced
antenna
element
length
where:
PC=A log [ N ( D+C )]
wherein A and C are constant coefficients for a given said fractal generator motit N is an iteration number, and D is a fractal dimension given by log(L)/log(r), wherein L and r are one-dimensional antenna element lengths before and after fractalization, respectively; and
wherein iteration N=3, PC is approximately 2.4, termination impedance is substantially 50Ω, and gain is substantially at least within 1 dB of unity.