23-02-2012, 12:14 PM
Small Fractal Antennas
1 Problem
Antennas for hand-held communication devices are necessarily small, and typically use wavelengths
λ that are large compared to the size of the antenna. This typically implies that
the magnitude of the antenna reactance X (= imaginary part of the antenna impedance) is
large compared to that of its radiation resistance Rrad (which is related to the time-average
radiated power P and the peak current I0 at the feedpoint by P = I2
0Rrad/2), so that it is
challenging to build an effective impedance matching circuit between the feedline and the
antenna. Furthermore, small antennas use small conductors so it may be that the Ohmic
resistance ROhm of the antenna is significant compared it radiation resistance,1 which lowers
the antenna efficiency, defined as
Antenna Efficiency =
Rrad
Rrad + ROhm
. (1)
It is possible to lower the reactance of an antenna by changing the shape of its conductors
without increasing the overall size of the antenna. If the length and complexity of the shape
of the antenna conductors is increased while keeping the overall area of the antenna constant,
we create what is sometimes called a fractal antenna.2 Details of the antenna reactance of
fractal antennas are best calculated with a numerical code such as NEC4. Here you are asked
to use relatively simple analytic arguments to discuss the radiation (and Ohmic) resistance
of a planar fractal antenna that fits within a square of edge length a λ
Show that the radiation resistance of a fractal loop antenna is smaller than that of a
simple loop antenna of the same extent a. Show that the radiation resistance of a dipole
antenna based on a dense (Hilbert) fractal pattern is essentially identical to that of a simple
linear dipole antenna of the same total height a λ, even if the total length L of the
conductor is of order λ.
Then, since the Ohmic resistance of a fractal antenna is necessarily larger than that of a
simple dipole or loop antenna of the same overall extent, the efficiency (1) of a fractal antenna
is lower than that of the simpler antenna. Nonetheless, in some cases the lower reactance of
the fractal antenna may provide a useful advantage in simplifying the feed electronics of the
antenna system.
2 Solution
We first discuss small fractal antennas as receiving antennas. This discussion will be somewhat
qualitative, so we follow it with more quantitative discussion of their behavior as
1The antenna impedance is Z = Rrad + ROhm + iX, where i =
√
−1.
2Fractal antennas are an outgrowth of meander antennas [1].
1
broadcast antennas. The antenna reciprocity theorem [2] guarantees that a good broadcast
antenna is also a good receiving antenna.
2.1 Remarks about Receiving Antennas
A receiving antenna can be considered as a 2-terminal device whose purpose is to produce a
voltage (that can be amplified externally, and demodulated to produce an audio signal, etc.)
in response to an electromagnetic wave. If the conductor of an antenna fits within a square
of edge a that is small compared to the wavelength λ of the electromagnetic wave that is to
be detected, then the electric and magnetic fields E and B have negligible spatial variation
over the antenna at any moment in time.
If the receiving antenna is a dipole, then it responds primarily to the electric field of the
wave. Clearly, the largest voltage drop across the antenna, is just the field strength E times
the largest spatial dimension of the antenna. That is
Vmax =
√
2aE (small dipole antenna), (2)
independent of the detailed arrangement of the conductor within the square of edge a.3 We
immediately infer that a small fractal dipole antenna cannot be superior to an ordinary small
dipole antenna if their overall spatial extents are the same.
In practice, the signal from a small dipole antenna is more like 1/2 of the maximal
voltage (2). This is because a signal in a dipole antenna is based on the induced electric
dipole moment p = qd, which depends on the distance d between the centers of each arm of
the antenna, which is typically half the distance between the tips.
A loop antenna responds primarily to the magnetic field of the broadcast wave, via
Faraday’s law. That is, the 2-terminal signal voltage is proportional to time rate of change
of the magnetic flux through the antenna, which is proportional to the area of the antenna,
V ∝ dΦ
dt
∝ ωBArea (small loop antenna), (3)
where ω = 2πf is the angular frequency of the (carrier) wave.4 Thus, if a loop antenna fits
within a square of edge a, the signal will be strongest if the shape is simply a square of edge
a. A fractal shape for the conductor reduces the area of the antenna (provided it still fits
within a square of edge a), and hence reduces its effectiveness as a small loop antenna.
The power extracted from the incident wave by an antenna depends on the effective
impedance Z of the combination of the antenna plus receiving circuit, according to P =
Re(V 2/2Z). If the total impedance of a small antenna + receiving circuit can be made
small, the small antenna can extract just as much power from the incident wave as the
large antenna. Hence, understand of antenna reactance is important for receiving as well as
broadcast antennas. This note, however, limits its further discussion to the real part of the
antenna impedance.
We now turn to a discussion of small antennas as broadcast devices.
3The maximal signal voltage can be achieved only with proper alignment of the antenna with respect to
the electric field of the wave; i.e., the arms of the dipole should be parallel to the electric field vector E.
4The maximal signal voltage in a loop antenna is achieved when the axis of the loop is parallel to the
magnetic field B of the wave.
2
2.2 Radiation Resistance of Small Linear and Loop Antennas
A simple measure of the performance of a broadcast antenna is its radiation resistance Rrad,
which relates the (time-averaged) radiation power P to the peak current I0 that drives the
antenna, according to
P =
1
2
I2
0Rrad. (4)
A higher radiation resistance is better, in that more power is radiated compared to the power
I2
0ROhm/2 lost to heating the antenna due to the ordinary resistance ROhm of its conductor.
2.2.1 Small Center-Fed Linear Dipole Antenna
Recall that the radiation resistance of a center-fed, linear dipole antenna of length a λ is
Rrad =
a
λ
2
197 Ω, (center-fed linear dipole), (5)
assuming that the current drops linear between the center of the antenna (the feed point)
and the tips (where the current must be zero)[3]. The radiation resistance of a small linear
dipole antenna of length a falls off as (a/λ)2.
2.2.2 Small Loop Antenna
Likewise, the radiation resistance of a small loop antenna of area A is [4]
Rrad =
A
λ2
2
31, 170 Ω, (loop), (6)
independent of the shape of the loop provided its longest diameter (or diagonal) is small
compared to λ. The radiation resistance of a small, square, loop antenna of edge a falls off
as (a/λ)4. For a <∼ λ/12, a loop antenna has lower radiation resistance than that of a linear
dipole antenna.
2.3 Small Fractal Antennas
Turning now to the question of the merits of a fractal antenna whose largest dimension a
is still small compared to the wavelength λ, we note that this condition implies that phase
differences are negligible between the radiation from different parts of the antenna. In this
case, it suffices to analyze the radiation in the dipole approximation. That is, all details of
the radiation pattern follow from knowledge of the electric and magnetic dipole moments of
the charge and current distributions in the antenna.
