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European Journal of Applied Sciences – Vol. 10, No. 6
Publication Date: December 25, 2022
DOI:10.14738/aivp.106.13699. Papageorgiou, C. D. (2022). Helical and Spiral Antennas as Electric Small Antennas. European Journal of Applied Sciences, 10(6).
587-605.
Services for Science and Education – United Kingdom
Helical and Spiral Antennas as Electric Small Antennas
Christos D. Papageorgiou
School of Electrical and Computer Engineering
National Technical University of Athens, Greece, Imperial college PhD
ABSTRACT
The simplest form of a classical antenna is the dipole antenna, which is essentially
two pieces of wire placed end to end with a feed point in the middle. The length of
this antenna is typically half the wavelength of the signal that is being received or
transmitted. An “electric small” antenna is defined as an antenna of much shorter
dimensions than the wavelength of the signals it is designed for. Small electric
antennas have an advantage when space is the most essential factor. Satellites and
mobile communication apparatuses for example can use small antennas in order to
free up more space for other components. The problem with classical electric small
antennas is that their bandwidth and radiation efficiency shrink as they get shorter.
Although such antennas have been in use for decades, they remain difficult to design
and limited in their applicability. However, the classic approach of studying metal
antennas using only Maxwell equations limits the real internal nature of them as
metal lattice structures and thus as quantum wells of free electrons. When
interpreted this way, free electrons are obeying the Schrodinger wave equation and
this view can give new ideas for designing effective curvilinear electric small
antennas. Such a class of electric small helical and spiral antennas is proposed in
the following paper.
Keywords: electric small antenna, antenna as quantum trap, curvilinear antennas, helical
antennas, spiral antennas.
INTRODUCTION
It is well known that given the electric current of any antenna for any operating frequency, the
electromagnetic field (near and far) can be calculated directly from Maxwell equations. Even if
we remain in the class of the simplest kind of antennas, which are the one-dimensional center- fed linear wire antennas of an arbitrary length, only numerical methods exist for finding their
approximate electric current distribution [1]. However, in the case that the length L of a thin
center-fed rectilinear antenna is equal to L=n*λ/2, where λ is the operating wavelength of the
electromagnetic wave with n an integer, its electric current can be accurately calculated and
being given by a sinusoidal standing wave of wavelength λ and frequency C/λ (where C is the
speed of light).
My personal view is that all the impressive progress in the field of linear antennas is mostly due
to laborious experimental work rather than a clear theoretical understanding of their internal
operation. This is the result of trying to describe this operation as the effect of action of
electromagnetic fields on them, considering the antennas as simple non active conducting
structures and thus ignoring their internal natural dynamics as quantum wells for free
electrons. In the present paper we focus on thin linear antennas comprising a standard metallic
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022
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lattice although an extension of the proposed method to more complicated structures is
possible.
It will be shown in the present report that the optimum operation of any type of metallic lattice
linear antenna can be achieved when the internal spatial waveforms of their free electron
density become resonant with the externally applied electromagnetic fields, following their
quantum mechanical behavior as described by the respective Schrodinger equation. An initial
presentation of antennas as quantum wells of free electrons has already been published by the
author [2].
Small electric antennas [3] are defined as the antennas that can operate effectively although
their dimensions are much smaller than the wavelength of their operation. It is suggested that
in order to design antennas for specific applications, like small electric antennas, we should
take into consideration their internal nature as quantum wells of free electrons governed by
the Schrodinger wave dynamics. A theoretical approach combining Schrodinger and Maxwell
equations in order to understand the properties of curvilinear antennas is presented in the
paper with the aim to design “small electric” antennas.
RECTILINEAR METALLIC ANTENNAS AS QUANTUM WELLS OF FREE ELECTRONS
Here, an example of a one-dimensional rectilinear antenna is studied as a quantum well of free
electrons. The most common antenna of this kind is a center-fed linear metallic antenna of
length L. Even the tinny metallic antennas of a few mm long have a tremendous number of free
electrons. Thus, we can consider all the metal lattice rectilinear antennas as wells of a cloud of
free electrons obeying the Quantum Mechanical laws as described by the respective
Schrodinger equation.
Let us consider a rectilinear antenna taken as very thin in comparison to its length. Thus, it can
be considered as a one-dimensional metal lattice structure along the x-axis from 0 to L. The
cloud of free electrons is obeying the stationary Schrodinger wave equation given by:
�!�(�)/�!� = −(� − �) ⋅ �(�) (1a)
(� − �) = !"
($/!&)! ∗ (� − �(�)) (1b)
V(x) is the electrostatic potential acting on the free electrons and E is the energy of the system
necessary to form the respective stationary Schrodinger wave. In fact, there is an internal
electrostatic action on free electrons arising from the ions of the metallic lattice, however, this
potential action can be omitted and the overall action due to the scattering of free electrons
with the positive lattice ions is usually replaced by an equivalent Ohmic resistance.
In the classic Born interpretation of the Schrodinger equation, the function |�(�)!| defines the
probability of an electron being at point x. Due to the extremely high number of free electrons,
we can assume that |�(�)!| is proportional to the number of electrons at x, thus this function
defines the electric charge along the conducting linear structure Q(x) and as a result, the
function �(�)��(�)/�� is proportional to the electric current of the linear antenna EC(x).
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Papageorgiou, C. D. (2022). Helical and Spiral Antennas as Electric Small Antennas. European Journal of Applied Sciences, 10(6). 587-605.
URL: http://dx.doi.org/10.14738/aivp.106.13699
The eigenvalues and eigenfunctions of linear antennas will then be given by the solutions of the
Schrodinger equation �!�(�)/�!� = −� ⋅ �(�), i.e. the functions without any external voltage
excitation. These eigenfunctions should be a set of eigenstates ��(�) with respective
eigenvalues εn defined by the geometric boundaries of the curvilinear structure, where the
respective electric current function I(x), is taken to be zero. Thus, at x=0 and x=L, either Y(x)=0
or ��(�)/�� = 0
Let us assume that the boundary conditions for a certain curvilinear antenna are Y(0)=0 and
Y(L)=0, hence the respective eigenstates are:
��(�) = �� ���(2���/2�)
while for boundary conditions Y(0)=0 and ��(�)/�� = 0 , the respective eigenstates should
be:
��(�) = �� ���((2� + 1)��/2�)
Thus, for both cases ��(�) = �� ���(���/2�) where n is an even or an odd integer the electric
current I(x) in the linear antenna will be given as �(�) = �� ∗ ��� (���/�) . For both cases the
respective eigenvalues i.e. the energy of the system for each eigenstate will also be given as εn=
(nπ/2L)^2. In a similar way, it can be proven that for ��(0)/�� = 0 and Y(L)=0 or ��(�)/�� =
0 the electric current and the respective energy eigenvalues remain the same as previously.
The fundamental eigenstate for the electric current is the one where the minimum energy
eigenvalue is achieved. Thus, the fundamental eigenstate is achieved for n=1, where the energy
is minimum and equal to �1 =(π/2L)^2, and the respective electric current in the rectilinear
antenna is I(x)=I1* ��� (��/�).
Thus, the lowest eigenstate for a linear center-fed antenna of length L is achieved by a semi
sinusoidal current wave that becomes zero at the endpoints and maximum at its center, where
we place the feed points of the antenna. This antenna will be tuned by an internal voltage source
(transmitter) or by an external Electromagnetic field (receiver), with a sinusoidal voltage of
angular frequency ω=k*C where k is equal to the wave number of its fundamental eigenstate
π/L and C=speed of light, at which state it will generate its highest output.
In general, a linear bipolar antenna of length L can be also tuned in higher frequencies ωn=n*ω
where tuning is achieved for higher harmonics, kn =(n*π)/L and ωn =kn*C. Any proper center
fed bipolar antenna is tuned with odd harmonics that give a maximum output at their centers,
while the even harmonics give zero output in their centers. Bipolar antennas are not used for
higher frequencies because they demand more energy (n2 times) to transmit and stronger fields
to receive.
Thus, the smallest tuned center-fed linear antenna has a length of half of the wavelength of the
electromagnetic wave, where it is designed to operate i.e. L=λ/2. In the case of a conducting
ground plane below the half of the antenna, its length could be L=λ/4. In a higher wavelength
range (smaller frequencies than f), the rectilinear bipolar antenna has a deteriorating operation
demanding far more energy in order to transmit or receive the respective electromagnetic
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022
Services for Science and Education – United Kingdom
signals. An electric small antenna is defined as one with substantially smaller dimensions in
relation to its operating wavelength band operating with relatively small energy.
CURVILINEAR ANTENNA AS QUANTUM WELL OF FREE ELECTRONS
Let us now consider a curvilinear one-dimensional, very thin metallic antenna inside three- dimensional free space. The geometrical shape of the antenna can be defined as a function of its
linear length s(x,y,z) as a parametric curve, where � ≥ � ≥ 0.
The geometry of the curvilinear metallic structure of the antenna defines the Schrodinger
equation of the cloud of free electrons inside its metallic lattice. More accurately the
Schrödinger wave equation for a one-dimensional (very thin) curvilinear conducting structure
developed along its parametric length s as has been proved [4] to be of the form
�!�(�)/�!� = − >
(!())
*
+ �? ⋅ �(�) (2)
Where σ(s) is the standard local curvature of the curvilinear one-dimensional antenna and
ε=
!"
($/!&)! � is the reduced energy of the cloud of free electrons inside the metallic lattice of the
antenna.
In equation (2) we have a boundary value problem for a second order ordinary linear
differential equation with variable coefficient because its curvature factor is a function of s,
however, there are several numerical methods that can tackle the problem.
In the present paper an effective technique named “resonance technique”, used in many similar
applications by the author [5,6] will be applied. The great advantage of the “resonance
technique” method is that it calculates initially the eigenvalues of the second rank differential
equation and afterwards the respective eigenstates i.e. the harmonics of the relative antenna
are readily obtained.
It should also be mentioned that the harmonics of the antenna are derived independently of the
properties (frequency f) of the acting electromagnetic field. However, for successful operation
of the antenna the frequency of the electromagnetic field should be adjusted accordingly.
The method is shortly presented as follows.
The original ODE is mathematically equivalent and is transformed to the following system of
two first order differential equations:
∂V(s) / ∂s = − j*A (s)*I(s)
∂I(s) / ∂s = − j*V(s)
Where the used functions are defined as: V(s)= j *∂Y(s) / ∂s, I(s)= Y(s) and A(s)= (
(!())
*
+ �)
These equations are equivalent to a set of two “spatial electric-lines” along s. The non-linear
homogeneous system of differential equations has solutions (eigenstates) only for special
values of the energy ε (energy eigenvalues), and the fundamental eigenstate is the one for the
minimum energy eigenvalue.