Mutual Impedance of Plasma Antennas
Bo Yin, Feng Yang
Bin Wang, Honggang Hao
School of Electronic Engineering,
University of Electrnic Scienic and Technology of China,
College of Electronic Engineering,
Chongqing University of Posts and Telecommunications,
Chengdu, 610054, China
Chongqing, 400065, China
E-mail: byin0520@163.com
Abstract-- the concept of plasma antennas array has been put
forward for a long time, but the study on the mutual coupling
between elements of the array is very little. In this paper, a model
for antennas array involving two plasma elements is set up,
where the density of element 1 is assumed to be very high and
could be used as metal, and the mutual impedance between
plasma elements is derived detailedly. Then, considering that the
density of the plasma element 2 is uniform or non-uniform
distribution, we analyse mutual impedance of the two plasma
column antennas which vary with the plasma density, collision
frequency inside element 2 and the distance of two elements. The
result of experiment shows, under normal circumstance, the
amplitudes of mutual resistance and reactance between two
plasma antennas are smaller than those between two metallic
antennas which has the same configure and size, and the plasma
density, collision frequency inside element 2 will impact the
impedance. This is very useful for beam forming and directional
reconfiguration among of plasma antenna’s application.
Keywords- mutual impedance; plasma column antennas; plasma
density; collision frequency
I．
INTRODUCTION
Plasma consists of electrons, ions and neurons. When
the density of ionized gas is very high enough, the
electromagnetic wave can not go deep into the plasma, and
the skin depth is quite small, then plasma exhibits properties
of a conductor. Therefore, plasma can be used as a radiator
element instead of metallic conductors. By using RF source,
the plasma antenna works immediately once it is energized,
and will stop instantly with the source removed. When
de-energized, plasma is non-conducting, and causes little
return wave to radar. Another advantage of plasma antenna is
that it can be reconfigured, to change the radiation pattern,
conveniently by electrical rather than mechanical control [1]
[2].
The previous experiments have demonstrated that the
directivity of monopole plasma column antenna is very weak
[3], and smaller than that of metallic antenna with the same
configuration. However, in the communication, especially the
communication from point to point, it needs the antenna with
the powerful directivity. This can accomplish through a
consideration of plasma antenna array [4]. Although studies
on plasma antennas have received considerable attention over
the past 30 years, a full set of plasma column antenna has not
been extensively studied experimentally and theoretically
[5][6]. To the best of our knowledge, there has been no report
on the study of the array properties of the plasma antenna so
far. Antenna engineers are frequently faced with the problem
of predetermining the input impedance of each element of a
directional array. In addition, they must often find the
interference pattern due to the parasitic antennas adjacent to
the fed antenna. The same problem has to be solved when we
use more than one plasma antenna at one time which the
distance between antennas can not be neglected. To determine
the impedance or determine the induced current on the other
plasma columns, it is necessary to find the mutual impedance
between elements. So the study on mutual impedance of
plasma column antennas is very important to analyse the array
properties of plasma antennas.
In this paper, a theoretical model of two center-fed
parallel plasma columns of any length is set up based on the
physical properties of the plasma. The mutual impedance
between plasma antennas is calculated for the uniform and
nonuniform distribution of plasma column, and the results are
compared with the mutual impedance of metallic antennas.
The data shows that the mutual impedance between two
plasma antennas is smaller than that between two metallic
antennas, whether the plasma distribution for the antenna 2 is
uniform or non-uniform. And the impedance curves are
altered as the plasma frequency or collision frequency varies.
II．
Determination of mutual impedance between plasma
column antennas
A. modeling
Referring to Fig.1, two center-fed plasma column
antennas 1 and 2 which lengths are 2l1 and 2l2 respectively, are
separated by a distance d. Such antennas are constructed from
insulting tubes filled with low pressure gases. In circuit theory,
The mutual impedance of two coupled circuits is defined as
the negative of the radio of the emf V21 in circuit 2 to the I1
flowing in circuit 1 with circuit 2 open[7]. The mutual
impedance of plasma antennas is
Z 21  
V21
I1
(1)
where V21 is the emf induced across the terminals of plasma
antenna 2 by the conference current I1 in the plasma antenna
1. The open-circuit voltage of terminal 2 which results from
voltage induced by the antenna 1 may be found.
V21 
1
I 2m

0
 l2
Ez 1 I 2 ( z )dz   Ez 1 I 2 ( z )dz
l2
0

(2)
density of antenna 1 is uniform and high enough to work as a
metallic antenna, the expression for the parallel component of
electric field in the vicinity of a radiating element 1, is given
as[13]
 2 j cos  l1e j r0 je j r1 je j r2 
Ez1  30 I1m 


 (6)
r
r
r
0
1
2


r1
dz
l1
then the impedance of plasma antennas is
r0
Z12  30   e (l2  z ) sin  (l2  z ) Rdz
 0
(7)
0
  ( l2  z )

 e
sin  (l2  z ) Rdz

 l2
2 j cos  l1e j r0 je j r1 je j r2
where R 


r0
r1
r2
l2
l2
d
Fig.1 Two parallel plasma column antennas
For signal frequencies well below the plasma frequency,
the surface wave will propagate along cylindrical plasma
column. However, the wave vector of surface wave should
vary with the plasma density inside the dielectric tube [8][9].
For a uniform density and infinitely long plasma column, we
may obtain the plasma surface wave dispersion relation by
Helmholtz’s wave equation and the boundary condition, that
is[10]
 rT0 I1 (Tp a) K 0 (T0 a)  Tp K1 (T0 a) I 0 (Tp a)  0 (3)
T  k   k and T  k  k a is the radius
of the plasma column, k0   / c is the wave number is
free space. I i () and Ki () denote modified Bessel
where
function
2
p
2
of
the
2
r 0
first
2
0
and
 r  1   / (  ivm )
2
pe
second
2
0 ,
kind
0.6
0.2
0.0
-0.2
0.0
0.5
is
the
1.5
2.0
(a)
is the plasma dielectric constant,
2
 pe
 nee2 / me 0
1.0
d/
respectively.
vm is the electron-neutral collision frequency. The
quantity
metal antnnas
plsama antennas
0.4
electron
0.1
(4)
For the plasma antenna 2, when the plasma density is
very high enough to be used as a radiator , the plasma antenna
surface current distribution may be assumed to be sinusoidal
and given by[11][12]
 I e (l2  z ) sin  (l2  z ) 0  z  l2
I 2 ( z )   0  (l2  z )
sin  (l2  z )  l2  z  0
 I 0e
0.2
plasma
frequency, where ne is the plasma density. The wave vector
propagating along the plasma column is a complex quantity,
and it can be expressed as
k    j .
metal antnnas
plsama antennas
0.3
(5)
where  ,  are the attenuation coefficient and the phase
coefficient of the surface wave, respectively. I0 is the
maximum current of antenna 2. If we consider the plasma
Im[Z21],ohms
and
2
B. The mutual impedance for uniform distribution of plasma
density in antenna 2
Re[Z21],ohms
r2
0.0
-0.1
-0.2
-0.3
-0.4
0.0
0.5
1.0
d/
1.5
2.0
(b)
Fig 2. Curves of (a) mutual resistance R21 and (b) mutual reactance X21 of two
parallel side-by-side antennas as function of distance between them
A basic dimension of antenna 2 is selected as follows:
the radius of plasma column a2= a1 =0.0125m, the plasma
column length l2=l1=0.5m, the plasma density nz =51017/m3,
the frequency of propagation signal is 150 MHz，and the
electron-neutral collision frequency vm  5  10 Hz . From
the equation (7), we can get the mutual impedance between
8
When the plasma column length l2= l1=0.5m, the antenna
1 keeps the postulation ahead and the antenna 2 keeps basic
parameters to be unchanged, then we can get the mutual
impedance between the two plasma column antennas which
vary with the plasma density of plasma 2 and the distance
between two antennas, as shown in Fig3.
antenna 2, then we can get the mutual impedance between the
two plasma column antennas which vary with the plasma
collision frequency inside antenna 2 and the distance between
two antennas, as shown in Fig 4.
0.8
3
18
3
vm=1x10 /m
0.2
0.0
-0.2
-0.4
0.0
0.5
1.0
d/
1.5
2.0
(a)
18
3
18
3
17
3
nz=5x10 /m
0.4
0.2
nz=1x10 /m
nz=5x10 /m
0.2
Im[Z21],ohms
Re[Z21],ohms
3
18
vm=5x10 /m
0.4
0.6
0.0
0.0
19
3
18
3
18
3
vm=1x10 /m
-0.2
-0.2
0.0
19
vm=1x10 /m
0.6
Re[Z21],ohms
the two plasma column antennas, and compare with that
between two metallic antennas, both of which have the same
configure and size, as shown in figure 2. Simulation results
indicate that the variation character of plasma antennas’
mutual resistance, with the variation of distances between two
plasma antennas, is similar to Bessel’s function distribution.
The mutual reactance of plasma antennas displays a
sinusoidal distribution and its amplitude decreases gradually.
And the amplitudes of plasma antennas’ mutual resistance and
reactance are smaller than that of metallic antennas. The lower
mutual coupling of plasma antennas is beneficial to built
antenna array.
vm=5x10 /m
0.5
1.0
1.5
vm=1x10 /m
2.0
d/
-0.4
0.0
0.5
1.0
d/
1.5
2.0
(a)
(b)
Fig 4. Curves of (a) mutual resistance R21 and (b) mutual reactance X21 of two
parallel side-by-side plasma antennas with different collision frequency in
antenna 2
Im[Z21],ohms
0.2
0.0
18
3
18
3
17
3
nz=5x10 /m
-0.2
nz=1x10 /m
nz=5x10 /m
-0.4
0.0
0.5
1.0
1.5
2.0
d/
(b)
Fig 3. Curves of (a) mutual resistance R21 and (b) mutual reactance X21 of two
parallel side-by-side plasma antennas with different plasma density in antenna
2
It is the collision frequency that is relevant to plasma’s
conductive and dielectric properties, which impact antennas’
characteristics with using plasma as a radiator. When the
plasma column length l2= l1=1m, the antenna 1 keeps the
postulation ahead and the antenna 2 keeps basic parameters to
be unchanged except the plasma collision frequency inside
From Fig.3 and Fig.4, we find that the mutual resistance
of plasma antennas exhibits a vibrating character similar to
Bessel’s function distribution and the mutual reactance shows
a sinusoidal distribution with the variation of distance
between two plasma antennas, and amplitudes of mutual
resistance and reactance decrease evidently with the plasma’s
density descending. This may be explained from the physical
character of plasma. With the increasement of concentration
for plasma, the imaginary part of wave vector for surface
wave would become small and even may be neglected, and
the real part approximates to the wave number in vacuum. So
the mutual coupling of high density plasma antennas is more
intensive than that of lower density antennas. While the
collision frequency has less influence on the wave number
than on the damping rate, collision frequency increasing will
also lead to the damping rate of surface wave become bigger,
then cause amplitude of mutual impedance decreasing,
C. The mutual impedance for nonuniform distribution of
plasma density in antenna 2
At present, the plasma antenna is always driven from one
end of the column by excitation of the plasma surface wave.
So the plasma density in the column is non-uniform, and
decreases in an approximately linear fashion along plasma
column [8]. Let the plasma column length l2= l1=0.5m, the
antenna 1 keeps the postulation ahead, and the plasma density
of antenna 2 is described as
nz 2  nr 
n0  nr
 l2  z  ,
l2
where n0 =21018 /m3 is the plasma density excited at the base
of the column 2, nr =51017/m3 is the minimum plasma
density required to keep surface wave propagating. The
other basic parameters of antenna 2 keep same as before.
Then we can get the mutual impedance between the two
plasma column antennas which vary with the plasma density
of plasma 2 and the distance between two antennas. As shown
in Fig 5, with the variation of distance between two plasma
antennas, the mutual resistance of plasma antennas exhibits a
vibrating character similar to Bessel’s function distribution
and the mutual reactance shows a sinusoidal distribution. Due
to the non-uniform distribution of plasma in antenna 2, the
curves of mutual resistance and reactance are not smooth and
vibrate tempestuously at the extreme value.
0.6
plsama in the antenna2 is
nonuniform distribution
plsama in the antenna2 is
uniform distribution
Re[Z21],ohms
0.4
0.2
0.0
-0.2
0.0
0.5
1.0
1.5
2.0
d/
(a)
0.6
plsama in the antenna2 is
nonuniform distribution
plsama in the antenna2 is
uniform distribution
Im[Z21],ohms
0.4
0.2
0.0
-0.2
-0.4
0.0
0.5
1.0
1.5
2.0
d/
(b)
Fig 5. Curves of (a) mutual resistance R21 and (b) mutual reactance X21 of two
parallel side-by-side plasma antennas with uniform and nonuniform plasma
distribution in antenna 2
III. Conclusion
As for plasma antennas, the curve of mutual impedance
is similar to that of mental antennas with the increasing of
distance between two plasma antennas, and the amplitudes of
plasma antennas’ mutual resistance and reactance are less than
that of metallic antennas’. Due to physical characteristics of
plasma, the density and collision frequency of plasma in
antenna 2 impact the mutual impedance of plasma antenna
array, and the nonuniform distribution of density inside
plasma column 2 induces unsmoothed variation curves of
mutual impedance. The low mutual coupling of plasma
antennas is convenient for the construction of antenna array.
Furthermore, it is helpful to realize the beam forming and
antennas reconfiguration through the change of plasma
characters.
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