Download In this paper, formulas for calculating impedance parameters among

1
Determination of Impedance Parameters among Antennas near Objects
Yoon Goo Kim1) and Sangwook Nam2)
1)
Hanwha Thales, [email protected]
Dept. of Electrical and Computer Engineering, INMC, Seoul National University
1 Gwanak-ro, Gwanak-gu, Seoul, 151-744, Republic of Korea, [email protected]
2)
Abstract—In this paper, formulas are derived for calculating impedance parameters among ports of
antennas near objects. Equivalent circuit for a receiving antenna near objects is derived and is used to
derive formulas for impedance parameters. To calculate impedance parameters, current distribution on a
transmitting antenna that is alone, total current at port, and electric field generated by open-circuited
antennas are required.
I. INTRODUCTION
I
N array antennas, it is necessary to determine the relation
among voltages and currents at ports of antennas when
performing impedance matching and beam forming. The
relation between voltages and currents at ports of antennas can
be described using scattering parameters, impedance
parameters, and admittance parameters. In wireless power
transfer, once we know one of these parameters, we can
calculate transferred power, maximum power transfer
efficiency and a load impedance that maximize the power
transfer efficiency. Therefore it is important to calculate
scattering parameters, impedance parameters, or admittance
parameters among ports of antennas in the analysis of array
antenna and wireless power transfer.
One method to calculate these parameters is a full-wave
simulation. Using a full-wave simulation, we cannot
understand factors that affect mutual coupling, so analytical
methods are needed. One analytical methods for calculating
scattering, impedance, and admittance parameters for finite
array is to use generalized scattering matrix based on spherical
waves [1]–[5]. In this method, when minimum sphere that
encloses an antenna overlap objects or minimum sphere of
another antenna, it is impossible to calculate network
parameters. Therefore, another method is required in this case.
In addition, time that takes to calculate network parameters
using generalized scattering matrix is long.
One analytical method to calculate impedance parameters for
finite arrays is induced EMF method [6], [7]. In the previous
induced EMF method, impedance parameters are calculated
when antennas are in free space. In practice, objects such as
ground and substrate exist near antennas. Therefore, to
calculate impedance parameters exactly, we should consider
objects near antennas. In addition, when self impedance is
calculated using induced EMF method, the result for the case of
lossy antenna is not exact while the result for the case of
lossless antenna is exact. Therefore, it is needed to improve the
induced EMF method.
In this paper, formulas for calculating impedance parameters
among ports of lossy or lossless antennas near objects are
derived. Throughout this paper, it is assume that an antenna has
one port.
II. EQUIVALENT CIRCUIT FOR AN ANTENNA
When an antenna is excited with a source at a feed port and
there is no incident field, current and voltage at a feed port can
be calculated using Thevenin equivalent circuit as shown in Fig.
1. In Fig. 1, ZA is the input impedance seen at the feed port of an
antenna; Zg is source impedance; Vg is the voltage of a source.
Suppose that there are an antenna terminated in a load and
objects and electromagnetic field generated by currents is
incident on an antenna and objects (Fig. 2). It is assumed that a
load is connected at an infinitesimal gap on conducting wire.
Current flowing at a load and voltage across a load for this
situation can be calculated using Thevenin equivalent circuit as
shown in Fig. 3. In Fig. 3, ZA is input impedance of an antenna
in the presence of objects and in the absence of currents, ZL is
load impedance, and Voc is open-circuit voltage. The formula
calculating open-circuit voltage Voc can be derived from
reciprocity theorem [6]. The formula calculating open-circuit
voltage is as follows:
1
V oc   t  Es (r )  J t (r)dv
(1)
I G
where Jt is electric current density of an antenna when its input
terminals are excited in the presence of objects and in the
absence of current; Es is electric field generated when objects
and currents are present and an antenna is absent; r is the
position of a point in currents of an antenna; It is total current at
input terminals when current density of an antenna is Jt. In [6],
open-circuit voltage was derived when antennas were in free
space.
When self- and mutual-impedances are derived, the
Thevenin equivalent circuit for a transmitting and receiving
antenna will be exploited.
III. INPUT IMPEDANCE OF AN ANTENNA NEAR OBJECTS
In this paper, objects that are present near antennas are
2
classified into two types. One kind of object will be called as
‘environment’ and the other kind of object will be called as
‘scatterer’. An environment is fixed objects and this always
exists in the process of calculating impedance parameters. In
this section, the difference between the input impedance of an
antenna in an environment in the absence of scatterers and the
input impedance of an antenna in the presence of environment
and scatterers will be derived.
When there are an antenna and environment, and there are
not scatterers, an antenna in transmitting mode can be modeled
using Thevenin equivalent circuit in Fig. 1. In this case, ZA is
the input impedance seen at input terminals of an antenna in the
presence of environment and in the absence of scatterers.
When an antenna in the presence of environment and
scatterers is excited with a source at its feed port, the
electromagnetic field transmitted from the antenna is scattered
by scatterers, and the scattered field is incident on the antenna.
In this case, the voltage due to the incident field as well as the
voltage due to a source is produced at the feed port. This can be
modeled such that a voltage source due to an incident field is
connected to ZA in the Thevenin equivalent circuit for a
transmitting antenna, as shown in Fig. 5. The voltage generated
by this voltage source is the same as the open-circuit voltage Voc
in the Thevenin equivalent circuit for a receiving antenna and is
calculated using (1). When Voc is calculated using (1), Es is an
electric field generated by scatterers in the presence of
environment (electric field generated by an antenna is not
included). Currents of scatterers are determined in the situation
where all antenna, environment and scatterers are present and
antenna is excited at feed port.
The voltage at a feed port of an antenna will be denoted by
Vp. Solving the circuit in Fig. 5, the current flowing at ZA is
obtained. Letting the current flowing at ZA be Ip,
V p  V oc
Ip 
(2)
ZA
The input impedance of an antenna is the ratio of the voltage
at a feed port to the current at a feed port. Therefore, the input
impedance of an antenna in the presence of environment and
scatterers, Z Ae , is
V pZ
Vp
(3)
 p Aoc
p
I
V V
Note that ZA is the input impedance of an antenna in
environment in the absence of scatterers. Calculating Z Ae  Z A ,
Z Ae 
V pZA
V oc Z A
V oc
(4)

Z


A
V p  V oc
V p  V oc
Ip
Substituting (1) into (4), a formula for the difference between
the input impedance of an antenna in environment in the
absence of scatterers and the input impedance of an antenna in
the presence of environment and scatterers is obtained as
follows:
1
Z Ae  Z A   p t  Es (r)  J t (r)dv
(5)
I I G
Note that Es is proportional to Ip and Jt is proportional to It.
Note that Ip and Es are independent of It and Jt. Es can be
Z Ae  Z A 
considered an electric field generated by scatterers and
environment when a current of Ip is applied to the input
terminals of an antenna.
Before calculating the input impedance of an antenna in the
presence of environment and scatterers ( Z Ae ), the input
impedance of an antenna that is present alone in environment
(ZA) should be calculated using a numerical or analytical
method.
IV. SELF IMPEDANCE FOR COUPLED ANTENNAS
The relation among voltages and currents at ports can be
described using impedance parameters as follows:
N
Vm   Z mn I n
(6)
n 1
where Vm is voltage at mth port, In is current at nth port, and Zmn
are impedance parameters. The impedance parameters can be
determined as
V
Z mn  m
when I i  0 for i  n
(7)
In
In this paper, Zmn is called a self-impedance when m = n. The
self-impedance Zmm is the same as the impedance seen looking
into an antenna at the feed port of mth antenna when all other
antennas are open-circuited. A self-impedance can be
calculated using the same method as used in the calculation of
the input impedance of an antenna near objects. All antennas
except mth antenna are open-circuited, and the input impedance
of mth antenna is calculated using (5) to obtain Zmm. Zmm is
given by
1
Z mm  Z Am 
E( m, m ) (r)  J tm (r)dv
(8)
I m I mt G
where Z Am is the input impedance of mth antenna that is alone
in an environment (i.e., in the absence of other antennas and
scatterers); J tm is the current density of mth antenna that is
alone in environment and excited at input terminals; I mt is total
current flowing at the input terminals of mth antenna in
transmitting mode when the current density is J tm ; and E ( m , m )
is the electric field generated by open-circuited antennas and
objects when mth antenna is excited with a current of Im at its
input terminals.
Using a reciprocity theorem [8, eq. (3-36)], (8) can be written
in another form as follows:
1
Z mm  Z Am 
Etm (r)  J ( m, m ) (r)dv
(9)
I m I mt G
where E tm is the electric field generated when mth antenna is
excited at feed port in a situation where only mth antenna and
environment exist and all other antennas and scatterers do not
exist; J( m, m) is the current density of objects and antennas
except for mth
antenna when all other antennas are
open-circuited and mth antenna is excited with a current of Im,
i.e., J( m, m) generates E ( m , m ) .
The self-impedance is calculated in a situation where all
3
antennas except one are open-circuited. If antennas are far
enough, magnitude of field scattered by open-circuited antenna
are small. Therefore, when antennas are far enough, the
self-impedance Zmm is similar to the input impedance of mth
antenna in the presence of objects and in the absence of other
antennas.
V. MUTUAL IMPEDANCE FOR COUPLED ANTENNAS
When m ≠ n, Zmn in (6) is called a mutual-impedance. In (7),
Vm is the same as the voltage at port of mth antenna with
open-circuited when nth antenna is excited at its port with
current of In and all other antennas are open-circuited. In (7), Vm
is the same as open-circuit voltage in the Thevenin equivalent
circuit for mth antenna in receiving mode and open-circuit
voltage can be calculated using (1). From (1) and (7), the
mutual-impedance, Zmn, is as follows:
1
Z mn  
E( m, n ) (r )  J tm (r )dv for m  n
(10)
I n I mt G
where E( m, n ) is the electric field generated by objects and
antennas except for mth antenna when all antennas except for
nth antenna are open-circuited and nth antenna is excited with a
current of In at its input terminals. Note that E( m, n ) is
proportional to In and J tm is proportional to I mt . Note that In
and E( m, n ) are independent of I mt and J tm .
Using a reciprocity theorem [8, eq. (3-36)], (10) can be
written in another form as follows:
1
Z mn  
Etm (r )  J ( m, n ) (r )dv for m  n
(11)
I n I mt G
where J ( m, n) is the current density of objects and antennas
except mth antenna when all antennas except nth antenna are
open-circuited and nth antenna is excited with current of In, i.e.,
J ( m, n) generates E( m, n ) . If antennas are dipole antennas and the
number of antennas is two, (11) is the same as equation (7.135)
in [7].
To calculate the mutual impedances using the method
presented in this paper, we should know the current
distributions of all antennas and objects. In general, the current
distribution is calculated using a numerical method in the
presence of all antennas and objects. However, in some cases,
we can predict the current distributions of antennas without
calculating the currents in the presence of all antennas and
objects. Unless antennas and objects are very close, the shape
of the current of a transmitting antenna in the presence of
scatterers and open-circuited antennas are similar to the shape
of the current of a transmitting antenna that is present alone in
environment. Therefore, in this case, the impedance parameters
among antennas can be calculated using only the current
distributions of transmitting antennas that is present alone in
environment.
VI. VALIDATION
We calculate impedance parameters among antennas using
the formulas presented in this paper and EM simulator FEKO,
and compare the results obtained with two methods. In
simulation, three dipole antennas are on infinite dielectric. Half
of space is free space and half of space is dielectric. Dielectric
constant and loos tangent of dielectric are 10 and 0.1,
respectively. Length of one dipole antenna is 15 cm, length of
another dipole antenna is 20 cm, and length of the other antenna
is 25 cm. Radius of wire in all dipole antennas are 0.1 mm. All
dipole antennas are made of copper. All dipole antennas are fed
at its center. Feed ports are ordered such that the first port is
port of 25 cm dipole antenna, the second port is port of 20 cm
dipole antenna, and third port is port of 15 cm dipole antenna.
The three dipole antennas are parallel and the line connecting
the feed ports of the three dipole antennas are perpendicular to
the three dipole antennas.
We calculated Z22 and Z32 from 500 MHz to 1.5 GHz in two
cases. In one case, the distance between the centers of two
dipole antennas (d in Fig. 7) is 1 cm. In the other case, the
distance between the centers of two dipole antennas (d in Fig. 7)
is 8 cm. For both cases, J2t and J3t in (8) and (10) were
calculated when the dipole antenna was alone in half space.
When d is 1 cm, E(2,2) in (8) and E(3,2) in (10) was calculated in
the case where 20 cm dipole antenna was excited at its feed port
and 25 cm dipole antenna and 15 cm dipole antenna were
open-circuited. When d is 8 cm, E(3,2) in (10) was calculated in
the case where 20 cm dipole antenna was alone in half space (i.e.
E(3,2) is electric field generated by J t2 ).
Fig. 8 (a) and (b) shows Z22 and Fig. 8 (c) and (d) shows Z32
when d is 1 cm. Fig. 9 (a) and (b) shows Z22 and Fig. 9 (c) and (d)
shows Z32 when d is 8 cm. In Fig. 8 and 9, Z22 and Z32 were
calculated with FEKO and the formula presented in this paper.
In Fig. 9 (a), the graph for Z22 calculated with FEKO and graph
for the input impedance of 20 cm dipole antenna when it is
alone in half space are shown. In Fig. 9, we can identify that Z22
is similar to the input impedance of 20 cm dipole antenna when
it is alone in half space. Furthermore, we can identify that
mutual impedance can be calculated with small error using only
current distribution of transmitting antenna that is alone in half
space.
VII. CONCLUSION
In this paper, formulas for calculating impedance parameters
among ports of antennas are derived. Equivalent circuit for a
receiving antenna near objects is derived and is used to derive
formulas for calculating impedance parameters. To calculate
impedance parameters, current distribution on a transmitting
antenna that is alone, total current at port, and electric field
generated by open-circuited antennas are required.
REFERENCES
[1]
J. E. Hansen, Spherical Near-field Antenna Measurements, London,
U.K.: Peregrinus, 1988
4
[2]
[3]
[4]
[5]
[6]
[7]
[8]
J. Rubio, M. A. Gaonzalez, and J. Zapata, “Generalized-scattering-matrix
analysis of a class of finite arrays of coupled antennas by using 3-D FEM
and spherical mode expansion,” IEEE Trans. Antennas Propag., vol. 53,
no. 3, pp. 1133–1144, Mar. 2005.
R. J. Pirkl, “Spherical wave scattering matrix description of antenna
coupling in arbitrary environments,” IEEE Trans. Antennas and
Propag., vol. 60, no.12, pp. 5654–5662, Dec. 2012.
W. Wasylkiwskyj and W. K. Kahn, “Scattering properties and mutual
coupling of antennas with prescribed radiation pattern,” IEEE Trans.
Antennas Propag., vol. 18, no. 6, pp. 741-752, Nov. 1970.
Y. G. Kim, “An analytical model for the scattering and coupling of
antennas and its application to wireless energy transfer,” Ph.D.
dissertation, Dept. Elect. Eng., Seoul National Univ., Seoul, Korea, 2015,
ch. 4.
E. C. Jordan and K.G. Balmain, Electromagnetic Waves and
Radiating Systems, 2nd ed., Englewood Cliffs, USA:
Prentice-Hall, 1968.
R.S. Elliott, Antenna theory and design, Revised Edition, Hoboken, N.J.,
USA: A John Wiley & Sons, 2003, ch. 7.
R. F. Harrington, Time-Harmonic Electromagnetic Fields, New York,
USA: McGraw-Hill, 1961.
Fig. 3. Thevenin equivalent circuit for a receiving antenna
(a)
Fig. 1. Thevenin equivalent circuit for a transmitting antenna
(b)
Fig. 4 Method for calculating open-circuit voltage (a) Situation
where Jt is determined (b) Situation where Es is determined
Fig. 2 Loaded antenna and object
electromagnetic field generated by current
illuminated
by
5
Fig. 5 Equivalent circuit for an antenna near objects
Fig. 7 Simulation configuration.
1600
FEKO
Formula
1400
(a)
Re(Z22) ()
1200
1000
800
600
400
200
0
500
600
700
800
900
1000
1100
1200
1300
1400
1500
Frequency (MHz)
(a)
1000
(b)
FEKO
Formula
800
600
Im(Z22) ()
400
200
0
-200
-400
-600
(c)
Fig. 6 Method for calculating mutual impedance Zmn (a)
Situation where impedance parameters among antennas are
determined (b) Situation where J tm is determined (c) Situation
where Em,n is determined
-800
500
600
700
800
900
1000
1100
Frequency (MHz)
(b)
1200
1300
1400
1500
6
500
500
FEKO
Formula
400
FEKO
Input impedance
400
300
300
Im(Z22) ()
Re(Z32) ()
200
200
100
0
100
0
-100
-200
-100
-300
-200
-300
500
-400
600
700
800
900
1000
1100
1200
1300
1400
-500
500
1500
600
700
800
Frequency (MHz)
900
1000
1100
1200
1300
1400
1500
Frequency (MHz)
(c)
(b)
200
60
FEKO
Formula
100
FEKO
Formula
40
20
0
Re(Z32) ()
Im(Z32) ()
0
-100
-200
-20
-40
-60
-300
-80
-400
-500
500
-100
600
700
800
900
1000
1100
1200
1300
1400
-120
500
1500
600
700
800
Frequency (MHz)
900
1000
1100
1200
1300
1400
1500
Frequency (MHz)
(d)
Fig. 8 Z22 and Z32 when d is 1 cm (a) real part of Z22 (b)
imaginary part of Z22 (c) real part of Z32 (c) imaginary part of
Z32
(c)
100
FEKO
Formula
80
60
1100
900
Re(Z22) ()
800
40
Im(Z32) ()
FEKO
Input impedance
1000
20
0
-20
700
-40
600
500
-60
400
-80
500
200
100
500
600
700
800
900
1000
1100
1200
1300
1400
1500
Frequency (MHz)
300
600
700
800
900
1000
1100
Frequency (MHz)
(a)
1200
1300
1400
1500
(d)
Fig. 9 Z22 and Z32 when d is 8 cm (a) real part of Z22 (b)
imaginary part of Z22 (c) real part of Z32 (c) imaginary part of
Z32