Handbook of Antenna Technologies
DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Linear Wire Antennas
Kazuhiro Hirasawa*
Institute of Information Sciences and Electronics, University of Tsukuba, Tsukuba, Ibaraki, Japan
Abstract
This chapter shows the basic characteristics of a linear wire antenna in time-harmonic electromagnetic
fields. The application of the method of moments is explained briefly to obtain the current distribution on
the antenna. For a transmitting antenna, input impedances, current distributions, and radiation patterns are
shown for four typical antenna lengths. For a receiving antenna, current distributions and reradiation
patterns are shown when the complex conjugate of the input impedance is loaded at the receiving point.
The received and reradiated power of the loaded receiving antenna is shown for three different plane-wave
incident angles. Also the application of the Thevenin equivalent circuit is discussed for the calculation of
the received power. Characteristics of a Yagi-Uda antenna are shown where the optimum gain of a
reactance-loaded Yagi-Uda antenna is compared with that of an end-fire array.
Keywords
The method of moments; Linear wire antenna; Transmitting antenna; Receiving antenna; Thevenin
equivalent circuit; Received power; Reradiated power; Scattered power; Reactance loading; Yagi-Uda
antenna; End-fire array; Optimization; Biquadratic programming method
Introduction
In this chapter time-harmonic electromagnetic fields are only considered. In section “History,” the brief
history of a linear dipole antenna development and analysis is introduced. The application of the method
of moments (MoM) to a linear wire antenna is explained in section “Analysis” where an electric-field
boundary value problem is solved to obtain the current distribution on the transmitting antenna. In section
“Transmitting Dipole Antenna,” transmitting antenna characteristics such as input impedances, current
distributions, and radiation patterns are shown for some antenna lengths. In section “Monopole Antenna,”
the equivalence between a monopole antenna on a large ground plane and a dipole antenna is explained. In
section “Receiving Dipole Antenna,” an MoM treatment of a receiving antenna with a plane-wave
incidence is shown where the input power, received power, and reradiated power are derived from the
incident voltage and the induced current on the antenna. Also the application of the MoM and the
Thevenin equivalent circuit is discussed for a receiving dipole antenna. In section “Received Power
and Reradiated Power,” the received power and the reradiated power are shown for a receiving antenna
when the complex conjugate of the input impedance is loaded. In section “Yagi-Uda Antenna,” characteristics of a Yagi-Uda antenna are presented where the optimum gain of a reactance-loaded Yagi-Uda
antenna is compared with that of an end-fire array.
*Email: hirasawa@ieee.org
Page 1 of 19
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# Springer Science+Business Media Singapore 2015
Fig. 1 A plate-loaded transmitting dipole antenna for Hertz’s experiment
Fig. 2 A transmitting monopole antenna on the earth for Marconi’s transatlantic wireless communication
History
In 1886 H. R. Hertz (Hertz 1962; Weeks 1968) used a plate-loaded dipole antenna (about 1.5 m long)
shown in Fig. 1 to transmit sparked electromagnetic waves in his experiment and showed the existence of
electromagnetic waves predicted in 1864 by J. C. Maxwell (1873). In 1901 G. Marconi built a fan-type
monopole antenna (about 45 m high) on the earth shown in Fig. 2. It consists of 50 vertical wires to
transmit the first transatlantic wireless signals from England to Canada (Weeks 1968). The wires are
supported by a horizontal wire with two vertical masts.
In 1906 de Forest invented the first triode vacuum tube that can continuously amplify higher-frequency
signals than those used by G. Marconi. Since then a half-wavelength dipole antenna has been used for
wireless communications. A Yagi-Uda antenna consisting of wires of about a half-wavelength is invented
by H. Yagi (1928) and S. Uda (1926).
E. Hallén, L. V. King, H. C. Pocklington, and R. W. P. King are early contributors to calculate the
current distribution on a linear wire antenna (Pocklington 1897; King 1937, 1956; Hallén 1938). The most
difficult part of the antenna analysis is to find the current distribution.
With the development of computers, frequency-domain numerical analysis methods such as the method
of moments (MoM) (Harrington 1993) have been widely used for the analysis of various antennas
(Hirasawa and Haneishi 1991; Balanis 2005; Stutzman and Thiele 2012; Kraus and Marhefka 2002).
Analysis
In this chapter time-harmonic electric and magnetic fields E and H are only considered and ejot is assumed
where o = 2p f (f, operating frequency). A transmitting straight wire antenna of length h and radius a in
free space (e 0, m 0) is considered as shown in Fig. 3. It is assumed that the antenna is lossless, a = 0.001l
(l is a free-space wavelength), and the feed gap is infinitesimal. Voltage V applied at the feed point
produces the current on the antenna. Once the current is known, it is easy to calculate the antenna
Page 2 of 19
Handbook of Antenna Technologies
DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Fig. 3 A dipole antenna on the z-axis with length h, radius a, and feed voltage V
characteristics such as input impedance and gain. A FORTRAN program (Hirasawa and Haneishi 1991)
may be used to obtain transmitting antenna characteristics shown in this chapter.
An equivalent theorem (Harrington 2001) is used to obtain the current distribution on a linear wire
antenna. In Fig. 4, a closed surface C is assumed just outside the antenna and the equivalent problem is
considered where E and H outside C are the same as the original problem and E and H inside C are forced
to be zero. This situation is realized by using impressed electric field Eimp and equivalent electric current
density Js
Js ¼ n^ H
(1)
^ is normal and outward to C. In Eq. 1, H is a magnetic field on C and Eimp on C is to
on C. The unit vector n
produce voltage V at the feed gap of the antenna. In the equivalent problem, the electric field inside C is
forced to be zero and conductors such as antennas inside C can be deleted. Therefore, Js can be in free
space and the free-space Green’s function can be used to solve the boundary value problem to obtain Js
with known Eimp.
In the equivalent problem shown in Fig. 4, the electric field just inside C becomes
¼0
E z ðJ z Þ þ E imp
z
(2)
is constant just inside and outside C. On
where it is assumed that Jz is an equivalent line current and Eimp
z
the other hand, just outside C the boundary condition becomes
(
E imp
z
¼
V
feed
d0
0 elsewhere
(3)
In Eqs. 2 and 3, the boundary condition is only considered on the z-component of the electric fields due to
is nonzero just outside the
the thin wire assumption (a = 0.001l). Also the impressed electric field Eimp
z
feed point, V is a known feed voltage, and d0 is the distance of an infinitesimal feed gap.
Next an antenna with one impedance load ZL is considered as shown in Fig. 5. The total electric field
becomes zero inside C as Eq. 2, but at the infinitesimal loading point E z þ E imp
is equal to ELz
E z þ E imp
z
z
due to ZL:
Page 3 of 19
Handbook of Antenna Technologies
DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Fig. 4 An original and the equivalent problem: both have the same electromagnetic fields outside region C
Fig. 5 A transmitting antenna with feed voltage V and load ZL
E z ðJ z Þ þ E imp
¼
z
n
ELz load
0elsewhere
(4)
Before the MoM is applied to the boundary conditions (Eqs. 2, 3, and 4), line current Jz with unknown
In is assumed as
0
J z ðz Þ ¼
N
X
I n J zn ðz0 Þ
(5)
n¼1
where N is the number of the expansion functions and the nth expansion function is
Page 4 of 19
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# Springer Science+Business Media Singapore 2015
Fig. 6 An expansion function for the current on the wire
8
sin k ðz0 zn1 Þ
>
>
zn1 z0 zn
>
>
< sin k ðzn zn1 Þ
J zn ðz0 Þ ¼
sin k ðznþ1 z0 Þ
>
zn z0 znþ1
>
>
sin
k
z
ð
z
Þ
>
nþ1
n
:
0 . . . . . . . . . . . . . . . . . . : elsewhere
(6)
and Dz ¼ h=ðN þ 1Þ, zn þ1 zn ¼ zn zn 1 ¼ Dz, k ¼ 2p=l (Fig. 6) (Stutzman and Thiele 2012).
Current coefficient In is a complex number and is determined to satisfy the boundary conditions in
Eqs. 2, 3, and 4.
For current Jzn, vector potential An(x,y,z) has only a z-component and electric field Ezn is expressed as
1 @ 2 Azn
E zn ¼ jom0 Azn þ
joe0 @z2
D
ðz
(7)
0
Azn ðx, y, zÞ ¼
Dz
J zn ðr0 Þejk jrr j 0
dz
4pjr r0 j
(8)
^ ^y, and ẑ are from the
Position vectors rð¼ xx^ þ y^
y þ z^
zÞ and r0 ¼ ðx0 x^ þ y0 y^ þ z0 z^Þ with unit vectors x,
0 0 0
origin to an observation point (x, y, z) on C and to a point (x , y , z ) on the line current, respectively. Electric
field Ezn (z) on the surface of cylinder C with radius a is due to current Jzn on the z-axis and becomes
jkrn1
j30
e
2 cos ðkDzÞejkrn ejkrnþ1
E zn ðzÞ ¼ þ
sin ðkDzÞ rn1
rn
rnþ1
(9)
where rn 1, rn, and rn+1 are the distances from points zn 1, zn, and zn+1 on the z0 -axis to a point z on the
cylinder C with radius a (Fig. 7) (Jordan and Balmain 1968).
An inner product
Dz
ð
hEn ðzÞ, Jw ðzÞi ¼
En ðzÞ Jw ðzÞdz
(10)
Dz
is chosen where En(z) is an electric field on C produced by current Jzn(z0 ) in Eq. 6. Jw(z) is a weighting
function on C.
Page 5 of 19
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DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Fig. 7 Electric field Ezn(z) on C (radius a) due to line current Jzn(z0 )
At first a transmitting antenna without impedance load ZL in Fig. 3 is considered to obtain the current
distribution on the antenna. A weighting function Jw(z) is chosen to be the same as the expansion function
in Eq. 6, and the MoM is applied to the boundary condition in Eqs. 2 and 3 with the inner product of Eq. 9.
Then, the simultaneous linear equations are derived as
N
X
Z mn I n ¼ V m ðm ¼ 1, 2, , N Þ
(11)
n¼1
where element Zmn of matrix [Z] is a mutual impedance between two small dipoles (section z n1 – z n+1 on
the z0 -axis and section z m1 – z m+1 on the surface C). The infinitesimal feed points are zn and zm. Mutual
impedance Zmn is obtained from Eq. 10 as
Z mn
ð zm
¼
zm1
sin k ðz zm1 Þ
E zn ðzÞdz sin k ðzm zm1 Þ
ð zmþ1
zm
sin k ðzmþ1 zÞ
E zn ðzÞdz
sin k ðzmþ1 zm Þ
(12)
Equation 12 can be calculated numerically by using a Gauss quadrature (Abramowitz and Stegun 1965),
and Eq. 11 can be written in a matrix form as
½Z ½I ¼ ½V (13)
where impedance matrix [Z] is N N and current matrix [I] and feed voltage matrix [V] are N 1.
Unknown current matrix [I] is obtained by solving the simultaneous linear equations.
Matrix [V] has only one nonzero element corresponding to a feed point. The feed point has to be chosen
to coincide with the peak of an expansion function (Fig. 6). Then, input impedance Zin becomes
Z in ¼
V
I
(14)
Page 6 of 19
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# Springer Science+Business Media Singapore 2015
where V and I are the elements of [V] and [I] in Eq. 13 corresponding to the feed point.
For a transmitting antenna with impedance load ZL shown in Fig. 5, the matrix equation is obtained
from the boundary conditions (3) and (4) as
½Z ½ I ¼ ½V þ V L
(15)
where ZL is loaded at the load point corresponding to the peak of an expansion function (Fig. 6). The mth
element of load voltage matrix [VL](N 1) has
ð zmþ1
L
E Lz J zm ðzÞdz
(16)
Vm ¼ zm1
¼ Z L I n ðn ¼ mÞ
(17)
where In is the nth element of the current matrix. From Eqs. 15 and 17, the matrix equation
Z þ Z L ½I ¼ ½V (18)
is derived where load matrix [ZL] is N N and the nth diagonal element ZLnn is only nonzero. In Eqs. 13
and 18, it is assumed that [V] is known and unknown [I] is obtained numerically.
Electric field Ey is the only component of the far electric field for the linear dipole (length h) located in
the z-axis direction and is obtained from Eq. 7 as
E y ¼ jom0 Az sin y
ð2
(19)
h
Az ¼
jkr
e
4pr
J z ðzÞejkz cos y dz
(20)
h2
Gain G is defined as
G ¼ 4pr
2 jE y j
2
0 P
(21)
pffiffiffiffiffiffiffiffiffiffiffi
In Eq. 21 0 ¼ m0 =e0, r is the distance from the antenna to the far electric field Ey and input power P of
an antenna is expressed by voltage V and current I at the feed point as
P ¼ ReðV I Þ
(22)
where Re and * denote “real part” and “complex conjugate,” respectively. Since Ey is proportional to 1/r
as shown in Eq. 20, G is constant with respect to r in the far field. Then, higher G gives higher |Ey| with
constant P as shown in Eq. 21.
Page 7 of 19
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Fig. 8 An antenna with a transmitter and the equivalent circuit where current I is at the feed point
Fig. 9 A dipole antenna with an offset feed
Transmitting Dipole Antenna
Figure 8 shows a transmitting dipole antenna with a transmitter and the equivalent circuit. Equivalent
circuits are often used to explain some antenna functions from the circuit point of view without using E
and H. The left-hand side of 1-10 is the Thevenin equivalent circuit of the transmitter that can be used to
obtain current I flowing terminal 1. Voltage Vg is the open-circuit voltage of the transmitter when output
terminals 1-10 are open circuited. Impedance Zg is the input impedance of the transmitter when the source
switch is off. Then, the complicated transmitter circuit is simply expressed by the voltage source and the
impedance if only current I flowing on the antenna is desired without knowing the details of the
transmitter. Impedance Zin (= Rin + jXin) is the input impedance of the antenna, and Rin is called the
radiation resistance closely related to the radiated power of the antenna. Thus, a transmitting dipole
antenna with feed voltage V is considered in this chapter as shown in Fig. 3 and the transmitter is omitted
for simplicity. Also it is recognized that voltage V is equal to Vg when Zg = 0.
An offset-feed dipole antenna (a = 0.001l) is shown in Fig. 9 to investigate the effects of the position
of the feed point. Length df is the distance to the feed from the bottom end of the dipole. The input
impedance of a half-wavelength dipole antenna is shown in Fig. 10 where df is changed from 0 to 0.5l. Rin
is very large near the end of the dipole due to the small current and becomes a minimum in the center due
to the current maximum. Xin becomes capacitive when the feed is near the end due to the large charge
accumulation at the end and inductive or the least capacitive at the center due to the current maximum.
Page 8 of 19
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Fig. 10 Input impedance Zin (= Rin + jXin) of a half-wavelength dipole antenna with an offset feed
Fig. 11 Input impedances Zin (= Rin + jXin) of a center-fed dipole antenna of length h
From the impedance matching point of view, it is recognized that the best position of the feed point is near
the center for the half-wavelength dipole, and this is one of the reasons why the center-fed halfwavelength dipole is widely used.
Figure 11 shows input impedance Zin of a center-fed dipole antenna (df = h/2, a = 0.001l) with
respect to h/l. The real part Rin is maximum around h = 0.9l and 1.9l. The imaginary part Xin is
capacitive when h is less than 0.5l. Xin becomes zero around 0.5l and 0.9l, 1.5l, 1.9l, and 2.5l. When Xin
is zero, it is called that the antenna is resonant. There are two kinds of resonance: series resonance (0.5l,
1.5l, and 2.5l) and parallel resonance (0.9l and 1.9l). For series resonance, Rin is not large, but Rin is very
large for the parallel resonance, and the dipole antenna is usually used around the series resonance,
especially around the first series resonance (h = 0.5l). Input resistance Rin is 70–100 [Ohm] around
h = 0.5l and good impedance matching can be obtained easily with the connected transmission line.
Figure 12 shows current distributions on dipoles (df = h/2) of h = 0.5l, l, 1.5l, and 2l where V = 1
[V]. The current amplitude has a peak every half-wavelength. When h = 0.5l and l, the phase is almost
constant except the feed point. When h = 1.5l and 2l, the phase change is about 180 [deg.] except
around the feed and the phase transition.
Figure 13 shows the normalized vertical radiation patterns |Ey| corresponding to the currents in Fig. 12.
The horizontal radiation patterns |Ey| are omnidirectional and are not shown here. When the phase of the
current is constant on a dipole antenna such as a half-wavelength or a one-wavelength dipole, there is only
one radiation peak, but there are more than one radiation peak when the current phase changes on dipoles
Page 9 of 19
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Fig. 12 Currents (amplitude and phase) on center-fed dipole antennas for h = 0.5l, 1l, 1.5l, and 2l (V = 1[V])
Fig. 13 Normalized vertical radiation patterns of the center-fed dipole antennas corresponding to the currents in Fig. 12
of h = 1.5l and 2l. Then, the radiated energy is distributed into more than one vertical direction. This
causes inefficient radiation in the desired direction and undesired radiation in the other directions.
Therefore, a half-wavelength dipole antenna is used widely due to its unidirectional vertical radiation
and easy impedance matching.
Page 10 of 19
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Fig. 14 A monopole antenna on an infinite ground plane and the equivalent dipole antenna
Monopole Antenna
A monopole antenna on a large ground plane such as the earth and building roofs can be used to make
antennas smaller without sacrificing communication quality. For the first transatlantic wireless communication, G. Marconi built a monopole antenna (45 m) where the earth is used as a large ground plane
(Fig. 2). The operating frequency of the transmitter is under 100 kHz and the antenna is still small in
wavelength (Weeks 1968).
A monopole antenna on an infinite ground plane is equivalent to a dipole antenna shown in Fig. 14. The
currents on the monopole and the electric field are the same as those of the dipole in the upper half-space.
Since the infinitesimal feed distance of the monopole is Dd and that of the dipole is 2Dd, voltage
imp
d
imp
is
V m ¼ E imp
z Dd at the monopole feed becomes half of V ¼ E z ð2Dd Þ at the dipole feed where Ez
d
the impressed electric field shown in Fig. 4. Thus, their input impedances Zm
and
Z
have
the
following
in
in
relation:
Zm
in
2V m V d Z din
¼ m ¼ d¼
2I
2
2I
(23)
Since V m = V d/2 and I m = I d, P m = P d/2 from Eq. 22. Then, from Eq. 21, the relation between
monopole and dipole gain G m and G d is obtained as
Gm ¼ 2Gd
(24)
Receiving Dipole Antenna
For a z-directed receiving antenna and the equivalent circuit shown in Fig. 15, it is assumed that a plane
wave Eimp is incident and impedance ZL is loaded at the receiving point on the antenna. Also Zin and Vo are
the input impedance and the open-circuit voltage of the receiving antenna, respectively. In Figs. 4 and 15,
the position of Eimp is on the feed point for the transmitting antenna and is outside for the receiving
antenna. Therefore, for the receiving antenna, the boundary condition is the same as Eq. 4 and the matrix
equation to obtain the current distribution becomes Eq. 18. It is assumed that the receiving point is
infinitesimal and corresponds to the peak of the expansion function (Fig. 6).
The incident plane wave is assumed as
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Fig. 15 A receiving antenna with a receiver and the equivalent circuit where current I is at the receiving point
Fig.
16 Currents (amplitude and phase) on center-loaded receiving dipole antennas for h = 0.5l, 1l, 1.5l, and
2l Z L ¼ Z in , E y imp ¼ 1½V=m, y ¼ 90
Eimp ðzÞ ¼ u^i ejki ^zz
(25)
where unit vector ûi specifies the polarization. Wave-number vector ki is in the direction of the incident
wave and jki j ¼ 2p=l. Then, the element of incident voltage matrix [V] becomes
ð zmþ1
J zm z^ u^i ejki ^zz dz
(26)
Vm ¼
zm1
Current distributions on center-loaded receiving dipoles of h = 0.5l, l, 1.5l, and 2l are shown in
Fig. 16 where it is assumed that electric field Ey imp (= 1 [V/m]) is normally incident (y = 90 ) and load
ZL = Zin* at the receiving point. Figure 16 can be compared to the current distributions of the transmitting
antennas in Fig. 12. The amplitude peak appears every half-wavelength that is similar to that of the
transmitting antenna for h = 0.5l, l, 1.5l. When h = 2l, there are only two amplitude peaks for the
receiving case instead of the four peaks for the transmitting one. This is because the current distribution as
well as the receiving characteristics is dependent on the incident wave and the load value at the receiving
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Fig. 17 Normalized vertical reradiation patterns of the center-loaded dipole antennas corresponding to the currents in Fig. 16
point (Hirasawa 1987). Figure 17 shows the normalized vertical reradiation pattern |Ey| corresponding to
the current in Fig. 16. If the reradiation and radiation patterns in Figs. 17 and 13 are compared, they are
similar for h = 0.5l and l where there is only one peak (y = 90 ). When h = 1.5l, the maximum is in
the y = 90 direction in Fig. 17 instead of y = 45 in Fig. 13. Also there are no deep nulls in the
reradiation pattern that are quite different from the radiation pattern. When h = 2l, there is a peak in the
y = 90 direction instead of the null in the radiation pattern. The horizontal reradiation patterns |Ey| are
omnidirectional and are not shown here. Also the receiving and the radiation patterns are the same for an
antenna due to the reciprocity theorem (Stutzman and Thiele 2012), and the receiving pattern is not
discussed here.
Open-circuit voltage Vo in Fig. 15 can be obtained from short-circuit current Is by the MoM. The
element of [I] corresponding to the receiving point in Eq. 18 becomes Is when the element of [ZL] is set to
zero:
V o ¼ Z in I s
(27)
The Thevenin equivalent circuit on the left-hand side of terminals 1-10 in Fig. 15 is valid only for the
calculation of the current and received power at ZL (Silver 1949). Therefore, in general, the power
consumed at Zin does not show the correct reradiated power. Similarly it is recognized that the electromagnetic fields in Fig. 14 are the same only above the upper half-space for the monopole and the dipole
problem. It is important to keep in mind the valid region of the equivalent problem.
The incident, received, and reradiated powers are respectively expressed as
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Pin ¼
N
X
Re V n I n
(28)
n¼1
Pre ¼ ReðZ L ÞjI j2
(29)
Pra ¼ Pin Pre
(30)
Reradiated power Pra can be also obtained by integrating power density (|Ey|2/0) over a large sphere. In
Eq. 29, I is the current at the receiving point where impedance ZL is loaded. Powers in Eqs. 28–30 are all
obtained by using the MoM. Also Pre can be obtained correctly by using the Thevenin equivalent circuit
shown in Fig. 15 as discussed above.
Received Power and Reradiated Power
Received and reradiated (or scattered) powers on the receiving dipole antenna have been investigated with
the relation to the MoM and the Thevenin equivalent circuit since 1994. The reradiated power and the
received power of a center-loaded dipole antenna and a Yagi-Uda antenna of load ZL = Zin* are
calculated by the MoM and the limitation of the Thevenin equivalent circuit is shown (Hirasawa
et al. 1994, 1997). Since then various discussions have been reported for a receiving antenna on the
Thevenin equivalent circuit and the reradiated power (Collin 2003; Onuki et al. 2007; Best and Kaanta
2009).
In this section, incident, received, and reradiated powers with respect to three incident angles y = 90 ,
60 , and 30 are shown for a center-loaded receiving dipole antenna of load ZL = Zin*. With this load
impedance, the maximum power is received at the load. It is also assumed that incident electric field
E y imp ¼ 1½V=m. Figure 18 shows received power Pre with respect to dipole length h. Also received
power Pre is obtained from the Thevenin equivalent circuit (Fig. 15) once the open-circuit voltage Vo is
obtained from Eq. 27. Figure 19 shows reradiated power Pra from Eq. 30 with respect to h. Pra can be also
obtained by integrating power density (|Ey|2/0) over a large sphere. In general Pra cannot be calculated
from the Thevenin equivalent since the circuit on the left-hand side of terminals 1-10 in Fig. 15 is
equivalent to the original problem only for the calculation of current I and Pre (Silver 1949; Collin
2003). Figure 20 shows Pre/Pra. The reradiated power becomes larger than the received power as
Fig. 18 Received power Pre with E y imp ¼ 1½V=m and ZL = Zin* for three incident angles y = 90 , 60 , and 30
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Fig. 19 Reradiated power Pra with E y imp ¼ 1½V=m and ZL = Zin* for three incident angles y = 90 , 60 , and 30
Fig. 20 Pre /Pra with Ey imp ¼ 1½V=m and ZL = Zin* for three incident angles y = 90 , 60 , and 30
h becomes larger than 0.9l, 0.6l, and 0.55l for incidence angles y = 90 , 60 , and 30 , respectively.
Figure 21 shows the current distribution (amplitude) for h = l and the current amplitude becomes almost
zero at the receiving point for y = 30 . Thus, the corresponding Pre and Pre/Pra becomes very small.
Figures 18, 19, and 20 show that the receiving antenna can receive and reradiate 50 % of the incident
power at best and receive less than 50 % of the incident power when h is larger than a certain length. Also
this can be explained by separating the current distribution on the receiving dipole antenna into the
coupling component and the non-coupling component. Then, the former contributes to the received power
at the load but the latter only contributes to the reradiated power (Onuki et al. 2007).
Yagi-Uda Antenna
A typical Yagi-Uda antenna of six elements (Yagi 1928; Uda 1926) is shown in Fig. 22. It consists of one
feed dipole, a reflector, and a director. The reflector is a wire located behind the feed dipole and is usually a
little longer than the feed dipole. The director consists of a few wires in front of the feed dipole, and they
are a little shorter than the feed dipole. The wire element spacing is 0.25l–0.375l where the mutual
coupling is strong between the elements. The reflector and director guide electromagnetic fields efficiently
in the x-axis direction. The antenna has a high gain in the x-axis direction with only one feed. The gain
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Fig. 21 Current amplitudes on the center-loaded receiving dipole antenna (h = 1l) with E y imp ¼ 1½V=m and ZL = Zin* for
three incident angles y = 90 , 60 , and 30
Fig. 22 A six-element Yagi-Uda antenna
Fig. 23 An equally spaced Yagi-Uda antenna with six half-wavelength elements and five reactance loads
increases gradually with the increase of the number of director wires. Due to the simple structure, it is
lightweight and easy to make. Thus, it is widely used as TVand FM receiving antennas in many countries.
Figure 23 shows a reactance-loaded Yagi-Uda antenna where the elements are equally spaced. Each
element is a half-wavelength wire with a = 0.001l. The reactance loads and the feed are located in the
center of the element. The current distribution is changed by loading a reactance component on a wire
except the feed wire. In Fig. 24 the optimum gain in the x-axis direction (y = 90 , f = 0 ) is compared
with that of the end-fire array shown in Fig. 25 where all elements are fed. The loaded reactance values for
the optimum gain are obtained by using a biquadratic programming method (Hirasawa 1980, 1987, 1988).
The difference between the optimum gains by reactance loads and feeds is 0.8 dB for element spacing
d = 0.35l as shown in Fig. 24. Figure 26 shows the normalized horizontal radiation patterns (y = 90 )
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Handbook of Antenna Technologies
DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Fig. 24 Optimum gains of a six-element Yagi-Uda antenna with optimum loads compared with those of an end-fire array
Fig. 25 An equally spaced end-fire array with half-wavelength elements
Fig. 26 Normalized horizontal radiation patterns of the optimum Yagi-Uda antenna and end-fire array when d = 0.35l
for a reactance-loaded Yagi-Uda antenna and the end-fire array when d = 0.35l. The sidelobes of the
end-fire array are a little bit lower than those of the optimized Yagi-Uda antenna and the slightly higher
gain is obtained by the end-fire array. Figures 27 and 28 show the input impedance and the loaded
reactance values for the optimum gain. As d gets smaller, Rin becomes smaller. Therefore,
d = 0.3l – 0.375l may be good in the point of gain and impedance matching. The loaded reactance
values for the optimum gain are all capacitive as shown in Fig. 28. It is quite difficult to realize the
optimum gain by adjusting six feed voltages accurately, but it is much easier to realize the optimum gain
of the Yagi-Uda antenna by using five capacitors.
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Handbook of Antenna Technologies
DOI 10.1007/978-981-4560-75-7_25-1
# Springer Science+Business Media Singapore 2015
Fig. 27 Input impedance Zin (= Rin + jXin) versus inter-element spacing d for the optimum gain of the Yagi-Uda antenna
Fig. 28 Optimum reactance loads for the optimum gain of the Yagi-Uda antenna
Conclusion
A brief history is shown for a linear wire antenna development and its frequency-domain analysis
methods. Then, the MoM is explained with the electric-field boundary conditions for a transmitting and
receiving linear wire antenna. By using the MoM, transmitting linear wire antenna characteristics such as
input impedances, current distributions, and radiation patterns are calculated. For receiving antennas,
current distributions, reradiation patterns, received power, and reradiated power are shown when a plane
wave is incident and the complex conjugate of the input impedance is loaded at the receiving point. Also
the application of the Thevenin equivalent circuit is discussed for the receiving antenna. At the end the
gain of a reactance-loaded Yagi-Uda antenna is shown and compared with that of an end-fire array.
Cross-References
▶ HF, VHF and UHF Antennas
▶ Impedance Matching and Baluns
▶ Numerical Modeling in Antenna Engineering
▶ Optimization Methods in Antenna Engineering
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