ON AN ELECTRON BEAM EXCITATION OF 
LINEAR IMPEDANCE ANTENNA 
S.D. Priymenko, Yu.F. Lonin, L.A. Bondarenko, I.N. Onishchenko
Institute for Plasma Electronics and New Methods of Acceleration, National Science Center 
“Kharkov Institute of Physics and Technology”,
1 Academicheskya St., Kharkov, 61108, Ukraine;
E-mail: spriyemenko@kipt.kharkov.ua
The excitation of linear impedance antenna by an electron beam in an infinite space is considered. The charge of 
beam is concentrated on one end of the antenna and excites frequency spectrum of its radiation. The system of inte-
gro-differential equations for symmetric and antisymmetric Fourier current components with respective boundary 
conditions is obtained. The solution of the system can be found by the averaging method. The Green function of in-
finite space and the Fourier transformation are used to calculate the strength of electric and magnetic radiation fields 
of the antenna.
PACS: 84.40.Ba:29.17.+w
Intensive  development  of  nonstationary  electrody-
namics [1] in last decades arouses considerable interest 
to impulse - radiating antennas (IRA) or ultra wide-band 
(UWB) antennas.  These  antennas  find  applications  in 
the  pulse  radio-location  [2],  ground  penetrating  radar 
(GPR) [3], [4], war developments [5], wide-band com-
munication [6]. Original and nonconventional variant of 
UWB antenna  is  the  antenna,  exited  by  the  electron 
beam [7].
The problem of UWB electromagnetic radiation for-
mation  includes  two components:  the  antenna  system 
and the method of its excitation. The curvilinear wire 
antenna is perspective type of UWB antenna system, be-
cause one  − wire line, which practically have not fre-
quency  dispersion,  is  its  basic  element.  The  electron 
beam  is  an  efficient  instrument  of  the  excitation  by 
charge the antenna. The advantages of the excitation by 
charge are the followings:
− excitation of the whole spectrum of frequencies, 
which are inherent to the antenna system (excita-
tion into the end-wall);
− creation of corresponding boundary conditions for 
the radiated antenna, which is opened at the end 
(maximum of charge).
The  linear  impedance  antenna  is  considered  as  a 
base model (Fig.).
Boundary conditions are 
)();0( 0 tQLxtQ δ=−== , (1)
0);0( =+== LxtQ  (2)
for the linear density of charge at the ends of an antenna 
with  consideration  for  the  time-space  distribution  of 
beam charge on the antenna surface. 
Using the inverse Fourier transformation by the time 
variable
dtetQQ tiωω −
+ ∞
∞−
∫= )()( , (3)
boundary conditions (1), (2) are rewritten as
0);( QLxQ =−=ω , 
0);( =+= LxQ ω . 
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-L +LO x
z
y
beam
Symmetric (s) and antisymmetric (a) parts on the 
space variable are extracted for the spectral component 
of the linear density of charge 
)),(),((2
1),()( xQxQxQ s −+= ωωω , (4)
)),()1(),((2
1),()( xQxQxQ a −−+= ωωω . (5)
Hence we have
,2
1)(2
1
),(),(
0
)()(
QQ
LQLQ ss
==
=+=−
ω
ωω
 (6)
.2
1)(2
1
),()1(),(
0
)()(
QQ
LQLQ aa
==
=+−=−
ω
ωω
 (7)
Taking into account the relation [8] (p. 57)
),(),( xQidx
xdJ
ωω
ω
= , (8)
the problem of excitation of the antenna by the arbitrary 
charge ( 0Q ) is reduced to the problem of excitation by 
the spectral component of partial derivative with respect 
to space variable for the electric current along the anten-
na.
As this takes place, the symmetric  ),()( xJ s ω  and 
antisymmetric  ),()( xJ a ω  spectral components of the 
current strength along the antenna are described by the 
system of integro-differential equations [9]
( ) ( )
[ [ ] [ ] [ ]
( ) ( ) ]
( ) ( )
[ [ ] [ ] [ ]
( ) ( ) ] )10(
)9(
,,1
,,~,
,,
,,1
,,~,
,,
)(
0
)()()()(
0
)()(
0
)(
2
2
2
)(2
)(
0
)()()()(
0
)()(
0
)(
2
2
2
)(2
xJZi
JxFJxFJxF
xJcdx
xJd
xJZi
JxFJxFJxF
xJcdx
xJd
a
ssasas
a
a
s
ssssss
s
s
ω
α
ω
ε µ
ωω
ω
α
ω
ε µ
ωω
⋅−+
+++=
=+
⋅−+
+++=
=+


ωεε
ωεε
where  µε ,  are,  respectively,  permittivity  and perme-
ability of  a  medium, in which an antenna is  situated; 
[ ] 1-)/2ln(2
1 aL=α  ( a  is the radius of a vibrator) is 
the  small  parameter  (α <<1); components  of  type [ ])()(0 , ss JxF  are the integro-differential operators; Z  
is the internal impedance of a vibrator ( a
ZZ s
pi2

= , sZ  
is the surface impedance of a vibrator).
This system is complemented by the boundary con-
ditions
( ) ( )
,2
1)(2
1
,,
0
)()(
QiQi
dx
LdJ
dx
LdJ ss
ωωω
ωω
==
=
+
=
−
(11)
( ) ( )
,2
1)(2
1
,)1(,
0
)()(
QiQi
dx
LdJ
dx
LdJ aa
ωωω
ωω
==
=
+
−=
−
(12)
which follow from (6), (7), (8).
The solution of system (9), (10) with boundary con-
ditions (11), (12) provides in the asymptotic approxima-
tion (with a precision to values of the order 2α ) a spec-
tral component of current strength along the antenna 
( ) ( ) ( )xJxJxJ as ,,, )()( ωωω += , (13)
where ( )xJ s ,)( ω  and ( )xJ a ,)( ω  describe a spectrum 
of symmetrical and antisymmetric oscillations of current 
strength.
The strength of electric and magnetic fields, which 
are radiated by the spectral component of current along 
the antenna [10] (p.10)
( )
,)(),(
)()(,
222
)(
2
2
222
xdzyxx
exJ
crdivrgradrE
L
L
zyxxci
′
++′−
′×
×



+=
∫+
−
++′−
ε µ
ω
ω
ε µ
ω
ω

 (14)
( ) ( )
.)(),(
)(1,
222
)(
0
222
xdzyxx
exJ
rrotirH
L
L
zyxxci
′
++′−
′×
×−=
∫+
−
++′−
ε µ
ω
ω
εεωω 
(15)
Thus, in the linear impedance antenna, which is exit-
ed by the beam, the time dependence of electric current 
is
ωω
pi
ω dexJxtJ ti∫+ ∞
∞−
= ),(2
1),( , (16)
and in the space-time coordinates radiated the electro-
magnetic field is
ωω
pi
ω derErtE ti∫+ ∞
∞−
= ),(2
1),(  , (17)
ωω
pi
ω derHrtH ti∫+ ∞
∞−
= ),(2
1),(  . (18)
Notice that in the case of a curvilinear antenna the 
system  (9),  (10)  becomes  coupled.  The  analysis  of 
curvilinear impedance antenna of two-dimentional and 
three-dimentional  space  configuration  can  be  reduced 
by the above-given method to the analysis of two and 
three coupled linear impedance antennas, respectively. 
This work was executed under the information sup-
port of the project INTAS 00-02.
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2004. № 4.
Серия: Плазменная электроника и новые методы ускорения (4), с. 24-26.25
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Russian).
8. G.T. Markov,  D.M. Sazonov.  Antennas.  M.: 
“Energy”. 1975, p.528 (in Russian).
9. S.D.Priyemenko,  L.A.Bondarenko  Linear 
impedance  vibrator  in  the  circular  waveguide // 
Telecommunication and Radio Engineering. 2002, 
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О ВОЗБУЖДЕНИИ ЭЛЕКТРОННЫМ ПУЧКОМ ЛИНЕЙНОЙ АНТЕННЫ 
С.Д. Прийменко, Ю.Ф. Лонин, Л.А. Бондаренко, И.Н. Онищенко
Рассмотрено возбуждение линейной антенны электронным пучком в неограниченном пространстве. За-
ряд пучка концентрируется на одном из концов антенны и возбуждает частотный спектр ее излучения. По-
лучена  система  интегро-дифференциальных  уравнений  для  симметричной  и  антисимметричной  Фурье 
компонент силы тока с соответствующими граничными условиями. Решение системы может быть найдено 
методом усреднения. Функция Грина неограниченного пространства и преобразование Фурье используются 
для получения напряженности электрического и магнитного полей, излучаемых антенной. 
ПРО ЗБУДЖЕННЯ ЕЛЕКТРОННИМ ПУЧКОМ ЛІНІЙНОЇ АНТЕНИ 
С.Д. Прийменко, Ю.Ф. Лонін, Л.О. Бондаренко, І.M. Оніщенко
Розглянуто  збудження  лінійної  антени  електронним  пучком  у  необмеженому  просторі.  Заряд  пучка 
концентрується  на  одному  з  кінців  антени  і  збуджує  частотний  спектр  її  випромінювання.  Отримано 
систему інтегро-диференційних рівнянь для симетричної і антисиметричної Фур'є компонент сили струму з 
відповідними граничними умовами. Розв’язок системи може бути знайдено методом усереднення. Функція 
Гріна  необмеженого  простору  і  перетворення  Фур'є  використовуються  для  одержання  напруженості 
електричного і магнітного полів, що випромінюються антеною. 
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2004. № 4.
Серия: Плазменная электроника и новые методы ускорения (4), с.5-7. 26