Some Thoughts About Transient Radiation by Straight Thin Wires Rafael Gdmez Martin, Amelia Rubio Bretones, and Salvador Gondlez Garcia Electromagnetic Group of Granada Dept. Fisica Aplicada Facultad de Ciencias University of Granada 18071 Granada Spain Tel: +34-958-243224 Fax: +34-958-243214 E-mail:
[email protected],
[email protected],
[email protected]
Keywords: Electromagneticradiation, transient radiation, physics, wire antennas
3. Fields created by a timedependent charge distribution Suppose a hounded distribution of time-dependent sources in free space is as shown in Figure 1. The retarded electric ( ( F , t )
1. Abstract
The primary goal ofthis paper is tutorial. It discusses, from a physical point of view, several basic features related to the transient radiation of electromagnetic waves by straight thin-wire
and magnetic P
= P(r,O,q)
A(r,t)
potentials at the observation point
and timet are given by
2. Introduction
N
owadays, there is a growing interest in the study of the radiation . ' and propagation of transient electromagneticwaves. It has avvlications in areas such as the develovment of radar tech.. niques with high-resolution target discrimination, electromagnetic compatibility (EMC) and interference (EMI) problems, and in the design of radio-communication equipment with a relatively large bandwidth [I]. Consequently, numerical techniques based on the solution of Maxwell's equations in the time domain have increased dramatically [Z]. However, although a time-domain approach can throw light on the problem in a way that is not possible using the conventional frequency-domain approach [3], very little attention has been paid to the theoretical hasis of transient electromagnetic wave radiation [4-81 and, therefore, all antenna theory rests on the assumption of harmonic waves [S-111. This situation is explained by the fact that the time-domain analysis is much more complicated than the harmonic. Nevertheless, as shown in this paper and in 1121, which presents ideas complementary to those given here, there we cases where a sirnplificd analysis is possible. The goal of this tutorial paper is to comment, from a physical point of view, on several basic features related to the radiation of transient electromagnetic waves by straight thin-wire antennas. With this purpose we initially obtain the expressions of the fields created by a hounded distribution of time-dependent sources in Gee space, and particularize them to find the Litnard-Wiechert expression of the fields due to a point charge in arbitrary motion. All the above will he used to study some basic properties of the electric field radiated by a traveling-wave thin-wire antenna (Section 4) and by a dipole thin-wire antenna (Section 5), excited by a transient cumnt. 24
1045-92431991$10.000 1999 IEEE
A(P,t)=- Po J-W, 151 412,. R
where 7 and p are the current and charge densities, respectively, 3 = P - 7 is the vector from the source point ( 7 )to the field point (P ), and, as is usual and as will he used henceforth, the square brackets (1 1) indicate that the magnitude inside them must be evaluated at the retarded time f' = t - R/c , where c is the velocity of light in free space. That is,
[PI = P(P',t') = p(i',t R / c ) ,
(3)
[TI = ?(?,t') = ?(7',t
(4)
~
-R/c).
Introducing Equations (1) and (2) in the expression of the fields a(7,t) and s(?,t)in terms of the retarded potentials, given by
B(7,t) = vx . q P , f ) ,
(6)
we obtain the fields from an integral of the source currents and charges over the volume V' as (see Appendix 1) %(P,t)=
iEEE A n t6' m a s and Propagation Magazine. V o l . 41, NO. 3. June 1999
r’
. R (t’)
=
F-F’(t’) Retarded and present
positions
‘e,”
.\
where the last term of each equation
of 4
S(t). trajectory of q Origin
Figure 2. A point charge in arbitrary motion.
where G = C ( J , t j is the velocity of the point charge, A - k ( ( t ) is specifies the radiated field in terms of the time variation of the current. Observe also that Equations (7) and (8) reduce directly to the Coulomb and Biot-Savart laws in the static limit, The fields created by a single point charge q can be obtained from Equations (7) and (8) by expressing [ p ] and [I]in terms of the delta functions as [I41
[PI=483 ( ? , - L q t ) , ) ,
(11)
[Il=4iG163(J-[S(t)l),
(‘2)
K=[l-p.li],
p=G/c,and y=(l-,b
)
1/2
Z= & / a t ,
.
The last term in Equation (13),
{( 8)x a},
i x ri Erad(“t) = 2 4 f f ~ o K ~ ‘ c2R
(15)
shows that the radiated far field is proportional to the charge acceleration.
where 7‘ = ‘ ( t ) is the radius vector describing the trajectory S ( t ) of the charge (see Figure 2). Then, Equations (7) and (8) simplify to the LiBnard-Wiechert fields [I51 created by a point charge in arbitrary motion (see Appendix 2):
e
the unit vector from the charge to the observation point,
IP
In the next section, Expressions (9) and (15) are used to explain some interesting features of the transient radiation field of straight thin-wire antennas. Although from a macroscopic point of view, Expression (9) might be sufficient to explain these features, some physical insight is gained by also considering Equation (15) to relate them to the more fundamental physical point of view of the charge acceleration. As is well known, Formulas (13-15) can be particularized for the two significant cases of synchrotron radiation, where the charge velocity and acceleration are perpendicular (point charge in circular motion), and bremsstvahlung radiation, where the acceleration is parallel to the velocity. In this case Equation (13) simplifies to AY
(iXi.)
Erad( F , t j = 4 1 7 1 4 f f ~ ~ K c~ R It is evident that for a straight thin-wire antenna, the velocity and acceleration of the charges propagating along the antenna are always parallel.
4. Traveling-wave thin-wire antenna
Figure 1. A bounded distribution of time-dependent current ( 7 )and charge ( p ) densities.
The radiation field, Equation (9), can be particularized for a traveling-wave thin-wire antenna with no reflection at the end and with the current approximated by an arbitrary transient pulse of
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25
~
then, we obtain the expression of the radiated field created by a traveling-wave antenna excited by an arhitrruy signal as
zf= 1
where qo sin% A ( 8 ) = -~ 4?ir 1- cos8 '
Z' t; = f - 71. and U ( t ) is the step function
U ( ( )=
0 for t < 0 1 for t > 0
z'=O Current at the feed point: I(t), 0 4 t I AT Figure 3. A traveling-wave thin-wire antenna.
t duration AT ( I ( t ) = 0 for t S 0 or t > A T ) that travels, with no deformation, along the wire at the velocity corresponding to the exterior medium. Then, the usual far-field limit, where R is much larger than the dimension o f the source (see Figure 3), allows us to approximate the retarded-time variation by a linear function of z', and the radiation field simplifies to [6-XI E,,d ( r , O , t ) Q =q- o s i n 8 I-dz'B, 'aI(z) 4?icr at I
0
is the impedance of free space,
2'
T = f'--
c
with
z'lc being the time that the current pulse takes in propagating a
distance t ' = f-- R
z' f
-
from r-z'cos8 c
c
time
T
at z'
the
origin
along
the
Figure 4. The electric field radiated by a traveling-wave antenna 1 m long, as a function of time, for three different observation angles.
wire,
, and I ( T ) is the current at the retarded
=0 .
Observe that the apparent spatial width (ASW) of the current, defined as the difference between the values of z' corresponding to ?=0 andr=AT.is
ASW=-
CAT 1- cos8 '
and depends on the angle 0 of observation. Integration o f Equation (17) can be very easily canied out taking into account Equation (18) and that
Figure 5. The critical angle versus the length, I, o f the wire, for a traveling-wave antenna excited by a current pulse of duration AT = 2.86 x 10-9 s. 26
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a) 1=0.25 m;
20°, 909 150'(always overlapping) Era,
b) 1=1 m;
20°, 81.84'
Elad
(ec), 150' Erad
c) 1=20m;
lo', 16.84' (Oc), 100'
Figure 6 . The amplitude of the radiated electric field for a traveling-wave antenna (normalized by qo/4ar) versus time (in units of 0.25/c) for different lengths of the antenna and different observation angles. The dashed lines are each of the two pulses defined in Equation (20), the sum of which gives the total radiated field. Expression (20) is very simple, and consists of the sum of just two terms, both multiplied by A ( O ) , and each one having the same
3.) The radiated-field shape alters from two pulses of amplitude A ( 0 ) to a single pulse when O AT
(26)
where g = 1 . 5 x l O Y s ~ ' and t , = 1 . 4 3 ~ 1 0 - ~ s Figure 4 shows the shape (for an angle 8 less than, equal to, or greater than 0,) of the electric field radiated hy an antmma 1 m long. Observe from Equation (24) that, for this case, 0, = 81.84" Figure 5 shows how B, decreases with the length of the wire. Notice that, for this Gaussian pulse, the maximum length for which there is always overlapping is 1, =0.429m. Figure 6 represents
s s ; according to Equation current of duration AT = 2 . 8 6 ~ 1 O ~ ~then, (24), there is no critical angle, and therefore there will always be overlapping. In the two remaining cases in Figure 6, the plots on the lefl show the radiated field for a 0 0,. It can be ohsewed again that, as the length of the antenna increases, the critical angle 8, decreases, while the distance between the two pulses increases in such a way that at the limit of I + m , there will only he a single pulse, due to the initial acceleration of the charges at the feed point. The question of why the critical angle exists can be explained by comparing the value of the paramckr ASW as a Cuiction of 0 (Equation (18)) to the length I of the antenna. Three possibilities arise (ASW greater than, equal to, or less than I), which are represented in Figures 7a, 7h, and 7c, respectively. In these figures, the lengths of the lines with mows are proportional to the ASW. The dotted lines show either the pari of the current pulse that has not yet entered the antenna, or that has already passed it. In each case, three different time steps are represented as follows. Figure 7a: 8 > 0, (ASW< f ) : In this case, after the positive pulse due to the initial charge acceleration, the current propagates along the antenna without radiating until it reaches the end of the wire, where the charges are decelerated and the radiation of the negative pulse takes place.
the amplitude (normalized by r70/(4nr))of the field radiated by the traveling-wave antenna plotted against time (in units of 0 . 2 5 1 ~ ) when we change both the length and the angle of observation. In all cases, the dashed lines represent the two pulses contained in
,. . . . . . . . . . . ASW = 1 z*= 0 8 =Bc
b)
> 7!
Figure 7h: 0 = 0, (ASW= 1): This case is similar to the previous one, hut now, just when the positive pulse finishes, the negative one begins.
z z*= 1
=n
7,
=*1
>
zf = 0
2,= I
. . . . . . . . . . ..>
4
,. . . . . . . . . . . . . ASW > 1 8
>.
2=0
z' = 1
z' = 0
z'=l
.............
.,
Figure 7. Diagrams representing a current pulse, with apparent space width ASW, propagating along the antenna at three different time steps: (a) The case with the observation angle B > 8,,(b) B = B, ,(e) 0 < 0,. 28
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z'=O
e Figure Sa. The mean power function of a traveling-wave antenna plotted against the observation angle, 6 , for 5 = Ig/c = 20 (where I is the antenna length and g is the Gaussian-pulse parameter).
Figure 9. The geometry of a dipole antenna of length L = 21, fed at its center.
Figure 7c: 0 < 0, (ASW> 0: Here, before the positive pulse has finished, the negative pulse begins, and there is an overlapping of the two pulses.
0 Figure 8b. The mean power function of a traveling-wave antenna plotted against the observation angle, 0 , for 5 = lg/c = 200.
e= lg/c
The mean power function of the traveling antenna, defined as [6, 71
(&
W
( r ,0))=
S E k ( r .O,i-)dr
(27)
has also been computed. It is represented as a function of 0 in Figure 8, for various values of the length of the antenna. It can he seen that the maximum radiated power always occurs at 6 = 6,
5. The dipole antenna
0 Figure 8c. The mean power function of a traveling-wave antenna plotted against the observation angle, 0 , for 6 = Ig/c = 1000.
The above ideas and results can be easily extended to study the field radiated by a center-fed dipole antenna of length L = 21, excited by a current pulse of duration AT (Figure 9). We can obtain the expression of such a radiated field by taking into account the reflections of the current pulse (deceleration and acceleration of charges) that take piace each time it anives at the end of each arm, and introducing a new reflection coefficient at the feed point that considers the losses occuning as the pulse retums from the end of the antenna to the feed point [6, 7, 181. Notice also that 0 changes to the complementary value when the current propagates in the -2 direction. Then, we arrive at the general expression of the radiated electric field [7]:
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29
0.01
Reflection Theory Numerical Solution - . -
A
~ ~ ~ ~
where A'(Q)=*-
sin 6 4ar (1 + cos6)
N is the number of reflections that have been taken into account,
0
1
(ro
2
1.5
2.5
Figure 11. The input snsceptance of a dipole antenna of length L = I m and radius a = 6.14 mm, plotted versus the electrical size L / A .
(30)
i odd
/&I)
i even
b )(i-l)/2
i odd
ry2rp
1
i even
(i-I)l/c
ri =
0.5
Llh
i odd
(i-l-cos(6))i/c
1
-0.01
and the following coefficients are defined
rb
where r, and are the reflection coefficients at the ends and at the feed point, respectively, which, strictly speaking, are frequency dependent variables. Nevertheless, there are several ways to approximate them by constant values [6, 7, 181. In this paper, for the example of a center-fed linear dipole antenna of length I = 1m and radius a = 0.00674m, we use the values given in [7], that is, r, = 0.9 and T b = 0.89, which give, on average, the best results in the frequency range analyzed. For this antenna, the conductance and susceptance have been computed by applying a Fourier transformation to the time-domain data. They are plotted in Figures 10 and 11, versus the normalized length o f the dipole antenna ( L I A ) . Figures 12 and 13 show the radiated fields at 6 = 90" (Figure 12) and at 0 = S O 9 (Figure 13). In all the figures, the solid lines repre-
4
Time - Uc
6
8
io
0.15
Refiection Theory Numerical Solution ~ ~
0.014 0.012
~
~
~0.1 .
.
I
E
0.01
z
0.05
dD
o
2 cr
-0.05
D -
.-
s
2
Figure 12. The radiated electric field of the dipole antenna plotted versus time normalized to Llc. The length of the antenna is L = 1m and the radius a = 6.74mm; it is excited by a Gaussian current pulse. The angle of observation is 6 = 90".
0.016
2
I
~0.25 0
0.006
0.004
~0.1
0.002 ~..,
0
I
,
0.5
1
1.5
2
2.5
3
Ll h
Figure 10. The input conductance of a dipole antenna of length L = 1m and radius a = 6.74 mm, plotted versus the electrical size, L / A . 30
-0.15
0
2
4
Time - Llc
6
8
10
Figure 13. The radiated electric field of the dipole antenna plotted versus time normalized to L l c . The length of the antenna is L = 1m and the radius a = 6.74mm; it is excited by a Gaussian current pulse. The angle of observation is 8 = SO".
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sent the results obtained with the simplified analysis described here, and the dashed lines give the results obtained by numerical analysis [19], which have been included for comparison. It can he seen that the results obtained by the simplified meihod agree quite closely with those obtained by numerical analysis. Notice, from Equation (ZS), that if somehow (e.g., loading the antenna with a resistive profile [ZO]) the current pulse is attenuated (with no reflection) along the wire, the dipole antenna will only radiate the initial feed-centered pulse, due to the initial acceleration of the charges at the feed point. The discussion presented in this section can be extended to the case of dipole antennas with the feed point located at an arbitrary position on the wire, and can also be applied to calculate the field scattered by a thin wire excited by a transient plane wave. Moreover, the results can be improved by using numerical techniques to calculate the reflection coefficients [18]. In any case, it should he pointed out that the approximation of the current in Equation (17) is based on neglecting the dispersion of the single current pulse as it propagates along the wire. The deformation of the pulses increases as the length of the antenna increases [SI. Therefore, the simplified analysis presented is acceptable provided that I is not large compared to C A T ,or if we are only interested in the early-time behavior of the fields.
Substituting Equations (31) and (32) into Equation ( 5 ) . we obtain
and finally Equation (7) by calculating [ V ' - j I as
9. Appendix B: Derivation of Expression (13) The fields created by a moving charge can be obtained from those due to a time-dependent p and f source distribution at a fixed location by substituting Equations (11) and (12) into Equation (7), which gives
6. Conclusions
This paper presented several basic features related to the radiation of transient electromagnetic waves by straight thin-wire antennas excited by a transient current. The analysis was based on the approximation that the current pulse propagates along the antenna with no deformation. Then, for a traveling-wave antennaand, by extension, for a center-fed dipole antenna-the analytical expression of the radiated field as a function of the angle of observation was obtained. For antennas that are not long compared to the ASW of the transient current, the results obtained using this simplified method agree quite closely with those obtained using numerical methods. To discuss the characteristics of the radiated field from a physical point of view, we first obtained the fields due to a timedependent hounded distribution of sources at a fixed location and from them, the Libnard-Wiecherl fields due to a point charge in arbitrary motion. Then, we used these results to analyze the shape of the radiated field as a function of the angle of observation, the length of the antenna, and the duration of the current pulse.
7. Acknowledgement
where A = I(/R is the unit vector from the position ofthe charge to the observation point (Figure 3). Because of thc delta function, the factors inside the integral in Equation (35) come outside and are evaluated at the retarded position and time, there only remaining the volume iniegral of the delta function, which simplifies to [21]
To obtain the Libnard-Wiechert fields, it is necessary to carry out the following derivatives, which take into account the implicit dependence of all the retarded terms on 7 and t :
at'
1
- 1
The authors thank Dr. E. K. Miller for his help in discussing some ideas behind this work. This paper was partially supported by CICYT through the project TIC-95-0783-C03-01 8. Appendix A: Derivation of Expression (7)
(37)
The gradient of the retarded charge, Equation (3), can be written as
and the continuity equation as IEttAntennos a n d P r o p a g a t i o n Magazine, V o l . 41. N o . 3. J u n e 1999
31
Then, from Equations (35) to (37) and after some laborious algebra, we obtain the Lienard-Wicchert fields, Equations (13) and (14). In this process, it can be observed that the radiation term, Equation (9). created by the bounded distribution of sources, when particularized to a point charge, not only produces the radiation teim in Equation (13), but also gives the following contribution to the near field:
14. D. J. Griffiths and M. A. Heald, “Time-Dependent Generalizations of the Biot-Savart and Coulomb 1aws;’Am. J Phys., 59, Fehmary 1991,pp. 111-117. 15. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, New York, Addison-Wesley, 1962.
16. E. K. Miller, “Radiation physics,” IEEE Antennas and Propagation Magazine, 38, June 1996, pp. 91-93.
which can be neglected if the velocity of the particle is low in comparison to c ( p
