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An Exact Time Domain Evaluation For Radiated Fields From A Hertz Dipole

A. ArifErgin* Department of Electronics Engineering,

Gebze Institute of Technology Gebze/Kocaeli

aergin@gyte.edu.tr

Abstract-This paper provides an exact derivation of the radiated

fields of a Hertz dipole in time domain. The usual derivation of the

frequency domain dipole fields involves neglecting the phase

variation across the dipole and hence, it is not exact. The time domain derivations presented here do not involve any

approximations. The relations between time domain and frequency

domain fields of an infinitesimal dipole are studied by taking the limit of the exact field expressions as the dipole length goes to zero.

Keywords - time domain; infinitesimal dipole; radiated fields, antenna.

I. INTRODUCTION

While dealing with the electromagnetic problems, computing the radiated fields due to the current distribution is one of the main concerns. This computation is necessary while analyzing the radiation pattern of an antenna or evaluating the radar cross section (ReS) of a platform. Infinitesimal dipoles are linear wire antennas whose length (tu.) and radius (r) are very smaIl compared to the wavelength; (tu.« 'A and r« 'A). As an antenna, infinitesimal dipoles are not very practical, nevertheless they have great importance while modeling any complex geometry. Due to this importance, most of the antenna or electromagnetic books include the subject. To the best knowledge of the authors, all the related documents handle the subject in the frequency domain (Le., phasors), and the phase variation over the dipole length is neglected. Therefore, the obtained field expressions are not exact even before the far-field approximations are applied. This paper presents an exact time domain derivation of the fields radiated by an impulsively excited Hertz dipole of length tu.. The dipole is assumed to carry constant current along its length. In order to verify the obtained results, the relation between the derived time domain and (customarily obtained) frequency domain fields are compared as the dipole length tends to zero. First, the geometry will be presented and the formulas for the radiated fields of a Hertz dipole will be derived. Then, the relation between time domain and frequency domain presentations will be established as the dipole length goes to zero.

This research is partially funded by the GEBIP program of the Turkish Academy of Sciences (TUBA).

Soner Karaca Department of Electronics Engineering,

Gebze Institute of Technology GebzelKocaeli

II. DERIVATION OF DIPOLE FIELDS IN THE TIME DOMAIN

It is important to examine the radiation properties of small or short dipoles, since any practical antenna which has a large size, may be considered as consisting of a large number of small or short dipoles. With this point of view, the infinitesimal dipoles become more valuable because they may be utilized as building blocks of complex geometries [1,2].

The general geometry for a Hertz dipole is shown in Figure-I. It will be important to remind that, if the dipole is vanishingly short, it is called an infinitesimal dipole. It is assumed that the medium surrounding the infinitesimal dipole is free space. The mid-point of the dipole is located on the origin, and it lies along the z-axis. The cylindrical coordinate system is preferred during the evaluation. The notations with the prime symbol are used for denoting the electric current locations and the unprimed vectors are used for denoting the observation points.

z t

·f····_·······

L1L ••

t ...

r- r'

r=pllp+ZIIZ r '=z' lIz

r - r' = P II p + (z-z') liz Ir - r'l = .J P 2 + (z_z,)2

observation point

Figure 1. Geometry for a Hertz dipole.

Using the vector magnetic potential, A, is a very common procedure to find the fields radiated by the electric current

978-1-4244-7243-7/10/$26.00 ©2010 IEEE

distribution, J. The time domain definition of the electric current distribution along the dipole is

J(r',t) = 11= 100(t)P(z',1ll)0(x')0(y� (1)

where the amplitude function, P, is defined as

Iz'I�� Iz'l>lllh (2)

The retarded vector magnetic potential, due to this electric current distribution is given by

o(t-�) A(r,t) = f.lo fJ(r',t) * c dr' .

4n v Ir- r'l

(3)

By using (1) in (3), a new time domain expression for retarded vector magnetic potential can be derived as

MJi o(t � p2 +(z_z,)2 )

A(r,t) = I1z f.lo 10 f � C dz' 4n -MJi p2 +(z_z,)2

(4)

After multiplying (4) by (Ill/Ill) and reorganizing the result, the vector magnetic potential can be written as

(5)

where Mo = 10 AL is the dipole moment

The only unknown part of this equation is the integral part, given by

MJi o(t _ � p2 + (z_z,)2 )

I(r,t) = f C dz' -MJi � p2 +(z_z,)2

(6)

The main effort is now computing I(r,t) while trying to understand the physics. By changing the variable, I (r,t) can be arranged as

b I(r,t) = f o(k)

dk a �(k - t)2 - (�i (7a)

where the condition that a < 0 < b should be satisfied. In fact a and b can be calculated as

71

Equation (7) is derived by assuming that the observer resides at a location for which z > III / 2 . Equation (7) can be evaluated easily by using the Dirac delta function properties and the integral tables, however the limits should be handled with care by considering their physical meaning.

The solution of (7) can be found as [3, 4]

.7(,,1)= �,, _

(%, )2 ;

(8)

c c

Note that I(r,t) = 0 outside the given temporal span.

By using (8) in (5), the time domain vector magnetic potential for a Hertz dipole can be written as

(9)

c c

The exact magnetic fields for the Hertz dipole in time domain can be easily derived using

1 H(r,t) =-V'xA(r,t). (10)

f.lo

III. COMPARISON WITH FREQUENCY DOMAIN FORMULAE

Plots of the z component of the vector magnetic potentials, which are evaluated for a constant observation point and different values of Ill, with respect to time are given in Figure-2. The location of the observation point is (p=1,¢,=0,z=2) in cylindrical coordinates, where the dipole is located at the origin and lies along the z axis. The length of the dipole varies from 0.050 to 0.925 by 0.125 increments. The highest amplitude is obtained for the shortest length and the lowest amplitude is obtained for the longest length. For shorter dipoles the temporal span converges to zero while the amplitude goes to infinity. Due to the great amplitude difference for the dipoles shorter than 0.050, it is difficult to plot the amplitude in the same figure. This is the only reason for selection of the dipole lengths between 0.050 and 0.925. The center of the peaks, which is the mid-point of the lowest and highest limit of the peaks, gives the retardation time.

•• Observation Point: (p= I, ;= O,z = 2), .8 Dipole Location : (p= 0, ¢= O,z= 0)' .7

� :: � � ..

.3 '

. ,

. ,

L1L = 0,050 -----:-±

°3�----�-----7------7-� TIME (in seconds) xlO.Q

Figure 2. Plots of the z component of the vector magnetic potential with respect to time.

After the examination of these plots, it is easy to see that vector magnetic potential has a character very similar to Dirac delta function while ill goes to zero. At that point, it is required to evaluate the area beneath the vector potential to check if it is constant or not for the limit of ill, where ill goes to zero. The mathematical formulation and the result of the process can be given by

1P'+(z+/ll,h)'

S = lim 8z . A(r,t) dt /ll,�0

P'+(z-/ll,h)' c

4n r

(11)

The result of the process given by (11) shows that, the area beneath the vector magnetic potential is constant and directly proportional to the dipole moment, Mo. This result and the vision, acquired by the plots, encouraged the authors to obtain an exact formula for an infinitesimal dipole, which has not been handled till this study. By focusing on the retardation time obtained from the plots and (11), the exact time domain vector magnetic potential for an infinitesimal dipole can be derived as

(12)

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The Fourier Transform of (12), which is a well-known formula for the frequency domain magnetic vector potential of an infinitesimal dipole and given in text books under the assumption that the phase variation is negligible across the dipole [1, 2], can be found as

A J.Lo M o -jkr A(r) = az --e .

4n r

(l3)

The derivation of the exact time domain formula can be verified by the consistency between the time domain and frequency domain. The magnetic field intensity radiated from an infinitesimal dipole in time domain can be given as

1 H(r,t) = -Vx A(r,t) Po

A Mo . 0 c c

[r5(t-�) 8 (r5(t-�»)1 =a¢--sm - .

4nr r 8r

(14)

To check the validity, the Fourier Transform of (14), which is given in text books [1, 2] as

M ( -jkr ) A • e . - 'kr H(r) =a¢-o-smO --+ Jk e ] , 4nr r

can be used.

IV. CONCLUSIONS

(15)

By this study, a novel exact time domain evaluation for radiated fields from a Hertz dipole has been acquired. To verify the obtained result, fields of an infinitesimal dipole are obtained by taking the limit as the dipole length tends to zero. The obtained vector potential and magnetic field expressions are consistent with the frequency.domain formulation given in text books and obtained under the assumption that the phase variation is negligible across the dipole [1, 2]. The novel exact time domain derivation can be thought as a unit impulse response and used for the different electric current variations in time.

REFERENCES [I] 1. D. Kraus and R. 1. Marhefka, Antennas: For All Applications, 3rd ed.,

McGraw-Hili: New York, 2003. [2] C. A. Balanis, Antenna Theory, 3rd ed., Wiley-Interscience: New

Jersey, 2005. [3] H. B. Dwight, Tables Of Integrals And Other Mathematical Data, 4th ed.,

Macmillan: New York, 1961. [4] 1. J. Tuma, Engineering Mathematics Handbook, 3rd ed., McGraw-Hili:

New York, 1987.