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MODELLING STUDY OF COAXIAL COLLINEAR ANTENNA ARRAY

John Litva, Yuan Zhuang and Andrew Liang

Communications Research Laboratory McMaster University

[ABSTRACT] A theoretical and experimental study of the coaxial collinear (COCO) antenna are presented. The research is concentrated on the analysis of large and planar arrays which consist of foam-filled cable with plastic housing. An equivalent transmission line model is used to establish the integral equation which suitably describes COCO antenna arrays. The Conjugate Gradient with Fast Fourier Transform (CG-FFT) method is used for solving the large linear equation. To study the effect of plastic housing, the Finite Difference Time Domain (FD-TD) method is used. The computer simulation and experimental results are presented, and they are in very good agreement.

1. INTRODUCTION

The coaxial collinear(COC0) antenna has been used successfully for many years in a number of applications, such as radar, windprofilers, and communication antennas[l]-[3]. A COCO antenna array consists of lengths of coaxial lines connected together with their inner- and outer- conductors transposed at each junctions, as shown in Fig.!. For most applications, the lengths of each section are equal to Xd2. where Xg denotes the wavelength in the coaxial line. There are a number of advantages of these types of arrays. First, by transposing the inner and outer conductors, the phase of the current on the outer conductor is forced to remain roughly constant. This enhances the radiation of the antenna because there is less destructive interference between current elements with opposing phases. Another major advantage of such array is an N x N COCO array has N feed points, whereas a N x N dipole array has feed points. In addition, COCO antennas are lightweight, portable, easily- erected and reasonably inexpensive.

Some theoretical descriptions of COCO antenna have been attempted[4]- [6]. But there is still a need for an accurate model of the antenna, especially for very large COCO arrays. In this paper, we develop a model for large COCO arrays by carrying out theoretical analysis, computer simulations and experimental studies. We have taken into account the following factors: the parasitic effects from the connections at the junctions between elements; the closed encasing foam and fibreglass radome; the reactive effects due to the termination at end of each linear COCO linear array; and the effect of a ground plane.

In order to efficiently solve the very large linear system, the CG- FFT method is implemented. This technique can save computer memory(O(N?)) and computing time (O(N+Nlog,(N))) , making it possible to analyze very large COCO planar arrays. We have simulated and experimentally studied a number of COCO antenna arrays, such as a 12 element COCO linear array and a 12 by 9 COCO planar array. The current distribution and radiation patterns have been measured on these arrays and are compared with analyzed results. Excellent agreement is obtained. By combining theoretical analysis, computer simulations and experimental study, we present an approach for calculating the input impedance of COCO arrays with the consideration of the effects mentioned above.

To investigate the effect from the enclosed foam and the fibreglass radome, the finite-difference time-domain (FDTD) method has been used. By discretizing the conductor and dielectric into some small grids, a antenna with multilayer dielectric housing can be analyzed directly. The input impedances of this kind of antenna are calculated and compared with that of the antenna without dielectric housing.

2. THEORY

2.1 EQUIVALENT TRANSMISSION LINE M O D E L OF COCO ANTENNAS

A major determining characteristic of a COCO antenna is introduced by the interchange of its inner and outer conductor at each junction. To represent the COCO antennas in a clear and easily-used way, a equivalent transmission line model has been developed (2). (See Fig.2.)

For a lossless COCO antenna with half-wavelength sections, it can be derived, using transmission line theory, that the voltage at each junction is

v, = v, = v, = ... = V"-,

The admittance at each junction is composed of two components., the radiation admittance y,, and a shunt admittance yin due to parasitic between the elements, yn=yrn+yIm. If a COCO is fed at its nth junction, the input impedance of this feedpoint is the sum of radiation admittance at the feed point y,. shunt admittance y, and circuit admittance of the coaxial line, Y,(left side) and Y,(right side)

yh=Y,+YJ+ y,+ yn

Y, and Y, are the sum of y,'s and y,'s at every junctions converted from left side and right side to the nth junction via transmission line. Therefore beginning from the first junction of both sides, we can obtain the circuit admittance junction by junction and eventually determine Y, and Y, recursively. For the fundamental COCO antenna, that is I , + ... =I,.,=hJ2, if the coaxial line is lossless, then

We assume all the shunt admittances for compensation are equal(because they are dominated mainly by the junction structure). The reactive radiation effect Yedad is involved in the value of the radiation admittances for the end junctions. When the current distribution along the outside of coaxial line is obtained using the CG-FFT method, the radiation admittance yc,, yr2, ....yrN., can be calculated using the current at each junction, . The shunt admittance needed to compensate for yj and radiation admittance ye- at end can be determined by experiment.

2.3.THE INTEGRAL EQUATION The basis of understanding the characteristic of an antenna is its

current distribution since the other parameters can be calculated from the current distribution with ease. Assuming a & d , where a is the radius of the COCO element h is the wavelength in free space, we can use Pocklinton integral equation,

where G(z.2') is the Green function in free-space or over ground; E,Yz) is the radiated field due to the current I(z') on the antenna surface, resulting from an impressed or source field E:(z) generated by a voltage V i applied at the antenna terminals. I, means to integrate along the wire antenna.

Conclusions are drown in the last section. CCECEKCGEI '93 0-7803-1443-3/93 $3.00 0 1993 IEEE 5 3 . 2

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We make two approximations to aid in the analysis: ( I ) the connections between the inner and outer conductors are of negligible length, and (2) the central source is a delta function generator of voltage V. Then the electrical field component parallel to the antenna axis about the centre connection can be expressed as:

The Vi's are the voltage sources at the connections, located at zi. N is the number of COCO antenna elements. For a COCO antennas with equal length elements, the integral equation becomes,

2.3. THE IMPLEMENTATlON OF THE CG-FFT M E l H O D We apply the Galerkin's procedure with piecewise

sinusoidal(PWS) functions as the basis functions for discretizing the integral equations. After expanding the current I(z') with the PWS basis functions and applying inner product to the integral equation with PWS test functions, the integral equation is changed into a linear system,

[zl 14 = I Y (7)

where Z is the generalized impedance matrix. V is the generalized voltage matrix, I is the current vector. Applying &gap source model, the generalized voltage matrix V is,

v - (0.0 ,..., 0,1,0 ,..., 0,1,0 ,..., o,oy

that, is the elements corespondent to the junctions being equal to one, elsewhere zero.

When the structure to be investigated is very large, it takes a lot of CPU time and computer memory to calculate and store the Z matrix and its inverse if MOM is used. The CG-FfT method can overcome the difficulties in terms of reducing time and memory requirement. If the above integral-differentiaI.equation is expressed as a operator equation as, AJ=Y, here J is the unknown to be solved, Y is the known excitation, and A denotes the integral-differential operator, the CG method develops the followings: tn=l/(llAPnll); J,+l=Jn+tnPn; Rn+I=Rn + t, AP,; bn=l/(l1A*Rn+J; P,,+l=P,-b,A'R,+l . The II I I in the above equations is defined as,

Here A' denotes the adjoint operator for A. and is defined by

The overbar represents the complex conjugate

It was observed that in the CG method described above, the bulk of the computation time is devoted to computing terms like AP,, A'P,. This is the motivation for our using the FFT in the analysis. The fundamental idea lying in combining FFT with CG is that, because the operation AR, appearing in the CG method, has the convolution nature, they can be easily evaluated, in an approximate way, by using the FFT algorithm, once the operator has be discretized. The computation for A'R can be done in a similar way except that the Green's function is replaced by its complex conjugate form. It was observed that by using FIT to compute the convolution integral, the computing time can be reduced greatly. Following the procedure described above, the iterations are carried out until the desired error criterion is satisfied. For instance, when IIAJ,-YIIAIYII < lo', then we stop the iteration.

2.6. ANALYSIS OF THE DIELECTRIC HOUSING USING FD-TD Even though CG-FFT can also be used to analyze a COCO

antenna with a dielectric housing, it is much complicated than that of the COCO without housing. One numerical scheme suitable for this problem is the Finite Difference Time Domain (FD-TD) method. The method is easily adapted to complex geometries, so simplified theoretical models are not required. The basic idea of the FD-TD method is that the time- dependent Maxwell's differential equations can be represented by a set of difference equations that can be solved numerically using a digital computer.

The Formulation of FD-TD for Straight Cvlindrical Antenna For a straight cylindrical antenna, the structure can be considered

to be rotationally symmetric and is to be excited by a rotationally symmetric source. Therefore, the electromagnetic field is independent of the cylindrical coordinate @ and only the rotationally symmetric TM mode field is excited. The relevant form of Maxwell's equations in cylindrical coordinates are,

After discretization, the equations take the form

With the electric field E, at the boundary A-A' (Fig.11) determined by the incident field, these equations are used in a time-stepping procedure to determine the EM field in the computational domain.

Absorbing Boundary Condition The Merewether's absorbing boundary condition is used in this

work . In this approach the filed near the boundary is assumed to have the functional form f(t-R/c)/R. and the tangential component of the electric field on the surface Se is computed from local values within V by interpolation. For instance, at the upper boundary, the boundary condition is,

.Source Model We use a delta-gap voltage source model for this antenna. This

model assumes that the field generated within the gap is uniform, it is a good approximation for small spacings. E,'=V,(t)/d. Here, Vi(t) is the voltage source across the A-A' . It could be a Gaussain excitation or a sinusoidal voltage excitation.

The Treatment of Dielectric Housing For homogeneous media range where the material constant, E, p,

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where vary continuously, no additional work needs to be done except specifying E, p at each grid point. But for heterogeneous media, such as an interface between two media, in which step changes of material constants occur at interfaces of adjacent homogeneous media, some special treatment are needed. Generally speaking, for the tangential components on the interface, the material constants can be chosen as the average of those in the adjacent two media. For the normal components on the interface, two value nodes are needed to be used.

3. NUMERICAL & EXPERIMENTAL RESULTS

3.1. EXPERIMENT SET-UP The measurements were carried out the roof of a cabin. The

measurement set-up is shown in Fig.3. We have measured two COCO arrays, a 12-element linear array and a 9 by 12 planar array.

3.2 SIMULATION A N D MEASUREMENT RESULTS

( I ) Linear array Figs.4.5 show the simulation and measurement results for the

magnitude-and phase of a 12-element COCO array. The two sets of results are in a very good agreement. The radiation pattern of this linear array is shown in Fig.6. A very good agreement is also observed. From these results we can find that the current distribution along two neighbouring elements of a COCO antenna is very similar to the current distribution on a full wavelength dipole. From the whole point of view, the COCO linear array is like a number of full-wavelength dipoles placed end to end. This is exactly what we expected. and the objective of designing a COCO antenna.

(2) Planar array A 9x12 planar arrays was studied. The lengths of all the

elements are hd2. The space between linear arrays is kd2. They are all fed with signals having same magnitude and phase. The current distribution is given in Fig.7. It is found that the current distribution on the linear array are almost identical. In some case there are small difference in their amplitude. It is also found, comparing these results with those of a linear array, that the current on the rows of planar array is only a little different from the current on the linear array. These results indicate that the coupling between the linear arrays is small, as long as the spacing between rows is sufficient large.

13) Encasing foam The encasing foam changes the attenuation of the signal because

of the dielectric loss and changes the velocity factor for the COCO cable. Usually, the attenuation factor of a coaxial line is very small (about the order of 10-3-10-4 dBlm). So it has small effect on the current distribution and the radiation pattern. The velocity factor mainly depends on the dielectric constant of the material in the coaxial line. For Teflon, ~,=2.1, v=lld2.1=0.69, the length of element becomes bO.34S)b. Fig.8 gives the result for 8-element linear array with velocity factor 4 . 6 9 . Because the length of the element is no longer a free-space half wavelength of free- space, the current and radiation pattern are changed to some extent.

16) The input impedance The calculated input admittance versus the number of elements

is given in Fig.9. From this figure, we find that the value of input admittance(rea1 and imaginary parts) are proportional to the number of the elements. In order to investigated the effect of a ground plane on the input admittance, the input impedances of COCOS at different height above the ground are calculated. Fig.10 gives the curve of input admittance(rea1 and imaginary) versus the distance. It show a pseudo- periodic behaviour versus distance . The period is about D2. The variation decreases with height and tends to a limiting value.

4.3. NUMERICAL RESULTS OF F D - T D A multilayer dielectric-housed antenna was simulated. The

geometry is shown in Fig.11. In this analysis, ~,,=1.26; ~,,=2.4; a=0.00115m, h=0.069m. We use a sinusoidal excitation with f=IGHz. The Er field distribution (at 4500 time step) is shown in Fig.12. A comparison of input impedance for an antenna with and without a housing is given in Fig.13. From the above results, we can find that the

dielectric housing effects the field distribution and input impedance are obvious and that we cannot neglect them in an accurate analysis.

S. CONCLUSIONS

In this paper, an comprehensive model for a COCO antenna is developed, based on theoretical analysis, computer simulations and experimental studies. The current distribution, radiation pattern and input impedance of linear and planar COCO arrays are analyzed. Two numerical techniques, C G - m and FD-TD method, are used in the analysis.

From the analysis, we found that COCO antenna is similar to a dipole array, composed of a series of dipoles placed end to end with very small gaps between the elements. The current (magnitude and phase) and the radiation pattern are all similar to the dipole array. This is just what we expected or say the goal to be reached when the COCO is designed. Based on this point, the other features of the COCO are similar to the ordinary dipole array, such as the effect of ground on the current distribution and radiation pattern, the effect on the input impedance, etc.

Through the investigation of dielectric housing with FD-TD method, it is shown that the dielectric housing effects the current distribution, radiation pattern and input impedance of a COCO antenna. Further studies are needed.

6. ACKNOWLEDGMENT

We wish to acknowledge the support of Andrew CanadaJnc. in this work.

[REFERENCES]

[ I ] B.B.Balsley and W.L.Ecklund. " A portable coaxial collinear antenna" IEEE AP V01.20, pp.5 13-5 16.1972

[2] B.B.Balsley, et al." The MST radar at Poker Flat, Alaska" Radio Science, Vol. I5.pp.213-223. Mar./Apr. 1980.

131 G.R.Ochs," The large 50 Mds dipole array at Jicamarca radar observatory," NBS Rep.8772, Boulder, CO, Mar. 1965.

[4] B.B.Balsley, et al. "The Coaxial collinear antenna: current distribution from the cylindrical antenna equation," IEEE AP Vo1.35, Mar., 1987, PP.988-996

[SI Thierry J.Judasz, Ben B.Balsley, "Improved Theoretical and Experimental Models for the Coaxial Collinear Antenna" IEEE AP V01.37, NO.3 March 1989, PP.289-296.

[6] Akihide Sakitani, Shigeru Egashira, " Analysis of coaxial collinear antenna: recurrence formula of voltage and admittance at connection" IEEE AP Vo1.39, No.1, Jan. 1990. PP. 15-20.

Figure.1 The structure of a COCO antenna.

Figure.2 The equivalent transmission line model of COCO antenna

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I 8 -71' ' t h

Figure.3 The experiment set-up Cuncnt distnbulion of the I 2 element COCO linear array

O t i 20 40 60 nn 100 120

-0s- '

solid line --- qimulalion. dash line --- mcawrenient

Figure.4 simulation and measurement for a 12-element COCO linear array.

Comparison of the current distribution(magnitude) between

Phase d~t r ihu l ion of the 12 e lcmmt COCO linear orrny r ~ ~ - ~ ~ j I ~ ~ .. r--------

100

1 -L .L

40 60 80 100 I20 "IL-- 20

solid line -- s~mulation. da5h linc i i i ~ a w r c m c n t

FIgure 5 simulation and measurement for a 12-element COCO linear array.

Comparison of the current distribution(phase) between

Radiation paltcm of the 12-clement COCO linear anay

-:: 10

1s

20

25

30

3s

40

-4s

20 40 60 80 IW I20 140 IM) 180

solid line --- measurement, dash line .-- simulation

Figure. 6 Radiation pattem of a 12-element COCO linear array.

Figure.7 Comparison of the current distribution between simulation and measurement for a 9 bv 12 COCO planar array.

~.________ ~~

CtiRJ'.bT D~slsuw~!o~. o>:- co.4qjM g,Llh'b'H yTF;$A S"C!I=O 000 I , A = 5 U10 Q ' A = 1005 E ! I = I 2 U? - '

-- AMPLIWDE +r-. P W S [

UlPLnUDl i

E TITHOUT GROUND

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ELECTRIC FIELD PATTERNS FOR THE COAXIAL COLINEAR ANTENNA

1,'- 6000 R/L= 0005 NL - 12 NW - I SPC/h OOW

Figure.8 COCO linear array with Teflon encasing foam.

Current distribution and radiation pattem of a 12-element

l n w t admittance of COCO with deffetent number of elements IO'.

I Y

2 4 6 a I O 12 1 4

number of elemenls

Figure.9 The input admittance of COCO antennas with different number of elements.

input admittonce of o COCO over ground. ne-2

1 5

-0 5

!/

n s I I 5 2

Figure.10 The change of input admittance of a COCO antenna with the height above a ground plane.

Figure.11 The geometry of a spright cylindrical antenna with dielectric housing (for FD-TD simulation)

Figure.12 The E, field distribution of the dielectric housed antenna.

h Inpu!impedanFe of dipole with~w$hout di+smc houtin '{"I -- '

without housing

... - 1

I M 198 400 '102 404 406 408 410 4 1 2 4 1 4 dipole Icnghh= 740mm(l wavclcngth at fD.405 4MHz).Prl=I 26.Er2=2 4

Figure.13 The comparison of input impedance of between antennas with and without dielectric housing.