IEEE TRANSACTIONS ON BROADCASTING, VOL. 39, NO. 2, JUNE 1993 273

PATTERN SYNTHESIS WITH A FEED NETWORK CONSISTING OF LOSSY TRANSMISSION LINE SECTIONS

COSTAS MERTZIANIDIS member IEEE TEI of Cavala

Ethnikis Antistasis 8C, CAVALA 654 03, GREECE

MICHAEL T. CHRYSSOMALLIS, GEORGE KYRIAKOU, members IEEE Department of Electrical Engineering

and JOHN N. SAHALOS, Senior member IEEE

Department of Physics University of Thessaloniki, THESSALONIKI 540 06, GREECE

University of Thrace, XANTHI 671 00, GREECE

ABSTRACT

A method for feeding the elements of an array with lossy transmission line sections is presented. The solution of the problem has a starting point at the lossless case, where we have relatively simple expressions. The procedure is completed by analyzing the antenna array in a frequency band around the central frequency. Several examples show the applicability of the method.

J. INTRODUCTION AND FORMIKA TION

It is well known that the radiation pattern of an antenna array can be controlled by several methods. Two of them are the most interesting. One has to do with reactively loading the terminal of the antenna elements ([1]-[5]) and the other with connecting them in parallel through transmission lines, [5] . Recently an analysis given by Y.- W. Wang and D. M. Pozar , 161, has shown that we can combine the above two methods. So, loading each element terminal with a reactive load and in series or parallel connecting each terminal to the single source with a transmission line we obtain the desired radiation pattern. A different feed configuration with only transmission line sections at specified characteristic impedance and length has also been given in [7] by the authors and in [9] by K. M. Lee and R. S. Chu.

In all the above works the transmission line losses were ignored. That was made with the assumption that the transmission line losses are very small and the solution for the lossless case is very close to the lossy one. As we will see later, the influence of the losses to the behavior of the system sometimes is very important.

The basic idea for the solution of the problem is the same with that presented in [7] and [8]. The feed configuration is given graphically in Fig 1. li denotes the physical lengths of the i"' transmission line sections. It is assumed that the characteristics of the element? of the array are known. With the procedure presented below we find first the lengths I , and I, of the two transmission line sections which connect the first two elements. Then we compute the current, the voltage and the impedance at the connection point A. This point is considered as a new load with impedance given by:

Point A is connected with the 3* element and the lengths 1- and l3 are found. The procedure is continued until the connection

T T Y"-' I

8 Fig. 1 The proposed feed configuration.

of the last element is reached. The impedance Z, at the last connection point P is the input impedance of the array.

The antenna design is completed by analyzing its characteristics in a frequency band around the design frequency, f,. In this procedure, the physical lengths l, remain constant, while the electrical lengths of the transmission line sections and consequently the input impedance and the radiation characteristics of the array are changed. So, the new radiation patterns are computed from the new excitations.

We begin the feed network design by computing the physical lengths I, of the transmission line sections. We start with the connection of the elements 1 and 2. Let Z,, Z, be the input impedances of the two elements, Z,,, Z,, the characteristic impedances of the two lines, 1,, 1, the physical lengths of the two sections - which are to be computed - and

the transmission coefficient of the lines. Looking at point A (see Fig. 1) we have the equivoltage condition which gives:

0018-9316/93$03.00 0 1993 TEEE

274

Combining (1) and (2) and separating the real and the

rIexp(a1 I 1 cos($: + P111 )+r;exp(-alIl) cos($; - PI 11 ) - A = 0 (3)

imaginary parts of the above equations, we have:

rIexp(alZ1) sin($:+PII1)+r;exp(-alZl) sin(@; -P1l1)-K=O (4)

where

A =A[r;exp(a&) cos(y+$i + P212)+

+r;exp(-Wdcos(~+ $; - P212)I (5)

+r;exp(-add sin(y+$; - P212)I (6) K=A[r;exp(a212) sin(y + $; + P&)+

The quantities A and K of the above two equations (5) and (6) depend on the length 12. Consequently we consider them as computable functions and we solve (3) and (4) for the quantities rfexp(kal11).

We suppose that h- --

and (7) and (8) are transformed to:

-cos z, &2 -sin

Putting

exp(x)= 1 K

Combining (13) into (10) and (11) we have:

We denote by

and equations (14), (15) become

Il(ja1 - -PI) = ~-jx+cos-'[Dexp(-jz)] (17)

-Zl(jal + P I ) =+tJx+cos-'[Dexp( jz)] (18)

In the above equations it is:

D exp(-jz) = cos(o) D exp( jz) = cos(w*)

where

w = p+jv (20)

Taking out I , from (17) and (18) we get the following real equation

a,(P-@) =P1(x-v) (21)

p = Re(c0s-l [Dexp(-Jz)]} (22)

v , v are defined by the expressions

v = Im{cos-'[D exp(-jz)]} (23)

l2 is the only unknown quantity in equation (21). This equation is nonlinear and is solved by using the classical perturbation method. We start from the solution of the lossless problem as it is described in [7] and [8] and by perturbing the solution we find the exact one for (21). We substitute Z2 in (1) and we solve (1) for 1,. With the help of l , , I , we proceed to the computation of the voltage V, and the current I, at the connection point A.

and

The above procedure is continued until all the elements are connected and thus the last connection point will be found. This point will be matched -by one of the available methods - with the transmitter.

The procedure is completed by analyzing the array in a band around the design frequency, f,. For this purpose we start from the computation of the new excitations of the elements and then the new radiation pattern.

11. APPLICATIONS

In [7] it has been shown the design procedure for feeding arrays with lossless cables. We will present here four lossy cases with dipoles arrays. The transmission lines which are used have the same velocity factor of 0.659 except one section with velocity factor of 0.84.

1" Example

We consider a uniformly excited antenna array with 3 col- linear dipoles in a frequency f0=200 MHz. The dipole length is

0.7 m, the distance between the dipoles is 1.125 m and their radius is 0.01m. Fig. 2 shows the feed configuration of the array.

4 I B

Main Feedinq Transmission Line

Fig. 2 The feed configuration of the I" exumple.

We use transmission lines with characteristic impedance Z, = 50 a.

The input impedances of the three dipoles are

Z, = 75.08 - j 8.11 Q Z, = 74.94 - j16.08 Q 5 = 75.08 - j 8.11 !2 .

Table I shows the lengths I , , I , , I , , I , for the cases of a=O and a=0.01284 Np/m. Looking at this Table we see that there is no "physical" solution for a=0.03475 Np/m. The lossless solution for I , and l2 gives 1,=2.228 m and 1,=2.266 m. (The values in Table I are multiplied by the velocity factor, 0.659).

TABLE I li a=O Np/m a=0.01284 Np/m a=0.03475 Np/m 1, 1.468 m 1.493 m I , 1.493 m 1.516 m There is no

1, 1.005 m 1.003 m "physical" Solution.

l4 1.502 m 1.521 m

The lossy solution for a=0.03475 Nplm (as well as for a=0.01284 Np/m) was expected to give similar values for I , , l , , I , , 14. However looking at Fig. 3a which shows the function:

F(M = a i ( ~ ~ - + ) - P i b - v )

(see equation (21)) versus 1, (in m), we see that there is no mathematical solution of (21) for 1,>2.0 m. If we accept the solution 1,=1.727 m (~0.659=1.138 m) we can see from Fig. 3b which shows the solutions of (1) that there is no solution for

We have analog condition for 1,=1.589 m and 1,=1.006 m, while for 1,=0.799 m and 1,=0.371 m the line sections I,+l,(x 0.659) are smaller than the distance between the dipoles.

Fig. 4 shows the corresponding curves for cables with a=O and a=0.01284 Np/m, where the existence of the solutions for I,, 1, is evident. It is also evident that there is a "small" difference

4.

275

between the two cases. The solution differs more as I , and I, increase.

We complete the design of this antenna by analyzing it in a frequency band around the frequency f0=200 MHz. Fig 5 shows the radiation patterns of the antenna.

First Example a=0.03475 Np/m

1 1 I

F(zJ 0.75

0.5

0.25

0

-0.25

n <

No "mathematical" solution A t t 4 solution

-.., 0 0.5 1 1.5 2 2.5 3 3.5 4

FM Example a=0.03475 Nplm

250

200

150

100

50

0 t\P 0 0.5 1 1.5 2 2.5 3 3.5 4

4 (m) @)

Fig. 3 The non-existence of "physical" solution for large losses.

Fig. 5a corresponds to attenuation a=O while Fig. 5b corre- sponds to a=0.01284 Np/m. As it is expected, the difference of the two groups of the patterns is insignificant and the basic characteristics of the radiation are maintained in a band *O.OSf, around f,.

Fig. 6 shows the normalized input impedance, &, and the VSWR as a function of the frequency around f,,. The input impedance is normalized with the impedance corresponding to f,. Z, does not appear remarkable changes for the cases a=O and a=0.01284 Np/m. It is found, that when a=O, the VSWR for 0.95f0 and l.O5f0 becomes equal to 1.36 and 1.11 respectively, while when a=0.01284 Nplm, the VSWR for the same frequencies is 1.42 and 1.17 respectively. If we examine the frequency bandwidth, in which S < 1.2, then for a=O we have Af=19.2 MHz, while for a=0.01284 Np/m we have Af=16.8 MHz. This means that the cable attenuation causes in this antenna a reduction of the frequency band about 12.5%.

27 6

First Example a=0.01284 Np/m

First Example a=0.01284 Nplm

t I I

I " -0.5 I

0 0.5 1 1.5 2 2.5 3 3.5 4

First Example Lossless

-0.5 '

0

-10 dB

-20

-30

-40

-50

-60

0 0.5 1 1.5 2 2.5 3 3.5 4

1, (m) (c)

0 0.5 1 1.5 2 2.5 3 3.5 4

4 (m) 0)

First Example Lossless

0 0.5 1 1.5 2 2.5 3 3.5 4

Fig. 4 Solutions for 1, and l2 for small (a, b) and zero (c, d) losses.

First Example LossleSS

._ .. .

0 30 60 90 120 150 180

Degrees

(a)

0

-10

dB

-20

-30

4 0

-50

First Example b Y

w

0 30 60 90 120 150 180 Degrees

(b)

Fig. 5 Radiation patterns of thejirst example for the lossless (a) and the lossy (b) case

277

First Example

- - - - Re(zJ, Lossy ...... Im(Zh), Lossy

length is 0.2345 m, the dipole distance is 0.25 m and the dipole radius is 0.01 m. The input impedances of the dipoles are:

Z, = 73.01 - j20.57 R Z, = 64.23 - j35.44 C2 Z, = 64.23 - j35.44 52 Z, = 73.01 - j20.57 52

Fig. 7 shows the feed configuration of our system. Table 11 shows the characteristic impedances and the lengths li

for the cases a=O as well as a=0.04910 Np/m (for &=50 0) and a=0.04079Np/m (for &=75 a).

1 J I

0.9 1 1.1 1.2 flfo TABLE I1

0.8

4 a=O a#O (4 &I (0) 0.138 m 0.139 m 0.157 m 0.158 m 0.145 m 0.145 m

0.277 m 0.276 m

50 1, 50 4 75 13

75 4 First Example

VSWR

0.8 0.9 1

@)

1.1 1.2 flf,

Fig. 6 Z, and VSWR around the design frequency f o of the first example.

2nd Example

In this example we will study a uniform array with four parallel dipoles (Fig. 7). The frequency f, is 600 MHz, the dipole

1 2 3 4

Main

Line

Fig. 7 The feed configuration of the 2"" example.

0.347 m 0.347 m 0.632 m 0.632 m

50 15

75 16

Second Example LQSSleSS

0

-10 dB

-20

-30

-50

-60 90 120 150 180 0 30 60

Degrees

(a)

Second Example LOSSY

0

-10 dB

-20

-30

-40

-50

-60 0 30 60 90 120 150 180

Degrees

@)

Fig. 8 Radiation pattents of the second example for the lossless (a) and the lossy (b) case.

27 8

Third Example

- Re(ZJ, Lossless - - - - Re(ZJ, Lossy

- - - Im(Z,), Lossless ...... Im(ZJ, Lossy

dB O i - F h ..... 0.95

-10 1.05 f,

0

-10

-20

dB

-30

-40

-50

-60

0 30 60 90 120 150 180

Degrees

(a)

0

-10

dB

-20

-30

-40

-50

Third Example

0 30 60 90 120 150 180 Degrees

@)

Third Example

-U"

0 30 60 90 120 150 180

Degrees ( 4

Fig. 11 Radiation pa t t em for the third example for the lossless (a), lossy (b) and "hybrid" (c) case.

From Fig. 12b we see that the curve for the VSWR is more smoothed for a z o than for a=O. The frequency bandwidth around the central frequency fa for S<1.2 is Af=0.014fa=4.2 MHz, wheq a#O, while it is Af=0.012fa=3.6 MHz when a=O. This means that it is observed an increase of the frequency bandwidth of about 16.7% when we take into account the attenuation of the transmission lines in designing this antenna.

I

5

ziu

3

0.8 0.9 1 1.1 1.2 flf,

(a)

Third Example VSWR

5 I

I - Lossy VSWR Lossless . . . . . . . * . t ,

* I I . , . . . . . I._.-

0.8 0.9 1 1.1 1.2 f/fo

(b)

Fig. 12 Z , and VSWR around the design frequency fo of the third example.

We must notice here, that this is not the rule, since in the first example we had a reduction of about 12 % , in the second example an increase of 7.25% and in this example an increase, too, of 16.7%.

If we apply the solution for a=O to the cables given in Table I11 with a # 0 we take the radiation patterns which are given in Fig. l lc. The result shows that we have not exactly a Chebyshev array. The sidelobes become unequal and the SLL from -30 dB becomes equal to -28 dB. So, it is evident that for a suitable design the transmission line losses must be taken into account.

4th Example

To see the significance of the transmission line losses we show one case with extremely sensitive pattern. We have an array with five 1112 parallel dipoles. The distance between the dipoles is d=0.062511 and the dipole radius is r=0.004h. We design a Dolph-Chebyshev array with SLL=-25 dB and we will use as a typical frequency f0=300 MHz. Fig. 10 shows the feeding network while Table IV shows the transmission line lengths. All lengths have Z0=50 Q with a=0.04382 Nplm except of 1, which has Zo=125 C2 with a=0.01322 Np/m.

For the lossless as well as the lossy case the lengths 1,=12 and 13=14 remain the same because of the symmetry. The radiation pattem for the lossless case of Table IV is shown in

279

Fig. 8 shows the radiation patterns of this array for the two cases (a=O and ago) around the frequency design. Comparing these diagrams we can say that the array maintains its general characteristics around the central frequency f,, and that the influence of the attenuation is not obvious. The same results are obtained if we examine the Z, and the VSWR (Fig 9) of the array.

Second Example

Im(Z,J, Lossless - - - -Re(Z,), Lossy ...... Im(Z3, Lossy

2

11.' I

0.8 0.9 1 1.1 1.2 ff f,

Second Example VSWR

Lossless . . . Lossy

0.8 0.9 1 1.1 1.2

(b)

f/f*

Fig. 9 2, and VSWR around the design frequencyfo of the second example.

The most remarkable thing is that the frequency bandwidth for S < 1.2 of the lossless case (a=O) is Af=0.138f0=82.8 MHz, while the same bandwidth for a#O is Af=0.148fo=88.8 MHz, i.e. we have an increase of the frequency bandwidth about 7.25% for this array.

3rd Example

As a third example we will study an array which is mentioned in [6] for comparison. It is a Chebyshev array of five h/2 dipoles, with SLL=-30 dB, distance between the dipoles d=0.7h and we will use as a typical frequency f,,=300 MHz. The radiation patterns are shown in Figs. 4a and 6a of [6] for series and parallel feed design correspondingly. The transmission line lengths of our proposed feeding method are shown in Table 111 for the two cases, a=O as well as a=0.04382 Np/m (for cables with &=50 a) and a=0.02682 Np/m (for cables with Z,,=75 a). The feed configuration for this array is shown in Fig. 10.

1 2 3 4 5

Main Feeding Transmission Line

Fig. 10 An example for the configuration feeding of a S-elements array.

TABLE I11 zo, (Q) 1, a=O C l Z O

50 1, 1.556 m 1.555 m 50 12 1.556 m 1.555 m 50 4 0.732 m 0.731 m

0.731 m 0.107 m

0.732 m 0.113 m

50 14

75 15

0.309 m 50 16

50 1, 0.293 m 0.303 m 0.643 m 75 1,

0.310 m

0.636 m

The lengths 1,=l2 and 13=14 were selected to be compatible with the array geometry. For the selected pairs of the lengths ( l , , 12) and (13, 14), in the case of the lossless problem - a=O - and using lines with Z0=50 R and 75 R, we have 32 different solutions. The same number we have for the lengths Is, I , , I T , I,.

A typical solution - which is the case shown in Table I11 - gives the radiation patterns of Fig. l la. These diagrams are the corresponding ones of Figs, 4a and 6a of [6]. Comparing the diagrams of Fig. 1 la with these of Figs. 4a and 6a of [6], we see that there are not significant differences. The main changes are focused at the SLL. We can say that our diagrams of Fig. l l a are closer to - and maybe better than - these of Fig. 6a of [6]. On the other hand, the diagrams of Fig. 4a of [6] are rather better - regarding the SLL - than ours of Fig. l la. In general, the array maintains its characteristics around the central frequency design f,, in a bandwidth Af= +0.05f0. Fig. 12 shows the normalized Z , (a) as well as the VSWR (b) versus frequency.

If we take into account the attenuation of the transmission lines used for the connection of the array elements (a5,=0.0438 Np/m and a,,=0.0268 Np/m) then the new lengths li are shown in the last column of the Table 111. Fig. l l b shows the corresponding radiation patterns for this case and Fig. 12 shows the Z, (a) and the VSWR (b) versus frequency. The difference between the corresponding diagrams of Fig. 11 is insignificant.

280

Fig. 13 (curve i). If we apply the solution for a=O to the lossy cables we take a pattern (see curve ii) which has nothing to do with curve (i) of Fig. 13. So it is evident that only a method that takes into account the losses can be used. It is believed that a case of feeding a so sensitive array with transmission line and Prouagation, vol. AP-15, pp. 502-515, July 1967. sections is given for the first time in the literature.

REFERENCES

[l] R. F. Harrington and J. R. Mautz, "Straight wires with arbitrary excitation and loadings", IEEE Trans. on Antenna5

[2] D. P. S. Seth and Y. L. Chow, "On linear parasitic arrays of dipoles with reactive loading", IEEE Trans. on Antennas and ProDaeation, vol. AP-21, pp. 286-292, May 1973. TABLE IV

z,, (a) li CX=O CX+O [3] R. F. Harrington and J. R. Mautz, "Pattern synthesis for

loaded N-port scatterers", IEEE Trans. on Antennas and 0.137 m 0.137 m 50 1, 50 12 0.137 m 0.137 m Prouaeation, vol. AP-22, pp. 184-190, Marh 1974.

0.183 m 0.183 m 0.183 m 0.183 m

50 1, 50 14

[4] R. F. Harrington, "Reactive controlled directive arrays", IEEE Trans. on Antennas and Prouagation, vol. AP-26,

125 15 0.237 m 0.192 m pp. 390-395, May 1978. 0.396 m 0.389 m

50 1, 0.191 m 0.203 m

50 18 0.414 m 0.407 m pp. 634-635, Dec. 1972.

50 4 [5] D. H. Sinnott and R. F. Harrington, "Simple lossless feed

networks for array antennas", Electron Lett., vol. 8, no. 26,

Fourth Example

-20

Lossless (i) ..... Lossy (ii) I I

6 0 ' I

0 30 60 90 120 150 180

Degrees

Fig. 13 Radiation patterns of the fourth example for the lossless (i) and the lossy (ii) case.

III. CONCLUSIONS

We studied in this paper a method for feeding antenna arrays by means of one unique source using lossy cable sections. We presented four examples and we studied the behavior of the systems in a frequency band around the design frequency. Besides the changes of the physical lengths we observed changes in the frequency bandwidth of the antennas. With the presented feeding configurations we can select among a lot of available solutions of the problem the geometrically compatible ones. Our method can be applied in extremely difficult cases as that given in the fourth example. When the transmission line sections have large attenuation constant, then the problem has not always solution. This is something important. It is also important to say that each transmission line length can not be increased by multiples of a full wavelength until the length becomes longer than the distance between the elements. This is one of the main differences between lossless and lossy cases.

[6] Y.W.Kang and D.M.Pozar, "Pattern Synthesis Using Reactive Loads and Transmission Line Sections for a Singly Fed Array", IEEE Trans. on Antennas and Propagation, vol. AP-37, no. 7, pp. 835-843, July 1989.

[7] C. Mertzianidis, I. Diamandi and J . N. Sahalos, "A Voltage-Matching Method for Feeding Antenna Arrays", IEEE Trans. on Broadcasting, vol. BC-38, no. 1, pp. 27-32, March 1992.

[8] A1 Christ", "A voltage-matching Method for Feeding Two-Tower Arrays", IEEE Trans. on Broadcasting., vol. BC-33, no. 2, pp. 33-40, June 1987.

[9] K. M. Lee and R. S. Chu, "Analysis of Mutual Coupling Between a Finite Phased Array of Dipoles and Its Feed Network", IEEE Trans. on Antennas and Propagation, vol. AP-36, no. 12, pp. 1681-1699, Dec. 1988.

Costas Merthnidis was born in Cavala, Greece, on February 2, 1945. He received the B.Sc. degree in Physics from the University of Thessaloniki, Greece, in 1968, the M.S. degree in Meteorology from University of Athens, Greece, in 1973, the M.S. degree in Electronics and the Ph.D. in Physics from the University of Thessaloniki, in 1975 and 1976 respectively.

From 1972 to 1977 he was with the University of Thessalonib. From 1977 to 1985 he worked as an engineer of telecommunications at the Hellenic Telecommuni- cations Organization. Since 1985 he has been with the Techno- logical Educational Institution (TEI) of Cavala as professor of Electronics. His research interests are in antenna and microwave engineenng.

28 1

Michael ChryssomaUis was bom in Dr. Sahalos is a professional Engineer and a consultant to Athens, Greece on September 1 1 , 1957. industry. He has been honored with the Investigation Fellowship He received the B.S.E.E. degree in 1981 of the Ministerio de Education Y Ciencia (Spain). Since 1985 he and the Ph.D. degree in Electrical Engi- has been a member of the Editorial Board of the IEEE tran- neering in 1988, all from the Demokritos sactions on Microwave Theory and Techniques. University of Thrace.

From 1982 to 1989 he worked as a Research Associate and since 1989 as a Lecturer at Microwaves Laboratory of Telecommunications and Space Science Sector of Electrical Engineering Depart-

ment. During these years he taught the courses of Telecommuni- cations, Microwaves, Antennas and Radiowave Propagation. His research interests are in the fields of microstrip lines analysis, Superdirective Antennas Arrays synthesis and Ionospheric propagation subjects.

He is a member of IEEE since 1988.

George A. Kyriacou was born in Famagusta, Cyprus on March 25, 1959. He received the Electncal Engineering Diploma in 1984 and the Ph.D degree in 1988, both from Demokritos Univer- sity of Thrace. Since January 1990 he is Lecturer at Microwaves Laboratory at the same University.

His current research interests include the modeling of Microwave Integrated Circuits, printed antennas especially on

anisotropic substrates and radiators over lossy earth. He is also working on Electncal Impedance Tomography, participating the European Community program COMAC-BME.

John N. Sahalos (Mt75-SM'84) was bom in Philippiada, Greece, in November 1943. He received the B.Sc. degree in Physics and the Diploma on civil engineering from the University of Thessaloniki, Greece, in 1967 and 1975, respectively and the Diploma of Post- Graduate Studies in electronics in 1975 and the Ph.D. in electromagnetics in 1974.

From 1971 to 1974 he was a Teaching Asststant of Physics at the University of Thessaloniki and was an Instructor there from 1974 to 1976. During 1976 he worked at the ElectroScience Laboratory, The Ohio State University, Columbus, as a Postdoctoral Univesity Fellow. From 1977-1985 he was a Professor in the Electrical Engineering Department, University of Thrace, Greece, and Director of the Microwaves Laboratory. During 1982, he was a Visiting Professor at the Department of Electrical and Computer Engineering, University of Colorado, Boulder. Since 1985 he has been a Professor at the School of Science, University of Thessaloniki, Greece.

During 1989 he was a Visiting Professor at the Universidat Politechnica de Madrid, Spain. His research interests are in the area of applied electromagnetics, antennas, high frequency methods and microwave engineering. He is the author of three books and more than 100 articles published in the scientific literature.

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