1989 02 Supergain Antennas and the Yagi and Circular Arrays.pdf

8/9/2019 1989 02 Supergain Antennas and the Yagi and Circular Arrays.pdf

1/9

178

IEEE TRANSACTIONS ON ANTENNAS AND

PROPAGATION,

VOL.

37,

NO. 2. FEBRUARY 1989

Supergain Antennas and the Yagi and

Circular Arrays

Abstract-The characteristics of end fire, Yagi and circular arrays of

dipoles are reviewed with special reference to their directional properties

and the possibility of supergain. The qu antum-mechan ical analog , which

suggested the investig ation, is described. It is no ted that circular arrays of

identical dipoles of which only one is driven have attractive directional

properties when the entire array is adjusted to resonance by a careful

selection of the length of the elem ents, the distance between them, and the

circumference of the array.

An

immediate application to beam scanning is

described.

I. INTRODUCTION

cally possible well over half a century ago, but for which no

successful design has yet been developed.

An absolute directivity or gain of an antenna is 47r times the

ratio of the radiation intensity Sr(O , ‘P) in the direction

( e m ,

am)

f its maximum divided by the total power radiated by the

antenna, i.e.,

1 )

4 r S r ( e m ,

a m )

D=

sz

d ’ ~r ~ ~ 0 ,

P)

sin 0 d e *

The relative directivity or gain is usually defined with respect

to a half-wave dipole for which D = 1.63 (or an electrically

short antenna for which

D = 1 . 5 ) .

Thus,

NTENNAS WHICH TRANSMIT an extremely narrow

beam are useful in point-to-point communication. Beams

which are narrowed by transmission from apertures-as with

horns and reflectors-are subject to diffraction and their field

A

patterns have many minor lobes. Large broadside arrays of

G = 10 loglo ( D / 1 . 6 3 ) .

(2)

dipoles have similar characteristics. The usual design of

endfire arrays can lead to negligible minor lobes but the main

lobe is generally very broad. An exception is the superdirec-

tive or supergain antenna which has been investigated for

endfire arrays of driven and parasitic elements.

A

quite

The efficiency w f an antenna is the total Power radiated

divided by the Power supplied to the antenna. Thus,

s z

d’P j

(e,P)

sin d e

different, very recently recognized possibility is the elliptical

w = O

9 ( 3 )

or egg-shaped closed loop of parallel parasitic elements

V i

GO

excited by a single driven-element at one- end. When large

enough with elements of proper length and correctly spaced,

this array can resonate with a very high Q and narrow radiated

beam. The resonant properties of a closed circular ring of

parasitic elements have been observed, but the circular array

does not have a narrow beam. Antennas with very narrow

where

Go

is the driving-point conductance

of

the antenna and

Vo is the root mean square (rms) driving voltage.

The directivity defined in

(1)

does not take account

of

the

ohmic losses. The effect of these can be included by

multiplying the directivity by the efficiency. Thus,

beams have many important applications. For example, for

communication by way of a satellite, highly directive antennas

are important. If a closed-loop array with resonant superdirec-

tive properties can be designed, the amplitudes of the radiating

currents will be limited by ohmic losses in the conductors. By

taking advantage of superconductivity-which recent advances

make possible at the temperature of liquid nitrogen-a

supergain superconducting array could be developed for use in

space where temperatures are nearly as low as that of liquid

nitrogen. This may lead to a realization of the Einstein

needlepoint radiation [ 1 ] - [ 3 ] ,which was known to

be

theoreti-

Manuscript received March 24, 1988; revised June 14, 1988. This work

was sponsored in part by the Joint Services Electronics Program under

Contract N00014-84-K-0465, in part by the Air Force Ele ctronics System

Division under Contract F19628-88-K-0024, and in part by the Strategic

Defense Initiative O rganization, Office of Innovative Sc ience and Technol-

( 4 )

In different parts

of

the following, the time dependences

e’*‘

and e-’”‘ are used. The relation j =

-

is maintained.

11. THESUPERGAINHEOREM

N D

ITSAPPLICATION

The supergain theorem states that it is theoretically possible

to design an antenna with arbitrarily small dimensions and a

directivity as high as desired. It follows from a proof by Oseen

[2] that the theorem is consistent with Maxwell’s equations.

The general consensus has been that superdirective antennas

are impractical because of critical tolerances, narrow band-

width, and very low efficiency

[ 4 ] , 5 ] .

An exception to this

point of view is found in the detailed investigation of Bloch,

Medhurst, and Pool [ 6 ]who state: “To dismiss superdirective

ogy,

and managed b y Harry Diamond Laboratories.

Cambridge, MA 02138.

aerials as impracticable [ 7 ] , 8 ] merely because some

ineffi-

cient superdirective current distributions have been found does

not seem justifiable.. .It appears that useful improvements in

The author is with the Gordon McKay Laboratory, Harvard University,

IEEE Log Num ber 88241 11.

OO18-926X/89/0200-0178 01

OO 989 IEEE


2/9


3/9

180

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 2, FEBRUARY 1989

2h

x

b/X

Fig. 4.

Gain in

dB of

eight-director Y agi array;

a/X =

0.00337

b/X

Insertion

loss

in

dB of

long array of cylinders of length 2h andig. 5 .

spacing b when terminated in a matched load.

maximum gain of 11.5. Also of interest in the diagram is the

fact that in the range 0.34 2 h / h .38, 0.1

b/X

5

0 .42 , the gain has values between 6 and 8 that are virtually

independent of b/X.

An important development in the study of long Yagi arrays

was the experimental work of Shefer [131. This demonstrated

that transmission losses-radiation and ohmic-along a Yagi

array of up to 101 elements long (over 16 wavelengths) are

virtually undetectable and independent of the length of the

array. Losses are smaller than in a hollow metal waveguide or

coaxial line.

A contour diagram of the insertion loss of a long array of

cylinders with a matched load as a function of 2 h

A

and b/X is

in Fig.

5 .

It is seen that the minimum loss occurs along a valley

in the range 0.35 h/X < 0.37. The loss is lower for small

spacings b/X than for larger ones, but the increase between

b/X

=

0.1 and 0.35 is quite small: 2.5 dB to 4.6 dB. It rises

steeply beyond

b/X

=

0.35.

Note that the conditions for

minimum insertion loss from Fig. 5 are the same as for a

forward gain independent of

b/X

from Fig. 4.

Important properties of a 19-director array of cylinders with

b/X = 0 .2 and a X

=

0.00635 are shown in Figs. 6-8 as a

function of 2 h / h . The driving-point admittance of the driven

element #1 is in Fig. 6. In the range of small insertion loss

(0.32

5

2h/X

5

0.38), the admittance is quite small; it grows

quite slowly with increasing lengths 2

h .

As 2

h/X

is increased

from 0.4 to 0.46, the admittance oscillates between maxima

and minima. At

2h/X

- 0.46, the elements become individu-

ally self-resonant when isolated. The currents in the elements

are shown in Fig. 7. With 2 h / h = 0.32 and 0.36, the currents

in all elements except the driven one are quite small and very

I Y I

EXP. POINTS

ms

0 1 I I I I I I I I

32 34 36 38

40

42 44 46 40 .

2h/X

Fig. 6 . Driving-point admittance

of

element #1 in 20-element array (data

of

Mailloux [12]).

2h

.A

-

.42

.408

.40

ELEMENT

NUMBER

Fig. 7. Currents at centers of dipoles in 20-element array

of

cylinders

=

0 32

;

-

=0.36

)

*

0.408

FIELD OF DRIVEN ELEMENT

ALONE

Far field of 20-element array; b/X

=

0.2,

a / h =

0.00635.

. _

-

Fig. 8 .

nearly constant in amplitude. There is a very low standing-

wave ratio

(SWR).

This indicates a very low reflection

coefficient

r .

Evidently, the electromagnetic wave travels

along the array and continues on into space with little

reflection at the end of the array. When 2 h / h is increased to


4/9

KING: SUPERGAIN ANTENNAS AND YAGI AND CIRCULAR ARRAYS 181

0.4,0.408, and 0.42, the SWR increases rapidly. When 2h/X

= 0.42, the current in successive adjacent elements alternates

from maximum to minimum in a resonant mode. The far-field

patterns generated by the currents in the 20-element array are

in Fig. 8. It is seen that there is always a maximum field in the

direction along the array. The minor-lobe level increases with

the SWR along the array.

It was also shown by Shefer [13] that the array can be bent

into a half-circle of sufficiently large radius with a decrease in

bandwidth but no observable increase in losses. The insertion

loss

of a semicircle of metal cylinders each of length 2h = 11

mm,

spacing

b

= 2.54

mm,

is of the order of 3 dB in a

frequency range from

8.5 to 10.5 GHz or wavelengths from

3.53 cm to 2.86 cm. In this range, 0.312

5

2h/h

5

0.385

and 0.0719 /X .0888; the losses are substantially

higher when

b

=

10.16

mm

and 0.288

5 b/X

0.355. It

appears that the correctly designed Yagi array has the

properties of an extremely low-loss transmission line with a

reasonable bandwidth. It follows that a properly dimensioned

closed loop of parallel conducting cylinders should be a

resonant high-Q circuit. This possibility was observed in the

complete analysis

of

the circular array of dipoles with one

element driven and the experimental verification of the

theoretical results

[

11, ch. 41, [141. It is stated on p. 105 of the

former work: Among the properties of circular arrays that

are revealed by a study of their self- and mutual admittances

are resonant spacings at which all of the elements interact

vigorously . In arrays containing only a few elements, the

resonant spacings are most important with half-lengths near h

= X/4; in larger arrays, they are most important for elements

with somewhat greater lengths. In Fig. 9 are shown the

theoretical self-conductanceand the mutual conductances for a

circular array of 20 elements each with half-length h

=

3X/8

and radius a = 0.007X; the distance between adjacent

elements

is b.

Only element #1

is

driven. The sharp maxima in

conductance occur at successive values of

b/X

such that the

circumference of the circle is near

nX

where

n

is an integer.

They denote resonant conditions around the array.

Fig. 9 is complete forb/h .2. For larger values of

b/h

only GII,GI*,and GI3are shown since GI4- are quite

small. A similar diagram for the susceptance is not shown.

The most interesting resonance in Fig.

9

is near

b/X

= 0.2

when the circumference of the circle is 4 h. The magnitude and

angle of the self- and mutual admittances, YI = GI

+

$31 k

= I Y l k lexp jOlk) ,

=

1, 2, e . . , N, are shown in Fig. 10

as a function of the element number. Since Yl

=

I I / VI,the

admittances are the normalized currents at the centers of the

dipoles. The corresponding graphs for an array with

N =

8

and

b/h

= 0.5 are

also

shown. For the eight-element array,

2h/X =

0.5;

for the 20-element array, 2h /X = 0.75.

The phase

e l k

of the currents in the several elements is quite

revealing. For the 20-element array, the currents in elements 2

and 20 lead the current in the driven element 1by about 80 .

Since the spacing is 0.2X, the driven element with each of the

adjacent elements forms a couplet with a maximum field

directed outward tangent to the circumference of the circle. In

effect, elements 1 ,2 , and 20 constitute a bidirectional unit that

transmits an electromagnetic field in both directions along the

2.0

+

b/A

Fig.

9.

Self- and mutual conductances o f circular array of 20 elements with

element #I driven;

a A =

0.007,

2 h / A =

0.75. (The mutual conductance

of antenna

n

referred to the driving voltage in antenna

1

is denoted by

G I ,

for

n <

10,

by

for

n 2 IO .

lY4

mS

0 81,

-90

-180'

1 2 3 4 5 6 7 8

ELEMENT

NUMBER

Fig.

10.

Self- and mutual admittances of circular array of

N

elements;

Y I

=

G l k+ j B I k

=

I , d V , ;k

=

1,

2

. . . , N .


5/9


6/9

KING:

SUPERGAIN AN TENNAS AND

YAGI

AND CIRCULAR ARRAYS

2

183

Fig. 14.

Circular array

of

N =

8

lements.

Since the lengths 2 h / h used in the calculations leading to

Figs. 9-12 are not in the range

0.35

h / h

5 0.37

for small

insertion loss as shown in Fig. 5, radiation from the circular

arrays may be substantial; however, the insertion loss of an

array with antiresonant elements has not been investigated.

The new insights into the properties of Yagi and circular

arrays which are outlined in the discussion up to this point are

based entirely on data selected from researches carried out

many years ago. They suggest a systematic further study

notably with the lengths of the elements in the circular array

changed from 2 h / h

=

0.75 or 2 h / h

=

0.5 to 2h/X

=

0.36,

the length for minimum insertion loss, as specified in Fig. 5.

With this value of 2 h / h , the traveling waves that propagate in

both directions around the circle should have larger amplitudes

and travel much greater distances around a larger circular

array before they are attenuated significantly. This should lead

to large currents in the elements diametrically across from the

driven element even in very large arrays. The ranges of the

traveling waves and standing waves around the circle will be

studied together with the associated field patterns in order to

learn how a highly directive pattern is generated.

IV. THEQUANTUM-MECHANICALNALOGF THE

YAGI-UDA RRAY

The motivation to investigate further the resonant properties

of large circular arrays of dipoles came from unrelated studies

in quantum mechanics. Specifically, a recent paper by

Grossmann and Wu [151 contains the following introductory

sentences: “It is the purpose

...

to study the quantum-

mechanical analog of one of the most intriguing and practical

phenomena in classical electromagnetism: the Yagi-Uda

antenna array. Although invented over half a century ago and

used almost universally for television reception, it has defied a

complete theoretical analysis despite the many excellent

papers on this topic.” The paper goes on to study the quantum-

mechanical analog, viz., an ideal polymer represented by an

infinite one-dimensional array of equidistant point interactions

(Fermi pseudopotentials). This is found to have resonances of

zero width. A linear polymer of finite length does not have

especially sharp resonances, due to radiation along its length.

However, “a very narrow resonance should occur if the ideal

polymer is bent into a closed loop

.

The present investigation

of the problem in quantum mechanics suggests strongly that

similar extremely narrow resonances must occur in various

antenna arrays. In particular, the circular array of dipole

antennas must exhibit such a resonance. It is curious that, in all

investigations of the circular array, the lengths and spacings

are such that narrow resonances fail to appear. It would be

very interesting to study, either numerically or experimen-

tally, a circular array with an extremely narrow resonance.”

A further step in the study of narrow resonances in dipole

arrays was taken by Wu [16] in an analysis of an array of the

simplest, nontrivial scatterer in the context of Maxwell’s

equations, the pseudo-dipole. This is shown to have infinitely

narrow resonances for the infinite array and “their existence is

the underlying reason for the excellent properties of the finite

array, includng the Yagi-Uda array.

”

V. CIRCULARRRAY

ITH

A LARGE UMBERF ELEMENTS

In order to investigate the possible and potentially important

applications of an egg-shaped array as a superdirective

microwave antenna, a systematic study must first be made of

the properties of a circular array when the number of elements

is large, one element is driven, and the dimensions are selected

to lead to large resonances. The circular array must be

analyzed first since, for it, the N simultaneous integral

equations for the currents in and admittances of the elements

can be replaced accurately by

N

independent integral equa-

tions for the Nphase-sequence currents. However, in order to

achieve extremely high directivity, a properly proportioned

egg-shaped array will probably be required. This will have to

be analyzed directly with only one element driven and not by a

superposition of Nphase sequences, each of which involves all

elements driven. The validity of such a method will be verified

with the circular array and then applied to the egg-shaped one.

Introductory aspects of the analysis of the circular array are

given below.

The circular array consists of

N

identical antenna elements,

uniformly spaced around a circle of radius

R .

Thus, the

angular separation between adjacent elements is 2 ?r/N. The

antenna elements may be circular disks, for example, or

parallel conductors of length 2 h and diameter 2 a . This latter


7/9

184

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31. NO. 2 , FEBRUARY

1989

N

arbitrary driving voltages V , including especially that of

primary interest here, when element 0 is driven and all others

exp [

1 4

case is illustrated in Fig.

14.

In this case, the set of N

simultaneous integral equations

for

the

N

currentsZJz’) in the

N

coupled elements is well known

[17].

t is

.NZ-k2~2- -

Ncosh-’

( N / 2 k R ) ] ]

2

1

=E

Cl cos kz+- V Isin klzl

CO

2

I = O 1 2, * e * , N - 1 ,

6 )

where V is the driving voltage,

O

= 1207r ohms, and the

kernel is

eikrlj

X d Z , Z ’ ) = - ; r0=J(z-z2‘)2+b:,, blI=a, (7)

‘0

where b, is the distance between element

I

and element

j .

The solution of the

N

simultaneous equations

6 )

has been

carried out by first reducing them to N independent integral

equations. This is accomplished by selecting N sets of driving

voltages If ”’), m = 0, 1 * * a ,

N

1 ( N phase sequences)

for

each

of

which the ratio

Zjm)(z’)/Ijm)(z‘)

s independent of

1. That is,

l :hZ(m)(z’)K(z ,’ ) dz‘

2

cos kz+- Vm )in k ( z l ,

(8)

where

With

1 = 0, (9)

becomes

r o j = J ( z - z ‘ ) 2 + b ~ j ,oo=a, 1 3 )

14)

Note that when m = 0, all currents are equal in magnitude and

in phase; when

m = N/2

(provided that

N

is even), eiZnm;IN

( - l)’, so

that the currents are all equal in

magnitude but alternate in direction, i.e. , with phases that

alternate between 0 and 7r. Because of symmetry, I ’ ) =

Z z‘) .

Of particular interest is the kernel of the integral equation

(9) with the phase-sequence currents (14).With z’ = z and

because of symmetry with respect to element 0,

with

IT) ’ ) / z$@(z)= eiZsmj/N.

=

-

Since

ka

4

this can be written as follows:

In the approximation to a circular antenna array by Fermi

pseudopotentials [15] , the behavior of the array when one

element is driven is determined by the right-hand side of

16) ,

except that the quantity

a

has to be reinterpreted, and depends

on the shape of the antenna elements.

For large Nand subject to the conditions m > kR, N m

> kR,

the imaginary part of

(16)

can be expressed as follows:

exp

{ 2 [ J m 2 - k 2 R 2 - m

cosh-’

(m/kR)]}

exp

[ 2 { J ( N - m ) 2 - k 2 R 2 - m

cosh-’

[ ( N - m ) / k R ] } ]

k

4J;;

m2

2R2)3/4

[(

N

-m ’

2 R 2 ]

’4

1 7 )

Here the exciting voltages and the currents have the form:

Let this be evaluated for the mth phase sequence with

alternating phases, i.e. , when

m

=

N/2

(which satisfies

m >

V(m)= V(m)eiZ*m(j-N , z ( m ) ( z / )=

~ j m ) ( ~) e i2 * m( j -

/ ) / N .

J I J

kR, N m > kR) ,

so

that exp ( i 2 a m j / N )

=

exp ( i a j )

=

27rb/h

-Nb/h=N/3 ,

19)

kb

kR =

2

sin

T / N ) - ~

in

( T / N )


8/9

KING: SUPERGAIN

ANTENNAS

A N D

YAGI

AND CIRCULAR

ARRAYS

L

n

.- 0.5.-

-f 5

E-.

Y

0

185

I

I

I

I

K Y ) k

I I

'.- 1 I

I /I

1 - 0

I

25

; I

I

I

I

I

I

I

I

I

k h z 1 . 5; z . 0 ; a/X=0.0033

b/X =0.33 --- b/A 0.033

8 2.0

0

20

(ml

k

K R

-

Phase

s e q u e n c e , m .

array of 150 elements.

Fig. 15.

Real and imaginary parts of kernel for each element in circular

where

b

is the distance between adjacent elements, it follows

that

With

N

=

150,

As a check,

(13)

was evaluated directly on a computer with

quadruple precision form =

N/2

and N =

150

with the result

7.1689 x which is in close agreement with (21). It is

seen that the imaginary part of the kernel is extremely small,

so

that

K z ,

z ) K R ( Z ,

z )

1 N-L

COS

kboj

=-+C

- 1 ) j - (22)

k k ka j = l kboj

Since ka is small, this quantity is large.

Graphs of K $ ) / kand Kk ) / k is defined in

(16)

are shown

in Fig.

15

as a function of the phase-sequence number m for

two values of b/h, viz., b/X = 1/3 in solid lines and b/X =

1/30 in broken lines. It is seen that K k m ) / ks almost the same

for all values of

m

nd quite large, viz., 49 for

b/X

=

1/3

and

61

for

b/X

=

1/30.

K j ) / k is vanishingly small for m in

a range near

m

= N/2 = 75. This range extends from 50 to

100 for b/h =

1/3,

from near 10 to 140 for

b/X = 1/30.

Outside of these ranges, K i ) / k oscillates about

1.5

for b/X

= 113 and about 16 for b/X = 1/30.

As

shown in the preceding discussion of arrays with N 5

20,

the properties of the circular array with only one element

driven depend critically on the length

2h

of the elements and

the distance b between adjacent elements; the radius a of the

elements is also an important parameter. Extensive further

calculations are in progress to examine the behavior of the

array with N much greater than

20

in various resonances

especially under conditions with minimum insertion loss.

VI. BEAM CANNING

A useful immediate application of the resonant circular

array is to provide a readily rotated or otherwise scanned

directional beam. This property is readily explained with

reference to Figs. 1 1 and

12

with N =

20.

It is evident from

Fig. 12 that a useful, quite narrow beam is available in the

direction 4 = 0 toward the single driven element. This

direction is readily changed electrically simply by switching

the applied voltage from the terminals of the antenna # 1 in Fig.

12 to the terminals of any other element. With 20 elements,

there are

20

increments of 18 . By increasing the number of

elements with the circumference of the circle fixed at four

wavelengths, the size of the increment can be reduced as

desired. With N = 36, the increments are 10 . A rotating

beam is obtained by continuously moving the applied voltage

from element to element around the circle. An oscillating

beam over any desired angle is also readily available. The

switching process must, of course, replace an effective short

circuit at the base of an element by the applied voltage and

reinstate the short circuit when the driving voltage is shifted to

another element. Clearly the circular array provides a very

simple scanning beam with no physically moving parts-

however, the switching circuits may be quite complicated.

VII. CONCLUSION

A review

of

available data from numerous experimental and

theoretical researches-many carried out over 25 years ago-

combined with a very recent quantum-mechanical investiga-

tion, has led to new insights into the possibilities of closed

loops of dipoles as highly directional arrays. The critical

newly emphasized feature is the remarkable high-Q property

of a correctly designed closed loop of coplanar dipoles when

only one element is driven and all dimensions-the length of

the elements, their cross-sectional size and shape, the number

of elements, and the circumference of the closed loop-are

correctly chosen. Extensive, highly precise theoretical and

experimental research is indicated to translate a challenging

possibility into a useful highly directive radiating system.

REFERENCES

A. Einstein, h e r die Entwicklung unserer Anschauungen uber das

Wesen U . die Konstitution der Strahlung,

Phys. Zfschr.

vol. 10, p.

817, 1909.

C. W . Oseen, Die Einsteinsche Nadelstichstrahlung und die Max-

wellschen Gleichungen,

An n . Phys.

vol. 69, p. 202, 1922.

H . Lottrup Knudsen, Superforstaerkningsproblemets Udvikling,

Saertryk Ingenioren

nr. 35, pp. 6 90-702 , P. Hansens Bogtrykkeri,

Copenhagen, Denmark, 1956.

T . T. Taylor,

A

discussion

of

the maximum directivity of an

antenna, Proc. IRE vol. 36, p. 1135, 1948.

L. J . Chu, Physical limitations of omi-dir ectional antennas,

J.

Appl.

Phys.

vol. 19,

p.

1163, 1948.

A.

Bloch, R. G . Medhurst, and S. D. ool,

A

new approach to the


9/9

186

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,

VOL. 37,

NO.

2,

FEBRUARY 1989

design of super-directive aerial arrays,” Proc. Inst. Elec. Eng. vol.

100, pt.

111,

p. 303, 1953.

[7] E. C. Jordan, Electromagnetic Waves and Radiating Systems.

New York: Prentice-Hall, 1950, p. 445.

[8] N. Yam , “A note on super-gain antenna arrays,” Proc. IRE vol. 39,

p. 1081, 1951.

[9]

H.

W . Ehrenspeck and H. Poehler, “A new method for obtaining

maximum gain from Yagi antennas,” IRE Trans. Antennas Propa-

gat. vol. AP-7, p. 379, 1959.

R.

W .

P. King, “Linear arrays: Currents, impedances, fields

I,”

IRE

Trans. Antennas Propagat.

vo l. A€’-7, p .

S440,

1959.

R. W . P. King, R. B. Mack, and S. S. Sandler, Arrays of Cylindrical

Dipoles.

R.

J.

Mailloux, “The long Yagi-Uda array,” IEEE Trans. Antennas

Propagat. vol. AP-14, pp. 128-137, 1966.

J.

Shefer, “Periodic cylinder arrays as transmission lines,” IEEE

Trans. Microwave Theory Tech.

vol.

MTT-11, pp. 55-61, 1963.

[ lo]

[ I l l

[I21

[13]

New York: Cambridge Univ. Press, 1968.

1141

[15]

R. B . Mack, “A study of circular arrays, Parts 1-6,” Cruft Lab. Tech.

Reps. 381-386, Harvard Univ., Cambridge, MA, 1963.

A. Grossmann and T. T. Wu, “ A class of potentials with extremely

narrow resonances,” Chinese

J .

Phys. vol. 25, pp. 129-139, 1987

(also Marseille CNRS Preprint CPT-Il/PE. 1291.,1981).

T. T. Wu, “Ferm i pseudopotentials and resonances in array s,” in

Resonances-Models and Phenomena: Proceedings Bielefeld

1984

S .

Albeverio, L.

S .

Ferreira, and L. Streit, Eds. Berlin:

Springer-Verlag, 1984, pp. 293-306.

R . W . P. King, Theory of Linear Antennas. Cambridge, MA:

Harvard Univ. Press, 1956, p. 351, eq. 7.

1161

[17]

Ronold W. P. King (A’3O-SM’43-F’53-LF’71), for a photograph and

biography please see page 1212 of the N ovember 1985 issue of this

TRANSACTIONS.

1989 02 Supergain Antennas and the Yagi and Circular Arrays.pdf

Documents

Transcript of 1989 02 Supergain Antennas and the Yagi and Circular Arrays.pdf