RESEARCH DEPARTMENT

METHODS OF. CALCULATING THE HORIZONTAL RADIATION PATTERNS

OF DIPOLE ARRAYS AROUND A SUPPORT MAST

Report No. E-OS2 (1958/10)

P. Knight, B.A., A.M.I.E.E. (w. Proctor Wilson)

This Bepor~ is ~he proper~y of ~he

Bri~lsh Broadcas~lng Corpora~lon and may no~ be reproduced in any form wl~hou~ ~he written permission of the Corporation.

(

(

Report No. ID-062

METHODS OF CALCULATING THE HORIZONTAL RADIATION PATTERNS

OF DIPOLE ARRAYS AROUND A SUPPORT MAST

Section

1

2

3

4

5

6

7

8

Title

SUMMARY ••••

INTRODUCTION 0

CARTER'S METHOD FOR CYLINDRICAL MASTS.

THE INFINITE-PLANE METHOD FOR LARGE POLYGONAL MASTS.

METHODS FOR POLYGONAL MASTS OF INTERMEDIATE SIZE •

4.1. The Induced-current Method 4.2. The Diffraction Method ••••••••••

COMPARISON OF CALCULATED WITH MEASURED PATTERNS. •

CONCLUSIONS. • •

ACKNOWLEDGMENTS. •

REFERENCES

APPENDIX I - CARTER v S FORMULAE FOR THE RADIATION PATTERNS OF DOUBLETS. • • • • • • • • • • • • •

APPENDIX II - THE EQUIVALENT RADIUS OF A MAST WHOSE CROSS-SECTION IS A REGULAR POLYGON • •

APPENDIX III - FORMULAE FOR CALCULATING RADIATION PATTERNS

1

1

1

3

3

3

4

6

7

14

14

15

19

BY THE INDUCED-CURRENT METHOD • • 0 • • • • 23

APPENDIX IV - SCHELKUNOFFv S FORMULAE FOR DIFFRACTION AT A WEDGE. 26

Report No. E-062

May 1958 ( 1958/10)

METHODS OF CALCULATING THE HORIZONTAL RADIATION PATTERNS

OF DIPOLE ARRAYS AROUND A SUPPORT MAST

SUMMARY

Most v.h.f. transmitting aerials consist of an array of dipoles mounted on a supporting masto Some theoretical methods which may be used to determine their horizontal radiation' patterns are described in this report, and their limitations discussed. Patterns calculated by these methods are compared with patterns measured using small-scale models.

1. INTRODUCTION

The currents flowing in the dipoles of a v.h.f. transmitting aerial induce currents in the supporting structure. The field associated with these induced currents usually modifies the radiation pattern to such an extent that the supporting mast must be considered as an integral part of the aerial system.

The supporting mast is usually of lattice construction and in the absence of screening its effect would be difficult to predict as it depends on the arrangement of the st,ructural memberso The mast is, however, usually covered by a screen of wires parallel to the plane of polarization of the radiated wave in order to reduce the fields within the structure. Its outer surface then behaves as a continuous sheet and may ,be treated as such'theoretically.

The cross-section of the mast usually takes the form of a circle, a s~uare or an e~uilateral triangle. The horizontal radiation patterns of arrays of vertical dipoles on masts of circular cross-section may be calculated exactly (see Section 2) but no such solution exists for masts of any other shape. or for horizontal dipoleso Approximate methods of treating these cases are described in this report.

2. CARTER v S METHOD FOR CYLINDRICAL MASTS

If a cylinder is placed near a Hertzian doublet currents will be induced on its surface and the radiation pattern of the doublet will be modified. The polar­i zatiol'l. of the resultant wave will depend on the relative orientation of the doublet and the cylinder and will in general be elliptical. The electric field can always be resolved into two orthogonal components normal to the direction of propagation; it is convenient to resolve the field into vectors perpendicular to, and lying in, the plane containing the axis of the cylinder.

2

Carter 1 has derived exact solutions for these two components by treating the doublet as a receiving" aerial. The voltage induced in the doublet is calculated in turn for incident waves polarizea in the directions of the two components, these voltages are, by the principle of reciprocity, proportional to the two components of the field which would be radiated if the doublet were driven.

Each of the incident waves is assumed to be plane and is resolved into a set of standing cylindrical waves whose axes are coincident with the axis of the cylinder. The field re-radiated by the currents induced on the surface of the cylinder can also be resolved into a similar set of cylindrical waves. Their amplitudes and phases may be determined by equating the sum of the two sets of waves to zero at the surface of the cylinder, since no tangential electric field can exist there. Both sets of waves are then completely specified and the voltage induced in a doublet having any position and orientation may now be calculated.

Radial doublet

Vortical doublet

Tangential doublet

Fig. I - Types of doublet

Carter's results are stated for doublets having the three orthogonal orientations shown in Fig. 1; the field due to a doublet having any other orientation can, of course, be obtained by resolving it into three component doublets. In this report attention has been confined to the fields set up by each of these arrangements of doublets in the plane perpendicular to the cylinder, which is assumed to be vertical. The expressions for these horizontal radiation patterns (h.r.p.s) are given in Appendix I; each takes the form of a Fourier series with complex coefficients and is closely related to corresponding formulae for the doublets in free space.

The formulae for the h.r.p.s of arrangements of vertical doublets apply exactly to similar arrange­

ments of vertical dipoles since the radiation patterns of all the elementary doublets which form a dipole are identical in shape. The h.r.p.s of horizontal dipoles mounted on a cylinder could be calculated by applying the formulae for tangential and radial doublets to the elementary doublets forming the dipoles but the necessary integration can only be performed numerically; this method is therefore impracticable. The pattern may, however, be calculated fairly accurately if each dipole is replaced by a doublet situated at its mid-point. A more accurate result is obtained if this substitution is used to calculate only the field radiated by the currents induced in the cylinder, the direct contributions from the dipoles themselves being deter­mined exactly by the usual geometrical method (this method was used for the patterns described in Section 5). Alternatively, tangential dipoles may be assumed to behave as dipoles bent into arcs of circles concentric with the cylinder. Although an exact solution exists for this arrangement, the use of this approximation for linear dipoles gives a less accurate result because the contributions from the dipoles them­selves are calculated only approximately.

Carter's method may be used to obtain an approximate solution for a mast of polygonal cross-section provided that the width of each face is sufficiently small compared with the wavelength. If the radiating elements are vertical, the diameter of the equivalent cylinder is such that it has the same inductance per unit lengthas the mast. The size of cylinder having the same inductance as a regular polygonal prism may be calculated exactly; the result is derived in Appendix II. But when the

3

radiating elements are horizontal the diameter of the equivalent cylinder depends not only on the shape of the mast but also on its orientation relative to the elements. For example, if the supporting mast were a thin strip it would have no effect on the h.r.p. of a dipole at A (Fig. 2), but a maximum effect for a dipole at B. For masts whose cross-sections are regular polygons, however, experience has shown that for horizontal radiating elements a good approximation is a cylinder of the same perimeter.

Comparison of measured and calculated patterns has shown that these approximations give reasonably accurate results for triangular masts whose sides are less than O'3A wide and for square masts less than o'5\. wide. B

3. THE INFINITE-PLANE METHOD FOR LARGE POLYGONAL MASTS

If the faces of square, triangular or other polygonal masts are so wide that the radiating elements may be mounted more than a wavelength from the corners, each face may be treated as an infinite plane and its effect calculated by the method of images, The total field in any direction may then be obtained by adding the contri­butions from those faces which are visible, with due regard to phase.

Fig.2-Horizontal dipoles near a support of "strip" cross-section

This method can be used only for masts whose faces are at least two wave­lengths wide. It fails to give accurate results for smaller masts because diffraction effects at edges cannot be neglected in such cases,

4. METHODS FOR POLYGONAL MASTS OF INTERMEDIATE SIZE

Square or triangula~ masts whose sides are less than O'5A or O'3A wide respectively may be replaced by equivalent cylinders and the patterns of dipoles mounted upon them calculated by Carter's method, described in Section 2, Alterna-tively, if the faces are at least two wavelengths wide the infinite-plane method described in Section 3 may be used. Both these methods fail if the width of the faces lies between those limits. Two methods which give more accurate results when the mast is of intermediate size are described in this section.

F

-----~------.-.-----------E-(atco) A 0 B -C (atoa)

Fig. 3 - A filament parallel to a conducting ribbon

4.1. The Induced-Current Method

In a recent paper 2 Moullin showed that the radiation pattern of a current filament F (Fig. 3) parallel to a ribbon AB could be calculated with little error by assuming that the current density at any point on the ribbon is the same as that at the corresponding point of a strip AB of an infinite plane EC. No minimum width for the ribbon was stated but it was shown that radiation patterns calculated by this method are in good agreement with measured patterns provided the ribbon is more than a

4

wavelength wide.

Moullin's method also applies to the distant field in a plane perpendicular to the ribbon if the filament is of finite length or if it is a dipole. In applying it to arrays of dipoles mounted on polygonal masts, each face which has a dipole associated with it is assumed to carry the current which would be induced on it if it were part of an infinite plane. The currents induced on the face by dipoles mounted on other faces are disregarded; these currents will be very small for square and triangular masts, since the face always lies in the shadow of the adjacent dipoles.* The radiation pattern of each face and the dipole upon which it is mounted is then calculated over the complete range of angles and the fields produced by all the dipoles added, with due regard to phase, to give the resultant h.r.p. The method of calculation is described in greater detail in Appendix Ill.

Fig. 1J. - Ribbon and line doublet.

A rrows show the di rection of current

flow

If the filament is replaced by a line doublet (Fig. 4) the currents induced on an infinite plane will change their direction through a right angle, but their amplitudes and phases will remain the same. It is reasonable to suppose that no great change in the distribution will occur if the plane is replaced by a ribbon more than one wavelength wide. This assumption has therefore been used as a basis for calculating the radiation patterns of horizontal dipoles mounted on polygonal masts, each dipole being treated as a line dOUblet.

It is shown in Appendix III that the radiation pattern of a line doublet and a ribbon may be derived from the corresponding pattern for a filament simply by multiplying by cos cp (Fig. 3l. If the radiating element is a horizontal dipole the filal)1ent pattern may instead be multiplied by the dipole radiation pattern. This alternative is thought to be slightly more accurate since the pattern of the dipole itself is not distorted. This method was therefore used for the calculations described in Section 5.

4.2. Tne Diffraction Method

A disadvantage of the method described in Section 4.1. is that the currents induced on the faces adjacent to the ones upon which the radiating elements are mounted are assumed to be zero and are therefore neglected. These currents, although small, nevertheless contribute to the distant field. Their presence can be dembn­str~ted by the fact that the h.r.p. of a single horizontal dipole mounted on a large mast does not in general exhibit zeros. Had the current been confined to the face parallel to the dipole two diametrically opposite zeros would have been present.

*Ir the mast has more than four faces, each face may be illuminated by dipoles OD adjacent faces and the resulting induced currents may not be small. Since, however, the crOss section will more nearly resemble a circle, Carter's method should be applicable to masts of greater cross­sectional area.

To overcome this difficulty, Schelkunoff's formulae 3 for the diffraction of a cylindrical wave by a wedge have been applied to the corners of triangular and square masts. Schelkunoff gives an expression for the radiation pattern of a filament F in the presence of a whole plane bent into a wedge A'OB (Fig. 5); it can be shown that a similar expression may be derived when the filament is replaced by a line doublet (see Appendix IV). If the field of the filament in the presence of the whole plane AOB is subtracted from the field of the

5

F •

A~ - - - - - - - O'---~---!Q~---- B

(at 10) (at co)

Fig. 5 - A current filament near a conducting wedge

filament in the presence of the wedge, the difference is equal to the diffracted field which is set up when the plane is bent. *

In applying this result to a mast we will assume that the field diffracted at 0 will be changed very little if the plane is bent again at Q and R although these bends will, of course, set up additional diffracted fields. This assumption is reasonable provided Q and R are a wavelength or more from 0; it is similar to Moullin's assumption that no change takes place in the current distribution near the edge of a half-plane when most of the half-plane is removed. Thus in calculating the pattern of an element at F the fields diffracted by the corners at 0 and Q are calculated separately by assuming that these corners are the corners of infinite wedges. The two diffracted fields are then added to the field which the element would set up over an infinite plane AB. The fields diffracted at all other corners are neglected.

When the radiating element is a vertical dipole the solution for a filament is used. When it is a horizontal dipole the diffracted fields appropriate to a line doublet are added to the infinite-plane solution for a dipole, as the diffracted f-ields for a dipole cannot easily be calculated. The error resulting from the use of this approximation is very small.

Horizontal radiation patterns calculated by the method described exhibit small discontinuities owing to the approximation made. For example, referring to Fig. 6, the field diffracted at 0 is necessarily assumed to be zero in the sector A'OB, and the field diffracted at Q is assumed to be zero in the sector B'QA. It follows that the total field calculated will in general be discontinuous along the lines OA, GB, RA' and RB'. In the ,case of horizontal polarization the value of the field will be discontinuous; for example the field diffracted at 0 will be finite up to the boundary (A'OB) of the sector in which it is zero. In the case of vertical polariz-

F • A--------v.-----e

"R I \

I \ I \

I \ I \

I \ I '. SI AI

Fig. 6 - A current filament near a conductor of triangular cross-section

ation the field diffracted at 0 will fall to zero at the boundary A'OB, so that the

*For simplioity the expression "the field diffracted at 0" is used to denote the change in field caused by bending an infinite plane at 0 to form a wedge.

6

discontinuity will be one of gradient only. In all h.r.p.s computed for vertical polarization the discontinuity in gradient has proved imperceptible. The discon­tinuitiesofboth types would vanish if account could be taken of multiple diffraction.

It is of interest to compare the patterns calculated by the two methods described in this Section in the special case when the "mast" is a thin strip. The pattern of this arrangement is calculated directly by the induced-current method and may be calculated by the diffraction method by bending the plane through 1800 at o and Q. The patterns for a filament distant 'A/4 from a strip one wavelength wide calculated by these two methods are compared in Fig. 7 and it will be seen that the agreement is quite good. Equally good agreement was obtained when the calculations

were repeated for a line doublet.

1'1

t:-::--1'0

9

08

0'7

6

E4>

O'l

0'2

O' I

o ---(). I

-().2

~ ~ I I

'- R'om,"'~'d' "-~ '\, 4r-A~1

,

'\ ,\, ImaainarY Dart

\,/ ~,

~,

/' ~ --~ Real Dart

./ ,/

~, ~ '-

./ ,\, "~ k::'= r'\..' .. 60· 90· --~30~" ~ " J 0° ''':J. o~o~ I ............. '-- ~ .. ---

et> I

---Radiation pattern calculated by the induced-current method

----- Radiation pattern calculated by the diffraction method.

The phose of Ihe field is referred ID Ihe mid-poinl of Ihe slrip.

In Ihe absence of Ihe strip its amplitude would be O'S

-==-__ MID'

Fig. 7 - The radiation pattern (in complex form) of a current filament parallel to a strip one wavelength wide

5. COMPARISON OF CALCULATED WITH MEASURED PATTERNS

Radiation patterns have been calculated by all the methods described in this report for horizontal and vertical dipoles mounted centrally on the faces of square and triangular masts. For both mast shapes the width of each face was one wavelength, this value being chosen because it lies within the range in which both of the appro~ imate methods described in Sections 2 and 3 fail. The object was to determine

7

whether the methods described in Section 4 gave more accurate results. Measurements of h.r.p. were therefore made with small-scale models. All the h.r.p.s have been normalized so that their maximum values are unity.

The h.r.p.s for a mast associated with a single dipole are compared in Figs. 8-11. Considering first the case of vertical polarization (Figs. 8 and 9) it will be seen that the h. r.p. s calculated by Carter's method show fairly good agreement with the measured results, but the infinite-plane method gives poor results because backward radiation is entirely neglected. The methods described in Section 4 both show better agreement.

In the case of horizontal dipoles (Figs. 10 and 11) the infinite-plane method is a better approximation than Carter's method, the latter giving too large a field in the backward directions. Both the methods described in Section 4 give reasonably good agreement. The discontinuities in the curve calculated by the diffraction method for the triangular mast (Fig. 10) should be noted. A sim:j.lar discontinuity exists at 90° in the corresponding curve for a square mast (Fig. 11),

but it is too small to be shown. The reason for these discontinuities was explained in Section 4.

Figs. 12 and 13 show the corresponding patterns for co-phased dipoles mounted on all faces of the masts. These patterns were calculated by combining the single dipole patterns with due regard to phase. The methods which gave the closest agreement with the measured patterns for single dipoles would also be expected to give the best agreement in this case. It will be seen that this is in fact true, the two methods of Section 4 showing quite good agreement throughout. All four methods show fairly good agreement with the measured pattern for vertical dipoles on a square mast. Carter's method is also approximately correct for vertical dipoles on a triangular mast. With these exceptions Carter's method and the infinite-plane method are not of sufficient accuracy to justify general use.

6. CONCLUSIONS

By using the more exact methods described in Section 4 the h.r.p. of a dipole mounted on a mast of polygonal crosE-section may be calculated when the size of the mast is such that neither Carter's method nor the infinite-plane method can be used. But in practice the calculation involved is long and tedious and the solution to the problem can usually be obtained more easily by making measurements with a model.

The fact that it is possible to calculate an accurate pattern when the mast is of intermediate size shows that the initial assumptions made about the current distribution were justified. A better understanding of the way in which the combination of mast and dipole radiates has therefore been obtained. Thus when a dipole is mounted at the centre of a face one wavelength or more wide, the currents induced on other faces are small and contribute little to the radiated field. With masts of this size a practical implication is that screening rods may be required only on those faces which have dipoles mounted upon them.

8

0·8

0·6

E4>

0·4

0·2

o o

1·0

0·8

06

Eq,

0·4

0·2

o o

-. f-.::..,:

--= ~

-~

~ .........

t.... .....

"- ..... , 1'--,

~ ~ irY' ~ ~J \~

\

l"" \ . , ,"" \

\

\ " "" \

~ \ \ "-\ r--\

....

-- H.R.P. calculated by Carters method. --- H.R.P. calculated by the infinite-

plane method

o Measured points (averaged over 360°)

~ .....

~ ~

I\~ A q-y <I ,-,.

h. \! ~~,

~ K " I

~ ~ I

' ... ~ ~ ~ -..,

~ .--""" ~ l.

1500

--H.R.P. calculated by the induced-current method. --- H.R.P. calculated by the diffraction method.

o Measured POints (aVf!taged over 360°)

Fig. 8 _ Theoretical and measured h.r.p.s of a vertical dipole spaced O'25A. from a triangular mast I'OA. wide

1-0

0-8

0-6

0-4

0-2

c

1-0

o-a

0-6

I

i 0-4 !

0-2

~ -- ... ~ ...

~ ....

-~

......

~ " ~ ~iW

~ IV

, ~

f--).-

\

\

" \ \

'" \ \

1\ ~ \ ~ \ ~

\ -....., p--...,

~ \ c 30

~

~

bO c

- H_R.P. calculated by Carter's method

H_R.P calculated by the infi nite­plane method

o Mq:-osured points (averaged over 360°)

'\, ~iW

~~ IV

->--

"",~-K', ~ :~

"' ~ .... --.::--~i -~

-- H_R_P. calculated by the induced - current method.

--- H.R.P. calculated by the diffraction method. o M easured pOints (Cveraged over 3600)

--

9

b.."....--c

! 1 1

I

l I

---

~ --r-_ b

Fig. 9 - Theoretical and measured h.r.p.s of a vertical dipole spaced O'25A from a square mast "OA wide

10

1·0 ~ ~,

"- , I\, ' \

, iW , , o·a

l\'h \f \ \ 0·6

\\ \ Et/>

Ecf>

0·4

02

1·0

0·8

06

0·4

o· 2

o o

~ ~

"-

, , ~ ~ >---

\ -r- "'" , , r--.-"-......

90 cl>

0 120 0 150 0

-- HR.P. calculated by Carter's method.

---- H.R.P. calculated by the infinite-plane method

o Measured points (9vrraged over 3600)

r"'\ \ ~.L W f\ V ,~

~~

'\, '\~

1

~ I> I

'Ii: ~ I

~ D

"" ~ ~- -- .-~'O_,-:" ~ IT- "'-

-HR.P. calculated by the induced-current method.

--- HR.P. calculated by the diffraction method.

o Measured points (averaged over 3600)

.".-

1/

:

, ... '"

::.-j..--

1800

Fig. 10 - Theoretical and measured h.r.p.s of a horizontal dipole spaced 0'25"- from a triangular mast I'O~ wide

1·0

08

0·0

0·4

0·2

o o

1·0

08

0·6

0·4

0·2

......

-...

11

~! ~!

~ !

I i

[\, Ql+y IV '\

\ !--}..-

1\ \ ~ 10-

'~~ V V r-. V " V , I'. / '-. "-I-"

.......... 30° 60° 9i" 1200 150° 1800

-H.R.P. calculated by Carter's method

---H.R.P. calculated by the infinite-plane method

o Measured points (averaged over 360~

~ " f\

\ i lilt=-)! 4·-,

I

I' I1 '\ l:':=}, ....:::::

~ i

~ '\ ~

\ ~ ~ ,

~'-

~ - - ~ ~ .. ~ ...- - -- ,.,...-.

'""-~

--H.R.P. calculated by the induced-current method. ---H.R.P. calculated by the diffraction method.

o Measured points @veraged over 360°)

Fig. II - Theoretical and measured h.r.p.s of a horizontal dipole spaced O'25A. from a square mast I'OA. wide

12

-- H.R.? calculated by Carters method --- H.R.I? calculated by the infinite-

plane method o Measured points (averaged over 3600

)

-- H.R.P. calculated by the induced-current method H.R.P.calculated by the diffraction method

o Measured pOints (averaged Over 360 0 )

Fig. 12 - Theoretical and measured h.r.p.l' of co-phased vertical dipoles spaced O'25A from triangular and square masts with sides I'OA wide

H.R.P. calculaUd by Carter~ method H.R.P. calculated by the infinite­

plane method

o Measured points (averaged over 3600)

--H.R.P. calculated by the induced-current method. --- H.R.P. calculated by the diffraction method.

o Measured points (averaged over 360°)

Fig. 13 - Theoretical and measured h.r.p.s of co-phased horizontal dipoles spacad O'25A from triangular and square masts with sides I'OA wide

14

If measurement is not possible and calculation becomes necessary, the most suitable method must be chosen. If the size of the mast is such that neither Carter's method nor the infinite-plane method is suitable one of the methods described in Section 4 must be used. These lead to results of similar accuracy, so that preference may be given to the method requiring the simplest calculation.

Radiation patterns may be calculated directly from tabulated functions by the diffraction method, but the calculation is long and tedious. With the induced current method a number of numerical integrations must be performed (unless the mast is large enough for the integrals to be replaced by asymptotic approximations) and the calculation of a single radiation pattern is no shorter. However, the functions which require to be integrated depend only on the spacing of the element from the mast face, and once these have been plotted the pattern'S for a variety of mast widths and element positions may be rapidly obtained by inrbegrating between different limits. Moreover when the field of an element and the face from which it is supported has been calculated, it may' be used for square, triangular or other polygonal masts without modification. A further advantage is that the pat-terns for horizontal dipoles may be derived from the corresponding patterns for vertical dipoles by a simple multiplication. Thus although the work involved in calculating an isolated pattern by either method is considerable, the induced current method is much more flexible in its application and is therefore to be preferred.

7. ACKNOWLEDGMENTS

The writer would like to record his thanks to Mr. A. Brown, who performed the measurements, and to Mr K. Hughes for drawing attention to the paper by Professor Moullin and suggesting that the methods describ ed therein could be appli·ed to the problems. discussed in this report.

8. REFERENCES

10 Carter, P. S., "Antenna Arrays around Cylinder's", Proc. LR.E, Vol. 31, No. 12, p. 671, (December 1943)

2. Moullin, E.B., "On the Current Induced in a Conducting Ribbon by a Current Filament Parallel to It", Proc. LE.E., VoL 101, Part IV, No. 6, p. 7, (February 1954)

3. Schelkunoff, S. A., "Advanced Antenna Theory", p. 181

4. Otake, T., "Calculation of Electrostatic Capacities of Long, Straight Cylinders with some Geometrical Cross Sections", Kyushu University College of Engineering Me~oirs, Vol. 1, No. 1, pp. 31-77, (1913)

15

APPENDIX I

CARTER'S FORMULAE FOR THE RADIATION PATTERNS OF OOUBLETS

Vertical

Carter has derived expressions* for the field radiated in any direction by one or more doublets placed near a cylinder. Simplified forms of these expressions used for calculating horizontal radiation patterns are stated in this Appendix. o Doublet

~~~-+----~~"at ~=o

·i The similarity between Carter's formulae and the corresponding expressions for the radiation patterns of doublets in free space should be noted. For example the h.r.p. of a vertical doublet D (Fig. 14) referred to a vertical axis A is

Fig. lij - Doublet and cylindrical mast

110

E.J. = eil'bcosc/> = J (j3b)+-2 L: J"nJ (j3b)cosnc:p '+' 0 n:l n

where j3 = 2rr/~ and I n is the Bessel function of the first kind of order n.

well-known Fourier series is convergent. This

If a conducting cylinder is present each coefficient in the series is modified by the addition of a complex term Mn , sometimes known as the "mast term", which is a function both of the cylinder radius a and the doublet spacing b. The expression for the h.r.p. then becomes

OIl

Ec/> = J o (j3b)+Mo+2 n~l r[Jn(j3b)+MnJcosnc:p

Thus the radiated field consists of two components, a primary field due to the doublets themselves and a secondary field due to the currents induced on the surface _of the cylinder.

The secondary field, which is given by

110

Mo +-2 n~l jl'Mncos n c:p

m~ be calculated separately if desired. This is sometimes an advantage when the cylinder is small and the doublet spacing large, because the series for the secondary field converges much more rapidly than the series for the total field and is therefore easier to calculate. A simple geometrical method is then used to calculate the primary field. The fields may also be calculated separately when the radiating elements are horizontal dipoles. As mentioned in Section 2, CarterYs method does not give an exact solution for this case, but in one of the approximations which may be used the primary field is calculated geometrically and the secondary field by assuming that the radiating elements are doublets.

* Care should be exercised in applying the formulae given on pp. 686 and 687 of Reference 1 as there are nU1Il8rou S 11.1 sprint s.

16

The simplified forms of Carter's formulae given below are normalized to the maximum field which would be set up if the array were replaced by a single doublet carrying the same current. They are given for arrangements of m similar doublets equally spaced around the cylinder, and driven either in phase or with a progressive phase change of 2rr radians around the ring. Formulae for arrangements driven with different methods of phasing, or for asymmetrical arrangements, may be built up from the expression for a single doublet Cm = 1). They are as follows:

(i) For rings of vertical doublets or dipoles

(a) Driven in phase co

Eq, = mCvo + 2 n~l jJlnVucosmnc;b]

(b) Driven in progressive phase rotation

In is the Besse1 function of second kind and order n.

(ii) For rings of tangential doublets

The formulae are similar to those used for vertical doublets but all the Bessel functions are replaced by their derivatives and the coefficients are denoted by Wn instead of Vn.

Thus J~ ({3b) - j Y~ ((3b)

Wn = J~({3b)-J~({3a) J~ ({3a) - j Y~ ((3a)

The derivatives may be calculated from the functions themselves by the well-known formulae

J' (z) = ¥J (z) -J (z)] , n n= 1 n+ 1

I'(z) = ~[I (z)-I (z)] n n D1 ,,+ 1

The expressions for the radiation patterns are therefore

(a) For doublets driven in phase

co Eq, = -jm[Wo +·2 "~1 j""W..,.cosmnc;b]

(b) For doublets driven in progressive phase rotation

co E = ejq,m[W - 2: JO",,(W e=j""q,-W ej"nq,J]

q, 1 ,,=1 ",,=1 ",,+1

(iii) For rings of radial doublets (these formulae are included for the sake of completeness although the calculation of patterns of radial elements is outside the scope of this report).

(a) Driven in phase

-j2m Er/> = 7Jb

co 2: j",n mn Bmnsinmn r:P

n= 1

Cb) Driven in progressive phase rotation

where Bn {3 '{3 J n C{3b)-jYn C{3b)

= J ( b) - J ( a) n n J~ ({3a) - jY~ ({3a)

17

If the doublets are mounted on the surface of the cylinder b = a and Bn simplifies to

-j 2

7T{3a[J~c{3a)-jy~{{3a)]

Although greater accuracy is obtained by calculation; the coefficients 1n and Wn (but not Bn) may also be determined by the following graphical construction.

If the expression for 1n is re-arranged in the form

j [J n ({3a)Y n C{3b) - J n ({3b)Y n ({3a)] 1n = ~,r-~----~{3~----~~--~----

J n ( a) - j Y n ({3a)

it is apparent that the phase of 1n depends only on -YnC,Bo)+jJn (,80)= jHn(JlC,sa)

Locus of V n when the doublet is a fixed distonce from the oxis and the cylinder radi us varies

the cylinder radius a and is equal to

Thus for a given cylinder radius the locus of 1n on the complex plane is a straight line as shown in Fig. 15.

If b, the distance of the doublet from the axis, is constant, Yn ({3a)/Jn ({3a)

may be considered as a real variable. Now we may also write 1n in the form

Locus of V n when t h. c:ylindn radius is constant and the distanc. of th. doubl.t from th. axis varies

Requir.d volu. of V n Centre of circl.

o

Fig. 15 - Graphical construction for Vn

j [Yn ({3b) - I n C{3b){Yn ({3a)/Jn ({3a)}]

1- j{Y n ({3a)/tTn ({3a)}

lB

This expression is a bi-linear function of the variable Yn Cj3a)/Jn Cj3a). It may be regarded as a transformation which m~ be applied to the real axis of the YnCj3a)/Jncj3a) plane (corresponding to all real values of Yn Cj3a)IJn Cj3a» into the Vn pl·ane. Ey virtue of the properties of the bi-linear transformation the locus of the real axis of the I n Cj3a)/J

hCj3a) plane in the V" plane becomes a circle

centre: i [In (j3bl +-jln Cj3bJ] = tni1) (j3b)

radius: ~ I I n Cj3b) - jI n Cj3bll = ~ la12) (j3bll = % I a~1) Cj3b) I

where nJ1) Cx), nJ2) (x) are the Hankel functions of the first and second kinds respe~ tively, of order n.

It follows that in the Vn plane the intersection of the straight line passing through the origin at an angle 7T/2+tanD1{YnCj3a)/JnCj3aJ} to the real axis, and the circle of c~tre~ni1) Cj3bl and radiustlai1) Cj3b) I represents the value of Vn for any given values of a and b.

The geometrical construction is therefore as follows:

Ci) Draw a straight line, passing through the origin of the complex plane, making an angle 7T /2+-tan° 1{Yn Cj3alIJn Cj3al} with the real axis.

Ciil Draw the circle, centre .;ni1) Cj3bl, which passes through the origin.

This corresponds to radius ~ I ai1) Cj3b) I. The intersection of the straight line Ci) and the circle Cii) gives the

required value Vn'

The method used to determine ¥n is similar, the derivatives of the Eessel functions being used in place of the functions themselves. ~he construction can­not be used for Bn since this function has different properties.

19

APPENDIX II

THE EOOIV ALENT RADIUS OF A MAST WHOSE CROSS-SECTION IS A REGULAR POLYOON

Within certain limits, described in Section 2, radiation patterns of vertical dipoles mounted on non-cylindrical masts may be calculated by assuming that the mast behaves as a cylinder having the same inductance per unit length. This Appendix describes a method which may be used to determine the radius of this cylinder when the cross-section of the mast is a regular polygon.

The inductance of an isolated conductor has no real meaning because there is no return path for currents flowing through it. It is therefore necessary to suppose that the straight conductor under consideration forms part of a closed circuit, a convenient form being a concentric transmission line, with a cylindrical outer conductor. The inductance for a polygonal inner conductor may be calculated and the size of cylindrical inner conductor having the same inductance may then be found. The radius of the equivalent cylinder is found to depend to some extent on the radius of the outer cylinder, but it tends to a limiting value when the latter becomes sufficiently large. conductor.

This limit is the value required for the problem of the ir:,.:>lated

A useful property of radio-frequency transmission lines is that, if two alternative inner conductors have the same inductance, their capacitances to the outer conductor will also be equal. Since it is more convenient in this case to calculate capacitance, the size of circular conductor having the same capacitance to the outer conductor as the polygon will be determined.

Fig. 16 shows part of the electrostatic field distribution for a square inner conductor and its equivalent cylinder. This case will be examined in det ail. Since the capacitance calculation is

Fig. 16 - Field distributions in concentric transmission 1 ines

a two-dimensional problem the field distribution between the polygonal conductor and the outer may be transformed conform ally into a distribution for which a solution is known.

The field of the square conductor has fou~fold symmetry and on~quarter 'of the capacitance exists in the region defined by ABCDE (Fig. 16). The field distribution in this region (part of the z-plane) may be mapped conform ally on another plane (the w-planel in such a manner that the line ABCDE becomes the real axis on the w-plane. The Schwarz-Christoffel method* is used. The field in the region

*See, for example, 'Statio and Dynamic Electricity" by 'f.R. Smythe, Chapter 4, Seotion lS.

20

!BCDE (Fig. 17) is then mapped on the upper haJ.f of the w-plane and the large cylindri­cal outer conductor is transformed into a cylinder of different size. The field distribution on the w-plane then corresponds to that between a charged strip and a concentric cylinder.

-<u - plane z - plane

Fi g. 17

It may be shown that the transformation required for the square conductor is

which can be evaluated generally only in terms of elliptic functions. All the vaJ.ues of z required for this problem may however be determined by simpler methods.

The points C and C' lie at the origins on the two planes. If D' is defined as the point 1+ jO on the w-plane its position on the z-plane will be given by

z = f 1 -=-==d=t = Vt 2 -1

o

the solution of which is known in terms of factorial or gamma functions. therefore write

We may

z = r o

pi

-I+JO l+jO c+jO

8' c' 0'

.w -plane

Fi go 18

Thus the length of the line CD on the z-plane is 1"198. It may also be shown that, if E' on the w-plane is the point -1+ jO, then the distance EC on the z-plane is also 1°198. Thus the length of ED is 2 0 396 and it transforms into a line of length 2 on the w-plane.

A semicircle of radius R on the upper half of the w-plane transforms into an arc of a circle centred on the origin and of radius 2Ii provided R is sufficiently large. This may be proved in the following way. Consider a point pi on the w-plane (Fig. 18)

21

whose distance from the origin is much greater than unity, and another point a' on the real axis distant a».l from the origin.

The co-ordinates of the point corresponding to p' on the z-plane are given by

and are independent of the path of integration provided it lies wholly in the upper half of the w-plane. If we take the path C'D'a'p' we have

JW o

at [1 at I a at [I> at tt2- 1 = J Vt 2- 1 + 'tIt 2-1 + J Vt 2-1

o 1 a

The three parts' of this integral can be evaluated separately as follows:

(i) We have already shown tha.t

(ii) If t is real and greater than unity

Thus when a» 1,

It-1 < l1t 2 -1 <It

fa Ja la at at at -> > -

1 It-1 1 'tIt 2 -1 1 It

2Ia-1> fa at "'-tt---> 2(/;-1)

1 t2-1

(iii) Provided the path taken from a' to p' is such that l'tl »1 we may write

Thus the complete integral is

J1II at 4~ ~·1°198 e~ii+2,,{;

o vJ;2-1

~. 2/'; if 11.111» 1

Thus the circle of radius R on the w-plane transforms into a circle of radius ~ on the z-plane, provided R» 1.

One quarter of the capacitance between a square conductor of side 2- 396 and a cylinder of radius 21i is therefore equal to one half of the capacitance between a

22

strip of width 2 and a cylinder of radius R. But a strip of width 2 is known to be equivalent to a cylinder of radius t. Thus the capacitance between the square conductor and its surrounding cylinder is equal to twice the capacitance between concentric cylinders of radii ~ and R, i.e. 4TTE/log 2R farads/metre. If r is the radius of the cylinder which is equivalent to the square conductor, we have

2TTE 4TTE =

21ii log 2R log-;;::-

:. r = 12

Thus a square conductor of side 2° 396 has the same capacitance and inductance per unit length as a cylinder of radius 12, a square conductor of unit side is therefore equivalent to a cylinder whose radius is 0°591.

A similar method may be applied to a conductor whose section is an equi­lateral triangle, using the transformation

J IQ dt

z = ~~-t-2---1 o

A triangle whose side dimension is unity is thus found to be equivalent to a cylinder whose radius is 0 0 422.

The general transformation for an m-sided regular polygon is

o

and this leads to the result* that the radius of the cylinder which is equivalent to a polygon of unit side is

m (-:)!

= = -------

If this function is evaluated for various values of m it is found that, as m increases, the equiValent cylinder tends to have the same cross-sectional area as the polygon. For values of m equal to or greater than 5 the difference between the radii of the cylinders which have the same area and the same inductance as the polygon is less

than ~~.

*This result, derived by a similar method, is given in a paper by T. Otake4

published in 1913.

23

APPENDIX III

FORMULAE FOR CALCULATING RADIATION PATTERNS BY THE INDUCED-CURRENT METHOD

In this method the current distribution induced on a mast face by a filament or a line doublet is assumed to be the same as that which would have been induced had the face formed part of an infinite plane. The field due to the current on the face is then obtained by integrating over the width of the face and is added to the field due to the radiating element itself.

The field radi ated by the current induced on a strip AB (Fig. 19) by a current filament at P carrying a current I is 2

'B" = r A

i e jf3z si n 4> dx %

A '8 I where i = . 771 d H (2) (f3P) z JJ:p 1

I----w ---..... -li and the phase of the field is referred to the point D. This integral cannot be solved analytically. A numerical integration must be performed for each value of ~ but the actual

Fig. 19 - A current filament parallel to a conducting strip

process of integration need only be performed within a range of 90° because the function is symmetrical. If the integration is carried out graphically the same curves m~ be used if the width of the strip or the position of the filament changes, provided the spacing d remains unal teredf only the limits are changed.

j f3d cos4> The field due to the filament itself referred to the point D is le

the total field is the sum of the two fields. In the region 90° < ~< ZlOo the two fields tend to cancel, giving a small residual field in the backward direction.

This method will now be extended to the case in which the radiating element is a line doublet. If the moment of the doublet is M per unit length the surface current density is

and the field radiated by the current induced in the strip is

B4> = cos~ JB ;, "ei f3z sin 4> dx

A This integral is identical with the previous result for a filament, since iz/N = i,.II. ,The field due to the doublet itself is M cos ~e jf3d coscP" It will therefore be seen that the radiation pattern of a doublet and a strip may be derived from the corresponding pattern for a filament and a strip simply by mUltiplying by cos~.

If AD and DB (Fig. 19) are greater than three-quarters of a wavelength, and if PD is less than one-quarter of a wavelength, the radiation pattern may be calculated

The field due to the currents in the part of the plane beyond B is therefore

E%2~ef~ ~dI{ -fF; eOjV+j27T/l-sinctl[~-ji-a(v)+j8(V)]} E r.1 may be evaluated in a similar way.

25

26

APPENDIX IV

SCHELKUNOFF'S FDRMULAE FDR DIFFRACI'ION AT A WEDGE

Fig. 20 - A current filament paraJJel to a

conducting wedge

Neglecting the factor

Schelkunoff gives 3 two formulae in tte form of infinite series for the field at a point P (Fig. 20) due to a current filament F near a wedge of external angle tf;. These two formulae are identical if P is equal to Po' When P is not equal to Po one series diverges and cannot be used, but the other converges. If P is a distant point P is greater than Po and the convergent series is then the expression for E:(P, <P), which is

+ '" 77 j3TJI Q) (2) Ez(P, '1-') = -----y;- "~1 J1/(j3Po lB1/ (j3P)sin1l<Po sin1l<p

wh~re 11 = m77~, TJ is the intrinsic impedance of the medium surrounding the wedge and I is the filament current.

In the present case we are only concerned with the field at a distant point and for this the asymptotic form of the Hankel function may be used. Thus as P ~ 00

which does not involve <p, and denoting the field by E z we have

Fig. 21 - A magnetic current fi lament para1 I el

to a conducting wedge

277 Q) • TT

E. = T. ~ J (j3p ) eJ 1/"2 sin1l<p sin 1l <p • 'I' a= 1 1/ 0 0

The amplitude of the field given by this expression is such that, if the wedge were removed, the field radiated by the filament would be of magnitude ~ in all directions. The phase term indicates the variation in the phase of the field at a large constant radius from the apex of the wedge.

The corresponding formula for a line doublet and a wedge is not given by Schelkunoff but it may be derived from his expressions for the field produced by a magnetic current filament (a source of cylindrical waves with the magnetic vector parallel to its length).

Fig, 21 shows a wedge, a magnetic current filament V and a non-radiating doublet AB. The electric field at the doublet may be defined by Ep and E~; the field in the direction AB is then E" = Ep cos<P-E~ sin<p. This will vary as V moves around the wedge and the current induced in the doublet will vary accordingly.

If the principle of reciprocity is applied, this variation of Ex will also describe the variation in the current induced in a receiving loop aerial placed at the position of V if the doublet is driven. If Po is then made sufficiently large the electric intensity at the loop will be perpendicular to the line joining it to the apex of the wedge and the variation of Ex will describe the far-field radiation pattern.

By expressing Maxwell' s equation E = (11 jCUE l curl H in polar form we obtain

1 OH E _ Z

P - jcuE P orp

where H z is the magnetic field parallel to the length of the magnetic current filament. Since P~ >P', Schelkunoff's expression for H-;(P, rpl must be used; this is

rrpv ro H-;(P, rp) = :.. -2'/' {H(2) (PP l J (PP) + 2 2: H.~2) (PP) J., (PP) cosvrp COSvrp}

'1'7) 0 0 0 m= 1 v 0 v 0

where v = 11/,rr IV; as before. Now when Po ~ co

V is the magnetic displacement current in the filament.

Neglecting the common phase term e·i{~Po·~) and denoting the constants outside the bracket by K we have

Then

Also

1 08% E -----=

P - jcuE P orp

_- -. K(3 ~ J·IJ'E. 2V p. A-. A-. L.- e 2 R JIJ (,.., p.) cos v '+"0 sin V,+"

JWE _:1 ,..,p

S.ubstituting these values in the expression

we obtain the electric field in the direction AB at the doublet~ re-arranging it may be shown that

28

Neglecting certain constants and denoting the distant field obtained when the doublet radiates by ErPo ' we have

271 ro 7T E = -:r. {J (f3P) sin cP· + L·

1 eiv!.Z'[J (f3P) sin(1I-1)cP+ J + (f3P) sin (11 + l)cP] COSllcP} rPo 'f' 1 .Ill = V· 1 V 1 0

The amplitude of the field given by this expression is such that, if the wedge were removed, the maximum field radiated by the doublet would be of magnitude f. The phase term indicates the variation in the phase of the field at a large constant radius from the apex of the wedge.

EGF MM

Printed by B.B.C. Researcb Department, Kingswood Warren, Tsdwortb, Surrey