Difference-modes in the scalar feed

Difference-modes in the scalar feed

Citation for published version (APA):

Lambrechtse, C. W. (1970). Difference-modes in the scalar feed. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1970

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\t..to'

bSe

j- ,-".. ",...'....-.''',0;,." _ ME ...._ _...,•••_,_.~

lSi

:"JOtEBIBLtOTH~LK

ELEKTROT

EC Hi',; I fK

E - HQOGBOUW

EINDHOVEN UNIVERSITY OF TECHNOLOGY

EINDHOVEN THE NETHERLANDS

DEPARTMENT OF ELECTRICAL ENGINEERING

DIFFERENCE-MODES IN THE SCALAR FEED

BY

C.W.LAMBRECHTSE.

ANTENNA LABORATORY OF THE GROUP THEORETICAL ELECTRICAL ENGINEERING

AUGUST 1970 ETA-) 5-1970

-Contents.

O. Summary

I. Introduction

2. The radiation pattern of the TMOv-mode

2.1 Introductory remarks

2.2 TMOv-mode in corrugated conical waveguide

2.3 Computation of the radiation pattern of the TM

Ov-mode

2.4 Experimental radiation patterns of the TM

av

-mode

3. The radiation pattern of the HEil)-mode

3.1 The HEil)-mode in corrugated

con~cal

waveguide

3.2

Calcula~ion

of the radiation pattern of the

HE~I)-mode

3.3 Experimental radiation patterns of the

HEi~)-mo~e

Acknowledgements. References

Summary

Radiation of difference-modes in corrugated conical horn antennas

is investigated.A theoretical explanation of the frequency independence property of the scalar feed is given.

Radiation patterns of the TMOv-mode are computed. The agreement between theory and experiment is found to be very close. Furthermore is proven that the radiated power patterns of the second order hybrid-mode

and of the TMOv-mode are identical for small values of the flare angle of the cone.

The scalar feed is a conical horn antenna with grooves perpendicular

to the wall of the horn(Fig.

1.1) .

..

Fig.I.] The scalar feed.

The propagation and radiation characteristics of corrugated cylinders and conical horns have been studied both theoretically and experimentally

[1-16] .

In the previously reported work, main attention is given to the

dominant hybrid mode. The application of the scalar feed in autotrack

·~

z

p

n'

I

SA

I

-

- -

-.r

I

e

_!

I

I

Fig. Z•J • J Conical horn antenna with coordinate system.

y

, / /'"

-VI

.

Autotracking is frequently accomplished by using a TMOI-mode in a dual

mode horn[l7] . The performance of a system using the TMOI-mode 1S

excellent when the polarization of the received signal is circular.

However, with elliptically polarized signals, cross-coupling occurs between

the two axes of the tracking antenna, and when the polarization 1S linear

the system cannot detect essentially the angle error orthogonal to the plane of polarization.

Nakahashi e.a. [18] contrived a method to overcome the defect of the above mentioned system. They proposed the use of the circular TEZI-mode.

The radiation pattern of the TEZI-mode from circular horns, both in its radial and circumferential components, is a S-curve, which is the indispen-sable characteristic for the difference signal pattern of the simultanous lobing system.

The present analysis shows that the spherical TMOv-mode can exist in corrugated conical horn antennas.

For tracking of sources which radiate linear polarized waves, the spherical

hybrid

HE~~2mode

is proposed.

Z. The radiation pattern of the TMOv-mode.

Z.I Introductory remarks.

When the field E' ,HI on the spherical-capped mouth SA (Fig.Z.I.I) of the scalar feed is known, then the intensity of the radiated field in the

point P can be found from the following representation theorem[20]

I-:(~)

CUrlpl

[E.'

x

I-:(~')J

G(r,r')dS + jce",

CUrlpCUrI~ [E.'X~(~'l(I'I')dS

SA SA with G(E.~') _4T[ - jkIr-r' I e -I I I iE--E- , ! and k=w(£ l.l )2 o 0 (2. 1. I )

In these expressions we have assumed that the outside of the horn antenna

7.

Furthermore we assume that the field at the mouth of the horn is that which would exist at that cross section if the horn extended to infinity. Then solutions of the wave equation in conical waveguide can be substituted

in (2. 1.I) •

2.2 TMOvJmode in corrugated conical waveguide.

The components of the TM -mode can be found by substituting the potential[211

Ov

A (r' e'

~')=A

p (cose') n(2)(kr')

r " v v

in the following expressions

(2.2.1) E r E,. E I

(1\

k2_{) Ar} H 0 + JWC:_o r Jr )2A I ,IA I r r jws r 3r3,J H rsinO ,it a (2.2.2) )2 A I ~,A r H

- - - -

r ~LL" a r s~nl, ,ir.i~

".

"

r 1\

-.--JWE Or

The results are

E

,= ---

v(v+l) p (cose') H(2)(kr') r ,2 v v JWE O r d P (cose') d

H~2)(kr')

v _--,-v-:-...-__ de' dr' (2.2.3)

It should be noted that primed coordinates are used for the description

of the electromagnetic field in the horn. For the electromagnetic field

in the grooves we use unprimed coordinates. Finally, the coordinate of a point outside the horn antenna will be unprimed again.

(2.2.4) The characteristic equation for vis derived from the condition that the longitudinal reactances of the waves in the cone and of the waves in the

slots are matched on the surface e'=e

o

From the expressions (2.2.3) the longitudinal reactance of the waves inside the cone is found,

X_r,C= jE_r,/H~,= v(v+l) P (cose')

'f' _v_ _~_

WE r'sine'P'(cose')

o v

, p'(cose') denotes d P (cosel )

'\I _...;.v _

d(cose')

The derivation of the expression for the longitudinal reactance of the waves inside the slots is simplified when the slots perpendicular on the

wall of the cone are approximated by spherical slots (Fig.2.2.1).

I

':;;-V

I

f

Fig.2.2.1 Spherical slot and spherical coordinate system.

In [19] is proven that in the spherical slot only TM-modes can exist, provided the width of the slot is smaller than half a wavelength.

We now assume that in the slots the TMOn-mode with the real value'of the degree n is excited only.

9.

The TMOn-mode in the groove can be derived from the potential [20

A

=kr(a j (kr)+b y (kr))(c P (cos6)+d Q (cos6))

r n n n n n n n n (2.2.5)

by means of the expressions (2.2.2).

The characteristic equation for n is derived from the condition E₆=O

on the boundaries I and II (Fig.2.2.1),

o

(2.2.6)

(2.2.7) Equation (2.2.6) has been solved in [19] where also was found, that no<kr. Making use of the boundary condition Er=O for 6=6

2 the following expression

for the longitudinal reactance of the waves inside the grooves is derived

X =jE /H~= n(n+1) Q (cos6

2)P (cos6) - P (cos62)Q (cos6)

r,s r 't' n n n n

WE rsin6 Q (cos6

2)P'(cos6) - P (cos62)Q'(cos6)

o n n n n

fx]

has been computed for two slots at a distance r

1 to the apex of

l

r, s 6=6

o

the cone of 3.74 em and 13.64 em respectively

The values of the other parameters are:6C~ 60 deg., b=2.6 rom and

o

r

I6?d=5.35 mm. The results are plotted in Fig.2.2.2.

The computations are restricted to values of the propagation coefficient for which the longitudinal surface reactance is capacitive and no surface waves are supported

The characteristic equation for l) ,

[x

r , c- Xr , s] 6 .. 6=6 =0

o

(2.2.8)

has been solved for the two cases given above. The results are plotted in Fig. 2 . 2• 3.

For the special case X=~ , which occurs if do<lA, has been computed as

r

a fuction of the flare ang1e6 .The results are given in Fig.(2.2.4).

o

The conclusion is that a large change in the value of the longitudinal surface reactance, caused by a large change in frequency, gives rise to a small change of the degr~e, '\1,..whidl implies a small change of the power distribution over

the aperture and explains the frequency-independence property of the corrugated conical horn antenna.

-x

Its. 100 50 '-250

\

\

\

\

\

\

\

\

\ \ \ \.

"

350

-

_-

_-

_-

_-

-450

.-:-~L

550 k, rad/m

Fig.2.2.2. Longitudinal surface reactance against propagation coefficient

___ r

1= 13.64 cm , ---- 1"1=4.0 cm

6

0=60 deg., rI61~d=5.35mID, b=2.6 mID.

\) 3.0 _

....

-

_-

_-

_-

_-

--

_-

-550 450 350

~~~

2.5 L - -

1_,

...

_,L

-250

<Fig.2.2.3. Solutions of the characteristic equation against the propagation coefficient

X

=00 ~ r =13.64 cm

X =X (k),

r - 1 r r

25 20 15 10 5

--JO

20 30

40

50 60 70 80 9.0

Fig.2.2.4. Solutions of the characteristic

- - - TM Ov-mode HE (1) -mode 2v equation; for X =00 r

2.3 Computation of the radiation pattern of the TMOv-mode.

The far field approximation of (2.1.1) gives

- ~

-jkr

e

r

f:5.(

4~.

A

{E~,

cos 0' - Zo"O' cos 0 } sin (¢ -

~')

+

(2.3.1)

+ {EO' + Zo".' cos 0' cos o}cos

(~-

.') + Zo".' sin 0 sin

0]

x

exp

[+

jkr' {cos 0 cos 0'

+

SLn 0 sin 0' cos (. - • ')}] (r')Z sinO'dO' d.'

=e-jkr

ikr[_{

E¢ (r,G,¢) r

~.

EO'

A

cos G

(2.3.2)

+ {- Zo"", + E., cos

0'

cos o} cos (. - .') + E., sinO sin

0]

x

exp

[+

jkr'{ cos () cos 0'

+

sinO sin 0' cos (. - .')}] (r')Z sinO'dO' d.'

Substituting (2.2.31 in (2.3.11 and using the relation

00

jkr' sinG sin G'

cos(~-¢')

= J (kr'sinGsin0')

+~

2jnJn (kr'sinG sinG')

e 0 ~ We obtain _-J'k_r E = -Z H(2)(kr,),ke _{r'Fo v(8,80} ,kr',v) 8 0 v J 2r

O

80 E

=

w:ere F (8,8

kr',v)=j

sin8' d

P

(cos8') [(I+COS8COS8')

oV 0 _~v.-

_

d8'

xjJZ(kr'sinasina') + sinaSina'Jo(kr'SinaSina')] exp(jkr'cosacosa')da'

(2.3.3)

(2.3.4) (2.3.5)

13.

maximal. From equation (2.3.4)

S ~s the value of S

m

follows the identity

for which

I

E S

I

is Es(S,So,kr',v) =1 Es(S ,s ,kr',v) m 0 F ( S, S ,kr',v) o F ( S

,s ,

kr' ,v) m 0

Then the relative radiated power is

(2.3.7) , dB F ( S , S ,kr' ,v) o 10 P (S,S ,kr',v)=20 log QV 0 F ( S ,S ,kr',v ) m 0

The function P has been computed for several values ofS and kr'.

QV 0

For v the values

ame

taken computed for the case Xr=oo .

From these computations the beamwidth has been derived as a function of kr'

(2.3.8) for S =15,30,45,60 and 75 deg •. The results are plotted in Fig.2.3.1 to

o

Fig.2.3.5 incl •.

In Fig.2.3.6 the relative power of the aperture field, 10

P~v(S',So,v)=20 log ES'(S',So'v)

ES'(S',S ,v)

m 0

, has been plotted for the case S = 60 deg., X =00.

o r

A comparison with the radiation patterns of two horns with the same flare

angle but with different lengths shows that for "long" horns the aperture field is a good approximation to the radiated field, providedS<S .

o

In Fig.2.3.7 and Fig.2.3.8 the computed radiation patterns of horns with

a

length r' of 13.64 and 4.0 cm respectively are shown for the frequencys

14.0, 19.1 and 23.9 GHz. At 14.0 GHz the depth of the slots is equal to a quarter of a wavelength. From these patterns can be concluded that the deviations in the beamwidth of the "long" horn due to the change of frequency are very small, but for the "short" horn the frequency-dependence may be of practical interest.

I

-e

70

:s

I

I:Q 60

so

40 30 20

~

10 20 30 o 40 I:Q ~ _ -10

so

o

3dB 60

o

3dB 70

o

lOdB 80

~20dB

15dB 0 10dB kr'

Fig.2.3.1. Beamwidth against kr' for

e

=15 deg.; dots indicate experimental results

o

.p-8 25dB 8 20dB 8 15dB 8 10dB 8 3dB

~

80dB 8 3dB 50 kr' 40 30 20

~

--

---~---.--_-­

8 10 JO 40 30 20

..

_{15 •} ..c: ~ ₇₀ 'U o,-l

~

0 Q) ₆₀ p:j 50

.

~ 80 'U

L

Fig.2.3.2. Beamwidth against kr' for

e

=30 deg.; dots indicate experimental

o

results obtained with the antennas 3,4 and 5 at a frequency of 14.5 GHzj diameter waveguide is 18 mm.

..

..c: ~ 70 OJ ~ 60 50 40 30 o 8 25dB 8 20dB 8 15dB 8 10dB 20 10 10 20 _{30 40 50} kr' Fig.2.3.3. Beamwidth against kr' for

e

~45 deg.j dots indicate experimental

o

results obtained with the antennas 6 7 8 9

, " and 10 at a frequency of 14.5 GHz.jdiameter waU"'011;..I_ :_

90

bil

80 G.l "t:l

..

,d ol-J 70 "t:l El:25dB

''''

!

_G.l El:20dB 60 !Xl 01jdB 0 8_10dB 50 8 3dB 40 0 30 20 10

~---o---e

-10 _{20 30} 40 _{50 kr'}

Fig.2.3.4. Beamwidth against kr' for 8 = 60 deg.; dots indicate

o

experimental results obtained with the antennas 12 and 13 at a frequency of 14.5 GHz.; diameter waveguide is 18 Mm.

17. 2-0 80

.

0 20dB bO Ql "0 ₀ 15dB

.d'

_{.u 70} 0 10dB "0 0 'rol

~

~

Ql ₆₀ !Xl

-

0 0 3dB 50 0 40 ₀ 0dB 30 20 o _o 10 10 20 30 40 50 kr' - - -

---Fig.2.3.5. Beamwidth against kr' for

e

=75 deg.; o

dots indicate experimental results obtained with the antennas 14 and IS at a frequency of

0

,

/ .

/

/

/

I

.

_.\ _\

I

_\

I

\ -10

\

I

\

!Xl

I

\

't:l

..

_I

1-1

\

~

0

I

\

(:l, Ql

I

_\

.~ ~

I

co

\

.-I -20 Ql

\

1-1

I

\ -30 0, deg.

Fig.2.3.6. Computed power patterns for_{. ..}

e

₀=60 deg.; --- aperture field

--- far field, kr'=12.5 •••••• far field, kr'=40.0 •

19. -10 I:Q '0

.

,... CIl :l 0 Q. CIl ₁ :>

i

•.-1 ol-l C\l -20

i

..-I CIl ,...

I

1

-30 36 72

e,

deg.

Fig.2.3.7. Computed radiation patterns for

8~=

60 deg., r'=13.64

cm. frequency 14.0 GHz

_,

_d/>"=0.25

---

_{19. 1 GHz}

"

=0.34

•

"

23.9 GHz

_"

_=0.43

...

Q -]0 -30

,

/

.

I : I :' I .... I .:

1/

,:

:1 .: I : I :' I : I . I I I I ,

,

.

,

I I I I I J I

,

~ '\ ,\-,

\

\.

\.

\..

\:. \.

""-

_\.

'\,

\'.

,.

"'-

'(.

\.

\.

\'. \'. \".

\'.

\ "

\

".

\ '

..

\ \ \ " ' - - _ - - - - ' -_ _- ' -_ _J . . - _ - . . L_ _---'----_ _. L -_ _- - - ' -_ _- - - L - - l 36 72

e,

deg. ]08

Fig.2.3.8. Computed radiation patterns for

e

= 60 deg, r'=4.0 cm~

o _______ frequency 14.0 GHz' d/A=0.25

"

"

19.1 GHz 23.9 GHz. " "0.34 " -0.43

21.

2.4 Experimental radiation patterns of the TMOv-mode.

The experimental verification of the theoretical curves of Fig.2.3.1 to Fig.2.3.5 incl. has been done by measuring the radiation patterns of the antennas listed, with their relevant dimensions, in table I.

All the slots are of the same depth and this was a quarter of a wave-length at the frequency of 14.0 GHz.It should be noted that the antenna

with

e

=15 deg. is an exception to this rule.

°

Table I antenna.

e

r1 _{d b} t °

_[en!]

@~

@~

_@mJ

I 1 15° 17.60 0.9 0.20 0.20 2 30° 3.50 0.535 0.26 0.04 3 300 4.20 0.535 0.26 0.04 4 30° 6.60 0.535 0.26 0.04 5 30° 9.00 0.535 0.26 0.04 6 450 2.78 0.535 0.26 0.04

.

_.

_.

_..

45° 7 3.71 0.535 0.26 0.04 8 45° 6.17 0.535 0.26 0.04 9 45° 8.66 0.535 0.26 0.04 10 45~ 13.57 0.535 0.26 0.04 1I 600 2.80 0.535 . 0.26 0.04

.

600 12 4.00 0.535 0.26 0.04 13 600 13.64 0.535 0.26 0.04

.

14 750 5.08 0.535 0.26 0.04 15 75° 10.23 0.535 0.26

_O'Oi

16 90° 10.64 0.535 0.26 0.0 _ . _ _ . _

-The radiation patterns of these antennas has been measured at the frequency of 14.5 GHz.The results are presented in the Fig.2.4.1 to

Fig.2.4.14 incl. The calculated patterns are also given. For v the value

was taken computed for the case X_r

=00.

The conclusion is that the experimental results are in good agreement with the theoretical predictions, especially for 8 <75 deg ••

0

23. I < \A-Fig. 2.4. 1 I

-2l

Antenna 1

I

Frequency 10.0 GHz

-4

~ +++++ computed

I

experimental

-6

-8

-10

-12

-14

+

-16

L

-18l

J

-20

l

₁₊

-22

-24

D m + Q) + CD ""D + """D

-26

+ lJ) ₊ c ₊ .... ₊ + II

-'-

-28

~ + Q) _{ill ::J +} 3 v- C ₊ 0 _{v- 0} 0.... _~ Q)

-30

D Q) v- ...c. + > _{~ +} .... ..,J + + 0

_-32

II + C + Q) + l.

....

+₊ +

-34

Q) ₊₊ ""D 0 _+++. 0 co E D I ill

-36

c (Y") 0 + I ~ \I I - L

-38

+ ~

-40

0

12

24

36

48

60

72

84

96

lhelo (degD)

o m (l) \ ) o (T") + II

96

84

+ + + + + + + + + + + + + + ++

72

+ + + + + + + + + + + + Fig.2.4.2 Antenna 3 Frequency 14.5 GHz ++++ computed ---- experimental

60

48

36

Q) \) 0 o E D I (T") C o + I ~ II ~ L + ~ -I' + + + + + + + + + + v- :J Lf) C (T")

a

(X) -.-> D Q) <.D ...c + -..J II C

-2~

-4

L

I

-6~

-8l

-1O~

-12

-14

-16

+

-18

-20

-22

-24

(l)

-26

-0 c

...

L

-28

(1) 3 0 Q..

-30

(1) >

...

..-.) 0

_-32

Q) L

-34

-36

-38

-40

0

12

24

'lhe'la (deg0)

25.

96

84

72

Fig.2.4.3 Antenna 4 Frequency 14.5 GHz ++++ computed ---- experimental

60

48

0 + 0 + D 0 +₊ C\J ₊ + + + II + L ..::s:. + + + + + + + + + + +

36

c m Q) -0 o (Y) + II Q) -0 o E I C o I :L I -+ v- :J Ln C (Y)

a

co ...> D Q) ill L + ...> II C

0

-2

-4

~

I

-6

-8

-10

-12

-14

-16

-18

-20

-22

-24

OJ

-26

-0 C .-< L

-28

Q) 3 0 Q... Q)

-30

> ...,J a

_-32

Q) L

-34

-36

-38

-40

0

12

24

lhela

(deg

D)

Fig.2.4.4

-2

_{Antenna 5} Frequency 14.5 GHz

-4

++++++ _computed experimental

-6

-8

-lOl

-12

-14

-16

-18

-20

-22

-24

a 01 Q) CO iJ \J

-26

CJ c (Y") + -< + \I + L

-28

.-.. + Q) _{"<T" :J} 3 U1 C ₊ 0 _m 0 Q... _{CO ....,J} + Q)

-30

c Q) >

c..o

...c + -< + ~ .-.J + 0

_-32

II

c

+ Q) L

_'"

-34

Q) ₊ iJ 0 0

c..o

+ E c + I

["'--36

c

C\J ₊ 0 + ₊₊ I :c II ++ ~ L ++

-38

+ ~ +

-40

0

12

24

36

48

60

72

84

96

lhela

(dsgCl)

0

27. F~g.2.4.5

-2

_{Antenna 6} Frequency 14.5 GHz

-4

_{++++ computed} ---- experimental

-6

-8

++ + + +

-10

++ + +

-12

+ + +

-14-

++ + + + +

-16

++ + + + ++

-18

++ ++ ++

-20

-22

-24

_c m Q.) a::l -0 \J

-26

lJ) c '<:t" ... ₊

-28

11 L ~ OJ (Y) :J 3 lJ) C 0 0 0 0.. _'<:t" -.J OJ

-30

c Q.) > '<:t" ..c. ... + .-> ..,J

a -32

II c OJ L 0..

-34

Q.) ""D 0 0 lJ) E c I CD

-36

c + 0 I L: II f - L

-38

+ ...:>t.

-40

0

12

24

36

48

60

72

84

96

lhela

(de9

Q)

96

84

72

Fig.2.4.6. Antenna 7 Frequency 14.5 GHz +++++ computed --- experimental

60

48

36

+ + + + + + + + + + + + + ++ Q) "D 0 o (T) E II I C o + I I : II I-- L + ~ c m Q) "D LD V' + II

...

(Y) :J U1 C o 0 ~ -.J c Q) ~ L + -.J II C

-2

-4

-6

-8

-10

-12

~

-14

-16

+

-18

-20

-22

-24

CD

-26

lJ c L

-28

ill 3 0 Q...

-30

ill >

....

.-oJ 0

_·-32

ill L

-34

-36

-38

-40

0

12

24

lhela

(de 9" )

0

29. Fig.2.4.7

-2

Antenna 8 Frequ~ncy 14.5 GHz

-4

+++++ computed experimental

-6

-8

-10

-12

-14

-16

-18

-20

-22

-24

D OJ Q) CD "'D ₊ -0

-26

Ul + C '<t"" ₊ ~ + + II L.

-28

.-0 + (I) (Y) :J + 3 Ul C + 0 0 0 ₊ 0- '<t"" ~

-30

+ (I) D Q) + > '<t"" L + .... + .-..) + -J ₊ 0

_-32

II c + (I) L. a.. t'

-34

Q) + "'D 0 0 C"- + E D + I CD

-36

₀c ₊ I L: II f - L.

-38

+ ..lIC

-40

0

12

24

36

48

60

72

84

96

lhelo

(de 9")

-40

"----'-o

12

lhelo

" ,

96

"" ... --l.._. _""_.""""_L__._._

72

64

Fig.2.4.8 Antenna 9 Frequency 14.5 GHz +++++ computed --- experimental

60

48

0 (Y) 0 ill C\J + + + II ++ L + ..Y (Y) :J LJl C o 0 -.r ~ o Q) -.r L + ~ II C o m Q) -0 LJl -.r + II Q) -0 o E I C o I L f -+

36

24

(degc)

-38

-36

-34

-2

-4

-6

-8

-10

+

-12

I

I

-14

~

-16

I I I

-

₁₈

l

+ ,

-20

l

-22

+

-24

OJ

-26

-0 c ,... L.

-28

iJ) 3 0 Q... iJ)

-30

> ...,...J o

-32

(l) L.

0, 31.

-2l

Fig.2.4.9 Antenna 10 I I Frequency 14.5 GHz

-4l

+++++ computedexperimental

-6

I I

-Sl

-10

-12

~

-14

~

+

-16

-18

-20

+

-22

-24

D

en

Q) o:l ~ ""0

-26

Ln c V" +

-28

II L ..-. Q) (Y') _~ 3 Ln C + 0

_a

_{0 +} 0- _{V" -.J} Q)

-30

D Q) > v-₊ .L_-.J ...,.J 0

_-32

II c Q) L 0-Q) +

-34

_{~ 0 +}

a

a

E- o I

-36

_a

c v-₊ I L II f-- L

-38

+ ~

-40

+

0

12

24

36

48

60

72

84

96

lhela

(de9

0)

-2l

Fig.2.4.10 Antenna 12

J

Frequency 14.5 GHz +++++ computed I

-6

~

experimental + + +

-8

+ + + +

-10

+₊

-12

+

-14

I

-16

+

-18

.

"

-20

-22

-24

+ D + m Q) CD "'0 -0

-26

0 c <.0 +

-28

"

L .-. Q) r--- _:J 3 lJ) _C 0 _{m 0} Q... _~

-30

_"

Q) Q) (T) _L > ,..., + ~ -....> 0

_-32

\I c Q) L 0..

-34

Q) "'0 0 0 E 0 I (\J

-36

0c ₊ I L: II ~ L

-38

+ ..x

-40

_{, 0}

12

24

36

48

60

72

84

96

lhela

(deg

c)

0

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

-24

CD

-2§

""D .' c i ~ ;l~~ '-

-28

(l) 3 a CL

-30

(l) > ~ ...,) 0

_-32

(l)

'--34

-36

-38

-40

0

12

24

lhela

(de 9

Q) Fig. 2.4.11 Antenna 13 Frequency 14.5 GHz ++++*computed --- experimental + D m Q) ""D + 0 <.D + II

...

₊ :J If) C + m 0 ...,) D Q) + (T) L + ...,) ₊ II + C + 0.. Q) ""D 0 ₊ 0 0 E 1:1 + I 0 ₊ C

«:t-o

+ + I ₊ I:: II ₊

...

'-+ ..¥. + + + + +

36

48

60

72

84

96

96

84

Fig .. 2.4.12 Antenna 14

72

+++++ computed Frequency 14.5 GH --- experimenta

60

48

'+++ ++ ++ ++++ + + + + + + + + + + + + + + + + -I-+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

36

... Q) ""D LJ") o V-E D I LJ") C

o

+ I ~ II ~ L + ..:s. D m Q) ""D Ul ("--. + II :J Ul C ("--. 0 ~ -.) a Q) (\J L + -.) II C 0 +'+ +

-2

++ + + + + +

-4

+ I ++

-6

~

+ + + I +

I

+

-8

~

+

-10

-12

-14

-16

-18

-20

-22

-24

(D

-26

"D c L

-28

(]) 3 0 0-(])

-30

> ~ 0

_-32

(]) L

-34

-36

-38

-40

0

12

24

lhela

(de9

0)

.-35. 96 Fig.2.4.13 .Antenna 15 Frequency 14.5 GHz ++++++ computed --- experimental __ L ...J.__ ... l ..L. __

48

60

72

84

+ + D + m ₊ Q) \J + Ul + r---+ II ... :J c 0 -.J Q) .L ....> (l) \J 0 o 0 E. D I C (Y) o + I ~ II f-- L + ~ Q C\J + II C _ _ _---'- ---'- .. _._l _ 36

0

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

-24

(lJ

-26

-0 c L

-28

(I) 3 0 0-(I)

-30

> ~ 0

_-32

(I) L

-34

-36

-38[

-40

0 12

24

lhela

(degu)

++ ++ Fig.2.4.14 +₊

-2

+ Antenna 16 + + + .requency 14.5 GHz

-4

+ + _{+++++ computed} + _experimental

-6

+ +

-8

+

-10

+

-12

+ +

-14

-16

-18

+

-20

+

·-22

-24

D m + Q) OJ ""D ""D

-26

₊

a

c

m '"' +

-28

II + L ... Q)

a

_:J 3

a

c

0

_a

0 + Q..

_a

--..) Q)

-30

D Q) > (\J L ... + --..) -..) 0

_-32

II ... C Q) L

....

-34

""DQ)

a

0

a

E D I

-36

_a

c

v-₊ I ~ II ~ L.

-38

+ ~

-40

0

12

24

36

48

60

72

84

96

108

lhelo

(de 9

01

(3.1.2) 37.

3. The radiation pattern of the

HE~~)-mode.

3.1 The

HE~~)-mode

in corrugated conical waveguide.

The

HE~~)-mode

can be understood as the sum of a TE-mode and a TM-mode.

The components of this hybrid mode can be found by substituting the potentials [21]

Ar=AIP~(cose')cos2~'H~2)(kr')

(3.1.1)

and

F =A

p2(cose')sin2~'H(2)(kr')

r

2

v

v

in expressions (2.2.2) and in the following expressions

E = 0 _H

...

_-.--

I _{( - - +}a2 _k2 _{) F} r _{r JW~o} ar2 r aF .,2F E G -I r H ". _I- l O r {3.1.3) =

r sinO a¢ G jw~o r

arac

I aFr _I a2F

E~

.

- - -

_{r ()O H~} = r

JW~o r sinG ard~

The results are

v(v*l) ~ ~(2) ~-~~-~ p~ (cos 0') cosl¢' H (kr') r,2 v v (3.1.4) _A_IZ_o_H_v_(-:-2)_(_k_r_'_) [ dP

1)

co

so' )

E , •

_o

_{r' dO'} sinG' ...,

I

p~

(eOSe')J (3.1.5) cosl.~, dr'

d~

(2)(kr') v A Z

Ii

(2)(kr') 1 0 v E~, • r'

[

1.

pl.-(eose') 1

~

sin G'

v

jkH

v

(2)(kr') A2 dP}(C080 ')] +

""i:Z

dl) , I 0 sinll/> ' (3.1.6)

I

p'l{

co sO') x sinO' v

d~

(Z)(kr,} v ~, dr' 0'Xf (3.1.9) dr'

J

(3.1.8) G') si~' (3.1.7)

d~

(Z)(kr') v

t

p1.

(cos sinG' v J kH (Z)(kr') v dG' dP'l(cosG' ) v dP~(cosO') 2,A_Z + -dO ' A Z_I 0 Z A \1(\1+1) p:l.. (cos G')

si~'H

(Z)(kr') o Z ,Z \I V \I r

A

_10\1Z H (Z)(kr')

[A

_Z r'

AZ

1 0 A Z_{I 0}

H

_v(Z)(kr')

t

= - - - - : - - - -

_r' Z H , == o 8 Z H , o r

and \I, and can be found after applying the boundary conditions

E~,=O and ZoH~,=O for 8'=8

0 [19] .From these conditions we find that

Al

In the expressions 3.1.4 to 3.1.9 incl. the unknown quantities are A

Z

(3.1.10)

AZ + 1

--

AIZ_o

So two modes with the same ~'-dependence, but different8'-dependence

can exist in the corrugated conical horn. The mode f~r which A

Z=ZoA1 is called the

HE~~)-mode,

while the other is the

HE~\I)-mode.

Substitution of (3.1.10) in the equation Z H ,=0 yields the characteristic

o ~

equations

(3.1.11)

and is a complex function

= 0

H

(Z)(kr') v dr' I jk

P~

(cosG')

~v

(kr'j E1==O o dH (Z)(kr') \I

k

sinG' The function ~v (k~ ~

[

dP~COSG')

dG'

±

Using the asymptotic expansion of

H

(Z)(kr') we see that

v lim ~v(kr')== -1

kr'--For this case we have calculated the mode patterns which are presented in the Fig.3.1.1 and Fig.3.I.Z.The calculations are given in the appendix. For the case that the hybrid mode is excited by the circular TEZI-mode the excitation of the HE

(Il

mode is to expect as can be concluded from

39.

---01 ______

--..._----~

---\

\_, \ \

\

, I

-- --

-====---=--

---o -07 -08

Fig.3. I • I • _{HE(I)-mode pattern for 6}

0

= 60 deg. 2v

(3.1.12)

41.

a comparison of the hybrid mode patterns with the TE

21-mode pattern.

The characteristic equation for the

HE~~)-mode

has been solved for

the lowest value v. The results are plotted in Fig.2.2.4.

Substitution of A 1Zo=A2 in (3.1.5) to (3.1.9) gives A Z

11

(2)(kr') 1 0 v £(1) (0') cosl¢' r'

2\1

..

E iP, Z H , o ¢ A Z

H

(2)(kr' ) 1 0 v £(1) (8') sin'!';" r"

2

v -vi' A Z

II

(2)(kr') 1 0 v £(1) (0') sin1,~'

r'

lv

.(.!I' (3.1.13) (3.1.14) (3.1.15) with £ (1)(8')

l

v dP'l(cos0' )

\'

+)...

p1.., (cosei) dO' sinn' v (3.1.16)

3.2 Calculation of the radiation pattern of the

HE~~)-mode.

Substituting (3.1.12) to (3.1.15) incl. 1n (2.3.1) and (2.3.2) and using the relation (2.3.3) we obtain

(3.2.1) and -jkr E = _ _k__e_ _ A1Z ¢ 4 r 0

H

(2)(kr') v (r,)2 sin,H F(O,Oo,kr') r ' )"y (3.2.2) with

~\}'.Oo.kr')

i

Oo

[{coso

+.

COS0'}{Jj(kr' sinO sm 0') + J.;(kr' sinO sinO')}+

+

{I

+ cos 0 cos o'}{Jt(kr' sin 0

+ 2j sinO sinO'

~(kr'SinOSinO')]

sin 0')

-~(kr'

sin

°

S1n 0')} +

\~I)(O')

exp[jkr'

cosocoso~

sinO'dO'

(3.2.3)

From the equation (3.2.1) and (3.2.2) we derive that

1\1

2

+1\1

2 is independent of ~, and that

E8 (8,8₀, kr ' ) , E 8(8 ,8 ,kr')m 0 E (8,8 ,kr') 1F2 (8,8 ,kr') =

p

0 v 0 E (8 ,8 ,kr') F 2 (8 ,8 ,kr') ~ m 0 v m 0 (3.2.4)

has been calculated

for the same for 8 =40 deg., kr'=20. The

o

Also in table II the computed pattern of the TMO~ode

values of the parameters 8 and kr' is given.

o

where 8

m ia the value of 8 for which IF2vl is maximal. The function p(11=20 10 log

F2~(8,80,kr')

2v

F_2v(8l!J.8₀,kr')

(3.2.5)

43.

Table II " .~ , , 6 i pO) _PO\l THE-A 2\1 ,dB dB

--+--:' _-12·701

-12.696

+.:.; _-6.915'6

-6.951

+6 -3'.891 -3.8A7 +1 .• Z •024

-2.021

.:1. .

-.885 -.883 • J.'~ -.ZIi.3

-.252

"':1./.~

-

') -.000 ·1'~I -.034

-.O3?

·1:~,

_-·280

-.281

·2~ -.6]3

-.672

."')

<. '

-1.158

-1.155

·2-

-1.696

-1.691

+2t; -2.271

"2.264

·22

-2..886

-2.877

... 3":

_-3.560

-3.550

• 3~l -4.316

-4.306

.

_I .~.' -5". t7'1

-5.164

·3t -6.1.44

-6.136

.3:l. -7.?::>A

-7.220

..

,

:\ _-8.4':':1

-S.3Q9

i

-9.647

.-

...

~;) _{-9'. 6::'~~} .4!;

_·to.9:H

-10.923

... 4t:.

.

-~?t92

-12.180

... 4(~! .•13.~?:'i

-13.375

·5'

_i -,'4.~~:,~

-14.484

.r::; ., _-1.'?5:n

_-15.504

.', .5,' -1f,.4~!

-16.455

·~h -t7.4;;~

-17.370

.5f, - 1. ).( •:.~ J ~~

-1S.2AO

·0,";

-:19.237

-19.2j.2

A comparison of the two patterns shows a very close agreement. This can

be explained by the agreement ex· .isting between the relative power distributions over the aperture of both modes, provided the flare angle is "small".

This can be shown easily by developing

f~ I~

in terms of

~

(cos e) Ilrid making

use of the recurrence relations

[2J]

d p~(cose) ~

sine \I = \lcose P (cose) - (\I+~)P\I~~cose)

de \I I

an<;l

p~+1

(cose)= (cose -I)-!

{(\I-~)cose p~(cose)

-

(\I+~)P~

I(cose)}

Then follows 2 + -sine' 2 P (cose')_v

=

p (cose') v sine' v(2+vcose')(v-l)cosfe

{----co-s2....e---,.-1- - - - v(2+vcose')(v+I)(v-l)

+v(v+2) (v-2)(v-l)cose' +v2_{(v+2)(v-2)cose'-v}2_{(v+2)(2v-3)cose'}}

(cos2e-I)(v-I)

+Pv-l (cose'){ -v(v+2)(2v-3)(2v-l)cos2e'+(2+vcose')(v+l) (v-I)(2v-l)co set

sine' +(v+2) (v-2)(v-l)(2v-l)cos2_{e'+v(v+2)(v-2)(2v-l)cos}2_e' (cos2e-I)(v-l) +v(v+2)(v-I)-v(2+vcose')(v-l)cose-(2+vcose')(v+l)(v-l) coset -(v+2) (v-2) (v-l)cos2_e' cos2_e'-1 } (3.2.7)

If v»J and cose'~l then (3.2.7) can be further simplified to

v3 {P (cose') - P "I (cose')} =v2_{d P (cose')}

sine' v v- _{--'-d:'""e""'l'''''--}v (3.2.8) (3.2.9) 2

e'=e'

_m de' d Pv(cos8') de' d P (cose') v

The difference between the values of the degree of the TM

O -mode

and of the

HEi~)-mode

has been found to be very small

espe~iallY

for small values of the flare angle (See Fig.2.2.4)

The relative power distribution of the

HEi~)-mode

over the aperture

is giVer

~t)

2

P(I)

,=

f 2v (e', eo' kr' ) 2v

f ( 1)

(e' e

kr ' )

2v m' 0 '

and is equivalent to the relative power distribution of the TMOv-mode, calculated in section 2.3.

The conclusion is that the computed beamwidth curves for the TM

O -mode

are a good approximation for the beamwidth curves of the

HEi~)-m~de

45.

3;3 Experimental radiation patterns of the

HE~I)-mode.

c

In Fig.(3.3.1) the measured radiation pattern of the

HE~~)-mode

of

antenna 10 is given. The cut-off frequency for the TE

21-mode 1n the

circular waveguide is 16.203 GHz. The measurement is done at a frequency of 16.5 GHz.

A comparison with the TMOv-mode pattern given in Fig.(2.4.9) shows that both patterns are identical as was predicted by theory.

47.

Acknowledgements

The author is particular indebted to Drs.M.E.J.Jeuken. who encouraged and provided many of the ideas for the development of the hybrid-mode technology presented in this report.

The solutions of the frequency dependent characteristic equation (2.2.8) are given by J.F.Mayer.

I.Ongers wrote the computer program for the drawings of the mode-patterns. The author wish to thank Prof.Dr.Ir.A.A.Th.M.van Trier for giving

References

Kay, A.F. :"The scalar feed",TRG report, contract AF 19(604)-8057, March 1964.

2 Minnett, H.C. and Thomas, B.MACA. :"A method of synthesizing radiation

patterns with axial symmetry",IEEE Trans.,1966,AP-14.

3 Jeuken, M.E.J. and Kikkert, J.S.,"A broadband aperture antenna with a

narrow beam", Alta Frequenza,38,1969,Numero Speciale pp.270-276.

4 Clarricoats, P. J •B. : "Analysis of spherical hybrid modes in a corrugated

conical horn",Electron.Lett.,1969,5,pp.189-190.

5 Clarricoats, P.J.B.and Saha, P.K.:"Theoretical analysis of cylindrical

hybrid modes in a corrugated horn", ibid.,1969,5,pp 187-189.

6 Clarricoats, P.J.B. and Saha,P.K.:"Radiation from wide-flare-angle

scalar horns", ibid.,1969,5,pp 376-378.

7 Jeuken, M.E.J.:"Experimental radiation pattern of the corrugated conical

horn antenna with small flare angle", ibid.,1969,5,pp 484-485.

8 Jeuken, M.E.J. and Lambrechtse, C.W. : "Small corrugated conical-horn

antenna with wide flare angle",ibid., 1969,5,pp 489-490.

9 Clarricoats, P.J.B. : "Similarities in the electromagnetic behaviour of

optical waveguides and corrugated feeds",ibid.,1970,6,pp 178-180.

I0 Booker ,D. D., and McInnes, P.A. : "Computer-predicted performance of

corrugated conical feeds using experimental primary-radiation patterns",ibid.,1970,6,pp 18-20.

49.

II Thomas, B.MACA.:"Bandwith properties of corrugated conical horns", ibid.,1969,5,pp 561-563.

12 Narasimhan, M.S., and Rao, B.V. : "Hybrid modes in corrugated conical hrns", ibid.,1970,6,pp 32-34.

13 Baldwin, R., and McInnes, P.A. : "Surface-Wave radiation from a corrugated horn", ibid.,1970,6,pp 259-260.

14 Clarricoats, P.J.B., and Saha, P.K.:"Attenuation ~n corrugated circular waveguide", ibid., 1970,6,pp 370-372.

15 Thomas, B.MACA. : "Prime-focus one- and two-hybrid mode feeds", ibid., 1970,6,pp 460-461.

16 Narasimhan, M.S., and Rao, B.V. : "Diffraction by wide-flare-angle corrugated conical horns", ibid.,1970,6,pp 469-471.

17 Cook, J.S., and Lowell, R.:"The autotrack system", BSTJ,42,No.4,pt.2, pp 1283-1307, July 1963.

18 Nakahashi, N., Kakinuma, Y., and Shirai, T.: "A mul timode Autotrack system employing the circular TEll and TE

21 modes", Journal of the radio research laboratories,14,No.73,pp 129-152, May 1967.

19 Jansen, J.K.M., Jeuken, M.E.J., and Lambrechtse, C.W. : "The scalar feed", T.H.Report 70-E-12,November 1969.

20 Flugge, S.:"Handbuch der Physik",Bd 25,S 238-240 Springer-Verlag, Berlin, Gottingen, Heidelberg, 1961.

21 Harrington, R.F.:"Time-harmonic electromagnetic fields",Chapter 6, Mc Graw-Hill 1961.

22 Smythe,W.R.":"Static and dynamic electricity", Chapter I,

APPENDIX

A.I Calculation of the

HE~~)-

and

HE~~)-conical

waveguide mode-patterns.

An

electric

field

is visualized by drawing the "lines of force" [22]

A line of force is a directed curve in an electric field such that the forward drawn tangent at any point has the direction of the electric intensity there.

It follows that if ds is an element of this curve,

dS=AE (A. I)

where A is a scalar factor. Writing out the components in polar coordinates, we have the differential equation of the lines of force

dp

pdcj>

E

=

-p-Eep (A.2)

The lines of force on the spherical aperture are projected on a plane perpendicular on the axis of the cone (Fig.A.I)

o

Fig.A.I. Coordinate system used for the calculation of the lines of force.

The normalized radius of the aperture is I . Then

Ep=EeCOSe=Ee"I-(PSineo)2 (A.3)

Substitution of the expressions for the aperture fields (3.1) for

the case

s

(kr')=-I in (A.2) gives

\)

dp = + cos2p dep (A.4)

- sin2ep

t=

and for the

p 1 I-t2 ~···2·_{sine 1+t}

o

Integration of equation (A.4) gives for the HE(l)-mode

2\1

t=F'-

C

~I

sin2rfJ

I

+ C

HE

(2)-mode 2\1 l-C

~

I

sin2rfJ

I

1+c~lsin2rfJl

P~ sine o 2 I-t 1+t2 51. (A.5) (A.6)

This are the equations giving the lines of force shown in the Figs. 3. 1• 1 and 3. 1.2.

2 8E'~ PI. T~fT~O, THITAO, " PHI, T, R,

:5 .LM!EGtB lI'fLL£P,J, "

4 8EAL ABei~ ¥C1RI<, YCIRI<, X,leX, VV, v[-400' 4001'

, LISB&iY ~~OTTE~,~~OT~I'Tuet,~~OT'¥I$,~LOT'¥11 2,¥"~CO 'w'V~,$C'~~'

6 8QQLE'~ ~PAPIe:I<, NUMERI[I<;

I I ,[ ·1 I , ~'::'\ ;,,;jr~t1 ".,10' ... til • l' GOIg NE.,.. V[ II, I, :'S60, 10, 0, 2077, -1, 1, 100, 2000,. h - l , 1, 100, 2000 • • V ~I'~~(IJI l1tl!.l.L. 60 QQ

4, X[I III li'1¥TC1, ., VII]11 ",~~(4)'

4, X[60 + 1ll' (1¥TC1, 4, VltlD • III' ",~CC.)'

4, X[12D • 11)1 l'1¥TC1, 4, VI120 • 11)1 'P,'~~C.l,

4. X[180 • Ill' l'1¥T<l, ., VI180 • 1III 'P,"~(4)1

4. X[241'1 • I III (1¥TC1, 4, Yl240 • Il" OP,.'tC41'

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begin ccmment program 06064164 c .w.lambrechtse

radiation pattern of the conical horn T HE(1)-2n-mode;

integer 1;

~ procedure gamma(x);~x;realx;

ge~18 i f x>1 then gamma:=gamma(x-1 }X(x-'I) else gamma:= l/x+ (-0.577191652+xX(0.988205891+xX(-0.897056937+xx(

• 1 20l>857 - -+xX(-0.756704078+xX(0.482199394+xX(-0.193527818+xXO.035868343))))))); ~ gamma; ~ procedure F(a,b,c,z,1,re);~a,b,c,z,re;~la,b,c,z,re;1nteger1; b;S1n real FG,t1,t2;FG:=t1 :=1;1:=1; herhaal:t :=t1x(a+1-1 )X(b+1-1 )xz/((c+1-1 )X1);FG:=FG+t2;t1 :=t2;1:=1+1;

!!

abs(t2/FG »re ~ ~to herhaal;

F:=FG ~;

/ (p1xgamma ( ( n-m+2 )/2) )

+f1xcos(theta)x(n+m)XS1n(p1X( n+rn)/2 )Y£amma( (n+m)/2)/ (p1y£amma( (n-m+1 )/

2) ))

~ procedure P(theta,rn,n,re);value theta,re,n,rn;~theta,re,m,n;

beg1li real p1,f1,f2;p1:~3.14159265359;

!!.

t eta~p1/4 ~ begin P :=(s1n(theta)lIm)xF( (n+m+1 )/2, (m-n)/2,m+1 ,s1n(theta)xs1n( theta) ,1,re )xgamma(

n+m+1

)f

( ( (-2)Am )y£amma.(rn+1 )Y£amrna ( n-rn+1 ) );

end

else begin f1 :=r((1-n-mJ!2,(n-m+2)/2,1.5,cos(theta)xcos(theta),1,re);

f2:=F((-- n-m)/2,(n-rn+1 )/2,0.5,cos(theta)

xcos(theta) ,1,re); P :=( (( 2)Am )><sqrt( p1)j (s1n( theta)lIm) )x(-f2X( (n+m-1 )/2 )XS1n(

p1X(n+m-1 )j2)y£amma( (n+m-1 )/2)

end

end.P;

-(-.00004 87613

rrugram UOU04-I04- p.e

real ~rocedure dP(theta,m,n,re);value theta,re,m,n;real theta,re,m,n;

---- - begIn dP:=-(n+l)Xcos(theta'XP(theta,m,n,re)-;rn-m+l)xP(theta,m,n+l,re)

~ dP;

real Er0cedure J2(x);value x;real

X;

---- begl9:!! x=O then J2:~lse J2:=2xJl (x)/x -JO(x)

end 2;

real procedure J3(x);value x;real

X;

---- begIn!! x=O then J3:=0 else

J3

:=4xJ2( x)/ x -Jl(x)

~ J3;

real procedure JO ( x ); value x; real x;

begin camnent A.J.M. Hitchcock, PO'iynan1al approximations to Bessel f'unct1ons

or

order zero and one and to related f'unct1ons ( 1957)" 86-88.;

real t; x := abs(x);

!! x

:5

4 ~ begin t :=

(xl

4)

A

2;

comment The maximum absolute error in the po~nam1alapproximation is 10_m-l0;

JO :=

if

x _{< 2D-6 then 1} . -- else 1 + t x (-3.99999 98721 + t x (3.99999 73021 + t x (-1.77775 60599

--.;;". ;., + t x (.44435 84263 + t x (-.07092 53492 + t x (.00767 71853 + t x (-.09050 1415))))))) end

~ begin ~ PO, QO, p14; t := (4/x)

A

2; p14 := .78539 81633 97448;

camnent The max:lInum absolute error in the po~nam1alapprox:lInat1ons

- f'or PO and QO are 12_{m-l0 and 25D-l0,} respect1ve~;

PO := if' x

>

0.5D6 then .79788 45608 02865

-

elSe

-

2 x (.39894 22793 + t x (- .00175 30620 + t x (.00017 34300 + t x + t x (.000173565 + t x (-.0000037043)))))); QO := 11' x

>

0.5D6 then -.79788 45608 02865/(8 x x) - else 8/ x x (- .01246 69441 + t x (.00045 64324 + t x (- .00008 69791

+t

x (.00003 42468 + t x (-.00001 42078 + t x (.00000 32312)))))); JO := (PO x cos(x - p14) - QO x s1n(x - p14))jsqrt(x) end JO(x); end

-

VI VI

real Erocedure J1(x); value

X;

real

X;

begin canmen! A.J.M. Hitchcock, -p(;iynarrial approximations to Bessel functions

of order zero and one and to related functions mtac(1957), 86-88.;

inte§er signx; real t; signx

:=

sign(x); x

:=

abs(x);

:!!

x ~ 4 ~ 2egin t := (x/4 )

A

2;

S21~The maximum absolute error in the polynanial approximation is 310-10;

J1 := i f x

<

2D-6 then signx x x/2 - else signx x x x .25 x (1.99999 99998 + t x (-3.99999 99710 + t x (2.66 666 60544 155 + t x (-.88888 39649 + t x (.17775 82922 + t x (-.02366 16773 + t x (.00220 69 + t x (-.00012 89769)))))))) + t x (-.00002 00920 + t x (.00000 42414)))))); Q1

:=

i f x

>

0.5»6 then .79788 45608 02865 x 3/(8 x x) - else 8/x x (.03740 08364 + t x (-.00063 90400 + t x (000010 64741 + t x r::oOOO3 98708 + t x (.00001 62200 + t x (- .00000 365°4))))));

J1 := signx x (P1 x cos(x - pi34) - Q1 x sin(x - p134))/sqrt(x)

x (.00005 80759

end J1 (x);

-end

else begin ~ P1, Q1, p134; t := (4/x)

A

2; pi3L. := 2.35619 L.4901 92345;

canment The max1mum absolute error in the polynanial approximations for P1 and Q1 are 15»-10 and 25_D-10, respectively;

P1

:=

i f x

>

0.5»6 then .79788 L.5608 02865 - else 2 x (.39894 22819 + t x (.00292 18256 + t x end (- .00022 32030 + t VI (1'1

.

\J1

-..J

.

begin real a,b,c,nnul,p,u,r,thetnul,thet,theta,n,pi,sk,dk,ek,k,thetanul,thetanulgrjinteger dia,sit, it,itsj_

a~ d,el1 :4],IEI,IER,IHR,IHI[0:180]j

camprex hank,q,s,j,tj

ca;r-ex arrar IH,IE[O :180] jlibrary INrIDRAL,ARG ,PlDI'PICTURE,SCALE,MARKED CURVE,RIDUIA FAlSI,PLOI'AXIS ,PLDrAXIS 2,

PLOTENj begin:a:=Oj

j :=can(O, 1)jpi:=3 .14159265359 jdia :=RFADjthetanulgr :=READjthetanul :=thetanulgrXpi/ 180jn:=READjr : =READjsk :=READjdk :=R

FADjek : =RFADj

e[1]:=READje[2]:=RFADje[3]:=READje[4]:=READjthetnul:=READjsit:=READj

for k :=sk stet dk until ek do tegin a :=OJ

for it:=sit ~ srt'Uliti1 the nu do ~ thet:=itxpi/180j IER[it] :=INrEGRAL(theta,re

r\dP(theta,2,n,e[2]XD~XP(theta,~n~eT2JXm-2))X(1X(1+cos(theta)xcos(thet))X(J1((p:=kXrXSin(theta)XSin(thet)) )-J3(p) ) +(cos(thet)+cos(theta))X(J1(p)+J3(p))+2XjXSin(theta)xsin(thet)xJ2(p))X(cos((u:=kXrxcos(theta)x cos(thet)) )+jXSin(u))) ,a,thetanul,d,e,false,false)jIEI[it]:=INTEGRAL(theta,tm ((dP(theta,2,n,eT2JX;-2J+2XP(theta,2,n,e[2]Xm-2))X(1X(1+cos(theta)xcos(thet))X(J1((p:=kXrXsin(theta)xsin(thet)))-J3(p)) +(cos(thet)+cos(theta))X(J1(p)+J3(p))+2XjXSin(theta)XSin(thet)xJ2(p))X(cos((u:=kXrxcos(theta)x cos(thet)) )+jXsin(u))) ,a,thetanul,d,e,false,false)jIE[it/sit]:=cam(IER[it],IEI[it])j

endjCARRIAGE(4)j~EXT{~corrugatedhorn HE(1)-2n-mode: THETANUL,n,r en k zijn respectievelijk:~)j

FIXT( 2,2,thetanulgr) jFIXT(2,8,nl jFIX'l,'(2,3,r) jFIXT(3 ,2,k) jCARRIAGE( 2)j

for it:=2 st~ 1 until thetul/lkb

iIfdtt

if mod(IE[it])~od(IE[it-1])then ~ to AAendjAA:its:=it-1 jPRINI'I'EXT(

<tlieta m (I/ror- Affi(I/IO) - 201n(I/IOH)jb:=-101 0 j - -

-for it:=1 step 1 until thetnul/sit do begin NLCRjFIXT(3,0,itXSit)jSPACE(10)jFIXT(1,5,(p:=mod(IE[it]jIE[its])))jSPA

~10)j -

-FIXT(1,3 ,Aro(IE[it]/IE[its] )) jSPACE( 10) jFIXT(2,3, (a :=20Xln(p)/ln( 10))) jSPACE( 10)j

~ c:=O,3,10,15,20,25,30,35,40 ~ begin!!( a<-cAb:>-c)V(a::::-cAb<-c) ~ FIXT(2,1,sitX(it-1-(b+c)/(a-b)))~jb:=aj

endjti' dia=O then befin

Pl'.DrPtCTURE(ixslt,ti' p:=20Xln(mod(IE[i]jIE[its]))/ln(10))>O then 0 else (if p<-40 then -40 else p),i,thetnul/sit

,2,-1,oo66,0,thetniiI,12, - - - -

-2540xthetnul/180,

<

theta (deg.)~,-40,0,2,2500,cl: relative power in dB~)j

PLOrTEXT(12,-38,90,30,0,true,2,<HE(1 )-2n-mode ,calculated ~);

PLOTTEXT(18,-38,90,30,0,tru~,0,<kr=~)jFIXPLOI'(0,0,90,30(0,false,0,3,2,kXr);

PI.DPI'EXT( 0,0,90,30,0 ,false

,0,<

thetanul=') jFIXPLOI' 0,~30,0,false,0,2,0,thetanulgr )j

PLOTTEXT(0,0,90,30,0,falSe,0,<deg.~)jENDPLODOC ---end ~ jp:=readj!! p.;otli~ got2 beginj ~ ~ pr{?§end

VI CD

.

begi? real a,b,c,nnul,p,u,r,thetnul,thet,theta,n,pi,sk,dk,ek,k,thetanul,thetanulgrjinteger dia,sit, 1t,1ts;array d,e 1:ri1';IEI,IER, nrn,IHI[O:1 SO]j

canplex hank,q,s,j,tj

2an!hi,j:

~ IH,IE[O:1SO] jlibrary INI'IDRAL,Aro ,PUJI'PICI'URE,SCALE,MARKED CURVE,RIDUIA FAlSI,PUJI'AXIS,PUJI'AXIS

,P ENj begin:a:=Oj

J:=cam(O,1)jp1:=3.14159265359jdia:=READjthetanulgr:=READjthetanul:=thetanulgrxp1j180in:=READjr:=READjsk:=READj dk :=READjek :=READj

e[1]:=READje[2]:=READje[3]:=READje[4]:=READjthetnul:=READjsit:=READj for k :=ak itet dk until ek do ~egina:=0j

for it:=ai ~ sit until tEe nu do bjSiin thet:=itXpij180j IER[it] :=INTIDRAL(theta,re

rctP(theta,O,n;eT2]XIl-mrrX( 1+cos(theta Xcos(thet) )XJ1 ((p:=kXrxsin(theta)xsin(thet» )+sin(theta)XSin(thet )XJO( p) )X( cos ( (u : =kxrXcos (theta )x

cos (thet) ) )+jXSin( u) ) )

,a,thetanul,d,e,false,false);IEI[it]:=INTEGRAL(theta,~

(dP(theta,O,n,e[2]XIl-2)X(jX(1+cos(theta)xcos(thet»XJ1((p:=kXrXSin(theta)XSin(thet»)+sin(theta)Xsin(thet)XJO( p»x(cos((u:=kxrxcos(theta)x

cos(thet»)+jXSin(u»)

,a,thetanul,d,e,false,false) jIE[it/ sit]:=ean( IER[it] ,IEI[it])j

endj CARRIAGE( ) jPRMTE5IT(:tCorrugated horn tm-01-mode: THErANUL,n,r en k zijn respectievel1jk:::J.)j

mr(2,2, thetanulgr) jFIXT(2,8,n) jFIXT(2,3,r) iFIXT(3,2,k) jCARRIAGE(2)j

for it:=2 st~ 1 until thetnu.l/sit do beg!!! if" mod(IE[it])<1nod(IE[it-1 ])then ~ to AAendjAA:1ts:=it-1 jPRINI'l'EXT(

<tlieta m (I/mr- Aro(I/tCS) -20ln(I/IO):J.);b:=-1l 10 j - -

-for it:=l step 1 until thetnul/sit do begin NLCRiFIXT(3,O,itXSit)jSPACE(10)jFIXT(1,5,(p:=mod(IE[it]jIE[its]»)jSP

ACE(10)j -

-FIXT(1 ,3,Aro(IE[itJ!IE[its]) )jSPACE(10)jFIXT(2,3,(a:=20X1n(p)/ln(10» )jSPACE(10)j

!2!:

c:=O,3,10,15,20,25,30,35,40 ~ begin!!( a<-cl\b>-c)V(a>-cl\b<.-c) ~ FIXT(2,1,sitx(it-1-(b+e)/(a-b»)~jb:=a jendjif" d1a=O then

Jlein

Pm!'PmrtJRE(iXS""'lt; p:=20Xln(mod(IE[iJ!IE[its)))jln(10))>O then

°

else (if p<-4o then

-40

else p),i,thetnul/sit,2,

-1,oo66,O,thetnul,rn, - - - -

-2540xthetnu.l/1SO,

.[ theta (deg.):J.,-40,O,2,2500,< rela~ive pot.Jer in dB:J.)j

p~(36,-38,90,30 ,O,true ,2,<IM-On-mode , n=:J.) jFIXPJ1Jl'(O,O,90,30,O ,false ,O,2,4,n)j

PLOTTEXT(48,-38,90,30,O,true,O,ikr=:J.)jFIXPJ1Jl'(0,O,90,30,O,false,O,3,2,kXr)j

PLOTTEXT(O,O,90,30,O,false,O,< thetanul=:J.)jFIXPJ1Jl'(O,~,90,30,O,false,O,2,O,thetanulgr)i

PLOTTEXT(O,O,90,30,o,faISe,O,~eg.»jENDPLODOC

-end

-end jp:=readj!! p!o ~ ~to beginj

end