eindhoven university of technology master corrugated ... · hybrid wave propagation in a circular...
TRANSCRIPT
Eindhoven University of Technology
MASTER
Corrugated coaxial horn antennas
Vokurka, V.J.
Award date:1973
Link to publication
DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Electrical Engineering
CORRUGATED COAXIAL HORN ANTENNAS
by
V.J. VOKURKA
. TECHNISCH:::: t-K'GESCHOOI..
E!i~OHOV~~N
_, STlID'FC-lT'1 .~,.,-q[r-I< ~r'tV fq"U_·'l._,,- •..• I-.../lll _ 'J
December 1972
ET(A) - 10 - 1972
This study is performed in fulfilment
of the requirements of the degree of
-----Mas~F of seienee{Ir. tcrt----tlw
Eindhoven University of Teehnology
under supervision of Dr. M.E.J. Jeuken.
---~
1
1.1
1.1.1.
1.2.
1.3.
1.3.1.
1.3.2.
1.3.3.
1. 3.4.
1.4.
1.5.
CONTIW'fS
Formulation of the problem 1
CHAPrER I
Introduction 3
Hybrid wave propagation in a circularcoaxial waveguide 4
General considerations 4
Coaxial waveguide with anisotropic boundaries 7
The dispersion equation 10
The case l.1/k < 1 10
The case t1/k > 1 12
The case IJ/k 0 14
The solution of the dispersion equation 14
The transverse fields in a circular coaxial waveguidewith anisotropic boundaries 24
Radiation from coaxial waveguide with anisotropicboundaries 36
CHAPTER II
2.
2.1.
2.2.
2.2.1
2.2.2.
2.2.3.
2.2.4.
2.3.
2.4.
Wave propagation in a circular corrugated coaxialwaveguide
Wave propagation in grooves
The dispersion equation
The case ~/k < 1
The case ~/k 0
The case ~/k > 1
The solution of the dispersion equation
The transverse fields, n = 1
The power radiation pattern of a corrugated coaxialwaveguide
52
53
57
57
61
62
65
72
83
2.5.
2.6.
2.7.
The power radiation pattern of a conical coaxialcor'rugated horn with smal1 flare angle
The experimental investigation
Conclusions and programme for fut.ure work
Aclmowl('dgements
References
Appendix A
93
100
111
113
114
1
Formulation of the problem
Corrugated antennas have been the subject of study for several years.
Their properties can be summarised as follows.
If the boundaries E. = H. = 0 are satisfied, antennas with a periodical
corrugated structure produce excellent circulary polarised waves.
Besides, corrugated conical feeds possess this important property
approximately in a frequency band 1 : 1.5.
It should be noted that these feeds also have a symmetrical radiation
pattern, which is of great advantage when such antennas are used, for
instance, in radar systems or radioastronomy. Comprehensive surveys of
the li terature on thi s subject can be found in /3 /.
However, in some cases one needs feeds which are capable of handling
two (or more) frequency bands located in the frequency spectrum at a
longer distance than 1 : 1.5, whereas the radiation pattern should be
symmetrical for the two frequency bands and the shape of the pattern
should be identical in the two bands. For instance, radioastronomical
investigations are sometimes carried out at two frequencies, while it
is not allowed to remove the feed in the parabolic reflector.
Another example is canmrllcatWn satellite traffic, which should be
capable of taking a place at, for instance, 10 and 20 GHz /19/.
In this case the bandwidth at both frequencies of about 10%, is also
of essential interest.
Other points of investigation are the antennas, which are in use
as feeds in parabolic reflector antennas and which give rise to
high aperture efficiency. This type of feeds produces a symmetrical
radiatri~1l pattern with a "dip"---i-n-the f~rward directi~T--._---
First, Ludwig /10/ proposed a circular waveguide with perfectly
conducting wall~ with four propagating modes, which gives a symmetrical
pattern with relatively high aparture efficiency. Another type of
multimode feed has been developed by Koch /11/. He used a coaxial
system with four modes, but in this case two modes were excited in
the central circular waveguide and the other two in the outer coaxial
guide.
2
Such a system has good radiation properties, but its bandwidth is
limited to about 6% /13/.
Probably the first feed which gives rise to high aperture efficiency
with two hybrid modes was introduced by Thomas and Cooper / 4/.
This feed has excellent radiation properties, but also in this case
in a narrow band.
Let us now return to the problem of dual-frequency antennas.
It is a natural choice to employ coaxial circular waveguides as radiators.
The central part of the feed is available for the higher frequency band,
whereas the outer part may be used for the lower band.
However, if the coaxial feed is fed by TEll mode, the radiation pattern
is asymmetric and exhibits high sidelobes in the E plane /7 /.
The application of a corrugated coaxial structure may probably improve
this situation. Hence, it seems worthwhile to study coaxial corrugated
waveguides.
The attractive properties of one-mode propagation in circular
waveguides and horns with corrugated boundaries suggest the study of
coaxial antennas with such a structure as a possibility to improve
properties of the feeds, which have been developed by Koch and others.
The purpose of this study is the investigation of the wave propagation
in coaxial corrugated waveguides and their radiation behaviour, both
theoretically and experimentally.
3
CHAPl'ER I
1. Introduction
In this chapter is first given the solution of the Maxwell's
equations for the propagation of the so-called hybrid modes in a
coaxial waveguide.
These modes are obtained as the sum of TE and TM fields.
Chapter I contains the study of wave propagation and the radiation
behaviour of the coaxial waveguide with frequency independent
anisotropic boundary conditions E; = H+ = O. First, the dispersion
equation has been solved. Further, the characteristics of the
transverse field and the power radiation pattern have been studied.
In Chapter II the frequency independence and symmetry properties of
corrugated coaxial antennas are described. Therefore, it is necessary
to solve the dispersion equation for the case where the depth of the
groove is taken into account. In this case too, the transverse field
and radiation characteristics have been dealt with.
In section 2.5. is also given the power radiation pattern of the
corrugated coaxial waveguide with quadratic phase distribution
across the aperture.
4
1.1. Hybrid wave propagation in a circular coaxial waveguide
1.1.1. General considerations
z
-7--.P+ko::-""~'~ -..L -- -- -- --
III- --- _.~~~~~t---------l
Fig. 1.1
A coaxial corrugated waveguide consists of three parts.
A central part (II) and two parts with uniformly spaced grooves
in the inner and the outer conduc tor, part I and part III, respecGi ve ly.
The calculation of the electromagnetic fields in such a system starts
with the observation that at the boundaries r = a1 and r = a2 there
exists an E and an H component. We shall start our study withz z
the electromagnetic field in the central part (II). The effect of the
grooves will at first be replaced by idealized boundary conditions,
which are given in section 1.2.
In the central part of the waveguide (II) propagation of only hybrid
modes is possible. This type of field is obtained as the sum of a TE
field and a TM field.
5
Suppose that the TE field has a generating function
(1.1 )
This function for the TM field is given by
~tJ. =: I ~2 J,., rKe; r) ..,...82~ (~crJ}co.s I?tjie ..·((,U-t-~z) (1.2 )
with 2 JZ _ A 2k - I< 0c
J (k r) is a Bessel function of the first kind and of the order n.n c
N (k r) is a Bessel function of the second kind and of the order n.n c
Then we can derive the components of the TE field from the following
expressions.
f rJ y{Er = - r -;;;::-
1 f
o
(1.3 )
For a TM field we find
~ ;)2 ~ -r c7~Er
:;
dr CJz/-Ir =- r eJr/14J ~o
~ -T i)2¥{HlP
~ ~(1.4)E p =
~¢ ~z.:
/tA.J~o,- dr
~ ~ .9Z~E;z = i:. :z. <If T- :J z ~) I-/z =- 0/t..JCo
6
Then we find for the field components of the hybrid field in part (II)
of the waveguide.
+ [: B, .M, (kc; r)/.J
k c ..82 /l1, fke r 1/~J ;?;+ I.4J e",
E.p ::: IL ~c. R;.2/(kcT)/3 n 2: (LcrJj.,..". lAJE. r .192
or L kG) ~ ; 62 ~ (~r!l) ,f/h n!4 ~ (i(e r) .,.
wEe>
i:2..C'2 = /~:o [ /7z ~ r'(r) To 32. AI" rKc r JJ ~.f nJi
fir ~ -1£ (N~." -tc h7,;:' r~r) To ~ ~ 0.'.{c r J/...
~/~~o Kc; ~ ~(cc;r) r : ~ A/,,(kc;r# .J·/h h~
(1.5 )
(1.6 )
(1. 7)
(1.8)
(1. 9)
(1.10 )
In (1.5) to (1.10) we have omitted the factor expo j("" t - t1z).
A1 , A2 , B1 and B2 are constants. The prime in In'(kcr) and in
N '(k r) means differentiating with respect to the argument.n c
7
1.2. Coaxial waveguide with anisotropic boundaries
E; H;O
z
Fig. 1.2
Next we assume the anisotropic boundaries at the inner and the outer
conductors to be independent of frequency.
It should be noted that the boundary conditions which are used in this
section do not represent the physical situation, but they serve to take
a first step towards solving our problem.
The boundary conditions are
E" = 0, H = 0
,; 0,tJ (1.11)
E H f 0z z
or
E = ZzH" wit-h 'lr-~--":'
z z
and (1. 12)
E; = Z..Hz wi th Z" 0
Now we choose
(a) A1 = Zo~' B1 = ZoB2 (1.13 )
(b) A1 =-ZoA2 , B1 =-ZoB2 (1.14 )
8
and try to satisfy the boundary conditions (1.11).
In the following section we shall observe that other solutions are also
possible, but only (1.13) and (1.14) give a symmetrical radiation pattern.
Substituting of (1.13) in equations (1.5) to (1~10) and using the
relations
Z -/,-.t;> £. and
we have
Er ... - ! ~ Zt;>[: 2rJcr) or : 1:, ~ ~",r)/,1-
-r 32. Z 0 [ ~ /0, (,fc r),I- -: kc A/,,)(kc r!jl CcJ nfIJ
£? l;'9z 2 0 [: ~.7n r4c '-) ~ ~G J;, }(kc. r J/r.82. 2
0,(:: fJ ,u" r.tor) .,. .to AI,,) r.t."'!l!.r,,,,,¢
(1.16 )
(1.17)
(1.18)
I-Ir = - ! rlz-j : .l: riGr) .,.. : 1:c 2.} (of, r)J-I-,.. ~L : M, (itt; r),I- : kG ~)(~c. r Jjj .r/n ;-If
IltjJ = - ! /1.zI: :. .J:. (J,;-) "" ~,.2..J (~c ry To-
.,. 3.£1 :: AI"r.l."') "'" Ko lJ"'rJ.,,r-$ t:ds ns"
(1.19)
(1. 20)
I-(z =
-- - - ---- ----
.3'2 ~ ~c.r)J .f"/h h~ (1.21)
In the same way we compute the field components for the case where
Ai =-ZoA2 and B1 =-ZoB2 •
Substitution of (1.14) gives
(1. 22)
9
E~ = f ~Zo/: J./;,('(r) - kc ~ (~,r)j'r
-I" ~ 2 0 /: : AI" ~e'-) -kc A/;,)ike r!J/$"/n n¢
.t/'== ~ L ,q, ~ ~c:: r) r g ~ (~c:: rJ/ (;.O.f n¢
(1 ')'"."-' ,:,) ,.'
( 1 ')4', .. -< - I
4.(1 n ;tin (ke,-)- ~c AI" ) (~G r!l) CoS ;'7~.,..r
h:zkG 2- L //2 .z. (ker) 8 2 ~(kcr.!.J.f/ f7 n;t.- /.4: +-
~ and B2 are constants. In (1.16) to (1.27) we have omitted the CGwmon
factor exp j (wt ...: 1.3 z).
10
1.3. The dispersion equation
1. 3.1. The case l3/k < 1
Our next step is to find the dispersion equation for the propagation
constant I.J.
The boundary conditions are
E~ == 0, H; == 0, at r == a 1 and r == a2 •
In the case of A1 == Zo~' B1 = ZoB2 we obtain from equations (1.17)
and (1.20) for r = a 1 ,
For boundary r == a 2 we find
//2/1 .;: J. (k, 4.) .. k, .h. h,.,.!f-3z. f': .,: M. rk,4.)'" i, M.;'"' "-j~ 0(1.29 )
For the other case with A1 =-ZoA2 and B1 =-Z
oB2 follows from equations
(1.23) and (1.26) for r = a 1•
/12.(: ~ .J;,f.(:(;q,,)_~c.7,.(kca,,:y ~ 3.zf1!' M,fkcq,)-kc M,ficQ;I= 0
(1.30 )
and for r == a 2
~If ;; ..7;(kc 02.) - .lc .4 jkc qa)/-,L 3,2Z1 Q: ~(,tc'&)~~~-M~ q;y': 0 -
(1.31)
We are now able to compute the dispersim equations,
For the case A1
== Zo~ and B1 = Zo
B2 we obtain from (1.28) and (1.29)
11
the dispersion equation in the following form
r, (~c. <3,) 9, fic. a,)
= 0 (1.32a)
~ (kc. CiZ ) ~z (Jc.CI..)
or
(1. 32b)
where
F; rite a,} ::0
C;, (-te.Q,) =
;:2 (Ne Q z) -=
~2. (otc Q z ) =
/Inr-;,~~.t t!1,
4.!!A: t1&
/Ini" "~
..l (Jt' 4,) or .Ie. ..z. '/Je. ",}
~ lie tI,) r Jc .(I" J'''e tI.,)
.4 (~e. "~) ..". k~ .z: f.i e ~~ )
)
~ 4 c "a)'" kc. It/n (.I.e. IJ~)
The solution for A1 =-ZoA2 and B1 =-Z
oB2 is given (using (1.30) and
(1.31)) by
-) /f; I Itt: a,)
= o (1. 33a)
or
(1.33b)
with
IVh (.tc. Q,) - kc. ~ )(~c. a,))
J;. (.Ie "ao) - Jc. ~ (.It' liz.)
!Un (~c "3.) - -te A..4/~t' (1.)
: £ :;;, /.4c. 13,)
/.I!lA: el,
!!!!~ <3~
/1 h
i "~
~ J(Je a,) ....)
Ci,(kc. a,) :"
~ )(Icc a ..) ::-)
4Z (I<c a ..) .=
12
1.3.2. The case ~!k > 1.
In this case the generating f~nction of the TE field in part (II) of the
waveguide is given by
rl 1 = [Al'In
(kc
'r) + B1
'Kn
(kc'r)] sin n~ e j (wt-4z)
whereas the TM field has the generating function
~21 = [A2 'I n (k c 'r) + B2'Kn(kc'r~cos n;
k c' = k ( /.S /k ) 2 - 1
(1.34)
(1.35)
Ai" ~', B1 ' and 82 ' are constants.
I and K are the modified Bessel functions.n n
lJsing the expressions (1.3) and 1.4) we can calculate again the components
of the electromagnetic field.
We find:
(1.36 )
~~ J.?, k'.. (~}) r w~. :: 8. J.4::.. (J:rd) .r'" "p
~ )2
~ ./~col)9,2 J.L:' (J:c]r) t- .6zJ
khf~/;..tl ~-J "-?--
(1.37)
(1.38)
(1. 39)
(1.40 )
(1.41)
13
In the expressions (1.36) to (1.41) we have omitted the factor
exp. j ( w t - 13 z) •
The boundary conditions E.;::= 0, HI>::= 0 have not changed.
We can now calculate the dispersion equation again.
After applying the boundary conditions we obtain for
Al ' ::= ZoA~, and B1 ' = ZoB;,
--- . ---
(1.42 )
(1.43 )
14
1.3.3. The case ~/k =0
At cut-off frequency ~/k = 0, k = k.c
Working out equations (1.32) and (1.33) for this case we obtain
J ) J )~ (,to,) 4 (kt:1z) - Mh rk 1:1,).:7;, (.I "2.) :: 0..
We see that in both cases the same relation is valid. We know that
equation (1.44) gives the cut -off frequency of TE mode in anm
coaxial waveguide with perfectly conducting boundaries at r a 1and r = a 2• The point ~ =0 is a common point for both modes.
1. 3.4. The solution of the dispersion equation
The dispersion equation has been solved for several values of a2/a1 ,
nand k.
The values of a 2/a1 are: 4, 3 and 2.
The ~/k curves are plotted as a function of 2a1/A.The dispersion equation gives the solution for two modes.
The mode ~ich corresponds with A1 = Zo~ and B1 = ZoB2 is called the
HE(+) mode, the other is the HE(-) mode.nm om
We observe that in the coaxial waveguide with anisotropic boundaries
slow and fast waves are possible.
For fast waves f3/k < 1. This means that the phase velocity of the wave
is greater then the velocity of light in free space.
For slow waves o/k> 1. Theil" velOG-i-ty is--~-thall thatMli-gh-:tin--
(1.44)
curves are plotted in figs. 1.3 - 1.11.
free space.
Further we observe that for ~/k
waveguide are zero.
Some IJ/k versus 2a1/A
1 all the field components in the
Fig. 1.3 The dispersion relation ( a 2/a1 =. 2 )
43
n = 1a,ja f = 2
LHEf+)
I 1m
_ :~ow_wbves- HE~;"I c:1::J C "va ves ---.-:;;;:;:.~===============-""- -
0.2
0.1
o~--1.-~--;----l--~----J..---L---l..----'----....l.-.----..L.--~1 2
0.4
OJ
1.3 1 ------------------------12
1.1r1/k
1.0
0.9t1\0.8 \
I0.7 I
0.6 II
0.5 II
Fig. 1.4 The dispersion relation ( a 2/a 1 = 3 )
2.0
n ::; 1aZ/a1=3
HE~~(-)
HE,m ---------------------
1.0
-,...../"/2
2 //
1/)
/IIII1
slow waves---------fast waves
1.3
12
1.1l1/k
1.0\ 10.9 1\
0.8 \I
\0.7
II
0.6 II
0.5 Ii
'I0.4
\OJ I0.2 I0.1
0
n = 1al.la1 = 4HE~+/n
HE~~-----
\1- _slow waves _\ fast waves
1\ 1\II
t3r--------------------------------.12
1.1rl/k
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
oI----L----I__"---J__~------L------J------!------!-----L-----L------!-----L-...J
Fig. 1.5 The dispersion relation ( aZ/a1
= 4 )
1.0 2.0
....00
HE (-1-)zm
4
-----------------------=-= - -slow waves
--- --faStwaves--r- - -
1.3,----------::_>------------12
1.1/k
1.0
0.9
0.8
0.7
0.6
0.5
0.4
OJ
0.2
0.1
o"-----...u.....--t,...L---I...--42L---L.-----:3f-----L--~4 !-----L.------I5l....-...---..L~2a1/}..
Fig. 1.0
r",o
-.
..
5
--- - --~~; ::~:- -::-::=-:=-~~===-=============~==:--
0.1
oL-.J----L-l_--L,L-WL--~2L---L---:3:-----'---~4~--l----t5 __-L-----I
2a1 fA
0.6
0.5
0.4
OJ
-----------:::;-----------11.3
12 .
1.1l1/k
1.0
0.9
0.8 1
Fig. 1.8
HE (.-)2m
2
___ slow waves a2/a f -=3
fastwaves -_---=-------===-=-====-=~====~~~~
1
OJ
0.2
0.1
a---.&--L.~1...L----L1--~2~----J.-~3_---J.......----;---l.--...l..-_~5
2a1/A
1.31------~---:::---:-----------12
1.1[3/ k
1.0
0.9
0.8
0.7
0.6
0.5
Fig. 1.9
E(+)
H- 2m
a/a,'"4
j
--_s_low_w~s -=-=..-.:=-=-----..:==========-===~~=-~~~=-====~fast waves
1.3112
1.1fJ/k
1.0
0.9
0.8
0.7
0.6
0.5
0.4
OJ
0.2
00.1 t_LLl---7-Lj-'--_~ __-'--:3~---'---t4---'--~5~:--2a1/A
Fig. 1.10
--slow waves _ _=-=--=-=--=-:::::.-=~====~============~- fast-waves- ========.
OJ
0.2
00.1 r-11-LJ--L--2---L--3-----'--4----'--!5~~~-L 3 4 2a,/~
0.9
0.8
0.7
0.6
0.5
1.3r-
12~
1.1f3/k
1.0
24
The conclusions are:
a. The branch for the HEi~) and HE~~) mode crosses the line ~/k 1
in a small frequency range.
This implies that these modes are not suitable for antenna applications.
The properties of these modes will be studied in chapter II in more
detaiL
b. For the case n = 1, i.e. a singular; - dependence, the HEi;) modp.
is probably suitable for our application. This mode is the lowest
that has the equivalent frequency behaviour in the fast wave region
as the HEi~) mode in a circular cylindrical waveguide with the anisotropic
boundary /16/.
1.4. The transverse fields in a circular coaxial waveguide with anisotropicbourldar i es •
After solving the dispersion equation we are able to investigate the
transverse fields in a coaxial waveguide. For choosing the modes for
antenna application it is necessary to know the behaviour of the field
components.
Further, we want to compare the field components of the HEi~), HE)~J,HF.i;) and HEi;) modes. .1<:<
Let us now investigate the transverse electric fields of the
REi:) modes.
It has been shown that for these modes the relations A1B1 = Z
oB2 are valid.
Using the recurrence relations /17/.
(1.45)
and.J
Z-. (2) ; I Z~-I (z) (1.46)
we obtain from (1.16) and (1.17) for n 1
25
with
;: (kc r) - ( -I P£ ~t!).70(i:c "')'" (i'- t )~ rJcr)
~ ( kc r) = (-I + :) A/o (4:, r) .". I!- ,(,./1)~ rl, r )
j' (kc r) = (17'- ~/1) Jo(~~ r) - (/- 1).7z. (.tc r )
ff.2/~c.r) ~ (1~ '1) A/o(kc~) - (i' - 1)~ (kc r)
In the same way we obtain equations for the field components when
A1 ~-Zo~ and B1 ~-ZoB2'
Using the relations (1.45) and (1.46) we obtain from equations
(1.22), (1.23), (1.25) and(1.26).
(1.47)
(1.48 )
(1.49 )
(1. 50)
(1. 51)
(1. 52)
(1. 53)
(1. 54)
26
1:, = Z2~' [ /12 '/)(.l:crJ 13z. ~) (i:c r)) CO./¢-r (1. 55)
£S- 2 0 icc ! Il. ) 132- ~} (.fcrij r~';,¢- 2 2 j'1 (k, r) -I" (1. 56)
2. I-/, Zo k c I ~ ;,) 132 ~)(kc. rij rl';,~o r = - 2 2. , (i, r ) -I- (1.57)
Zo'H{J= -2 0 kc. f 112 j/ (kcr) 32 J: (~rjco.{~ ,Z +- (1.58 )
'Where
rf J(kcr);1
I ) .70 (k, r) - 1/;£ '1).7; (ic r).: ("j"- (1. 59)
(z) (ok,r) /jI) Alo (kc. r ) (1-1- 1) ~ rk,r):=. (~ - - (1.60 )
} (I - f) .70 (kc r) (~ ,l- t) ~ (ier)jf (.1:(; r) = + (1.61)
) (t- t) Yo (icr) ,L (1 -I- 1) ~ (~c')j:L ~c.r) - (1.62 )
respectively.
B2The values of DC =
A2
We now define
Er
can be found from equations (1.28) and (1.30),
(1.63 )
(1.64)
(+)We have computed the functions Er(rel) and E~(rel) of the HEll '
REi;), REi;), and ~;) modes for several types of coaxial waveguides for
which the dispersion equation has been solved.
In choosing the modes sui table for antenna application, the following
points are to be considered.
:!7
1. The radiation pattern must have max. radiation intensity in the
forward direction.
2. The H and E plane patterns should be almost identical.
3. Sidelobe level should preferably be kept as low as possible.
In chapter II we shall prove that only HE~:) modes radiate power with
maximum intensity along the guide axis. If the boundary conditions
E; = 0, H; = 0 are satisfied and the feeds excite modes with m = 1
only, the HE~;) mode probably produces the best radiation pattern.
From equations (1.47) to (1.50) incl. we may conclude that the electric
field lines of the HE(+) mode are of the same form as the magnetic field12lines apart from a rotation in ; of 900
•
A sketch of the field lines of the HEl;) mode is given in fig. 1.12.
(a) TE12 mode
Fig. 1.12
(b) HE(+) mode12
The cut-off frequency of the H4;+mode is the same as that ~fth;TE12 mode in coaxial waveguide with perfectly conducting boundaries.
The field lines of this mode are given in fig. 1.12.
We may conclude that the figs. 1.13 - 1.).9 show much
analogy with the case of a circular waveguide with anisotropic boundary.
Properties of the HE~;) mode in a coaxial guide correspond with those
of the HElr)mode in a circular waveguide.
Jeuken 11 I indicated that a rather large value of E at the boundaryr
r = a produces high sidelobes in the E plane of the radiation pattern.
Also in our case probably a large value of E will produce a higherr
sidelobe level than a small value.
We can conclude that for the large-size aperture the values of E (r reI)and E;(rel) are nearly identical. In that case we may expect a minimal
eidelobe level.
Further we observe that the values E;(rel) of the same a2/a1 do not
change for different dimensions of the aperture.
For higher a2/a1 the maximum value of E;(rel) and Er(rel) respectively,
moves as expected, in the direction of the inner conductor. This is
shown in fig. 1.18.
TRANSVERSE FIELDS (n=l)
-1
(l)
~-2w
o
~---==- ---'- - - - -
o
-1
oa,
2 3 4 5! () 7 8 9 10rho 02
a 1 2 3 4 .5 6 7 a 9 10a1 rho a~
Fig.l.13
TRANSVERSE FIELDS (n:1)
a, = 6.6cm
~ f = 8.63 GHz~1w
o
a2 /a 1 '" 211-1
HE'2mode
oa,
2 3 4 5 6 7 8 9 10rho a2
Fig.1.14
o 1 2 3 4 5 6 7 8 9 10a~ rho a,
TRANSVERSE FIELDS (n=1)
az/a, = 2~)
HE1'1mode
a1 =2.25cmf = 8.46GHz-~
QJ
~1w
0----A.--L
6 7 B 9 10 0 2 3 4 5 6 7 8 9 10rho a2 a, rho a2
Fig. 1.15
a 2 3 42,
TRANSVERSE FIELDS (n:1)
o
oat
2 3 4 6 7 8 9 10rho a2
C1J
~1w
oat
Fig. 1.16
at = 2.25cmf = 12.68GHz
2 3 4 5 6 7 8 9 10rho a 2
TRANSVERSE FIELDS (n=1)
o
oa,
~-<U
~1w
o
2 3 4 5 6 7 8 9 10 0rho a:z a,
Fig.l.17
alia, =3(-1-)
HE l2-mode
2 3 4 5 6 7 8 9 10rho a2
TRANSVERSE FIELDS (n=l)
Q)
~1w
o
az /a 1 -4(f.)
HE ,2 mode
o 1 2 3 4a1
6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10rho a2 a1 rho a2
Fig. 1.18
TRANSVERSE FIELDS (n=l)
cu~1---~~------w
o
-~1~
UJ
-1 --------- ------- -- - -
o 1 2 3 4 $ 6 7 8 9 10a, rho a2
o 1 2 3 4 5 6 7 8 9 10a, rho a2
Fig. 1.19
36
1.5. Radiation from coaxial waveguide with anisotropic boundaries.
z ---------
Fig. 1.20
p
In this chapter we shall study the power radiation pattern of a
~oaxial waveguide with boundaries E; = 0, H; = O.
The power radiation pattern will be calculated with the Kirchhoff-Haygens method.
In our case we assume that the aperture of the waveguide is an equiphase plane.
The electromagnetic fields ()fuacoaxta~ radiator can De fofuidufromtJie
following representation theorema/1 /.
E (c'; = c.:.trlp I ,!' x E--(~).I t; r~ ~ ~) eIS +
s..
CU,./I" c",r~~ x ~f~)I C;(~: ~) a'S
s...
cur~II~ J( !I(~).1 ~ (~~J ~) dS-S'"
c""~1" ~tl"1/2 x §~).1 c; rr/ C) d S
.r""
(1.65 )
(1.66 )
37
llith
-.lie /~J-C I
o (~)./ .!'" ) = ~~ 7T7E J- rj
and
If we assume that the diameter of the aperture is at least a few
wavelengths and the distance r is long, we obtain after some calculationo
'221T
~"":"'..'1T-·4"'-J e -.;'J:gflE;.
(/, "
Qa 211"
e -.iJ:rJII/fE.,; CoJ e - 2 D #r) &.o.f (¢J_ ~) -
q, tJ
(1.67)
J.wit.h HI""~
In these expressions the aperture fields are unprimed and written in
circular ~i-nateB. Th~ primed radiftiion fields are given-in
spherical coordinates (fig. 1.20).
In the case of constant phase distribution across the aperture it i~
-jwr2allowed to neglect the factor e for the calculation of the electric
field in the far-zone region.
'fhe aperture fields for Ai = Zo~ and Bi =ZoB2 and for %< f have already
been computed in a previous section. We write these equations again.
38
ErZo ~c [:
g2 Ii ~c rj}t:QJ"~== - 2 H'z ~ I'~cr) .,4 (1.69)
E:; Zo -ic. I ,q ~ ,g2./a (k<: r!l .r"n '¢::: 2 2l' /e.cr) .,4 (1. 70)
ZoHr =- Z2~c I~; I'~cr) ~ ~ ~ fA,rj r/h r; (1.71)
Z o h'p=_2,2kc !~2 j,(k,r) ~ 8 2 /20 (ic. rf Co,/"? (1. 72)
with
I; r;,r) =
(z. {~,r} =
j" (.l, rj =
(1.73)
(1.74)
(1.75)
where
(f ~ f) A/o (-':c") - (f
it, r:l ~ r If - (~) 2.
(1. 76)
Substitution of the expressions (1.69) to (1.1~ incl. in the formulae
(1.61) and (~~Ji8) and usingtM relaticms
~r
j ~·t:lco.r(fi,)-';}s-/".p .r/h r¢) - <I)e d.p =
orq-
/
~~eos{fi)-~J,r/h ~ t:.::JW J - ¢ j e tIIp =
o
(1. 77)
(1. 78)
(1. 79)
392n-
/
,./ae~~/,p,}-~)tOJ.) ,f/'" (¢ - ~) e a'~ =
~
nere 4 - k. ,.. or I'" 6'
(1.80 )
we obtain the following expressions for the electric field in the
far-zone region
(1. 81)
(1.83 )
1::-,,; = ~,.t:}2. 2 0
.~ _/./:.,.. J
..L ,~.,..~ e S",,'n~J
2 ~,..,}
~',l '~r J
T.:z p~2. .82 20 .c:- e -;/ .r/h ¢ ,} (1.82 )
~rJ
where
da.
J;", ::I L p = /f2 (?'~ ~/J C<).r&~ ric r))':. /'kr.rI'nc:J-,;{a,r);I/lrr"."., I.,..q, J'
.,.,2(;.4 ~ co.r&)j.z fie r)..7c: (I: r .r1'nt9) .,.. y;'(kcr ) ..:{r~ r.r/~ 8!11rdr
and
dZ
.J28 = .L 2.{> =1/2(1 Hor 19) [;I/o (.I.r) J. f,l r" /n <7).-. M,,~, -;).%firs/n~
4,
... .2(I" / ""'61)jAJ,,(k,,) .7.(i n/h 11) - -4z (.+,,,).% II< r d",y:ll r ,yry (1.84)
40
The integral solution of the form /Z, t;-)( ) ~ ~)1)()( tt/x
we find in /17/.
We can now write the functions Ee and E, in a closed form.
Substitution of the integral solution in the expressions (1.81)
and (1.82) gives us the results for the field components
EIP = E• (r' , 9, _,) and Ef' = E; ( r ' , t?, ;').
First we shall write down the expression 118 and 12e in a closed form.
(1.86 )
41
The radiated power as a function of 8 relative to the value on the
axis of the waveguide is
(1.87)
in the E plane
in the H plane
(1.88)
{1.89}
From equations (1.86) and (1.87) follows:
(1. 90)
(1. 91)
42
From equations (1.86), (1.87), (1.00) and (1.91) we conclude that
FE == PH == PER"
We observe that the coaxial waveguide with anisotropic boundaries
produces a symmetrical radiation pattern.
However, the sidelobe level is higher than in the case of the
corrugated conical antenna.
We have calculated the power radiation pattern for several values
of frequency and a 2/a1• The results are plotted in figs. 1.21 - 1.29,
The excited mode is the HEi~) mode. For the calculation of PER we
need the values of %. These have already been computed in
section 1.3.
If the ~;) mode is excited we obtain a ~etrical radiation pattern
with a dip in the forward direction.
10
~o
a
o 30
43
60
Fig. 1.2t
PATIERN NO. DATE
PROJECT
ENGRS.
REMARKSa2/a t = 2
f = 8.46 GHz
at = 2.25 em
90angle (degrJ
120
10
20
to
o 30
44
60
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKSa2/a1 = 2
f = 8.88 GHz
a1
= 2.25 em
90angle (degrJ
120
Fig. 1.22
45
12090angle (degr.)
REMARKSa
2/a
1= 2
f = 9.43 GHz
a1
= 2.25 em
PATTERN. NO. DATE
PROJECT
ENGRS.
60Fig. 1.23
30
\\\
10
o
20
10
46
12090angle (degr.)
REMARKSa2/a1 == 2
f == 10.15 GHz
PATTERN NO. DATE
PROJECT
ENGRS.
60
Fig. 1.24
30
30
20
10
o
Leu~oQ.
eu>
47
PAnERN NO. DATE
PROJECT
ENGRS.
REMARKS
a2/a1 = 3
f = 8.48 GHz
a 1= 2.25 em
10
L.QJ
Soa.QJ>
20
12090angle (degr.)
60Fig. 1.25
30
II
t --...L._.J------'--_---1.-~__~o
30
10
,..."
en"'C-. L
(1)
~oa..
(1)
>
20
48
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKSa2!a1 = 3
f = 8.74 GHz
a1
= 2.25 em
30
o 30 . 60
Fig. 1.26
90angle (degr.)
120
10
---.CD.""C
20
49
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKSa2/a t == 3
f = 9.03 GHz
at = 2.25 em
30
o 30 60
Fig. 1.27
90angle (degrJ
120
10
20
50
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKSa2/a1 := 3
f c;:: 9.36 GHz
30
a 30 60Fig. 1.28
90angle (degrJ
120
10
.......00-0......
'(lJ
~oa.(lJ
>
20
51
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKS
a2/a1 = 3
f = 9.72 GHz
a1
= 2.25 em
30
o 30 60Fig. 1.29
90angle (degrJ
120
52
CHAPTER II
2. Wave propagation in a circular corrugated coaxial wavegui de.
In the first chapter has been given the solution of the dispersion
equation and of the power radiation pattern of a coaxial waveguiJe
with anisotropic boundaries.
However, this solution is valid as far as we can satisfy the boundary
conditions E; = 0, H; = 0, independent of frequency.
The purpose of this chapter is to study the properties of the
coaxial corrugated radiator. Therefore, it is necessary to find
t..'1.e solution of the dis~ersion equation for the case where the
depth of the grooves is taken into account.
First we shall study the fields in the grooves. The following step
is to find the dispersion equation. When the dispersion equation is
solved, we shall be able to investigate the transverse fields and
to compute the power radiation pattern of the E and the H planes.
53
2.1. Wave propagation in grooves
III
II
I
z-Fig. 2.1
We assume that the distance between two consecutive grooves is short and
that there are many grooves per wavelength. ThiR implies that it is
possible to ignore the periodic structure of the waveguide.
We observe that region I between r = a 1 and r = b1 and also region
III between r = a 2 and r b2 are in fact the radial waveguides which
are short-circuited at r b1 and r = b2, respectively.
The fields which can exist in a radial waveguide are TE and TM fields with
raspect to the z-axis.
The TM fields can be derived from the generating function.
f3 ~ (~Gr)",... 7T z (2.1 )= COf,n ~ CoJ
~.2.
with )2~2. k2. (h7F= -Co ~z.
Using (1.4) we can derive the field components of the TM field.
E. ""r- 1
54
(2.2 )
Ec; ==1 !J.
r(2.3 )
(2.4)
(2.5)
Ht! = co.! (2.6 )
The boundary conditions for z
E = 0r
o and z = t 2 are
(2.7)
(2.8)
These conditions are satisfied in both cases.
The TE fields can be derived from the generating function
(2.9)
55
Using (1.1) we obtain for the field components of the TE field:
0- : In fier) $IJ, n¢"., ii 2
= or/;,t 2
E?i,(h (,Ie r)
COJn~;M ;; z:
= f/ndr t2.
; = 0
A~I ;n7l" air''' fk~ r)
nfji;"., liZ
= C'O,{ CO.fjt-Yr-o t z dr ~2.
/-I~~ "., h J-) (h IJ:~ r) .$i'n ;,~ C-D.J'
;0" ;;- Z::: -./ t-Yr-_ C% r ~.2.
(2.10)
(2.11)
(2.13)
(2.14)
(2.15)
We see that the boundary conditions (2.8) at z = 0 and z = t 2 are satisfied
again.
For waves propagating in the positive r-direction we have to take
(2.16 )
and for waves propagating in the negative r-direction
(2.17)
H1 and It- are Hankel functionsn n
J (k r) is a Bessel functionn c
N (k r) is a Neumann functionn c
From the expressions (2.2) to (2.15) incl. we conclude that the
dominant mode is the TM mode with the components Ez ' H; and Hr'
For the dominant mode we find k = k.c
56
If we choose tz < ....3-2 , then only the dominant mode can propagate.
Under these conditions and using (2.16) and (2.17) we finally obtain
for the field components.
a. In region I.
(2.18)
b. In region III
£2. = / /lor ~ rkr) ,.. .t'96 N h r.irJ/ coor;'7~
f-? =: n f19.r ~ rk.r) -I' R, ~ (4rJj ~""'n ;-r ~~.~,..-.-
H .. =: k r.?J,r ~ )//.:.;-) 7' ,.q~ ~ ~..I:r)lco.rH~.,. /.~,.,.,.,<I L -5'
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
A3
, A4
, A5
and A6
are constants.
The prime in J ' and N ' means differentiating with respect to the argumentn nkr.
57
2.2. The dispersion equation
2.2.1.'rhe case IJA < 1
We Shall first write again the expressions for the field components of
the hybrid field of part II of the waveguide.
These are (1.5) to (1.10) incl.
/J
(2.24 )
(2.25)
E ==°2 (2.26 )
: .8z A41'~<""# s,"n H,t>
~ ~:(kc;-;J~
(2.27)
(2.28)
(2.29)
We now need the boundary conditions.
58
They are:
1- E = 0 at r = b1 and r = b2z
The results are (from 2.18 and 2.21)
~+ h'3J; rA:. 6~)
a. :: - (2.30 ).11.1,., (J: 6,)
b. ,tt, ,c}.r..7n (''" 6a,J
(2.31)-A..I~(I.: ~)
2. E = 0 at r = a and r = a2; 1
These conditions give, from (2.25),
3. We have to match the transverse field components H~
of part II and of the grooves at the boundaries r =
Thus, H~ is continuous at r = a 1 and r = a2•
The results are:
and Ez
a 1 and r = a2 •
(2.36 )
(2.37)
59
4. Ez is continuous at r = a1 and r = a 2•
This gives:
Elimination of A3 , A4 , A5 and A6 from equations (2.30) to (2.39)
inc!. gives us the dispersion equation in matrix form.
ror a non-trivial solution the determinant of the following matrix
must be equal to zero.
(2.38)
(2.39)
I<G~};g I? kc. A.4) !L !J. N,
WE; ;;;:J, wE. Q, f
417 k.c c;rt} ~lZAI, 1<(. q(.2)TQ~, Ie 4,
::::0 (2.40)
J ~ !2. J k.c N./ !!...nlJkc~ iAJ £. 4.l 2- ~£o eta :.z
/JJ?.z 1:(. C; (J),4/7
J. c r/+J I4' 4; 2. :;: 4;~
where)
Go.J¢'. J,) - ~ ...£.2G{I} :=
k. ' wE. e, '1
}
G,(2) ~·M)- ~ c, N.- Ie. '1 (.,AJ£o C, I
)
~J,) - 2s.. t:.z.z4(3) - 2.- k 2. Go.J£ 0 C2
}
C7 ('IJ wrw_ AI J ~ Cz. ~(2.41),. z - c...J ~o Czk
C, -= .7~ ,J:.o,j - 4 AI" (.t. q,j
60
C2.. = .z, (.tqa) - .D3, A/" (A: az.)
J } JC f .::: J; (J:q, ) - ~ AI", (1:.Q,)
C2 ) = ..h,) fkq.t) -)
.2)2 M, fiQz )
~~ f'k/',)
:=
~(J: b~)
4 ..... J:, (~6.2.)(2.41
A4 (J:.6~)
..~ ~ (i: c 4,) .z} .J.... , =.z (k~Q,)J
~ /~&q~)} }M '=" ~ = 4 (keQ,)
J
~ ~) = )
= ~ I'.I:(' ".I) ~ (kc Q~).I
~ I~C4z.)) 4. )(J~a~~ = A!.? ='
J
The dispersion equation can also be written in the follo~ing fo ;'j(.
Det A = 0 gives
where
FfI) ::;:(.2) =F (3) =:
;r:(7') ==
T (.r) =:
'"7' )A I ) _ M:Z)J, /~ ,:2(0 ~ - M~)(A/~~ - ~ J;);Vj~ ) - .z ~)M~ - Z.J~
~~ "0.JZ
(2.42
(2.43)
61
The relations (2.41) are still valid.
The equation (2.42) has been solved numerically for several values of
the parameters a 1 , a2 , b1 , b2, k and n = 1.
2.2.2. The case P/k = 0
At cut-off ~/k = 0, k = k.c'rhe matrix (2.40) can be written in the following way
~.7,J 0 ~~J 0
0 kqfl) 0 ~4(2)
~A{I= 0 (2.44)
.t~J 0 0
0 J: qf:J) 0 ~ 4(~)
If k = k then the expressions in the matrix 2.44 are the same as inc
equations(2.41).
Thus, J 1' = I n (kca1) ~Jn(ka1)" etc.
Working out yields:
The expressions (J2 I N2 - N21~2) and (Nt 'J1 - J11N1) are non-zero
(Wronskian).
Thus we only have to solve:
(2.45)
o (2.46 )
and
~ ri:.6~) Mnrk6,) - ~ (k.h~) N.., (1<62 ) = o (2.47)
We know that the condition (2.46) gives the cut-off frequency gf
TE mode in coaxial waveguide with perfectly conducting boundariesIUD
at r = a 1 and r = a2• The equation (2.47) gives the cut-off frequency
of the TM mode in a perfectly conducting coaxial waveguide withIUD
radius b1 and b2, respectively.
62
2.2.3. The case (J/k>l
For the propagation in part II of the waveguide the same expressions
for the electromagnetic fields are valid as in the case of anisotropic
boundary conditions.
We write the equations (1.36) to (1.41) inc 1. again.
(2.48)
(2.49 )
(2.50)
= - .fr...!!.../ 1 ,..",..-.
(2.51)
(2.52)
where
(2.53 )
)
~ =:----
" /(~L_.2_,----( _
63
A1 " A2 ', B1 ' and B2 ' are constants.
I and K are the modified Bessel functions.n n
The field components in a groove have not changed. Thus, the
expressions (2.18) to (2.23) incl. are still valid.
The boundary conditions are
i. Ez
= 0 at b1
and b2
, respectively
2~ Et' = 0 at a1 and a2
, respectively
3. H~ is continuous at a1 and a 2, respectively
4. E is continuous at a1 and a 2 , respectivelyz
Applying the boundary conditions we obtain the dispersion equation in matrix
form:
J,.'r J1.3 .!?-r 1:.,) K/ -1 ;: kll
4:. -/ 1M ~. Q, -f "J~::>
4hZ ~: /-Iff) 4" k, kc.) 1-1(2.)--:[ i; I IL '"-.= [1 (2.54)
It. J 7 ) /J - /<.)/<) /J !2. k z_ !J,.1.2-Co 2 wC.., -.1 ,..., l. 'I" 42,
/in- K,' f{3) 11" k<, J J-f,)--I -; 11.1. K.2Ie "z-2
w;i. th
,k,',
1-1 (I)W~o#I z,J- ~ ~I- " we.. C, f
t.v,u .. k/ - i,' co' Kf1/(2) - -= c,"'- (,oJ £0(2.55 )
/-I(J)w,-. _J Kc ' CJ,.)
I"=
.L,2W€o Cz. 2
It, ')
1-/(1,0)W,A:f· 1<.)- C.l. I<z.<2
k 3 c.v £.~ Cz
64
c~::: .I;,f,io,) -
)C, =I;, (kQL)
7)-JI (kp,) -
J
In f1<.4~) -
.z. (,J in)
KJI (K6,)
1;, (~6~)
KI>t (4~)
K .. (.(:112.)
)
k,., (./:. 0,)}
I<h (~Q~)
) ) ..z; == In (It~~,)
) ) J
1<, = K .. (k<Q#)
(2.55 )
The primes in I I and K ' mean differentiating with respect to then n
argument.
65
2.2.4. The solution of the dispersion equation
The dispersion equation has been solved.
The following table gives values of a1, a2 , b1 and b2 •
ThE" solutions for n = 1 are given in figs. -2.2 - 2.7.
Antenna a 1 a2 a2/a1 b 1 b2
1 22.5 45.0 2 13.5 54.0
2 22.5 67.5 3 13.5 76.5
3 22.5 90.0 4 13.5 99.0 nan
4 66.0 132.0 2 57.0 141.0
5 44.0 132.0 3 35.0 141.0
() 33.0 132.0 4 24.0 141.0
We notice that the curve of the dispersion relation has several
branches.
It has been shown that for the case of l3/k = 0 we find two different.
solutions.
These relations are
= 0
and
(2.57)
The m()cJ.e of which the cut-jlffisdetennined by (2.56) is call-ed. the HE~) made.
The other mode, which corresponds with expression (2.57), is the HE(l) mode.nm.
From the ~/k versus 2a1/A curves we conclude that slow and fast waves are
p()ssible.
The branch of the HEi~) mode and of the HE~~) mode crosses the line ~/k = 1.
//
2/I
//
//
a2/a1 = 2
n = t
at = 2.25 em
at - b i = b2 - a2 = 0.9 em
lIE(2)1m
-- - - - lIEU)1m
2
- - - - =- -== --==-~--=-~--=- ~ - -------/'
//I
1J
J
IIIIII
1.3
12
1.1fJ/k SLOW WAVES
1.0FAST WAVES
0.9
0.8
0.7
0.6
0.5
0.4
OJ
0.2
0.1
0
Fig. 2.2
----
2
2;
---~---=-~ "=----====- = -= -=-~----~~.---
~
//
1/IIIIIII
1
1.3r------------------;-~-----~~--
12(2) a2/al = 3
----HE1m n = 1
- - --- HE(l) a l = 2.25 em1m
a l - bl = b2 - a 2 = 0.9 em
0.9
0.8
0.7
0.6
0.5
0.4
OJ
0.2
OJo~_"___L____J..._..1._L____JL_ _--.1..1"---
1
1.1rJ/k
1.0 SLOW ~S_
FASf WAVES
Fig. 2.3
2
II
1.31--------~---____;-----~----12 a2/al = 4____ HE(2)
1m n = 1
rJ/k 1.1 -- - - - HE~;) a l = 2.25 em
SLOW WAVES a l - bl = b2 - a2 = 0.9 em
1.0 FA-sr-WA-VE-S - - - - -==-=---=---==----~~~-=---=-- ......-.-._----- ----0.9 ~
1/ ----0.8 ;/ 2 /~
/2/ 3 /
0.7 I0.6 I //
I /0.5 I I0.4 I I0.3 I I
I I0.2 I I0.1 I I
I Io J-.-_---L..L._.!-..--1L.....!.....~J.L...-.----l.-.-.-.~,.----L----l---..l..-.----L----l..---....J.......------J--.J
Fig. 2.4
1.3 ~-
12 HE(2) a 2/a1 = 21m n = 1
1.1______ HE(1) a2 = 13.2 em
rl/k1m
1.0 --a 1 - b1 = b2 - a2 = 0.9 em
------ - ~----=...--=--=----- - - - - ---0.9 -- --...-- ----
./" --1( -
0.8 -------I 2 ./
0.7I
2/ 0)
7co
0.6 /IJ
/0.5 I j
I I0.4 I IIOJ I f
0.2 I II I
0.1 I I0
1 2 3 4 5Fig. 2.5
2a1 /)..
2
---------,/'
/i
//
I
/II
1.31-----------~
12
-----~a:-21./a:-1-=~3;:--~--------_____ HE(2)
1m n = 1
1.1 -- - - - HE(1) a2 = 13.2 em~/k 1m1.0 SLO.!J!AVE_S _ a 1 - b 1 = b2 - a2 = 0.9 em.
--------FAST WAVES -- - _.- -=---=.-.---- -
0.9 1 /---
0.8 ;/
0.7 /
0.6 I0.5 I
I0.4I
0.3 I
0.2 I0.1 Io-_..........._-L-.L....L_-r-L-.-L~_..l-
Fig. 2.6
2
a2 = 13.2 em
a 1 - b1 = b2 - a = 0.9 em
-4------ 2! ..,.,.- --- -- - - - -~ ...=:::=:=' -=" .=..~~ ---- - -
/1 --- ---.--1/ I ___
/~ ,-/"
2// /
// V/ /I I/ I/ II II I
SLOW WAVESFASfWAVES
1
1.31---~-""!-------------;"~-----12
(2) a2/a1 = 4_____ HE1m n = 1
l1/k 1.1 - -- - -HE~)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
OJ
0.2
0.1
O.&--....I.-....L....J..~--L-...I..-.---r-L-_L-.-
12
This implies that there is a change from fast waves ('%. <. f ) into
slow waves (~ > 1 ).If the depth of the grooves is a quarter of a wave length (H = 0),
the branch of the HEi~) mode crosses the curve of the HEi;) :ode.
for this case we may write that
and
For all other values of the frequency more general relation is
used
~ = D'1 ZoA2 , B1 = "'2 ZoB2 and B2/~ = DC
The values of "1' DC. 2 and CIG. can be computed from relation (2.40).
Thi s is given in some detai I in A.ppendix A.
Finally we observe that if H. = 0, the matrix (2.40) can be written
in a simple form as was done in chapter I for equations (1.32.b) or
(1.33.b.).
2.3. The transverse fields, n 1
In this section we want to investigate the transverse fields of a
corrugated coaxial waveguide.
We are interested in the behaviour of the field components of the
HE(1)and HE(2) modes as a function of the frequency, especially of1m (1) 1m (2)
the HEU and HE12 mode.
First we assume that
These expressions are given in Appendix A.
In the same way as in the preceding section we find for the
transverse fields in the case of IJA, <: f for a singular 9 dependence:
(2.58 )
(2.59)
73
with
+
(2.60)
(2.61)
(2.62)
(2.63 )
(2.64)
(2.65)
(2.66 )
(2.67)
(2.68)
(2.69)
(2.70)
and
We define
a. = Jk. r If _ ('%.)'2.
G,..1:,..1"") = I G'rl,,"4xJ I
a.nd
E~rht)£,p
= l--e~ tWIn) I
(2.71)
(2.72)
Er(rel) and E~(rel) has been computed for several types of antemlas
as a function of frequency.
We observe that if the boundary condition H; = 0 (depth of the
grooves is approximately ~ ) is sati sfied, than"'1 = "'2 = 1-
In that case the expressions (2.59) to (2.62) incl. are equal to the
expressions (1.47) to (1.50) incl. or to the equations (1.55) to
(1.58) incl. We conclude that we can find a common frequency value where
the HEi:) and HEi:) modes and also the HE~) and HE~;) modes have the
SWMe properties.
74
On the following pages are plotted some Er(rel) and E~(rel) curves,
We observe that changing from the boundary H~ = 0 to H" '1= 0 means a
larger value of Er(rel) at the boundaries r = at and r = a2 ,
The same effect has been found by Roumen /~/ for the corrugated ~ircular
waveguide.
TRANSVERSE FIELDS
Q)
~2w
a2/a1 = 2.625
a2 = 6 em
a 1 ~ b1 = b2 - ~=gmHE( ) - mode
119.5 GHz
1
o
oa,
- - - - --~-e:::t~
23'" 5 6 7 8rho
1
o
oa,
Fig. 2.8
2 3 4 5 6 7 8 9 10rho. a2
TRANSVERSE FIELDS
(1)
~2w
82/81 = 2.625
8 2 = 6 C2Il
8 1-b1 = b2 - 8 2 = 0.9 em
10.0 GHz
HE(1) _ mode11
1 -----
a
- - - - - =-----=::::t- - -- - -1
a
012345i6a1
7 8 9 10rho a2
a 1a1
2 3 4 5 6 7 8 9 10rho a2
Fig. 2.9
TRANSVERSE FIELDS
-~2~
LJJ
1 - - - - -- ===---=::r- - ~---
-~Q)
~2w
1
a2/a1 == 2.65
a2
= 6 em
a 1-b1 == b2 - a2 == 0.9 em
HE1~) - mode
10.5 GHz
0 0I I I I I • • I
0 1 2 3 4 5 6 7 8 10 0 1 2 "'I 4 5 6 7 8 9 10.)
a1 rho a2 a1 rho a2Fig. 2.10
TRANSVERSE FIELDS
2 3 4 5 6 7 8 9 10rho a2
o
-~<U
~1w
0__J,--J
2 3 4 5 6 7 B 9 10 0 1rho a2 a,
Fig. 2.11
a2/a1 = 2
~ = 13.2 em
HE(2) _ mode12
9.59 GHz
TRANSVERSE FIELDS
o
2 3 4 5 6 7 8 9 10rho a 2
HE(2) _ mode12
9.87 GHz
a2/a1 = 2
a 2 = 13.2 em
oa1
2 3 4 5 6 7 8 9 10rho a2
oa1
o
Fig. 2.12
TRANSVERSE FIELDS
00o
HE~;) - mode10.18 GHz
a2/a1 = 2
a 2 = 13.2 em(l)
L- 1L:::W
-----~-- ....--....::--
0 0I I I I , I i ' I , ....J.........-.J I I I ~
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10a, rho a2 a, rho a2
Fig. 2.13
TRANSVERSE FIELDS
HE12 - mode
10.52 GHz
a2/a1 = 2
a 2 = 13.2 em
---_.~~,-------
0
.......L-...-L. , , I I -0 2 3 4 5 '6 7 8 9 10 0 2 3 4 5 6 7 8 9 10a1 rho a2 a, rho a2
Fig. 2.14
TRANSVERSE FIELDS
a2/a1 = 2
a 2 = 13.2 em
HE12 - mode
10.91 GHz
2 3 4 5 6 7 8 9 10rho a2
Q)
~1w
0 0
..L-J0 2 3 4 5 :6 7 8 9 10 0a, rho a2 a,
Fig. 2.15
83
2.4. ~hc power radiation pattern of a corrusated coaxial waveguide
x
a
//
z.-
Fig. 2.16
.- - ---.p
The power radiation pattern will be calculated by the Kirchhoff-Huygens
method.
First we assume that the aperture is an equiphase plane.
The expressions for Es and Ed are~ 2r .,
E61 = t":,... e -/",..//[f0- .,. Z ~ COS8) COS (¢J_¢) "I-
a, 0
I 7 I~ r .s"-" iii c.O~ I"~ ~- t;J) -/A"" ~7""(£9' -.4 HrCO.fOl) J/~ f?}-P); e tf? r d'r d¢
(2.73)
84
ilia :lIT"
e _/Arj!/iE; ~,u8 _ 2 0 h'r)c,:,,rft/J)- ,p) -
ii, 0
with
iV-
For the calculation of the electric field in the far-zone region,
in the case of constant phase distribution it is allowed to neglect_jwr2
the factor e •
Assuming the singular I dependence we write down the expressions for
aperture fields (~ ~ f)
E. :=- Ke:2 Z o [ It (ker) +- ()(, ~ (~er)JCOJ';.r
Er} .1:''1 z.[ p (ker) ()(. ~ (~er!l Slh?:: r
2 0 1-1,.. ;; - .l:e ~2 Zo[J'~ (i:~r) +- c<; /2. (k, r!l .r/h rP
ZoHtP~ /(~ /9 2 Zi ex. j'¥ {~';?l COS l'2 73 (k.e,r) +-
with
J.t1 ) ..zr~~r)11-----
;: (~r) :- (et, + 7- (a, - J;"")..7; rokcr)
(#'.2. r-/] (ot2 - : ) AJ2. (ke "')
~ (~er) = k ) Nofi:c r) .;.
/J /.l
(3 (.l:er) = (d- f +T ) hiker} - (O<., - ~) J; (,ic r )
/.J (7Ilf fier) .:: (dz -I- T) No (i,r)- (o<.z - :r) A.J:2, (/.c,r)
(2.75)
(2.76)
(2.77)
(2.78)
(2.79)
85
(2.79)
where
(2.80)
Using the relations (1.77) to (1.80) incl. we find for E and E., fA
after some calculation:
4z.
£tp = -~2.2o kC"CO.$~lffif(I'" ,,11 ~.LapJ:.(i:.rJI""&)-~~(i:I'.J/I'7r:!}~4,
.". (1 -I- Cos &)Ix Yo (J:r.J/~ s) +- .h ~(k.r.t/n,;;Jyf i"" elr
~ J
- 8~ 2 ~ I<. 7J cosI'I!". (f. : GdS"1,y~ :J. (~m rl~) - ""';:;;, (k~S;rl~0,
oilc/J + co,r 61)[No .7c.rlc.,..riH&) ~ ~ ~(k"J,.·H :};/~. rc/r
Qz
tEl> = 192 Z ~ kc 17""1 'I(Dt, (/ ~ COS 61J/.70 7"rbS'rl '") + J. .:t, (A r","~
86
~2
+ .82 2 0 ~cT.r,,,,p,) I !~z. (l!'" c<)J~)/"A.{, Jork.,...o"6)..;; ~ .J;,r.t"',J',"61!l~
,. If". 1- =, ..) /A/•.7..1',iN'·~"J - ~ :{(k. n,-,. # "dr
(2.81)
where
The integral of the form / ~ ("ruN) ~ /)/)() x I¥'x
can be written in following way /17/
Using this relation we can write the functions E. and E~ in aclosed form.
The results are
EfP :::: - ,fC}2 2'0 I:c7l'COJ~' f 4, /1' ~ 1 £.OJ e1)L ..F (~J - IH/)~
- -JJ.l L o ~c1J'co.r¢~! tJ(.2 (I.j. 1COs e)/.1 (J):- =t (.2?l~
"" Ix r ~~~ / -L (1) ". L-R!J} (2.82 )
~ = ~ 2 0 i c 7T j''h$d! tX, /1 ~ t:o.rt9J/Iff) .,.. It'l!?
~ (I r- 'i Cd.rc9)/.LI?) _ It/!/} ~
" 3.2 Zc "-.- c''''''1e(. (: ' tAO.9) / T I:J) r Z(?'-Y.,l
... (.-... 1 «>s ,g)/ ..l/3) - £(.2$ (2.83)
87
with
I (.2) :::
z (.1) =
q~
/ Z (kcr) .J; (krJI''''W) r dr0,
I"~ rl,,) J; rb",·"",) ,d,.~1
/ ". A/,. rJ. ,..) :T.I'b .N""J ,. d,..
(2.84)
(2.85)
(2.86 )
1;.,q,
I (+) = / :/0(1<_ 11-) J: (kr.r,'~.)ra'_
tI.,
(2.87)
We observe that the expressions (2.83) and (2.82) E9
and E;
are equal only if d. 1 =d, 2 = 1 or tlG1 = «'2 = -1.
This condition is satisfied if H~ = E; = O. In that case the
equations (2.82) and (2.83) are equal and the radiation pattern
is symmetricaL
Further we define the radiated power as a function of 9 relative to
the value on the axis of the waveguide.
in the E plane P = 2010 log! Eli} (~' el, 0) /
E l E ~ (r,' 0, 0)
10 I E ~ (,.' &~ ~) Iin the H plane PH = 20 log 1:1
E.- (r/ o~ ~~)
The functions PE and PH give the power radiation pattern in the
E plane and in the H plane, respectively.
T-h;- values of i ,~i~h~re necessary for the calculation
of PE
and PH were already computed in the preceding section.
The expressions for «. l' cf. 2 and Df. can be found in Appendix A.
The results, which are given in figs.2~7-2.~have been computed(2)for the HE12 mode.
(2.88 )
(2.89 )
88
6045angle (degrJ
-- - - H planea 2!a
1= 2 a2 = 13.2 em
_____ E plane
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKS
\\\\\\\~
//IIIIII
---- -
IIII
I,,I
f = 9.59 GHz
30Fig. 2.17
"'-,I \
/ \
I \I \I \I \I \
I I \I I ,I I \I I \
\\ I \\ I I\
I
\
I
15
~
'\\\\\\\
\\ (\\ I \
\ f \\ I \
\ : \\ \ I \
\ I
~ I\
\\I11II\1
l~
~
I1
I)
10
)
•oJ
20
30
o
u>>):>..
.....o:J--
89
-----------------1::r~;;~--____;;DATEPATIERN NO.
PROJECT
ENGRS.
REMARKS
E planeI H plane
2 a = 13.2 ema2/a1 ~ 2
Fig. 2.18
f - 9.87 GHz
30
........"I \10 I \I
.....I (0I (:J.-I
I11,)
\ I>I I>
::>I~
IIJ I I>
\ II:;tl
~ IJ
J
20 \I\I\
90
PAITERN NO. DATE
PROJECT
ENGRS.
REMARKS
f = 10.18 GHz
E plane
---- - H planea2/a1 ~ 2 a2 = 13.2 an
6045angle (degr)
30
Fig. 2.19
15
\
o
10
)
L
i\I
20 1III
\
\
\
III~
30
91
60
E plane
H planea2 = 13.2 em
45angle (degrJ
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKS
I I, II
30
f = 10.52 GHz
,,-,\
I \I \
\\\\\\\\\\
\\
15
, ....I \I \
I \I \I \
I \\
I I \I I \\ I \
\ I \\ I \\ I I\ I \ I
I I fI \ I
II I II I ', I II
II III \~ \I\ If
~I
o
20
30
10
Fig. 2.20
92
DATEPATTERN NO.
PROJECT~ENGRS.\
REMARKS
,\
E planeI
H plane
If = 10.91 GHz
a 2/a1 = 2 a = 13.2 em
\
2\\
\,\
10 \I -,I f \....I I \
Q:J \ I \...
I I \I I \u
I>I I \>I IJ
\:l... II I
\I I -"u
\ I \>I,
I \::;
i i \ I'0
\,.,
II\ I
l)
\20 I I {
\I I III\ I
,~ I I \I I I \,
I I \I I \ -"\ I
I ,
\ I ,I I ,\I \ I __ j1I \I
\I\30
I\I I\,
LI I
\I,\I,\II
45 6030angle (degr.>
150
Fig. 2.21
aperture.
3
2.5. The po~r radiation pattern of a conical coaxial corrugated
horn with small flare angle
Fig. 2.22
If we restrict our considerations to conical horn antennas
wi th flare angle OIl 2 < 15 0, the theory di scussed in the preceding
chapter can be used.
In this case we assume a quadratic phase distribution across the
_/..,,, zThe expressions (2.73) and (2.74) are still valid, but the term ~
cannot be neglected.
Thus, the expressions for E1 and Es are
Oa 2fT
e -/",-".1//(EjP Col'& - 21:> ~) ?oJ(t,i'- ¢) 7"
q, "
. z-.I .... r
rPf'~d¢
(2.90)---------------------
94
q.. 2/T
E 2 .L£ e -./-,r/ / [fEr ~ Zo#; ~s&) CAf (~)-") ~t9 ~1?r' / L I
q~ Q
If we suppose a quadratic phase distribution, the expression
for w is given by / 1/
kw=kd+-2r '
We assume again that the term ~r,in this expression is negligible.
For the term kd, the following relation is valid
In this case it is not possible to write the functions PE and PH'
Which represent the power radiation pattern in the E plane and
the H plane in a closed form. The functions PE and PH can be found
~ applying the method of numerical integration.
Thus, in this case, we write for the electric field components Ee
and E~
tIa.
£9. E~ (r/ 6J~ f/J) = -,I1z 2.,. kc; 11" cos.p ffi, ((totC<J.I CJto :1.,('" ,.,//1'1 s) -
- y. :t.(~'~'''''!f f1 • «0.<.0)/x :r.,;~N , .. '") l' ~ .7.. r"~~'H:I-iN';. >'4,.-_
0l.
- 3z
zQ kc7lCC~.p).Jld. z / ~r'- ! ~~g?)I AID .7'o(~N'·HJI) -~ .:'z("'n~·;t~
..< (/..<- '0';.0;~07.. (inIH<9) ,,~~ r.h.riH",#e ;iNr~ dr
(2.92)
95
"aEf/) Ie E; (,.,' t9, t;') #: ~zo ~c T J'/h¢!14,(t l' co.rgpJ:,rJn""ti1j"'.:tJ;(.1:"',r,,,~
III,
with
J = J (k r)o 0 c
N = N (k r)o 0 c
J 2 = J 2 (kcr)
N2 = N2 (kcr)
The power radiation pattern can be found from:
in the E plane
10 /Es(rl
, Q, 0))p = 20 log ------E E,,(r', 0,0)
in the H plane
in dB
in dB
(2.94)
(2.95)
96
45 60angle (degr.)
----- H planea 2/a1 = 2 a2 = 13.2 em
-- E plane
PATIERN NO. DATE
PROJECT
ENGRS.
REMARKS
30
f = 9.59 GHz
15
(\f \
\ I \\ I \I I l
\ II II ,\-II I\III\
r\
/ \I \III
o
10
20
30
Fig. 2.23
97
98
PATTERN NO. DATE
PROJECT
ENGRS.
REMARKS
E plane
f = 10.18 GHz ---- Hplanea2/a1 = 2 a 2 = 13.2 em
10 {rll
~i
Ii
~~
I ~ II, I I20
. ~\ I
I I
I }I r
\II
30 II III
\I I
0 15 30 45 60angle (degr.)
Fig. 2.25
99
DATEPATTERN NO.
PROJECT
ENGRS.
REMARKS
E plane-~ H planef == 10.52 Gllz
a /a =2 a = 13.2 em22 1,I
10 \(\I.....
\I \m
"CI \- .IL..
\
~Q)
5\
I0~
IClJ\>
;:::;
\ I~oJ
1)-I20
\
I II I\ IIIlJ
\30 II
\I II ILII45 60
angle (degr.)150
Fig. 2.26
100
2.6. The experimental investigation
In section 2.2 it has been shown that the HE(i~ mode produces a
symmetrical radiation pattern. This implies that for the experimental
investigation of this mode the launching of the TE12 mode in a
coaxial waveguide with perfectly conducting boundaries will
probably give the best results.
However, the solution of the dispersion equation in the case
where the depth of the grooves is not a quarter of a wavelength,
gives us the possibility of investigating the HE~~) mode.
In this case the mode in the circular waveguide is the TEll mo~e.
The picture of the X -band model of this antenna is shown in fig. 2.27.
The dimensions of the measured antenna are given in the following
table
69.0 mm
Some results of these measurements are given in figs. 2.28 - 2.30.
However, in this case, too, the coupling between the cir~ular
waveguide and the corrugations is a difficult task.
'l'he--4ime-nBi~ in- regiens Iand--ff{figs. 2.28 - ~3&t-areto be choosen
very carefully. A good agreement has been found only in the
small frequency band. In the large frequency band the excitation
of higher modes is possible which makes the agreement between the
calculations and the measured results unsatisfactory.
101
Another investigation has been carried out with corrugated
coaxial horns with wide flare angle.(Photographs 2.1 and 2.2~
A good broadband behaviour has been observed in the frequency
band 1.5 - 17 OOz.
Probably because of the excitation of the TEll there is
non-symmetry in the E and H planes in the region between
a = ~ 250
(figs. 2.31 - 2.33).
From the experimental results we may conclude that the coupling
between the coaxial waveguide and corrugated section will be
more difficult than in the case of the corrugated conical horn.
If we assume a quadratic phase distribution across the aperture
(wide angle, or large aperture), then good results in a relatively
large frequency band can be expected.
x - Band - Model
IT
Fig. 2.27
H- PLANE
!.. '" 9.5 GHz
experiment- - - - calculated
60 90angle (degr:)
30
\\\\I\ (\I J \
I I \\ I \I I
\/ \~ \
\\\ /'-\
\
____.....1......-_........\ -L-.../_.i-\_\L...--__~_~120
0 0E- PLANE
f = 9.5 GHz
experiment---- calculated
.........(]) .........
CD~10 ~10L.. L..Q) Q)
~ ~0 0a. Cl.
Q) <LJ> >;0 +-'ro~
-..J
Q)L.. L..
20 20IIlIII
30 I \ 30\\
"\0 30 60 90 120 0angle (degr.)
Fig. 2.28
120
H- PLANE
f == 10.0 GHz
-- experiment- -- calculated
\60 90
angte (degr.)30
\\\
\
\/'"\
I \\ I \i I \I I \, I \I I \\I \II \II \
O~-------------,...---------I
30
20
L..Q)
~o0.
Q)
>
-en~10
Fig. 2.29
120 0
\
\
\\
E - PLANE
f == 10.0 GHz
experiment- --- calculated
/-I \
\\\\
\
60 90angle (degr.)
\\\\\\\\ I\ II I\I\ I
I'
~
\\\\\
\
\
\I\ /
II~
30
~
~\
~ I\ I\ I\ I\ ,\I\
I
o
Qrre:.-------------,--------,
20
30
en~10
-L..Q)
~o0.
Q)>
19Q)L..
120
H- PLANEf =' 10.5 GP."z
experiment-- - - - calculated
60 90angle (degr.)
30
\
\
\
I /\I / \
I I \
I ,I \I \
1/ \II \
!I \II \II I
o~...---------~----------,
30
20
L..<lJ
~o0-
<lJ>
......rtI---Q)L..
CD~10
120 0Fig. 2.30
E- PLANEf =' 10.5 GHz
experiment-- - - calculated
60 90angle (degr.)
r\/ \I \
\\\\\\
30
\\\\ r\
\ / \\ I\ \\ I \II \\' \
II \~ \
\ I1 I\ II J
II
o
O~---------r-'----------.
30
20
CD~10
120
f = 15 GHz
H- PLANE
60 90angle (degr.)
30
Or-----......,-=::--------r---------r
30
Fig. 2.31
120 0
f = 15 GHz
E - PLANE
60 90angle (degr.)
30o
O..----~:::-------_r_------__.
30
-ro -ro~10 :g10L-
L-Q)<LJ
5 50 00. 0.
<LJ (1J> >
19 - .....~ 0Q)Q) (Xl
L..L-
20 20
120
f = 16 ~
H- PLANE
60 90angle (degr:)
30
o~--~~-----~---------.
30
Fig. 2.32
120 0
:r = 16 GIIz
E- PLANE
60 90angle (degr.)
30o
Or------,c::------~------_.,
30
-(l) -(l)~10 :g10
L.. L..OJ OJ~ ~0 0c- o.OJ OJ> >
...- ...-I-".!9 ro 0~ coOJ OJL...
L..
20 20
120
f == 17 GHz
H- PLANE
60 90angle (degr.)
30
30
Fig. 2.33
120 0
f == 17 GHz
E - PLANE
30o
Or----~~----~------__.
30
- -CD~10
L..L..Q) Q)s s
0 0a. a.Q) Q)> >
+-' .....1li ctl .....--' --' 0ClJ Q)L..
L..
20 20
111
Conclusions Ilnd programme for future work ••
It has been shown that in a corrugated waveguide the propagation
of two types of hybrid modes is possible. Both types of modes
give a symmetrical radiation pattern.
This phenomenon does also exist in the circular waveguide with the
same boundary conditions.
For n = 1, i.e. a singularf-dependence, the first two modes
which give a satisfactory radiation pattern are the HE~~) and HEg)modes. The HEi~) mode produces a maximum of radiated power along
the guide axis. It should be noted that these modes are higher
modes, whereas the HE(~i mode is the dominant mode; the first higher
mode is the HEi~)mode.The dispersion curves of the HEi~) mode cross the line ~/k = 1.
This implies that the balanced hybrid conditions for this mode
occur in the slow wave region.
From the results which have been found in chapter II we may
conclude that the coaxial corrugated antennas with small flare
angle are essentially narrow-band feeds.
To obtain good radiation properties in the broad frequency band,
the dimensions of the aperture should be very large.
It has been shown experimentally that corrugated coaxial feeds
with wide flare angle have satisfactory broadband properties.
In this case the measured bandwidth was about 20%.
This implies that theoretical analysis of spherical hybrid modes
propagation in a coaxial corrugated conical horn is necessary.------ ~- ~--
The results foriarge a;?a~~~e certainly promising; this problem
requires some theoretical and experimental investigation.
~rom the calculated radiation pattern we conclude that a coaxial
corrugated antenna is not suitable as part of dual frequency feed.
High sidelobe level reduces the aperture efficiency to about 60%.
However, the multimode operation in a coaxial horn is still
possible. It would be worthwhile to make some theoretical and
experimental investigation on this subject.
The properties of corrugated coaxial waveguides imply that there is
a possibility of developing a hybrid-mode feed of the same
112
geometry as has already been done by Koch 16 I. However
in our case we need only two hybrid modes. And also, what
is more important, for the same field distribution in the
aperture of the coaxial part of the feed we need only one
hybrid mode. Such a feed should probably have better broadband
properties than the system where four modes are used. This
problem should also be investigated theoretically, in particular
the maximum of the aperture efficiency to be achieved and the
bandwith in so-called balanced hybrid conditions.
The essential point in using coaxial corrugated feeds is th~
coupling between the uncorrugated waveguide and the corru~ated
region.
The optimal dimensions of the grooves for the inner and the
outer conductor can be computed from expressions for Cl' and
C2 ', respectively (equations 2.41 page 60).
113
Acknowlediements
The author appreciates the assistance of Mr. I. Ongers for having
wriiten some of the computer programmes.
The antennas have been constructed by Mr. R. Atema and Mr. A. Neyts.
The help of Mr. M. Knoben in the experimental investigation is also
greatly appreciated.
References
/ 1/ Jeuken, M.E.J.
/ 2/ Jeuken, M.E.J. andRoumen, H.P.J.M.
/ 3/ Clarricoats, P.J.B. andSaha, P.K.
/ 4/ B.MacA. Thomas andCooper, D.N.
/ 5/ Jeuken, M.E.J. andVokurka, V.J.
/ 6/ Koch, G.F.
/ 7/ Scheffer, H.
/ 8/ Barlow, H.M.
/ 9/ Harrington, R.F.
/10/ Ludwig, A.C.
/11/ Koch, G.F. andScheffer, H.
/12/ Thielen, H.
/13/ Seher, K.
/14/Marcuwitz, N.
/15/ Takeichi, Y.
114
Frequency independence and symmetryproperties of corrugated conical hornantennas with small flare angles.Ph.D. Thesis 1970, Eindhoven Universityof Technology, Netherlands.
Broadband corrugated conical antennaswith small flare angles. Presented atInternational $ymposium on Electromagnetic Wave Theory (DRSI) , Tbilisi 1971.
Propagation and radiation behaviourof corrugated feeds. Proc. I.E.Eovol. 118 no. 9, 1971.
Two hybrid mode feeds for radiotelescopes. NachrichtentechnischeFachberichte, Band 45 1972.
The corrugated coaxial antenna.Nachrichtentechnische Fachberichte,Band 45 1972.
A new feed for low-noise paraboloidantenna. I.E.E. Conf. Publ. no. 21.
Die Strahlung der mit H-Wellenangeregten, offenen Koaxialleittmg.AEU Band 22, 1968, Heft II, p. 514-~18.
Screened surface waves of the rlipolefamily in a coaxial waveguide.J. Phys. D: Appl. Phys., Vol. 5, 1972.
Time-harmonic electromagnetic fields.New York, 1961.
Radiation pattern synthesis for circularaperture horn antennas, I.E.E.E.,vol. A. P. 14, no. 4, July 1966.
Coaxial radiator as feed for low noiseparaboloid antennas, p. 166-173.Nachrichtentechnische Zeitschrift 1969,fIeft 3.
Mehrmoden - Koaxialerreger.Nachrichtentechnische Zeitschrift 1~71,
Heft 6, p. 307-313.
\l.ehrmoden-Koaxialerreger fur Parabolantennen mit grossem Offnungswinkel,F.T.B., Juli 1972. Darmstadt.
Waveguide Handbuok.
The ring-loaded corrugated waveguide.I.E.E.E.-G.M.T.T. symposium, 1971.
/16/ Roumen, H.P.J.M.
/17/ Jahnke I E. andEmde F. andLosch F.
/18/' S·l1 ver,. S.
/19/ Podraczky, E. andElbert, B. andMagenheim B.
115
Corrugated conical horn antennas withsmall flare angle, Eindhoven Universityof Technology, E.T.A.-17-1970.
Tafeln hoherer Funktionen,Stuttgart, 1966.
Microwave antenna theory and design,Mac Graw-Hill, New York, 1949.
Trends in earth Terminal requirements,report of the E.S.P.S., October 1971.
A-1
APPENDIX A
The dispersion equation can be written in the following form (2.40).
A1xl1 A2x12 B1x13 B2x14
A1x21 A2x22 B1x23 B2x24A = A1x31 ~x32 B1x33 B2x34
A1x41 ~x42 B1x43 B2x44
(A.1)
Where A1 , ~, B1 and B2 are constants.
For a non-trivial solution of this set of equations the
determinant of A must be zero, det A = O. This gives the
dispersion equation.
Further we define
A1 = «1Zo~ B1 = 0(2 ZoB2' o(.~ = B2
Thus we have to solveA1 B1 B2 (A.2)
"1 = Zo~, Cl(2 = ZB ' 0(= A
o 2 2
For the determination of A1, 0(2 and at we need only the first three
equations of '(A.1)
A1xl1 ~x12 B1x13 B2x14 = 0
A1x21 ~x22 B1x23 B2x24 0 (A.3)
A{X31 ~x32 Bt x33 ~x34 = -O--~-
Using Kramer's rule we can easily compute A1/A2, B1/B2 and
B2/A2 •
We know that
A1 : A2 : B1 B2 det A1 (- det ~) det B1 (-det B2)
(A.4)
A-2
where
X12 x13 x14
det A1 x22 x23 x24
x32 x33 x34
x11 x13 x14
det A2 x 21 x23 x24
x31 x33 x34
(A.5)x11 x
12x14
det B1 x21 x22 x24
x31 x32 x34
x11 x12 x13
det B2 x21 x22 x23
x31 x32 x33
The results are
, AI 'det A
1= Zo flc" Zn Ic/J (AI,' - ~c .£L N,)( Z.4
( ~ C, <:I,
(A.6 )
(A.7)
A-3
(A.8)
(A.9 )
with
J t J (k at) J ' J '(k at)n c t n c
J 2 = J (k a2
) J I J '(k a2 )n c 2 n c(A.tO)
Nt N (k at) N' = N '(k at)n c t n c
N2 N (k a2
) N' N '(k a2 )n c 2 n c
Ct In(ka t )
In(kbt )Nn(kat )- Nn(kb't)
C ' J '(ka ) -I n (kb t )
Nn ' (kat)t n t Nn
(kbt
)
A-4
We can now easily compute 0/. 1' oc.2 and 0(,.
These are given by
«-1det A1
(- det ~)
det B1~. = (- det B2) (A.H)
()(.det B2det A2
We observe that 0(;1' D(.2 and aGo are functions of the
d~ensions and of the frequency.