COMPUTER AIDED DESIGN OF STEPPED IMPEDANCE TRANSFORMERS
REALIZED IN COAXIAL TRANSMISSION LINE
Gabor MATAY and Zoltan VERES*
Department of Microwave Telecommunications Technical University of Budapest
H-1521 Budapest, Hungary email: t-matay@nov.mht.bme.hu
s-veres@nov.mht.bme.hu"
Abstract
This paper describes computer programs that have been developed to design quarter-wave and short-step impedance transformers. In the literature dealing with these transformers usually there are tables to make designing easier, but in these the fringing capacitances occurring at the steps between lines of different impedances are not taken into considera- tion. The computer programs can calculate these capacitances and compensate for. their effects. For short-step impedance transformers they can design dielectric supports as well.
These programs are able to design transformers up to ten sections in a structure where the outer conductor has constant diameter and the inner conductor is alternating. At the end of this publication the different matching structures are compared on some examples.
1. Introduction
Quarter-wave transformers are widely used to obtain an impedance match within a specified tolerance between two Hnes of different characteristic impedances over a specified frequency band.
The computer programs are able to design Chebyshev and maximally flat quarter-wave transformers consisting of up to ten sections. The meth- ods used by the programs were developed by FELDSTEJN [1]. The quarter- wave transformers are useful at microwave frequencies, but for applications involving frequencies much below 1000 MHz the size of step transformers can become impracticably large.
In this case we can use short-step impedance transformers. The com- puter program designs such transformers in two structures. One structure is of the form of a conventional low-pass filter structure. Such procedures are approximate but can give very good results if the design is worked out carefully.
The other structure is designed directly from the distributed-element impedance transformers and treated on an exact basis.
The short-step structures are realized in coaxial form, and they consist of short lengths of line sections having various impedances, where relatively
154 G. MATAY and Z. VERES
high impedance transmission lines alternate with relatively low impedance lines. The line-section lengths are 1 = >"0/16 or >"0/32. The computer program is useful for synthesizing and analyzing these quarter-wave and short-step transformers.
2. Quarter-wave Transformers
2.1. Design of Quarter-wave Transformers
The formulas to design Chebyshev and maximally fiat quarter-wave trans- formers are obtained from FELDSTEJN [1]. The transducer attenuation of Chebyshev transformers is given by
(1) where Tn(x) is the Chebyshev polynomial, h is a parameter depending on the maximum refiection coefficient in the passband, S is a scale factor,
e
is the electrical length of the section, and for maximally fiat transformers
L = 1
+
Q2 cos2ne,
(2)where Q is a factor that sets the scale of attenuation.
The formulas given by FELDSTEJN [1] are not exact but they are comparatively simple, easily programmable and give very good results by means of mutual compensation of approximations. Such transformers are useful about and over 1000 MHz frequency bands because in these cases the total length of transformers will be of favourable size. At lower RF frequencies the total size of a transformer can be several meters or even more, thus they are too long for practical realization and the short-step impedance transformers described in the next chapter give acceptable size.
Since transmission lines and impedance transformers are designed for the dominant mode, higher frequencies are limited due to the appearance of higher-order modes. Usually the use of Chebyshev transformers is more practical; we can realize them with fewer sections and shorter sizes. The maximally fiat transformers are favourable concerning phase and group delay characteristics.
2.2. Corrections for Fringing Capacitances
At the steps between lines with different impedances due to these discon- tinuities the electric and the magnetic fields become deformed and higher- order modes occur (Fig. 1).
Fig. 1.
The changes in the section are accounted for by a lumped discon- tinuity admittance shunting the lines at the junction (shunt capacitance) [2]. Using these shunt capacitances for the discontinuities of stepped trans- formers, the equivalent circuit in Fig. 3 will be obtained.
I
YI Y2 Y3 Y4Yn-I
YD YD+I
I-'- - -
==~
ifJI ifJ2 ifJ D-2 ifJ D-I
Fig. 2.
Fig. 3.
Although the changes in the section in case of quarter-wave trans- formers are smaller than in short-step transformers, we can see from thp results of analysis for the design examples, at the end of this article, that it is important to compensate their effect. Reference [3] gives corresponding procedure for compensating the fringing capacitances. The compensation
156 G. MATAY and Z. VERES
technique obtained by using scattering matrix analysis of the equivalent circuit gave good results in computing input reflection coefficient. This pro- cedure compensates for these capacitances by making small adjustments in the physical lengths of the various line sections of the impedance trans- formers.
2.3. Analysis
The computer program uses some approach formula for the synthesis proce- dure and takes advantage of mutual compensations. Corrections for fring- ing capacitances take only phase modification effects into consideration.
After having transformers designed, there is a possibility to analyse them and compare the results of analysis with the specifications. The analysis is free from approaches, takes the fringing capacitances into consideration so it can give correct results. The program calculates step by step the value of input admittances at different planes of equivalent circuit, and from this, it determines the input reflection coefficient function.
-l
~~ 2'..,
Yo-l
where
and
t
c• c~ = Zot C£':>,z .. ,
Y~,-<
Y .. -<
Fig. 4.
Y£ni(W)
=
YOiYi(W)+
jYOi tanj3l, Yi-l(W)=
Yini+
jwCi,Yoi = 1 ZOi,
r =
YOg - YoYOg
+
Yo YOg = 1ZOg.
The results of analysis can be found at the end of this article.
(3) (4) (5)
(6)
(7)
3. Short-step Transformers
3.1. Design of Short-step Transformers
It is often necessary to design impedance transformers for frequencies below 1000 MHz. In this frequency band the length of quarter-wave transformer is unfavourably long. The length of this transformer with many sections may be several meters. For this case the practical solution is the use of a short- step impedance transformer (Fig. 5).
Fig. 5.
The computer program is able to design these transformers with two kinds of procedure, namely from lumped element low-pass filter form and in an exact way using the theory of distributed element networks.
The first procedure uses an approximate equivalent circuit of the transmission line, and the capacitances of lumped element low-pass-filter are replaced by transmission lines of small characteristic impedances and the inductances by transmission lines of large characteristic impedance. In this procedure the tangent functions describing the transmission lines are approximated by their arguments, because the length of the lines is small.
In case of l =
>../16
the results are only approximate.The other procedure to design short-step transformers is treated on an exact basis. It makes directly the synthesis of distributed-element network.
This procedure has another advantage, namely the impedance ratio at the steps between lines of different impedance is smaller and easier to realize. In order to eliminate the periodic functions of distributed-element impedance transformers, and to simplify the synthesis problem, Richards makes use of the mapping functions
p
=
tanhas2 ,
(8)158 G. MATAY dnd Z. VERES
where 8
=
'U+
jn is the complex-frequency variable for the transmission- line circuit and p=
(f+
jw is the frequency variable of the mapped transfer function. The parameter a is defined bya I 7r
2"
= ;; = 20.1/ 4' (9)where 11 is the velocity of propagation and 0.1/ 4 is the radian frequency for which 1
= >"m/4.
CD
Fig. 6.
The attenuation characteristic in Fig. 6 can be obtained by mapping the at- tenuation characteristic of a conventional Chebyshev low-pass filter (which
• 11 1/ 11
has an equal-rIpple pass band from W = 0 to W = W1 ) by use of the mapping function
2 2
W' =
W~A(Ww2 ~~o),
(10) . where w' is the sinusoidal frequency variable for the conventional low- pass filter, w~ is the cut-off frequency of the conventional filter, and W is the frequency variable for the circuit of the form in Fig. 6. The parameter Wo is as defined in Fig. 6:2 2
2 Wa
+
WbWo =
2 (11)
, where Wa and Wb are limit frequency of pass band. In this way the desired reflection coefficient function is obtained by using two mapping functions.
The voltage reflection coefficient function is
r(p)
=
K(p - jWa)(P+
jwo:)(p - jWf3)(p+
jWf3) . .. (12) (p - pI)(p - P2)(p - P3)(p - P4) ...where K is a constant. The design steps are: Specification data is set: fre- quency band, maximum of reflection coefficient absolut value in the pass- band and parameters of transmission lines connecting to the ends of the transformers. First the degree of transfer function is obtained (the degree is equal to the number of sections), then the reflection coefficient function of conventional Chebyshev low-pass filter from FANo's paper [4] is calculated.
Next the poles and zeros of this function are mapped to obtain the reflec- tion coefficient function for the desired form of network, from which the input impedance function is formed. Finally the circuit is synthesized by removing one unit element at a time by successive application of RICHARDs method [5]. In this manner the impedance values Zl,Z2, etc. are obtained, and these impedance values are also the characteristic impedances of the line sections.
3.2. Dielectric Supports
At the realization of short-step impedance transformers the diameter of the inner conductor of coaxial lines with the large characteristic impedances may be such small that it is a problem to hold it in the axis of the outer conductor. Usually the transformers consist of air-filled coaxial transmis- sion lines for the sake of transmitting large power. In this case dielectric supports are used to hold the inner conductor coaxially. There are dielec- tric supports in several forms. The form used in this computer program and its equivalent circuit are shown in Fig. 8.
Fig. 7.
Dielectric support modifies some properties of the transmission line, decreases its characteristic impedance due to changing the dielectric con- stant and increasing the electrical length of the line sections.
(13)
160
T
CA
1
==:::0_
ClG. MATAY and Z. VERES
T T
~':ro:o =~=I =C:o:_IL....o::==z ..
=2Fig. 8.
q,i = {31iy'€;i, (14)
where: Eri is the dielectric constant of the section of the transmission line,
<Pi is the electrical length of the section, lk is the physical length of section, ai is the inner diameter of transmission line, bi is the outer diameter of transmission line.
We can compensate for the change in the characteristic impedance by decreasing the inner conductor diameter of these sections. At the de- crease of inner conductor's diameter fringing capacitances occur in both ends of support. These capacitances and the increase in the electrical length caused by support are compensated by decreasing the physical length of transformer section.
3.3. Corrections for Fringing Capacitances
It is very important to compensate for the effects of fringing capacitances in short-step impedance transformers. The compensation of transformers designed from lumped element form is comparatively simple. Every even- numbered line section with lower characteristic impedance was obtained by replacing capacitances of lumped element form of low-pass-filter. So we can subtract the fringing capacitances occurring at both ends of these sections of transformers from these capacitances, and we will obtain the reduced capacitance value from which we can calculate the compensated lengths of the line sections.
In the other case, when the synthesis is achieved directly on the distributed-element networks, we use the procedure mentioned at com- pensation of quarter-wave transformers. We can add the capacitances gen- erated by the dielectric supports to the fringing capacitances occurring at the steps between lines of different impedances and compensate them to- gether. We can take the coaxial-line step discontinuities into consideration as lumped elements only if the distance of two discontinuities is at least as large as the diameter of the outer conductor.
3.4. Analysis
This part of the program is similar to the part used for quarter-wave transformers because in this case transmission line sections with different impedances alternate too, but the sections containing dielectric supports will consist of three transmission line sections.
4. Comparison and Evaluation of the Results
Three different computer programs were made to design transformers of different types.
All of them are able to design transformers built from coaxial air-filled, lossless transmission line sections.
After entering the specifications, the programs will check that data. If the discontinuities are very close to each other, or higher-order modes can occur because the mean diameter is large enough, brings it to the user's attention and will ask for new data.
Next the program calculates the data of desired transformers and displays them on the computer screen in the format that will be shown in the examples. If the user wishes, it can perform the analysis of the transformers. If we have a look at the examples, we can see that the compensation for fringing capacitances is less important in case of quarter- wave transformers, than in case of short-step transformers.
The first two examples present quarter-wave Chebyshev (Figs. 9-10) and maximally fiat (Figs. 11-12) transformers with and without compen- sation. We can see, that the number of sections for the Chebyshev trans- former is smaller than for the maximally fiat.
The next example (Figs. 13-14) presents a short-step transformer, which was designed directly by the help of synthesis of distributed-element network because this procedure gave a better solution.
Measurements were achieved on a short-step transformer built some years ago at the Department of Microwave Telecommunications. The re- sults are shown in Fig. 15. The calculated results by the computer program using synthesis and analysis are presented for uncompensated (Fig. 13.) and compensated (Fig. 14.) transformers.
The results obtained by calculations and measurements show good agreement. The small differences mainly originate from inaccuracies in geometrical sizes of manufacturing. Further investigation will be carried out to obtain answer for the tolerance sensitivity of short-step transformers.
162 G. MATAY and Z. VERES
Ch.r~ct.ristic. iHPadan~Oi.~t.r of in. U.lu. o~ fringing Length of sect./H~/
C8Si of sections /at.rv": c::o-nductors ~/ CitP4lCit"·Ui8$ / p f / : zO(0):49.9774
:::0<.1>=53.8.190 zO(2):::61.2503 zO(3)=69.7077 zO(4):7'3.0659
~ ~.OG2
,
\ ,
O.p55
0.048
0.041 I '.
10. 0~4\
I
0.027
.(0)='9.1.300 a(1)=9.:5G39 a(2)=7.S662 .(3)=6.571.4 211(4)=6.01.00
d1=0.OO1B7
<12=0.00464 d3=0.00715
<14=0.00323
1(1)=21.3700 1 <2>=21.3700 1 (3)=21.3700
i
--===========================d
I
Print / 1 / Next /Space/Fig. 9a.
Print / 1 / ; h~xt /spac~/
Reflection coefflClent of Cheb~shev transfor~er
I
Ii I I
~I
~ I: I : I ... 0.:.9.2 ] /
,
1/:
I:",
\
\\ /
\
/
\/
4.~5 s.:Oo fr-eq.
Fig. 9b. Quarter-wave Chebyshev transformer without compensation
CtMrac:tariatic:. i...,..d.., cas of 54tCt ion5 /00"/:
zO(0)=49.9774 zO(J.)=53.8J.9O zO(2)=6J. .2503 zO(3)=69.7077 zO(4)=75.0659
--j-;;-;;;[--- ---
Oi.,.,.t .... of'" in. U.1u. of'" I'"I"'ln91n9 L.ength of" sect ./r.,M./
conductors. /""'" caD~it~as / p f / !
,,(0):9 .1.3DD dJ.=D.OO1S7 1 (1.)=21. .2399
a<1.)=8 .5638 d3=D.OO464 1(2)=21.0913
.(2)=7.5662 d3=D.OO?15 1(3) =21. 3930 a(3)=6.5'714 d4=D.OO333
a(4)=6.01oo
1=4
---::---:---;---[---
\ Print / 1 / : Hext /Space/
Fig. 10. a.
Print /J.~ ; Next /Spac~/
~,,_ .. __ "'0'-' ., _ "_'0,'
freq.
Fig. 10. b. Quarter-wave Chebyshev transformer with compensation
164
Characteristic. iMDadan cas of sections ~:
:0(0):49.9'774 zO(1)::$I.6J.68 zO(2)=53.9'3S:S :0(3)=61.2503 zO(4)=69.3532 :0(5)=74.1176 :0(6)=7'5.0659
--k~1
iO•012
10.006
G. MATAY and Z. VERES
i ) l - t . r of 1n. Velutt of" t'ringlng ~th of ...,t • ..--...
c:onduclto... ..--... ... I t _ /"gf"/:
8(0)=9.1300 d1:Q.OOOOO 1(1)=:21.4286 8(1)=9 .1300 csa:Q .00197 1(2)::31.4386 .. (3);;8.lM67 cI3:Q.OO4::iO 1(3)::31.4386 8(3)=7.5662 d4:Q.00?3a 1(4)::31.4286
a(4)=6.~ cSS:Q.00109 1(5)=31.4286 ,,(5)=6.105? d6:Q.00343
aCS)=6.0UlO
fa:345S
:,---,---~3---E---l--
I
Pr1nt /1./ ; Hex t ,l"$pac:a/Fig. 11. a.
Print /1./ : Next /Spac::~
""'."00 '0""""" o. 0 •• ' .. "" .,., ""\0'0"'"
.. 0,9.2 ...•..
/ !
-igg~Q~g~g~~2~.po~---2~.7/5~----~==~3~.5;O;:=---.4~.~~~5~---~5-.~ i Fig. 11. b. Quarter-wave maximally flat transformer without compensation
Characteristic. l~dan ~~ ... t . r of 1n. U.lUII of' f'ring irnl L.angth o~ aect • .IMft/
cea of' ..at icww /CliIhft.I': ~tora~ c~ltanses /pf/:
&0(0) .... '.9774 .(O)~.~ d1=O.OOOOO 1(1)=:21.1777 ZOU,):SJ .6.1.68 .. (U~.~ cSa=O .00197 1(2)=:21.3399 zO(2)=:I3.938:5 .. (2)=&.:5467 d3=O.~ 1(3)=21.1.1.1:1 zO(3)=61..Z503 .(3)=7.:5662 d4=O.OO73a 1(4) =21. .761l2 zO(4)=69.5:132 .(4)=6 .:1SS3 d5=O .00l.O9 1 (5)=21.5579 zO(5)=T4.1.176 .. (5)=6.1057 cI6=O .D0006
zO(6)='75.0659 ,,(6)=6.0100
~
--t~:~1
--- ---,---,---.. ---E---l--I
Print .IV ; Haxt .Ill...,.Fig. 12. Q.
Print .IV : Hext Reflection coefficient o¥ naxi"al1y flat
... q.,.q~ ... ... .
0.006
Fig. 12. b. Quarter-wave maxim ally flat transformer with compensation
166
Char_t. i - - ' -
of .... t 1 _ 'chV:
,.0(0)=4'.9"174 zDU)_.S19S zOCa)::a6._?
.. 0(2):14,.1'1'47 .. 0 ( 4 ) = = . _ ,.O(S):.!.4:I.UilU aO(6):4:I.:I61:1 zO(?):?S.DG159
z(O)
0.31.5
0.276
0.23'7
0.1.59
0.1.1.6
0.079
G. MATAY and Z. VERES
a i _ t a r of 1n.
_ t o ... ~ a(O)=' • .\.3DQ .U.)=4.~
.(a):1:I._
.(2):1. 'I'3D3 .(4):.1.:1.8339
.(S):1.~J,8
a(6)a.\O • .1J1144 .(7):6.0100
D i _ t .... the .... t . ~th of _ t . " - '
w1th swPDOr~:
ak(a)=11.~ 1(1)=:16.0881 . . (4)=11.468 1<a>=:J4.2911i 81«6)=1'.36' 1(3):36.0881
' - - t h of .... t 1 _ w1th - " " - ' : 11< ( a ) . 4.DOI) 11«4)0: 4.DOI) 11< ( 6 ) . 4.DOI)
1(4)=:J4.2911i 1<:5)=:16.0881 1(6)=:J4.2911i
Fig. 19. a.
Absolute value of reflection coefficient
... ~ ... . ...
0.039
Fig. 13. b. Short-step transformer without corn pensation
Cha.-.ct. il"lD.cIanca Of . . . . t ions "oRV:
zO<0)=49.9'774 zO<U=EI6.lSL9S ,,0<2)_.2047 zO<3)=L49.7747 zO<4)::25.0483 zO<lS)=L43 • .l6l5L zO<6)=43.36L3
zO<7)=7l5.~
:CU z(O)
0.038
0.025
O.DL3
O i _ t e r of in.
conductor. tfNV aCO)=9.L300 .<1.)=4._
a(2)=L3.1S6S9 .(3)=1.7303 .(4)=L3.83a9 aClS)=1..9318 .(6)=1O.1._
D(7)=6 .0100
O l _ t e r ~ ... t. '--th of .... t."'-"
.. lth SUDPOrt...fIwV:
~(2)=1.1..~ 1CL)=39.066l5
~<4)=U.468 l(2):a7.3?20
I.8n9th of ....,t 1 _ with support "'-":
lie <2). 4.000 lie <4). 4.000 1Ie<6)= 4.000
H3) =37 • 34O:S H4):a7.0434 1<lS) =37 • 9GOL 1(6)_.3883
, , ( 7 )
Fig. 14. a.
L:Prlnt :...,. :Hext
Absolute value of reflection coefficient
Fig. 14. b. Short-step transformer with compensa.tion
168 G. MATAY and Z. VERES
CHi RFL 11n MAG 100 mU/ REF 0 U :!l; 31 813 mU
l1!iil 435. 99 5CO 101HZ
Cor
I I~
64 ;; 02 mU20.2, 7101Hz
MAR ER 4
...
I7Q ml4F5. E 995 MH~ 33.4 1 101Hz
3: 9~~61 6 2101Hz mU
1
I I
I I
11
I 11
...
~ /
~
~~ h4
START .300 000 101Hz STOP 1 300.000 000 101Hz
CHi RFL 11n MAG
l1!iil
' -
"'-
""-
"'-..
I I
CH2 RFL 200 mU F'S
STOP
785 MHz
Fig. 15. a.
25 mU/ REF 0 U
"'"
"'JI-., 43
-
V----
~ 32 077 mU 435. 90 OC 0 104Hz
1 ~::>~4::>! iil5MI!l~
::> 59 24 ml.J
'33.3-9 Mt-p 3. 63 I:!S.JllJ
/
~
,
1 ... ./2
62.36
.1; 6~ • ~779my 620.~66 104Hz
f-L-,-~.,.---!-,,<:;L;-l-..l...-JL...4-,' 5~ oil ~§S3 my 33.349 101Hz . 63.776 mU 157.92 • 294.002 104HZ
START .300 000 104Hz STOP 725.CCO 000 M~Z
Fig. 15. b.
References
1. FELDSTEJN, A. L. - JAVICH, L. P.: Synthesis of Four-Pole and Eight-Pole Networks on Microwaves, Svjaz INC, Moscow, 1965., pp. 120- 165. (In Russian).
2. WHINNERY, J. R. - JAMIESON, H. W. - ROBBINS, T. E.: Coaxial- Line Discontinu- ities, Proc. IRE, Vol. 32, pp. 695-709, November 1944.
3. JACHIMOVITS, L. : Matrix Analysis of Stepped Impedance Transformers, Journal on Communications, Vol. XXI, pp. 33-39, February 1966. (In Hungarian).
4. FANo, R. M.: A Note on the Solution of Certain Approximation Problems in Network Synthesis, J. Franklin Inst., pp. 189-205, March 1950.
5. RICHARDS, P. 1.: Resistor-Transmission-Line Circuits, Proc. IRE, pp. 217-220, Febru- ary 1948.
6. MATTHAEI, G. 1.: Short-Step Chebyshev Impedance Transformers, IEEE Trans. on MTT, pp. 372-383, August 1966.
7. MATTHAEI, G. 1.: Tables of Chebyshev Impedance-Transforming Networks of Low- Pass Filter Form, Proc. IEEE, pp. 939-963, August 1964.
8. HARASZTOSI, L.: Discontinuities in Coaxial Transmission Line, Diploma Work, TUB, Department of Microwave Telecommunications, Budapest, 1990. (In Hungarian).
9. VERES, Z.: Computer Aided Design of Stepped Impedance Transformers Realized in Coaxial Transmission Line, Diploma Work, TUB, Department of Microwave Telecommunications, Budapest, 1993. (In Hungarian).
