sensor and simulation notes note 301 22 may 1987...
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Sensor and Simulation
Note 301
22 May 1987
REVIEIJ OF CHARACTERISTIC
Notes
,
IMPEDANCE
OF TWO CONDUCTOR TRANSMISSION LINES
by
N. Ari
BBC Research Center, ~aden, S~iitzerland
and
D.V. Giri
Pro-Tech, 125 University Ave., Berkeley, CA 94710, U.S.A.
Abstract
The objective of this note is to revieki the impedance of two cGnductGr
transmission I:nes includii:: TEII ceils with the inner septum placed
symmetrically and with sutme vercic:’~ offset. The required computer programs
were developed and implemented on the computer system at the BBC Research
Center.
as well”
Some ~.,,=,-~”,a-..-.,..~.- !-ssd?:s a“re oresented here in figu!-ss i+”ndtabular form
for future reference.
\
.
.wz-EPR-sslf-3ol
%9. .
Sensor and Simulation Notes
Note 301
22 May 1987
REVIEW OF CHARACTERISTIC IMPEDANCE
OF TWO CONDUCTOR TRANSMISSION LINES
by
N. Ari
BBC Research Center, Baden, .Switzerland
and
D.V. Giri
Pro-Tech, 125 University Ave., Berkeley, CA 94710, U.S.A.
Abstract
The objective of this note is to review the impedance of two conductor
transmission lines including TEIIceils with the inner septum placed
symmetrically and with some vertic:l offset. The required computer programs
were developed and implemented on the computer system at the BBC Research
Center. Some example results are presented here in figures and tabular form
as well for future reference.
conductors, transmission lines, THI
,
CONTENTS
Section
1.
11.
111.
IV.
Introduction
Two-Conductor Transmission Lines
Numerical Results for the Characteristic Impedance
of Symmetric and Vertically Offset TEM Cells
Summary
References
Page
3
5
13
29
30
“o2
.
.
lbI. Introduction
“ Two conductor transmission lines used in propagating
electromagnetic (EM) waves has many forms. Some examples of
such transmission lines are i) two-wire line [I], ii) conven-
tional coaxial lines with one round conductor inside another
[1], iii) parallel plate lines [2] and iv) rectangular coaxial
lines [3, 4 and 5]. The dominant mode of electromagnetic wave
propagation on such transmission line systems is the transverse
electromagnetic (TEM) mode. The TEM fields are such that the
r “–”:??f-lcs along the propagation direction are identic~lly-----
zer~ and the non-zero components are present in the transverse
plane. This fundamental property of the TEtl mode permits the
following:
a) the analysis may be carried out in a transverse
plane, usually by employing conformal mapping
techniques while recognizing that the relative
field distribution is the same in all transverse
planes’
b) in a practical situation, meas~jrement of the
axial components of the electric or the magnetic
fields results in the detection of the presence
or absence of non-TEM modes.
In addition to the field distribution, the TEM mode is also
characterized by its impedance, Zc which i: the ratio of voltage
and current corresponding to TEM fields. This impedance Zc is a
function of the cons’titutive parameters (u, E, p) of the trans-
mission medium and the geodetical parameters of the transmission
Zc = if9
(1)
line, as expressed below
3
where
.Zc ~.,characteristic impedance of the TEM mode (Q)
n s characteristic impedance of the medium (Q)
-.{-= qzometrical factor for the impedance “s
It is observed that equation (1) simplifies to
(2)
for the case of free space medium with Qo=377n.
The present interest is in the impedances of rectangular
coaxial. transmission lines (TEM cells) with vertically offset
inner conductor. Strictly speaking, the expressions used here
[3 and 5] apply to the central parallel section of a conventional
TEM cell with tapered lines on either side. However, if the
tapered lines are long (- angles of 20° or less], the same expres-
sions can be used to yield results within a few percent accuracy.
In Section 11, we have lised the two-conductor transmission
lines considered in this review, along with impedance expressions
available in the literature. Section 111 presents the numerical.results for the case of symmetric and vertically offset TEM cells.
4
,
@—
Wii)
II. Two-Conductor Transmission Lines
The object of this section is to list commonly used two-
conductor transmission lines and review their impedance
characteristics. The transmission lines. considered are:
1)
2)
3)
~)..2 ,)
6)
7)
8)
9)
10)
11)
12)
13)
14)
two wire line of equal radii [1]
two wire line of unequal radii [6]
single wire above a ground plane [1]
coaxial cable [1]
eccentrjc coaxial cable [7]
parallel strip line ~ [6]
parallel plate line [2]
single wire in a square enclosure [6]
single wire in a trough [6]
two wires enclosed by a circular shield [6]
two wires enclosed by a rectangular shield [6]
two wires in a trough [6]
symmetric TEM cell [3, 5]
vertically offset TEM cell [3, 5] “
—
k.)
The impedance expressions listed are for free space medium.
For other simple dielectric media with Pr = 1, the impedance
expressions sho,uld be divided by <.
5
Zc = ()120 cosh-l ~
()z 120 Rn -$Q
(if D>>d)
Figure 1. Balanced two wire line with equal radii
z. = 60 COSh ‘l(N)
N[
= 1 4D2 ‘1 ‘2“—-~-q2 ‘ld2 1
Figure 2. Balanced two wire line with unequal radii
Figure 3. Single wire above a ground plane
6
1.
c“ ●4L
Figure”4.
Figure 5.u
Coaxial Cable
@
‘1
‘2
E=E Eor
Eccentric CoaXial Cable
D‘2
‘1
E=E E (3
0 r /+-D-E\
Line Par~meters
DL! =
()
‘2~f!n—
‘1
SK
?c=:’—c
Line Parameters.—..
;
;
c’ = ()‘227fZJ?,n —
‘1
()
‘2l.n —
‘1
,
,
+
/[ (222
c’ = 211E-1 ‘l+r2-D
cosh2r1r2
)
+cosh-’(r%i_D2)“
Note: For D-~0 , .O.ll-lfi,,j ()‘2
–-— ZLn —
‘1
b----w-+
Figure 6. Parallel - strip
~> /
Note
The reader is referred to reference [2]
for the expressions for the impedance.
The accompanying table of impedances is
excerpted from [21 by Baum, Giri &
Gonzalez.
Figure 7. Parallel plate line
8
.
for (w/L) <0.1
Zc = 377(;)
rb/a
0.1667
0.4068”
0.5
0.7
0.9
1.0
1.23526
1.6
2.0
L
2.5
3.0
6.99”
Zc $2
(free space)~
50.240
99.961
115.439
143.927
167.595 ●‘ 178.058
199.896
227.859
252.848
278.407
299.593
399.722
b
.
“T A = D/d
H’-d
A= (1 + 0.405 A-4)/(1 - 0.405 A-4)
alD B= (1 + 0.163 A-8)/(1 - 0.163 A-8)
c= (1 + 0.067 A-lZ)/(l - 0.067 A-12)
.+ ~ L
h ‘~{
z s [60 ~n (A) + 6.48 - 2.34 A - 0.48 B - 0.12C]E
~;:.:y~ g. Single wire in a square enclosure
od
+w/2 —+ w/21t
h
For d << h
d <<w
Zc = 60 En[4W tanh ~h/w
nd ‘1
Figure 9. Single wire in a trough
—
LJ9
igure 10. !ko Wires Enclosed by a
Circular Shield
Assumption:
&
‘i<< :1,2, (r~-Di) , i.e.,
the conductors are separated
sufficiently from each other
and the shield as well,
Line Parameters
:C’ )=+itj
where
1)1
J(D1D2)2 -F r: - 2D1D2 r: COS(612)
,. ‘,.. (DID*) 2 + D: - 2D2D; Cos(elz) )
,(J24
2DlD2 r: cos(612)r2_D2
-h‘1 (D1D2) + ‘s -~
]()
h d
(DlD2)2+D~ - 2D2D~ COS(612)r; r
1
..
* For KD, W, h,
T [(Zo= (276/*ut)lWIO
4h trmh (TD/2A)
)—
( )1–-glog,, ~,%(1 m
r ‘ ‘
D~z.
~
+ Lk
where~ ~2 z sinh (zZ)/2h)
++ .-isinh (zD/2h)
;— ‘“= cosh(rm+%) ‘m= Sil)ll(W?@2L).
Figure 11. Two wires enclosed by a rectangular shield
For J<<D, w,h,
2,= (276/d~2) log&2u/7cl(A’]2)]
where
A= cos&(TD/w) i- cosech’ (Z5+’w)
Figure 12. Two wires enclosed by a trough
9d 11
.
.
0
2b
2a
Zc()T
2W
Figure”13. Symmetric
‘?72lb.-.
a=
K(o L
= I((a’)
where ‘m=;
K(k’)
.=a
TEM cell [3]
.Y k’= m
[K”(i3) (l.- B2AI) + 2AIE’(S)l=
4[K(f3){l + AI(2 - 6Z)I - 2A1E(13)I
where
‘1 =
f3 = sin (ww/2a)
. . ()nb ()]mh
cosh T-Costl ~
-1sihn (rob/a)
Figure 14. Vertically offset K“(B) = K(BI)
= E(BI)TEM cell [5]
E’($)
,
III. Numerical Results for the Characteristic Impedance of
Symmetric and Vertically Off;et TEM Cells
The two cases of symmetric (figure 13) and vertically offset
(figure 14) are considered in this section for numerical results.
The expressions from references [3 and 51 are programmed on the
computer system at the BBC Research Center. The required complete
elliptic integrals and elliptic functions were evaluated with
subroutines developed on the basis of equations in [8].
The results of the symmetric case are presented in figures
15 Z;! L5 with accompanying Tables 1 and 2. For the case of
vertically off,set cell, vertically offsets of h/b = 0.1; 0.3,
0.5, 0.7 and 0.9 are presented in this paper in figures 17, 18,
19, 20 and 21. For ~hese four figures, the results are also
tabulated in accompanying Tables 3, 4, 5, 6 and 7.
13
10°
..
10 -
10 “
W/a = O*Q50910
0.30
0.50
0.70
o~qo0.95
2b
(b/a)
101
Figure 15. Characteristic impedance of the dominant TEII
mode in a symmetrical rectangular coaxial line
(tabulated in the next page)
*TABLE 1. Normalized characteristic impedance (Zc/I-l) of a symmetric
%cell (plotted in Figure 15)
\
w/a
b/a
0.10000.11220.12590.14130. 15850. 177au. 1995o.2z39
0..3ss:0.44670.50120.56230.63100.70790.79430.89131.00001.12201.25B91.41251.58491.77831.95532.23972.51192.81843.16233.54813.98114.46685.0L195.62346.30967.07957.94338.912510.0000
0.05
0.26660.28350.30060.3180(3.33550.35320.37100.38890.4C169ij.~~~o
:.4s31..4512C!.4792i3.49710.51470.53180.54810.56330.57700.58900.59920.60730.61350.61810.62110.62310.62430.62490.62520.62540.62540.62550.62550.62550.62550.62550.62550.62550.62550.6255.0.6255
0.10
0.17350.18770,20240.21770.23350.24970,26620.28310.30030.3176U.3351i3.3!i2r0.37040.30800.40530.42220.43830.45330.46690.47080.48880.49690.50310.50760.51060.51260.51370.51430.51470.51480.51490.51490.51490.514$30.51490.51490.51490.51490.51490.51490.5149
0.30
0.07260.08030.08850.09750.1o710.11750. 12850.1403il.15280. lb60L).L7SJa0.15610 .20890.22390.23900.25390.2683(3.~8170.29400.30470.3L380.32110.32660.33070.33340.33520.33630.3368o.33fl0.33730.33730.33730.33730.33730.33730.33730.33730.33730.33730.33730.3373
0.50
0.04590.05100.05670.0628
~ 0.06950.07690.08480.09340. 1028o.1~~70.1?33o. 13c5O. L462O. 15830.17040, 18240.1941o.~05(30.21500.2237(3,Q31O0.23690.24140.24470.24690.24830.24910.24960.24980.24990.25000.25000.2500Q.25000.25000.25000.25000.25000.25000.25000.2500
0.70
0.03360.03?40.04170.04630.05140.05710.06320.06990.0771cl.oa480.0930C!.1016IJ.L1050.11960.12860. 13750.14600.15390. 16090.16710.17220.L763o. 17940.18160.1s310.1841o.la470.18500, 18520.18520.18530.18530.18530.1853CL.18530. la53O. 18530.18530. 18530. 18530.1853
0.90
o.o~64
0.02340.03270.03630.04020.04450.04900.05390.05900,0643O.clGGa0.07540.08100.08650.09180.096EJ0.1(3150.10570.1o940.11250.11500.L1700.11E160.11960.12040. 1201J0.12110.12>:J0.12130.12140.12140.12!.40.12140.12160.12140.12140.12140.1-2140.12140.12140.1214
0.95
0.02480.02’/50.02U50.(]3370.03710.04CIBrl.lJ4n7O.o.luuo.05310.0s7.7O.nvl+0,(16!;30.U7CIG13.074H(-).07880.08250.08590.08CJ90.0916o.053ao .0555o.09ii9i).09fj0iJ.l19H70. I’19!J2[1.[39:)5().()9:1/IJ.[)9S81).IJ!J$J!)1).[1!J,,J9t).[1:1!)[91).1)’J>)2(}.09$.)91],lJ$J!lgII.i19i)9!).[)9!)9o. [Ju:)$l1).[):).)9o .os!)90.09!19[J.~J9:JiJ
od
15
.
.
()Zc 10
T
.
I id
2b 2W.
(w/a) “’
I I I ! 10.00 >0.20 o.~o 0.6iI 0.80 1.00
Ficjure 16. TEM impedance of a symmetric cell (tabulated below)
TABLE 2. Normalized impedance of a symmetric cell (plotted above)
\
(b/a)
(w/a)
(I.I31OO0.02000.03000.04000.05000.06000.07000.00000.09000.10000.11000.12000.13000.14000.15000.16000.1700
0.125 I0.250
o.55io ‘-’0.44130.37770.33330.29950.27250.25030.23150.21540.20150.18930.17840.16880.16010.15230. 14520.13B8O. 13290.12750.1225
0.6611CI.551O‘0.48670.44130.40620.37770.35390.33330.31540.29950.28530.27250.260!30.25030.24050.231!$0.22320.21540.20820.2015
o“.500
0.77020.65990.59550.54980-51440.48550.46110.44000.42150.40500.39000.37650.36400.35260.34190.33200.32270.31400.30580.2981
1:000
0.0553”0.74500.68050.63470.59920.57010.5456
. 0.52440.50560.48860.47370.45980.44710.43530.42430.41400.40440.39530.3867.o.37i35
2.000
0.88050.77020.70560.65980.S2430.59520.57070.54940.53060.51370.49850.404(50.47100.4599
. o.44i3a0.43040.42870.41!350.41070.4025
TI
o.BEll-/ L1.tiul?0.7714 0.77140.7060 0. /otie0.6610 rJ.61jlo0.6255 0.62550.59640.571H0.5505 .0,53170.5149(7.49970.485-/0.4729().4(;100, 4!JOCI0.4396o,429f.1
0.!5964o.5718
“ 0.55050.53170,5149
“(-).49’37il.413570.4729L).4f5111
16
f’-’● ●& 4? ..
,oco~~~oooooooo 0000000000000000000000000000 oooooooooooa@o 0000000000000000000-0000. . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..”” ““”.” ““””wtoamamwmof-ommm mmmmwmm~<<+~++ +~<mmmmmmmmcnm mmuim”mmmmutIQm*~m~UN*O~n~ mWbWW-OOD+UJW& cJNPOmCD*OIUIbWN. OIDCC-JChUiAWNt-
;.Lt.hb$huuwf.IU@ tJU@Q~N~~NN~NN8$;:UI
GOOooooOOOOooO 0000000000000000000000000000 00000@ooGOOCJ(3(”I GoOCIOOOOOOOOOO 000000000WJ.alJWoWmQmmb (JtiWo@M*I?)O&UN I-
00o~OOo,OOOGOoO 0000000000000000000000000000 QOOOOOOOOOOLJ<,CB 00000000000000000000000.
—-- .... .. ....—
0000000000000000000000000000 00000000000(30(3 Oooooooooooc$oo 00000000000000000000000. . . . . . . . . . . . . . . . . . . . . ..0. . . . . . . . . . . .,,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ● $.”” ..”.Ooocoooooooo 000000000000000000000000 000000000000 WVwl--w+-r-t-w-- ---$--------- ~--$-~--~~amv.(nmmmcnmmm mcnmmmmmmm-44.44 <+uti<+timmmmffim ~mwwwawwlfloooo OCIP.FI-NNNt.)G)(JtJ w~bb(nutmmtnwumm tfJ(J<W*t..IAUIm<mOOF N(.dbDUI@<DwOWMU (JbUm*UIIOOFJ(J &LI)m.J@o~WbM-JIDONJ.m +IDI-.(.)mtiGtC&UONW meOmNUIGA@UmU~ Whobwwum+tir.o<da W*N.I-U(D<WbLJNt-0 wmmmmmm@oP N&m apbmNmoutp4hb.m +mmutmaemom-om tmwwtiawt-umuta~u N
o.
mo0
Iv.
000
.b
.000
cc. .
Z.G o GC o 000. . .-r--U-J<
o 00. .F*
(DIDu-(Jc
Ooc,...-h-l(OL9P2..JCbwu.
o =:. .
!.JWr>c ?.@LCc(iCC.
oi2. .
=:. .mlf.1Wb10*Jq to
--NmulQ
wL1mo
N
k&
12-.4
,,
.-,
e.(-~L’t.L,lJG
OLGINPJN
— - —--- ..-— —.—. ..——-
nI
1Ov
10
10
. .
-1
[..
.
I (w/a)-2 1 t I I
0.00 t3.2cJ 0.40 0.-60 0.80 1,00
Figure 17. TEM impedance of a vertically offset cell (~abula~ed
below) (h/b) = 0.1
TABLE 3. Normalized impedance of a vertically offset cell
XEE().01000.02000.03000.04000.05000.06000.07000.00000.09000.10000.11oo0.12000.13000.1400Q. 15000,16000.170001 laoo0.19000.20000.21000.22000.23000.24000.2500
0.65930.54910.48470.43900.40370.37500.35080.32990.3115
‘ 0.29520.28050.26720.25510.24390.23360.22410.21520.20690, 19910.19100.1850o.~7a50.17240.16670.1612
0.77040.66010.59570.55000.51450.48560.461.20.440L0.42160.40500.39000.37640.36390.352-40.34170.33180.32250.3137
- 0.30550.29770.29030.20330.27670.27040.2643
0.667
0.81020.69990.63540.58970.55420.52520.50080.47960.46100.44430.42920.41550.40290.39130.38040.3703 .0.36000.35190.34350.33560.32800.32000.31400.30740.3012
1.0
0.85400.74360.67910.63330.59700.56000.54430.52300.50430.48750.47230.4585c3.445B0.43400.42300.41270.40310.39400.38540.37720.36940.36200.3549 .0.34020.3418
2.0
o.aao30.76990.70540’.65960.62400.59500.57040.54910.53030.51350.49830.48430.47150.45960.44860.43820.42J340,41920.41050.402-20,39430.38680.37960.37260.3660
4.0
LI.BHL)cJ.771ao.7c)iih0.6(51110.62550.5964CJ.571ti0.5:,050.5:51.!o,5i49o .45)9j0.48570.4729O.’lfil{lu ,4$JUUU.439tj0.429HfJ.42Lll)0.411Vo.4rJ3110,3951cJ.3tiRl0,38090.37400.3673
6.0
0,B817{J,771dLJ.706h1).6610il.u?-bs1).59!)41).571H{!.s!511:,11.5311IJ.’I1.ILJ[1..l9iJ71).4rfli7IJ.,17.’”JII.ml{.,lU,1..l!jl)()Il..1.’1,1!2Il..lLJ,IH1)..l2i1l\!I..ll I.Jlj,,i~l!(;11..!J.)!,/
18
.b
b“
*
● ✎✎LJ
.
TABLE 3 (continued)
Kb/a
w/a
0.2600 .O;27Q00.28000.29000.30000.3LO00.32000.33000.34000,35000.36000.37000.30000.39000.400(30.41000.42000,43000.44000.45000.4600o.~~~~0.45CZ0.4s020.50G00.5100o.5~cJ(3
0.53000.54000.55000.56000.5700c).58#cJ
0.59000.60000.61000.62000.63000.64000.65000.66000.67000.68000.69000.70000.71003.72003.73003.74003.7500).7600J.77003.760Q3.7900>.8000).81001.0200).83001.84001,s5001.6600).8700).88001.8900).9000).9100).9200).9300).9400).9500>.96001.9700).9800
I0.99001,0000
0.2
0.15610.16120.14660.14220.13800.13400.13020.12660.12320,11990.11680.11390.11100.10830.10570.10330.10090.09860.09650.09440.09250.09060.08800.08710.Q8540.08380.08230.08090.07950,07810.07690,07560.0745
6.07330.07220.07120.07020.06920.06030.06740.06650.06570.06490.06410.06330.06260.Q6LR0.06110.06050.05980.05910.05050.05790.05720.05660.05600.05540.05400.0S420,05350.05290.05230.05160.05090.05020.04950.04070.0479 ‘0.0469”0.04590.04470,0*330.04140.03860.0177
0..5
0.25850.25300.24770.24270.23780.23310.22860.22430.22010.21610.21220.2ti850.20480.20130.19790.19460.19140.18630.18530.18240.17950.17682.174L0.17150. lbH9O. 16640.16400.16160.15930.15700.15480.15270.1505
0. 14850.14640.14440.14240. 140sO. 13860.13670.13490. 13300.13120.12940.12770.12590.12420.12240.12070.11900.11730.11560.11380.11210.11040.10860.10690.10510.10330.10140.09950.09760.09560.09350.09140.06910.08680.08420.08150.0705o.a75~0.07120.06630,05940.0211
0.667.,
0.29520.28940.28390.27860.27340.26850.26370.25910.25470.25030.24620.24210.23820.23430.23060.22700.22350.22000.21670.21340.21020.2071Q.2Q4Lo.2ClLlo.19B20.19530.19250.18980.18710.18450.18190.17930. 176%
O.I?*o.17i9O. 16950.16710.16480.16240.16020.15790.15560.15340.15120.14900.14680.L446
. 0.14250.14030.13810.13600.13380.13160.12940.12720.%250O. 12280.12050.11820.11580.11340.11100.10850.10580.30310.10030.09740.09420.09080.08710.08300.(37820.07240.06420.0217
1.0
0.33540.32940.32360.31800.31260.30730.30220.29730.29250,28790.28340.27900.27470.27050.26640.26240.25050.25470.25100.24730.24370.2402o.~3f57
o .2333CJ.23000.22670.22350.22030.2L720.2141om~lll
0.20000.2051
0.20210.19920.19640.19350.19070.18790.18510.18240.17960.17690.17420.17150.16880.16610.16340.16080.15810.15540.J5270.15000.14730.14450.14170.13900.13610.13330.13030.12740.12430.12120.11800.11470.11120.10760.10380.09970.09530.09040.08480.07800.06850.0221
2.0
0.35960.35340.34740.34160.33610,33060.32540.32020.31530.31040.30570.3o110.29660.29220.28790.28360.27950.27540.2715Q-26760.26370.25990.25620,2525~.~49~0.24540.24190.23850.23510.2317‘0.22840.22510.2218
0.21860.21540.21220.2091(3.~Q590.20280.19980.19670.19370.19060. 16760. 18460.18160. 17860. 175(3.0.17260. 16950.16650.16350. 16050.L5740.15430.15120.14810. 14490.14170.13850.13520.13180.12030.12470..12100.11720.11320.10900.10450.09570,09430.08820.08090.07080.0224
4.0
0.360Y.0.35470.34870.34290.33730.33190.32660.32150.31650..31160.30690.30230.29760.29330.28900.28480.28060.27660.27260.26860.2648o.~6~o0.2573o.~535
0.25000.24640.24290.23940.23600.23260.22930.22600.2227
0.21950.21630.21310.2C1990.20680.2G360.20050.19750.1944o.1~140.18630.18530, 18220. 17920.17620.17320.1702o.lb710.16410.161UO. 15800. 15490.1517o.14dbo. 14540.14220.13890.13560.1322o,i~870.12510.12140.11750.1135cl.10930.10480.09990.09460,08840.08100.07090.0224
6.0
U.>tat)il0.35470.34870,3429(3.33730.3319o.3~~~
0.32150.3165cl.311bo.3(Jii$l0.3(12:1u .2378i).2$I:J3o.2ti9uo.2ki48(].~~”(j0.27b60.2726o.2b87U.264E!0.261UlJ.2573U.25.!C1. .u .?5CIUo.24ti4U.2429(.1.~3:J5[“1.23[;011‘vqv~
0.2293().226U~.z2z7
cl.2195o.~l(j3
. :::::;0.2C16S0.20360.2006c).1’3750. 19440.19140. LBti.3Il.Itj.).l0. lB:’Ud. 17!12
. l-).1/{>2[).l-/:j2l).17!}2[).11,/1Il.1041II.161L3II.Ibrlll(1.lr).19,1.1517IJ.14}!(>l).14’,411.1422l).luiliJIl.13>($l).12<20. 12)170.12:,1U.12140.11/5r).11.150. lrli):jc).10480.0999CJ.09460.0884O.OH1O0.0709o.t322j
19
.
10° . I I I Y i
10
10
-1 :()z c—.n
(w/a)
0.00 0.60. -.
Figure 18. TEM impedance of a vertically offset cel. .. . .
TABLE 4.
\
b/a
w/a0.01000.02000.03000.04000.05000.06000.07000.08000.09000.10000.11000.12000.1300”0.14000.15000.16000.17000.18000.1900 .0.20000.2100!3.22000.23000.24000.2500
below) (h/b) = 0.3
TEM impedance of a vertically offset
0.2
0.65180.54150.47710.43150.39620.36750.34330.32240.30410.28780.273L0.25980.24770.23660.22630.21680.20790.19960.19190.18460.17730.17140. 16!540.15970.1543
0.5
0.75650.64620.581813.!i36L0.5006o.471R0.44740:42630. 407a0.39120.37630.3627 .0.35-03o.33aa0,3202 .0.31830.30900.30030.29210.20440.277L0.27020.26360.25740.2514
0.667.
0.79!570,68540.62090.57520.5398o.s~oa0.48640.46520.44660.43000.4L500.40130.30870.37710.3663“0.35630.3469o.33ao0.32970.32180.31430.30720.30040.29390.2878
1.0
0.84220.73180.66730.62150.58600.55700.53250.5113
-0.49260.47580.46070.44690.43420.42240.41150.4013Q.39170.38260.37410.36600.35820.35090.34390.33720.3307
cell (p
2.0
0.87770.76740.7Ck2B0.65700.62150.59240.56790.54660.52780.5L090.4957o.4ala0.46900.45710.44610,4357‘0.42600.41680.40810,39980.39190.38440.37720.37020.3636
1.00
(tabu
otted
ated
hove)
4.0
0.8nlb0.771:50.7U680,66100.62540.59640.57180,55050.5317O.!ilJls)0.49960,48570.47290.46100..44990.43950.429t30.42060.411U0.60350:39560.3BBIo.38nD0.37390.3673
6.0
l],adt7i3.7714[1.YGt>u0.6610~.62.55,iI.59640.57180.55050.53170,51490.49970.48570.472s!0.46100.45000.43960.4?5s0.4206O.alluil.
w0.. i’t).:
4 TABLE 4 (continued)
w/a .
0 0.2457 0.2818 0.3245 0.3572 0.36080.1443 0.2403
d.3bti90.2762
~.~O~o 0.13970.3L86
0.2351.0.3510 0.3547 0.3547
0.2707 0.3129 0.3451 0.348-/0.2900 0.1354 0,23CIl
o.34ti70.2655 0.3073
0.3000 0.1312 0.22S30.3393 0.3429 (3.34?9
0.2605 0.3020 0.3338 0.3373 0.33730,3100 0, 1273 0.2207 0.2556 0.2969 0.3204 0.33190.3200 0.1236 0.2163
0.33)90.2510 0,2919 0.3231 0.3266 CI.321i6
0.3300 0.1200 0.2121 0.2465 0.2870 0.3180 0.32150.3400 0.1166 0.2080 0.2421 0.2823 0.3130
(J.32150..3165 0.3165
0.3500 0.1134 0.2041 0.2379 0.2778 0.3082 0.3116 0.31160.3600 0.1103 0.2003 0.2338 0.2733 0.3035 0.3069 cl.3uti90.3700 0.1074 0.1966 0.2299 0.2690 0.2989 0.30220.3800 0.1046 0.1931
0.30230.2260 0.2648 0.2944 0.2977 0.2978
0.3900 0.1020 0.1897 0.2223 0.2608 0.2900 0.2933 0.29330.4000 0.0994 0.1864 0.2187 (3,2568 Q.~~58 0.28s+0 ~.2~9(30.4100 0.0970 0.1832 0.2152 0.2529 0.2816 0.28470.4200 0.0947 0.1801
0.28480.2118 0.249L 0.2774 0.2806
0.4300 0.0925 0.17720.2806
0.2085 0.2454 0.2734 0.2765 0.27660.4400 0.0904 0, 1743 0.2053 0.2417 0.2695 0.272!5 0.27260.4500 0.0884 0.1714 0.2021 0.2382 0.2656 0.26e6 o0.4600 0.0865 0, 1687
.2b870.1991 0.2347 0.2618 0.2648 u ._’ti.l.5
0.4700 “ 0.0047 0.1661 0.1961 0.2313 0.25%0 o,~610 o.~ti~~0.4800 0.0829 0.1635 0.1932 0.2200 0.2543 0.25720.4500
0.25730.0813 0:1610 0.1903
0.50000.224? 0.2507 0.2536
0.0797u .?S36
0..1586 0.18760.51G2
0.2215 0.2471 o.~50,3 !“10.0781, 0. 1562
.25:~rlo. 1040 0.21R3 0.243[S o. 2.l(in
0.52G21).,-,<,,4
:,0767 0. 1539 0. 1822 o,~i52 “ 0.2<01 ~,~,~~:j !;0.53G3
2.1::1G.0753 5.1517 cl.1796 0.2121 c).~367 0.2J9q (J..?~>lh
0.5.4GCJ 0.0739 0. 1495 0.L77CI 0.2091 0.2333 n.23Ci(J ?~,jol).-0.5500 0.0727 0.1473 0. 1745 0.2062 0.2300 0. 2:j2h (),~~~~0.5600 0.0714 0.1453 0.1721 0.2032 0.2267 0.22930.5700
u.22:~*0.0703 O. 1432 0.1697 0.2003 0.2234 L1.22btJ
0.5800 0.0692 0.1412L).22U0
O. 1673 0.1975 0.2202 0.222’/ [).22270.5900. 0.0681 0.1393 0.1650 0.1947 0.2170 D.2194 ().21:J5
0.6000 0.0670 0.1374 0.1627 0.19190.6100
0.2130 o.~J/52 I).Zl,i30.0661 0. 13s5 0.1604 0.1892 0.2107 iJ.213u ~J.2J.~1
0,6200 . 0.06S.1 0.1337 0. 15820.6300
0.1865 0.2076 o.2(19b 0.20990.0642 0.1319 0.1560 0.1838 0.2045 o.~067 I.]2f168
0.6400 0.0633 0.1301 0.15380.6500
0.1811 0.2014 0,2113Ci ,!.2{13{>0.0625 0.1204 0.1517 0.1765 0.1983 ~.~(,o:j !1.21Jrl(,
0.6600 0.0617 0. 1266 0. 1496 0,1758 0, 1953 0 .197.10.6700 0.0609 0.1249 0. 1475 Cr.1732
Il.1:)/50.1923 0.1944
0.6800 0.0601 0.1233 0.1454 0.1707,).1:}.!4
0.6900 0.0594 0.12160. ln93 0.191:!
0.1434().l~Jld
o.16el 0. 18630.7000 “ 0.0587
0. ItlR:f0.1200
Il.ltlt!~o. 1413 0.1655 0. 1&33 0. ltjs:i
0.7100 0,0580 0.1184~1.1U!)3
0.1393 0.1630 0. 1803 0.11122 [).16x0.7200 0.0.573 0.1168 0.13730.7300 0.0567
0. 1604 0.1774 0.1/920.1152
u. 17!120.1353
0.74000.1579 0.1744 0.1762
0.0561 0.1136cJ.17(j?
0.1333 0,1553 0. 1714” 0.17320.7500 0.0554 0.1121
0.17320.1313 0.1520 0.1684 0.1701
0.7600 0.0548 0.11050.17(-)?
O. 1293 0.1503 0. 1655 0.16710.7700 0.0543 0.1089
0.16710. L273 0.1477 0. L625 0.1641
0.78000.1641
0.0537 0.1074 0.1253 0.1451 0.1595 0.1610 0 .16100.7900 0.0531 0-1058 0.1233 0.1426 0. 1564 0.15790.8000 0.0526 0. 1042
0.15800.1212 0.1400 0. 1534 0.1548
0.8100 0.0520 0.1026 0.1192 0.13730. L5d9
0.02000. 1503
0.05150.1517
0.1010 0.1171 0.13470.1517
0.03000.1472 0.1486
0.0509 0.0994 0.11!50L1.14tib
O. 13200.0400
0.1441 0.14540.0504 0.0977 0.1129
0.14540.1293 0. 1409 0.1422
0.6500 0.0490 0.0961 0.11070.142:,
0.1266 0. L377 0.1389 u. l.tB!j0.8600 0.0493 0.0943 (3.1005 0.1237 0.1344 0.13560.8700 0.0487 0.0926
[).lU:>OO. 1062 0.1209 0.1311 0.1321 0.1322
0.8000 0.0481 0.0900 0.1039 0.1179 0, 127(50.8900 0.0475
0.12870.0889 0.1015 0.1149
rl.12H70.1241 0.1251
0.9000 0.0469 0.0870o.~~~]
0.0990 0.1117 0. 1204 0.12140.9100 0.0462
().12140.0649 0.0964 0.1084 0.1167 0.1175
0.9200 “ 0.0456 0.0827G.ll?s
0.0936 0.1050 0.1127 0.11350.9300 0.0448 0.0804
0.11350.0907 0.1013 0.1085 0.1093 cl.lu93
0.9400 0.0440 0.0779 0.0076 0.0974 O.1O4L 0.1048 cl.lU’JB0.9500 0.0431 0.07S2 . 0.0841 0.0932 0.0993 0.09990.9600 0.0420 0.0721
i~.09990.0803 0.0805 0.0940 0.0945
0.9700 0.0407 0.06850.0946
0.0758 0.0831 0.0879 0,0884 o.L)tiB40.9800 0.0391 0.0639 0.07,03 0.0765 0.0806 O.CIEIIIJ0.9900 0.0366 0.0574
[J.U81O0.0625 0.0674 0.0706 0.0709 0.07(1%
1.0000 0.0173 0.0208 0.021S 0.0220 0,0223 0,0224 0.0224
21
10°
“()Zc
ii-I,
I1
, 4
10
10
*I
ml+,
“!0, I
I
0.20 0.40 0.60 0.80 1.00
Figure 19. TEM impedance of a vertically offset cell (tabulated ,
below) {h/b) - 0.50
TEM impedance of a vertically offset cell .(plotted above)TABLE 5.
\
b/a
wlaO.Q1OO0.02000.0300
0’.5 0.667 1.0 2.0 4.0 6.00.2
0.s1370.7034
0.63660.52640.4620
0.76420.65390.5894
0.06790.75760.69310.64730.61170.58270.55810.!5.369o.51810.50130.4B610.47220.45940.44760.43660.42620.41650.40740.39870.39U50.38260.37520.36800,36110.35460.34820,34210.33620.33050.32’50
O.Bell0.77000.70620.66040.62490.59580.57120.54990.53110.51430.49910.48520,47230.46050.44940.43900.42930.42000.41130.40300.3’351o.3Ei7Gc1.3t3Q30.37340.36680, 315030.354?o.34f120.34240.33tiJi
lJ.BEJ170,7713II./tJb8O.bblu0.62540 .!5L)f>4o,571tiU.5Si)>(J.531-O.51AU.49LIfl,4Ei5.ll,f172~1).4Ellu:0,4499 ILI.439610. f12!J840.42U6$0.4119!0.4035[CJ.J9S7+
0.72730.61700.55250.5069i3.4715!3.44260.41830.39730.37880.36230.34750.33400.32160.31030.29970, 28990.28080.27220.26410.25650.24940.24260.23620,.23010.22430.21BB0.21350.20850.2037o.i991
0.63090.59320.55770.52%70.50430.48310.46440.44700.43270,41900.40640.3947o.3a390.37370.36420.35530.3469o.33a90.33130.32410.31730.31070.30440,29840.29260.2871o.2aL70.2766
0.04000.05000.06000.07000.08000.09000.10000.11(300.1200.0.13000.1400-0.15000.16000.1700
0.4164. 0..38110.35240.32B20.3074o.2a910.27280.25820.24490.2329o.221a0,21150.20210.1933
0.54370.50830.47940.45500.43390.41540.39000:30390.37030.35780.34630.33570.32570.31640.30770.29~0.291.0.20440.2774o.270a0.26450.25850. 252a0,24730.24210.23700.2322
.] o.la51
o0,
000000
19002000210022002300240025.00
0.17740.170?O. 163S0.15710.15110.14550. 14t320.13520.13040.12590.12160.L176
2600IiJ.2700o.2aob0.29000.3000
22
. .
h
h’/a. .
0.3100 cl.1137 0.1947 0.2276 0.2716 0.3197 o.33id lJ.3.3190.3200 0.1101 0.1905 0.2231 0.2660 0.3145 0.3261 u.32660.3300 0.1066 0.1864 0,2188 0.2622 0.3095 0.3210 “1).32150.3400 0.1034 0.1825 0.2147 0.2577 0.3046 o.3tFill o.31G50.3500 0.1002 Q. 170a 0.2107 0.2534 0. 299ti rJ.3111 11.31160.3600 0.0973 0.17!52 0.2069 0.2492 0,2952 o.30ti4 IJ.306Y0.3700 0.0944 . (1.171a 0.2032 0.2451 0.2907 0.3018 ().30230+3800 0.0910 0.1685 0. L996 0.2411 o.~863 0.2973 0.2s7”10.3900 0.0892 0.1653 0.1961 0.2372 0.2820 0.2928 ,,.~9330.4000 0.0068 0.1622 0. 192s 0.2335 o.277e o. 2(+0s l).2e9cJ0.4100 0.0845 0. L593 0.1095 0.2299 0.2737 0.2t44:+ lJ.2fl.\~0.4200 0.0823 0.1564 0.1864 0.2263 0.2697 o.2~$J:) ~).2806Q.4300 0.0802 0.1536 0.1833 0.2228 0.2657 0.2161 0.27650.4400 0.0782 0.L510 0.1004 0.2195 0,2619 0.272i ().27260.4500 0.0763 ‘. O. 1484 0.1775 0.2162 0,25B1 o,~(,fj~ II.268b0.4600 0.0745 0.1459 0.1747 0.2129 “ 0.2543 0.2ii41i 11.2b4&0.4700 0.0727 0.1435 0.1720 0.2098 0.2507 0.2blIt> IJ.26100.4800 0.0711 0.1412 0.1693 0.2067 0.2471 0. 256t3 IL.25730.4900 0.0695 0.1389 0.1668 0.2037 0.2436 0.2532 ().25360.5000 . 0.0680 0. L368 0.1643 0.2008 0.2401 0.2490 u ‘5tJL10.5100 0.0666 0.1347 0.1619 0.1979 0.2367 . 0.2460 (3:;4~40.5200 . 0.0653 0. 1326 0.1595 0.1950 0.2333 o.~~~~ ~.~4?9
(3.5300 0.0640 . 0.1306 0.1572 0.1923 0.2300 0.2390 Q.~394
0.5400 0.0628 0..1287 0.1549 0.1895 0.2267 0.235($ ().23600.5530 0.0616 0.1268.569S
O. 1527 0.1669 o.~235 0.2322 ~.~3~6
o 0.0605 0.1250 0.1506 0.1842 0.2203 0.2289 [J.229.3o,=y;~ 0.0594 0.12320.53:3
0. 1484 0.1816 0.2171 ~e2~5& u.2?r>o3.0596 0.1215 0.1464 0.1791 0.2140
0.59C0 0.0574
(’JT7-)Y---- 0. 22270.1198 0.1444 0.1766 ~.~~Qg cJ.2isll LJ.?l:lb
0.60G0 0.0565 0.1181 0.14240.6100
0.1741 o.207a 0.21590.0556 0.1165 0.1404 0.1717
0.2162o.~048 0.2127 ~.~>J:
0.6200 0.0548 0.1150 0. 13s5 O. 1693 0.2018 0.2096 o.~[)g~
0.6300 0.0540 0.1134 0.1366 0.1669 0.1988 0.2064 (3.2~~B
0.6400 0.0532 0.1L19 0.1348 0.1645 0.1959 0.2033 o.z~36
0.6500 0.0525 0.1105 0.1330 0.1622 0.1929 0.2002 ~.~~~s0.6600 0.0510 0.1090 0.1312. 0. L599 0.1900 0.19720.6700 0:0511 0.1076
0.19750.1294
0.60000.1576
0.05040.1871 0.1941
0.10620. 1944
0.1277 . 0.1553 0.1842 0.1910 0.19130.6900 0.0498 0.1049 0.1260 0.1531 0.1813 (3.18800.7000 0.0492
0.18830.1035 0.1242 ‘ 0.1508 0.1785
O.?loo 0.0406 0.10220. laso I-J.1853
0.1226 0.1486 (3.1756 0.1020 0. 18220.7200 0.0481 0..1009 . 0.1209 0.1464 0.1728 ~ 0.1790 0.17920.7300 0.0476 0.0996 0.L192 0.1442 0.1699 0.1759 0.17620.?400 0.0470 0.0983 0.1175 0.1420 0.1671 0.1729 0.17320.7500 0.0465 0.0970 0.1159 0.1398 0.1642 0.16990.7600 0.0460 0.0958 0.1142 0. 1376
o.17ill0.1613 0. 1669 0.1671
0.7700 0.0456 0.0945 0.1126 0.1354 . 0.1585 0.1638 0.1641o.7Eiao 0.0451 0.0933 0.1109 0.1331 0.1556 0.16080.7900 0.0447
C1.161O0.0920 0.1093 0.1309 0.1527 0. 1577 0.1579
0.8000 0.0442 0.Q907 0.1076 0.1287 0.1498 0.1546 (i.15.$90.8100 0.0438 0.0895 0.1060 0.1264 0.1468 0.1515 0.15170.8200 0.0434 0.0882 0.1043 0.1241 0.1439 0.141.14 ().14&Jti0.8300 0.0429 0.0869 . 0.1025 0.1218 0.1409 0.1452 0. 14540.8400 0.0425 0.01356 0.1008 0.1194 0. 137a 0.1420 0.14220.8500 0.0421 o,oa43 0.09 0.1170 0.1347 0.1307 Q. 13890.8600 0.0417 0.0829 o.09*- 0.1146 0.1316 0.1354 0.13560.0700 0.0412 0.0815 0.0954 0.1121 0. L283 O. 1320 0. 1322o.8aoo 0.0400 0.0001 0.0935 0.1095 0.1250 0.1285 0.12870.8900 0.0403 o.07a6 0.0915 0.1069 0.1216 0.1249 0.12510.9000 0.0399 0.0770 0.0894 0.1041 0.1181 0.12120.9100 0.0394 0.0754 0.0073
0.1214O.1OI.2 . 0.1145 0.1174 0.1175
0.9200 0.0389 0.0737 o.oa50 0.0982 0.)107 0.1134 .0.11350.9300 o.03a3 0.0718 o.oa26 0.0950 0.1066 0.1092 0.IG930.9400 0.0377 0.0698 0.0799 0.0916 0.1023 0.,1047 cl.1U4S0.9500 0.0370 0.0676 0.0770 o.oa78 0.0977 0.0998 0.CJ9!J90.9600 0.0362 0.0651 o.073a 0.0836 . 0.0925 0.0945 0.09453.9700 0.0353 0.0621 0.07003.9000
o.078a o.oa67 o.oa03 o.lJsil-10.0340 0.0583 0.0653 0.0729 0.0795 0.0810 LJ.d81!J
3,9900 0.0321 0.0529 0.0585 0.0646 0.0697 0.0708 0. (1’/()!31.0000 . 0.0162- 0.0202 0.0210 0.0217 0.0223 0.0224
1
0.0224
23.
.—. - ---- . ..- -.. . . . . .
,
.-
10°
.
{(w/a)
1.0.20. -
I0.40 0.60,.. 0.s0 I-nfi
Figure 20. TEM ~mpedance of a vertfciil!y offset cell (ta-bii~ted’below) (h/b) -= 0.7 0 i
TAGLC 6. TEM jm~edance of a ver~ica...
b]a
‘\wia
O.o1oo
0.02000.03000.04000.05000.06000.07000.08000.09000.1000O.lioo0.12000.1300i).14000. 15G00.1600
0.19000.20000.21000.22000,23000;24000.25000.26000.27!30o. 200(30.29000.3000
0.2
0.61360.5(3340.43900.39340.35820.32!350.30530.28450.26630.25000.23550.22230.2103ct.19930.18910.17970.17100.16290. 15!53o.14a20.14160.13540.12950.1240o.l~aao.11390.10920.10490. Loo?0.0960
0.5
0.67980.56950.50510.45950.424L0.3954.0.37110.3!502o.331a0.31540.30070.2073.0.27S10,26390, 2!5350.24390,23490.22850.21870.21130.20440.19780.19160.10500. ~ao2o. 17500.17000,16520.16070.1564
24
0.667 .
0.7100 -0.59970.53530.48960.45420.42540.40110.30010.36170.34520.33040.31700..30470.29340.28290.27320,26410.25560.24760.24010.23300.22630.2200o.21ao0.2002o.202a0.19770.19270. lBao0.1335
Iy Offse
1.0
-0.64680.58240.53670.!50120.47240.4.4800.42690.40840.39180.37690.36330.35090.33940.32SBo.31a90.30960.30090.29270.28500.27770.27080.2642Q.25790.2520Q.2463o.240a0.23560.2306~,2259
cell (plotted abo\
2.0
U.B3340.7231o.65a5o.612a0.57730.54830.523t30.50260.40390.46720,45200.43030.42560.41390.40290.3920t3.3B320.37420.36570.3576o.34~90.34260.33560.32900.32260.3165O.31OG0.30490:2994o.294~
4.0
0.E!7.44
0.76400.69950.65370.6182”0.58910.56450.54330.52450.507?0.49240.47850.46570.45390.442ti0.43250-42270.41360,40490.396b0.38870.38120.3740o.3t)710.3GU!=I0.3541o.34k!ll0.3421(J.33t3:j0, 33QB
LI.77L130.70570.b5990.6244U.5953o.57cJ7U.54940.531)Ul),51J130..I9H6(1..is.ll)(1..17IB0.45.99[1..1489().43115
[1 . .;267
11.4195
iJ.411Ja
0.4025IJ.J9.J151)..3t3/l,].~y<jgu. i“/,!9[’l.J61;3l),,j!~’iIl.!Il..& I—1] . .+ ‘)
l), . .
-..
●✎L.)
TABLE
T
b/a
w/a0.31000.32(300.33000.34000.3!5000.36000.37000.30000.39000.40000. 41000.42000.43000.440!30.45000.46000.47000.4800Q. 49000.5000(3.51000.52000.!530C0.54Gt30.550Q
c).~;.;:,
d . . -.5:4Go.6ci G6(3. i31Gf20.62GL30.63000.64000.65000.66000.67000.68000.69000.7000o.71ao0.72000.73000.74000.7500 .0.76000.770(30.78000.79000.00000.81000.82000.83000.04000.85000.86000.87000.08000.09000.90000.91000.92000.93000.94000.95000.96000.97000.9,0000.99001.0000
(continu
O*2
0.09310.08960,08630.06310,00020.07730.07470.07210.06970.06750.065.30.06330.06130.05950.05770.05610.05450.05300.0.5160.05030.04900.04780.04670.04560.04460.0437G.0428G.c1419G. O-J1l0.0403 .0.03960.03%90.03530.03760.03710.03650.0360.0.03550.03500.03460.03410.03370,0333.0.0330CJ.03260; 03230.03200.03170.03140.03110.03000.03050.03030.03000.02970.02950.02920.02900.02870.02850.02820.02790.02760.0273o.o~690.02650.02600.02530.02420.0139
>d)
o.*5
0.1523
0:14840, 14470,14110. 13770.13450.13130.12840.12550, 12280.12020.11770.11530.11300.11080.10860.10660.10470.10260.10100.09920.09760.09600.09440.09290 .09150.09010.0E18BG. 08750.08620.08500.08390.08280.00170.0S060.07960.07860.07760.07670.07570.07480.07390.07310.07220.07140.07060.06970.06890.06810.06730.06650.06580.06500.06410.06330.06250.06170.06080.05990.05890.05790.05690.05580.05450.05310.05160.04970.04720.04360.0187
0.667
0.17930.17520.17120:16750.16390.16040.15710.15390.15080.14790. 14510.14230.13970.13720.13470.13240.13010.12790.12580.12380.12180.11990..11810.1163 .0.11450.1129o.l~lzo. 10s70.10810.10660.10520.10380.10240.10100.090.096.0.09720.09600.09480.09360.09240.0913000
.000000.
090108000879086808570846083508250814
0.08030.07920.07810.07690.07580.07460.07330.07210.07080.06940.06790.06630.06460.06270.06050.05790.05460.04980.0197
25
1.0
0.22130; 21690.21260.20850.20460.20080.19720.19360.19020.18690.18370.18060.17760.17470. L719O. 16920.16650.16400.16150.15900.15670.15440.15210.14990.14780.1457c. 14360.14160.13970.137$0.13590.13410.13220.13050.12870.12700.12530.12360.12200.12030.11870,11710.11550.11390.11240.11080..10920.10760.10610.10450.10290.10130.09960.09800.09630.09460.09280.0910o.oe9~o.oe72o.oe510.00290.0806o.07e~0.07s30.07220.06860,06400.05750.0208
2.0
0.28910,28410.27930.27470.27020, 2659.0.26160.25750.25350.24960.24580,24200.23840.23490.23140.22800.22470.22140.21820.21510.2120.0.20900.20600.20310.20020.19740. 1946o.191d0.16910.18640. le38o.le12O. 17860. 17600.17350.17090.16840.16600.16350.16100.15860.15610 .15370.1513o.14B8O. 14640,14390.14150.13900.13650.13400.13150.12890.12630.1237 .0.1210o.lle20.11540.11240.10940.10630.10300.09940.09s70.09160.0871o.oei80.07550.06660.0219
4.0
0.325*10.32020.31510.31020.30540.30070.29610.29170.28730.28300.2789o.274e0.2-/080,26690.26300.25920,25550,25190,24830.24470.24120.23780.23440.2311o.2~7H(3.~~4,~.~~~~o.~~~~0.21490.211J3o.2oe70.20560.20250. 19950. 19b50.19350.19050.18760.1846o.~e~70.17870 ,17580.17290.16990.16700.16410.1611(3.1581o. 15520.15220.14910.14610.14300. 139!30.13670.13340.13010.12670.12330.11970.11590.11200.1 L17LJo. 103513.09~70.09350.08750.08020.0?030.0223
6.0
().331)0o.325b
0.32050.31550.31(170.30600.30130. 2g6eo.z9240.2881o. 2&390.27970 .2757~.~7J7o.2blao. ~~~o0.26(i20.25650.25280.24920.24560.74210,23870.2353CJ.23J9{),22a GU. 22>.?u .,’~2ij---0 .:>ILIH“.~~’Jf,1).21:~4!0.209> ;().20()1 \o.~~J,) ;L).199$J1
0.1969 ~o.193fj I0.1908 Io.le78o.~e470.1 EJ17[).17F17LI.1757iJ.1/27(J.IU..J7il.]0,>/u. 11>.JO[1.1(,,1(.,LJ.15/~.Il.15-15[1.lf>la1). 141J2Il.14,,(1().1.11}{II.IJti(>i}.1:)’)2~1.l:\l{J1). 12(14.1.IJ.lB,J.1..1111.11.’<[1.I1 J.i!).II J<I1
Il. 1010”11.11 !l. )/
IJ. IJ9 .14
lJ. l}ti[i~
(J. {JH(J:)
[). U?(JLIU. fJ2/4 I
. .. -.
10°
. ‘lo
Figure
-1
-, I I
0.00 0.20 0.40 O*6O 0.80 1.00
21. TEM impedance of a vertically offset cell (tabulated
below) (h/b) = ~.?
TARIC 7 TFM imDedance of a vertically offset cell (plotted abo>e)I nULL r . .-., ,-..r–
~:ja0.0100 0.58230.0200 0.4721
n.7kIll0.4991 . 0.s131 0.5359
t).:i.l1410.6016 0.6BOH
0.0300 0.4CJ77 0.4347lJ./-Jll
0.4488 0.4745 0.5372 0.6163 1’).t)!>,i~dI0.0400 0.3622 0.3892 0.4032 0.4289 0.49150.0500 0.3270
o.51Llti ,J.,,l”FI0.3539 0.3679 0.3936 . 0.4562
0.0600 0.29830.5352 l).-,{~.j
0.3252 0.3392 0.3649 0.4274 o.50 Li20.0700 0.2742
()..)4[1s,0.3011 0.3150 0.3407 0.4030 0.4818
0.0000 0.2!535U.:12lt!
0.2803 0.2942 0.3198 0.38210.0900 0.2353
o,4blI/ II.!jCI.’l>0.2620 0.2759 0.3015 0.3636
0.1000 0.21910.442LJ lJ..1H1:)
0.245% . 0.2597 0,2052 0.3472 0.42540.1100 0.2047
lJ.01,/<! 0.2313 0.2451
0.12000.2705 0.3324
0.19160.4104 Il..!!><1
0.2181 “ 0.2319 0.25730.1300
0.31890.1796
CJ,39~7 11.431130.2061 0.2199 0,2451 0.3066 0.3842
0. 140Cl 0.1687 0.1951 0.20%8 0.2340(J.42!;G
0.2953 0,37260.1500 0.1587
IJ.41J$JO. 1850 0.1906 0.2237 0.2848 0.3618
0.1600 0.14940.40,10
0.1756 0.1892 0,2142 0.2750o.170Q
0.351!30.1408
0.39LBO .L669 ‘ 0.1805
0.18000.2054
0.13280.2659 0.3424
0.1588 0.1723 0.19710.3832
0.1900 0.12540.2574 0.3336
0.1512 0.1647 0.18940.3742
0.2000 0.11640.2494 0.3252
0.1442 0.1576 0.1021o.3b37
0.2100 0.1J190.2419
0.1376 0.15090.3L74
0.17530,35.7(>
0.22000.2348
0. 105B0.3099 0.3499
0.1313 0.1446 0.1689 0.22HL 0.30280.2300 0.1001
0.J4°O. 1255 0.13s7 Q. 1629 o,221fl 0.2961
0.2400 0.0947 @1).3”
0.1200 0.1331 0..1572 0.21580.25D0 0,0897
0.28!26 LI..JL0.1148 . 0.1279 0.1518 0.2100 0.2835
0.2600 0.08500.3
0.1099 0.1229 0.1467 0.2046 0.27760.2700 o.oao5
u,;?0.1053 ‘ 0.1182 : 0.1419
0, 28000. 1994 0.27LLI
0.0763 0,1010C1.:3)i)b
0.113B 0, 1373 0.19450.290CI 0.0723
0,26650. 09i39 O. 1096 0.1330
0.3U4CIo. l139e il.26L3
0.3000 4 0.0686 0.0930 A 0.1056 0: L289(J,2994
o. ln531
0.25G3 0.2CIA2
.
26,.. . . . . ..>_-, . . -------- o—.- ..... . .. ..------ ...
.. . .
! TfiRIF 7 [rnnt;nltod]
o
Inwbt. I \wv**-v*.uti-j
ti/a
0.3100 0.0651 0.0093 ‘ 0.1019 0.1250 0.1810 0.25150.3200 0.0618
0.28910.0058 0.0983 0.1212 0, 1769 0.2469 o.2b.Jl
0.3300 0.0586 0.00250.3400
0.09490.0557
0.1177 0.17290.0794 0.0917
0.2.$24 0.27940. 1143 0 .1692 0.238J 0.274”?
0.3!500 0.0529 0.0765 0.0887 0.1111 0.1655 ,0.2339 0.27020.3600 0.0503 0.07370.3700 0.0478
0.0850 0.10s1 0.1621 0.2299 0.26590.0710 - o.oa3L 0.1052 0. L587 0.2260 0.261b
0.3800 0.04!54 0.0605 0.00050.3900 0.0432
0.1024 0.15’55 0.2222 0.25750.0661 0.0700 0.0998 0.1525
0,4000 0.04120.21E15 0.2535
0.0639 0.0?57 0.0973 0.1495 0.2149 0.~496o. 4100 “ 0.0392 0.0618 0.0734 0.0949 0.1467 0 ,2115 o. 245a0.4200 0.0373 0.0s97 o.0713 0.0926 0.1439”0.4300 0.0356
0.2081 D. 24200.0578 0.0693 0.0904 0. 1413 0.2048 0
0.4400.~~~~
0.0340 0.0S60 0.0674 0.0FIR3 0.1387 OO~(Jl~ u .~~qgD.4500 0.0324 0.0543 0.06 0.0863 0.1363 .0.4600 0.0310 0.0527
il.1965 u. 23140’.06-- 0.0844 0. 1339 0.1955 0.22FJU
9.4700 0.0296 0.0511 0.0623 0.0826 0.1316 0.14263.4000 0.0283 0.0497 0.0607 0.0009
0.22470.1294 0.1897 11.2214
D. 4900 0.0271 0.0483 0.0592 0.0792 0.1273 0.1869 cl.21823.5000. 0.0260 0.0470 0.0578 0.0777 0.1252 0.1842 0.21519.5100 0.0250 0.0458 0.056S 0.0762 0.1233 0.1815 0.21203.5200 0.0240 0.0446 0.0553 0.0747 0.1213 0.1789 o.~oQQ
3.5300 0.0230 0.0435 0.0541 0.0733 0.1195 0.1763 0.2(3603.5400 0.0222 0.042S 0.0S29 0.0720 0. 1177 0. L-?38 , 0.20313.5500 0.0213 0.0415 0.0519 0.0708 0.1159 0 .1713 o.~~~~
3.5609’ 0.0206 0.0405 0.0500 0.0696 0.1142 0. 1689~.~;~.: ~,o~99 ?3.0397 0.0499 0.0684
0.197:0.1125 0. 1606 u. l!$.16.
2.55:5 0.0192 o.03a8 0.0489 0.0673 0.1110 0.1643 c) ‘,,,!42.53L.,J o.ola6 0.0381. . 0.0481 0.0662 0.1094 cl.lti21”1 Il.1s.’):3.6000 0.0180 0.0373 0.0472 0.0652 0.1079 0 .1591 [1.181,43.6100 0.0175 0.0366 0.0464 0.0643 0.1065 0.1575 ().lH {B7.62C0 0.0170 0.0359 0.0457 0.0633 0.1050 0.15533.6303
L1.luli’0.0165 0.03s3 0.0450 0.0624 0.1036
3.64300. 1532 I’I.I?ilC>
0.0161 0.0347 0.0443 . 0.0616 0.1023 0.1511 1),170(-19.6500 0.0156 0.0342 0.0436 0.0607 0.1009 0. L49CI {).17153.6600. 0.0153 0.0336 0.0430 0.06003.6700
0.0996 0.1469 (1.17100.0149 0.0331 0.0424 0.0592 0.0984 0.1449
3.6800 0.0146il.lt,’i4
0.0327 0.0419 0.0585 0.0971 0. 1429 II,Ifi!;[lD:6900 0.0143 0.C1322 0.0413 0.0577 0.0959 0.14090.7000
l).1(;350.0140 0.0318 0.0400 0.0571 0.0947 0.1309 ,1.Lt;ll-)
r).71t30 0.0130 0.0314 0.0403 0.0564 0.0935 0. L369D. 7200 0.0135
IJ.IS-6.0.0310 0.0399 0.0557 0.0923 0.1349 [J.L,b,,l
9.7300 0.0133 0.0307 0.0394 0.0551 0.0912 0.13309.7400 0.013L 0.0303
0.12$7- 0.0390 0.0545 0 .0901 0.1310 L).151.3
0.7500 . 0.0129 0.0300 o.03a6 0.0539 0.0889 0.1291 l).l-ltib0.7600 0.0128 0.0297 0.0382 0.0s33 0.08780.7700
0.1271 0. 14t)40.0126 0.0294 0.0378 0.0528 0.0867 0.12s2 ().1439
0.7800 . 0.0125 0.0291 0.0374 0.0522 0.0856 0.1232 0. J4L5D. 7900 0.0123 0.0288 0.0371 0.0!517 0.0845 0.1212 0. 12909.8000 0.0122 o.02a6 0.0367 0. 0511 0.0034 0.11933. aloo
o. 13LS50.0121 0.0283 0.0364 0.0506 0.0823 0.1173
0.8200 0.0120 0.02810.1340
0.0361 0.0501 0.0812 0.1153 0.1315o.a300 0.0119 0.0278 0.0357 0.0496 0.0801 0.1132D.a400 o.olla 0.0276
0. 12890.0354 0.0490 o.07a9 0.1112
o.a500 0.0117 0.02740.1263
0.0351 o.04a5 0.0770 0.1091D.a600
o. 12370.0117 0.0272 0.0348 o.04ao 0.0766 0.1069 0.1210
0.8700 0.0116 0.0269 0.0344o.a800
0.0474 0.0754 0.1o47 o.lld20.0115 0.0267 0.0341 0.0469 0.0741
0.0900 0.01140.1024 n. 1154
0.0265 0.0338 0.0463 o.072a0.0000 0.0114 0.0263 0.0334
0,1oo1 0.1125
0.91000.0457 0.0715 0.0977
o.o L13I-J.1094
0.02600.9200
0.0331 0.0451 0.0701‘ 0.0113
0.0951 0. lUti30.0258
0.93CI00.0327 0.0444 0.0686
0.01120.0924 [).1113U
0.0255 0.0323 0.0437 0.0669 0.08960.9400 0.0111
u,ll~J940.0252” 0.0319 0.0429 0.0652 0.0b65
0.9500 0’.01100.0557
0.0249 ‘ 0.0314 0.0421 0.0632 o.oa3~ ().o$llb0,9600 . 0.0110 0.0246 0.0300 0,0411 0.06100.9700 0.0109
0.07940.02.41 0.030L
11.otJ71o.039a o.05a4 0.075U
0.9800 0.0107u.oe)H
0.0235 0.0292 o,03a2 . 0.055L 0.06960.9900 0.0105
U.[J7550.0226 0.0277 o. 035a 0.0502 . 0.0620 0.0666
1.0000 0.0000 0.0133 0.0150 0.o171 0.0198 0.0214 0.0219
.
27
Illustrative Example for BBC GTEH-1509
With reference to figure 14, showing the vertically offset
TEM cell, the experimental parameters for the ~TEH-1500 are
w/a = 0.6333
b/a = 0.667
h/b = 0.5
h/a = 0.333
Frcm figure 19 and the accompanying Table 5, the valqe of (Zc/n) =
~.~~~ for the ~b~~~e set of parameters. Consequently, the
C71E-:~~~~~s~jc impsdance of air-filled GTEN - 150(I is
z= z 377 X 0.136 = 51.27 Q
and the capacitance per unit length is●
c’= Comch) = 65 pF/m
Considering the approximations involved in applying the
readily available expressions from [3, 53 to the case of GTH?-1509,
the above results are seen to be in good agreement with the
experimental values.
28
0
—
d
IV. Summary
The objective of this note was to review the impedance of
two conductor transmission lines including TEN cells with the
inner septum placed symmetrically and with some vertical offset.
The required computer programs were developed and implemented
on the computer system at the BBC Research Center. Some example
results are presented here in figures and tabular form as well,
for future reference.
—
.
29
References
1.
2.
3.
4.
5.
6.
7.
8,
S. Ramo, J.R. Whinnery and T. Van !luzer,
in Communication Electronics, John Wiley
Second Edition, 1984
Fields and Waves
and Sons,
C.E, Baum, D.V. Giri and R.11. Gonzalez, “Electromagnetic
Field Distribution of the TEM Mode in a Symmetrical Two-
Parallel-Plate Transmission Line”, Sensor and Simulation
Note 219, 1 April 1976
J.C. Tippet, “Modal Characteristics of Rectangular Coaxial
Transmission Line”, Ph.D. dissertation, University of
Colorado, Boulder, 1978
M.L. Crawford, “Generation of,Standard EM Fields Using TEil
Transmission Cells”, IEEE Transactions on Electromagnetic
Compatibility, Vol. EMC-16, pp 189-195, N“ovember 1974
J.C. Tippet and D.C. Chang, “Characteristic Impedance of a
Rectangular Coaxial Line with Offset Inner Conductor’i,
IEEE Transactions on Microwave Theory and Techniques,
Vol . MTT-26, pp 876-883, November 1978
Reference Data for Radio Engineers, Published by Howard W.
Sams & Co., Inc., New ‘fork, 6th Edition, Chapter 24, 1975
EMP Handbook, Principles, Tectiniques & Reference Data,
Edited by K.S.H. Lee, AFNL~TR-80-402, December 1980, also
published by the Dept. of Electrical and Computer
Engineering, University of New Mexico, July 1983
,.Handbook of Mathematical Functions, Edited by .M. Abramowitz
and I.A. Stegun, Dover Publ~cations, New York, 1965 ‘
30
●