The N-wire exponential transmission line

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Masters Theses
Student Research & Creative Works
1968
The N-wire exponential transmission line
Vernon Ray Stanley
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Stanley, Vernon Ray, "The N-wire exponential transmission line" (1968). Masters Theses. Paper 5193.
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THE N
-:'ITRJ~ '~:XPONEN'I'L"lL
BY
'l6·!.NON H.. STANLliT.I
\C1'f~
A
THl~'3IS
subr:J5.ttcd to the :fc.cul"i:,y o:f
in pr:r·tl::>.l :fu.1.fillqcmt of the
Degree of
1968
t'(
. .L
. ~----·~
. ~
. -·-·-----c~c}~~sor)
requircT~J.ent::;
:for the
ii
ABSTHACT
This paper deals 1-r.i..th the n-"rire exponentially tapered
transmission line.
A ch..:'lracteristic im)ecbnce is defined, and the
results of terminating in this
im"~edance
are
inver;tigated.
A very
long line and the condition of no reflect:!.0ns are considered, and the
conclusions are com:9ared.
Finally, three specific 3-•·Tire line problems are 1rorked 'tvith a
ch~racteristic
impedance termin'1tion and a unit s·tep
input~
\
iii
ACKNOi·11EDGEHZNT
The author 1·Tishes to express his sincere tlPllrcciation to his major
. adVisor, Dr. EdHard C. Bertnolli, for his valuable guidance during
the
i·Jri ting
of this thesis.
iv
TABLE OF CONTENTS
Page
ABSTR.IiCT ·
. ii
AC:KNO'h'LEOOENEI:--TT
iii
LIST OF FIGURES
v
PROLOGUE TO THESIS
1
THE N-i·HRE EXFONENTIAL
3
TR.li.NSHISSION LINE
INTRODUCTION
3
II.
DEFINING CHARACTERISTIC IHPEDANCE
3
III.
INPUT IHPgD.ANCE 01? VJYRY LONG LINE
5
LINE TERT:JINATED IN CHA..RACTERISTIC HLPEDANCE
6
CONCEPTS OF 'l'R..tt!.rELING 1.JAVES
7
EXANPLE 3-1·1IRE LINE CALCULATION
8
I.
IV.
V.
VI.
VITA
EXA!·PLE
2
12
EXAJ:.!PLE
3
15
VII.
CONCLUSIONS
36
VIII.
REFE.RE.tiCES
37
38
v.
___ __
_;._
LIST OF FIGUR.ES
1
2n + 2 Terrri.na1 Ta!)ered Line of Length
2
Equivalent Circuit of Length
3
x of n-1-rlre
2n + 2
21
Terminal Exponential Line of Length d ••••••••
d \·Jith Z
0
Termination ••.••.••••••••••
~r~ple
1. • . . • . . . . . • . . • . . . . • . . . . . . . . • • •
Six Terminal LC Ljne With C3
7
Unit Step. Response of a 3'-'t·:rire LC Line.
'Exar:lpl e 1.
1.
a
= 1......................................... 25
a-::. 1 ....... .......... r....................
26
a :::. 1 ,. . • . . • • . . • . .. . • • . . • . • • • • • • • • • • • .. • • • • •
27
n
=- 0 . . . . . . • . . . . . . • . • ..... ~ . . . • . . • . • • • • • . • •
28
Unit Sten Re >ponse Q' a. 3-1·::irc LC Line.
E..~~pl e
1.
a
= 1. . . . ................................ .} ~
29
Unit Step Res:~;onse cif a 3-~rire LC Line.
ExPJn.ple 1.
1h
2l~
Unit Step Res~~onse of a 3-vrire LC Line.
E::v.Jr~Ple 1 .
13
a :::. 1. • . . . • . . • . • . • . • . . • . . . • .. . . • . • • . . . . • • •
23
Unit Step Response o.f a 3-H:i.re LC Line.
~""l,T,1ple 1.
12
0 ••.••••••••••••••••••
Unit Step Response of a 3-·uire J...C Line.
Exa.tnple l.
ll
~
22
Unit Step Response ~ a 3-~rirc LC Line.
~aJrrple
10
22
Six Ter:minal Line ShoH·ing Capacitance and
6
9
20
20
Inductance of
B
d n ••••••••••••
Distributed Line.....................................
Line of I.-ength
5
b.
11
n
=0 .. •.. ,. . .. . . • . • . • • • . • . • • . • . . . . . • • • • .. . • . .
30
Unit Step Response of a 3-wire LC Lire •
Exarrrr.>J.. e 2 •
a ::::. 1. ,. . • • • • • • • • • • • . • • • • • • • • • • • .. • • • • .. • • • • • • 31
vi.
Pnge
Figures
15
Unit ~)tep Res~)onse of a 3-'l·r.i.re LC Line.
J!;x3.i"'1plc 2.
a = 1.
..... ·• ................................... . 32
a ,. ._ J................ " •••••••.••••••• " • • . .. .. • • • • • • • .. • • 33
17
Unit Step Ttcsl_:lOnse of a 3->·Jire LC Lire •
F'x.,-unple 2 •
18
a
= 0 .•
• • • • • • • • a • • • • • • • • • • • .. • • a • e • • • • •
111
•
•
•
•
•
•
•
•
•
•
•
•
•
34
Unit Stop Response of a 3->·rire LC Line.
S;C;s.rrrple 2.
a
0 ••••••
'I·
•
•
..
•
•
•
•
•
•
•
•
.. •
•
•
•
•
•
•
•
..
•
•
•
•
•
..
•
•
•
•
35
J..
THESIS PROLOGUE
T'nis thesis is concerned vli th tne analysis of the N-1rr.ire
tial tr;msmission line.
The
<mal~:rsis
e:~:ponen-
that folloviS is considered
import;:mt because of the increased al')'l_pha.sis on mul tila'rer, micrvo-electronic, and thin-film circuits
~-~hich
make use of uniform ::rod non-
uniform distributed parameter structures.
Figure 1 shm;rs a pictorial representation of a t~ered N-I·Jire
~
line.
The exponential tr,ms,.,:lssion line differs from its classical
counternart by the fact that the per unit ir:r:Jedc:.nces and ad.mi ttances are not const.:nt but var-<J as e 2 a.x
The taper const;lllt
11
and e:.. 2 a.:x:
,
an is an arbitrary real constrmt.
can be 2ccom:plished by
physic~l·
respectively.
This variation
:construction.
T'ne distributed -parmneter aP''roach to 2Ilalysis is to subdivide the
net-vrork length;-dse j_ntc eler..ent,"'J_ sections of length
2 sh01·Js such a section located
at position x.
f:l x.
Figure
The functions
zi (x)
and Yi (x) shm.m in fig1.rre 2 specify se~ics line .~-:1:ped.ances per meter
and shunt admitta.._'l1.ces 9er meter, res!_Jectively.
A
stl"ai~htforl.·rard
application of Kirchhoff's voltage and current laws follovrod by a ljJnit
as 6 x approaches zero yields the follo1·r.ing transformed Telegraphist 1 s
Equations:
iJ V(x,s)
-o'X
~(x)f(x,s)
iJ I(x,s)
"'!..(x)y_(x, s)
ax-
•·rhcre· the above vectors and matrices are defined as follo"l'rs:
(1)
(2)
2.
V(x,s)
-.
!_(x,s)
~(x,s)
!_(x,s)
a
col(Vi(x,s))
(3)
=
col(Ii(s,s))
(4)
[ Zij(X) ]
(5)
=
·=
[ ..v·l.J·(;-)
·- ]
i , j : ~,2, •••.•• ,N
(6)
The above Telegraphists Equations (1) and (2) can be vl!'itten
in a more compact manner as
a
ax
Y._(x,:)
0
!(x)
Y,.(x,s)
(7}
-
I(x,s)
!(x)
!_(x,s)
0
The solutions to the Telegraphist's
Equations are used as a basis
for the analysis that :tollo1·1S Hi th ~~(x~ and
2
-.
proportional to e2ax and e- axrespectively.
l.!<xi being set
3.
THE 'N-:•n:RE EXFONENTIAL
THAHSlliSSION LINE
______ ,.
I. -Introduction
This thesis is concerned i·rith the analysis of the N-•·r.ire exponentr<msmission line.
The transfer matrix, Hhich relates nct'I-JOrk
input quantities to output
quC~ntities,
of the general N--vr.ire.
·~::;ponential
line, Has ~;ublished recently by Dertnolli and Vandivort(1,2).
This
transfer matrix is used as the basis for this ?Jlalysis.
~e
purpose of this thesis is to define a characteristic
i.mpeclan.ce and to find solutions for three specif.'ic
transient analysis problems.
3-~nre
line
The effect of terminating the
N-~-rlre
line
in its characteristic impedance is investigated.
II. Dcf:tning
Ch:1racteristic Impedallcc
The solution of the Telegraphist 1 s Ec1uations v-.'ith ~(x)
z e2ax
and
!(x)
=
.!
e- 2ax
=
, the :m...'l.trices Z and Y being
constant,· is found in the following forr11:
col(V(x,s),I(x,s)) ~
-
1rrhere
~
-
T(x,s)•col(V(O,s),I(O,s))
-
-
(8)
XJ.o) -
col(V21 (s),V22(s), ••••• ,v2n(s))
(9)
!(o; -
col(I2l(s),I22(s), ••••• ,r2n(s))
(10)
are the variables at the receiving end of the line as shown in Figure.
1.
The tenn !_(x, s), called the t,r<U'lsfer matrix, relates neh·rork
i..'1put quantities
to output quantities, and is eiven by the
.follouing equation: (1,2)
-1
cosh(rx) -a. rsinh(rx)
0
T(x,s)
r sinl:.(f.'X}
-T
~·
-
(ll)
E e-a."'<:
0
.
't·.rhere E is the u.11.it matrix,
and
z
f
a.
::::o
r T represents the trro1spose
!
~!
of
_,
!r )
rsinhux
I
-r.)
coshlfx t a
Jrr"
.r sinhJ:x
-T
+ a 2f_,
..r.
It ca..tl be seen from the above equations that vrhen the ta:pering
constant tra 11 equals zr::ro, the line reduces to the const<lllt pareJneter
case considered by Rice(3).
If the above matrix equations are re1r.1ritten in term.s of ex:xmential
functions the .follm-ring equations result:
(12)
and
{~ - a ..('-.,.)~0-ly2 + · (E + a_['-T)!-2}
-1/2e-axerTx {C~ - af'-T)_?.o-~ - (!£ - ar-T)!L}
!(x)= l/2e-a-Xe..r .,-x
The impedance matrix
(13)
~e2ax is arbitr<'U'ily defined to be. the line's
c~qra.cteristic L~)cdance
and is oaual to
(14)
5.
III.
Inl!ut ImPedance of'
~
Long
~
Now consider a line of length c as shoHn in Figure 3.
A
s:i.Inple change of variables will refer position to the sending end.
Let
x =d - y, where y is the
dist~ce
from the sending end.
Rearranging the above equations gives
-1
!.
where
! -1 (y,s)
~]. (y)
(y ,s)
(15)
is the inverse matrix of !,(y,s)
As the length of the line d gets very large,
!
2 and! 2 will
•
approach zero.
end by setting y
I
Kaking this approximation and considering the sending
= o,
results in the following equations:
0
0
Carrying out the indicated multiplication gives
(16)
6•
. ....
--~
and since d is very large, the ·follo~rlng approximation results
(18)
Tnis equation indicates that the input impedance is
.
z
.
:m = y-1( r ! a!)e2ad = ~e2ad
(19)
Therefore, the input impedance of a very long tapered l:ine · is
equal to the characteristic impedance, and the characteristic :impedance
could be defined to be the input impedance of a very long line.
IV.
JJine · Terminated in Characteristic Impedance
Figure
4 shm1s a line of length d, terminated at x = 0
axJ x.::.o
characteristic impedance, Z 0 e 2
-
Clearly the relationship
!
2
=
in its.
=Z-o •
Z 0 !, 2
holds and the follo'Wi.ng
equ.,."ltion may be developed from (12)
(20)
Therefore, a cho"tracteristic impedance termination has elj.minated
the e- Cx term•
In a si.rnilar manner from {13)
r 'T..
e x~-a.X]:
-2
(21)
and from equations(20) and (21)
T
V(x)
-
e rx~ 0 e 2axe-!:x!,(x)
(22)
7.
Equation (22) can be shov-m to reduce to the form
.
V(x) = Z e 2 axr(x)
-o
This equation indiqates that the voltagc. an<;!. current are
e 2a:x.
related through Z
-o
Conce'\")ts
V.
It'
~
(23.)
a.eyvmere along the line.
Traveling Waves
the or5.ginal. solution to the exponential line problem equation
(8) is re·written in terms of exponentials the follo1·ring equations Hill
result:
=l/2erx
e-ax·y_(x)
·+ 1I 2e-rx
-
{c~- a.r-')y
2+r-• ! ! 2}
r.(
~!
2
J
l
{
+ a r-•)!
cr ) ! !
-
r-• ! !
2
1
}
(24)
}
+ (E! + a\1· ) - l'e
I
}
(f T~
)Y J.. 2- (~- aff':T)- I.z
T-f
[ 01 T
2
(25)
Ii' the exponen-tial terms in equations {24) and (25) are considered
to be traveling
HaVeS,
on the line and e-
elimina~e
~·Jhen
Y.
2
r
X
the e
rx term "tti.ll travel from left to right
term Hill travel from right to left.
reflections, set the coefficient cf
e-C x
equal to zero •
.this is done, the f'ol.lovr.i..ng relationship results:
= (f +a!)-l Z f
V,.,=
Z
I_
2 =Y-l
(f! a!)!_
2
To
8.
It is, therefore, se:n that in order to
el~~ate
reflections,
the line must be terni.nated in its characteristic L-:1pccb.nce, and a
stipuJ.ation of zero reflections cmLld be used to define the
characteristic impedD.:nce.
This agrees with the previous section 1-Jhich
e-rx
concluded there nns no
ter:rn in' the current or voltage equations
1rfucn the tcrm.in..--:ttion uas the characteristic :i.rapeda.nce.
It is therefore
concluded that the concepts of tr2.veling i·raves, a very long line, and
characteristic termination a..:,'e all three compatible in defining the
character5 st,ic
VI.
in1ped..:mce: of a tapered line.
Exa..rn.ple .3-i'Iire JJine Calculation
The most general case is considered first, and the circlut
to be analyzed is that o:f Figure
capacitances and
farads/raeter
induct~ces
L 1= L 2
=
s.
In this case, the distributed
are chosen to be C 1
L .3
=1
= c
2
==
:c
.3
=
1
henr;:.rs/meter.
~ere.f9re,
z
=
r
s
[2 . -ll
and.
-1 2J
(27)
2.
is fO'lL11d using the Va1ues above as :
2.
.f=ZY+ a
2
E
=
(2~)
Since one input is an open circuit this forces I 12
eqtml zero.
The voltage V 11 is chosen to be a
vu~it
to
step or
1/s in the Laplace tr<:.ns:form dona:in.
Since the netuork is terninated in its chnracteristic i..-,pedance
the voltage ru1d current are related
~treugn
Z e 2ax I(x)
-o
x::: d at the sending end this gives
Y,.(x) ::
Since
(30)
!(d)
Substituting the value o.f
f
Vu
and I 12 and evaJ.uating
0 yields the follmrl.ng equa.tion:
1/s
v
(29)
0 2ads
-a+
12
.J a+2 3s2'
2
-1
-1
2
Iu
(31)
0
Simple algebra yields the solution:
(32)
v 12
=
-l/2s
Attention is no1-r turned to the receiving end, and the reader
(33)
10.
is referred to Figure
follolr.ing
eqtritiO~l..
5.
Again, because of the ~
0
termination, the
can be ~r.ritten:
(34)
and a straight for..rard substitution yields the folloi·Ting solution:
I ~2
=0
(35)
I 21
=
(36)
T
=
21
Now that =.
2 [ I ..
L22
!
J
has been .fotmd,
2 can be found through the relationship:
v
-2
z
= -o
1
-2
=[vn]
v
(37)
22
This gives the solutions:
v
21::: e
-ad
2 + 3s 2'
e -d)a
(38)
s
v 22 = -V2 21
The solutions in the time domain are (4,5):
(39)
ll.
rlhere J 1 (:x) in a Bessel function of the first Jd_"!d a..Dd
order, and U(t) is the
stc~)
first
i'tmctj_on.
For the case Hhere a ::. 0
the folloirlng sin.pJ.ific:.::.ticns
result:
_ t;'ds
Ill
v 21
I21
v
- -e.- -
22 -
the other. V2.I'iables rcr;tc:ti.11inG the s:;r'le.
In the tine clon<-dn
I 1l (t)::: ~3/2 U(t)
V 21. (t) -
I 21 (t) :-:: D/2 U(t-{3 d)
7 22(t) =. -1/2 U(t
U(t -
fJ'
d)
-6 d)
12.,
Since the time domain solutions could not be found in a closed
finite form, a va.J..ue o:r the tapering constant
nan must ·1;>e chosen so
the ~.ntegration can be carried out nrunericaJJ.y.
A value of 1 ·is chosen
for "a" and the plot of the responses is sho1-m in f-igures 7 ,8,9 ,and
10.
t\Tith a= 0
, the constant parmnetor case, the responses are sho-vm
in figm--es 11,12, and 13.
There is a l!k'U'ked cU.fference betHe en the
constant parruneter and 'k"lpered current response, the tapered current
response being unbom1def.
It should also be pointed out· that the
tapered line acts as a transformer.
The tapered line . introduced time
delay as did the constant line case, but
j.n
the t.:=tpercd case the load
responses are di:r£erent in ma&.nitude from the source quantities as
opposed to the constant r,1agni tude in·'.the constant line case.
As·: seen
from the ana.lytic solutions, for very large time the source and load
variables -vdll be approximately eqt"k"il.
E:x:anmle 2
The above problem is repeated Hi th the parameters changed to
C 1 =- C 2
1 2
=
=
farad/meter and 1 1 = 1 3
1, C 3 = 0
0 he:nr;Vs/metcr.
=
1,
The circuit is agnin terminated in its
characteristic impedance.
Hith these
ne..r parameters the Z nnd Y matrices take on the
values
z
!.
1
8[
0
·a]
1
13.
?..
and
r is i'Ol.md using the above values as
ZY + a
2 .
E
(h5)
AgainJ the ch.2.1'acteristic 'llapeclance is foundJ and it turns out to.
be a particuJ.arly simple form.
~0
(h6)
and Hhen the analysis is carried out in a proccdm·e parallel to the
previous e);:,::;rpleJ the
follo~ring
solutions result:
In
(4?)
v
v 12
22:::. 0
(h8)
I 21
I 12
et~9)
I 22 -:-
(50)
0
_d
v
21 :::: e
-ad
,\J<l
I 2
e
---s
+
21
s-
The solutions in the tirne_ domc:d.n are:
(51)
·
.LLJ..
= e-2ad [~at
(52)
)r 21 ( t:::
e -2ad L-e at
(53)
I ll (t)
(54)
Again a value o:f
11
a.U must be chosen, a : ;: 1, and the integration
carried ou:t numerical~y.
and 16.
The results are
sho'tm in .figures
The ca~>e for a ::.0 is sh01,m in figures 17 and 18.
14, 15,
Notice· ·
should again be taken of the unbounded nature of the response currents
for .the a~ 1 case, and of the ti·ansformer effect betueen load and so1trce varia1)1es.
Novr considering the r.8.r:::e
a ::. 0
I l l = 1/s
v 12 = v
I 2l
=-
22 -.:.. 0
=
I 12::. I 22
v
22
v 21 =
1
e-sd
s
0
e -sd
a
e -sd
s
Or equivalently i..'rl the time dornain
gives the solutions:
15 •.
I 11 -::: U(t)
U(t - d)
U(t - d)
v
21
(t)
U(t - d)
Exanrole 3
-~--
This eY. .a.mple represents a prG.ctical situation in which the
parrunoters to.lce on the values C 1 :::. C 2
meter &"ld L 1 ::::: L 2
=-
~
1, C 3 :::. 0
farads/
L 3 =. 1 hel:1...rys/meter.
'rhe circuit is sh01·m in figure 6 J 'Hhere it is ngain terminated in
its characteristic inpedance.
~L'he
characteristic :i.mped<.mce must. again be found but it
presents more op a oroblem this
diagonal.
To find~
eigenva.lues of'
r 2.
0
tir~e
since some of the matrices are not
Sylvester's Theorem (6) Hill be used.
a.rc found as f'ollo-vTs :
0
The
.J..o.
which gives the folloHing solutions:.
s
2"
+
a
2
a
2
(56)
In this special case Sylvester t s Theorem may be H:ri tten as
cr
-;t
2.
(57)
Tr
'j::.!
Jtt:
~
Where f(
L ) is
c>.:ny function of
r
r
2
and -\ :repreGents the resrective
~
eieenv<liue of _[' •
fu.n.ction of
r
The cllaracte:ristic inpedance is expressed ·as a
?...
in the follovring r:lD...'I'Jller:
(58)
Applic~tion
of Sylvester 1 s Theorem;yields the follo-vr:ing for! o•
( ~ a2
+ s2'-t ~a2 -t- 3s2 '
-2a)
(
~a2+
2s
z
s 2'
-i a 2+ 3s 2
I }
2s
-
(59)
-o
(~ a2+ s 2' -Ja2+ 3s 2 ')
2s
( Ja2+
s21
+-4 a2 +
2s
3s2' - 2a)
17.
The circuit is again .:malyzed using the same !"'rocedure as. used
in the previous e:xa.mple. ·· The sending end is. malyzed first,, <2nd ·the
follo1;·r.Lng values are found:
2 8 -2ad
In
(60)
v 12
(61)
Again, going back. to the originitl cqrotion and using the
relation'!.. 2
~ 0
I 2 -vre get:
(62)
Since "t·re r..no't·T the va~ue of the m.."':.trix !(d), vTe need to find
!
2•
This is fotm.d r.tost easily by mldl:lg usc d: Sylvester's 'l'heoren
.
.
.
.
~~
to take the inverse of the (!IJ.<;-mtity -vrhich is a fi.mction of I
~·Jii.cn
this is carried out, the folloHing solution results:
ad
I 21
=
e
.
(e-d\r a2+ s2 + .
~a2 ~
s 2 '+
~ a2 +
e-d
3s2
~) + 3s2'
.)
(63)
18.
e-d~ a.2.,_
e-ad
I 22
~~ 2~ s2 1
+
s2, -
e..d
~ a2~-3;2'
~2:;_---37'
(64)
-2a
The matrix:! 2 in aga:i..n ffnmd using the rr3lation
v,
-.c_2
·=
[vv J
21
22
(65)
-
i"Thich eives,
I 2
\
-d
( '\1 a + 3s2 - a) e
~a2+3s 21·
2a)
and
v
(~~2 + 3s2
22
_ a)
~-d·!a2 ; 3l
-2a )
If He again assume a
case.
= o,
th:is reduces to the constant
The ab9ve solutions then reduce to the follo-vr.ing
simvlified forms:
2
1
I 11= - - -
l+IT
V 12 I ').,
s
(iY - 2 ) 1/s
1
e -sd+
6
-YJ'd.s
par~eter
19.
1
e
I 22
11 I')'
v 21
l+"fJ'
v 22
ru1d
1
s
ff e-D~d
s-
_13' e~ 13' sd
e -sd
1+ lJ
s
the characteristic impedance Z
-
(1 t YJ)
z
-o
e - 13"Sd
e-sd+
l
--
-sd
-
-0
·is resistive,
.
(1 - {3)
(68)
1/2
(1-t
f3
1
)
20.
r'll(s)
+o-----=~---~
vll(s)
+o
-+
I12(s)
Q~---'3>------7
ifln\s)
v
•.;.o----;;;>::;;:...---1
(S)
Zh
X
<-J.
·--------------~
1 J.(x+l'q:,s)
Axz,(x)
+~·-------~----~~
. ( X+
l).tx;s')
~------~----~---~---~+
V:1 l ~ : ·~ )
t\ ~, S )
.,__ _ _--1----::T----=:J---~
+
V'2 <z, s >
~-----~--~----r---~+
•
•
•
•
•
•
•
•
Ax Zn(x)
-~o------------V\1\i\(---------T...1-t....,...... , .... ,....
#""\
~
.... .t--~-~,_......,..a.,.
1"1:.:.-- ...
.I..L.
-L'
T - ...... _...._,__
X~-1
'"'----.../!'a-
'tr.l-~-
+
21.
1
d
<.:
t-
(7\-
V11cs)
Yz I ( s)
-t-
+
Vu(s)
V,'l.(s)
f!J-
+- r}--·----------------~---:-----o +-·
Ven (5)
---·-c-
-<$:-----~1.
)::.
1='1~-
3
2
n ·r-
0 F
?-·
'[EI~MtNAL
U:.. N G; IH
d •-
E.'i-PO!-.fE.l\J\iAL
. II
22.
.IZI(,/1
r--·-l--:-,-,(~-----------·-__A_I\\E.I::_I~l?~.~---· -
o--==-=-------··--------·-------·----.::::-:..:..7
___t c---~
.
t
Vzlh)
Vii(S)
l
-------------b----~
~~-~-----·-
1-
-0
------<.:~
..'•
•
0-----------------------------------FiG. 4·
r
------··· .
Li N E. .
L
I
d
0 F
L 1:: l\1 G, \ ~ ·
--·(j·--.
W I n~
d
I
M•=.-.·r:.r=<s
~,;.-,
----··--·-··
.
-""'"""--r----·1- -.- .....
:)~
1
.
.
L_
. _· ..•... _.rv"""'"-·-··j··------:'l---·- ..•.. -----c·------1--..o----· ......
.
----------
-----c----..-----
C
II
~-r-
V;?l
'-;.
r
··-:>-
L lz= o
. cz '"T-
.
Tc3
Vn
l.....J
f
• -"'1:'"\f"\1""\-
FtC,.
.
,• ...
r
·------""0
Ct\Pi\C..ITANCE.
5
f.\ tJ D
i
1'-.1
D U C. T 1\ 1~ C f:_
.0 F
'!=_ 1-. 1\ M P L
E..
i .
'-...J•
Zo
Tor:m~na
S>----
:~'igure
6. Six Terminal LC Line
~{ith
c3.,=- o.
t:i n
<.)
J
,.J
\
fl!.
$
f
{q
.:!.
tL
0
uJ
(J}
2
0
n.
~
tr
Q.
U1
II
~
•
"\"I
1:1
J
a.
·:i
<(
.,t.
JJ}
!::
z
:::;:,
,....
.
<!J
lJ..
~
2
0
<J
\-
U}
«)
-t4
w
2
J
l.!.l
V)
.....,
uJ
~
i=
25.
J
.J
LJ
Jt
-
s
I
f!)
<
IJ..
0
w
!/)
":2.
.....\
II
rd
•
~
uJ
.J
CL
a.
JJl.
<(
Ct.
IJ..I
0
!,})
lt
lJ.l
1(/)
z:::>-
1-
~
~
,......
tlj
0
•
"Z
'Z
uJ
V)
v
0
(.)
tiJ
..J
ul
:i:
!=
~
t.S
IL.
•
;!1'l
J
.J
tJ
~
<:....
'
rf)
.
>((.
u.
0
w
~
tl
rd
liJ
z
0
1)..
tJ)
-.-1
uJ
•JJ
Q!
-'
a.
0..
<:(;
•.tJ
0
.,
~
~
l- w
{I)
1- w
z
z
::> J
.
0"
6
iC
ul
.s.:
f-
.
0
"-I.
(.)
....l
0
-
,j
w
oc
3r
11)
1:)
,...
\)
r.:!:
""'II
I!.
rd
Q
(ll
0
"Z
Ill
<J
z
0
-
U)
Ill
0
(/)
uJ
~
0
a.
"i-
1.:1
:Jl
a;
i=
(!..
1:1
l-
<I)
<)
>Q
2
:::>
.
...
~
..."
....
u!
J
0..
z4.
~
uJ
~
J- uJ
0
0
•
(}1
G:
z
...1
28.
2
3
·seconds
t
~XL'X:i p 1 e
At<\PS
1.
11
a 11 :=
Q
tJN\T STE.'? 'Rf:.SPCI\lS\:..
0\=
LI~E.~
·a= o,
E:.i.A~<~PI-F_ 1~
I21 ( t}
1
.866
1
2
t
seconds
3
4
11.
3- \ 1JiP-E
LC
.
0
0
uJ
to
...l
11
a:: r;l.
$ 0
I
!Y)
_....
(/')
D
<(
:z
u...
(..)
0
c
w
tl}
u.!
U)
z
0
!l..
':/}
~
"
.o
....
uJ
..J
c.
w 2
cr <(
0.
:><
IU tr.l
1-
(/)
,_
-z
td
:z
:::> _l
0
30.
r21 ( t)
VoLT.S
1 .
. 866
1
2
t
3
seconds
E:<o.l:Jple 1.
"a"=
0
v22 ( t)
\/oL-TS
Fl~.
13
t)N\T
STl:.P RE.f)PONS\:. OF
Ll Nt. ..
2
-.5
t
seconds
Exnmple 1.
Q •.
3
- 1
11
au
=0
A 3·-WIRE.
t
•]
I
I
I
!,-,.
'ill
II)
<J
!7.
10
J
l
!
-J
I.!J
:;I]
'v
IJj
lui
0::
'$ ......
I
("()
.:(
I.L
()
ll
rd
•
<\l
t_,j
I~
I
0
\.'\
V)
z
ul
1)..
:I}
D..
0
uJ
~
11.
u.l
J-
V)
1.::
z
::>
_j
:z
<(
·,(,
1.!.1
., 0
I~
.
w
-z
J-
.
()
.
rt)
-..1:-
cl-
u..
.
Q
N
u
-l
U.l
ct
~
I
ft)
•
<!
-M
IJ..
II
0
rU
!IJ
<I)
<Q
(\f
U)
uJ
J
r.L
Ill
~
n.
tr.J
,.,..
0.
:E
<!
-.,.<:
fJ]
1r.J ......
lJ)
1-
::z
::;)
..
f}~
ul
2
"'l;:::.
....J
0
"Z
tJ
u!
fl)
o....,
.
b)
0
.r.-
0
<.)
.-I
ul
(t
-
<
I
.
fY)
.<:£
..-I
11'
lL
0
w
~
t.IJ
{\!
0
u.l
:z
Q...
<fl
w
~
,....
U')
...I
0
"Z
0..
z
0
c)
<:!
·><
a. IU
u.l
~
l!.l
'-J
!-
u.l
(J')
1"'Z
::>
l::
,:
t.J
2
.J
.
()
~
.s
.
c.5
l.l.
'.n
..:.
Ill
(\).
H
a.
'f:
<(
....9
\
~·
'
l_
n
.)LJ.e
,.
o ~~ I • I
/'·'.
'1
r----------------------------------------------------
2
1
3
seconds
t
Example 2.
"~"::::.
0
UNIT ST£~ RE'SPOI'tSE. O!=" A 3-'f./IRE
fiC1. l7
LtNE.. ..
E~AMPLE.
1
3
2
1
t
seconds
2..
3
::=.0.,
LC
VII.
Conclusions
The soJ_utior.s found in this thesis are :for the e:-Qonentiruly
tapered tral"l.smission line and can be used .for a. constant parameter line
by setting the taper constant "a11 equal to zero.
Although the
ex.a.mples Harked here 'iere .for LC lines, the general equations
developed in the first part of fue thesis a})l')ly equally l·rell to
eh~onential
axrr
line.
It is ap-parent that anc"llysis of the problem is much simplified
.
-
'1-Jhcn the J.ine is terminated in the characteristic irape<'l.ance,
since this €limina:tes the reflec"Wd uaves of both current and vol ta.ge •
It should be noted from the e::camples, that the current responses
increased without bonnd in the tapered cases, but 1rere bounded for the
constant pnrameter cases.
A"l.other
..
i.rn.portant dif.ference beti·reen the
tapered line and the constcmt para."neter line is that the ta:x:1red line
acts as a transformer.
The responses at
t.~e
in time, but the ma.gni tude is also different.
load are not only delayed
':Jhen the tc;.pering
constant nan equals zero, this rednccs to the constant parameter line
case and it is seen that the
transforr:~er
between the source <.>nd load variables.
effect no longer holds
VIII.
Hcferences
1.
C.A. Vondivort and B.C. J3ertnolli, 11 De'termining the
Trr~ns:fer !!:·; trix of Tapered ~·:ul ti ~·Tire Trnns:-nl ssion
Lines", 10 th Midwest S:mposium qn Circuit Theory,
Purdue University, 1'-~ay 1967
2.
E. 0. Bertnolli, "Anr:tlynis of the t:r-;nre l'~.x:ponentinl
Line", IE3E Proceedings vo1.55,No.?,pp.l225,July,l967
.3.
Rice, s.o., "Steady State Solutions of TransmisGion
Line Equations", The Bell. 8,;"8te~ Technicn..1 Journal,
VOL. 20, No. 2,pp. 131-17", April 1941
4.
P.A. ~··cc.)1lum, B.F. Brow·n, '"Loplace Trnnsfor::1 Tables
a:-:d The;)rems, Holt, H.ienhart, ·.nnston,l965
5.
G.E. Roberts,H. ·Kaufman, Table of Lenlnce Trannforms,
:?. Saunders Co., 1966
-
1;{.
6.
P. i·L Do russo, R ...J. :~oy, St2. te Vs.ri :1 bles for .El~f:ineers,
John Wiley ond sons, Inc.,l96~
7.
J. J. K8rak~, sh, Tr~.nsmission
Hncr.tillan Co., 1950
8.
Fra: er, Dunc1'=ln, 9.nd Coll~~r, ].le•nentnry ~1atrices,
Ca:u•.~ridge:
Cambridge University Fress ,. 195'7
L~nos ~
Filter
~:ct;-rorks,
132976 .
_./V•
Vita
The author nas born on January 19, 19h3 in Oak Hill, H. Va.
He received his primary and
~econdary
education at Fayetteville,
·H. Va., end then enroJJ.ed in ':lest Vj_rginia Ll"lstitute of
Technology in 1961 at
Hontgome~J, 1·1,. Va~
He received his
fuchelor of Science in Electrical Engii1eering in Juno 1966.
has been a f'u1.1 tirae student at the University of r·1issom"'i
at Rolla since Se1")tember 1966.
He