Lossy Transmission Lines Terminated by Parallel Connected RC-Loads and in Series Connected L-Load (I)

International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.3, Issue.3, May-June. 2013 pp-1410-1418 ISSN: 2249-6645
www.ijmer.com 1410 | Page
Vasil G. Angelov, 1
Andrey Zahariev2
1
Department of Mathematics, University of Mining and Geology” St. I. Rilski”, 1700 Sofia, Bulgaria
2
Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv, Bulgaria
Abstract: The present paper is the first part of investigations devoted to analysis of lossy transmission lines terminated by
nonlinear parallel connected GC loads and in series connected L-load (cf. Fig. 1). First we formulate boundary conditions
for lossy transmission line system on the base of Kirchhoff’s law. Then we reduce the mixed problem for the hyperbolic
system (Telegrapher equations) to an initial value problem for a neutral system on the boundary. We show that only
oscillating solutions are characteristic for this case. Finally we analyze the arising nonlinearities.
Keywords: Fixed point theorem, Kirchhoff’s law, Lossless transmission line, mixed problem for hyperbolic system, Neutral
equation, Oscillatory solution
I. INTRODUCTION
The transmission line theory is based on the Telegrapher equations, which from mathematical point of view
presents a first order hyperbolic system of partial differential equations with unknown functions voltage and current. The
subject of transmission lines has grown in importance because of the many applications (cf. [1]-[9]).
In the previous our papers we have considered lossless and lossy transmission lines terminated by various
configuration of nonlinear (or linear) loads – in series connected, parallel connected and so on (cf. [10]-[16]). The main
purpose of the present paper is to consider a lossless transmission line terminated by nonlinear GCL-loads placed in the
following way: GC-loads are parallel connected and a L-load is in series connected (cf. Fig. 1).
The first difficulty is to derive the boundary conditions as a consequence of Kirchhoff’s law (cf. Fig.1) and to
formulate the mixed problem for the hyperbolic system. The second one is to reduce the mixed problem for the hyperbolic
system to an initial value problem for neutral equations on the boundary. The third one is to introduce a suitable operator
whose fixed point is an oscillatory solution of the problem stated. In the second part of the present paper by means of by
fixed point method [17] we obtain an existence-uniqueness of an oscillatory solution.
The paper consists of four sections. In Section II on the base of Kirchhoff’s law we derive boundary conditions and
then formulate the mixed problem for the hyperbolic system or transmission line system. In Section III we reduce the mixed
problem to an initial value problem on the boundary. In Section IV we analyse the arising nonlinearities and make some
estimates which we use in the second part of the present paper.
Fig. 1. Lossy transmission line terminated by circuits consisting of RC-elements in series connected to L-element
II. DERIVATION OF THE BOUNDARY CONDITIONS AND FORMULATION OF THE MIXED PROBLEM
In order to obtain the boundary conditions we have to take into account that if  is the length of the transmission
line then, LCLCT  )/1/( where L is per unit length inductance and C – per unit-length capacitance
Lossy Transmission Lines Terminated by Parallel Connected
RC-Loads and in Series Connected L-Load (I)

In accordance of Kirchhoff’s V-law (cf. Fig. 1) we have to collect the currents of the elements pG and pC after
that to collect the voltage of ppCG with the voltage of )1,0( pLp . But we deal with nonlinear elements, that is,
)1,0(,)(
1
)(
 

piriR
m
n
np
np and 


m
n
np
np iliL
0
)(
)( , )()(
~
,.)(.)(
~
0
)(
uCuuCiliiLiiL pp
m
n
np
npp  

;
);1,0(,
)(
)(
))(()(
~






 p
di
idL
iiL
dt
di
dt
iLid
dt
iLd p
p
pp
).1,0(,
)(
)(
))(()(
~






 p
du
udC
uuC
dt
du
dt
uCud
dt
uCd p
p
pp
One can formulate boundary conditions corresponding to Fig. 1: for )),0((0 0000
tiiiix CGCG 
)()(),0(
),0(
)),0((
)),0((
),0(
),(),0()(
)(
0000
00
0
000
00
000
00
000
00
tEtutu
dt
tdi
tiL
di
tidL
ti
uGti
dt
du
uC
du
udC
u
CG
CG
CG
CG
CG
CG
CG
CG




















(1)
And for )),(( 1111
tiiiix CGCG  :
).()(),(
),(
)),((
)),((
),(
)(),()(
)(
1111
11
1
111
11
111
11
111
11
tEtutu
dt
tdi
tiL
di
tidL
ti
uGti
dt
du
uC
du
udC
u
CG
CG
CG
CG
CG
CG
CG
CG























(2)
Here we consider the following lossy transmission line system:
0),(
),(),(
0),(
),(),(












txRi
x
txu
t
txi
L
txGu
x
txi
t
txu
C
(3)
         ,0,0,:,),( 2
txtxtx
Where ),( txu and ),( txi are the unknown voltage and current, while L, C, R and G are prescribed specific
parameters of the line and  > 0 is its length.
For the above system (3) might be formulated the following mixed problem: to find ),( txu and ),( txi in  such
that the following initial conditions
  ,0),()0,(),()0,( 00 xxixixuxu (4)
And boundary conditions (1) and (2) to be satisfied.
III. REDUCING THE MIXED PROBLEM TO AN INITIAL VALUE PROBLEM ON THE BOUNDARY
First we present (3) in the form:
0),(
),(1),(
0),(
),(1),(












txi
L
R
x
txu
Lt
txi
txu
C
G
x
txi
Ct
txu
(5)
And then write it in a matrix form

0),(
),(),(






txBU
x
txU
A
t
txU
(6)
Where













































x
txi
x
txu
x
txU
t
txi
t
txu
t
txU
txi
txu
txU
),(
),(
),(
,
),(
),(
),(
,
),(
),(
),( , .
0
0
,
0
1
1
0


























L
R
C
G
B
L
C
A
In order to transform the matrix A in a diagonal form we have to solve the characteristic equation: 0
/1
/1





L
C
whose
roots are ,/11 LC LC/12  . For the eigen-vectors we obtain the following systems:
0
11
0
11
21
21




LCC
LLC
And
0
11
0
11
21
21




LCC
LLC
.
Hence        LCLC ,,,,, )2(
2
)2(
1
)1(
2
)1(
1  
Denote by  the matrix formed by eigen-vectors










LC
LC
H and its inverse one














LL
CC
H
2
1
2
1
2
1
2
1
1 . It is known
that 1can 
 HAHA , where










LC
LC
A
/10
0/1can
.
Introduce new variables Z = HU, (or U = H-1
Z)























),(
),(
,,
),(
),(
txi
txu
U
LC
LC
H
txI
txV
Z .
Then
),(),(),(
),(),(),(
txiLtxuCtxI
txiLtxuCtxV


(7)
or
).,(
2
1
),(
2
1
),(
),(
2
1
),(
2
1
),(
txI
L
txV
L
txi
txI
C
txV
C
txu


(8)
Replacing ),(),( 1
txZHtxU 
 in (6) we obtain
      01
11





 

ZHB
x
ZH
A
t
ZH
.
Since 1
H is a constant matrix we have

    0),(
),(),( 111





 
txZBH
x
txZ
AH
t
txZ
H .
After multiplication from the left by H we obtain     011





 
ZBHH
x
Z
AHH
t
Z
, i.e.
  01





 
ZHBH
x
Z
A
t
Z can
. (9)
But




































LL
CC
L
R
C
G
LC
LC
HBH
2
1
2
1
2
1
2
1
0
0
1






































L
R
C
G
L
R
C
G
L
R
C
G
L
R
C
G
2
1
2
1
2
1
2
1
.
Applying Heaviside condition
C
G
L
R
 we obtain







































































0
0
),(
),(
0
0
),(
),(
1
0
0
1
),(
),(
txI
txV
L
R
L
R
x
txI
x
txV
LC
LC
t
txI
t
txV
. (10)
The new initial conditions we obtain from (4):
)()()()0,()0,()0,( 000 xVxiLxuCxiLxuCxV  ,   ,0x (11)
)()()()0,()0,()0,( 000 xIxiLxuCxiLxuCxI  ,   ,0x . (12)
One can simplify (10) by the substitution:
),(),(
),(),(
txIetxJ
txVetxW
t
L
R
t
L
R


Or
),(),(
),(),(
txJetxI
txWetxV
t
L
R
t
L
R




. (13)
Substituting in (8) we obtain
).,(
2
1
),(
2
1
),(
),(
2
1
),(
2
1
),(
txJe
L
txWe
L
txi
txJe
C
txWe
C
txu
t
L
R
t
L
R
t
L
R
t
L
R




(14)
Then with respect to the variables ),( txW and ),( txJ (10) looks like:
.0
),(1),(
,0
),(1),(












x
txJ
LCt
txJ
x
txW
LCt
txW
(15)

The mixed problem for (1) - (4) can be reduced to an initial value problem for a neutral system. The neutral system
is a nonlinear one in view of the nonlinear characteristics of the RGLC-elements.
From now on we propose two manners to obtain a neutral system for unknown voltage and current functions.
First manner: The solution of (15) is a pair of functions  vtxtxW W ),( and  vtxtxJ J ),( , where W and ФJ
are arbitrary smooth functions. From (14) we obtain
    
    .
2
),(
2
),(
vtxvtx
L
e
txi
vtxvtx
C
e
txu
JW
t
L
R
JW
t
L
R




(16)
Hence
 
 .),(),()(
),(),()(
txuCtxiLevtx
txiLtxuCevtx
t
L
R
J
t
L
R
W


(17)
For x we obtain
 
 .),(),()(
),(),()(
tuCtiLevt
tiLtuCevt
t
L
R
J
t
L
R
W


(18)
Let us put  vTTtvttvtvt /'/''  and then replacing t by Tt ' in the first equation of (18) we get
 )',()',()'(
)'(
TtiLTtuCevt
Tt
L
R
W 

.
For the second equation of (18) we put
 vTTtvttvtvt /''/''''  And then we have
.)](Λ)(Λ[)''(
)''(
,t''-TuC-,t''-TiLevt
Tt
L
R
J


So we obtain
 ),(),()(
)(
TtiLTtuCevt
Tt
L
R
W 

(19)
 ),(),()(
)(
TtuCTtiLevt
Tt
L
R
J 

. (20)
From (16) by x = 0 we have
    
    .
2
),0(
2
),0(
vtvt
L
e
ti
vtvt
C
e
tu
JW
t
L
R
JW
t
L
R




(21)
Substituting )( vtW  and )(vtJ from (19) and (20) into (21) we obtain:
         









TtuCTtiLeTtiLTtuCe
C
tu L
TtR
L
TtR
,,,,
2
1
),0(
)()(
         









TtuCTtiLeTtiLTtuCe
C
ti L
TtR
L
TtR
,,,,
2
1
),0(
)()(
.

Substitute the above expressions into the boundary conditions (1), (2) we have
          ,
)(
~
)(
)(
~
2
,,,,)(
00
000
000
00
000
0
0
)(
0
)(
00
CG
CG
CG
CG
CG
L
TtR
L
TtR
CG
du
uCd
uG
du
uCd
Z
TtuTtiZeTtiZTtue
dt
tdu




         
         
,
/)),0((
~
2
)(2)(2,,,,
,,,,
2
1
000
0000
)(
0
)(
0
)(
0
)(
0
CG
CG
L
TtR
L
TtR
L
TtR
L
TtR
ditiLd
tEtuTtuTtiZeTtiZTtue
TtuTtiZeTtiZTtue
dt
d
Z























11111
11111
/)(
~
)(),()(
CGCG
CGCG
duuCd
uGti
dt
tdu 
 ,
.
/)),((
~
)()(),(),(
111
111
CG
CG
ditiLd
tEtutu
dt
tdi




Let us put Tt  .Then we arrive at a system that we cannot formulate an initial value problem.
Second manner
We proceed from (14) and obtain
),0(
2
1
),0(
2
1
),0(
),0(
2
1
),0(
2
1
),0(
tJe
L
tWe
L
ti
tJe
C
tWe
C
tu
t
L
R
t
L
R
t
L
R
t
L
R




And
).,(
2
1
),(
2
1
),(
),(
2
1
),(
2
1
),(
tJe
L
tWe
L
ti
tJe
C
tWe
C
tu
t
L
R
t
L
R
t
L
R
t
L
R




Substituting in (2.1) and (2.2) we obtain
00000
00000
/)(
~
2
))((2),0(),0()(
CGCG
CG
t
L
R
t
L
R
CG
duuCdL
tuGLtJetWe
dt
tdu 


;
000
00000
/)),0((
~
)(2)(2),0(),0(
),0(),0(
CG
CG
t
L
R
t
L
R
t
L
R
t
L
R
ditiLd
tELtuLtJeZtWeZ
tJetWe
dt
d 












;
11111
11111
/)(2
))((2),(),()(
CGCG
CG
t
L
R
t
L
R
CG
duudCL
tuGLtJetWe
dt
tdu 


;
111
11100
/)),((
~
)(2)(2),(),(
),(),(
CG
CG
t
L
R
t
L
R
t
L
R
t
L
R
ditiLd
tELtuLtJeZtWeZ
tJetWe
dt
d














.
But
),(),0(),,(),0(),(),0(),,(),0( TtJtJtWTtWtJTtJTtWtW  .
We choose ),()(),(),0( tJtJtWtW  to be unknown functions and then the above system becomes

.
/)),((
~
)(2)(2)()(
)()(
;
/)(2
))((2)()()(
;
/)),0((
~
)(2)(2)()(
)()(
;
/)(
~
2
))((2)()()(
111
1110
)(
0)(
11111
111
)(
11
000
000
)(
00)(
00000
000
)(
00
CG
CG
t
L
R
Tt
L
R
t
L
R
Tt
L
R
CGCG
CG
t
L
R
Tt
L
R
CG
CG
CG
Tt
L
R
t
L
R
Tt
L
R
t
L
R
CGCG
CG
Tt
L
R
t
L
R
CG
ditiLd
tELtuLtJeZTtWeZ
tJeTtWe
dt
d
duudCL
tuGLtJeTtWe
dt
tdu
ditiLd
tELtuLTtJeZtWeZ
TtJetWe
dt
d
duuCdL
tuGLTtJetWe
dt
tdu

































(22)
We notice that if (22) has a periodic solution
 )(),(),(),( 1100
tJtutWtu CGCG
Then functions )(),( tJetWe
t
L
R
t
L
R

are oscillatory ones and vanishing exponentially at infinity. Therefore we put
)()(),()( tJetJtWetW
t
L
R
t
L
R



and then we can state the problem for existence-uniqueness of an oscillatory solution
vanishing at infinity of the following system (we denote by )(),( tJtW

again by )(),( tJtW ) and obtain:
);;[,
/)),((
~
)(2)(2)()()()(
);;[,
/)(2
))((2)()()(
);;[,
/)),0((
~
)(2)(2)()()()(
);;[,
/)(
~
2
))((2)()()(
111
11100
11111
11111
000
00000
00000
00000

















Tt
ditiLd
tELtuLtJZTtWZ
dt
TtdW
dt
tdJ
Tt
duudCL
tuGLtJTtW
dt
tdu
Tt
ditiLd
tELtuLTtJZtWZ
dt
TtdJ
dt
tdW
Tt
duuCdL
tuGLTtJtW
dt
tdu
CG
CG
CGCG
CGCG
CG
CG
CGCG
CGCG
(23)
,)(,)( 11110000
T
CGCG
T
CGCG uTuuTu  ],0[,
)()(
,
)()(
),()(),()( 00
00 Tt
dt
tdJ
dt
tdJ
dt
tdW
dt
tdW
tJtJtWtW  .
IV. ANALYSIS OF THE ARISING NONLINEARITIES
First we precise the definition domains of the functions:
)1,0(,
)).(()(
~
 p
dt
uuCd
dt
uCd pp
Where
  h
p
h
pp
h
p
p
p
u
c
u
c
uC





/1
)( , ]3,2[,0,0  hc pp are constants and  100 ,min  u . We have to
show an interval for u where
h
p
h
pppp
u
u
cuCuuC

 )(.)(
~
Has a strictly positive lower bound.
First we calculate the derivatives

  ;
)( 1
h
h
p
h
ppp
u
h
c
du
udC 



   h
h
p
h
ppp
u
h
h
h
c
du
uCd 21
2
2
1)( 



 ;

du
udC
uuC
du
uCd p
p
p )(
)(
)(
~
      hpp
h
pphp
h
pphp
h
pp uu
h
h
cu
h
u
cuc
1
11
11 1 











 
 ;
   
 
 
.
1221
1
11)(
~
1
2
22
2
1
2
1
1
2
2
hp
p
h
pp
hpphp
h
pp
p
u
uhhh
h
c
u
h
u
h
h
u
h
h
c
du
uCd






























 





 

Since  










 10100
1
,
1
min,min
h
h
h
h
u 
the derivative
0
)(
~

du
uCd p
.
Then   .0ˆ,min].[:)(
~
min
0
0
0
0
0
0
00 















 p
h
p
h
pp
h
p
h
pp
h
p
h
ppp C
c
ccuuC







Further on we have
 
 
 
    
 
.
12212)(
~
21
0
2
0
2
1
0
21
1
0










 h h
p
p
h
pp
hp
h
pp
hp
h
ppp
h
huhc
h
ch
u
h
c
du
uCd



    
 
 
 
.
212 )1(
21
0
2
0
21
0
2
00
p
h h
p
p
h
pp
h h
p
p
h
pp
C
h
hc
h
hhc 












For








 1000 ,,
1
min
h
h
u  it follows
   
)1(
01
0
1
ˆ11)(
~
pp
h
h
p
h
pp
p
h
h
p
h
ppp
C
h
hc
u
h
h
u
c
du
uCd





 








 



 


.
Therefore
 
 
,ˆ
)1(
)(
~
],[:)(
~
min 1
1
0
0
000 p
hp
p
h
pp
pp C
hh
h
c
CuuC 







 
 
 
 
)2(
1
2
0
0
22
21
2
22
22
2
122122)(
~
p
hp
p
h
pp
hp
p
h
ppp
C
hhh
h
c
u
uhhh
h
c
du
uCd 










.
For the I-V characteristics we assume )1,0(,)(
1
)(
 

piriR
m
n
np
np and 


m
n
np
np iliL
0
)(
)( then
..)(.)(
~
0
)(



m
n
np
npp iliiLiiL
For )(
~
iLp we get 



m
n
np
n
m
n
np
n
m
n
np
np
pp
ilnilinliiL
di
idL
i
di
iLd
1
)(
1
)(
1
1)(
)1()(
)()(
~
.
Assumptions (C):
 
;
2
),0( 0
00
0





JWe
tu
T
 
;
2
),( 0
00
0





JWe
tu
T
 
0
0
00
0
2
),0( 


Z
JWe
ti
T
,
 
0
0
00
0
2
),( 


Z
JWe
ti
T
.
Assumptions (L): ,0ˆ)1(
)(
~
)1(
1
)(
0  

p
m
n
np
n
p
Liln
di
iLd
i

)2(
1
1
1
0
)(
2
2
)1(
1
0
)(
)1(
)(
~
,)1(
)(
~
p
m
n
np
n
p
p
m
n
np
n
p
Llnn
di
iLd
Lln
di
iLd 
 




.
Assumptions (G): .)()(,)()(
1
11
)1(
111
1
00
)0(
000 


m
n
n
LRnLR
m
n
n
LRnLR igiGigiG
V. CONCLUSION
Here we have investigated lossy transmission lines terminated by circuits different from parallel or in series
connected RGLC-elements. It turned out that in this case one obtains more number of equations which leads to more
complicated boundary conditions at both ends of the line. First difficulty is to find independent unknown functions −
voltages and currents and to obtain a system of neutral differential equations. We show that just oscillatory solutions are
specific for the lossy transmission lines and in the second part of the paper we formulate conditions for existence-uniqueness
of an oscillatory solution. They can be easily applied to concrete problem because they are explicit type conditions − just
inequalities between specific parameters of the line and characteristics of the circuit.
REFERENCES
[1] G. Miano, and A. Maffucci, Transmission lines and lumped circuits (Academic Press, NY, 2-nd ed., 2010)
[2] D. Pozar, Microwave engineering (J. Wiley & Sons, NY, 1998).
[3] C.R. Paul, Analysis of multi-conductor transmission lines (A Wiley-Inter science Publication, J. Wiley &Sons, NY, 1994).
[4] S. Ramom, J.R. Whinnery, and T.van Duzer, Fields and waves in communication electronics (J. Wiley & Sons, Inc. NY, 1994).
[5] S. Rosenstark, Transmission lines in computer engineering (Mc Grow-Hill, NY, 1994).
[6] P. Vizmuller, RF design guide systems, circuits and equations (Artech House, Inc., Boston - London, 1995).
[7] P.C. Magnusson, G.C. Alexander, and V.K. Tripathi, Transmission lines and wave propagation (CRC Press. Boka Raton, 3rd ed.,
1992).
[8] J. Dunlop, and D.G. Smith, Telecommunications engineering (Chapman & Hall, London, 1994).
[9] V. Damgov, Nonlinear and parametric phenomena: Theory and applications in radio physical and mechanical systems (World
Scientific: New Jersey, London, Singapore, 2004).
[10] V.G. Angelov, and M. Hristov, Distortion less lossy transmission lines terminated by in series connected RCL-loads. Circuits and
Systems, 2(4), 2011, 297-310.
[11] V.G. Angelov, and D.Tz. Angelova, Oscillatory solutions of neutral equations with polynomial nonlinearities, in Recent Advances
in Oscillation Theory − special issue published in International Journal of Differential Equations, Volume 2011, Article ID 949547,
2011, 12 pages.
[12] V.G. Angelov, Periodic regimes for distortion less lossy transmission lines terminated by parallel connected RCL-loads, in
Transmission Lines: Theory, Types and Applications, (Nova Science Publishers, Inc., 2011) 259-293.
[13] V.G. Angelov, Oscillations in lossy transmission lines terminated by in series connected nonlinear RCL-loads, International Journal
of Theoretical and Mathematical Physics, 2(5), (2012).
[14] V.G. Angelov, Various applications of fixed point theorems in uniform spaces. 10Th
International Conference on Fixed Point Theory
and its Applications. Cluj-Napoca, 2012.
[15] V.G. Angelov, Lossless transmission lines terminated by in series connected RL-loads parallel to C-load. International Journal of
Theoretical and Mathematical Physics, 3(1), 2013.
[16] V.G. Angelov, A method for analysis of transmission lines terminated by nonlinear loads (submitted for publication).
[17] V.G. Angelov, Fixed points in uniform spaces and applications (Cluj University Press “Babes-Bolyia”, Romania, 2009).

Lossy Transmission Lines Terminated by Parallel Connected RC-Loads and in Series Connected L-Load (I)

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Lossy Transmission Lines Terminated by Parallel Connected RC-Loads and in Series Connected L-Load (I)