P R O C E E D I N G S O F P L U M E E 2 0 1 5 [619170]

P R O C E E D I N G S O F P L U M E E 2 0 1 5

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ON MODELING THE LONG TRANSMI SSION LINES THROUGH
EQUIVALENT Π CIRCUITS

GHEORGHE HAZI1, ANETA HAZI1, SORIN VERNICA1

1 Université “Vasile Alecsandri” du Bacău, Calea M ărășești 156,Bac ău, 600115, Roumanie

Abstract : In this paper the authors present the results obtained in modeling a long
transmission line through a sequence of equivalent  circuits. An equivalent  circuit is
model with transverse elements capacities. Are shown the res ults obtained for idle regime
and for various load regimes. Are compared the results obtained with those theoretical
given by the long lines equations. Conclusions are drawn about the usefulness of the model
used in the laboratory.

Keywords: telegraph e quation, long line equation, laboratory stand

1. INTRODUCTION

For the long transmission lines, with the scheme given in figure 1, we can write the function equations [1]:

Fig. 1. The explanation of long lines

22
00 2
22
00 2u i iRLx x t x
i u uGCx t t t         
          (1)

In sinusoidal regime, the solution of these equations is:

 Corresponding author email: [anonimizat]
12l
x dx
i (t) i(x,t) i(x+dx,t) i (t)
11 2
u (t)1u (t)2 u(x,t) u(x+dx,t)

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1
1( ) ( )()()() ()c
cch x Z sh xU x Ush xch x I x IZ
              (2)
or

2
2( ( )) ( ( ))()( ( ))( ( )) ()c
cch l x Z sh l xU x Ush l xch l x I x IZ
                (3)
where U1 and I1 are the voltage and the current at the beginning of the line, and U2 and I2 at the end of the line.

2. PRESENTATION OF THE STAND USED

The plant scheme is given in figure 2.

Fig. 2. The stand scheme used for tests

~220V
Q101S101 Q101
S102Q101 Q101
T101 1 2 14 15
C L X L R L

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The elements from figure 2 have the following significance:
 T101 – Power transf ormer with voltage 230/10 V ac, 65 VA;
 Q101 – Contactor 230 V ac;
 S101 – Connecting button;
 S102 – Disconnecting button;
 1,2,..,15 – Equivalent  circuits with the following characteristics
o Z=3.9+j∙39 Ω
o C=2 x 0.47 μF (B=295.3 μS)
o a Π circuit models 100 km line
 XL – Inductive receptor (coil), 9.7 Ω, 1.152 H;
 RL – Resistive receptor (potentiometer), with domain 135÷730 Ω;
 CL – Capacitive receptor (0.8 -30) μF
An equivalent Π circuit models 100 km line. Characteristic sizes Z0 (specific impedance), Y0 (specifi c
admittance), Zc (characteristic impedance) and γ (phase constant) are:

0
0
0
0
06
53
039 0.39 /
2.953 10 /
50.0100
.359 10 1.075/100
363.86-j18.148
= 10cj km
SZZ
Y j B j km
ZZY
ZY j 

  


  


3. OBTAIN RESULTS

3.1. Idle regime
The regime has drawn for 1000 km, to avoid the 1500 km resonance, which is dangerous for the laboratory plant.
For d rawing the theoretical curve it is utilize the (2) relation, with l=1000 km. From the second relation in (2),
taking account that I(l)=0, results:

1
1()(x l)()csh lUIZ ch l
   (4)

1 1()( ) ( ) ( )
csh xI x U ch x I lZ     (5)
In figure 3 are shown the calculated and measured values for voltages, and in figure 4 the calculated and
measured values for currents.

Fig. 3. Measured (Um) and calculated (Uc) voltage variation for l=1000 km, idle

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Fig. 4. Measured (Im) and calculated (Ic) current variation for l=1000 km, idle

3.2. The regime with load equal with natural power
For drawing the theoretical characteristic, we use the (3) relation, in which U2/I2=Zc.
In figure 5 are shown the calculate d and measured values for voltages.

Fig. 5. Measured (Um) and calculated (Uc) voltage variation for l=1500 km, natural power

4. CONCLUSIONS

From the obtained values results the following:
 The correspondence between experimental and theoretical resul ts is very good in all cases, the
differences are less than 6%.
 The idle regime highlights the resonance phenomenon at l=1500 km, the overvoltage value at 1000 km
being greater than 2.
 For natural load regime, are obtained approximately constant voltage an d current, a small decrease is
due pure resistive load ( Zc=364 Ω).
 The stand allows for a wide range of samples.

REFERENCES

[1] Renato Orta, Transmission Line Theory , Department Of Electronics and Telecommunications, Politecnico di
Torino, November 2012
[2] Göran Andersson -Modelling and Analysis of Electric Power Systems , EEH – Power Systems Laboratory,
ETH Zürich, September 2008