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CHAPTER 2<br />
<br />
TRANSMISSION LINE MODELS<br />
<br />
As we have discussed earlier in Chapter 1 that the transmission line parameters<br />
<br />
include series resistance and inductance and shunt capacitance. In this chapter we shall<br />
<br />
discuss the various models of the line. The line models are classified by their length. These<br />
<br />
classifications are<br />
<br />
• Short line approximation for lines that are less than 80 km long.<br />
<br />
• Medium line approximation for lines whose lengths are between 80 km to 250 km.<br />
<br />
• Long line model for lines that are longer than 250 km.<br />
<br />
These models will be discussed in this chapter. However before that let us introduce the<br />
<br />
ABCD parameters that are used for relating the sending end voltage and current to the<br />
<br />
receiving end voltage and currents.<br />
<br />
2.1 ABCD PARAMETSRS<br />
<br />
Consider the power system shown in Fig. 2.1. In this the sending and receiving end<br />
<br />
voltages are denoted by VS and VR respectively. Also the currents IS and IR are entering and<br />
<br />
leaving the network respectively. The sending end voltage and current are then defined in<br />
<br />
terms of the ABCD parameters as<br />
<br />
VS = AVR + BIR (2.1)<br />
<br />
S CVR DIR I = + (2.2)<br />
<br />
From (2.1) we see that<br />
<br />
=0<br />
<br />
=<br />
<br />
R R I<br />
<br />
S<br />
<br />
V<br />
<br />
V A (2.3)<br />
<br />
This implies that A is the ratio of sending end voltage to the open circuit receiving end<br />
<br />
voltage. This quantity is dimension less. Similarly,<br />
<br />
=0<br />
<br />
=<br />
<br />
VR R<br />
<br />
S<br />
<br />
I<br />
<br />
V B Ω (2.4)<br />
<br />
i.e., B, given in Ohm, is the ratio of sending end voltage and short circuit receiving end<br />
<br />
current. In a similar way we can also define<br />
<br />
=0<br />
<br />
=<br />
<br />
R R I<br />
<br />
S<br />
<br />
V<br />
<br />
I C mho (2.5) <br />
<br />
1.31<br />
<br />
=0<br />
<br />
=<br />
<br />
VR R<br />
<br />
S<br />
<br />
I<br />
<br />
I D (2.6)<br />
<br />
The parameter D is dimension less.<br />
<br />
Fig. 2.1 Two port representation of a transmission network.<br />
<br />
2.2 SHORT LINE APPROXIMATION<br />
<br />
The shunt capacitance for a short line is almost negligible. The series impedance is<br />
<br />
assumed to be lumped as shown in Fig. 2.2. If the impedance per km for an l km long line is<br />
<br />
z0 = r + jx, then the total impedance of the line is Z = R + jX = lr + jlx. The sending end<br />
<br />
voltage and current for this approximation are given by<br />
<br />
VS =VR + ZIR (2.7)<br />
<br />
S R I = I (2.8)<br />
<br />
Therefore the ABCD parameters are given by<br />
<br />
A = D =1, B = Z Ω and C = 0 (2.9)<br />
<br />
Fig. 2.2 Short transmission line representation.<br />
<br />
2.2 MEDIUM LINE APPROXIMATION<br />
<br />
Medium transmission lines are modeled with lumped shunt admittance. There are two<br />
<br />
different representations − nominal-π and nominal-T depending on the nature of the network.<br />
<br />
These two are discussed below.<br />
<br />
2.2.1 Nominal-π Representation<br />
<br />
In this representation the lumped series impedance is placed in the middle while the<br />
<br />
shunt admittance is divided into two equal parts and placed at the two ends. The nominal-π<br />
<br />
representation is shown in Fig. 2.3. This representation is used for load flow studies, as we<br />
<br />
shall see later. Also a long transmission line can be modeled as an equivalent π-network for<br />
<br />
load flow studies. <br />
<br />
1.32<br />
<br />
Fig. 2.3 Nominal-π representation.<br />
<br />
Let us define three currents I1, I2 and I3 as indicated in Fig. 2.3. Applying KCL at<br />
<br />
nodes M and N we get<br />
<br />
s R R<br />
<br />
s R<br />
<br />
V I Y V Y<br />
<br />
I I I I I I<br />
<br />
= + +<br />
<br />
= + = + +<br />
<br />
2 2<br />
<br />
1 2 1 3<br />
<br />
(2.10)<br />
<br />
Again<br />
<br />
R R<br />
<br />
s R R R R<br />
<br />
V ZI YZ<br />
<br />
I V Y V ZI V Z V<br />
<br />
+<br />
<br />
= +<br />
<br />
+<br />
<br />
= + = +<br />
<br />
1<br />
<br />
2<br />
<br />
2 2<br />
<br />
(2.11)<br />
<br />
Substituting (2.11) in (2.10) we get<br />
<br />
R R<br />
<br />
s R R R R<br />
<br />
I YZ V YZ Y<br />
<br />
V I Y V ZI Y YZ I<br />
<br />
+ +<br />
<br />
= +<br />
<br />
+ +<br />
<br />
+<br />
<br />
= +<br />
<br />
1<br />
<br />
2<br />
<br />
1<br />
<br />
4<br />
<br />
2<br />
<br />
1<br />
<br />
2 2<br />
<br />
(2.12)<br />
<br />
Therefore from (2.11) and (2.12) we get the following ABCD parameters of the<br />
<br />
nominal-π representation<br />
<br />
= = +1<br />
<br />
2<br />
<br />
YZ A D (2.13)<br />
<br />
B = Z Ω (2.14)<br />
<br />
1 mho<br />
<br />
4<br />
<br />
= +<br />
<br />
YZ C Y (2.15)<br />
<br />
2.2.1 Nominal-T Representation<br />
<br />
In this representation the shunt admittance is placed in the middle and the series<br />
<br />
impedance is divided into two equal parts and these parts are placed on either side of the<br />
<br />
shunt admittance. The nominal-T representation is shown in Fig. 2.4. Let us denote the<br />
<br />
midpoint voltage as VM. Then the application of KCL at the midpoint results in <br />
<br />
1.33<br />
<br />
Furthermore the sending end current is<br />
<br />
S M R I = YV + I (2.19)<br />
<br />
Then substituting the value of VM from (2.16) in (2.19) and solving<br />
<br />
R R RI YZ I YV<br />
<br />
= + +1<br />
<br />
2 (2.20)<br />
<br />
Then the ABCD parameters of the T-network are<br />
<br />
= = +1<br />
<br />
2<br />
<br />
YZ A D (2.21)<br />
<br />
Ω<br />
<br />
= +1<br />
<br />
4<br />
<br />
YZ B Z (2.22)<br />
<br />
C = Y mho (2.23) <br />
<br />
1.34<br />
<br />
2.3 LONG LINE MODEL<br />
<br />
For accurate modeling of the transmission line we must not assume that the<br />
<br />
parameters are lumped but are distributed throughout line. The single-line diagram of a long<br />
<br />
transmission line is shown in Fig. 2.5. The length of the line is l. Let us consider a small strip<br />
<br />
∆x that is at a distance x from the receiving end. The voltage and current at the end of the<br />
<br />
strip are V and I respectively and the beginning of the strip are V + ∆V and I + ∆I<br />
<br />
respectively. The voltage drop across the strip is then ∆V. Since the length of the strip is ∆x,<br />
<br />
the series impedance and shunt admittance are z ∆x and y ∆x. It is to be noted here that the<br />
<br />
total impedance and admittance of the line are<br />
<br />
Z = z ×l and Y = y ×l (2.24)<br />
<br />
Fig. 2.5 Long transmission line representation.<br />
<br />
From the circuit of Fig. 2.5 we see that<br />
<br />
Iz<br />
<br />
x<br />
<br />
V V Iz x = ∆<br />
<br />
∆ ∆ = ∆ ⇒ (2.25)<br />
<br />
Again as ∆x → 0, from (2.25) we get<br />
<br />
Iz<br />
<br />
dx<br />
<br />
dV = (2.26)<br />
<br />
Now for the current through the strip, applying KCL we get<br />
<br />
∆I = ( ) V + ∆V y∆x =Vy∆x + ∆Vy∆x (2.27)<br />
<br />
The second term of the above equation is the product of two small quantities and therefore<br />
<br />
can be neglected. For ∆x → 0 we then have<br />
<br />
Vy dx<br />
<br />
dI = (2.28)<br />
<br />
Taking derivative with respect to x of both sides of (2.26) we get<br />
<br />
dx<br />
<br />
dI<br />
<br />
z<br />
<br />
dx<br />
<br />
dV<br />
<br />
dx<br />
<br />
d =<br />
<br />
1.35<br />
<br />
Substitution of (2.28) in the above equation results<br />
<br />
0 2<br />
<br />
2<br />
<br />
− yzV = dx<br />
<br />
d V (2.29)<br />
<br />
The roots of the above equation are located at ±√(yz). Hence the solution of (2.29) is of the<br />
<br />
form<br />
<br />
x yz x yz V Ae A e<br />
<br />
− = 1 + 2 (2.30)<br />
<br />
Taking derivative of (2.30) with respect to x we get<br />
<br />
x yz x yz A yz e A yz e<br />
<br />
dx<br />
<br />
dV − = 1 − 2 (2.31)<br />
<br />
Combining (2.26) with (2.31) we have<br />
<br />
x yz x yz e<br />
<br />
z y<br />
<br />
A<br />
<br />
e<br />
<br />
z y<br />
<br />
A<br />
<br />
dx<br />
<br />
dV<br />
<br />
z<br />
<br />
I −<br />
<br />
= −<br />
<br />
= 1 1 2 (2.32)<br />
<br />
Let us define the following two quantities<br />
<br />
= Ω which is called the characteristic impedance<br />
<br />
y<br />
<br />
z ZC (2.33)<br />
<br />
γ = yz which is called the propagation constant (2.34)<br />
<br />
Then (2.30) and (2.32) can be written in terms of the characteristic impedance and<br />
<br />
propagation constant as<br />
<br />
x x V Ae A e γ −γ = 1 + 2 (2.35)<br />
<br />
x<br />
<br />
C<br />
<br />
x<br />
<br />
C<br />
<br />
e<br />
<br />
Z<br />
<br />
A<br />
<br />
e<br />
<br />
Z<br />
<br />
A I γ −γ = − 1 2 (2.36)<br />
<br />
Let us assume that x = 0. Then V = VR and I = IR. From (2.35) and (2.36) we then get<br />
<br />
VR = A1 + A2 (2.37)<br />
<br />
C C<br />
<br />
R Z<br />
<br />
A<br />
<br />
Z<br />
<br />
A I 1 2 = − (2.38)<br />
<br />
Solving (2.37) and (2.38) we get the following values for A1 and A2.<br />
<br />
2 and 2 1 2<br />
<br />
R C R R C R V Z I A V Z I A − = + =<br />
<br />
1.36<br />
<br />
Also note that for l = x we have V = VS and I = IS. Therefore replacing x by l and substituting<br />
<br />
the values of A1 and A2 in (2.35) and (2.36) we get<br />
<br />
R C R l R C R l<br />
<br />
S e<br />
<br />
V Z I<br />
<br />
e<br />
<br />
V Z I V γ − −γ +<br />
<br />
+ = 2 2 (2.39)<br />
<br />
R C R l R C R l<br />
<br />
S e<br />
<br />
V Z I<br />
<br />
e<br />
<br />
V Z I I γ − −γ − + = 2 2 (2.40)<br />
<br />
Noting that<br />
<br />
l e e l e e l l l l<br />
<br />
γ γ<br />
<br />
γ γ γ γ<br />
<br />
cosh<br />
<br />
2<br />
<br />
sinh and 2 = + = − − −<br />
<br />
We can rewrite (2.39) and (2.40) as<br />
<br />
V V l Z I l S R C R = coshγ + sinhγ (2.41)<br />
<br />
I l<br />
<br />
Z<br />
<br />
l I V R<br />
<br />
C<br />
<br />
S R γ γ cosh sinh = + (2.42)<br />
<br />
The ABCD parameters of the long transmission line can then be written as<br />
<br />
A = D = coshγl (2.43)<br />
<br />
B Z l C = sinhγ Ω (2.44)<br />
<br />
ZC<br />
<br />
l C sinhγ = mho (2.45)<br />
<br />
Example 2.1: Consider a 500 km long line for which the per kilometer line impedance<br />
<br />
and admittance are given respectively by z = 0.1 + j0.5145 Ω and y = j3.1734 × 10−6<br />
<br />
mho.<br />
<br />
Therefore<br />
<br />
= ∠ − ° Ω<br />
<br />
° − ° ∠<br />
<br />
× = × ∠ °<br />
<br />
∠ ° = ×<br />
<br />
+ = = − − −<br />
<br />
406.4024 5.5<br />
<br />
2<br />
<br />
79 90<br />
<br />
3.1734 10<br />
<br />
0.5241<br />
<br />
3.1734 10 90<br />
<br />
0.5241 79<br />
<br />
3.1734 10<br />
<br />
0.1 0.5145<br />
<br />
6 6 6 j<br />
<br />
j<br />
<br />
y<br />
<br />
z ZC<br />
<br />
and<br />
<br />
0.6448 84.5 0.0618 0.6419<br />
<br />
2<br />
<br />
79 90 0.5241 3.1734 10 500 6<br />
<br />
j<br />
<br />
l yz l<br />
<br />
= ∠ ° = +<br />
<br />
° + ° = × = × × × ∠ − γ<br />
<br />
We shall now use the following two formulas for evaluating the hyperbolic forms<br />
<br />
( )<br />
<br />
( ) α β α β α β<br />
<br />
α β α β α β<br />
<br />
sinh sinh cos cosh sin<br />
<br />
cosh cosh cos sinh sin<br />
<br />
j j<br />
<br />
j j<br />
<br />
+ = +<br />
<br />
+ = +<br />
<br />
Application of the above two equations results in the following values <br />
<br />
1.37<br />
<br />
coshγl = 0.8025 + j0.037 and sinhγl = 0.0495 + j0.5998<br />
<br />
Therefore from (2.43) to (2.45) the ABCD parameters of the system can be written as<br />
<br />
2.01 10 0.0015<br />
<br />
43.4 240.72<br />
<br />
0.8025 0.037<br />
<br />
5 C j<br />
<br />
B j<br />
<br />
A D j<br />
<br />
= − × +<br />
<br />
= + Ω<br />
<br />
= = +<br />
<br />
−<br />
<br />
∆∆∆<br />
<br />
2.3.1 Equivalent-π Representation of a Long Line<br />
<br />
The π-equivalent of a long transmission line is shown Fig. 2.6. In this the series<br />
<br />
impedance is denoted by Z′ while the shunt admittance is denoted by Y′. From (2.21) to<br />
<br />
(2.23) the ABCD parameters are defined as<br />
<br />
+ ′ ′ = = 1<br />
<br />
2<br />
<br />
Y Z A D (2.46)<br />
<br />
B = Z′Ω (2.47)<br />
<br />
1 mho<br />
<br />
4<br />
<br />
+ ′ ′ = ′ Y Z C Y (2.48)<br />
<br />
Fig. 2.6 Equivalent π representation of a long transmission line.<br />
<br />
Comparing (2.44) with (2.47) we can write<br />
<br />
l<br />
<br />
l Z<br />
<br />
l yz<br />
<br />
l l zl<br />
<br />
y<br />
<br />
z Z Z l C<br />
<br />
γ<br />
<br />
γ γ γ γ<br />
<br />
sinh sinh ′ = sinh = sinh = = Ω (2.49)<br />
<br />
where Z = zl is the total impedance of the line. Again comparing (2.43) with (2.46) we get<br />
<br />
sinh 1<br />
<br />
2<br />
<br />
1<br />
<br />
2<br />
<br />
cosh + ′ + = ′ ′ = Z l Y Z Y l C γ γ (2.50)<br />
<br />
Rearranging (2.50) we get<br />
<br />
( ) ( ) ( )<br />
<br />
( )<br />
<br />
( )<br />
<br />
( ) 2<br />
<br />
tanh 2<br />
<br />
2<br />
<br />
2<br />
<br />
tanh 2<br />
<br />
2<br />
<br />
tanh 2 tanh 2 1<br />
<br />
sinh<br />
<br />
1 cosh 1<br />
<br />
2<br />
<br />
l<br />
<br />
Y l<br />
<br />
l yz<br />
<br />
yl l l<br />
<br />
z<br />
<br />
y l<br />
<br />
l Z<br />
<br />
l<br />
<br />
Z<br />
<br />
Y<br />
<br />
C C<br />
<br />
γ<br />
<br />
γ<br />
<br />
γ γ γ<br />
<br />
γ<br />
<br />
γ<br />
<br />
=<br />
<br />
= = = − = ′<br />
<br />
(2.51) <br />
<br />
1.38<br />
<br />
where Y = yl is the total admittance of the line. Note that for small values of l, sinh γl = γl and<br />
<br />
tanh (γl/2) = γl/2. Therefore from (2.49) we get Z = Z′ and from (2.51) we get Y = Y′. This<br />
<br />
implies that when the length of the line is small, the nominal-π representation with lumped<br />
<br />
parameters is fairly accurate. However the lumped parameter representation becomes<br />
<br />
erroneous as the length of the line increases. The following example illustrates this.<br />
<br />
Example 2.2: Consider the transmission line given in Example 2.1. The equivalent<br />
<br />
system parameters for both lumped and distributed parameter representation are given in<br />
<br />
Table 2.1 for three different line lengths. It can be seen that the error between the parameters<br />
<br />
increases as the line length increases.<br />
<br />
Table 2.1 Variation in equivalent parameters as the line length changes.<br />
<br />
Length of Lumped parameters Distributed parameters<br />
<br />
the line<br />
<br />
(km) Z (Ω) Y (mho) Z′ Ω Y′ (mho)<br />
<br />
100 52.41∠79° 3.17×10−4<br />
<br />
∠90° 52.27∠79° 3.17×10−4<br />
<br />
∠89.98°<br />
<br />
250 131.032∠79° 7.93×10−4<br />
<br />
∠90° 128.81∠79.2° 8.0×10−4<br />
<br />
∠89.9°<br />
<br />
500 262.064∠79° 1.58×10−3<br />
<br />
∠90° 244.61∠79.8° 1.64×10−3<br />
<br />
∠89.6°<br />
<br />
∆∆∆<br />
<br />
2.4 CHARACTERIZATION OF A LONG LOSSLESS LINE<br />
<br />
For a lossless line, the line resistance is assumed to be zero. The characteristic<br />
<br />
impedance then becomes a pure real number and it is often referred to as the surge<br />
<br />
impedance. The propagation constant becomes a pure imaginary number. Defining the<br />
<br />
propagation constant as γ = jβ and replacing l by x we can rewrite (2.41) and (2.42) as<br />
<br />
V V x jZ I x = R cos β + C R sin β (2.52)<br />
<br />
I x<br />
<br />
Z<br />
<br />
x I jV R<br />
<br />
C<br />
<br />
R β β cos<br />
<br />
sin = + (2.53)<br />
<br />
The term surge impedance loading or SIL is often used to indicate the nominal<br />
<br />
capacity of the line. The surge impedance is the ratio of voltage and current at any point<br />
<br />
along an infinitely long line. The term SIL or natural power is a measure of power delivered<br />
<br />
by a transmission line when terminated by surge impedance and is given by<br />
<br />
C<br />
<br />
n Z<br />
<br />
V SIL P<br />
<br />
2<br />
<br />
0 = = (2.54)<br />
<br />
where V0 is the rated voltage of the line.<br />
<br />
At SIL ZC = VR/IR and hence from equations (2.52) and (2.53) we get<br />
<br />
j x<br />
<br />
R<br />
<br />
x<br />
<br />
R V V e V e γ − β = = (2.55) <br />
<br />
1.39<br />
<br />
j x<br />
<br />
R<br />
<br />
x<br />
<br />
NPTEL<br />
<br />
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