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