TRANSMISSION LINE MODELS

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TRANSMISSION LINE MODELS

TRANSMISSION LINE MODELS

NPTEL

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

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)

= +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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