Test: Calculation of Inductance of Transmission Line - Electrical Engineering (EE) MCQ

Concept:
GMR is defined as the effective distance over which self magnetic flux linkages occur
For a solid conductor with radius r,

GMR = r' = 0.7788r
GMR is less than the physical radius of the conductor.
In the given figure, standard conductor with four identical strands touching each other is given with equal radius r.


= 1.722 r
Since each strands are identical.
∴ GMR of conductor will be equal to GMR of strand i.e.,
GMR_cond = GMR_a1
= 1.722 r
Therefore, correct option is (d)

Effect of Unsymmetrical Spacing of 3ϕ Line Conductors:

1. Due to unsymmetrical spacing, flux linkage with all three phases is different. Because of this, there will be an unequal voltage drop in all three phases. Thus the voltage at the receiving end will not be the same for the three phases. 
2. Due to unsymmetrical spacing, the magnetic field external to the conductor is not zero. Thereby causing the induced voltage in adjacent electric circuits, thus producing the interference in communication lines.
3. Unsymmetrical spacing configurations cause line interference.

​The above-mentioned problems can be eliminated by the transposition of the 3ϕ line.

Transposition is the periodic swapping of positions of the conductors of a transmission line, in order to reduce interference and voltage induction in the 3ϕ transmission line.

The inductance of a transmission line is given by,

Where, d = distance between the conductors
r’ = 0.7788r
r = radius of conductors
Therefore, if we increase the spacing between the phase conductors, the value of line inductance will increase.

Concept:

• The inductance of a transmission line is independent of the height of the transmission tower.
• The capacitance of the transmission line is dependent on the height of the transmission tower.
• If tower height increases then the distance between the current carrying wire and ground increases, and line-to-ground capacitance decreases, as capacitance is inversely proportional to the distance between the line and earth.
• While there is no effect on line to line capacitance.

Calculation:
Self GMD or GMR:

• Self GMD is also called GMR. GMR stands for Geometrical Mean Radius. 
• GMR is calculated for each phase separately.
• self-GMD of a conductor depends upon the size and shape of the conductor
• GMR is independent of the spacing between the conductors.

GMD:

• GMD stands for Geometrical Mean Distance.
• It is the equivalent distance between conductors.
• GMD depends only upon the spacing 
• GMD comes into the picture when there are two or more conductors per phase.​

The inductance per phase is given as,

GMD = Mutual Geometric Mean Distance = D
GMR = 0.7788r
r = Radius of the conductor​
Calculation:
Given conductor configuration,

r = 0.03 m
d = 35 cm = 0.35 m
D = 4 m

The inductance per phase is given as,

L = 0.8041 mH/km
Note: According to the Official UPRVUNL JE EE 2014 Previous Year Paper, then select option was 2.2 mH/km. Because of calculation, We correct the option accordingly

Concept:
The inductance of a three-phase line having its conductors at the corners of an equilateral triangle is given by:

Calculation:
Given, a = 1m

L = 9.21 H/m
L = 9.21 × 10^-1 mH\km
L = 0.921 mH\km

Concept:
The inductance of a transmission line consists of two parallel conductors is given by,

Where, d = distance between the conductors
r’ = 0.7788 r
r = radius of conductors
Calculation:
Given
d = 1 m
Diameter (D) = 1.25 cm
r = 0.625 cm = 0.625 × 10-2 m
r' = 0.7788 r

L = 21.3 × 10^-4 Henry/ km

D_ab = D_bc = D_ca = 1m
GMD = 1 m
GMR = 0.01 m
frequency (f) = 50 Hz

= 9.21 × 10^-7 H/m
X_L = 2πfL
= 2π × 50 × 9.21 × 10^-7
= 0.2893 × 10^-3 Ω/m
= 0.2893 Ω/km

When single phase,


 
When converted to three phase,

Solving , we get

Consider a 3-ɸ line with conductors A, B, and C; each of radius r metres. Let the spacing between them be d₁, d₂ and d₃ and the current flowing through them be I_A, I_B and I_C respectively.
The inductances for the above arrangements are given by,

Thus we see that when the conductors of a 3-phase transmission line are not equidistant from each other, i.e., unsymmetrically spaced, the flux linkages and inductances of various phases are different which cause unequal voltage drops in the three phases and transfer of power between phases (represented by imaginary terms of the expression for inductances) due to mutual inductances even if the currents in the conductors are balanced.

The unbalancing effect on account of irregular spac­ing of conductors is avoided by the transposition of conduc­tors, as shown in the figure.

The effect of transposition is that each conduc­tor has the same average inductance, which is given as:

Therefore, Statement (I) and Statement (II) are individually true and statement (II) is the correct explanation of statement (II)