Inductance of Overhead Transmission Lines

In this article we will discuss about the determination of inductance for overhead transmission lines.

Inductance is the property by virtue of which a circuit opposes changes in the value of a varying current flowing through it. While the resistance of a circuit opposes the flow of both steady as well as varying currents, the inductance causes opposition only to varying currents. Inductance does not cause any opposition to steady or direct current.

In case of transmission and distribution lines, the current flowing is varying or alternating current, the effect of inductance, in addition to that of resistance, is therefore to be considered. The opposition to the flow of varying current owing to inductance is viewed as a voltage drop.

It is well known fact that a current carrying conductor is surrounded by concentric circles of magnetic lines. In case of ac system this field set up around the conductor is not constant but changing and links with the same conductor as well as with other conductors. Due to these flux linkages the line possesses inductance, defined as the flux linkages per unit current. Thus for determination of inductance of a circuit, determination of flux linkages is essential.

ADVERTISEMENTS:

Flux Linkages of a Conductor:

A long straight cylindrical conductor carrying a current is surrounded by a magnetic field. The magnetic lines of force will exist inside the conductor as well as outside the conductor. Both of these fluxes contribute to the inductance of the conductor.

i. Flux Linkages of a Conductor due to Internal Flux:

Consider a long straight cylindrical conductor of radius r metres and carrying a current of I amperes, as shown in Fig. 4.1.

In overhead lines it may be assumed without appreciable error that the current is uniformly distributed.

The current inside a line of force of radius x,

Field strength inside the conductor at a distance x from the centre,

Flux density, B_x = µ₀ µ H_x Wb/m² or T where µ₀ = 4 × 10^-7

and µ is the relative permeability of the medium and for non-magnetic materials µ = 1

Now the flux through a cylindrical shell of radial thickness dx and axial length one metre,

But this flux links with only the current lying within the circle of radius x i.e. with

Total flux linkages from centre of the conductor and up to the surface of the conductor,

ii. Flux Linkages of a Conductor due To External Flux:

Consider two points 1 and 2 distant d₁ and d₂ from the centre of the conductor. Since the flux paths are concentric circles around the conductor, whole of the flux between points 1 and 2 lies within the concentric cylindrical surfaces passing through these points 1 and 2.

The field strength at any distance x from the centre of the conductor (x > r),

So the flux through a cylindrical shell of radial thickness dx and axial length one metre,

Now flux linkages per metre is equal to d ɸ since flux external to conductor links all the current in the conductor once and only once,

Total flux linkages between points 1 and 2,

Inductance of Single Phase Overhead Lines:

Inductance of a Two-Wire Single Phase Line:

Consider a single phase line consisting of two parallel conductors A and B of radii r₁ and r₂spaced d metres apart (d being very large as compared to r₁ and r₂). Conductors A and B carry the same current (i.e., I_A = I_B) in magnitude but opposite in directions, as one forms the return path for the other.

The inductance of each conductor is due to internal flux linkages and external flux linkages and the following points are to be noted regarding external flux linkages:

1. A line of flux produced due to current in conductor A at a distance equal to or greater than (d + r₂) from the centre of conductor A links with a zero net current, as the current flowing in the two conductors A and B are equal in magnitude but opposite in directions.

2. Flux lines at a distance (d – r₂) link with a current I and those between (d – r₂) and (d + r₂) link with a current varying from I to zero.

As a simplifying assumption, it can be assumed that all the flux produced by current in conductor A links all the current up to the centre of conductor B and that the flux beyond the centre of conductor B does not link any current.

The above assumption simplifies the calculations and results obtained are quite accurate especially when d is much greater than r₁and r₂, as is usually the case in overhead lines.

Based on the above assumption flux linkages of conductor A due to external flux can be determined from Eq. (4.2) by substituting d₂ = d and d₁= r₁. Thus flux linkages of conductors A due to external flux only,

Flux linkages of conductor A due to internal flux only,

Total flux linkages of conductor A,

Total inductance of conductor A,

The product (r₁ e^-1/4) is known as Geometric Mean Radius (GMR) of the conductor and is equal to 0.7788 times the radius of the conductor. Let it be represented by r’₁ where,

r’₁ = 0.7788 r₁

If r’₁ = r’₂ = r’, the loop inductance of the line is given as:

The idea of replacing the original conductor of radii r by a fictitious conductor of radii r’ is quite attractive because streamlined equations for inductance can be developed without bogging down in accounting for the internal flux.

Flux Linkages of One Conductor in a Group of Conductors:

Consider a group of parallel conductors 1, 2, 3, … n carrying currents I₁, I₂, I₃ … I_nrespectively, as illustrated in Fig. 4.4. Let it be assumed that the sum of the currents in various conductors is zero i.e. I_x + I₂ + I₃ + … I_n = 0

Theoritically, the flux due to a conductor extends from the centre of the conductor right up to infinity but let us assume that the flux linkages extend up to a remote point P and the respective distances are as marked in Fig. 4.4.

The current in each conductor sets up a certain flux due its own current. The sum of all these fluxes is the total flux of the system and the total flux linkages of any one conductor is the sum of its linkages with all the individual fluxes set up by the conductors of the system.

Now let us determine the flux linkages of conductor 1 due to current I₁ carried by the conductor itself and flux linkages to conductor 1 due to currents carried by other conductors (2, 3, … n)

The flux linkages of conductor 1 due to its own current I₁ (internal and external), up to point P,

The flux linkages of conductor 1 due to current in conductor 2,

Flux due to conductor 2 that lies between conductors 2 and 1 does not link conductor 1 and therefore the distances involved are d_2p and d₁₂.

Thus the expression for flux linkages of conductor 1 due to currents in all conductors can be written as:

The above equation may be written as:

To account for the total flux linkages to conductor 1, the point P must approach infinity and in this condition,

This simplifies the Eq. (4.8) and the equation for the flux linkages to conductor 1 becomes,

Inductance of Composite Conductor Lines—Self and Mutual GMDs:

Consider a single phase line consisting of two parallel conductors A and B, conductor A consisting of x and conductor B of y strands, as illustrated in Fig. 4.5.

Let the conductors A and B carry currents I and – I respectively (since conductors of a 2-wire line carry the same current but in opposite directions).

Assuming uniform current density in both the conductors the current carried by each strand of conductor A will be I/x while that carried by each strand of conductor B will be –I/y.

Using Eq. (4.9) the flux linkages of strand 1 in conductor A can be written as:

Inductance of strand 1 of conductor A,

In above expression numerator of argument of log_e is written y th root of distances d_11’, d_12′, d_13′, d_14′ …, d₁y multiplied together where distances d_11′, d_12′, d_13′, d_14′ …, d_1y are the distances of strands 1′, 2′, 3′, 4′ …., y (all segments of conductor B) from segment 1 under consideration.

The denominator of argument of log_e is x th root of distances d₁₂, d₁₃, d₁₄ …d_1x and r’ multiplied together where distances d₁₂, d₁₃, d₁₄ …. d₁x are the distances of strands 2, 3, 4, ….., n (all strands of conductor A) from strand 1. r’ can also be represented by distance d₁₁ and the expression for inductance for conductor A becomes,

Similarly the expression for inductance for strand 2 can be written as:

Thus we see that the different strands of a conductor have different inductances.

∴ Average inductance of strands of conductor A,

Since x such strands of conductor A are electrically in parallel,

Therefore inductance of conductor A,

In the above expression the numerator of argument of log_e is called the GMD (often called the mutual GMD) between conductors A and B and the denominator of argument log_e is called GMR (often called self GMD). GMD (Geometric Mean Distance) and GMR (Geometric Mean Radius) are denoted by D_m and D_s respectively.

Inductance of Three-Phase Overhead Lines:

With Unsymmetrical Spacing:

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 flux linkages of conductor A due to its own current I_Aand other conductor currents I_B and I_c,

If the system is balanced,

I_A = I_B = I_c = I (say) in magnitude

Taking I_A as a reference phasor, the currents are represented, in symbolic form as

I_A = I; I_B = I (- 0.5 – j 0.866) and I_c = I (- 0.5 + j 0.866)

Substituting these values of I_B and I_c in the expression for Ψ_A we get,

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 transposition of conductors, as shown in Fig. 4.10. In practice the conductors are so transposed that each of the three possible arrangements of conductors exists for one-third of the total length of the line.

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

(ii) If the conductors are equi-spaced (let the spacing be equal to d), as shown in Fig. 4.11, the inductance of each conductor will be same and can be obtained by substituting d₁= d₂= d₃ in above expression.

So inductance of each conductor:

For stranded conductor r’ will be replaced by D_s (self GMD).

(iii) When the conductors of three-phase transmission line are in the same plane, as shown in Fig. 4.12.:

In this position, d₁ = d, d₂ = d and d₃ = 2d

Substituting d₁ = d₂ = d and d₃ = 2 d

in general Eqs. (4.15), (4.16), and (4.17) for L_A, L_B and L_c respectively we get,

(iv) When the conductors are at the corners of a right-angled triangled, as shown in Fig. 4.13.:

In this position,

d₁ = d₂ = d and d₃ – √2 d

Substituting d₁ = d₂ = d and d₃ = √2 d in general Eq. (4.15), (4.16) and (4.17) for L_A, L_B and L_c respectively we get,

Inductance of Three Phase Lines with More than One Circuit:

It is usual practice to run 3-phase transmission lines with more than one circuit in parallel on the same towers, because it gives greater reliability and a higher transmission capacity. If such circuits are so widely separated that the mutual inductance between them becomes negligible, the inductance of the equivalent single circuit would be half of each of the individual circuits considered alone.

But in actual practice the separation is not very wide and the mutual inductance is not negligible. GMD method is used for determination of inductance per phase by considering the various conductors connected in parallel as strands of one composite conductor.

It is desirable to have a configuration that provides minimum inductance so as to have maximum transmission capacity. This is possible only with low GMD and high GMR. Therefore, the practice is to have the individual conductors of a phase widely separated to provide high GMR and the distance between the phases small to give low GMD.

Thus in the case of a double circuit in vertical formation the arrangement of conductors would be as illustrated in Fig. 4.16 (a) and not as illustrated in Fig. 4.16 (b) because the arrangement of conductors given in Fig. 4.16 (a) results in low inductance in comparison to that given by the arrangement illustrated in Fig. 4.16 (b).

Inductance of a 3-Phase Double Circuit Line with Symmetrical Spacing:

Consider a 3-phase double circuit connected in parallel-conductors A, B, C forming one circuit and conductors A’, B’, C’ forming the other one, as illustrated in Fig. 4.17.

Flux-linkages of phase A conductors:

Similarly inductance of remaining conductors can be worked out, which will be the same as L_A. This is due to the fact that the conductors of different phases are symmetrically placed.

Since conductors are electrically in parallel, inductance of each phase:

Inductance of a 3-Phase Double Circuit with Unsymmetrical Spacing but Transposed:

Now consider a 3-phase double circuit connected in parallel—conductors A, B and C forming one circuit and conductors A’, B’ and C’ forming the other one, as illustrated in Fig. 4.18. (Conductors unsymmetrically spaced and transposed).

Since the conductors are thoroughly transposed, the conductor situations in the transposition cycle would be, as illustrated in Figs. 4.18 (1), 4.18 (2) and 4.18 (3).