Week 5-7 Inductance of TLs

EE356: Power Transmission,
Distribution and Utilization
Farhan Mahmood, PhD

Department of Electrical Engineering
UET, Lahore
May 23, 2016

Outline
• Transmission line conductors

• Stranded conductors
• Transmission line parameters
• Resistance of overhead lines
• Inductance of overhead lines
• Inductance of long cylinderical conductor
• Inductance of single-phase transmission line
• Flux linkages in a group of conductors
• Inductance of composite conductor
Page 2
Transmission Line Conductors
• Copper conductors
˗ High electrical conductivity
˗ High tensile strength
˗ High current density
˗ High cost
˗ Low availabilty
• Aluminium conductors have replaced copper conductors;
˗ Low cost
˗ Light weight
˗ High availability
Page 3
Stranded Conductors
• Stranded conductor consists of multiple small strands (or sub-conductors) which

group together to form a single conductor.
Solid Round Conductor
Stranded Conductor
Page 4
Stranded Conductors
• Advantages:
˗ Flexibility
˗ Easy manufacturing
˗ Low resistance and reactance
˗ Better mechanical strength
• Size: 130 sq. mm,19/2.9 mm
• No. of strands in an n-layer stranded conductor:
• Overall diameter:
• Where d = diameter of one strand
Page 5
Stranded Conductors
• Types:
˗ All Aluminium Conductor (AAC)
˗ Aluminium Conductor Steel Reinforce (ACSR)
˗ Aluminium Conductor Alloy Reinforce (ACAR)
˗ All Aluminium Alloy Conductor (AAAC)
Aluminum
Steel
Page 6
Transmission Line Parameters
• All transmission lines in a power system exhibit the electrical properties of

resistance, inductance, capacitance and conductance.
• Inductance and capacitance are due to the effects of magnetic and electric
fields around the conductor.
• These parameters are essential for the development of the transmission line
models used in power system analysis.
• The shunt conductance accounts for leakage currents flowing across insulators
and ionized pathways in the air.
• The leakage currents are negligible compared to the current flowing in the
transmission lines and may be neglected.
Page 7
Resistance of Overhead Transmission Line
• The dc resistance of a solid round conductor at a specified temperature is
l
Rdc 
Where : A
ρ = conductor resistivity (Ω-m),
l = conductor length (m) ;
A = conductor cross-sectional area (m2)
• Significant effect:
˗ Generation of I2R loss in transmission line.
˗ Produces IR-type voltage drop which affect voltage regulation.
Page 8
• Conductor resistance is affected by three factors:-

˗ Temperature
˗ Spiraling
˗ Skin effect (‘frequency’)
Page 9
Temperature
• As the temperature increases, the conductor resistance also increases. This change in
temperature can be considered linear over the normal operating temperatures
according to,
T  t2
R2  R1
Where: T  t1
R1 = conductor resistances at t1 in Cº
R2 = conductor resistances at t2 in Cº
T = temperature constant and represents the temperature at which conductor will
exhibit zero resistance. It depends on conductor material and is obtained by
extrapolation
Page 10
Spiraling
• For stranded conductors, alternate layers of strands are spiraled in opposite
directions to hold the strands together.
• Spiraling makes the strands 1 – 2% longer than the actual conductor length.
• DC resistance of a stranded conductor is 1 – 2% larger than the calculated value.
Page 11
Frequency – Skin Effect

• When ac flows in a conductor, the current distribution is not uniform over the
conductor cross-sectional area and the current density is greatest at the surface of
the conductor.
• This causes the ac resistance to be somewhat higher than the dc resistance. The
behavior is known as skin effect
• The skin effect is where alternating current tends to avoid travel through the center of
a solid conductor, limiting itself to conduction near the surface.
• This effectively limits the cross-sectional conductor area available to carry alternating
electron flow, increasing the resistance of that conductor above what it would
normally be for direct current
Page 12
Page 13
Resistivity and Temperature Constant of Conductor Material
ρ20ºC T
Material Resistivity at 20ºC Temperature Constant
Ωm×10-8 Ωcmil/ft ºC
Copper
Annealed 1.72 10.37 234.5
Hard-drawn 1.77 10.66 241.5
Aluminum
Hard-drawn 2.83 17.00 228
Brass 6.4 – 8.4 38 – 51 480
Iron 10 60 180
Silver 1.59 9.6 243
Sodium 4.3 26 207
Steel 12 – 88 72 – 530 180 – 980
Page 14
Exercise
Problem 2:
A solid cylindrical aluminum conductor 25 km long has an area of 336,400 circular mils.
Determine the conductor resistance at (a) 20º C and (b) 50º C. The resistivity of
aluminum at 20º C is 2.8 × 10−8 Ω-m.
Page 15
Exercise
Page 16
Design Exercise
Problem 3:
• A three-phase transmission line is designed to deliver 190.5-MVA at 220-kV over a
distance of 63 km. The total transmission line loss is not to exceed 2.5 % of the rated
line MVA. If the resistivity of the conductor material is 2.84 × 10−8 Ω-m, determine
the required conductor diameter and the conductor size in circular mils.
Page 17
Design Exercise
Page 18
Field Lines Produced by Transmission Lines
Page 19
Inductance of Overhead Transmission Line
• A current-carrying conductor produces a magnetic field

around the conductor.
• For non-magnetic material, the inductance L is the
ratio of its total magnetic flux linkage to the current I,
given by,
where ψ =flux linkages, in Weber turns.

• The inductance of the conductor can be defined as the
sum of contributions from flux linkages internal and
external to the conductor.
Page 20
Outer Flux
Inner Flux
Page 21
• The inductance of a magnetic circuit that has a constant permeability µ can be

obtained by determining the following:
1) Magnetic field intensity H, from Ampere’s law
2) Magnetic flux density B (B = µ.H)
3) Magnetic flux φ (φ = B.A)
4) Flux linkages ψ ( Ψ = N. φ)
5) Inductance L from flux linkages per ampere (L = ψ / I)
Page 22
Inductance of Long Cylinderical Conductor
Inductance Due to Internal Flux Linkages
Page 23
dx
dφx
Page 24
• Internal inductance can be express as follows:-
0 1
Lint   107 H / m
8 2
Where
µo = permeability of air (4π x 10-7 H/m)
• The internal inductance is independent of the conductor radius r
Page 25
Inductance Due To External Flux Linkages

• External inductance between to point D2 and D1 can be express as follows:
7D2
Lext  2  10 ln H /m
D1
Page 26
Inductance Due To External Flux Linkages

• External inductance between to point D2 and D1 can be express as follows:
7D2
Lext  2  10 ln H /m
D1
Page 27
Inductance of Single-Phase Transmission Line
• The Inductance of a single-phase lines can be expressed as below
Page 28
Inductance of Single-Phase Transmission Line
• If the two conductors are identical, then inductance per phase per metre length pf the
line is given by,
• In the above equation, the first term is only a function of the conductor radius. This
term is considered as the inductance due to both the internal flux and that external to
conductor 1 to a radius of 1 m.
• The second term of above equation is dependent only upon conductor spacing. This
term is known as the inductance spacing factor.
• The above terms are usually expressed as inductive reactances at 50/60 Hz and are
available in the manufacturers table in English units.
Page 29
Exercise
Problem 4:
A single-phase line has two conductors 2 m apart. The diameter of each conductor is
1.2 cm. Calculate the loop inductance per km of the line.
Page 30
Flux Linkages in terms of Self and Mutual
Inductances
• A single-phase line can be viewed as two magnetically coupled coils.
Page 31
Flux Linkages in terms of Self and Mutual
Inductances
• The flux linkages of a conductor in a group of conductors can be expressed as,
Page 32
Inductance of Composite Conductor
Page 33
Inductance of Composite Conductor
• The inductance of composite conductor is given by,
Page 34
Exercise
Problem 5:
• Conductor X has 3 strands
whereas conductor Y has 2-
strands. The radius of conductor
a, b, c is 2.5 cm while that of d, e
is 5 cm. Calculate LX, LY, LLOOP.
Page 35
Exercise
Problem 6:
• Calculate GMR of an ACSR conductor with r being the radius of each conductor.
Page 36
Inductance of 3-Phase Line
Symmetrical Spacing
• Inductance per phase of three-phase line with symmetrical spacing (equilateral
spaced soild conductor) is given by,
Page 37
Symmetrical Line Spacing – 69 kV
Page 38
Exercise
Problem 7:
• In a 3-phase transmission line, the conductors are placed at the corners of an
equilateral triangle of each side 2.5 m. If the radius of each conductor is 0.8 cm, find
inductance per phase per km length of the line.
Page 39
Unsymmetrical Spacing
• The practice of symmetrical spacing between the phases
is not convenient.
˗ Symmetry is lost – unbalanced conditions
˗ Horizontal or vertical configurations are most popular
• Restore balanced conditions by the method of
transposition of lines.
˗ Each phase occupies each position for the same
fraction of the total length of the line
˗ Average inductance of each phase will be the same
Page 40
• The average inductance per phase is given by,
Page 41
Page 42
Exercise
Problem 8:
• In a 3-phase transmission line, three conductors are placed at the corners of a
triangle of sides 1.5 m, 3 m and 2.6 m respectively. If the diameter of each conductor
is 1.4 cm and the conductors are regularly transposed, calculate the inductance per
phase per km length of the line. (Ans: 1.206 mH)
Problem 9:
• A 3-phase overhead transmission line has equilateral spacing of 8 feet. It is decided
to rebuild the line with flat horizontal spacing (the line being transposed) such that
the line has same inductance as in the original design. Calculate the distance
between the adjacent conductors.
Page 43
Inductance of 3-Phase, Double Circuit Line
• A three-phase double circuit line consists of two identical

three phase circuits.
• Double circuits are used where greater reliability is
needed.
• This method of transmission enables the transfer of more
power over a particular distance.
• The circuits are operated with a-a’, b-b’ and c-c’ in parallel.
• Because of the geometrical differences between
conductors, voltage drop due to line inductance will be
unbalanced.
• Types:
˗ Flat vertical spacing
˗ Hexagonal arrangement
Page 44
Flat vertically spaced Hexagonally arranged
Page 45
Flat vertically spaced
Page 46
Flat vertically spaced
Page 47
Hexagonally arranged
Page 48
Exercise
Problem 10:
• Calculate the inductance per km of a transposed double circuit 3-phase line as
shown in figure. Each circuit of the line remains on its own side. The diameter of the
conductor is 2.532 cm.
Page 49
Inductance of 3-Phase, Bundled Conductor Line
• In transmission lines (voltage level ≥ 220 kV), it is

common practice to use more than one conductor
per phase, a practice called bundling.
• Conductors are connected in parallel in each phase.
• Bundled conductors are commonly used
˗ To reduce the electric field strength at the
conductor surface, hence minimize corona power
loss.
˗ To reduce the inductance and increase the
capacitance of line
˗ To improve the voltage regulation and efficiency
of power line.
Page 50
Page 51
• AC current shows the property of skin effect. This reduces the effective cross-section
of conductor and hence resistance becomes high. To take this effect to its minimum,
same cross-section is used, but in form of bundled wire of small cross-sections.
• This increases the number of outer surfaces and hence has more surface area which
in turn would allow more current to flow. As our current here prefers to flow through
skin of the conductor. This way by making bundled conductors, the transmission line
is utilized better.
Page 52
Page 53
Phase A Phase B Phase C
Page 54
Page 55
Page 56
765 kV Bundled Conductor Transmission Line
Page 57
Exercise
Problem 9:
• Calculate the inductance per km per phase of a single circuit 500 kV line using two
bundle conductors per phase as shown in figure. The diameter of each conductor is
5 cm.
Page 58
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EE356: Power Transmission,

Distribution and Utilization

Farhan Mahmood, PhD

May 23, 2016

• Transmission line conductors

• Stranded conductor consists of multiple small strands (or sub-conductors) which

Solid Round Conductor

• Where d = diameter of one strand

• All transmission lines in a power system exhibit the electrical properties of

• The dc resistance of a solid round conductor at a specified temperature is

• Conductor resistance is affected by three factors:-

Frequency – Skin Effect

• A current-carrying conductor produces a magnetic field

where ψ =flux linkages, in Weber turns.

• The inductance of a magnetic circuit that has a constant permeability µ can be

Inductance Due to Internal Flux Linkages

• Internal inductance can be express as follows:-

Inductance Due To External Flux Linkages

Inductance Due To External Flux Linkages

• The Inductance of a single-phase lines can be expressed as below

• A single-phase line can be viewed as two magnetically coupled coils.

• The flux linkages of a conductor in a group of conductors can be expressed as,

• The inductance of composite conductor is given by,

Symmetrical Line Spacing – 69 kV

• The average inductance per phase is given by,

• A three-phase double circuit line consists of two identical

Flat vertically spaced Hexagonally arranged

Flat vertically spaced

Flat vertically spaced

• In transmission lines (voltage level ≥ 220 kV), it is

Phase A Phase B Phase C

765 kV Bundled Conductor Transmission Line

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