Name: transmission line inductance in power cables
Uploaded: 2024-11-21T14:54:23.000Z
Duration: 1 min 28 s
Channel: JoVE

Consider single-phase two-wire and three-phase three-wire transmission lines with equal phase spacing.
The single-phase line consists of two solid cylindrical conductors, x and y, each carrying a phasor current.
Given that the sum of currents is zero, the total flux linking conductor x is computed, followed by its inductance.
Similarly, the total flux linking conductor y is computed and its inductance is determined.
The loop inductance, also known as the total inductance of the single-phase circuit, is then calculated.
If the radii of both conductors are equal, the total circuit inductance can be simplified.
Now, the three-phase line consists of three solid cylindrical conductors, a, b, and c, of equal radii and phase spacing.
Assuming balanced positive-sequence currents such that the sum of currents equals zero, the total flux linking phase a conductor is calculated.
From this, its inductance is determined. Due to symmetry, the same result applies to phases b and c.
However, for the balanced operation of this line, only one phase needs consideration since each phase&#39;s flux linkages have equal magnitudes and 120-degree displacement.

inductance: single-phase and three-phase line

Inductance: Single-Phase And Three-Phase Line

Understanding the inductance of transmission lines is crucial for efficient design and operation in electrical power systems. This discussion delves into the inductance characteristics of single-phase two-wire and three-phase three-wire transmission lines with equal phase spacing.
Single-Phase Two-Wire Line:
A single-phase line consists of two solid cylindrical conductors, denoted as x and y. Each conductor carries phasor currents ix and iy, respectively. Given that the sum of these currents is zero, we can calculate the total flux linking conductor x. This flux linkage is given by:
<img alt="Static equilibrium, formula &#955;x=2&#215;10^-7ln(D/r'x) Wb-t/m, equation, scientific analysis." src="/files/ftp_upload/16142/16142_Equation%201.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Where rx' is the effective radius of conductor x, accounting for the proximity effect.
The inductance per conductor, Lx of conductor x, is then:
<img alt="Static equilibrium formula, Lx = &#955;x/ix, with logarithmic component, illustrating energy distribution." src="/files/ftp_upload/16142/16142_Equation%202.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
Similarly, for conductor y, the flux linkage and inductance are computed. The total inductance of the single-phase circuit, also known as loop inductance, is:
<img alt="Electromagnetic inductance equation L=Lx+Ly=4x10^-7 ln(D/sqrt(rx'ry')) H/m; physics formula." src="/files/ftp_upload/16142/16142_Equation%203.svg" style="height: 46px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
Three-Phase Three-Wire Line:
For a three-phase line, there are three solid cylindrical conductors, a, b, and c, each with equal radius r and equal phase spacing D. Assuming balanced positive-sequence currents, the total flux linking phase a is:
<img alt="Magnetic flux equation &#955;&#8336;=2&#215;10&#8315;&#8311;i&#8336;ln(D/r') Wb-t/m, mathematical concept." src="/files/ftp_upload/16142/16142_Equation%204.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
From this, the inductance of phase a is derived:
<img alt="Static induction, formula: La=&#955;a/ia=2&#215;10^-7ln(D/r'), equation for electromagnetic research." src="/files/ftp_upload/16142/16142_Equation%205.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation5" />
Due to symmetry, the same inductance applies to phases b and c. For a balanced three-phase operation, only one phase needs consideration since each phase's flux linkages are equal in magnitude and displaced by 120 degrees.
Understanding these inductance calculations helps in the design and analysis of transmission lines, ensuring minimal losses and optimal performance.

Consider single-phase two-wire and three-phase three-wire transmission lines with equal phase spacing.
The single-phase line consists of two solid ...

inductance-single-phase-and-three-phase-line

Education

JoVE Core

Electrical Engineering

Unit 1

Transmission Line Parameters

KEY TERMS AND CONCEPTS

transmission line inductance in power cables

Transmission line inductance helps explain how current-carrying conductors behave in power systems. It is especially important when studying single-phase two-wire lines and three-phase three-wire lines with equal phase spacing. These line types are common models for efficient transmission design and operation.

A single-phase line uses two solid cylindrical conductors, labeled x and y. Each conductor carries a phasor current, ix and iy, and the sum of these currents is zero. The total flux linking conductor x can then be found, and the effective radius rx' is used to account for proximity effect. From the flux linkage, the inductance per conductor, Lx, is obtained.

The same process applies to conductor y. Its flux linkage and inductance are calculated in the same way. The inductance of the full single-phase circuit is the loop inductance, which combines the two conductors into one circuit value.

A three-phase line uses three solid cylindrical conductors, a, b, and c. They have equal radius r and equal phase spacing D. For balanced positive-sequence currents, the flux linking phase a can be determined from the current pattern in the line. The inductance of phase a follows from that flux linkage.

Because the system is symmetric, phases b and c have the same inductance as phase a. In balanced three-phase operation, one phase is enough to analyze because all three phases have equal-magnitude flux linkages shifted by 120 degrees. These inductance results are useful for line design and analysis, where reducing losses and improving performance are important.

Transmission Line Inductance in Power Cables

transmission-line-inductance-in-power-cables

transmission line design considerations

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Magnetic flux linkage in a solid cylindrical conductor is found by combining the field inside the wire with the field around it. The method uses a 1-meter section of a non-magnetic conductor with radius r. It also assumes the current is spread uniformly and ignores end effects. Inside the conductor, Ampere's law is used to find the magnetic field intensity H. The field is concentric and constant within the conductor. From H, the magnetic flux density B inside the wire is calculated. The...

Transmission line inductance helps explain how current-carrying conductors behave in power systems. It is especially important when studying single-phase two-wire lines and three-phase three-wire lines with equal phase spacing. These line types are common models for efficient transmission design and operation. A single-phase line uses two solid cylindrical conductors, labeled x and y. Each conductor carries a phasor current, ix and iy, and the sum of these currents is zero. The total flux...

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Three-Phase Line Impedance Matrix

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capacitance: single-phase and three-phase line

capacitance-single-phase-and-three-phase-line

Transmission Line Capacitance in Power Systems

Transmission line capacitance affects how power lines store charge and behave in electrical power systems. It is an important part of line modeling because it helps describe efficient operation. For a single-phase line, the key idea is the charge on the conductors and the voltage between them. A single-phase transmission line uses two wires with equal spacing. One conductor carries a uniform positive charge, and the other carries an equal negative charge. From the voltage between the...

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Image Method for Line Capacitance

Image conductors help model line capacitance in transmission lines above Earth. A uniformly charged conductor induces an equal and opposite charge on the ground. This creates electric field lines between the conductor and the Earth. The method of images replaces the Earth with a matching conductor placed symmetrically below the line. The image conductor has the same radius and the same charge magnitude as the original conductor. That setup keeps the electric field and voltage above the Earth...

key terms and concepts

Learn how inductance is calculated for single-phase two-wire and three-phase three-wire transmission lines, including flux linkage, loop inductance, and balanced phase behavior.

Transmission Line Inductance in Power Cables | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

Transmission Line Inductance in Power Cables

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