Name: three-phase line impedance matrix
Uploaded: 2024-11-21T14:54:23.000Z
Duration: 1 min 27 s
Channel: JoVE

Para calcular las impedancias en serie, incluidas las resistencias y las reactancias inductivas, considere una red de conductores trifásicos y N-neutros conectados a tierra regularmente para una línea aérea trifásica.
Usando las leyes de Kirchhoff, se deriva una ecuación integro-diferencial para la red.
Las corrientes de fase desequilibradas pueden inducir corrientes de retorno a través de los cables neutros y la tierra, siguiendo el camino de menor impedancia.
Los conductores de retorno a tierra, que reflejan la corriente negativa del conductor aéreo, pueden reemplazar la tierra.
Tienen el mismo radio medio geométrico y resistencia que sus homólogos aéreos.
Los conductores aéreos se renumeran, comenzando con los conductores de fase, seguidos de los neutros.
En una línea de transmisión, la corriente total a través de los conductores es cero, definiendo el flujo que une cada conductor aéreo.
Usando la representación del circuito de una sección de un metro, se calcula el vector de caídas de voltaje a través de los conductores.
Las ecuaciones se dividen y resuelven, lo que produce vectores de caída de voltaje a través de conductores de fase y corrientes de fase y su relación.
Esto proporciona la matriz de impedancia de fase en serie de tres por tres para comprender el comportamiento de la línea.
En el caso de las líneas totalmente transpuestas, el promedio de los elementos diagonales y desdiagonales ofrece una vista de comportamiento de línea generalizada.

impedancias en serie: línea trifásica

Series Impedances: Three-Phase Line

El cálculo de impedancias en serie para una línea aérea trifásica implica evaluar resistencias y reactancias inductivas en una red con conductores trifásicos y múltiples neutros conectados a tierra a intervalos regulares.
Utilizando las leyes de Kirchhoff, se deriva una ecuación integrodiferencial para la red. Esta ecuación tiene en cuenta las corrientes de fase desequilibradas, que pueden inducir corrientes de retorno a través de los cables neutros y la tierra, buscando la ruta de menor impedancia. Los conductores de retorno a tierra pueden reemplazar a la tierra real en los cálculos, reflejando la corriente negativa de los conductores aéreos y teniendo un radio medio geométrico (GMR) y una resistencia idénticos.
Los conductores aéreos se renumeran, comenzando con los conductores de fase, seguidos por los conductores neutros. La corriente total a través de todos los conductores en una línea de transmisión es cero, una condición que define la relación de flujo para cada conductor aéreo. El vector de caídas de voltaje a través de los conductores se puede calcular representando una sección de un metro del circuito.
Las ecuaciones derivadas de las leyes de Kirchhoff se dividen y resuelven para obtener vectores de caída de tensión en los conductores de fase y las corrientes de fase correspondientes. Estas relaciones permiten la formulación de una matriz de impedancia de fase en serie de tres por tres. Esta matriz encapsula el comportamiento eléctrico de la línea y proporciona información fundamental sobre las interacciones entre los conductores de fase y sus características de impedancia.
En el caso de líneas totalmente transpuestas, en las que las posiciones de los conductores se rotan regularmente para equilibrar la inductancia y la capacitancia, la matriz de impedancia se puede simplificar aún más. Al promediar los elementos diagonales y fuera de la diagonal, se obtiene una visión generalizada del comportamiento de la línea. Este enfoque ayuda a predecir el rendimiento de la línea de transmisión en diversas condiciones de funcionamiento, lo que garantiza una transmisión de energía eficiente y confiable.
Comprender las impedancias en serie en una línea trifásica a través del modelado detallado y la resolución de ecuaciones complejas es crucial para diseñar y operar sistemas de transmisión de energía eficientes.

Impedancias en serie: línea trifásica

Para calcular las impedancias en serie, incluidas las resistencias y las reactancias inductivas, considere una red de conductores trifásicos y N-neutros ...

Education

JoVE Core

Electrical Engineering

Unit 1

Transmission Line Parameters

KEY TERMS AND CONCEPTS

three-phase line impedance matrix

Three-phase line impedance matrix calculations describe how resistance and inductive reactance behave in an overhead power line. The model includes three phase conductors and multiple neutral conductors that are grounded at regular intervals. It also accounts for unbalanced phase currents, which can send return current through the neutral wires and the earth along the lowest-impedance path.

Kirchhoff’s laws are used to build an integro-differential equation for the conductor network. To make the calculation practical, the earth can be replaced by an earth return conductor in the model. That conductor carries the negative current of the overhead conductors and is given the same geometric mean radius, or GMR, and resistance.

The overhead conductors are then renumbered. The phase conductors come first, and the neutral conductors follow. The total current across all conductors in the transmission line is zero, and that condition is used to define the flux linkage for each overhead conductor.

A one-meter section of the circuit is used to form the vector of voltage drops across the conductors. The equations from Kirchhoff’s laws are then partitioned and solved for the phase conductors and their currents. These results lead to a three-by-three series-phase impedance matrix that describes how the phases interact electrically.

For fully transposed lines, the conductor positions are rotated in a regular pattern to balance inductance and capacitance. In that case, the impedance matrix can be simplified by averaging the diagonal and off-diagonal elements. This gives a general picture of line behavior under different operating conditions and helps support efficient, reliable power transmission.

Three-Phase Line Impedance Matrix

transmission line design considerations

Overhead Power Line Materials and Safety

Overhead transmission lines rely on the right materials, insulation, and support hardware to move electricity safely and efficiently. Aluminum is now the preferred conductor material because it is abundant and cost-effective. Copper was used more often in the past, but aluminum has become the common choice for modern lines. The most widely used conductor is aluminum conductor, steel-reinforced, or ACSR. It uses aluminum strands around a steel core for added strength. Other conductor types...

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How Transmission Lines Lose Energy

Transmission lines lose energy through resistance and conductance. DC resistance depends on a conductor’s resistivity, length, and cross-sectional area. Resistivity is a material property, and annealed copper is the international standard used for measurement. Material choice and conductor shape both matter. Hard-drawn aluminum at 20 degrees Celsius has a resistivity equal to 61% of the standard conductivity of annealed copper. Stranded conductors also gain a little extra resistance because...

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Magnetic Flux Linkage in Solid Conductors

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...

inductance: single-phase and three-phase line

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

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...

series impedances: three-phase line

Three-phase line impedance matrix calculations describe how resistance and inductive reactance behave in an overhead power line. The model includes three phase conductors and multiple neutral conductors that are grounded at regular intervals. It also accounts for unbalanced phase currents, which can send return current through the neutral wires and the earth along the lowest-impedance path. Kirchhoff’s laws are used to build an integro-differential equation for the conductor network. To make...

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...

shunt admittances

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 Kirchhoff’s laws, neutral conductors, and earth return paths are used to build a three-by-three impedance matrix for a three-phase overhead line.

Three-Phase Line Impedance Matrix | 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.

electrical engineering

Three-Phase Line Impedance Matrix

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