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

Para calcular impedâncias em série, incluindo resistências e reatâncias indutivas, considere uma rede de condutores trifásicos e neutros em N aterrados regularmente para uma linha aérea trifásica.
Usando as leis de Kirchhoff, uma equação integro-diferencial para a rede é derivada.
Correntes de fase desequilibradas podem induzir correntes de retorno através de fios neutros e da terra, seguindo o caminho de menor impedância.
Os condutores de retorno à terra, espelhando a corrente negativa do condutor aéreo, podem substituir a terra.
Eles têm o mesmo raio médio geométrico e resistência que suas contrapartes aéreas.
Os condutores aéreos são renumerados, começando com os condutores de fase, seguidos pelos neutros.
Em uma linha de transmissão, a corrente total entre os condutores é zero, definindo o fluxo que liga cada condutor aéreo.
Usando a representação do circuito de uma seção de um metro, o vetor de quedas de tensão entre os condutores é calculado.
As equações são particionadas e resolvidas, produzindo vetores de queda de tensão entre condutores de fase e correntes de fase e sua relação.
Isso dá a matriz de impedância de fase em série de três por três para compreender o comportamento da linha.
Para linhas totalmente transpostas, a média dos elementos diagonais e fora da diagonal oferece uma visualização de comportamento de linha generalizada.

impedâncias em série: linha trifásica

Series Impedances: Three-Phase Line

O cálculo de impedâncias em série para uma linha aérea trifásica envolve a avaliação de resistências e reatâncias indutivas em uma rede com condutores trifásicos e neutros múltiplos aterrados em intervalos regulares.


 Usando as leis de Kirchhoff, uma equação integro-diferencial para a rede é derivada. Esta equação considera correntes de fase desbalanceadas, que podem induzir correntes de retorno através de fios neutros e da terra, buscando o caminho de menor impedância. Os condutores de retorno à terra podem substituir a terra real nos cálculos, espelhando a corrente negativa dos condutores aéreos e tendo raio médio geométrico (GMR) e resistência idênticos.


 Os condutores aéreos são renumerados, começando com os condutores de fase, seguidos pelos condutores neutros. A corrente total em todos os condutores em uma linha de transmissão é zero, uma condição que define a ligação de fluxo para cada condutor aéreo. O vetor de quedas de tensão nos condutores pode ser calculado representando uma seção de um metro do circuito.


 As equações derivadas das leis de Kirchhoff são particionadas e resolvidas para produzir vetores de queda de tensão em condutores de fase e as correntes de fase correspondentes. Essas relações permitem a formulação de uma matriz de impedância de fase em série três por três. Essa matriz encapsula o comportamento elétrico da linha, fornecendo insights críticos sobre as interações entre condutores de fase e suas características de impedância.


 Para linhas totalmente transpostas, onde as posições dos condutores são regularmente rotacionadas para equilibrar a indutância e a capacitância, a matriz de impedância pode ser ainda mais simplificada. Ao calcular a média dos elementos diagonais e fora da diagonal, uma visão generalizada do comportamento da linha é obtida. Essa abordagem ajuda a prever o desempenho da linha de transmissão sob várias condições operacionais, garantindo uma transmissão de energia eficiente e confiável.


 Entender as impedâncias em série em uma linha trifásica por meio de modelagem detalhada e resolução de equações complexas é crucial para projetar e operar sistemas de transmissão de energia eficientes.

Impedâncias em Série: linha trifásica

Para calcular impedâncias em série, incluindo resistências e reatâncias indutivas, considere uma rede de condutores trifásicos e N-neutros aterrados ...

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...

resistance and conductance

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...

inductance: solid cylindrical conductor

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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