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

Per calcolare le impedenze in serie, comprese le resistenze e le reattanze induttive, si consideri una rete di conduttori trifase e N-neutro messi a terra regolarmente per una linea aerea trifase.
Utilizzando le leggi di Kirchhoff, viene derivata un'equazione integro-differenziale per la rete.
Le correnti di fase sbilanciate possono indurre correnti di ritorno attraverso i fili neutri e la terra, seguendo il percorso di minor impedenza.
I conduttori di ritorno a terra, che rispecchiano la corrente negativa del conduttore aereo, possono sostituire la terra.
Hanno la stessa media geometrica, raggio e resistenza delle loro controparti aeree.
I conduttori aerei vengono rinumerati, iniziando con i conduttori di fase, seguiti da quelli neutri.
In una linea di trasmissione, la corrente totale attraverso i conduttori è zero, definendo il flusso che collega ciascun conduttore aereo.
Utilizzando la rappresentazione del circuito di una sezione di un metro, viene calcolato il vettore delle cadute di tensione attraverso i conduttori.
Le equazioni vengono partizionate e risolte, producendo vettori di caduta di tensione attraverso i conduttori di fase e le correnti di fase e la loro relazione.
Questo fornisce alla matrice di impedenza trifase in serie tre per tre di comprendere il comportamento della linea.
Per le linee completamente trasposte, la media degli elementi diagonali e fuori diagonale offre una visualizzazione generalizzata del comportamento delle linee.

impedenze di serie: linea trifase

Series Impedances: Three-Phase Line

Il calcolo delle impedenze in serie per una linea aerea trifase, comporta la valutazione delle resistenze e delle reattanze induttive, in una rete con conduttori trifase e più neutri, collegati a terra a intervalli regolari.

Utilizzando le leggi di Kirchhoff, si ricava un'equazione integro-differenziale per la rete. Questa equazione tiene conto delle correnti di fase sbilanciate, che possono indurre correnti di ritorno attraverso i fili neutri e la terra, attraverso il percorso a più bassa impedenza. I conduttori di ritorno a terra, possono sostituire la terra effettiva nei calcoli, rispecchiando la corrente negativa dei conduttori aerei e avendo identici raggio medio geometrico (GMR) e resistenza.

I conduttori aerei vengono rinumerati, iniziando dai conduttori di fase, seguiti dai conduttori neutri. La corrente totale attraverso tutti i conduttori in una linea di trasmissione è zero, una condizione che definisce il collegamento di flusso per ogni conduttore aereo. Il vettore delle cadute di tensione attraverso i conduttori, può essere calcolato rappresentando una sezione di un metro del circuito.

Le equazioni derivate dalle leggi di Kirchhoff, vengono suddivise e risolte per determinare i vettori di caduta di tensione attraverso i conduttori di fase e le corrispondenti correnti di fase. Queste relazioni, consentono la formulazione di una matrice di impedenza di fase della serie tre per tre. Questa matrice incapsula il comportamento elettrico della linea, fornendo informazioni critiche sulle interazioni tra i conduttori di fase e le loro caratteristiche di impedenza.

Per linee completamente trasposte, dove le posizioni dei conduttori vengono ruotate regolarmente per bilanciare induttanza e capacità, la matrice di impedenza può essere ulteriormente semplificata. Mediando gli elementi diagonali e fuori diagonale, si ottiene una visione generalizzata del comportamento della linea. Questo approccio aiuta a prevedere le prestazioni della linea di trasmissione in varie condizioni operative, garantendo una trasmissione di potenza efficiente e affidabile.

La comprensione delle impedenze in serie in una linea trifase attraverso una modellazione dettagliata e la risoluzione di equazioni complesse, è fondamentale per progettare e gestire sistemi di trasmissione di potenza efficienti.

Impedenze di Serie: Linea Trifase

Per calcolare le impedenze in serie, comprese le resistenze e le reattanze induttive, si consideri una rete di conduttori trifase e N-neutri ...

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

Description

Transcript

Tags