# Coupled Lines (Three-Phase)

Magnetically couple three-phase lines

**Library:**Simscape / Electrical / Passive / Lines

## Description

The Coupled Lines (Three-Phase) block models three magnetically coupled lines. Each line has a self-inductance, series resistance, and parallel conductance. In addition, there is a mutual inductance and mutual resistance between each pair of lines.

Use this block when the magnetic coupling in a three-phase network is nonnegligible. These effects are most prominent when:

The lines are parallel and close together.

The self-inductances of the lines are high.

The AC frequency of the network is high.

To model magnetic coupling of a single pair of lines, use the Coupled Lines (Pair) block. To model capacitive coupling between the lines, use the Transmission Line block.

### Equivalent Circuit

The equivalent circuit shows the coupling between two arbitrary phases
*i*, and *j*. The block models the magnetic
coupling using such an equivalent circuit between each of the three phases
*a*, *b*, and *c*.

Here:

*R*and_{i}*R*are the series resistances of lines_{j}*i*and*j*, respectively.*L*and_{i}*L*are the self-inductances of lines_{j}*i*and*j*, respectively.*R*is the mutual resistance between the two lines. You can use this parameter to account for losses in a common return path._{m}*L*is the mutual inductance between lines_{m,ij}*i*and*j*, respectively.*G*and_{i}*G*are the leakage conductances of lines_{j}*i*and*j*, respectively.*V*and_{i}*V*are voltage drops across lines_{j}*i*and*j*, respectively.*I*and_{i}*I*are the currents through the resistors_{j}*R*and_{i}-R_{m}*R*, respectively._{j}-R_{m}

### Equations

The defining equation for this block is:

$V=\left[\begin{array}{ccc}{R}_{a}& {R}_{m}& {R}_{m}\\ {R}_{m}& {R}_{b}& {R}_{m}\\ {R}_{m}& {R}_{m}& {R}_{c}\end{array}\right]I+\left[\begin{array}{ccc}{L}_{a}& {L}_{m,ab}& {L}_{m,ac}\\ {L}_{m,ab}& {L}_{b}& {L}_{m,bc}\\ {L}_{m,ac}& {L}_{m,bc}& {L}_{c}\end{array}\right]\frac{dI}{dt},$

where:

$V=\left[\begin{array}{c}{V}_{a}\\ \begin{array}{l}{V}_{b}\\ {V}_{c}\end{array}\end{array}\right],$

$I=\left[\begin{array}{c}{I}_{a}\\ \begin{array}{l}{I}_{b}\\ {I}_{c}\end{array}\end{array}\right].$

*I _{a}*,

*I*, and

_{b}*I*are, in general, not equal to the currents in lines

_{c}*a*,

*b*, and

*c*. These terminal currents make up the vector:

${I}_{total}=I+\left[\begin{array}{ccc}{G}_{a}& 0& 0\\ 0& {G}_{b}& 0\\ 0& 0& {G}_{c}\end{array}\right]V.$

### Inductive Coupling

To quantify the strength of the coupling between the two lines, you can use a coupling
factor or coefficient of coupling *k*. The coupling factor relates
the mutual inductance to the line self-inductances:

${L}_{m,ij}=k\sqrt{{L}_{i}{L}_{j}}.$

This coupling factor must fall in the range $-1<k<1$, where a negative coupling factor indicates a reversal in
orientation of one of the coils. The magnitude of *k* indicates:

$\left|k\right|=0$ — There is no magnetic coupling between the two lines.

$0<\left|k\right|<0.5$ — The two lines are loosely coupled and mutual magnetic effects are small.

$0.5\le \left|k\right|<1$ — The two lines are strongly coupled and mutual magnetic effects are large.

### Mutual Resistance

If the three lines share a common return path, you can model the resistance of
this return path using the **Mutual resistance** parameter
*R _{m}*. This workflow is equivalent to
setting the

**Mutual resistance**to zero and explicitly modeling the return path resistance

*R*, as shown in this diagram.

_{m}If the three lines do not share a common return path, set the mutual resistance parameter to zero and model each of the return resistances explicitly.

## Ports

### Conserving

## Parameters

## Extended Capabilities

## See Also

**Introduced in R2018a**