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Magnetically couple three-phase lines

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

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.

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}

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}*,

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

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.

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

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.