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Magnetically couple two lines

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

The Coupled Lines (Pair) block models two 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 the two lines.

Use this block when the magnetic coupling between the two lines 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 three-phase line, use the Coupled Lines block.

The figure shows the equivalent circuit for a pair of coupled lines.

Here:

*R*and_{1}*R*are the series resistances of lines 1 and 2, respectively._{2}*L*and_{1}*L*are the self-inductances of lines 1 and 2, respectively._{2}*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 the two lines._{m}*G*and_{1}*G*are the leakage conductances of lines 1 and 2, respectively._{2}*V*and_{1}*V*are voltage drops across lines 1 and 2, respectively._{2}*I*and_{1}*I*are the currents through the resistors_{2}*R*and_{1}-R_{m}*R*, respectively._{2}-R_{m}

The defining equation for this block is:

$V=\left[\begin{array}{cc}{R}_{1}& {R}_{m}\\ {R}_{m}& {R}_{2}\end{array}\right]I+\left[\begin{array}{cc}{L}_{1}& {L}_{m}\\ {L}_{m}& {L}_{2}\end{array}\right]\frac{dI}{dt},$

where:

$V=\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right],$

$I=\left[\begin{array}{c}{I}_{1}\\ {I}_{2}\end{array}\right].$

*I _{1}* and

${I}_{total}=I+\left[\begin{array}{cc}{G}_{1}& 0\\ 0& {G}_{2}\end{array}\right]V.$

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:

${L}_{m}=k\sqrt{{L}_{1}{L}_{2}}.$

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 two lines share a common return path, you can model the resistance of this
return path using the **Mutual resistance** parameter. This
workflow is equivalent to setting the **Mutual resistance** to zero
and explicitly modeling the return path resistance
*R _{m}*, as shown in this diagram.

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