# Transmission Line (Three-Phase)

Three-phase transmission line using lumped-parameter pi-section line model

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• Simscape / Electrical / Passive / Lines

## Description

The Transmission Line (Three-Phase) block models a three-phase transmission line using the lumped-parameter pi-line model. This model takes into account phase resistance, phase self-inductance, line-line mutual inductance and resistance, line-line capacitance, and line-ground capacitance.

To simplify the block-defining equations, Clarke’s transformation is used. The resulting equations are:

${V}_{1}^{\prime }-{V}_{2}^{\prime }=\left[\begin{array}{ccc}R+2{R}_{m}& & \\ & R-{R}_{m}& \\ & & R-{R}_{m}\end{array}\right]{I}_{1}^{\prime }+\left[\begin{array}{ccc}L+2M& & \\ & L-M& \\ & & L-M\end{array}\right]\frac{d{I}_{1}^{\prime }}{dt}$

${I}_{1}^{\prime }+{I}_{2}^{\prime }=\left[\begin{array}{ccc}{C}_{g}& & \\ & {C}_{g}+3{C}_{l}& \\ & & {C}_{g}+3{C}_{l}\end{array}\right]\frac{d{V}_{2}^{\prime }}{dt}$

${I}_{1}^{\prime }=T\prime {I}_{1}$

${I}_{2}^{\prime }=T\prime {I}_{2}$

${V}_{1}^{\prime }=T\prime {V}_{1}$

${V}_{2}^{\prime }=T\prime {V}_{2}$

$T=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& \sqrt{2}& 0\\ 1& -1}{\sqrt{2}}& \sqrt{3}{2}}\\ 1& -1}{\sqrt{2}}& -\sqrt{3}{2}}\end{array}\right]$

where:

• R is the line resistance for the segment.

• Rm is the mutual resistance for the segment.

• L is the line inductance for the segment.

• Cg is the line-ground capacitance for the segment.

• Cl is the line-line capacitance for the segment.

• T is the Clarke’s transformation matrix.

• I1 is the three-phase current flowing into the `~1` port.

• I2 is the three-phase current flowing into the `~2` port.

• V1 is the three-phase voltage at the `~1` port.

• V2 is the three-phase voltage at the `~2` port.

The positive and zero-sequence parameters are defined by the diagonal terms in the transformed equations:

${R}_{0}=R+2{R}_{m}$

${R}_{1}=R-{R}_{m}$

${L}_{0}=L+2M$

${L}_{1}=L-M$

${C}_{0}={C}_{g}$

${C}_{1}={C}_{g}+3{C}_{l}$

Rearranging these equations gives the physical line quantities in terms of positive and zero-sequence parameters:

$R=\frac{2{R}_{1}+{R}_{0}}{3}$

${R}_{m}=\frac{{R}_{0}-{R}_{1}}{3}$

$L=\frac{2{L}_{1}+{L}_{0}}{3}$

$M=\frac{{L}_{0}-{L}_{1}}{3}$

${C}_{l}=\frac{{C}_{1}-{C}_{0}}{3}$

${C}_{g}={C}_{0}$

The figure shows the equivalent electrical circuit for a single-segment pi-line model using Clarke’s transformation.

To increase fidelity, you can use the Number of segments parameter to repeat the pi-section N times, resulting in an N-segment transmission line model. More segments significantly slows down your simulation.

To improve numerical performance, you can add parasitic resistance and conductance components. Choosing large values for these components improves simulation speed but decreases simulation accuracy.

## Ports

### Conserving

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Electrical conserving port corresponding to ground connection at `~1` end of the transmission line.

Electrical conserving port corresponding to ground connection at `~2` end of the transmission line.

## Parameters

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### Main

Length of the transmission line.

Resistance of the transmission line per phase per-unit length.

Self-inductance of the transmission line per phase per-unit length.

Line-line mutual inductance per-unit length. Set this to `0` to remove mutual inductance.

Line-line capacitance per-unit length.

Line-ground capacitance per-unit length. The default value is `0``μF/km` (no line-ground capacitance).

Line-line mutual resistance per unit length. The default value is `0``Ohm/km` (no line-line mutual resistance).

Number of segments in the pi-line model.

### Parasitics

Resistance value, divided by the number of segments, that is added in series with every capacitor in the model.

Conductance value, divided by the number of segments, that is added in parallel with every series resistor and inductor in the model.