Coupled Lines (Three-Phase)

Magnetically couple three-phase lines

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Libraries:
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.

In this figure:

• R_i and R_j are the series resistances of lines i and j, respectively.

• L_i and L_j are the self-inductances of lines i and j, respectively.

• R_m,ij is the mutual resistance between lines i and j, respectively.

• L_m,ij is the mutual inductance between lines i and j, respectively.

• G_i and G_j are the leakage conductances of lines i and j, respectively.

• V_i and V_j are voltage drops across lines i and j, respectively.

• I_i and I_j are the currents through the resistors R_i-R_m,ij and R_j-R_m,ij, respectively.

Equations

The defining equation for this block is:

$V=\left[\begin{array}{ccc}{R}_a& R_{m,ab}& R_{m,ac}\\ R_{m,ab}& R_b& R_{m,bc}\\ R_{m,ac}& R_{m,bc}& R_c\end{array}\right]I+\left[\begin{array}{ccc}{L}_a& L_{m,ab}& L_{m,ac}\\ L_{m,ab}& L_b& L_{m,bc}\\ L_{m,ac}& L_{m,bc}& L_c\end{array}\right]\frac{dI}{dt},$

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_b, and I_c are, in general, not equal to the currents in lines 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_m, as shown in this diagram.

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.

Faults

Since R2025a

To model a fault in the Coupled Lines (Three-Phase) block, in the Faults section, click Add fault next to the fault that you want to model. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

The Coupled Lines (Three-Phase) block allows you to model these types of fault:

• Single-phase-to-ground fault (a-g, b-g, or c-g)

• Two-phase fault (a-b, b-c, or c-a)

• Two-phase-to-ground fault (a-b-g, b-c-g, or c-a-g)

• Three-phase fault (a-b-c)

• Three-phase-to-ground fault (a-b-c-g)

You can model a fault at one of the two three-phase connection ports or at both ports at the same time.

This figure shows how the block models the faults at the two three-phase ports:

For more information about adding faults to blocks and specifying fault triggers, see Introduction to Simscape Faults.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Ports

Conserving

~1 — Lines positive terminals

electrical

Expandable three-phase electrical conserving port associated with the positive terminals of lines a, b, and c.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports

~2 — Lines negative terminals

electrical

Expandable three-phase electrical conserving port associated with the negative terminals of lines a, b, and c.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports

a1 — Line a positive terminal

electrical

Electrical conserving port associated with the positive terminal of line a.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

a2 — Line a negative terminal

electrical

Electrical conserving port associated with the negative terminal of line a.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

b1 — Line b positive terminal

electrical

Electrical conserving port associated with the positive terminal of line b.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

b2 — Line b negative terminal

electrical

Electrical conserving port associated with the negative terminal of line b.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

c1 — Line c positive terminal

electrical

Electrical conserving port associated with the positive terminal of line c.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

c2 — Line c negative terminal

electrical

Electrical conserving port associated with the negative terminal of line c.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports

Parameters

Main

Electrical connection — Whether to model composite or expanded three-phase ports

Composite three-phase ports (default) | Expanded three-phase ports

Whether to model composite or expanded three-phase ports.

Composite three-phase ports represent three individual electrical conserving ports with a single block port. You can use composite three-phase ports to build models that correspond to single-line diagrams of three-phase electrical systems.

Expanded three-phase ports represent the individual phases of a three-phase system using three separate electrical conserving ports.

Parameterization — Line impedance parameterization

Balanced impedance (default) | General impedance

Specify how to parameterize the impedance of the three lines:

• Balanced impedance — Specify the same series resistance, series inductance, and parallel leakage conductance for all lines.

• General impedance — Specify the series resistance, series inductance, and parallel leakage conductance separately for each line.

Line a inductance — Line a self-inductance

1e-3 H (default) | positive scalar

Self-inductance of line a.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line b inductance — Line b self-inductance

1e-3 H (default) | positive scalar

Self-inductance of line b.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line c inductance — Line c self-inductance

1e-3 H (default) | positive scalar

Self-inductance of line c.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line inductance — Self-inductance

1e-3 H (default) | positive scalar

Self-inductance of lines a, b, and c.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Line a-b mutual inductance — Mutual inductance between a-b

3e-4 H (default) | positive scalar

Mutual inductance between lines a and b. If you know the coupling factor, set this value to $k\sqrt{L_a{L}_b}$. To have a physically realizable mutual inductance, this value must satisfy:

$$-\sqrt{L_a{L}_b}<L_{m,ab}<\sqrt{L_a{L}_b}.$$

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line b-c mutual inductance — Mutual inductance between b-c

3e-4 H (default) | positive scalar

Mutual inductance between lines b and c. If you know the coupling factor, set this value to $k\sqrt{L_b{L}_c}$. To have a physically realizable mutual inductance, this value must satisfy:

$$-\sqrt{L_b{L}_c}<L_{m,bc}<\sqrt{L_b{L}_c}.$$

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line a-c mutual inductance — Mutual inductance between a-c

3e-4 H (default) | positive scalar

Mutual inductance between lines a and c. If you know the coupling factor, set this value to $k\sqrt{L_a{L}_c}$. To have a physically realizable mutual inductance, this value must satisfy:

$$-\sqrt{L_a{L}_c}<L_{m,ac}<\sqrt{L_a{L}_c}.$$

Dependencies

To enable this parameter, set Parameterization to General impedance.

Mutual inductance — Mutual inductance

3e-4 H (default) | positive scalar

Mutual inductance between each pair of lines. If you know the coupling factor, set this value to kL, where L is the series inductance of each of the lines. To have a physically realizable mutual inductance, this value must satisfy:

$$-L<L_m<L.$$

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Resistance

Line a resistance — Line a series resistance

0.001 Ohm (default) | nonnegative scalar

Series resistance of line a.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line b resistance — Line b series resistance

0.001 Ohm (default) | nonnegative scalar

Series resistance of line b.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line c resistance — Line c series resistance

0.001 Ohm (default) | nonnegative scalar

Series resistance of line c.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line a-b mutual resistance — Mutual resistance between a-b

0 Ohm (default) | nonnegative scalar

Since R2025a

Mutual resistance between lines a and b.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line a-c mutual resistance — Mutual resistance between a-c

0 Ohm (default) | nonnegative scalar

Since R2025a

Mutual resistance between lines a and c.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line b-c mutual resistance — Mutual resistance between b-c

0 Ohm (default) | nonnegative scalar

Since R2025a

Mutual resistance between lines b and c.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line resistance — Series resistance

0.001 Ohm (default) | nonnegative scalar

Series resistance of lines a, b, and c.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Mutual resistance — Mutual resistance

0 Ohm (default) | nonnegative scalar

Mutual resistance between each pair of lines. Use this value to account for losses in a common return path.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Line a leakage conductance — Line a parallel leakage conductance

1e-9 1/Ohm (default) | nonnegative scalar

Parallel leakage conductance of line a.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line b leakage conductance — Line b parallel leakage conductance

1e-9 1/Ohm (default) | nonnegative scalar

Parallel leakage conductance of line b.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line c leakage conductance — Line c parallel leakage conductance

1e-9 1/Ohm (default) | nonnegative scalar

Parallel leakage conductance of line a.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line leakage conductance — Parallel leakage conductance

1e-9 1/Ohm (default) | nonnegative scalar

Parallel leakage conductance of lines a, b, and c.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Faults

Port 1 connection fault — Option to add fault to port 1

Add fault

Since R2025a

Option to add a fault to the three-phase connection port ~1 of the block.

To add a fault, click the Add fault hyperlink.

Fault type — Type of fault

Single-phase to ground (a-g) (default) | Single-phase to ground (b-g) | Single-phase to ground (c-g) | Two-phase (a-b) | Two-phase (b-c) | Two-phase (c-a) | Two-phase to ground (a-b-g) | Two-phase to ground (b-c-g) | Two-phase to ground (c-a-g) | Three-phase (a-b-c) | Three-phase to ground (a-b-c-g)

Since R2025a

Type of fault to model. The visibility of the related parameters depends on the fault model.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter.

Faulted phase-neutral resistance — Faulted resistance between phase connection and neutral point

1e-3 Ohm (default) | positive scalar

Since R2025a

Resistance between the phase connection and the neutral point when the fault is active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter, and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Faulted neutral-ground resistance — Faulted resistance between neutral point and electrical reference

1e-3 Ohm (default) | positive scalar

Since R2025a

Resistance between the neutral point and the electrical reference when fault is active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter, and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase to ground (a-b-c-g)

Unfaulted phase-neutral conductance — Unfaulted conductance between phase connection and neutral point

1e-6 1/Ohm (default) | positive scalar

Since R2025a

Conductance between the phase connection and the neutral point when the fault is not active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Trigger type — Trigger type

Always on (default) | Timed | Additional triggers | Conditional | Manual

Trigger type, specified as one of these options:

• Always On — The fault injects at the start of the simulation.

• Timed — The fault injects when the simulation time reaches the value you specify for the Trigger fault at time parameter.

• Additional triggers — The fault injects as a result of additional triggers not available in the block dialog box. Select this option and click the Open fault properties hyperlink to set the Trigger type to one of these values in the Property Inspector:

  • Conditional — The fault injects as a result of a condition that reflects a behavior associated with a signal. Conditionals evaluate the Boolean expression in the Condition parameter at each time step. To learn more, see Create and Manage Conditionals.

  • Manual — The fault injects or clears when you toggle the status in the Fault Dashboard pane. To learn more, see Manually Trigger Faults in Models.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter.

Trigger fault at time — Time before entering faulted state

0 s (default) | nonnegative scalar

Simulation time at which the block enters the faulted state.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter and set Trigger type to Timed.

Trigger configuration — Option to enable additional trigger types

Open fault properties

Option to enable additional trigger types not available in the block dialog box. To set the Trigger type parameter to Conditional or Manual, click the Open fault properties hyperlink and set the Trigger type in the Property Inspector.

To enable this parameter, click the Add fault hyperlink for the Port 1 connection fault parameter and set Trigger type to Additional triggers.

Port 2 connection fault — Option to add fault to port 2

Add fault

Since R2025a

Option to add a fault to the three-phase connection port ~2 of the block.

To add a fault, click the Add fault hyperlink.

Fault type — Type of fault

Single-phase to ground (a-g) (default) | Single-phase to ground (b-g) | Single-phase to ground (c-g) | Two-phase (a-b) | Two-phase (b-c) | Two-phase (c-a) | Two-phase to ground (a-b-g) | Two-phase to ground (b-c-g) | Two-phase to ground (c-a-g) | Three-phase (a-b-c) | Three-phase to ground (a-b-c-g)

Since R2025a

Type of fault to model. The visibility of the related parameters depends on the fault model.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter.

Faulted phase-neutral resistance — Faulted resistance between phase connection and neutral point

1e-3 Ohm (default) | positive scalar

Since R2025a

Resistance between the phase connection and the neutral point when the fault is active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Faulted neutral-ground resistance — Faulted resistance between neutral point and electrical reference

1e-3 Ohm (default) | positive scalar

Since R2025a

Resistance between the neutral point and the electrical reference when fault is active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase to ground (a-b-c-g)

Unfaulted phase-neutral conductance — Unfaulted conductance between phase connection and neutral point

1e-6 1/Ohm (default) | positive scalar

Since R2025a

Conductance between the phase connection and the neutral point when the fault is not active.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter and set Fault type to one of these options:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Trigger type — Trigger type

Always on (default) | Timed | Additional triggers | Conditional | Manual

Trigger type, specified as one of these options:

• Always On — The fault injects at the start of the simulation.

• Timed — The fault injects when the simulation time reaches the value you specify for the Trigger fault at time parameter.

• Additional triggers — The fault injects as a result of additional triggers not available in the block dialog box. Select this option and click the Open fault properties hyperlink to set the Trigger type to one of these values in the Property Inspector:

  • Conditional — The fault injects as a result of a condition that reflects a behavior associated with a signal. Conditionals evaluate the Boolean expression in the Condition parameter at each time step. To learn more, see Create and Manage Conditionals.

  • Manual — The fault injects or clears when you toggle the status in the Fault Dashboard pane. To learn more, see Manually Trigger Faults in Models.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter.

Trigger fault at time — Time before entering faulted state

0 s (default) | nonnegative scalar

Simulation time at which the block enters the faulted state.

Dependencies

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter and set Trigger type to Timed.

Trigger configuration — Option to enable additional trigger types

Open fault properties

Option to enable additional trigger types not available in the block dialog box. To set the Trigger type parameter to Conditional or Manual, click the Open fault properties hyperlink and set the Trigger type in the Property Inspector.

To enable this parameter, click the Add fault hyperlink for the Port 2 connection fault parameter and set Trigger type to Additional triggers.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2018a

R2025a: Model faults at the three-phase connection ports

You can now model a phase-to-ground or a phase-to-phase fault at each of the two three-phase connection ports of the block. To specify the value of the faulted resistances and the unfaulted conductance, use the Faulted phase-neutral resistance, Faulted neutral-ground resistance, and Unfaulted phase-neutral conductance parameters.

R2025a: The Modeling Option parameter has changed

The Modeling option parameter has been replaced with the Electrical connection parameter inside the Main section in the block dialog box or Property Inspector.

If a model you created in an earlier release contains this block, the software automatically sets the value of the Electrical connection parameter to the value you specified for the Modeling option parameter.