Coupled Lines (Pair)

Magnetically couple two lines

Libraries:
Simscape / Electrical / Passive / Lines

Description

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.

Equivalent Circuit

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

Here:

R₁ and R₂ are the series resistances of lines 1 and 2, respectively.
L₁ and L₂ are the self-inductances of lines 1 and 2, respectively.
R_m is the mutual resistance between the two lines. You can use this parameter to account for losses in a common return path.
L_m is the mutual inductance between the two lines.
G₁ and G₂ are the leakage conductances of lines 1 and 2, respectively.
V₁ and V₂ are voltage drops across lines 1 and 2, respectively.
I₁ and I₂ are the currents through the resistors R₁-R_m and R₂-R_m, respectively.

Equations

The defining equation for this block is:

$V = [\begin{matrix} R_{1} & R_{m} \\ R_{m} & R_{2} \end{matrix}] I + [\begin{matrix} L_{1} & L_{m} \\ L_{m} & L_{2} \end{matrix}] \frac{d I}{d t},$

where:

$V = [\begin{matrix} V_{1} \\ V_{2} \end{matrix}],$

$I = [\begin{matrix} I_{1} \\ I_{2} \end{matrix}] .$

I₁ and I₂ are, in general, not equal to the currents in line 1 and line 2. These terminal currents make up the vector:

$I_{t o t a l} = I + [\begin{matrix} G_{1} & 0 \\ 0 & G_{2} \end{matrix}] V .$

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:

$L_{m} = k \sqrt{L_{1} L_{2}} .$

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:

$| k | = 0$ — There is no magnetic coupling between the two lines.
$0 < | k | < 0.5$ — The two lines are loosely coupled and mutual magnetic effects are small.
$0.5 \leq | k | < 1$ — The two lines are strongly coupled and mutual magnetic effects are large.

Mutual Resistance

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.

Ports

Conserving

expand all

1+ — Line 1 positive terminal
electrical

Electrical conserving port associated with the positive terminal of line 1.

1- — Line 1 negative terminal
electrical

Electrical conserving port associated with the negative terminal of line 1.

2+ — Line 2 positive terminal
electrical

Electrical conserving port associated with the positive terminal of line 2.

2- — Line 2 negative terminal
electrical

Electrical conserving port associated with the negative terminal of line 2.

Parameters

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Parameters

Parameterization — Line impedance parameterization
`Balanced impedance` (default) | `General impedance`

Specify how to parameterize the impedance of the two lines:

Balanced impedance: Specify the same series resistance, series inductance, and parallel leakage conductance for both lines.
General impedance: Specify the series resistance, series inductance, and parallel leakage conductance separately for each line.

Line 1 inductance — Line 1 self-inductance
`1e-3` `H` (default)

Self-inductance of line 1. This value must be greater than zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line 2 inductance — Line 2 self-inductance
`1e-3` `H` (default)

Self-inductance of line 2. This value must be greater than zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line inductance — Self-inductance
`1e-3` `H` (default)

Self-inductance of line 1 and line 2. This value must be greater than zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Mutual inductance — Mutual inductance between lines
`3e-4` `H` (default)

Mutual inductance between line 1 and line 2. If you know the coupling factor, set this value to $k \sqrt{L_{1} L_{2}}$ . To have a physically realizable mutual inductance, this value must satisfy:

$- \sqrt{L_{1} L_{2}} < L_{m} < \sqrt{L_{1} L_{2}} .$

Line 1 resistance — Line 1 series resistance
`0.001` `Ohm` (default)

Series resistance of line 1. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line 2 resistance — Line 2 series resistance
`0.001` `Ohm` (default)

Series resistance of line 2. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line resistance — Series resistance
`0.001` `Ohm` (default)

Series resistance of line 1 and line 2. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Mutual resistance — Mutual resistance between lines
`0` `Ohm` (default)

Mutual resistance between line 1 and line 2. This value must be greater than or equal to zero. Use this value to account for losses in a common return path.

Line 1 leakage conductance — Line 1 parallel leakage conductance
`1e-9` `1/Ohm` (default)

Parallel leakage conductance of line 1. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line 2 leakage conductance — Line 2 parallel leakage conductance
`1e-9` `1/Ohm` (default)

Parallel leakage conductance of line 2. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to General impedance.

Line leakage conductance — Parallel leakage conductance
`1e-9` `1/Ohm` (default)

Parallel leakage conductance of line 1 and line 2. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Extended Capabilities

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

Version History

Introduced in R2018a

Coupled Lines (Pair)

Description

Equivalent Circuit

Equations

Inductive Coupling

Mutual Resistance

Ports

Conserving

1+ — Line 1 positive terminal electrical

1- — Line 1 negative terminal electrical

2+ — Line 2 positive terminal electrical

2- — Line 2 negative terminal electrical

Parameters

Parameterization — Line impedance parameterization Balanced impedance (default) | General impedance

Line 1 inductance — Line 1 self-inductance 1e-3 H (default)

Dependencies

Line 2 inductance — Line 2 self-inductance 1e-3 H (default)

Dependencies

Line inductance — Self-inductance 1e-3 H (default)

Dependencies

Mutual inductance — Mutual inductance between lines 3e-4 H (default)

Line 1 resistance — Line 1 series resistance 0.001 Ohm (default)

Dependencies

Line 2 resistance — Line 2 series resistance 0.001 Ohm (default)

Dependencies

Line resistance — Series resistance 0.001 Ohm (default)

Dependencies

Mutual resistance — Mutual resistance between lines 0 Ohm (default)

Line 1 leakage conductance — Line 1 parallel leakage conductance 1e-9 1/Ohm (default)

Dependencies

Line 2 leakage conductance — Line 2 parallel leakage conductance 1e-9 1/Ohm (default)

Dependencies

Line leakage conductance — Parallel leakage conductance 1e-9 1/Ohm (default)

Dependencies

Extended Capabilities

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

Version History

See Also

1+ — Line 1 positive terminal
electrical

1- — Line 1 negative terminal
electrical

2+ — Line 2 positive terminal
electrical

2- — Line 2 negative terminal
electrical

Parameterization — Line impedance parameterization
`Balanced impedance` (default) | `General impedance`

Line 1 inductance — Line 1 self-inductance
`1e-3` `H` (default)

Line 2 inductance — Line 2 self-inductance
`1e-3` `H` (default)

Line inductance — Self-inductance
`1e-3` `H` (default)

Mutual inductance — Mutual inductance between lines
`3e-4` `H` (default)

Line 1 resistance — Line 1 series resistance
`0.001` `Ohm` (default)

Line 2 resistance — Line 2 series resistance
`0.001` `Ohm` (default)

Line resistance — Series resistance
`0.001` `Ohm` (default)

Mutual resistance — Mutual resistance between lines
`0` `Ohm` (default)

Line 1 leakage conductance — Line 1 parallel leakage conductance
`1e-9` `1/Ohm` (default)

Line 2 leakage conductance — Line 2 parallel leakage conductance
`1e-9` `1/Ohm` (default)

Line leakage conductance — Parallel leakage conductance
`1e-9` `1/Ohm` (default)

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