Coupled Lines (Three-Phase)

Magnetically couple three-phase lines

Library:
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.

Here:

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 is the mutual resistance between the two lines. You can use this parameter to account for losses in a common return path.
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 and R_j-R_m, respectively.

Equations

The defining equation for this block is:

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

where:

$V = [\begin{matrix} V_{a} \\ \begin{array}{l} V_{b} \\ V_{c} \end{array} \end{matrix}],$

$I = [\begin{matrix} I_{a} \\ \begin{array}{l} I_{b} \\ I_{c} \end{array} \end{matrix}] .$

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:

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

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:

$L_{m, i j} = k \sqrt{L_{i} L_{j}} .$

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:

$| k | = 0$ — 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 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.

Ports

Conserving

expand all

`~1` — Line 1 positive terminal
electrical

Expandable three-phase electrical conserving port associated with the positive terminals of lines a, b, and c. To use this composite port, right-click the block and select Simscape > Block choices > Composite three-phase port.

`~2` — Line 1 negative terminal
electrical

Expandable three-phase electrical conserving port associated with the negative terminals of lines a, b, and c. To use this composite port, right-click the block and select Simscape > Block choices > Composite three-phase port.

`a1` — Line a positive terminal
electrical

Electrical conserving port associated with the positive terminal of line a. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

`a2` — Line a negative terminal
electrical

Electrical conserving port associated with the negative terminal of line a. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

`b1` — Line b positive terminal
electrical

Electrical conserving port associated with the positive terminal of line b. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

`b2` — Line b negative terminal
electrical

Electrical conserving port associated with the negative terminal of line b. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

`c1` — Line c positive terminal
electrical

Electrical conserving port associated with the positive terminal of line c. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

`c2` — Line c negative terminal
electrical

Electrical conserving port associated with the negative terminal of line c. To use this expanded port, right-click the block and select Simscape > Block choices > Expanded three-phase port.

Parameters

expand all

Main

`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)

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

Self-inductance of line c. 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 lines a, b, and c. This value must be greater than zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

`Line a-b mutual inductance` — Mutual inductance between a-b
`3e-4` `H` (default)

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:

$- \sqrt{L_{a} L_{b}} < L_{m, a b} < \sqrt{L_{a} L_{b}} .$

Dependencies

To enable this parameter, set Parameterization to General impedance.

`Line b-c mutual inductance` — Mutual inductance between b-c
`3e-4` `H` (default)

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:

$- \sqrt{L_{b} L_{c}} < L_{m, b c} < \sqrt{L_{b} L_{c}} .$

Dependencies

To enable this parameter, set Parameterization to General impedance.

`Line a-c mutual inductance` — Mutual inductance between a-c
`3e-4` `H` (default)

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:

$- \sqrt{L_{a} L_{c}} < L_{m, a c} < \sqrt{L_{a} L_{c}} .$

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

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)

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

Series resistance of line c. 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 lines a, b, and c. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

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

Mutual resistance between each pair of lines. This value must be greater than or equal to zero. Use this value to account for losses in a common return path.

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

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

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

Dependencies

To enable this parameter, set Parameterization to General impedance.

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

Parallel leakage conductance of line a. 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 lines a, b, and c. This value must be greater than or equal to zero.

Dependencies

To enable this parameter, set Parameterization to Balanced impedance.

Documentation

Coupled Lines (Three-Phase)

Description

Equivalent Circuit

Equations

Inductive Coupling

Mutual Resistance

Ports

Conserving

~1 — Line 1 positive terminal electrical

~2 — Line 1 negative terminal electrical

a1 — Line a positive terminal electrical

a2 — Line a negative terminal electrical

b1 — Line b positive terminal electrical

b2 — Line b negative terminal electrical

c1 — Line c positive terminal electrical

c2 — Line c negative terminal electrical

Parameters

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

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

Dependencies

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

Dependencies

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

Dependencies

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

Dependencies

Line a-b mutual inductance — Mutual inductance between a-b 3e-4 H (default)

Dependencies

Line b-c mutual inductance — Mutual inductance between b-c 3e-4 H (default)

Dependencies

Line a-c mutual inductance — Mutual inductance between a-c 3e-4 H (default)

Dependencies

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

Dependencies

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

Dependencies

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

Dependencies

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

Dependencies

Line resistance — Series resistance 0.001 Ohm (default)

Dependencies

Mutual resistance — Mutual resistance 0 Ohm (default)

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

Dependencies

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

Dependencies

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

Dependencies

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

Dependencies

Extended Capabilities

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See Also

Simscape Electrical Documentation

Support

`~1` — Line 1 positive terminal
electrical

`~2` — Line 1 negative terminal
electrical

`a1` — Line a positive terminal
electrical

`a2` — Line a negative terminal
electrical

`b1` — Line b positive terminal
electrical

`b2` — Line b negative terminal
electrical

`c1` — Line c positive terminal
electrical

`c2` — Line c negative terminal
electrical

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

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

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

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

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

`Line a-b mutual inductance` — Mutual inductance between a-b
`3e-4` `H` (default)

`Line b-c mutual inductance` — Mutual inductance between b-c
`3e-4` `H` (default)

`Line a-c mutual inductance` — Mutual inductance between a-c
`3e-4` `H` (default)

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

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

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

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

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

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

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

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

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

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

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Generate C and C++ code using Simulink® Coder™.