Coupled Lines (Three-Phase)
Three-phase lines with magnetic coupling.
Description
The Coupled Lines (Three-Phase) block models three magnetically coupled lines. Each line is characterised by inductance, series ohmic resistance and parallel conductivity. In addition, there is mutual inductance and mutual resistance between each pair of lines.
Use this block when the magnetic coupling in a three-phase network is not negligible. These effects are most noticeable when:
-
The lines are parallel and close together.
-
The intrinsic inductance of the lines is high.
-
The frequency of alternating current in the network is high.
To model capacitive coupling between lines, use the block Transmission Line (Three-Phase).
Equivalent circuit
The equivalent circuit shows the coupling between two arbitrary phases and . The block models magnetic coupling using the following equivalent circuit between each pair of phases , and .
In this circuit:
-
and are the ohmic resistances of the lines and .
-
and - eigenvalue inductances of the lines and .
-
- is the magnetic resistance between the two lines. This parameter can be used to account for losses in the common neutral (see below).
-
- mutual inductance of the lines and .
-
and - leakage conductivity of the lines and .
-
and - voltage drop on lines and .
-
and - currents through resistors - and - .
Equations
The defining equation for this block is:
(dI)/dt,]
where:
, and , are generally not equal to the line currents of , and . The line currents form the following vector:
Inductive coupling
To quantify the magnetic coupling between two lines, we can use the coupling coefficient or coupling factor . The coupling coefficient relates the mutual inductance to the line’s own inductance:
This coefficient should be in the range , where a negative value indicates a change in the orientation of one of the coils. The value indicates:
-
- there is no magnetic coupling between the two lines.
-
- the two lines are weakly coupled and the mutual magnetic effects are small.
-
- two lines are strongly coupled and the mutual magnetic effects are large.
Mutual resistance
If three lines share a common neutral, the resistance of this return path can be modelled using the Mutual resistance parameter . This approach is equivalent to setting Mutual resistance to zero and explicitly specifying the neutral resistance , as shown in the diagram below.
If the three lines do not share a common neutral, set Mutual resistance to zero and explicitly add each of the neutral resistances to the model.
Parameters
Main
parameterization - parameterization of line impedance
Balanced impedance (by default) | General impedance | Sequence impedance
Specify how to parameterize the impedance of the three lines:
-
Balanced impedance- the same series impedance, series inductance and parallel leakage conductance is used for all lines. -
`General impedance' - uses series impedance, series inductance and parallel leakage conductance separately for each line.
-
`Sequence impedance' - uses three phase balanced inductive and resistive impedance with mutual coupling between phases. This provides a more convenient way of entering system parameters in terms of forward and zero sequence resistances and inductances than intrinsic and mutual resistances and inductances.
Line inductance - line inductance
1e-3 Gn (by default).
Own inductance of lines a, b and c. This value must be greater than zero.
Dependencies
To use this parameter, set parameterization to Balanced impedance.
Mutual inductance - mutual inductance
3e-4 Gn (by default).
The mutual inductance of each pair of lines. If you know the coupling coefficient, set this value to , where is the intrinsic inductance of each line. To be physically realisable, this value must satisfy the inequality:
Dependencies
To use this parameter, set parameterization to Balanced impedance.
Line a inductance is the self-inductance of the line a
1e-3 Gn (by default).
Line self-inductance . This value must be greater than zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line b inductance - self-inductance of line b
1e-3 Gn (by default).
Line self-inductance . This value must be greater than zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line c inductance is the self-inductance of the c line
1e-3 Gn (by default).
Line self-inductance . This value must be greater than zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line a-b mutual inductance - mutual inductance between a and b
3e-4 Gn (by default).
Mutual inductance between lines and .
If you know the coupling coefficient, set this value to . To have a physically realisable mutual inductance, this value must be satisfied:
Dependencies
To use this parameter, set parameterization to General impedance.
Line b-c mutual inductance - mutual inductance between lines b and c
3e-4 Gn (by default).
Mutual inductance between lines and .
If you know the coupling coefficient, set this value to . To have a physically realisable mutual inductance, this value must be satisfied:
Dependencies
To use this parameter, set parameterization to General impedance.
Line a-c mutual inductance - mutual inductance between lines a and c
3e-4 Gn (by default).
Mutual inductance between lines and .
If you know the coupling coefficient, set this value to . To have a physically realisable mutual inductance, this value must be satisfied:
Dependencies
To use this parameter, set parameterization to General impedance.
Positive-sequence inductance - forward sequence inductance
50e-3 Gn (by default).
Direct-sequence inductance.
Dependencies
To use this parameter, set parameterization to Sequence impedance.
Zero-sequence inductance - zero-sequence inductance
100e-3 Gn (By default).
Zero-sequence inductance.
Dependencies
To use this parameter, set parameterization to Sequence impedance.
Resistance
Line resistance - series resistance
`0.001 ohms (by default)
Series resistance of the lines , and . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to Balanced impedance.
Mutual resistance - mutual resistance
`0 ohms (by default).
The mutual resistance of each pair of lines. This value must be greater than or equal to zero.
Use this value to account for losses in the common neutral.
Dependencies
To use this parameter, set parameterization to Balanced impedance.
Line leakage conductance - parallel leakage conductance
1e-9 1/ohm (by default).
Parallel leakage conductance of lines , and . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to Balanced impedance.
Line a resistance - the series resistance of the line a
`0.001 ohms (by default).
Series resistance of the line . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line b resistance - series resistance of line b
0.001 ohms (by default).
Series resistance of the line . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line c resistance - series resistance of line c
0.001 ohms (by default).
Series resistance of the line . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line a leakage conductance - parallel line a leakage conductance
1e-9 1/ohm (by default).
Parallel line leakage conductance . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line b leakage conductance - parallel b line leakage conductance
1e-9 1/ohm (by default).
Parallel line leakage conductance . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Line c leakage conductance - parallel c line leakage conductance
1e-9 1/ohm (by default).
Parallel line leakage conductance . This value must be greater than or equal to zero.
Dependencies
To use this parameter, set parameterization to General impedance.
Positive-sequence resistance is the series resistance of the direct sequence
`2.0 ohms (by default).
Positive-sequence series resistance.
Dependencies
To use this parameter, set parameterization to Sequence impedance.
Zero-sequence resistance is the zero-sequence impedance in series
`4.0 ohms (by default).
Zero-sequence series resistance.
Dependencies
To use this parameter, set parameterization to Sequence impedance.