Multiphase Distributed Parameter Line

R2026b

Multiphase transmission line with distributed parameters

Since R2026b

Libraries:
Simscape / Electrical / Passive / Lines

Description

The Multiphase Distributed Parameter Line block models a multiphase (M-phase) transmission line using distributed parameters. An M-phase distributed parameter line is an electrical transmission line with multiple conductors whose electrical characteristics are spread continuously along the line.

This block allows you to model either a balanced or an unbalanced line. To choose which line to model, set the Line type parameter to either Balanced lines or General lines.

This figure shows the equivalent circuit for an M-phase distributed parameter line in the modal domain.

Equivalent circuit diagram for an M-phase distributed parameter line in the modal domain

Balanced M-Phase Lines

For balanced M-phase lines, the resistance, inductance, and capacitance matrices per unit length in the phase domain are

$\begin{array}{l} R_{phase} = [\begin{matrix} R_{s} & R_{m} & \dots & R_{m} \\ R_{m} & R_{s} & \dots & R_{m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ R_{m} & R_{m} & \dots & R_{s} \end{matrix}] \\ L_{phase} = [\begin{matrix} L_{s} & L_{m} & \dots & L_{m} \\ L_{m} & L_{s} & \dots & L_{m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ L_{m} & L_{m} & \dots & L_{s} \end{matrix}] \\ C_{phase} = [\begin{matrix} C_{s} & C_{m} & \dots & C_{m} \\ C_{m} & C_{s} & \dots & C_{m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ C_{m} & C_{m} & \dots & C_{s} \end{matrix}] \end{array}$

where:

R_s is the self resistance of the transmission line per phase per unit length.
R_m is the line-line mutual resistance per unit length.
L_s is the self inductance of the transmission line per phase per unit length.
L_m is the line-line mutual inductance per unit length.
C_s is the self capacitance of the transmission line per phase per unit length.
C_m is the line-line mutual capacitance per unit length.

To transform these matrices from the phase domain to the modal domain, the block defines a transformation matrix T_v or T_i. This matrix is a canonical orthonormal modal transformation for an M-phase transposed and perfectly symmetric line.

$[T] = [\begin{matrix} \frac{1}{\sqrt{M}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \dots & \frac{1}{\sqrt{J (J - 1)}} & \dots & \frac{1}{\sqrt{M (M - 1)}} \\ \frac{1}{\sqrt{M}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \dots & \frac{1}{\sqrt{J (J - 1)}} & \dots & \frac{1}{\sqrt{M (M - 1)}} \\ \frac{1}{\sqrt{M}} & 0 & - \frac{2}{\sqrt{6}} & \dots & ⋮ & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & \dots & - \frac{(J - 1)}{\sqrt{J} (J - 1)} & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & \dots & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & \dots & ⋮ \\ \frac{1}{\sqrt{M}} & 0 & 0 & \dots & 0 & \dots & - \frac{(M - 1)}{\sqrt{M} (M - 1)} \end{matrix}] .$

In this matrix, M denotes the number of phases and J is an index from 2 to M that defines the successive orthogonal modal vectors. For example, if M is equal to 5, the transformation matrix is:

$[T] = [\begin{matrix} \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{12}} & \frac{1}{\sqrt{20}} \\ \frac{1}{\sqrt{5}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{12}} & \frac{1}{\sqrt{20}} \\ \frac{1}{\sqrt{5}} & 0 & - \frac{2}{\sqrt{6}} & \frac{1}{\sqrt{12}} & \frac{1}{\sqrt{20}} \\ \frac{1}{\sqrt{5}} & 0 & 0 & - \frac{3}{\sqrt{12}} & \frac{1}{\sqrt{20}} \\ \frac{1}{\sqrt{5}} & 0 & 0 & 0 & - \frac{4}{\sqrt{20}} \end{matrix}] .$

By applying the transformation matrix to the resistance, inductance, and capacitance matrices, the block calculates these diagonal matrices in the modal domain:

$\begin{array}{l} R_{mode} = [\begin{matrix} R_{z e r o} \\ R_{p o s} \\ ⋱ \\ R_{p o s} \end{matrix}] \\ L_{mode} = [\begin{matrix} L_{z e r o} \\ L_{p o s} \\ ⋱ \\ L_{p o s} \end{matrix}] \\ C_{mode} = [\begin{matrix} C_{z e r o} \\ C_{p o s} \\ ⋱ \\ C_{p o s} \end{matrix}] \end{array}$

where:

$R_{z e r o} = R_{s} + (M - 1) R_{m}$ is the individual zero-sequence resistance.
$R_{p o s} = R_{s} - R_{m}$ is the individual positive-sequence resistance.
$L_{z e r o} = L_{s} + (M - 1) L_{m}$ is the individual zero-sequence inductance.
$L_{p o s} = L_{s} - L_{m}$ is the individual positive-sequence inductance.
$C_{z e r o} = C_{s} + (M - 1) C_{m}$ is the individual zero-sequence capacitance.
$C_{p o s} = C_{s} - C_{m}$ is the individual positive-sequence capacitance.

By applying the transformation matrix to the phase voltages V_phase and phase currents I_phase, the block calculates the mode voltages V_mode and mode currents I_mode:

$\begin{array}{l} V_{phase} = T_{v} \cdot V_{mode} \\ I_{phase} = T_{i} \cdot I_{mode} \end{array}$

As a result, by applying the transformation matrix, the block transforms the original M-phase coupled transmission line in the phase domain into an M-phase decoupled transmission line in the modal domain. Then, the block can solve the equations corresponding to these decoupled transmission lines as single-phase equations.

In the modal domain, the block models the lossless distributed parameter line using its characteristic impedance Z_c and delay τ. These equations define the impedance and delay for each mode:

$\begin{array}{l} Z_{c} = \sqrt{\frac{L_{mode}}{C_{mode}}} \\ τ = \frac{length}{v} \end{array}$

where length is the transmission line length and $v = \frac{1}{\sqrt{L_{mode} C_{mode}}}$ is the propagation speed.

To introduce losses, you can connect N delay-based losses lines in series through a set of resistors:

Equivalent circuit diagram showing N delay-based components connected in series via a set of resistors

where N is an integer greater than or equal to 1 and $r = \frac{R_{mode} \cdot length}{N}$ . For more information about modeling delay-based lines, see Transmission Line.

General M-Phase Lines

For multiphase untransposed lines, the diagonal and off-diagonal elements of the resistance, inductance, and capacitance matrices per unit length are not necessarily identical. If they are not identical, then set the Line type parameter to General lines and individually specify the resistance, inductance, and capacitance matrices per unit length.

The general lines equations are the same as the balanced lines equations, but the block must derive the transformation matrices T_v and T_i from the eigenvalue and eigenvector theory. For more information, see the Section 4.1.5 of the Electromagnetic transients program (EMTP) theory book [1].

You can specify electrical quantities like the resistance, inductance, and capacitance matrices by using the block parameters or you can use a built-in MATLAB^® script to calculate the parameter values. To open the script, click the Open live script button next to Calculate parameters at the top of the block mask. The script shows how to calculate the parameters using electrical properties that manufacturers typically provide for the conductors and geometric data that describes the conductors and transmission tower. Enter numerical values in the edit fields and run the script to calculate the parameters. To apply the parameters to the block, click the Apply Parameters button at the bottom of the script.

Assumptions and Limitations

The 1 and 2 ports of this block represent multiple nodes as single ports. The number of nodes inside each port depends on the value of the Number of phases parameter. To access each connection port on both sides of the block, use the Array Connection block.
The Multiphase Distributed Parameter Line block does not support energy accounting. If you try to get energy information for this block using the getEnergyInfo function, the function generates an error message.

Ports

Conserving

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1 — End 1 of transmission line
electrical

Electrical conserving port associated with one end of the transmission line.

2 — End 2 of transmission line
electrical

Electrical conserving port associated with the other end of the transmission line.

Parameters

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Calculate parameters — Open live script
button

Click this button to open a MATLAB script that calculates block parameter values from data that describes the conductor and transmission tower geometry. You can apply the parameters directly to the block by clicking the Apply Parameters button at the bottom of the script.

Line Type — Option to specify line type
`Balanced lines` (default) | `General lines`

Option to specify the line type, specified as one of these values:

Balanced lines — Model the transmission line as a balanced line. In a balanced line, the diagonal and off-diagonal elements of the resistance, inductance, and capacitance matrices per unit length are identical. If you select this option, then you must specify the distributed parameters of the line, including the self and mutual resistances, self and mutual inductances, and self and mutual capacitances.
General lines — Model the transmission line as an unbalanced line. In an unbalanced line, the diagonal and off-diagonal elements of the resistance, inductance, and capacitance matrices per unit length are not necessarily equal among themselves. If you select this option, you must specify the resistance, inductance, and capacitance matrices per unit length of the transmission line.

Line length — Total length of transmission line
`100` `km` (default) | positive scalar

Total length of the transmission line.

Frequency used for rlc specification — Frequency used for RLC specification
`60` `Hz` (default) | positive scalar

Frequency that the block uses for the R, L, and C specification, where:

R is line resistance per unit length.
L is the line inductance per unit length.
C is the line capacitance per unit length.

Number of phases — Number of phases
`3` (default) | positive scalar greater than 1

Number of phases in the transmission line.

Number of segments — Number of model segments
`2` (default) | positive scalar greater than or equal to 1

Number of model segments that the block uses to represent the transmission line.

Self resistance per unit length — Self resistance per phase per unit length
`0.02` `Ohm/km` (default) | positive scalar

Self resistance per phase per unit length of the transmission line. This parameter represents the value of each of the diagonal elements, R_s, of the resistance matrix per unit length.

The value of this parameter must be greater than the value of the Mutual resistance per unit length parameter.