Compute RLC parameters of radial copper cables with single screen, based on conductor and insulator characteristics

`power_cableparam`

For a set of N cables, `power_cableparam`

computes
the self- and mutual impedances, the phase-to-screen, and screen to
ground capacitances of radial cables with screen.

The `power_cableparam`

function assumes that
a cable consists of an inner copper phase conductor with an outer
screen conductor, using cross-linked polyethylene (XLPE) insulator
material.

The following figure shows a typical high-voltage cable.

The variables used in the equations are:

*N*: The number of cables

*n*: the number of strands contained in the
phase conductor.

*d*: the diameter of one strand (m)

*f*: the nominal frequency of the cable application

*r*: the radius of the phase conductor

µ*r*: the relative permeability of phase
conductor

*rint*, *rext*: the internal
and external radius of the screen conductor

GMD: Geometric mean distance between the phase conductors.

ρ: Resistivity of the screen conductor

ɛ*rax*: Relative permittivity of the
phase-screen insulator

ɛ*rxe*: Relative permittivity of the
outer screen insulator

*dax,Dax*: the internal and external diameter
of phase-screen insulator

*dxe,Dxe*: the internal and external diameter
of the outer screen insulator

The self-impedance of the copper phase conductor is calculated as follow

$$\begin{array}{cc}{Z}_{aa}={R}_{\varphi}+{R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GM{R}_{\varphi}}\right)& \Omega /\text{km}\end{array}$$

The DC resistance of phase conductor is given by

$$\begin{array}{cc}{R}_{\varphi}={\rho}_{Cu}\frac{1000}{{S}_{Cu}}=(17.8e-9)\frac{1000}{n\pi {(d/2)}^{2}}& \Omega /\text{km}\end{array}$$

The resistance of earth return is given by

$$\begin{array}{cc}{R}_{e}={\pi}^{2}\cdot {10}^{-4}\cdot f& \Omega /\text{km}\end{array}$$

The frequency factor is given by

$$\begin{array}{cc}{k}_{1}=0.0529\cdot \frac{f}{0.3048\cdot 60}& units\text{}(\Omega /\text{km})\end{array}$$

The distance to equivalent earth return path is given by

$$\begin{array}{cc}{D}_{e}=1650\sqrt{{\rho}_{e}/\left(2\pi f\right)}& m\\ {\rho}_{Cu}=17.8e-9& \Omega /m\end{array}$$

The geometric mean radius of phase conductor is given by

$$GM{R}_{\varphi}=r\cdot \mathrm{exp}\left(-\frac{{\mu}_{r}}{4}\right)$$

The self-impedance of the screen conductor is calculated as follow

$$\begin{array}{cc}{Z}_{xx}={R}_{N}+{R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GM{R}_{N}}\right)& \Omega /\text{km}\end{array}$$

The DC resistance of the screen conductor is given by

$$\begin{array}{cc}{R}_{N}=\rho \frac{1000}{S}& \Omega /\text{km}\end{array}$$

The geometric mean radius of the screen conductor is given by

$$GM{R}_{N}={r}_{\mathrm{int}}+\frac{\left({r}_{ext}-{r}_{\mathrm{int}}\right)}{2}$$

The mutual impedance between the phase conductor and its corresponding screen conductor is calculated as follow

$$\begin{array}{cc}{Z}_{ax}={R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{{D}_{n}}\right)& \Omega /\text{km}\end{array}$$

*Dn* corresponds to the distance between
the phase conductor and the mean radius of the screen conductor.

If more than one cable is modeled (N>1), the mutual impedance between the N phase conductors is calculated as follow

$$\begin{array}{cc}{Z}_{ab}={R}_{e}+j{k}_{1}\mathrm{log}\left(\frac{{D}_{e}}{GMD}\right)& \Omega /\text{km}\end{array}$$

In general, the Geometric Mean Distance (GMD) between the phase conductors of a given set of cables can be calculated as follow

$$GMD=\sqrt[n]{{\displaystyle \prod _{1}^{n}{d}_{xy}}}$$

where* n* is the total number of distances
between the conductors. However the GMD value is not calculated by
the function and needs to be specified directly as an input parameter.

The capacitance between the phase conductor and its corresponding screen conductor is calculated as follow

$$\begin{array}{cc}{C}_{ax}=\frac{1}{0.3048}\left(\frac{0.00736{\epsilon}_{rax}}{\text{log}({D}_{ax}/{d}_{ax})}\right)& \mu F/\text{km}\end{array}$$

The cross-linked polyethylene (XLPE) insulator material is assumed in this equation.

The same equation is used to calculate the capacitance between the screen conductor and the ground

$$\begin{array}{cc}{C}_{xe}=\frac{1}{0.3048}\left(\frac{0.00736{\epsilon}_{rxe}}{\text{log}({D}_{xe}/{d}_{xe})}\right)& \mu F/\text{km}\end{array}$$

The capacitive effect between the phase conductors is negligible and therefore not computed by the power_cableparam function.

`[r,l,c,z] = power_cableparam(CableData)`

computes
the impedances and capacitances of a given set of cables with screen
conductor. The conductor and insulator characteristics are given in
the `CableParam`

structure with the following fields

Field | Description |
---|---|

| the number of cables |

f | the frequency in hertz to be used to evaluate RLC parameters |

rh0_e | the ground resistivity (in ohm.meters) |

n_ba | the number of strands contained in one phase conductor |

d_ba | diameter of one strand (in m) |

rho_ba | DC resistivity of conductor in ohms*m. |

mu_r_ba | relative permeability of the conductor material. |

D_a | phase conductor outside diameter (in m) |

rho_x | DC resistivity of the screen conductor in ohms*m. |

S_x | Total section of screen conductor (in m^2) |

d_x | screen conductor internal diameter (in m) |

D_x | screen conductor external diameter (in m) |

GMD_phi | Geometric Mean Distance between the cables. |

d_iax | phase-screen insulator internal diameter (in m) |

D_iax | phase-screen insulator external diameter (in m) |

epsilon_iax | relative permittivity of the phase-screen insulator material. |

d_ixe | outer screen insulator internal diameter (in m) |

D_ixe | outer screen insulator external diameter (in m) |

epsilon_ixe | relative permittivity of the outer screen insulator material. |

The output arguments are of the form of structure variables with the following fields

Variable, Field | Description |
---|---|

| Self resistance of phase conductor, in Ohm/Km |

r.xx | Self resistance of screen conductor, in Ohm/Km |

r.ab | Mutual resistance between the phase conductors, in Ohm/Km |

r.ax | Mutual resistance between phase and screen conductors, in Ohm/Km |

l.aa | Self inductance of phase conductor, in Henries/Km |

l.xx | Self inductance of screen conductor, in Henries/Km |

l.ab | Mutual inductance between the phase conductors, in Henries/Km |

l.ax | Mutual inductance between phase and screen conductor, in Henries/Km |

c.ax | Capacitance between the phase conductor and its screen conductor, in Farad/Km |

c.xe | Capacitance between the screen conductor and the ground, in Farad/Km |

z.aa | Self impedance of phase conductor, in Ohm/Km |

z.xx | Self impedance of screen conductor, in Ohm/Km |

z.ab | Mutual impedance between phase conductors, in Ohm/Km |

z.ax | Mutual impedance between phase and corresponding screen conductors, in Ohm/Km |

These computed resistances, impedances, and capacitances need
to be organized into 2N-by-2N matrices that can be directly used in
the Cable block. See the `power_cable`

example for
an example on how to build a block that represents a 4-Cables with
Screen block.

The RLC matrices are defined as follows (the example is given for a 3-cable configuration):

$$\begin{array}{cc}R=\left[\begin{array}{cccccc}{r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}\end{array}\right]& L=\left[\begin{array}{cccccc}{l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}\end{array}\right]\end{array}$$

$$C=\left[\begin{array}{cccccc}{c}_{ax}& -{c}_{ax}& 0& 0& 0& 0\\ -{c}_{ax}& {c}_{ax}+{c}_{xe}& 0& 0& 0& 0\\ 0& 0& {c}_{ax}& -{c}_{ax}& 0& 0\\ 0& 0& -{c}_{ax}& {c}_{ax}+{c}_{xe}& 0& 0\\ 0& 0& 0& 0& {c}_{ax}& -{c}_{ax}\\ 0& 0& 0& 0& -{c}_{ax}& {c}_{ax}+{c}_{xe}\end{array}\right]$$

`power_cableparam`

command opens a user interface (UI) that is used to
specify the cable parameters and to compute the electrical R, L, C cable parameters.

**Number of cables**Specify the number of cables. A cable consists of an inner phase conductor, an outer screen conductor, and insulator. This parameter determines the dimension of the R,L, and C matrices as follows: 2N-by-2N, where N is the number of cables.

**Frequency**Specify the frequency in hertz to be used to evaluate RLC parameters.

**Ground resistivity**Specify the ground resistivity in ohm.meters.

**Geometric mean distance between cables**Specify the Geometric Mean Distance (GMD) between the cables. Set this value to zero if the Number of cables parameter is set 1.

**Comments**Use this window to type comments that you want to save with the line parameters, for example, voltage level, conductor types, and other information.

**Number of strands**Specify the number of strands contained in the phase conductor.

**Strand diameter**Specify the diameter of one strand (in mm, cm, or m).

**Resistivity**Specify the DC resistivity of conductor in ohm*m.

**Relative permeability**Specify the relative permeability of the conductor material.

**External diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**Resistivity**Specify the DC resistivity of conductor in ohm*m.

**Total section**Total section of screen conductor (in mm^2, cm^2, or m^2).

The screen total section value is sometimes provided in datasheets. If you do not know this value, you can compute it as follows:

Total section = pi*

*r_out*^2 – pi**r_in*^2where:

*r_out*is the external radius of screen conductor*r_in*is the internal radius of screen conductor**Internal diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**External diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**Relative permittivity**Specify the relative permittivity of the phase-screen material.

**Internal diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**External diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**Relative permittivity**Specify the relative permittivity of the outer-screen material.

**Internal diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**External diameter**Specify the phase conductor outside diameter (in mm, cm, or m).

**Load typical data**Load the default cable parameters provided with Simscape™ Power Systems™ software. Opens a browser window where you can select the

`DefaultCableParameters.mat`

file, which represents the four-cable configuration used in the`power_cable`

example.**Load user data**Opens a browser window letting you select your own cable data. Select the desired

`.mat`

file.**Save**Saves your cable data by generating a

`.mat`

file that contains the GUI information and the cable data.**Compute RLC matrices**Computes the RLC matrices for a given cable. After completion of the parameters computation, results are displayed in a new window, entitled Display RLC Values. See Display RLC Values GUI for more details on this window. The obtained results are of the form of 2N-by-2N RLC matrices that can be directly used in the cable block. For an example, see the 4 Cables with screen block in the

`power_cable`

example.

See the `power_cable`

model
for an example using the `power_cableparam`

function.

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