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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 below 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
µr: the relative permeability of phase conductor
rint, rext: the internal and external radius of phase-screen insulator
GMD: Geometric mean distance between the phase conductors.
ρ: Resistivity of the phase-screen insulator
ɛrax: Relative permittivity of the phase-screen insulator
ɛrxe: 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
$$\begin{array}{cc}{D}_{e}=1650\sqrt{{\rho}_{Cu}/\left(2\pi f\right)}& m\\ {\rho}_{Cu}=17.8e-9& \Omega /m\end{array}$$
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 phase-screen insulator is given by
$$\begin{array}{cc}{R}_{N}=\rho \frac{1000}{S}& \Omega /\text{km}\end{array}$$
The geometric mean radius of phase-screen insulator is given by
$$GM{R}_{N}=\frac{{r}_{ext}-{r}_{\mathrm{int}}}{2}$$
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 phase-screen insulator.
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 need to be specified directly as an input parameter.
Capacitance Between the Phase and Screen Conductors
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 |
---|---|
N | 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 resistance 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 resistance 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 | Specify the outer screen insulator external diameter (in m) |
epsilon_ixe | Specify the 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 |
---|---|
r.aa | 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 phase and screen conductor, in Henries/Km |
l.ax | Mutual inductance between the phase conductors, in Ohm/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 follow (the example is given for a 3-cables configuration):
$$\begin{array}{cc}R=\left[\begin{array}{cccccc}{r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}& {r}_{ab}& {r}_{ab}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{aa}& {r}_{ax}\\ {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ab}& {r}_{ax}& {r}_{xx}\end{array}\right]& L=\left[\begin{array}{cccccc}{l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}& {l}_{ab}& {l}_{ab}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{aa}& {l}_{ax}\\ {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ab}& {l}_{ax}& {l}_{xx}\end{array}\right]\end{array}$$
$$C=\left[\begin{array}{cccccc}{c}_{ax}& -{c}_{ax}& 0& 0& 0& 0\\ -{c}_{ax}& {c}_{xe}& 0& 0& 0& 0\\ 0& 0& {c}_{ax}& -{c}_{ax}& 0& 0\\ 0& 0& -{c}_{ax}& {c}_{xe}& 0& 0\\ 0& 0& 0& 0& {c}_{ax}& -{c}_{ax}\\ 0& 0& 0& 0& -{c}_{ax}& {c}_{xe}\end{array}\right]$$
power_cableparam command opens a graphical user interface (GUI) that is used to specify the cable parameters and to compute the electrical R, L, C cable parameters.
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.
Specify the frequency in hertz to be used to evaluate RLC parameters.
Specify the ground resistivity in ohm.meters.
Specify the Geometric Mean Distance (GMD) between the cables. Set this value to zero if the Number of cables parameter is set 1.
Use this window to type comments that you want to save with the line parameters, for example, voltage level, conductor types, and other information.
Specify the number of strands contained in the phase conductor.
Specify the diameter of one strand (in mm, cm, or m).
Specify the DC resistance of conductor in ohms/m.
Specify the relative permeability of the conductor material.
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the DC resistance of conductor in ohms/m.
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^2
where:
r_out is the external radius of screen conductor |
r_in is the internal radius of screen conductor |
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the relative permittivity of the phase-screen material.
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the relative permittivity of the outer-screen material.
Specify the phase conductor outside diameter (in mm, cm, or m).
Specify the phase conductor outside diameter (in mm, cm, or m).
Load the default cable parameters provided with SimPowerSystems™ 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.
Opens a browser window letting you select your own cable data. Select the desired .mat file.
Saves your cable data by generating a .mat file that contains the GUI information and the cable data.
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.