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# Line Generator | Introduction

- The Line Generator tool computes the resistance, inductance, and capacitance matrices of an arbitrary arrangement of conductors of an overhead transmission line.
- Hypersim calls a module to calculate the R, L and C matrices of the lines. Then, the functions of the CLAPACK (user’s guide, third edition, SIAM, 1999, library), are used to transform the R, L and C phase parameters into mode parameters, sequence parameters and Ti transformation matrix used by Hypersim.
- For a set of N conductors, it computes three N-by-N matrices: the series resistance and inductance matrices [R] and [L] and the shunt capacitance matrix [C]. It also computes the Ti matrix, sequence and mode parameters.
- For two coupled conductors i and k, the self and mutual terms of the R, L, and C matrices are computed using the concept of Figure 1–3 image conductors:

**Self and mutual resistance terms:**

**Self and mutual inductance terms:**

**Self and mutual potential coefficient terms:**

Where:

µ_{0} | permeability of free space=4.10-4 H/km |
---|---|

ε_{0} | permittivity of free space=8.8542.10-9 F/km |

ri | radius of conductor i |

d_{ik} | distance between conductors i and k |

D_{ik} | distance between conductor i and image of k |

h_{i} | average height of conductor i above ground |

R_{int}, L_{int} | internal resistance and inductance of conductor |

ΔR_{ii}, ΔR_{ik} | Carson R correction terms due to ground resistivity |

ΔL_{ii}, ΔL_{ik} | Carson L correction terms due to ground resistivity |

- The conductor self inductance is computed from the magnetic flux circulating inside and outside the conductor, and produced by the current flowing in the conductor itself.
- The part of flux circulating inside the conducting material contributes to the so called internal inductance Lint which is dependant on the conductor geometry. Assuming a hollow or solid conductor, the internal inductance is computed from the T/D ratio where D is the conductor diameter and T is the thickness of the conducting material.
- The conductor self inductance is computed by means of modified Bessel functions from the conductor diameter, T/D ratio, resistivity and relative permeability of conducting material and specified frequency [1].
- The conductor self inductance can be also computed from parameters that are usually found in tables provided by conductor manufacturers: the Geometric Mean Radius (GMR) or the so called "Reactance at one-foot spacing."
- The GMR is the radius of the equivalent hollow conductor with zero thickness, thus producing no internal flux, giving the same self inductance. The conductor self inductance is then derived from the GMR using the following equation:

For a T/D ratio=0.5, the GMR is given by:

**Where d =spacing between conductors bundle**

- The GMR obtained from the above equation assumes a uniform current density in the conductor.
- This assumption is strictly valid in DC. In AC, the GMR is slightly higher. For example for a three cm diameter solid aluminum conductor (Rdc = 0.040 /km), the GMR increases from 1.1682 cm in DC to 1.1784 cm at 60 Hz. Manufacturers usually give the GMR at the system nominal frequency (50 Hz or 60 Hz).
- The reactance Xa at 1-foot spacing (or 1-meter spacing if metric units are used) is the positive- sequence reactance of a three-phase line having one foot (or one meter) spacing between the three phases and infinite conductor heights. The reactance at one-foot spacing (or 1-meter spacing) at frequency f is related to the GMR by the following equation:

The conductor resistance matrix at a particular frequency depends on the DC resistance of the conductor corrected for skin effect and ground resistivity. In fact, both the resistance matrix and the inductance matrix are dependent on the ground resistivity and frequency. Correction terms for the R and L terms as developed by J.R. Carson in 1926 [2] are implemented in the Line Generator.

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