EE369

POWER SYSTEM ANALYSISLecture 4Power System Operation, Transmission Line ModelingTom Overbye and Ross Baldick

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Reading and HomeworkFor lectures 4 through 6 read Chapter 4we will not be covering sections 4.7, 4.11, and 4.12 in detail,We will return to chapter 3 later.HW 3 is Problems 2.42, 2.44, 2.45, 2.47, 2.49, 2.50, 2.51, 2.52; due Thursday 9/18.HW 4 is chapter 4 case study questions A through D, and Problems 2.31, 2.41, 2.43, 2.48, 4.1, 4.3, 4.6, due Thursday 9/26.HW 5 is Problems 4.9, 4.11, 4.13, 4.18, 4.21, 4.22, 4.24, 4.25 (assume Cardinal conductor and look up GMR in Table A.4); due Thursday 10/2.*

Development of Line ModelsGoals of this section are:develop a simple model for transmission lines, andgain an intuitive feel for how the geometry of the transmission line affects the model parameters.

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Primary Methods for Power TransferThe most common methods for transfer of electric power are: Overhead ac Underground ac Overhead dc Underground dcThe analysis will be developed for ac lines.*

Magnetics ReviewMagnetomotive force: symbol F, measured in ampere-turns, which is the current enclosed by a closed path,Magnetic field intensity: symbol H, measured in ampere-turns/meter:The existence of a current in a wire gives rise to an associated magnetic field. The stronger the current, the more intense is the magnetic field H.Flux density: symbol B, measured in webers/m2 or teslas or gauss (1 Wb /m2 = 1T = 10,000G):Magnetic field intensity is associated with a magnetic flux density.*

Magnetics ReviewMagnetic flux: symbol measured in webers, which is the integral of flux density over a surface.Flux linkages measured in weber-turns.If the magnetic flux is varying (due to a changing current) then a voltage will be induced in a conductor that depends on how much magnetic flux is enclosed (linked) by the loops of the conductor, according to Faradays law.Inductance: symbol L, measured in henrys:The ratio of flux linkages to the current in a coil.*

Magnetics ReviewAmperes circuital law relates magnetomotive force (the enclosed current in amps or amp-turns) and magnetic field intensity (in amp-turns/meter):

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Line IntegralsLine integrals are a generalization of standard integration along, for example, the x-axis.Integration along thex-axisIntegration along ageneral path, whichmay be closedAmperes law is most useful in cases of symmetry, such as a circular path of radius x around an infinitelylong wire, so that H and dl are parallel, |H|= H is constant,and |dl| integrates to equal the circumference 2x. *

Flux DensityAssuming no permanent magnetism, magnetic field intensity and flux density are related by the permeability of the medium.*

Magnetic Flux*

Magnetic Fields from Single WireAssume we have an infinitely long wire with current of I =1000A.Consider a square, located between 4 and 5 meters from the wire and such that the square and the wire are in the same plane. How much magnetic flux passes through the square?

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Magnetic Fields from Single WireMagnetic flux passing through the square?

Easiest way to solve the problem is to take advantage of symmetry.As an integration path, well choose a circle with radius x, with x varying from 4 to 5 meters, with the wire at the center, so the path encloses the current I.

*Direction of H is givenby the Right-hand Rule

Single Line Example, contdFor reference, the earths magnetic field is about 0.6 Gauss (Central US)*H is perpendicularto surface of square

Flux linkages and Faradays law*

InductanceFor a linear magnetic system; that is, one where B= m H,we can define the inductance, L, to be the constant of proportionality relating the current and the flux linkage: l= L I,where L has units of Henrys (H).*

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