Fundamentals研发培训,设计培训ppt课件

1、Review of Fundamentals Haran KarmakerAugust 4, 2005Topics Vectors and ScalarsPartial Differential Equations Electromagnetic Concepts Maxwells Equations Electric and Magnetic Calculations Principles of Machine Operation Mathematical ModelingMagnetic FieldsTwo-Reaction TheoryVectors and Scalars Vector

2、s and scalars are mathematical representations of physical quantities. Vectors have both magnitudes and directions. Scalars have only magnitudes, no directions.For example, location of a point in space from the origin is defined by a position vector with three components along the x, y and z axes. T

3、he distance of the point from the origin is a scalar quantity without any direction.Vector Representation In the example, the flux density in the air gap will have components in the radial, tangential and axial directions and is a vector.The air gap flux over an area is a scalar. Another example of

4、a scalar is the room temperature which has no direction. The gradient of a scalar is the rate of change of the scalar along a direction. The gradient is a vector quantity.Vector Representation The divergence of a vector quantity represents the net flow of the quantity through a volume.The divergence

5、 is a scalar function of a vector field.The curl of a vector is a measure of circulation of the vector field. Stokes theorem relates the integral of a vector field along a closed path to the integral of the curl of the field on the surface defined by the closed path.The three vector functions, gradi

6、ent, divergence and curl describe the nature of variation of all physical quantities in the universe.Partial Differential Equations The nature of distribution of a physical phenomenon is governed by equations including the change in multiple variables called partial differential equations.The most c

7、ommon governing equations are Laplace, Poisson, diffusion and wave equations.Solution of most engineering problems can be formulated as boundary value problems, which require governing equations and boundary conditions for their solutions.Once the boundary value problem has been formulated, it can b

8、e solved by analytical or numerical methods.Electromagnetic Concepts Electrostatic fields are caused by stationary electric charges.In 1785, Coulomb investigated the nature of force between two charged bodies and formulated the following equation from experiments.F = Q1*Q2/(4*r2)where F = Force in N

9、ewtonsQ1, Q2 = Charges in Coulomb = Permittivity of the medium in farads/mr = Distance between charges in metersElectromagnetic Concepts Electric field intensity due to a charge Q is defined asE = Q/(4*r2)Electric field intensity is a vector whose magnitude is in units of Newtons per Coulomb, which

10、can be converted to the units of volts/meter.Therefore, the vector E can be treated as a force field that acts on a charge. It can also be treated as a gradient of a voltage.Electromagnetic Concepts Electric flux density is defined asD = EIt has units of charge per area or Coulombs per square meter.

11、According to Gausss law, the integral of electric flux density over a closed surface is equal to the free charge enclosed by the surface.Electromagnetic Concepts The governing equations for electric field problems are often described by a potential function defined as E = - VMathematical representat

12、ion of Gausss law gives . D = where is charge density.Therefore, - . ( V) = V = - / Poissons equation V = 0 Laplaces equation22 Faradays Law Faradays Law states that a changing magnetic field will induce an electric field.The electric field exists in space regardless of whether a conductor is presen

13、t or not. When a conductor is present, a current will flow.The differential form of Faradays Law is Faradays Law Induced voltage around a stationary closed contour C linked by a changing magnetic field is given by the line integral of electric fieldThe magnetic flux isThe integral form of Faradays L

14、aw is Amperes Law A vector called magnetic field intensity is defined asThe differential form of Amperes Law is Amperes Law Integral form of Amperes Law is obtained by applying Stokes theoremAmperes Law in integral form states that the line integral of field intensity around a contour is equal to th

15、e net current enclosed by the contour. Maxwells Equations Maxwells equations describe the theory of electromagnetism by unifying all laws. div D = div B = 0D = E Symbols in Maxwells Equations Permeability is a physical property of a material relating flux density B to field intensity H.Permeability

16、of free space is 4x1E-7 Henry/m.For magnetic steel, permeability varies with flux density or field intensity.Electric conductivity is reciprocal of resistivity and varies with temperature.Electric and Magnetic Calculations For many magnetic applications, good approximate solutions can be obtained by

17、 a circuit analysis similar to that of a d.c. circuit composed of series and parallel combinations of resistors. For example, consider a toroid with N turns carrying current I. Electric and Magnetic Calculations The magnetic field intensity in the toroid is continuous. The flux density is much great

18、er inside than outside because of the permeability of the magnetic material.Electric and Magnetic Calculations Applying Amperes Law around the circular path C in the interior of toroid,H = N*I/(2*d)Amps/mwhere N*I is called the MMF (Magneto Motive Force) analogous to EMF (Electro Motive Force or Vol

19、ts) in electric circuit.Similarly, magnetic flux is analogous to electric current.Electric and Magnetic Calculations Ohms Law for electric circuit,E = I * Rwhere R = electric resistance = l/( * A)Similarly, for magnetic circuit,MMF = * Rwhere R = magnetic reluctance = l/(* A)Electric and Magnetic Ca

20、lculations To understand the concept of inductance, consider two circuits magnetically coupled.Any change in current results in a change in magnetic field. When current changes, the flux linking the circuits change and voltages are induced in the circuits.Electric and Magnetic Calculations The self

21、inductance is defined as the ratio of flux linking the circuit to the current in the same circuit.L11 = 11/I1The mutual inductance is defined as the flux linking the circuit by a second circuit to the current in the second circuit. L12 = 12/I2Magnetic Forces The magnetic force on a conductor of leng

22、th L in magnetic field B isF = I*L x BNewtonsThe force density on a conductor is F = J x BNewtons/m3where J is the current density, Amps/m2Principles of Machine Operation If an N turn coil is linked by a flux whose time rate of change is dF/dT, the terminals of the coil will have a voltage induced a

23、ccording to Faradays LawPrinciples of Machine Operation Consider a series of magnets moving past a coil. As each magnet moves past the coil, the coil sees an increase of flux linkage, hence an increase of voltage followed by a decrease of flux linkage hence a decrease of voltage. If we further arran

24、ge it such that the magnets alternate polarity, the resulting voltage will rise above “zero volts to a maximum and decrease below “zero volts to a minimum.Principles of Machine Operation Consider now another stationary coil fixed adjacent to the first one. Another similar alternating voltage will ap

25、pear on its terminals. Principles of Machine Operation The wave shape of the second voltage will be identical to the first coil, however the timing will be different. The amount of this “phase difference will depend on the speed of the magnetic poles and on the distance between the two stationary co

26、ils. Principles of Machine Operation Fundamental Flux and Voltage in a machine is given by Symbols in Voltage Equation f = fundamental flux per pole (Wb)kVll = line-line kVN = number of turns in series per phasef = frequency, HzKp = pitch factorKd = distribution factorKs = skew factor Symbols in Vol

27、tage Equation Pitch factor in voltage equation is defined as n is harmonic order and PUP is per unit pitch (ratio of span to pole pitch) Symbols in Voltage Equation Distribution factor in voltage equation is number of coils (slots) per pole per phase reduced to lowest termA+B/C Symbols in Voltage Eq

28、uation Skew factor in voltage equation is skew angle is shown below Slot Combinations The term slots when used in electrical design is generally interchangeable with coils. Since each slot has two coil legs, the number of coils and the number of slots are identical.The basic art of the design of mac

29、hines is that of selecting the number of coils, turns and circuits to provide the optimum design.For economic reasons, all coils are usually made identical, that is the same number of turns per coil, the same span, and the same skew. Slot Combinations Consider a winding of Np poles, and Ns coils. Th

30、e number of coils per pole per phase For example, 324 slots, 24 poles and 3phases gives Slot Combinations This is the average number of coils of one phase under each pole. To optimize the distribution, the actual number under each pole will vary with the pole, but will be as close as possible to thi

31、s number.Hence in this case, pole #1 may have 4 phase a coils while pole #2 may have 5 phase a coils and so on around the unit. Slot Combinations The pattern of phase a coils with respect to pole number would be Slot Combinations The number of coils per pole In this example, D + E/F is 13 + 1/2.We p

32、ut 13 coils under one pole and 14 under the next pole. Slot Combinations The winding pattern is the sequence of numbers that represents the phase a, b and c coils as they are laid out. In this example Coils and Poles Per Circuit Each circuit should have the same number of coils and should cover an i

33、ntegral number of poles. These assumptions may be violated resulting in an unbalanced winding.Circuits are laid onto the winding pattern after the pattern has been sketched around the machine. Single and Double Layer Windings Single layer windings have each armature slot containing only one coil leg

34、. Double layer windings have two legs per slot and are most common for large machines. Armature Reaction Besides the magnetic fields generated by the rotor poles, additional fields are caused by the currents in the stator coils. The stator currents set up a distribution of mmf (with harmonics), the

35、fundamental of which rotates at synchronous speed. Armature reaction affects the air gap flux in various ways depending on the winding arrangement, the phase currents, the reluctance of the main flux path and the power factor. Two-Reaction Theory The air gap flux in a salient pole synchronous machine is distorted due to the non-uniform air gaps ov