东南大学《电机内的电磁场》课件--绪论

Two Forms of Maxwell’s Equations
• Integral form
– Describe EMF in a space region
• Differential form
– Describe EMF of some space point
• Integral form=Differential form+boundary conditions
• Scalars
– ρ…电荷密度 charge density C/m3
EMF Laws (1)
• 全电流定律(Total current law)
∇ × H = J c + ∂D / ∂t = J
H ⋅ dl = ∫ ( J c + ∂D / ∂t ) ⋅ ds ∫
l s
– ∇ … Hamilton operator
– Basic theory of magnetostatic field – Basic theory of time varying field
• Numerical Method of EMF
– Mathematic foundation of numerical methods – 有限差分法（Finite Difference Method, FDM） – 有限单元法（Finite Element Method, FEM）
S2
B2
EMF Laws (4)
• 高斯定律(Gauss law)
D ⋅ ds = ∫ ρdv ∫
s v
∇⋅ D = ρ
EMF Laws (5)
• 电流连续性定律(Current continuity law)
∇⋅J = 0 J ⋅ ds = 0 ∫
s
Maxwell’s Equations
• Maxwell’s equations ---Basic equations
Research Objects of EMF in EM
• • • • Magnetic field in air-gap Magnetic field in iron-core Leakage field in the ends of EM Leakage field in slot and between salient poles • Magnetic field in solid rotors
Sort of EMF in EM
• Magnetostatic field • Time varying field
– Low frequency , neglecting displacement current 似稳场
Analysis Methods for EMF
• Analytical method
– Analysis of EMF is the key to design, analyze and optimize all kinds of EMs
Importance of the Course(2)
• This course is one of the courses for Master’s degree
– JC，conducting current – JD，displacement current – JD is neglectable
EMF in Electrical Devices
• If field quantities are sinusoidal
ωD ε = 2πf J D / JC = σ σE
• Accurate description of EM phenomenon
– Describe complex EM phenomenon in nature with simple formulae
• Great theoretical discovery
– Know the nature and change the world – One of the most important theoretical discoveries of human being
Lectures and Examination
• Lectures
– 3 hours lecture each week，10~13 weeks totally
• Examination
– Read papers and write a report on numerical analysis of electromagnetic fields in electrical machines – Workpaper in team group
In Electrical Machines
• B is more important than E
B
E
– When f<1010, JD/JC<10-7 in metals
Maxwell’s Equations in Low Frequency
∇ × H = Jc ∇ × E = −∂B / ∂t ∇⋅B = 0 ∇⋅D = 0
Curl of Vector
i ∂ ∇×H = ∂x Hx j ∂ ∂y Hy k ∂ ∂z Hz
Energy Density in Air
• Emax=3×106 V/m
We = 39.8J / m
• B=1T
5
2
Wm = 3.98 × 10 J / m
2
Energy of Fields
• Electric field
1 2 We = εE 2
• Magnetic field
1 2 W m = µH 2
∂H z ∂H y ∂H x ∂H z =( − − )i + ( )j ∂x ∂y ∂z ∂z ∂H y ∂H x )k +( − ∂y ∂x
Divergence of Vector
∂B x ∂B y ∂Bz + + ∇⋅B = ∂z ∂x ∂y
Gradient of Scalar
∂φ ∂φ ∂φ ∇φ = i+ j+ k ∂x ∂y ∂z
EMF Laws (3)
• 磁通连续性定律(Magnetic flux continuity law)
∇⋅B = 0 B ⋅ ds = 0 ∫
s
EMF Laws (3)
• 磁通连续性定律(Magnetic flux continuity law)
S1 S2
B1
S1
B1 ⋅ ds = ∫ B2 ⋅ ds ∫
Importance of the Course(1)
• EMF is the key of energy conversion in electrical machines
– Performance of EM depends on the distribution of EMF
• Parameter、loss、electromagnetic force
– Laws (1)～(4) – (1)and (3) determine the ∇× and ∇⋅ of magnetic field – (2 ) and (4) determine the ∇× and ∇⋅ of electric field
Constitutive Relations
• 本构方程
– 分离变量法, 复变函数法
• 图解(diagrammatize)方法 • 模拟法 • Numerical method
– FDM、FEM、边界单元法BEM、棱边 元法Edge Element、积分方程法IEM
Status of EMF Analysis
• Two dimensional problems基本解决
Transformation between the two forms
• Gauss theorem
∇ ⋅ Adv = ∫ A ⋅ ds ∫
v s
• Stocks theorem
(∇ × A) ⋅ ds = ∫ A ⋅ dl ∫
s l
EMF in Electrical Devices
• Low frequency
H…磁场强度 magnetic intensity A/m B…磁感应强度 magnetic flux density T E…电场强度 electric field intensity V/m J…电流密度 current density A/m2 D…电位移 electric displacement/ electric flux density C/m2
B = µ H J = σ E D = εE
– µ，磁导率 permeability 4π×10-7 H/m – σ，电导率 conductivity – ε，电容率/介电常数，permittivity /primary electric constant 8.85 ×10-12 F/m
Significance of Maxwell’s Equations
– 模型、程序简单，已大量应用于工程
• Three dimensional problems有所应用
– 恒定磁场问题难度不算大 – 似稳电磁场有一定困难 – 瞬变电磁场难度较大 – 模型、程序复杂，工程应用困难
• Vectors
– – – – –
Quantities of EMF and Their Units
Text Books and References
• Text book
– 汤蕴璆《电机内的电磁场》73.211
• References
– – – – 胡之光《电机电磁场分析与计算》73.117/5 陈丕璋《电机电磁场理论与计算》73.117/18 盛剑霓《工程电磁场数值分析》 73.117/43 俞宏生《工程电磁场 Laws (1)