Chapter 5

Chapter 5 - 5-1 Chapter 5 Magnetic Circuits Exit 2001 by N....

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Unformatted text preview: 5-1 Chapter 5 Magnetic Circuits Exit 2001 by N. Mohan Print TOC " ! 5-2 Magnetic Field ❑ Magnetic field, H, produced by current carrying conductor ❏ Ampere’s Law dl H " ∫ closed path Exit 2001 by N. Mohan H d! = ∑i i3 i1 i2 TOC " ! 5-3 H in a Toroid i rm ID OD ID OD 1 ID + OD Mean radius, rm = 2 2 lm = 2π rm Ni Ni = Ampere's Law ⇒ H m = 2π rm lm Exit 2001 by N. Mohan TOC " ! 5-4 Flux Density B ❏ Units:Weber / meter 2 [Wb / m 2 ] or Tesla [T ] henries µo = 4π × 10 −7 ❏ In air B = µo H , m ❏ Ferro-magnetic materials Bm Bm µo Bsat µm µo Hm Hm x Linear approximation Bm = µm H m x Bsat ~ 1.6 - 1.8 Tesla x In saturation µm approaches µo Exit 2001 by N. Mohan TOC " ! 5-5 Flux, Flux Linkage, and MMF ❏ Flux fm [Wb] [assuming uniform flux density] Am φm = Bm Am φm Ni Bm = µ m H m and H m = !m Ni Ni F ∴ φm = Am µ m = = ! m ! m ℜm µ A m m !m µ m Am ❏ Reluctance ❏ Flux Linkage λm = Nφm ❏ MMF Exit ℜm = F = Ni 2001 by N. Mohan TOC " ! 5-6 Magnetic Structures with Air Gaps Hm !m + H g ! g = N i Bm = µ m H m , Bg = µo H g φm = φ g = φ i Bg Bm !m + !g = N i µm µo !g N φ = Am Bm = Ag Bg φ Am Bm = Bg = φ Ag !g !m )= N i φm ( + Am µ m Ag µo #$% # % $ ℜm φm = Exit 2001 by N. Mohan To account for fringing Ag = ( w + ! g )(d + ! g ) ℜ = ℜm + ℜ g ℜg F ℜ TOC " ! 5-7 Inductance i φm Am N λm = Lm i i N × lm Hm µm × ( µm ) Bm × ( Am ) N2 × Lm = lm µm Am φm ×( N ) λm N2 N2 λm N Lm = = µ m Am N = = i lm lm ℜ µ m Am • For linear magnetic conditions inductance depends only on magnetic circuit ❏ Energy stored in magnetic circuits ❏ Energy density Exit 2001 by N. Mohan W 1 2 w= Bm = volume 2 µ m 1 1 2 W = Li 2 = Bm Amlm & 2 2 µm volume TOC " ! 5-8 Faraday’s Law - Induced Voltage ❏ Induced voltage dλ dφ e= =N dt dt ❏ Current direction is into positive polarity voltage → flux direction φ (t ) i (t ) + e(t ) N − ❏ Lenz’s law: Polarity of induced voltage x When current and flux directions are consistent (a current as indicated would create a flux as indicated), then the voltage should be labeled positive where the current enters the coil. Exit 2001 by N. Mohan TOC " ! 5-9 Coil in Sinusoidal Steady-State ❑ Induced voltage under sinusoidal steady-state Given φ (t ), i (t ) e(t ) ˆ φ (t ) = φ sin ω t e (t ) = N i (t ) t dφ ˆ = N φ ω cos ω t dt φ (t ) + e(t ) N − ❑ Relating e(t ), φ (t ), and i (t ) λ Nφ L= = i i N ⇒ i (t ) = φ (t ) L dφ(t) & e(t) = N dt Exit 2001 by N. Mohan di (t ) ⇒ e(t ) = L dt TOC " ! 5-10 Leakage and Magnetizing Inductances φm i + ⇒ e − i + e − i (t ) e (t ) − + l + + Ll φl e(t ) − N φ! λ = Nφ = N φm + & & λ! λm φ = φ m + φ! λ λm λ! = + i i i ⇒ Lself = Lm + L! λ = Lself i = Lm i + L! i em (t ) − Lm φm R + v (t ) − Ll i (t ) + el (t ) + em (t ) e(t ) − − di di di e = Lm + L! = em + L! dt dt dt & & em Exit 2001 by N. Mohan e! TOC " ! 5-11 Transformers ❏ Tightly coupled coils (low leakage inductance) ❏ Essential for power transmission and distribution ❏ Helpful in understanding induction machines Exit 2001 by N. Mohan TOC " ! 5-12 Transformers - Development ❏ Single coil Assuming zero resistance and zero leakage inductance e1 = N1 φm + e1 − N1 dφm dt φm determined completely by 1 applied voltage: φm = N ∫ e1 dτ 1 im depends on Lm ❏ Two coils dφm dt e (t ) N ⇒ 1 = 1 e2 (t ) N2 e2 (t ) = N 2 Exit 2001 by N. Mohan & e1 (t ) = N1 dφ m dt + e1 im Lm − φm + e1 − N1 N2 + e2 − TOC " ! 5-13 Transformer Model + im + e1 Lm e2 − − N1 N 2 #' % $' Ideal Transformer ❏ Dot polarity ❏ Magnetizing inductance Exit 2001 by N. Mohan TOC " ! 5-14 Transformer with Secondary Loaded ❏ φm determined by e1 alone hence i2 in secondary induces φm i1 (t ) + e1 − N1 N2 i2 (t ) + e2 i2 ' in the primary such that − ′ N1 i2 = N 2 i2 i1 (t ) i′ N ⇒ 2 = 2 i2 N1 + i1 (t ) = i2 '(t ) + im (t ) #$% # $' ' % relflected load current Exit 2001 by N. Mohan i2 (t ) i2 '(t ) magnetizing current im + e1 Lm e2 − − N1 N 2 #' % $' Ideal Transformer TOC " ! 5-15 Real Transformers i1 (t ) ❏ Add leakages + ❏ Core loss v1 - hysteresis − - eddy currents ❏ Winding resistances R1 i2 '(t ) Ll1 + e1 Ll2 R2 i2 (t ) + Lm − v2 − Rhe + e2 im − TOC " N1 N 2 #' % $' Ideal Transformer Real Transformer ❏ Laminations to reduce eddy current loss i φm circulating currents circulating currents φm Exit 2001 by N. Mohan ! 5-16 Determining Transformer Model Parameters i1 (t ) R1 i2 '(t ) Ll1 + + v1 e1 − − Real Transformer i2 (t ) + v2 − Lm + e2 im Rhe R2 Ll2 − N1 N 2 #' % $' Ideal Transformer ❏ Open circuit test x Core loss, Rhe x Magnetizing inductance, Lm ❏ Short circuit test x Winding resistance, R1 , R2 x Leakage inductance, Ll1 , Ll2 Exit 2001 by N. Mohan TOC " ! 5-17 Open Circuit Test ❏ Secondary unloaded (open circuit) ❏ Rated voltage applied to primary ❏ Measure x To find Rhe Rhe I oc 2 Voc = Poc x To find Lm + Voc jX m Rhe − V = oc Rhe jX m I oc Exit 2001 by N. Mohan TOC " ! 5-18 Short Circuit Test ❏ One winding shortened small voltage applied to other winding ❏ Measure VSC , and I SC , and PSC x To find R1 and R2 I1 1P R2 = SC 2 2 I SC N R1 = R2 1 N2 R1 + 2 2R2 + j2 X l 2 = 2001 by N. Mohan + VSC − − N2 I SC 2 N jX l1 2 + jX l 2 N1 VSC I SC 2 I SC E2 − N1 N = Xl2 1 X l1 N2 R2 + E1 x To find Ll1 and Ll2 Exit jX l 2 jX l1 2 N R1 2 + R2 N1 + VSC − N1 N2 TOC " ! 5-19 Permanent Magnets ❏ Typically used in smaller motors ❏ Applicable power range increasing due to new materials ❏ In simplest analysis, treated simply as a source of magnetic flux Bm (T) −1.4 −1.2 −1.0 e -F Nd -B Sm Co −0.8 −0.6 Alnico 0.4 Ferrite | 1000 | 800 | 500 − H m ( kA / m ) | 200 Figure 5-20 Characteristics of various permanent magnet materials. Exit 2001 by N. Mohan TOC " ! 5-20 Summary ❏ What is the role of magnetic circuits? Why are magnetic materials with very high permeabilities desirable? What is the permeability of air? What is the typical range of the relative permeabilities of ferromagnetic materials like iron? ❏ Why can "leakage" be ignored in electric circuits but not in magnetic circuits? ❏ What is Ampere's Law and what quantity is usually calculated by using it? ❏ What is the definition of the mmf F? ❏ What is meant by "magnetic saturation"? ❏ What is the relationship between φ and B? ❏ How can magnetic reluctance ℜ be calculated? What field quantity is calculated by dividing the mmf F by the reluctance ℜ ? Exit TOC " 2001 by N. Mohan ! 5-21 Summary ❏ In magnetic circuits with an air gap, what usually dominates the total reluctance in the flux path: the air gap or the rest of the magnetic structure? ❏ What is the meaning of the flux linkage λ of a coil? ❏ Which law allows us to calculate the induced emf? What is the relationship between the induced voltage and the flux linkage? ❏ How is the polarity of the induced emf established? ❏ Assuming sinusoidal variations with time at a frequency f, how are the rms value of the induced emf, the peak of the flux linking a coil, and the frequency of variation f related? ❏ How does the inductance L of a coil relate Faraday's Law to Ampere's Law? ❏ In a linear magnetic structure, define the inductance of a coil in terms of its geometry. Exit TOC " ! 2001 by N. Mohan 5-22 Summary ❏ What is leakage inductance? How can the voltage drop across it be represented separate from the emf induced by the main flux in the magnetic core? ❏ In linear magnetic structures, how is energy storage defined? In magnetic structures with air gaps, where is energy mainly stored? ❏ What is the meaning of "mutual inductance"? ❏ What is the role of transformers? How is an ideal transformer defined? What parasitic elements must be included in the model of an ideal transformer for it to represent a real transformer? ❏ What are the advantages of using permanent magnets? Exit 2001 by N. Mohan TOC " ! ...
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