13,開關(guān)電源變壓器與設(shè)計(jì)磁芯及骨架magnetics basis b

1、MAGNETICSAstec Custom PowerLecture OutlineMagnetic Quantities and UnitsBasic ConceptsTransformers and Transformer CoresThe B-H CurveHysteresis Loops and Volt-Second BalanceTransformer LossesDesign ConsiderationsMagnetic Variables and UnitsAmperes LawA current-carrying wire generates a magnetic field

2、 perpendicular to the current, with the magnetic field lines following the Right Hand Rule.Amperes Law relates magnetic field intensity and current density, thus: lHdl = AJdA = ITIntegrating the magnetic field intensity around a closed magnetic path gives the total current following through the wire

3、. Therefore, magnetomotive force is given as mmf = NI.Closed pathClosed pathFaradays LawFaradays Law is the complement of Amperes Law and relates electric field intensity and magnetic flux density, thus:lEdl = -/t ABdAThe voltage induced in a conductor is proportional to the change in magnetic flux

4、passing through the surface enclosed by the conductor.v = Nd/dtv+-AlB(t)Current and Magnetic FieldsCurrent flowing through a closed path generates concentric “flux rings” perpendicular to the conductor.The loops of a current-carrying coil are said to be “l(fā)inked” by these magnetic “flux rings”. These

5、 flux lines constitute a magnetic field, with a FLUX DENSITY of: B = f/AeffFlux density is more or less proportional to the MAGNETIC FIELD INTENSITY H by a factor called PERMEABILITY: m = B/HExample: B around a long wireQuestion: What is the flux density around a very long straight conductor (surrou

6、nded by air) carrying a current IT?The current will generate concentric flux rings around the conductor with constant H along each flux ring.From Amperes Law: lHdl = IT Since H = B/, and = o for air, we have:ringHdl = (B/o)(2r) = IT B = oIT /2r teslarExample: B in a single-turn coilQuestion: Given a

7、 single-turn coil of radius r with a current Icoil flowing through it, what is the flux density inside the coil?From Amperes Law: lHdl = IT Since one “flux ring” is approximately circular for a single-turn coil, integrating along the flux ring with the same radius as the coil, we have:ringHdl = (B/)

8、(2r) = Icoil B = oIcoil /2rFor an N-turn flat coil, B = oNIcoil /2r teslarrB = 0.63NIcoil/r gaussExample: B in a long coilQuestion: Given an N-turn coil whose length is large compared to its radius (length 30r), with a current Icoil flowing through it, what is the flux density inside the coil?Again,

9、 from Amperes Law: l (B/)dl = IT Now IT = NIcoil , and the total length of the flux path is approximately twice the coil length. This yields: (B/)(2length) NIcoil B NIcoil /length teslawhere is the permeability of the inductor core.lengthB = 0.63NIcoil/lengthgaussExample: B in a toroidal coreQuestio

10、n: Given an N-turn coil wound around a toroidal core with inner radius r and outer radius R, with a current Icoil flowing through it, what is the flux density of the core?We can deduce from Amperes Law that the flux density increases as the radius decreases. To get BAVE, we can use an effective radi

11、us of rEFF = (R+r)/2.Thus, BAVE can be calculated as: B = EFFNIcoil /(R+r) teslawhere EFF is the effective core permeability.RrB = 1.26NIcoil/lengthgaussTransformersThe magnetic flux path is provided by high-permeability material (e.g., iron), thereby thereby forcing the flux to flow through both wi

12、ndings.Any change in voltage in the primary will induce a specific amount of flux to flow through the core which, in turn, induces a proportional change in voltage in the secondary. (Amperes & Faradays Law)The total power handling capability of the transformer is related to its physical size. This c

13、an be seen by applying Amperes law to the window area AW and Faradays Law to the core cross section AC:AP (area product) = AWAC = (NI/J)(Vt/NB) VI = PACAWDevices made up of two or more coils coupled together through a common magnetic flux path.Transformer CharacteristicsOutput voltage depends on tur

14、ns ratio: VP/VS = nP/nSOutput is isolated from input. Current at the low voltage winding is a limiting factor:(ideally) PP = PS VHIGHILOW = VLOWIHIGH IHIGH dictates wire size and window sizeCore SaturationFrom Amperes Law, the current in a transformer winding is proportional to the magnetic field in

15、tensity in the core. H IIncreasing the magnetic field intensity has the effect of aligning the molecules in the magnetic material, thereby increasing the number of flux lines through the material.There eventually comes a point when all the molecule dipoles e aligned and no amount of current increase

16、 will cause a significant rise in magnetic flux density B.The operating point is thus said to have entered the SATURATION region.IIlinearsat 0IdealActualEffect of Core SaturationDuring saturation, the change in flux per change in current approaches zero.Knowing from Faradays Law that V = Ndf/dt , if

17、 df/dt = 0 then it follows that V = 0.An effective short circuit occurs at the secondary.If the input voltage is cyclical, severe cyclical peak currents will be observed at the primary.IptHysteresis and Coercive Force After field intensity H is increased by a specified amount DH with a corresponding

18、 increase in flux density B, decreasing H by the same amount DH will yield a resultant flux density greater than the original (B Bo).This phenomenon is called HYSTERESIS.A certain amount of negative field Hc is necessary to return B to zero from its zero-H (remanence) value.This is called the COERCI

19、VE FORCE.B-H CurveFor practical applications, permeability is still equal to m, when operating far from the saturation region.Permeability m = B/H.(HMIN, -BSAT)(HMAX, BSAT)The operating point will follow the hysteresis curve at right (blue and red) if the core is driven at an H between HMIN and HMAX

20、. BREM-BREM-HCHCMinor B-H LoopsWhen the core is driven at a level lower than saturation, the resultant B-H curve follows a MINOR LOOP:(HSAT, BSAT)(-HSAT, -BSAT)(HMAX, BMIN)(HMIN, BMIN)The minor loop curve is symmetrical about the origin if the winding is driven identically in both polarities (+ and

21、-).Hysteresis loss, corresponding to the area of the loop, is lower for smaller loops.Hysteresis Loop DriftSince the operating point tends not to return to the original flux density level Bo, it is possible for the operating loop to drift up or down the hysteresis curve.If loop drifts up (or down) t

22、oo far, it may reach saturation.To avoid this, make sure that the transformer is “reset” (brought to its starting point) after every cycle.Always try to maintain Volt-second balance: Vdt = Nd = constant“Current Walk”When volt-second balance is not maintained, minor loops may drift up (or down) the h

23、ysteresis curve:This may cause the core to saturate.(HSAT, BSAT)operating pointEnergy & Power RelationsThe energy in an inductor is as follows:W = Pavet = VaveIavet = (LPI/t)(Ippeak/2)t = LS(ISpeak)2The energy stored in an ideal transformer is given by:WSTORED = LP(IPpeak)2 = LS(ISpeak)2Power is sim

24、ply W/t or Wf for the case of cyclical signals.Throughput power (output power) is therefore: Pout = (W2-W1)f where W2 = peak energy level, W1 = minimum cyclical energy levelPout = LP(IPpeak22 - IPpeak12)f = LS(ISpeak22 - ISpeak12)fEnergyThe energy in a transformer can also be shown as W = HdB denote

25、d by the area bounded by the BH curve and the B axis. Ideally (w/o hysteresis & saturation), the energy is simply W = HB (i.e., the triangular area between the ideal curve and the B axis)However, the area enclosed by the BH curve with hysteresis implies that energy is expended within the core during

26、 each cycle.Power loss associated with hysteresis.Core LossesHysteresis Loss - loss corresponding to the work required to reorient the magnetic domains within the core material.Select core with narrowest B-H characteristic (smallest hysteresis loop area)Trade-off: Cost VS Loss VS Frequency Eddy Curr

27、ent Loss - loss caused by the flow of circulating magnetic currents within the material caused by rapid transitions in magnetic flux density.Depends on core shape and cross-sectional areaUse laminated cores or high reluctance coresEddy Current LossSince magnetic cores like iron are also conductors o

28、f electricity, from Faradays Law, it can be seen that the varying magnetic flux through the core sets up electric field lines along small concentric “current loops” perpendicular to the flux. Resistance of the core material leads to I2R losses in these “current loops”.Laminating the core reduces the

29、 number of “current loops”, minimizing eddy current loss. Using low resistivity material also reduces eddy current loss.dV/dtd/dtI = dB/dHGapping a CoreAdding an air gap to the core decreases the effective permeability , making it more difficult to enter saturation:If both are operated to saturation

30、, hysteresis losses are unaffected by the airgap.Notice that, since the coercive force (Hc) is dependent on core material, it remains the same for either core -gapped or ungapped.If the H limits are the same, area of loop will decrease, and hysteresis losses will be less.The Gapped CoreA gapped core

31、 (with lower eff)needs more turns to attain the required inductance: L = N(df/di) effSince L N2, however, the change required in N is smaller than the resulting allowable current swing. e.g., if eff decreases to the original, the number of turns N need only be doubled to have the same amount of curr

32、ent swing. ReluctanceConstricting the flux to flow through a specified magnetic path is analogous to making current flow through an electric circuit.With a specified path, Amperes Law reduces to: Hl = NIGiven that H = B/m and B = /Acore, we have: (l/mAcore) = NIDefining reluctance = l/mAcore , we ha

33、ve = NI = mmf.This is analagous to Ohms Law: IR = V = emf.Reluctance is therefore the “resistance” to the flow of flux, and can be treated (connected in series or parallel) in the same way as resistance. Effective Relative Permeability of Gapped CoresThe total reluctance of a gapped core is as follo

34、ws:total = core + gapltotal/mOmReffAeff = lcore/mRcoremOAcore + lgap/mRgapmOAgapSince mRgap1 (for air) and mRcore1 (mRcore1000 for Fe) ltotal/mReffmOAeff lgap/mOAgapLetting Aeff = Agap, we have: mReff ltotal/lgapTherefore, the effective relative permeability of a gapped core is equal to the ratio of

35、 the total path length to the air gap.lcorelgapThe “Distributed” GapIron powder cores effectively have a gap that is “distributed” throughout the core.Since the core contains numerous tiny pockets of air, these contribute to the lowering of the permeability considerably (m 100).Molybdenum-permalloy

36、cores are effectively gapped by the non-magnetic particles within the magnetic core.Given the effective air gap, the effective relative permeability of any core is simply the ratio of the total path length to the effective air gap.mReff = lTOTAL/lGAPeffInductanceWe are familiar with the equation V =

37、 LdI/dt. From Faradays Law, V = Ndf/dt. Thus, LdI = Ndf.Inductance is therefore the ratio of the total flux linkages Nf in a coil to the current I which they link:L = Nf/ISince f = BaveAcoil , L = NBA/I , where NBA = flux linkageExample: Inductance in a long coilQuestion: What is the inductance of a

38、n N-turn coil whose length is large compared to its radius r?It has previously been determined thatB = NIcoil /length teslawhere is the permeability of the inductor core.Since L = NBA/I, where A = r2 L = r2N2/lengthlengthExample: L of a toroidal coilQuestion: What is the inductance of an N-turn coil

39、 around a high permeability core with inner radius r, outer radius R, and a very small air gap lGAP?It has previously been determined thatB = EFFNIcoil /(R+r) teslawhere EFF = oReff and Reff (R+r)/lGAP.Since L = NBA/I L oAcoreN2/lGAP If the core cross section is circular, Acore = rcore2L o rcore2N2/

40、lGAPRrlGAPrcoreExample: L of a C-core inductorQuestion: What is the inductance of an N-turn coil around a high permeability core with inner radius r, outer radius R, and a very small air gap lGAP?We know that for a toroidal coil: L oAcoreN2/lGAP This is also consistent with the equation derived for

41、a long coil: L = r2N2/length , if highly permeable material is used inside the coil (Reff = 2/1)Since the former equation is independent of total path length, we can apply this equation to any high- core where the cross sectional area is practically constant.Thus, for the C-core, L oAcoreN2/lGAPAcor

42、elcorelgapThis is also applicable to EE-cores, where Acore and lgap pertain to the center leg.Self and Mutual InductanceSelf inductance L can be defined as the sum of flux linkages Nf in a coil when a current I of 1A flows through it: L = Nf/IIn an transformer, mutual inductance M can therefore also

43、 be defined as the amount of flux linkages N2f2 established at the secondary due to a primary current I1 of 1A: M = N2f2/I1I12leakCoupling CoefficientLeakage inductance is dependent on the level of coupling between two coils. If coupling is zero, the coils act like two separate inductors leakage ind

44、uctance is at a maximum. The coupling coefficient K of a transformer is defined as: K = _M_ 1 , where L1 = self inductance of the primary L1L2L2 = self inductance of the secondary M = mutual inductance between the coilsThe relationship between the leakage and magnetizing inductances (Lleak & Lmag) a

45、nd coupling coefficient K is given by: LL = (1 K2) LM For an efficient transformer, K must be very close to 1 (i.e., LL is minimized).Transformer Equivalent CircuitLmag = magnetizing inductanceLPleak = primary leakage inductanceLPleak = secondary leakage inductanceRP = primary winding resistanceRS =

46、 secondary winding resistanceRcore = equivalent core resistanceLPleakLSleakRcoreRSRPCPSLmagCPCSTidealCP = primary capacitanceCS = secondary capacitanceCPS = primary-to-secondary CTideal = ideal transformerTransformer Design ConsiderationsCore Material, Geometry and SizeNumber of Turns and Wire SizeW

47、inding MethodCore MaterialFerrite - most widely used for commercial applicationsMolybdenum Permalloy - higher losses, often used at f 20kHzCore GeometryPot cores - good magnetic shielding but higher winding temperature than with open air cores- for low power levels, low input & output current levels

48、 (because of temperature), and low input voltage (because of narrow notch)EE, ETD or EC cores - lower cost but large EMI (w/o belly band)- mean turn length is shorter for ETD and EC coresRM or Square cores - compromise between Pot and EE.UU or UI cores - large window permits large NI for high voltag

49、e or very high power applications.Core SizeThe area product (ACAW) is more or less proportional to the power handling capability of the transformer: AC AW NI , where I wire size E = Pdt = VIdt = (Nd/dt)Idt = NId AC AW NI AC AW ECore Gap ConsiderationsTo be able to store significant amounts of energy

50、 in the core, an air gap is required (e.g., flyback transformer)If a core gap is required, it is best to use a core with a gapped center leg than an ungapped core whose halves are separated with insulating shims:More cost effective.More stable.Produces less EMI.If the core gap is too large, it will

51、be hard to control the inductance.Number of TurnsWinding inductance is proportional to the square of the turns: L = Vdt/dI = (Ndf/dt)dt/dI = Ndf/dI or Nf/I since f = NI , we have L = N2/ = momrN2A/l since AL = 1/ , we have L = ALN2 in any case: L N2Winding MethodFactors which enhance electrical perf

52、ormance:Tight magnetic coupling of the windings to the corereduces leakage inductance.Tight magnetic coupling between all the windingsaids in coupling high frequency switching edges.Inter-winding coupling can be optimized by using multi-filar windings.Low voltage difference between adjacent turns mi

53、nimizes effect of interwinding capacitance.All windings without high dielectric voltage isolation requirement should be bifilar-wound, otherwise, interleaving may be best.Copper LossesSkin Effect - caused by eddy currents induced in a wire by magnetic field of current in the wire itself.Proximity Effect - caused by eddy currents induced in wire by varying magnetic fields from adjacent conductors or coil layers.Skin EffectCentral current is c