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Improved Expression for Estimation of Leakage Inductance in E Core Transformer Using Energy Method 635715

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Compute leakage inductance
  Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2012, Article ID 635715, 6 pagesdoi:10.1155/2012/635715 Research Article ImprovedExpressionforEstimationofLeakageInductancein E   CoreTransformerUsingEnergyMethod SivanandaReddyThondapu, 1 MangeshB.Borage, 2  YashwantD.Wanmode, 1 andPurushottamShrivastava  1 1 Pulse High Power Microwave Section, Raja Ramanna Centre for Advanced Technology, Indore 452013, India  2 Power Supplies and Industrial Accelerator Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, India Correspondence should be addressed to Sivananda Reddy Thondapu, 31 December 2011; Revised 13 April 2012; Accepted 29 April 2012Academic Editor: Pavol BauerCopyright © 2012 Sivananda Reddy Thondapu et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work isproperly cited.This paper proposes a simpler and more accurate expression for estimation of leakage inductance in  E   core transformer, which isthe most widely used transformer structure. The derived expression for leakage inductance accounts for the flux extending intoair. The finite element method (FEM) analysis is made on the secondary shorted transformer to observe the  H  -field pattern. TheresultsobtainedfromFEManalysisareusedforapproximatingthefieldthatisextendingintoairtoderiveanexpressionforleakageinductance. This expression is experimentally validated on prototype transformers of di ff  erent core dimensions. 1.Introduction Transformer is one of the basic building blocks of many power converters. The following are some of the cases whereaccurate estimation of leakage inductance is required.(i) Di ff  erent resonant converter topologies, discussed in[1–5], use parasitics of transformer as a part of reso- nant tank network. For designing power converterwith such topologies, one requires accurate estima-tion of leakage inductance.(ii) In hard switched converters, in every cycle the energy stored in the parasitics appears as loss in converter. Inestimation of e ffi ciency of such converters, one needsto estimate leakage inductance before hand.(iii) For designing snubber circuits to limit device voltageduring turn-o ff   transients [6–8], one needs to esti- mate leakage inductance. These turn-o ff   transientsmainly occur due to energy stored in the leakageinductance of the transformer.Methods that are usually employed for estimation of leakage inductance are (i) energy method [8–13] and (ii) method of mutual fluxes.In energy method, the energy stored in magnetic field of the secondary shorted transformer is calculated and equatedto (1  /  2) L leak  I  2  p  where  L leak   is the leakage inductance of thetransformer when referred to primary, and  I   p  is currentflowing through primary.The  H  -profile inside the coil is calculated using Ampere’slaw. The energy stored in magnetic field is calculated by evaluating the volume integral in (1): E stored  = µ 2   H  2 dv  = 12 L leak  I  2  p .  (1)The expression derived for leakage inductance usingenergy method is independent of frequency. Hence, it doesnot consider any frequency-dependent e ff  ects on leakageinductance. The energy method is used for comparing leak-age inductance, in di ff  erent winding configurations.On the other hand, method of mutual fluxes uses Max-well’s equations to predict the leakage inductance moreaccurately at high frequencies. As this method accounts forfrequency-dependent e ff  ects like eddy current losses andaltered flux pattern due to eddy currents, it gives moreaccurate results, particularly at high frequencies. In [14], afrequency-dependent formula is presented to find leakageinductance in a toroidal core transformers.  2 Advances in Power Electronics x  y z  (a) Perspective x  y z  (b) Front view  z x  y  (c) Top view  z  x  y  (d) Side view  Figure  1: Simulation result of FEM analysis. In this paper, an expression for leakage inductance isderived using energy method. Expression derived accountsforthefluxthatextendsintoairinasecondaryshortedtrans-former, which is not considered earlier [8, 10, 11]. Therefore, the estimated leakage inductance using this expression hasbetter accuracy. 2.EstimationofLeakageInductanceon E  CoreTransformer FEM analysis on secondary shorted transformer is madeusing CST Ver 2008.5 (with magneto-static solver). Thesimulation result of FEM analysis is shown in Figure 1. Thesimulation result is the  H  -field pattern of secondary shorted E  core transformer. This gives an idea of profile of   H  -fieldfrom the surface of the core to the outer surface of the wholewinding. The profile of   H  -field of a  E  core transformer anditswindingconfigurationisshowninFigures2and3,respec- tively.Energy stored in the magnetic field in the secondary shorted transformer is equated to energy stored in leakageinductor, which is given in (1).The volume represented with purple color in Figure 4 isconsidered for volume integration. This constitutes (i) thevolume occupied by the coils and (ii) the volume formedby extrusion of the coils along the centre limb of the core,excluding the volume of the core. The flux extends intoair shown in Figure 5. The volume shown in Figure 4 is divided into four parts, whose top view is shown in Figure 3.The profile of   H  -field is calculated using Ampere’s law, thesame profile is observed in FEM analysis. The  H  -profile isexpressed mathematically in (2): H  ( z  ) =  H  m z h 1 0  < z < h 1 , H  m  h 1  < z < h 1  + t  , H  m  h 1  + h 2  + t  − z h 2   h 1  + t < z < h 1  + h 2  + t. (2)Using  H  -profile given in (2) to evaluate volume integralgiven in (1), we get expression of the leakage inductance: L leak   = 13 µ o N  21 F  2  ( h + 2 t  )[ FC   + B ( E  + 2 h )], (3)where  h  =  h 1  +  h 2  +  t   is total thickness of the windingmeasured from the surface of the core to the outer surface of the outer winding,  t   is total thickness of the insulation usedbetween the layers.  B ,  C  ,  E ,  F   is are the dimensions of the  E core shown in Figure 6.  Advances in Power Electronics 3 t h 1  h 2 H  ( z  ) z H  m Figure  2: Profile of   H  ( z  ) from the surface of the core. Part 3Part 1Part 2 Ex z C  Part 4Center limbof the coreSecondary winding (S)Primary winding (P)Interwindinginsulation Figure  3: Top view of   E  core transformer with windings.  y x z  Figure  4: Volume around the core considered for integration.      (     A     /   m     ) 15.413.511.59.617.695.773.841.920 z  x  y  Figure  5: Scaled side view to show the flux that extends into air.  4 Advances in Power Electronics Table  1: EE core transformer with interwinding insulation.Sample no. 1 2Core type EE42/21/15 EE65/38/13 F   14.45mm 22.65mm C   15.20mm 13.45mm B  21.10mm 32.59mm E  12.05mm 19.77mm h 1  thickness of primary 3.20mm 3.81mm h 2  thickness of secondary 1.90mm 1.55mm t   thickness of insulation 1.27mm 2.00mm N  1  34 48 N  2  17 24Photograph h 1  h 2 PS t  1 Windingconfiguration Leakage inductance  L leak,earlier  from (5) 11.91 µ H 22.52 µ HLeakage inductance  L leak   from (3) 15.32 µ H 28.21 µ HLeakage inductance  L exp  at 10kHz measured 14.13 µ H 26.76 µ H% deviation of   L exp  from calculated  L leak,earlier  15.71% 15.84%% deviation of   L exp  from calculated  L leak   8.38% 5.12% EF BD AC  Figure  6:  E  core dimensions given in data sheets. It is seen in [8, 11–13], by sandwiching the windings, leakage inductance reduces by   p 2 times, where  p  is the num-ber of interfaces between primary and secondary [8]: L leak   = 13  p 2 µ o N  21 F  2  ( h + 2 t  )[ FC   + B ( E  + 2 h )] .  (4) 3.ExperimentalResults Three di ff  erent samples are made to validate the derivedexpression. The results are compared with the expressionsavailable in literature. The expression given in [8, 10, 11] is replaced with the dimensions of the core (shown in Figure 6)and the dimensions of windings will give (5): L leak,earlier  = 13  p 2 µ o N  21 F  2  ( h + 2 t  ) F  ( C   + E  + 2 h ) .  (5)Sample no. 1 is made with  E  core (EE42/21/15) woundwith stranded conductor made with three conductors of 24SWG. With interwinding insulation of thickness 1.27mm isused. Standard winding configuration as shown in Table 1 isused to wind primary and secondary.Sample no. 2 is made with  E  core (EE65/38/13) woundwith stranded conductor made with three conductors of 24SWG. With interwinding insulation of thickness 2.00mm.Standard winding configuration as shown in Table 1 is usedto wind primary and secondary.Sample no. 3 is made with  E  core (EE42/21/15) woundwith stranded conductor made with two conductors of 24 SWG. With total interwinding insulation of thickness0.46mm + 0.26mm = 0.72mm is used. Sandwiched winding
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