Accurate and Computationally Efficient Modeling of Flyback Transformer Parasitics and their Influence on Converter Losses

Transcription

1 Accurate and Computationally Efficient Modeling of Flyback Transformer Parasitics and their Influence on Converter Losses D Leuenberger, J Biela Laboratory for High Power Electronic Systems - ETH Zurich Physikstrasse 3, Zurich, Switzerland leuenberger@hpeeeethzch Abstract Emerging renewable energy applications, such as PV micro inverters, demand for high step-up isolated DC-DC converters with high reliability and low cost, at high efficiency Thanks to its low part-count the flyback converter is an optimal candidate for such applications To achieve high efficiency over a wide load range, a decent transformer design must be performed, considering also the effects of the transformer parasitics Therefore this work analyzes the influence of the transformer parasitic capacitances and leakage inductance in such a way, that it presents a complete tool to consider the transformer parasitics in the flyback-converter design process A loss-analysis is performed for all three operation modes of the flyback-converter and methods for modeling the parasitic elements are discussed To model the frequency dependence of the leakage inductance a new method is proposed The applied models are explained in-depth and verified with measurements on prototype transformers 1 Introduction Flyback converters feature an isolated DC-DC conversion at lowest possible part count and are one of the very basic, standard power electronic circuits Despite that fact, the flyback converter is still subject of various recent research publications aiming to improve the performance of the flyback converter in various ways, such as advanced soft switching techniques [1], online efficiency optimized control or additional clamping circuits [2] This continuing interest in the flyback converter is caused by emerging renewable energy applications, such as PV micro inverters, which demand for isolated high step-up converters with high reliability and low cost at high efficiency Though PV converters must feature high efficiency over a wide load range At low load, different loss components become dominant and the operation mode of the flyback converter might need to be changed, eg from boundary (BCM) to discontinuous conduction mode (DCM) Choosing an optimal transformer design, which optimizes the efficiency over the whole load range, becomes the main challenge A detailed analysis of the different loss components of the converter, including the transformer, is necessary to perform this task Core losses, winding losses and losses caused by the transformer leakage inductance are usually considered for the converter design [3],[4] However, also the parasitic transformer capacitance causes losses, which are often omitted When minimizing the leakage inductance by means of interleaving of layers or lower layer distances, the parasitic capacitance increases Hence there exists a tradeoff between leakage inductance and parasitic capacitance, also shown in [5] for the example of a high-voltage flyback-transformer In order to find the optimal winding arrangement of the transformer it is therefore necessary to include the influence of the parasitic capacitances in the design process Besides the work in [5], there exist various publications that deal with modeling the parasitic capacitances of transformers in general Effects of parasitic capacitances in flyback converters are only discussed in [6] and [7], which derive the adapted operation mode, but do not consider the influence on losses This work aims to analyze the influence of both transformer parasitics in such a way, that it presents a complete tool to consider the transformer parasitics in the flyback-converter design process This includes not only the lossanalysis, but also the more involving part of modelling the transformer parasitics Figure 1: a) GaN/SiC high step-up flyback-converter prototype for PV micro-inverter, b) Two-port transformer, general parasitics equivalent circuit, [8]

2 Starting from the general equivalent circuit for a two winding transformer, the general equivalent circuit for a flyback transformer is derived step by step in section 2 This allows then to investigate the influence on the operation modes and to derive formulas for the losses caused by the parasitic elements Section 3 deals with fast and accurate methods to model the parasitic elements, suitable also for model based optimization A new method is proposed, to more accurately consider the frequency dependence of the leakage inductance Finally, the derived tools are applied for a model-based optimization of a DC-DC flyback-converter in section 4 2 Non-ideal Flyback Transformer The parasitic elements of any two port transformer can be split into an electrostatic and a magnetic part The electrostatic behaviour of the transformer can be described by the equivalent circuit proposed in [8] and [9], consisting of six equivalent capacitances, the 6C-model The magnetic part consists of the transformer magnetizing inductance L mag and the leakage inductances between the primary and the secondary winding, referred to as the T-equivalent circuit [1] The complete general equivalent circuit is shown in fig1b) 21 General Flyback Transformer Equivalent Circuit The general transformer equivalent circuit in fig1b) can be simplified for flyback-operation, by taking into account the specific operating conditions of the converter Equivalent Parasitic Capacitance: In flyback converter operation, the transformer is subject to a bipolar voltage pulse V pulse on the primary winding Furthermore in most applications there is a constant isolation voltage V iso between the negative rail of the primary and the secondary, see fig2a) C 4 C 6 I pulse C 1 V pulse V p nv p 1:n C 2 V p C eq,p 1:n nv p C 5 V iso C a) b) I iso Figure 2: a) Electrostatic model of flyback converter operation, b) general input-equivalent circuit Under this operating condition, the total electrostatic energy stored in the transformer can be expressed from the 6C-model by W tra f o = 1 2 V 2 p [C 1 +C 2 n 2 +C 4 (N 1) 2 +C 5 n 2 +C 6 ] V 2 iso[c 3 +C 4 +C 5 +C 6 ]+ 1 2 V pv iso [C 4 (n 1) +C 5 n C 6 ] The first part of this equation contains the energy, that the source V pulse must deliver to charge the primary input to V p, if V iso = const Its structure allows for derivation of the energy equivalent capacitor C eq,p = C 1 +C 2 n 2 +C 4 (n 1) 2 +C 5 n 2 +C 6 (2) The second part in (1) contains the energy delivered by the source V iso, to charge C 3,C 4,C 5,C 6 to V iso, under the condition that V p = V The last part of (1) is the additional energy delivered by V iso, when V p V It is caused by the charging currents of C 4,C 5 and C 6 flowing through V iso The equivalent capacitance C eq,p fully describes the dynamic behaviour as well as the capacitive energy seen from the flyback transformer primary and secondary input There is a high frequency current I iso flowing through the source V iso, though this current does not cause losses in the voltage-source V pulse The losses caused by I iso can only be determined on a system-level context For the flyback converter analysis, the terms in (1) containing V iso are therefore not relevant and the electrostatic model can be simplified to a one capacitor model as shown in fig 2b) L-Type Magnetic Model: The flyback converter operates the transformer as a coupled inductor Figure 3a) shows the equivalent magnetic circuit under these operating conditions At turn-off of switch S in, the current commutates from the input to the output-side However due to L σ,1 and L σ,2 the current can not commutate immediately A resonance takes places through the loop Com loop marked in fig 3a), which leads to an overshoot of the blocking voltage at switch S in Changing the model to the simpler L-equivalent circuit, ( fig 3b), does not influence this resonance, as the inductance L σ is still within the same loop The difference in the two models is the current through L mag The T-equivalent reproduces a distortion in i L,mag caused by the resonance (1)

3 The L-equivalent can not account for the magnetizing current distortion For low leakage inductance, L σ L mag, this distortion has a negligible influence on converter operation and transformer losses As a flybacktransformer must fulfill this requirement for performance reasons anyway, the L-equivalent can usually be applied to model the magnetic behavior V in + - Com Loop L σ,w1 L mag L σ,w2 ideal: 1:k V out + - L σ C eq,p L mag ideal: 1:k a) S in D out b) Figure 3: a) Schematic of T-equivalent magnetic circuit in flyback operation b) General equivalent transformer circuit for flyback converter operation, with k n for L σ L mag Finally, the general flyback transformer equivalent circuit is derived from the simplified electrostatic and magnetic model The equivalent circuit referred to the primary side is shown in fig3b) The equivalent capacitance C eq,p is in parallel to L mag, to correctly reproduce the resonance taking place between L σ and C eq,p 22 Flyback Converter Operation under Non-ideal Conditions and Losses caused by the Parasitics The parasitic elements of the flyback transformer influence the operation of the converter and cause additional losses, depending on the operation mode In the following subsections, the operation of the non-ideal flyback converter is analyzed for the different operation modes and analytical formulas are derived to calculate the losses caused by the parasitic elements Figure 4: Flyback Operation in BCM with valley switching: V in = 2V, V out = 2V, n = 2, L mag = 2µH, P = 4W 221 Boundary Conduction Mode (BCM) Figure 4 shows simulated current and voltage waveforms of a DC-DC flyback converter operating in BCM Its switching period is split into four intervals The first interval, [t,t 1 ], and the third interval [t 2,t 3 ] correspond to the operation-intervals of an ideal flyback converter The additional two intervals are due to the charging and discharging of the parasitic transformer capacitance Figure 4, at the bottom, shows the current paths for the four intervals in the circuit diagram The main current path, driven by the magnetizing current I LM, is denoted with a solid line At the transition between some of the intervals, damped high frequency resonances dissipate energy

4 stored in the parasitic elements These high frequency resonance currents are denoted with dotted lines in the circuit diagram 1) In interval [t,t 1 ] the current builds up in L σ and L m 2) When switch S 1 is turned off at t 1, the capacitor C eq,p is discharged by I Lm and reaches the negative value V out at t 2 Simultaneously the energy stored in L σ is dissipated in a damped resonance starting at t 1 through the elements L σ, C oss,s1 and R wdg,tot,p This resonance further causes an over voltage at switch S 1, which will in most cases require a dissipative clamp circuit to protect the switch In most practical cases the resonance is sufficiently damped, such that the energy in L σ is dissipated within one switching period Under this assumption the losses caused by the leakage inductance can be expressed as: P L,lkg = 1 2 L σi LM (t 1 ) 2 f switch, (3) where I LM (t 1 ) is the current through L m at the turn-off of S 1 and f switch the instantaneous switching frequency 3) During the interval [t 2,t 3 ] the current is flowing through the output side and I LM decreases, reaching zero at t 3 4) In the following interval [t 3,t 4 ], a resonant circuit is formed by L m, C eq,p, C oss,s1 and C oss,d The voltage of C eq,p being initially V out /n, changes its polarity At this instant two distinct cases need to be distinguished: valley switching and zero voltage switching If V in > V out /n, the resonance can not charge C eq,p to V in and switch S 1 is turned on at its minimal blocking voltage V S1,valley = V in V out /n Figure 5a) shows zoomed in waveforms of this valley switching case Once the switch is turned on, C eq,p is charged to V in by the damped resonance circuit formed by L σ, C eq,p and S 1 The surplus-energy dissipated in this charging process, is the energy stored in L σ during the first half-cycle of the resonance Based on a standard LCR resonant-circuit these losses can be derived as: P C,eq,ValSw = 1 2 C eq,p( V in V out /N ) 2 f switch, (4) Figure 5: Flyback Operation in BCM a) Valley switching, b) Zero voltage switching The other case is for V in V out /N, where the resonance charges C eq,p to V in, see figure 5b) In this case, V S1 reaches V at t 4 and the antiparallel diode of S 1 begins to conduct While the diode is conducting, S 1 can be turned on under ZVS conditions During the charging of C eq,p, the input side current I prim consists of two overlaid resonance currents First, the main charging current I prim,chg at the frequency ω chg = 1/ L m (C eq,p +C oss,s1 + n 2 C oss,d ) (for L σ Lmag ) Second, the damped resonance current I prim,res at ω res = 1/ L σ 1/(1/C eq,p + 1/C oss,s1 ) Note that the switch output capacitance C oss,s1 is actually nonlinearly dependent on the blocking voltage, which would require numerical methods to solve the involved second order differential equations for V S1 For the following analysis the capacitance is assumed to be constant, with C oss,s1 equal to the energy equivalent capacitance at the maximal blocking voltage The charging current amplitude results from the standard LCR resonant-circuit as V out Î prim,chg = C oss,s1 ω chg, (5) n and its value at the instant t 4, the end of the resonance, is [ ( )] Vin I prim,chg (t 4 ) = Î prim,chg sin arccos (6) V out /n The high frequency resonance is caused by the fact, that the voltage across L σ at the beginning of the charging interval is V L,σ (t 3 )=V Though that V S1 can keep track with the charging of C eq,p, the initial voltage would need to be V L,σ,Nores (t 3 ) = L σ di prim,chg (t 3 ) dt = L σ C osss1 ω 2 chg V out (7) n

5 Consequently a resonance settles this difference, exciting the high frequency current I prim,res This resonance is actually also present for valley switching, however its amplitude is much lower for the given example Generally, the amplitude of this resonance is negligible small, such that its losses are not considered Once switch S 1 turned on at t 4, the magnetizing current starts to increase As result of the foregoing charging interval, I Lm and I prim do not initially match A damped resonance through the elements L σ, S 1 and C eq,p settles this difference The losses related to this resonance are given by P on,zv S = 1 2 L σ[i prim,chg (t 4 ) I Lm (t 4 )] 2 f switch, (8) ( ( )) V out I Lm (t 4 ) (C eq,p +C oss,s1 )ω chg N sin Vin arccos V out /N (9) Besides of directly causing converter losses, the parasitic capacitance also influences the modulation The resonance interval [t 3,t 4 ] lowers the switching frequency Furthermore, because the primary current is negative in the interval [t 3,t 4 ], and for ZVS case also during [t,t 1 ], there is energy fed back to the input This lowers the transferred energy and will require an increased peak-current to maintain the same power transfer as in ideal operation Whether these two effects have a relevant influence on the converter losses, depends on the actual transformer design and the operating point 222 Discontinuous Conduction Mode (DCM) Figure 6: Flyback Operation in DCM: V in = 2V, V out = 8V, n = 2, L mag = 2µH, P = 2W The discontinuous conduction mode is very similar to the BCM Simulation waveforms are shown in fig 6 The current paths are the same as for BCM, see fig 4 The first three intervals [t,t 1 ], [t 1,t 2 ], [t 2,t 3 ] are identical to BCM and so is the calculation of the losses caused by the leakage inductance (3) In the following resonance interval [t 3,t 4 ] the switch is not turned on after the first resonance cycle, but C eq,p keeps oscillating with ω res = 1/ L m (C eq,p +C oss,s1 + n 2 C oss,d ) Under the assumption of a critically damped RLC series resonance, the initial resonance amplitude ˆV eq,p (t 3 ) = V out /n decays with time by ˆV eq,p (t) = V out /ne α(t t 3) (1) where α is given by α = R wdg,1 /(2L m ) and R wdg,1 is the primary side winding resistance at the resonance frequency If the resonance did not fully decay within the interval [t 3,t 4 ], valley switching can be applied at the turn-on of S 1 in the same way as for BCM The losses involved to charge C eq,p to V in are given by P C,eq,DCM = 1 2 C eq,p( V in ˆV eq,p (t 4 ) ) 2 f switch (11) 223 Continuous Conduction Mode (CCM) The continuous conduction mode consists of three intervals, see fig7 The turn off of S 1 at t 1 and the following interval [t 1,t 2 ] is equivalent to BCM and hence the losses caused by the leakage inductance can be calculated by (3) After a fixed time S 1 is turned on at t 3 and a high frequency damped resonance with ω res = 1/ L σ 1/(1/C eq,p + 1/C oss,s1 ) charges C eq,p from V out /N to V in and builds up the current in L σ to I Lm The losses involved in this resonance can be derived by combining (4) and (8) P on,ccm = 1 2 C eq,p(v in +V out /N) 2 f switch L σ(i Lm (t 3 ) 2 f switch (12)

6 Figure 7: Flyback Operation in CCM: V in = 2V, V out = 8V, n = 2, L mag = 2µH, P = 15W 3 Modeling of the Parasitic Elements The influence of the parasitic elements, as derived in section 22, must be considered for an optimal and viable transformer design Therefore, the parasitic elements of the equivalent circuit L σ and C eq,p need to be determined based on mathematical models A transformer design-procedure, by means of a model-based optimization algorithm, further requires the models to feature fast calculation time and applicability to arbitrary geometries 31 Equivalent Parasitic Capacitance Figure 8: Procedure for parasitic capacitance calculation In literature, various approaches for predicting parasitic capacitances of transformers can be found Whereas some approaches, eg [11], apply time consuming finite element simulations, most approaches are based on analytical formulas A comprehensive review of these analytical approaches is given in [12] Among them, the method in [13] is most suited for low power flyback-transformers It allows to predict the capacitances of multi-layer transformers with interleaved windings, based on a parallel plate approach Another approach for capacitance prediction, is the charge simulation method [14],[15] This method is, unlike the analytical approaches, not restricted to certain windings and core geometries Furthermore, the core-to-winding capacitances, which are commonly neglected in the analytical approaches, can easily be taken into account In this work, the charge simulation method is the approach of choice, since its calculation time showed to be fast enough for model based optimization For applications where calculation time is more critical, the approach [13] would be an alternative

7 The procedure for calculating C eq,p with the charge simulation method is schematically shown in fig 8 First an electrostatic test-voltage V test is applied to the transformer terminals Assuming a linear decrease of the potential along the turns of a winding, the potential of each conductor Φ 1 Φ nw in the winding window is determined Second the 2-D charge simulation method is used to calculate the 2-D E-field in the winding window Each conductor is represented by four simulation charges q j q j+3 arranged circularly inside the conductor For each of these simulation charges a corresponding contour-point C j C j+3 is defined on the surface of the conductor, having the reference potential Φ j of the considered conductor If the conductor is not a round-conductor but a Litz-wire, it is approximated as round-conductor with the same outer-diameter as the actual Litz-wire Foil-windings can be treated in the same manner as round-conductors, just with more simulation charges and contour-points placed along the long side of the foil The walls of the winding window are replaced by a parameterizable number n core of simulation charges, placed at a distance d q,core from the winding window surface, as shown in fig8 Again, for each simulation charge a contour-point is defined on the surface of the winding-window The contour-points have the reference potential Φ core This potential can either be specified directly or assumed as unknown, for the common case of a floating core It results a linear equation system for a total number of n tot = n w + n core unknown simulation charges and the unknown core potential Φ core ([14] section II, [15] section 22), which can be expressed in matrix-form: p 1,1 p 1,2 p 1,ntot 1 p 1,ntot p 2,1 p 2,ntot p nw 1,1 p nw 1,n tot p nw,1 p nw,2 p nw,n tot 1 p nw,n tot p nw +1,1 p nw +1,2 p nw +1,n tot 1 p nw +1,n tot 1 p nw +2,1 p nw +2,n tot p ntot 1,1 p ntot 1,n tot 1 p ntot,1 p ntot,2 p ntot,n tot 1 p ntot,n tot 1 k 1 k 2 k ntot 1 k ntot q 1 q 2 q nw 1 q nw q nw +1 q ntot 1 q ntot Φ core = Φ 1 Φ 2 Φ nw 1 Φ nw (13) The rows 1n w of this matrix-equation define the potential of all conductor contour-points Rows n w +1n tot set the potential of all contour-points on the core-surface to be equal to Φ core The geometrical coefficients p i, j are given by ([14], expression 23): p i, j = 1 2πε ln ( ) 1 (xi x j ) 2 + (y i y j ) 2 where ε is the electrical permittivity, (x i,y i ) the position of the contour-point C i and (x j,y j ) the position of the simulation-charge q j The last row of (13) accounts for the fact, that the core is floating Therefore, the sum of all core simulation charges must be zero, setting k 1 k nw = and k nw +1k ntot =1 In the third step in fig 8, the 2D electrical field within the winding window is calculated Due to the 2-D nature of the applied charge-simulation method, the resulting electrical field E is per unit length and can be calculated at an arbitrary position (x,y) inside the winding window with: E x = n tot j=1 q j 2πε n x x tot j (x x j ) 2 + (y y j ), 2 E y = j=1 q j 2πε (14) y y j (x x j ) 2 + (y y j ) 2 (15) In case of an RM-core with a cylindrical center-leg of diameter a, the actual E-field E is given by E = E a ) 2π( 2 + l w,bal (16) assuming rotation symmetry In case of a non-linear energy-distribution, the mean turns length can not be used to calculate E An energy weighted mean turns length l w,bal must be used instead (see fig8, step 3), which can be numerically determined by: lw,bal Y E 2 dydx = E 2 da/2 (17) The last, straight forward step in fig 8 involves the calculation of the equivalent capacitance from the total electric energy in the winding window 32 Leakage inductance Finite element simulations are commonly applied to determine the leakage inductance, based on 2D H-field calculation However, FEM suffers from long computation times and difficult parametrization [16] Alternatively, there

8 exist methods based on 1D H-field-calculation, which allow for an analytical calculation of the leakage inductance Among them, the method in [17] is the most accurate, considering also the frequency dependence of the leakage inductance caused by eddy currents However, this method showed to be inaccurate (deviation above 3%) for the investigated flyback transformers of typical core size similar to RM12, low number of layers and low filling factor Therefore, a new approach has been developed It combines the analytical methods with a numerical calculation of the H-field This results in a method applicable to arbitrary geometries and is more accurate for the above mentioned winding arrangements Figure 9: Procedure for leakage inductance calculation The new semi-numerical method is based on a 2D H-field calculation applying the mirroring method [1],[18] and consists of four steps, schematically shown in fig9 First, the test-currents I test and I test /n are applied to the primary and the secondary winding Second, the field outside of the conductors is calculated numerically using the mirroring method [18] Note that the calculated 2-D H-field H is per unit length Winding types others than round-conductor are treated as follows Litz-wire windings are taken into account by treating each strand as separate round-conductor Foil-windings are transformed into aligned paralleled round-conductors with the equivalent copper-area and the same height as the original foil-winding The H-field inside the conductors is attenuated at higher frequencies due to Eddy currents, which makes the leakage inductance frequency dependent The mirroring method, being a low frequency-approximation method [1], can not consider this effect To accurately model the H-field attenuation, the H-field inside the conductors is calculated analytically by using the formulas given in [19] for a cylinder exposed to an external transverse H-field: where H r (r,ϕ) = 4µ 2 H e j 3 2 k 3 1 J 1 ( j 2 kr) cos(ϕ), H ϕ (r,ϕ) = 1 F r r 2 j 3 2 k 4µ 2 H e 1 [J j 3 1 ( j 2/3 kr) J 1 ( j 2/3 kr)]sin(ϕ), 2 k F r (18) F r = (µ 1 +µ 2 )J ( j 3 2 kr cond )+(µ 1 µ 2 )J 2 ( j 3 2 kr cond ); k = (2π f )ρ 1 µ 1 µ 1 magnetic permeability of the conductor material ρ 1 conductivity of the conductor material µ 2 magnetic permeability of material around the conductor H e sinusoidal transverse magnetic field vector with amplitude H e and ϕ He f frequency of H e The external transverse field H e is derived at the center of the considered conductor using the mirroring method, see step 21 and 22 in fig9 The third step in fig9 involves the calculation of the magnetic-field H inside the winding window over a given x-y grid Note that the H-field is obtained per unit-length, because of the 2-D field calculation In the same way as for the E-field, the magnetic field per unit length is transformed using the energy weighted mean turns length: H = H a ) 2π( 2 + l w,bal (19) assuming rotation symmetry Where l w,bal is the energy weighted mean turns length (see fig9, step 3) numerically determined by: lw,bal Y H 2 dydx = H 2 da/2 (2) Finally, the inductance L σ,p is obtained from the total magnetic energy in the winding window

9 33 Model Validation The models for the parasitic elements described in the first part of this section are verified on a set of three flyback transformer prototypes The transformer prototypes consist of an RM12 core, foil windings on the primary and round wire on the secondary The turns-ratio is n 2 : n 1 = 55 : 5 The three transformers differ in the interleaving of the primary and the secondary winding The interleaving varies from single to full interleaving, as shown in fig1a)-c) The leakage inductance can be derived with an impedance-analyzer by measuring the impedance on the secondary side, while shorting the primary-side (see fig 3c) The constraint that L m >> L σ is fulfilled for all three transformers The new developed model for the leakage inductance shows good accordance with the measurements Up to 2MHz, the deviations are below 15% and reach maximally 22% at higher frequencies The model slightly underestimates L σ, because of not considered effects, such as the wiring to the transformer ports and imperfect geometric winding arrangement The parasitic capacitance referred to the primary side is measured in open-circuit mode by measuring the impedance on the primary side above the resonance frequency given by ω res = 1/ L mag C eq,p The modelled capacitances exhibit good accordance with the measurements The deviations stay below 18% Figure 1: Flyback transformer prototypes: a) Single interleaving b) Multiple interleaving c) Full interleaving 4 Application Example - Flyback DC-DC Converter Figure 11: Wide-Load-Range DC-DC Flyback Converter Design: a) Converter specification b) Optimized winding structure c) Loss-Components over load-range, in percentage of the total converter-losses at the respective output-power A high step-up DC-DC flyback-converter is designed by model-based optimization, applying the analysis and models derived in section 2 and 3 The converter is optimized for a wide load-range according to the Europeanefficiency Figure 11 shows the converter with specifications, the winding-structure of the optimal transformer and the loss-distribution The loss-component L σ in fig11c) contains the losses directly caused by the leakageinductance The loss-component C par includes the losses caused directly by C eq,p as well as the losses caused indirectly by the increased rms-current, due to the additional resonance-intervals The transformer parasitics increase the converter losses by 1-4%, depending on the output-load It is remarkable, that even though the converter operates in BCM from 1W down to 1W P out, the parasitic capacitance still causes increased losses due the increased rms-current Without considering these losses, a winding-structure with full-interleaving, eg as in fig1c), would mistakenly be selected as the optimal winding structure 5 Conclusion In this work, the parasitic elements of flyback transformers are thoroughly discussed Starting from the T-equivalent circuit, the general transformer equivalent circuit for a flyback-converter is derived Depending on the operation

10 mode, the parasitic elements are found to cause additional losses and influence the basic operation-waveforms by additional resonance-intervals Even for an optimized converter design, these additional losses are 1-4%, depending on the output-load The influence on the converter-performance is twofold First, losses are directly caused by the energy stored in the parasitic elements Second, the rms-current increases due to the additional resonance intervals, increasing conduction and switching losses To consider these losses in the design-process, computationally fast models for the parasitic elements are needed A short overview on the existing methods for modeling the parasitic elements is given and the applied models are explained in-depth For modeling the frequency dependence of the leakage inductance a new method is proposed The accuracy of the models is shown by measurements on prototype transformers The modeling errors are below 2% The analysis and the applied models enable to consider the effect of the parasitics in the design process and to perform a model based optimization of a whole flyback converter, including the transformer windings In addition an application example is given for such an optimization It clearly shows, that the losses caused by the parasitics must be considered in the optimization References [1] B Mahdavikhah and A Prodic, A digitally controlled dcm flyback converter with a low-volume dual-mode soft switching circuit, in APEC, March 214, pp [2] W-S Choi, J-W Park, S-J Park, C-F Jin, and D-S Jo, A new topology of flyback converter with active clamp snubber for battery application, in ICIT IEEE, Feb 214, pp [3] S H Kang, D Maksimovic, and I Cohen, Efficiency optimization in digitally controlled flyback dc dc converters over wide ranges of operating conditions, Power Electronics, IEEE Transactions on, vol 27, no 8, pp , 212 [4] P Suskis, I Galkin, and J Zakis, Design and implementation of flyback mppt converter for pv-applications, in Electric Power Quality and Supply Reliability Conference (PQ), June 214, pp [5] H Schneider, P Thummala, L Huang, Z Ouyang, A Knott, Z Zhang, and M Andersen, Investigation of transformer winding architectures for high voltage capacitor charging applications, in IEEE,29th Annual Applied Power Electronics Conference and Exposition (APEC), March 214, pp [6] H Kewei, L Jie, H Xiaolin, and F Ningjun, Analysis and simulation of the influence of transformer parasitics to low power high voltage output flyback converter, in ISIE, IEEE, June 28, pp [7] S-K Chung, Transient characteristics of high-voltage flyback transformer operating in discontinuous conduction mode, IEE Proceedings-Electric Power Applications, vol 151, no 5, pp , 24 [8] B Cogiore, J-P Keradec, and J Barbaroux, The two winding ferrite core transformer: An experimental method to obtain a wide frequency range equivalent circuit, in Instrumentation and Measurement Technology Conference (IMTC IEEE), May 1993, pp [9] F Blache, J-P Keradec, and B Cogitore, Stray capacitances of two winding transformers: equivalent circuit, measurements, calculation and lowering, in Industry Applications Society Annual Meeting (IEEE), Oct 1994, pp vol2 [1] A Van den Bossche and V Valchev, Inductors and Transformers for Power Electronics CRC Press, Taylor and Francis Group, 25 [11] M Xinkui and C Wei, More precise model for parasitic capacitances in high-frequency transformer, in IEEE, 33rd Annual Power Electronics Specialists Conference, PESC, 22, pp vol2 [12] J Biela and J Kolar, Using transformer parasitics for resonant converters - a review of the calculation of the stray capacitance of transformers, in Conference Record of the Industry Applications Conference, 4th IAS Annual Meeting, vol 3, Oct 25, pp [13] T Duerbaum and G Sauerlaender, Energy based capacitance model for magnetic devices, in IEEE, 16th Annual Applied Power Electronics Conference and Exposition, APEC, 21, pp vol1 [14] H Singer, H Steinbigler, and P Weiss, A charge simulation method for the calculation of high voltage fields, IEEE Transactions on Power Apparatus and Systems, vol PAS-93, no 5, pp , Sept 1974 [15] N H Malik, A review of the charge simulation method and its applications, IEEE Transactions on Electrical Insulation, vol 24, no 1, pp 3 2, Feb 1989 [16] P R Wilson, Frequency dependent model of leakage inductance for magnetic components, Advanced Electromagnetics, vol 1, no 3, pp 99 16, 212 [17] P Dowell, Effects of eddy currents in transformer windings, Proceedings of the Institution of Electrical Engineers, vol 113, no 8, pp , 1966 [18] J Muehlethaler, J W Kolar, and A Ecklebe, Loss modeling of inductive components employed in power electronic systems, in Proc IEEE 8th IPEC (ECCE Asia), 211, pp [19] J Lammeraner and M Stafl, Eddy Currents, I B LTD, Ed SNTL Publisher of Technical Literature, 1966