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

D. Leuenberger, J. BielaLaboratory for High Power Electronic Systems - ETH Zurich

Physikstrasse 3, Zurich, Switzerlandleuenberger@hpe.ee.ethz.ch

Abstract

Emerging renewable energy applications, such as PV micro inverters, demand for high step-up isolated DC-DCconverters with high reliability and low cost, at high efficiency. Thanks to its low part-count the flyback converter isan optimal candidate for such applications. To achieve high efficiency over a wide load range, a decent transformerdesign must be performed, considering also the effects of the transformer parasitics. Therefore this work analyzesthe influence of the transformer parasitic capacitances and leakage inductance in such a way, that it presents acomplete tool to consider the transformer parasitics in the flyback-converter design process. A loss-analysis isperformed for all three operation modes of the flyback-converter and methods for modeling the parasitic elementsare discussed. To model the frequency dependence of the leakage inductance a new method is proposed. Theapplied 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 verybasic, standard power electronic circuits. Despite that fact, the flyback converter is still subject of various recentresearch publications aiming to improve the performance of the flyback converter in various ways, such as ad-vanced soft switching techniques [1], online efficiency optimized control or additional clamping circuits [2]. Thiscontinuing interest in the flyback converter is caused by emerging renewable energy applications, such as PV microinverters, 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 componentsbecome dominant and the operation mode of the flyback converter might need to be changed, e.g. from bound-ary (BCM) to discontinuous conduction mode (DCM). Choosing an optimal transformer design, which optimizesthe efficiency over the whole load range, becomes the main challenge. A detailed analysis of the different losscomponents 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 forthe converter design [3],[4]. However, also the parasitic transformer capacitance causes losses, which are oftenomitted. 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 capaci-tance, also shown in [5] for the example of a high-voltage flyback-transformer. In order to find the optimal windingarrangement of the transformer it is therefore necessary to include the influence of the parasitic capacitances inthe design process. Besides the work in [5], there exist various publications that deal with modeling the parasiticcapacitances of transformers in general. Effects of parasitic capacitances in flyback converters are only discussedin [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 completetool to consider the transformer parasitics in the flyback-converter design process. This includes not only the loss-analysis, 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 para-sitics equivalent circuit, [8]

Starting from the general equivalent circuit for a two winding transformer, the general equivalent circuit for a fly-back transformer is derived step by step in section 2. This allows then to investigate the influence on the operationmodes and to derive formulas for the losses caused by the parasitic elements. Section 3 deals with fast and accuratemethods 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 areapplied 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. Theelectrostatic behaviour of the transformer can be described by the equivalent circuit proposed in [8] and [9], con-sisting of six equivalent capacitances, the 6C-model. The magnetic part consists of the transformer magnetizinginductance Lmag and the leakage inductances between the primary and the secondary winding, referred to as theT-equivalent circuit [10]. The complete general equivalent circuit is shown in fig.1b).

2.1 General Flyback Transformer Equivalent Circuit

The general transformer equivalent circuit in fig.1b) can be simplified for flyback-operation, by taking into accountthe specific operating conditions of the converter.

• Equivalent Parasitic Capacitance:In flyback converter operation, the transformer is subject to a bipolar voltage pulse Vpulse on the primarywinding. Furthermore in most applications there is a constant isolation voltage Viso between the negative railof the primary and the secondary, see fig.2a).

1:nC1

C6

C5

C3

C4

C2

Vpulse Vp nVp

Viso

a) b)Iiso

Ipulse

+-

1:nCeq,p

Vp nVp

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 fromthe 6C-model by

Wtra f o =12

V 2p [C1 +C2n2 +C4(N−1)2 +C5n2 +C6]+

12

V 2iso[C3 +C4 +C5 +C6]+

12

VpViso[C4(n−1)+C5n−C6].

(1)

The first part of this equation contains the energy, that the source Vpulse must deliver to charge the primaryinput to Vp, if Viso = const. Its structure allows for derivation of the energy equivalent capacitor

Ceq,p =C1 +C2n2 +C4(n−1)2 +C5n2 +C6 (2)

The second part in (1) contains the energy delivered by the source Viso, to charge C3,C4,C5,C6 to Viso, underthe condition that Vp = 0V . The last part of (1) is the additional energy delivered by Viso, when Vp 6= 0V . It iscaused by the charging currents of C4,C5 and C6 flowing through Viso. The equivalent capacitance Ceq,p fullydescribes the dynamic behaviour as well as the capacitive energy seen from the flyback transformer primaryand secondary input. There is a high frequency current Iiso flowing through the source Viso, though thiscurrent does not cause losses in the voltage-source Vpulse. The losses caused by Iiso can only be determinedon a system-level context. For the flyback converter analysis, the terms in (1) containing Viso are thereforenot 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 equivalentmagnetic circuit under these operating conditions. At turn-off of switch Sin, the current commutates fromthe input to the output-side. However due to Lσ,1 and Lσ,2 the current can not commutate immediately. Aresonance takes places through the loop Comloop marked in fig. 3a), which leads to an overshoot of theblocking voltage at switch Sin. Changing the model to the simpler L-equivalent circuit, ( fig. 3b), does notinfluence this resonance, as the inductance Lσ is still within the same loop. The difference in the two modelsis the current through Lmag. The T-equivalent reproduces a distortion in iL,mag caused by the resonance.

The L-equivalent can not account for the magnetizing current distortion. For low leakage inductance, Lσ�Lmag, this distortion has a negligible influence on converter operation and transformer losses. As a flyback-transformer must fulfill this requirement for performance reasons anyway, the L-equivalent can usually beapplied to model the magnetic behavior.

b)a)

ideal:1:k’

Ceq,p

Lσ

Lmag

ideal:1:k

Lσ,w1 Lσ,w2

LmagVin

Sin

Vout

Dout

+- +

-ComLoop

Figure 3: a) Schematic of T-equivalent magnetic circuit in flyback operation b) General equivalent transformer circuit forflyback converter operation, with k′ ' n for Lσ� Lmag.

Finally, the general flyback transformer equivalent circuit is derived from the simplified electrostatic and magneticmodel. The equivalent circuit referred to the primary side is shown in fig.3b). The equivalent capacitance Ceq,p isin parallel to Lmag, to correctly reproduce the resonance taking place between Lσ and Ceq,p.

2.2 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 additionallosses, depending on the operation mode. In the following subsections, the operation of the non-ideal flybackconverter is analyzed for the different operation modes and analytical formulas are derived to calculate the lossescaused by the parasitic elements.

Figure 4: Flyback Operation in BCM with valley switching: Vin = 20V , Vout = 20V , n = 2, Lmag = 20µH, P = 40W

2.2.1 Boundary Conduction Mode (BCM)

Figure 4 shows simulated current and voltage waveforms of a DC-DC flyback converter operating in BCM. Itsswitching period is split into four intervals. The first interval, [t0, t1], and the third interval [t2, t3] correspond tothe operation-intervals of an ideal flyback converter. The additional two intervals are due to the charging anddischarging of the parasitic transformer capacitance. Figure 4, at the bottom, shows the current paths for the fourintervals in the circuit diagram. The main current path, driven by the magnetizing current ILM , is denoted with asolid line. At the transition between some of the intervals, damped high frequency resonances dissipate energy

stored in the parasitic elements. These high frequency resonance currents are denoted with dotted lines in thecircuit diagram. 1) In interval [t0, t1] the current builds up in Lσ and Lm. 2) When switch S1 is turned off at t1, thecapacitor Ceq,p is discharged by ILm and reaches the negative value Vout at t2. Simultaneously the energy stored inLσ is dissipated in a damped resonance starting at t1 through the elements Lσ, Coss,S1 and Rwdg,tot,p. This resonancefurther causes an over voltage at switch S1, which will in most cases require a dissipative clamp circuit to protectthe switch. In most practical cases the resonance is sufficiently damped, such that the energy in Lσ is dissipatedwithin one switching period. Under this assumption the losses caused by the leakage inductance can be expressedas:

PL,lkg =12

LσILM(t1)2 fswitch, (3)

where ILM(t1) is the current through Lm at the turn-off of S1 and fswitch the instantaneous switching frequency.3) During the interval [t2, t3] the current is flowing through the output side and ILM decreases, reaching zero at t3.4) In the following interval [t3, t4], a resonant circuit is formed by Lm, Ceq,p, Coss,S1 and Coss,D. The voltage of Ceq,pbeing initially −Vout/n, changes its polarity. At this instant two distinct cases need to be distinguished: valleyswitching and zero voltage switching.If | Vin |>| Vout/n |, the resonance can not charge Ceq,p to Vin and switch S1 is turned on at its minimal blockingvoltage VS1,valley =|Vin | − |Vout/n |. Figure 5a) shows zoomed in waveforms of this valley switching case. Oncethe switch is turned on, Ceq,p is charged to Vin by the damped resonance circuit formed by Lσ, Ceq,p and S1. Thesurplus-energy dissipated in this charging process, is the energy stored in Lσ during the first half-cycle of theresonance. Based on a standard LCR resonant-circuit these losses can be derived as:

PC,eq,ValSw =12

Ceq,p(|Vin | − |Vout/N |)2 fswitch, (4)

Figure 5: Flyback Operation in BCM a) Valley switching, b) Zero voltage switching

The other case is for | Vin |≤| Vout/N |, where the resonance charges Ceq,p to Vin, see figure 5b). In this case, VS1reaches 0V at t4 and the antiparallel diode of S1 begins to conduct. While the diode is conducting, S1 can be turnedon under ZVS conditions. During the charging of Ceq,p, the input side current Iprim consists of two overlaid reso-nance currents. First, the main charging current Iprim,chg at the frequency ωchg = 1/

√Lm(Ceq,p +Coss,S1 +n2Coss,D)

(for Lσ�Lmag ). Second, the damped resonance current Iprim,res at ωres = 1/√

Lσ ·1/(1/Ceq,p +1/Coss,S1). Note thatthe switch output capacitance Coss,S1 is actually nonlinearly dependent on the blocking voltage, which would re-quire numerical methods to solve the involved second order differential equations for VS1. For the followinganalysis the capacitance is assumed to be constant, with Coss,S1 equal to the energy equivalent capacitance at themaximal blocking voltage. The charging current amplitude results from the standard LCR resonant-circuit as

Iprim,chg =Coss,S1ωchg|Vout |

n, (5)

and its value at the instant t4, the end of the resonance, is

Iprim,chg(t4) = Iprim,chg sin[

arccos(

Vin

−Vout/n

)]. (6)

The high frequency resonance is caused by the fact, that the voltage across Lσ at the beginning of the charginginterval is VL,σ(t3)=0V. Though that VS1 can keep track with the charging of Ceq,p, the initial voltage would need tobe

VL,σ,Nores(t3) = Lσ

dIprim,chg(t3)dt

=−LσCossS1ω2chg|Vout |

n. (7)

Consequently a resonance settles this difference, exciting the high frequency current Iprim,res. This resonance isactually 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 S1 turned on at t4, the magnetizing current starts to increase. As result of the foregoing charginginterval, ILm and Iprim do not initially match. A damped resonance through the elements Lσ, S1 and Ceq,p settlesthis difference. The losses related to this resonance are given by

Pon,ZV S =12

Lσ[Iprim,chg(t4)− ILm(t4)]2 fswitch, (8)

ILm(t4)' (Ceq,p +Coss,S1)ωchg|Vout |

Nsin(

arccos(

Vin

−Vout/N

))(9)

Besides of directly causing converter losses, the parasitic capacitance also influences the modulation. The reso-nance interval [t3, t4] lowers the switching frequency. Furthermore, because the primary current is negative in theinterval [t3, t4], and for ZVS case also during [t0, t1], there is energy fed back to the input. This lowers the trans-ferred 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 transformerdesign and the operating point.

2.2.2 Discontinuous Conduction Mode (DCM)

Figure 6: Flyback Operation in DCM: Vin = 20V , Vout = 80V , n = 2, Lmag = 20µH, P = 20W

The discontinuous conduction mode is very similar to the BCM. Simulation waveforms are shown in fig. 6. Thecurrent paths are the same as for BCM, see fig. 4. The first three intervals [t0, t1], [t1, t2], [t2, t3] are identical toBCM and so is the calculation of the losses caused by the leakage inductance (3). In the following resonanceinterval [t3, t4] the switch is not turned on after the first resonance cycle, but Ceq,p keeps oscillating with ωres =

1/√

Lm(Ceq,p +Coss,S1 +n2Coss,D). Under the assumption of a critically damped RLC series resonance, the initialresonance amplitude Veq,p(t3) =Vout/n decays with time by

Veq,p(t) =Vout/neα(t−t3) (10)

where α is given by α=Rwdg,1/(2Lm) and Rwdg,1 is the primary side winding resistance at the resonance frequency.If the resonance did not fully decay within the interval [t3, t4], valley switching can be applied at the turn-on of S1in the same way as for BCM. The losses involved to charge Ceq,p to Vin are given by

PC,eq,DCM =12

Ceq,p(|Vin | −Veq,p(t4) |)2 fswitch. (11)

2.2.3 Continuous Conduction Mode (CCM)

The continuous conduction mode consists of three intervals, see fig.7. The turn off of S1 at t1 and the follow-ing interval [t1, t2] is equivalent to BCM and hence the losses caused by the leakage inductance can be cal-culated by (3). After a fixed time S1 is turned on at t3 and a high frequency damped resonance with ωres =1/√

Lσ ·1/(1/Ceq,p +1/Coss,S1) charges Ceq,p from −Vout/N to Vin and builds up the current in Lσ to ILm. Thelosses involved in this resonance can be derived by combining (4) and (8)

Pon,CCM =12

Ceq,p(Vin +Vout/N)2 fswitch +12

Lσ(ILm(t3)2 fswitch. (12)

Figure 7: Flyback Operation in CCM: Vin = 20V , Vout = 80V , n = 2, Lmag = 20µH, P = 150W

3 Modeling of the Parasitic Elements

The influence of the parasitic elements, as derived in section 2.2, must be considered for an optimal and viabletransformer design. Therefore, the parasitic elements of the equivalent circuit Lσ and Ceq,p need to be determinedbased on mathematical models. A transformer design-procedure, by means of a model-based optimization algo-rithm, further requires the models to feature fast calculation time and applicability to arbitrary geometries.

3.1 Equivalent Parasitic Capacitance

Figure 8: Procedure for parasitic capacitance calculation

In literature, various approaches for predicting parasitic capacitances of transformers can be found. Whereas someapproaches, e.g. [11], apply time consuming finite element simulations, most approaches are based on analyticalformulas. 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 transformerswith interleaved windings, based on a parallel plate approach. Another approach for capacitance prediction, isthe charge simulation method [14],[15]. This method is, unlike the analytical approaches, not restricted to certainwindings and core geometries. Furthermore, the core-to-winding capacitances, which are commonly neglected inthe 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 fastenough for model based optimization. For applications where calculation time is more critical, the approach [13]would be an alternative.

The procedure for calculating Ceq,p with the charge simulation method is schematically shown in fig. 8. First anelectrostatic test-voltage Vtest is applied to the transformer terminals. Assuming a linear decrease of the potentialalong the turns of a winding, the potential of each conductor Φ1..Φnw in the winding window is determined. Secondthe 2-D charge simulation method is used to calculate the 2-D E-field in the winding window. Each conductor isrepresented by four simulation charges q j − q j+3 arranged circularly inside the conductor. For each of thesesimulation charges a corresponding contour-point C j −C j+3 is defined on the surface of the conductor, havingthe 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 canbe treated in the same manner as round-conductors, just with more simulation charges and contour-points placedalong the long side of the foil. The walls of the winding window are replaced by a parameterizable number ncoreof simulation charges, placed at a distance dq,core from the winding window surface, as shown in fig.8. Again,for each simulation charge a contour-point is defined on the surface of the winding-window. The contour-pointshave the reference potential Φcore. This potential can either be specified directly or assumed as unknown, for thecommon case of a floating core. It results a linear equation system for a total number of ntot = nw +ncore unknownsimulation charges and the unknown core potential Φcore ([14] section II, [15] section 2.2), which can be expressedin matrix-form:

p1,1 p1,2 . . p1,ntot−1 p1,ntot 0p2,1 . . . . p2,ntot 0. . . . . . 0. . . . . . 0

pnw−1,1 . . . . pnw−1,ntot 0pnw,1 pnw,2 . . pnw,ntot−1 pnw,ntot 0

pnw+1,1 pnw+1,2 . . pnw+1,ntot−1 pnw+1,ntot −1pnw+2,1 . . . . pnw+2,ntot −1

. . . . . . −1

. . . . . . −1pntot−1,1 . . . . pntot−1,ntot −1pntot ,1 pntot ,2 . . pntot ,ntot−1 pntot ,ntot −1

k1 k2 . . kntot−1 kntot 0

·

q1q2..

qnw−1qnw

qnw+1..

qntot−1qntot

Φcore

=

Φ1Φ2..

Φnw−1Φnw

000000

. (13)

The rows 1..nw of this matrix-equation define the potential of all conductor contour-points. Rows nw+1..ntot set thepotential of all contour-points on the core-surface to be equal to Φcore. The geometrical coefficients pi, j are givenby ([14], expression 23):

pi, j =1

2πε· ln

(1√

(xi− x j)2 +(yi− y j)2

)(14)

where ε is the electrical permittivity, (xi,yi) the position of the contour-point Ci and (x j,y j) the position of thesimulation-charge q j. The last row of (13) accounts for the fact, that the core is floating. Therefore, the sum ofall core simulation charges must be zero, setting k1..knw=0 and knw+1..kntot =1. In the third step in fig. 8, the 2Delectrical field within the winding window is calculated. Due to the 2-D nature of the applied charge-simulationmethod, the resulting electrical field ~E ′ is per unit length and can be calculated at an arbitrary position (x,y) insidethe winding window with:

E ′x =ntot

∑j=1

q j

2πε·

x− x j√(x− x j)2 +(y− y j)2

, E ′y =ntot

∑j=1

q j

2πε·

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 ′ ·2π

(a2+ lw,bal

)(16)

assuming rotation symmetry. In case of a non-linear energy-distribution, the mean turns length can not be used tocalculate ~E. An energy weighted mean turns length lw,bal must be used instead (see fig.8, step 3), which can benumerically determined by:∫ lw,bal

0

∫ Y

0E ′2dydx =

∫E ′2dA/2 (17)

The last, straight forward step in fig. 8 involves the calculation of the equivalent capacitance from the total electricenergy in the winding window.

3.2 Leakage inductance

Finite element simulations are commonly applied to determine the leakage inductance, based on 2D H-field calcu-lation. However, FEM suffers from long computation times and difficult parametrization [16]. Alternatively, there

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 leakageinductance caused by eddy currents. However, this method showed to be inaccurate (deviation above 30%) for theinvestigated 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 calculationof the H-field. This results in a method applicable to arbitrary geometries and is more accurate for the above men-tioned 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 [10],[18]and consists of four steps, schematically shown in fig.9. First, the test-currents Itest and Itest/n are applied to theprimary and the secondary winding. Second, the field outside of the conductors is calculated numerically usingthe mirroring method [18]. Note that the calculated 2-D H-field ~H ′ is per unit length. Winding types others thanround-conductor are treated as follows. Litz-wire windings are taken into account by treating each strand as sepa-rate round-conductor. Foil-windings are transformed into aligned paralleled round-conductors with the equivalentcopper-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 leakageinductance frequency dependent. The mirroring method, being a low frequency-approximation method [10], cannot consider this effect. To accurately model the H-field attenuation, the H-field inside the conductors is calculatedanalytically by using the formulas given in [19] for a cylinder exposed to an external transverse H-field:

Hr(r,ϕ) =4µ2|~He|

j32 k

1Fr0

J1( j32 kr)r

cos(ϕ), Hϕ(r,ϕ) =−12

j32 k

4µ2|~He|j

32 k

1Fr0

[J−1( j2/3kr)− J1( j2/3kr)]sin(ϕ),

(18)

where

Fr0 = (µ1+µ2)J0( j32 krcond)+(µ1−µ2)J2( j

32 krcond); k =

√(2π f )ρ1µ1

µ1 magnetic permeability of the conductor material

ρ1 conductivity of the conductor material

µ2 magnetic permeability of material around the conductor

~He sinusoidal transverse magnetic field vector with amplitude He and ϕHe

f frequency of He.

The external transverse field ~He is derived at the center of the considered conductor using the mirroring method,see step 2.1 and 2.2 in fig.9. The third step in fig.9 involves the calculation of the magnetic-field ~H ′ inside thewinding window over a given x-y grid. Note that the H-field is obtained per unit-length, because of the 2-D fieldcalculation. In the same way as for the E-field, the magnetic field per unit length is transformed using the energyweighted mean turns length:

~H = ~H ′ ·2π

(a2+ lw,bal

)(19)

assuming rotation symmetry. Where lw,bal is the energy weighted mean turns length (see fig.9, step 3) numericallydetermined by:∫ lw,bal

0

∫ Y

0H ′2dydx =

∫H ′2dA/2 (20)

Finally, the inductance Lσ,p is obtained from the total magnetic energy in the winding window.

3.3 Model Validation

The models for the parasitic elements described in the first part of this section are verified on a set of three flybacktransformer prototypes. The transformer prototypes consist of an RM12 core, foil windings on the primary andround wire on the secondary. The turns-ratio is n2 : n1 = 55 : 5. The three transformers differ in the interleavingof the primary and the secondary winding. The interleaving varies from single to full interleaving, as shownin fig.10a)-c). The leakage inductance can be derived with an impedance-analyzer by measuring the impedanceon the secondary side, while shorting the primary-side (see fig. 3c). The constraint that Lm >> Lσ is fulfilledfor all three transformers. The new developed model for the leakage inductance shows good accordance withthe measurements. Up to 2MHz, the deviations are below 15% and reach maximally 22% at higher frequencies.The model slightly underestimates Lσ, because of not considered effects, such as the wiring to the transformerports and imperfect geometric winding arrangement. The parasitic capacitance referred to the primary side ismeasured in open-circuit mode by measuring the impedance on the primary side above the resonance frequencygiven by ωres = 1/

√LmagCeq,p. The modelled capacitances exhibit good accordance with the measurements. The

deviations stay below 18%.

Figure 10: 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 andmodels derived in section 2 and 3. The converter is optimized for a wide load-range according to the European-efficiency. Figure 11 shows the converter with specifications, the winding-structure of the optimal transformerand the loss-distribution. The loss-component Lσ in fig.11c) contains the losses directly caused by the leakage-inductance. The loss-component Cpar includes the losses caused directly by Ceq,p as well as the losses causedindirectly by the increased rms-current, due to the additional resonance-intervals. The transformer parasitics in-crease the converter losses by 10-40%, depending on the output-load. It is remarkable, that even though theconverter operates in BCM from 100W down to 10W Pout , the parasitic capacitance still causes increased lossesdue the increased rms-current. Without considering these losses, a winding-structure with full-interleaving, e.g. asin fig.10c), 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-equivalentcircuit, the general transformer equivalent circuit for a flyback-converter is derived. Depending on the operation

mode, the parasitic elements are found to cause additional losses and influence the basic operation-waveformsby additional resonance-intervals. Even for an optimized converter design, these additional losses are 10-40%,depending on the output-load. The influence on the converter-performance is twofold. First, losses are directlycaused by the energy stored in the parasitic elements. Second, the rms-current increases due to the additionalresonance 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 formodeling the parasitic elements is given and the applied models are explained in-depth. For modeling the fre-quency dependence of the leakage inductance a new method is proposed. The accuracy of the models is shown bymeasurements on prototype transformers. The modeling errors are below 20%.The analysis and the applied models enable to consider the effect of the parasitics in the design process and toperform a model based optimization of a whole flyback converter, including the transformer windings. In additionan application example is given for such an optimization. It clearly shows, that the losses caused by the parasiticsmust be considered in the optimization.

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