Transformer Winding Loss Caused by Skin and Proximity Effects Including

www.ietdl.org

Published in IET Power Electronics Received on 31st January 2010 Revised on 14th July 2010 doi: 10.1049/iet-pel.2010.0040

ISSN 1755-4535 Transformer winding loss caused by skin and proximity effects including harmonics in pulse-width modulated DC–DC ﬂyback converters for the continuous conduction mode D. Murthy-Bellur1 N. Kondrath2 M.K. Kazimierczuk3 1School of Engineering, Pennsylvania State University, The Behrend College, Erie, PA 16563, USA 2Department of Electrical and Computer Engineering, University of Minnesota Duluth, Duluth, MN 55812, USA 3Department of Electrical Engineering, Wright State University, Dayton, OH 45435, USA E-mail: [email protected]

Abstract: High-frequency (HF) transformers used in pulse-width-modulated power converters conduct periodic non-sinusoidal HF currents, which give rise to additional winding losses caused by harmonics. This study presents expressions for winding power losses in a two-winding transformer with periodic non-sinusoidal primary and secondary current waveforms for ﬂyback converters operating in the continuous conduction mode (CCM). Dowell’s equation, which takes into account both the skin and proximity effects, is used to determine the primary and secondary winding resistances as functions of frequency. Fourier series of the primary and secondary current waveforms and the winding resistances are used to determine the winding power losses at harmonic frequencies. The harmonic primary and secondary winding loss factors FRph and FRsh are introduced for CCM operation. The theory is illustrated by performance evaluation of the transformer in a ﬂyback converter in CCM over the entire range of converter operation. Power losses in the primary and secondary transformer windings are illustrated as functions of the output power and the DC input voltage.

1 Introduction [16], the AC resistances for a number of arbitrary waveforms were determined using the rms values of the current High-frequency (HF) transformers are an integral part of pulse- waveforms and the rms values of its derivatives. Methods to width-modulated (PWM) converters, which are widely used in design inductors and transformers for PWM converters are many applications such as power-factor correctors, DC–DC available in [30–32]. However, the transformer design in [30] power converters and electromagnetic interference (EMI) filters and other publications do not account for winding losses [1–5]. For low-power applications, flyback PWM DC–DC caused neither by HF skin and proximity effects nor by converter is a popular choice because of its low parts count harmonics. Until now, the Fourier expansion of the flyback when compared with other isolated DC–DC converter transformer primary and secondary current waveforms in topologies. The HF transformer in a flyback converter is used PWM converters operating in the continuous conduction to provide galvanic isolation and also to store the magnetic mode (CCM) as shown in Fig. 1 has not been presented in the energy required for power conversion. Operating the flyback literature. converter at high switching frequency reduces the size of the The objectives of this paper are (i) to develop winding power transformer, but increases the transformer winding losses loss equations for a two-winding transformer conducting caused by skin and proximity effects. Additionally, since periodic non-sinusoidal HF currents containing DC transformers in isolated PWM converters conduct periodic non- components and harmonics such as those in flyback PWM sinusoidal currents in steady state, the spectra of the primary converters [33–35] operating in CCM, using Dowell’s theory and secondary current waveforms of the transformer are very and the methods developed in [36–38], (ii) to give a step-by- rich and the losses caused by the harmonics further increase the step procedure to design a two-winding transformer for winding power losses. The harmonic losses in transformer flyback converter operated in CCM, using the area product windings at HF also increase the operating temperature. method and (iii) to study the effect of harmonics on winding HF effects in windings and cores were studied in several losses in the transformer of a flyback PWM DC–DC papers [6–29]. The winding AC resistance caused by the skin converter in CCM over the entire range of converter operation. and proximity effects for sinusoidal current in a winding Sections 2 and 3 present the derivations of core area was presented in [6–8]. The theoretical analyses presented in product and winding power loss equations for a flyback [6–8] were further extended in [9–15]. In another method transformer in CCM conducting periodic non-sinusoidal

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 363 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011 www.ietdl.org

where Awp is the bare wire cross-sectional area of the primary winding in m2. The window area limited by the maximum current density in both primary and secondary windings is given by

N A + N A = p wp s ws Wa (4) Ku

where Ns is the number of turns in the secondary winding, Aws is the secondary winding bare wire cross-sectional area in 2 m and Ku is the window utilisation factor. Assuming the winding allocation is such that NpAwp ¼ NsAws, the window area is

2N A = p wp Wa (5) Ku

where the window utilisation factor is deﬁned as

2N A = ACu = p wp Ku (6) Wa Wa

where ACu is the total copper cross-sectional area in the window in m2. The maximum energy stored in the = 2 magnetic field of the transformer is Wm (1/2)LmILm max, where Lm is the magnetising inductance on the primary side and ILm is the current through the magnetising Fig. 1 Flyback converter circuit and two-winding flyback inductance Lm. Assuming that Lp ≃ Lm and Ipmax ≃ transformer current waveforms for CCM operation ILm(max), and using (2), (3) and (5), the area product of the a Flyback PWM DC–DC converter circuit core is defined as b Primary winding current ip c Secondary winding current is 2L I 2 4W A = W A = p p max = m (7) p a c K J B K J B current waveforms, respectively. The procedure to design a u m pk u m pk two-winding transformer for a flyback PWM DC–DC converter operated in CCM is given in Section 4. In Section In (7), Bpk , Bs, where Bs is the core saturation flux density at 5, the performance evaluation of the designed flyback the highest operating temperature. The area product Ap is a transformer is given. Experimental and simulation results measure of energy handling capability of a core and can be are presented in Section 6. Finally, conclusions and used to select a suitable core using the manufacturer’s discussions are given in Section 7. datasheets.

2 Core area product for a two-winding 3 Copper power loss for flyback transformer flyback transformer conducting periodic non-sinusoidal current in CCM The instantaneous magnetic flux linkage of the primary winding is given by In transformers used in isolated PWM DC–DC power converters in CCM, the current waveforms through the l(t) = N f(t) = N A B(t) = L i (t) (1) primary and secondary windings are pulsating. Therefore p p c p p the amplitudes of the harmonics are high, resulting in high winding losses. This situation is different in non- where f(t) is the instantaneous magnetic flux linking Np turns isolated PWM converters in CCM because the current of the primary winding of the transformer, Ac is the core 2 waveform of the inductor is nearly constant and the cross-sectional area in m , Lp is the inductance of the amplitudes of harmonics are low as compared to the DC primary winding, ip(t) is the instantaneous current in the component [38]. primary winding and B(t) is the instantaneous magnetic flux density in T. Equation (1) for the peak values becomes 3.1 Primary winding power loss l = N f = N A B = L I (2) pk p pk p c pk p p max Assume that the primary winding current for CCM is a rectangular waveform with a duty ratio D as shown in The maximum current density of the primary winding wire is Fig. 1b. The primary winding current waveform is given by I ≤ = p max ≃ Ip max, for 0 , t DT Jm (3) ip (8) Awp 0, for DT , t ≤ T

364 IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-pel.2010.0040 www.ietdl.org

The DC component of the primary winding current waveform harmonic loss factor is deﬁned as the total primary winding for CCM is loss consisting of DC and AC power losses normalised with respect to the primary winding DC power loss given 1 T I DT by I = i dt = p max dt = DI (9) pdc T p T p max 0 0 P 1 2 = wp = + sin(npD) FRph 1 2 FRpn (15) The Fourier series of the current waveform in the primary Pwpdc n=1 npD winding is The skin depth of the winding conductor at the nth 1 sin(npD) harmonic is i = I 1 + 2 cos(nv t) (10) p pdc npD s n=1 d d = √w1 (16) wn n and the amplitudes of the fundamental component and the harmonics of the primary current waveform for CCM is = where dw1 r/pm0f is the conductor skin depth at the fundamental frequency f, which is equal to the 2Ip max 2Ipdc I = sin(npD) = sin(npD) (11) converter switching frequency fs. The winding conductor pn np npD thickness normalised with respect to the conductor skin depth is The AC resistance factor F obtained from Dowell’s one- R √ dimensional solution in Cartesian coordinates [7] is √ = h = h n = An A n (17) + dwn dw1 = Rw = + = sinh(2A) sin(2A) FR Fs Fp A − Rwdc cosh(2A) cos(2A) where the normalised winding conductor thickness at (12) 2(N 2 − 1) sinh(A) − sin(A) the fundamental frequency for a rectangular conductor is + l [39] 3 cosh(A) + cos(A) h w where the ﬁrst term represents the skin effect winding AC A = (18) d p resistance factor FS and the second term describes the w1 proximity effect winding AC resistance factor FP. The power loss in the primary winding for CCM is for a square conductor is [39] ⎡ ⎤ 2 1 1 R I = h h P = R I 2 ⎣1 + wpn pn ⎦ A (19) wp wpdc pdc dw1 p 2 n=1 Rwpdc Ipdc ⎡ ⎤ 2 and for a round conductor is [39] 1 1 I = P ⎣1 + F pn ⎦ wpdc 2 Rpn I n=1 pdc p 3/4 d d A = (20) 1 2 4 dw1 p = + sin(npD) Pwpdc 1 2 FRpn n=1 npD In (18), (19) and (20), w/p, h/p and d/p are called the layer = porosity factors, w is the width of the rectangular bare PwpdcRwpn (13) conductor, h is the thickness of the square and rectangular bare conductors, d is the diameter of the round bare where the normalised primary winding resistance at the conductor and p is the distance between the centres of fundamental frequency and harmonic frequencies obtained two adjacent conductors, called the winding pitch. from (12) is 3.2 Secondary winding power loss R √ = wpn = + = FRpn FSn FPn A n Assume that the secondary winding current for CCM is a Rwpdc √ √ rectangular waveform as shown in Fig. 1c. The secondary + winding current is given by × sin(2A√n) sin(2A √n) − cosh(2A n) cos(2Ap n) 0, for 0 , t ≤ DT √ √ i ≃ (21) 2 s ≤ 2(N − 1) sinh(A n) − sin(A n) Is max, for DT , t T + l √ √ (14) 3 cosh(A n) + cos(A n) The DC component of the secondary winding current waveform for CCM is The factor FRpn takes into account the skin and proximity effects in the winding. Note that FR and FRpn are different T T = 1 = Is max = − as the latter is dependent on the nth harmonic of the Isdc isdt dt (1 D)Is max (22) primary winding current waveform. The primary winding T 0 T DT

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 365 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011 www.ietdl.org

Table 1 Two-winding transformer design for ﬂyback PWM DC–DC converter in CCM

Step Parameter Equation Value number

Converter speciﬁcations

1 input voltage VI 50 + 10 V

2 output voltage VO 24 V

3 switching frequency fs 100 kHz

4 output power POmax(POmin) 30 W (5 W) Design equations for the transformer in ﬂyback converter 2 5 maximum load resistance = VO 115.2 V RL max PO min 6 minimum DC voltage transfer function = VO 0.4 MVDC min VI max 7 maximum DC voltage transfer function = Vo 0.6 MVDC max VI min ∗ 8 transformer primary-to-secondary turns ratio = nconvDmax 1.5 nT − (1 Dmax)MVDC max 9 minimum duty cycle = nTMVDC min 0.4 Dmin + nTMVDC min nconv 2 − 2 10 minimum magnetising inductance for CCM ≃ = nTRL max(1 Dmin) 466 mH Lp Lm( min ) 2f1 pick 500 mH L 11 secondary winding inductance = p 222 mH Ls 2 nT − 12 maximum peak-to-peak primary winding current = = nTVo(1 Dmin) 0.43 A DiLp( max ) DiLm( max ) fsLp

13 maximum DC input current = MVDC maxIO max 0.83 A II max hconv − 14 maximum peak primary winding current = IO max + nTVO(1 Dmax) 1.84 A Ip max − nT(1 Dmax) 2fsLD Area product

15 window utilisation factor Ku 0.3

16 peak ﬂux density Bpk 0.31 T 2 17 maximum current density of the winding wire Jm 6A/mm 18 maximum energy stored in the transformer magnetic = 1 2 0.85 mJ Wm LpIp max ﬁeld 2 4 19 core area product (calculated) = = 4Wm 0.61 cm Ap WaAc KuJmBpk Core selection 20 selected core magnetics ferrite p-type round-slab (RS) core 0P-43019 4 21 area product of the selected core Ap 0.63 cm 2 22 core cross-sectional area Ac 1.23 cm 2 23 core window area Wa 0.51 cm

24 mean magnetic path length lc 4.56 cm

25 mean length of single turn lT 6.05 cm 3 26 core volume Vc 5.61 cm

27 core permeability mrc 2500 + 25%

28 core saturation ﬂux density Bs 0.49 T = a b 29 coefﬁcients in core power loss density equation (for p- Pv k(fs in kHz) (10Bm) type ferrite material and 100 kHz ≤ fs , 500 kHz) 29(a) k 0.0434 29(b) a 1.63 29(c) b 2.62 Wire selection r (208C) d = cu 30 skin depth of copper wire (at 208C and fs ¼ 100 kHz) w1 0.209 mm pfsm0

31 selected copper wire AWG No. 26

32 bare wire diameter of the strand wire dis 0.40 mm

33 insulated wire diameter of the strand wire dos 0.45 mm 2 34 bare wire cross-sectional area of the strand Awsi 0.12 mm 2 35 insulated wire cross-sectional area of the strand Awso 0.16 mm

36 DC resistance of the strand per unit length Rwdc/lw 0.1345 V/m Primary winding design I 2 37 cross-sectional area of the primary winding wire = pmax 0.30 mm Awp Jm Continued

366 IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-pel.2010.0040 www.ietdl.org

Table 1 Continued

Step Parameter Equation Value number A 38 number of strands in the primary winding = wp 2.38 sp Awsi pick 2

39 number of turns of the primary winding = Ku(Wa/2) 29.82 Np SpAwsi pick 30

= 40 total length of the primary winding wire Iwp NplT 181.5 cm pick 187 cm Primary winding losses 41 primary strand DC and low-frequency resistance = Rwdc 251.5 mV Rwpdcs lwp lw R 42 primary winding wire DC and low-frequency resistance = wpdcs 125.8 mV Rwpdc Sp = 2 43 primary winding DC and low-frequency power loss Pwpdc RwpdcIs max 0.087 W 44 primary winding harmonic loss factor FRph (Use (15) @ VImin, POmax) 5.195 = 45 primary winding total power loss Pwp FRphPwpdc 0.453 W Secondary winding design = 46 maximum peak secondary winding current Is max nTIp max 2.81 A 2 47 cross-sectional area of the secondary winding wire = Is max 0.46 mm Aws Jm 48 number of strands in the secondary winding = Aws 3.63 Ss Awsi pick 4

N 49 number of turns of the secondary winding = p 19.88 Ns nT pick 20

= 50 total length of the secondary winding wire Iws NslT 121 cm pick 130 cm Secondary winding losses 51 secondary strand DC and low-frequency resistance = Rwdc 0.174 V Rwsdcs lws lw 52 secondary winding wire DC and low-frequency = Rwsdcs 43.71 mV Rwsdc resistance Ss = 2 53 secondary winding DC and low-frequency power loss Pwsdc RwsdcIO max 0.068 W 54 secondary winding harmonic loss factor FRsh (Use (26) @ VImin, POmax) 5.19 = 55 secondary winding total power loss Pws FRshPwsdc 0.354 W Total winding losses = + 56 total DC and low-frequency power loss in both Pwdc Pwpdc Pwsdc 0.155 W windings = + 57 total winding power loss Pw Pwp Pws 0.808 W Core losses m A N2 58 air-gap length = 0 c p − lc 0.26 mm lg Lp mrc

59 maximum peak magnetic flux density m0NpIp max 0.25 T B = pk l + (l /m ) g c rc m N Di 60 magnetic flux density AC component = 0 p Lp 0.024 T Bm + lg (lc/mrc) 2 = a b 3 61 core power loss density Pv k(fs in kHz) (10Bm) 2.07 mW/cm = 62 core loss Pc VcPv 10.99 mW Total losses, temperature rise and efficiency = 63 total transformer power loss Pcw PcPw 0.819 W 2 64 total core surface area At 31.95 cm 65 surface power loss density P 0.025 W/cm2 c = cw At 66 temperature rise of the transformer 21.798C DT = 450c0.826 (N s + N )A 67 window utilisation factor (recalculated) = p p s wsi 0.35 Ku Wa P 68 transformer efficiency at full power and VImin = O 97.34% ht + PO Pcw

∗ Assume converter efﬁciency h ¼ 0.9 and Dmax ¼ 0.5

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 367 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011 www.ietdl.org

Table 2 List of symbols

A winding conductor thickness normalised with respect to conductor skin depth 2 Ac transformer core cross-sectional area in m

ACu total wire cross-sectional area in the core window in m2 4 Ap area product in m

Awp bare wire cross-sectional area of the primary winding in m2

Aws bare wire cross-sectional area of the secondary winding in m2

Bpk peak magnetic ﬂux density in T

Bs saturation magnetic ﬂux density in T D converter duty cycle

FR winding AC resistance factor

FRph primary winding harmonic loss factor

FRsh secondary winding harmonic loss factor

ILm magnetising inductance current in A

Ip primary winding current in A

Is secondary winding current in A 2 Jm winding wire maximum current density in A/m

Ku window utilisation factor

Lm magnetising inductance of the transformer in H

Lp inductance of the primary winding in H

Ls inductance of the secondary winding in H

Np number of turns of the primary winding of the transformer

Ns number of turns of the secondary winding of the transformer

Pwp primary winding power loss in W

Pws secondary winding power loss in W T time period of one switching cycle of the converter in s 2 Wa core window area in m

Wm maximum energy stored in the magnetic field of the transformer in J dw skin depth of the winding conductor in mm l magnetic flux linkage in Wb-turns w magnetic flux in Wb

The Fourier series of the current waveform in the secondary winding is

1 − = + sin[np(1 D)] is Isdc 1 2 − cos (nvst) (23) n=1 np(1 D)

Fig. 3 Winding power losses as functions of DC input voltage VI and output power PO

a Primary winding loss Pwp b Secondary winding loss Pws c Total winding loss Pw

and the amplitude of the fundamental component and the harmonics of the secondary winding current waveform for CCM is

= 2Is max − = 2Isdc − Isn sin[np(1 D)] sin[np(1 D)] Fig. 2 Primary winding harmonic loss factor FRph and secondary np np(1 − D) winding harmonic loss factor F as functions of the DC input Rsh (24) voltage VI

368 IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-pel.2010.0040 www.ietdl.org

Fig. 4 Core loss, total transformer loss, transformer temperature rise and transformer efficiency as functions of DC input voltage VI and output power PO a Core loss PC b Total transformer power loss Pcw c Flyback transformer temperature rise DT d Transformer efficiency ht of the flyback converter in CCM

The power loss in the secondary winding for CCM is In (26), FRsn can be computed using (27) for the secondary winding and using (18), (19) and (20) for rectangular, 1 2 R I 2 square or round secondary winding wire, respectively. P = R I 2 1 + wsn sn ws wsdc sdc 2 R I n=1 wsdc sdc 4 HF transformer design for a flyback 2 2 converter in CCM = + 1 Isn Pwsdc 1 FRsn 2 n=1 Isdc The above theory will be illustrated by the case study of the transformer used in the flyback converter operating in 1 2 sin [np(1 − D)] CCM. The following specifications of the flyback converter = P 1 + 2 F wsdc Rsn − are used: DC input voltage V ¼ 50 + 10 V, DC output n=1 np(1 D) I voltage VO ¼ 24 V, maximum output power POmax ¼ 30 W, = PwsdcFRsh (25) minimum output power POmin ¼ 5 W and switching frequency fs ¼ 100 kHz. Using the procedure for the flyback = where the secondary winding DC power loss Pwsdc converter design [33], a step-by-step design procedure of the 2 RwsdcIsdc. The secondary winding harmonic loss factor is flyback transformer is given in Table 1. A list of symbols is defined as the total secondary winding DC and AC power given in Table 2.InTable 1, steps 1–4 are the flyback loss normalised with respect to the secondary winding DC converter specifications. The design equations for the power loss given by transformer used in flyback converters for CCM operation are given in steps 5–14 [33]. Steps 15–19 are used in the P 1 sin[np(1 − D)] 2 calculation of area product. In steps 15 and 16, the values of F = ws = 1 + 2 F (26) window utilisation factor K and peak flux density B are Rsh P Rsn np(1 − D) u pk wsdc n=1 assumed to be 0.3 and 0.31 T, respectively. The typical values of Ku for transformers and inductors are 0.3 and 0.4, where respectively. The typical range of values of saturation flux density Bs for ferrite core material at high temperatures is R √ = wsn = + = 0.3–0.35 T. In step 17, the value of current density Jm FRsn Fsn Fpn A n 2 Rwsdc is assumed to be 6 A/mm . The typical range of Jm for power √ √ converter applications is 0.1–8 A/mm2. Using the value + × sinh(2A√n) sin(2A √n) of area product calculated in step 19 and using the − cosh(2A n) cos(2Ap n) manufacturer’s datasheets, the parameters of the selected core √ √ are listed in steps 20–29. In order to mitigate the influence of 2(N 2 − 1) sinh(A n) − sin(A n) skin effect, several strands of copper wires are used to make + l √ √ (27) the primary and secondary winding wires. The diameter of the 3 cosh(A n) + cos(A n) bare strand is taken to be twice the value of skin depth dw.

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 369 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011 www.ietdl.org

Fig. 5 Spectra of primary and secondary currents, winding resistances and winding power losses a Amplitudes of the fundamental component and the harmonics of the flyback transformer primary winding current in CCM at minimum DC input voltage VI and maximum output power PO b Spectrum of the primary winding AC resistance of the flyback transformer c Spectrum of the primary winding power loss of the flyback transformer at minimum DC input voltage VI and maximum output power PO d Amplitudes of the fundamental component and the harmonics of the flyback transformer secondary winding current in CCM at minimum DC input voltage VI and maximum output power PO e Spectrum of the secondary winding AC resistance of the flyback transformer f Spectrum of the secondary winding power loss of the flyback transformer at minimum DC input voltage VI and maximum output power PO

The value of skin depth dw which is a function of converter 64–66 are used to estimate the temperature rise of the switching frequency fs is calculated in step 30. The parameters transformer. In step 66, an empirical formula is used to of the selected strand wire are listed in steps 31–36. estimate the rise in temperature of the transformer [30].The Equations in steps 37–40 and 46–50 are used to determine assumption of the value of Ku in step 15 is rechecked in step the number of turns and the number of strands of primary and 67. The efficiency of the transformer estimated at full power secondary windings, respectively. Steps 41–45 and 51–55 and minimum DC input voltage is given in step 68. are used to estimate the DC resistances and the power losses of primary and secondary windings, respectively. The total DC power loss and the total winding power loss are estimated 5 Characteristics of a HF transformer for using steps 56 and 57, respectively. Steps 58–62 are used to flyback converters in CCM estimate the transformer core loss. In step 61, the core power loss density is estimated using the Steinmetz equation In this section, the computed characteristics of the designed [40, 41]. Step 63 gives the total transformer power loss. Steps flyback transformer are presented for a wide range of

370 IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-pel.2010.0040 www.ietdl.org

Fig. 6 Spectra of primary and secondary currents and winding power losses a Amplitudes of the fundamental component and the harmonics of the flyback transformer primary winding current in CCM at maximum DC input voltage VI and maximum output power PO b Spectrum of the primary winding power loss of the flyback transformer at maximum DC input voltage VI and maximum output power PO c Amplitudes of the fundamental component and the harmonics of the flyback transformer secondary winding current in CCM at maximum DC input voltage VI and maximum output power PO d Spectrum of the secondary winding power loss of the flyback transformer at maximum DC input voltage VI and maximum output power PO

Fig. 7 Simulation results of the amplitudes of the fundamental component and the harmonics of the flyback transformer winding currents in CCM a Converter operating at minimum DC input voltage VImin and maximum output power POmax. Top trace: Ipn for primary winding current waveform. Bottom trace: Isn for secondary winding current waveform b Converter operating at maximum DC input voltage VImax and maximum output power POmax. Top trace: Ipn for primary winding current waveform. Bottom trace: Isn for secondary winding current waveform operating conditions of the flyback converter in CCM, that is, and the width of the primary winding current waveform is over the entire range of the load current and the DC input low and contains many significant harmonics. It is also seen voltage. Fig. 2 shows the primary winding harmonic loss that the maximum value of FRsh occurs at minimum input factor FRph and the secondary winding harmonic loss factor voltage. This is because at minimum input voltage, the duty FRsh as functions of input voltage VI computed using (14), cycle is high and the width of the secondary winding (15), (20), (26) and (27) at Nl ¼ 2, d/dw1 ¼ 1.9378, current is low and contains many significant harmonics. For d/p ¼ 0.8 and n ¼ 100. It is clearly seen that the maximum CCM operation, FRph and FRsh are independent of the load value of FRph occurs at maximum input voltage. This is current. The value of FRph and FRsh at the minimum input because at maximum input voltage, the duty cycle is low voltage for a specific case of the CCM flyback transformer

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 371 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011 www.ietdl.org is predicted to be 5.195. Using FRph and the primary winding at full load, obtained from Saber Sketch simulation. The DC power loss obtained in step 45, primary winding loss Pwp theoretical values of the amplitudes of the fundamental as functions of input voltage and output power is plotted in component and the harmonics of the primary and secondary Fig. 3a. The maximum primary winding power loss occurs current waveforms of the flyback transformer predicted in at full load and minimum input voltage. Using FRsh and the Figs. 5 and 6 were in excellent agreement with the secondary winding DC power loss obtained in step 55, simulated waveforms of Fig. 7. secondary winding loss Pws as functions of input voltage Measurement of the winding losses is a challenging and output power is depicted in Fig. 3b. The maximum problem, especially with the current harmonics present in secondary winding power loss occurs at full load and the primary and the secondary windings. A major difficulty minimum input voltage. The plot of total winding power is to separate the winding losses and the core loss. Also, the loss Pw of the transformer as functions of input voltage and other component losses have to be considered while output power is shown in Fig. 3c. The maximum total measuring the converter efficiency. Therefore the theory winding power loss occurs at full load and minimum input proposed in this paper is validated by presenting the voltage. experimental results of the winding DC and AC resistances The variation of core loss PC as a function of input measured at the fundamental frequency and several voltage is shown in Fig. 4a. The core loss is calculated harmonics for the windings of the constructed transformer using steps 12, 29 and 58–62 in Table 1. The total power along with simulation results depicting the spectra of loss Pcw of the transformer as functions of input voltage currents in primary and secondary windings of the two- and output power is shown in Fig. 4b. The maximum total winding transformer used in a flyback converter. power loss of the transformer occurs at full load and minimum input voltage. Fig. 4c shows the temperature rise DT in the flyback transformer as functions of input voltage 7 Conclusions and output power. The plot of transformer efficiency ht as functions of input voltage and output power is Expressions for the primary and secondary winding power shown in Fig. 4d. As expected, the minimum efficiency of losses for periodic non-sinusoidal transformer current the transformer occurs at full load and minimum input waveforms have been derived. A transformer has been voltage. designed for a flyback converter in CCM and its Figs. 5a–c show the spectra of the primary winding performance has been evaluated as functions of the output current, the primary winding AC resistance and the primary power and DC input voltage. The resistances of both winding power loss computed using (11), (14) and (13), windings have been measured at selected frequencies. It has respectively, for the flyback transformer in CCM at full been shown that the primary winding harmonic loss factor load and minimum input voltage. Figs. 5d–f shows the FRph increases as the DC input voltage VI increases because spectra of the secondary winding current, the secondary the duty cycle decreases with increase in VI. In contrast, the winding AC resistance and the secondary winding power secondary winding harmonic loss factor FRsh decreases as loss calculated using (24), (27) and (25), respectively, for the DC input voltage increases because the width of the the flyback transformer in CCM at full load and minimum secondary current increases with increase in VI. Both these input voltage. Figs. 6a and b show the spectra of the factors are independent of the output power. The winding primary winding current and the primary winding power power losses caused by the harmonics of the primary and loss computed using (11) and (13), respectively, for the secondary currents are not negligible for CCM operation of flyback transformer in CCM at full load and maximum isolated PWM converters. This is because the spectra of the input voltage. Figs. 6c and d show the spectra of the winding currents are very rich as opposed to the inductor secondary winding current and the secondary winding current spectrum in non-isolated PWM converters. The power loss using (24) and (25), respectively, for the flyback maximum power loss in each winding occurs at the transformer in CCM at full load and maximum input minimum DC input voltage VI and the maximum output voltage. The winding resistance is independent of both the power PO for the CCM flyback converter. A similar input voltage and the load current. procedure to evaluate the winding power losses in transformers can be used for other isolated converters. 6 Experimental and simulation results The designed transformer was built and the winding DC and 8 References AC resistances were measured at selected frequencies using 1 Tan, G.H., Wang, J.Z., Ji, Y.C.: ‘Soft-switching flyback inverter with Hewlett Packard 4275A multi-frequency LCR meter. The enhanced power decoupling for photovoltaic applications’, IET Proc. AC resistances of the primary winding of the transformer Electr. Power Appl., 2007, 1, (2), pp. 264–274 measured at 100, 200, and 400 kHz were 0.386, 0.861 and 2 Lai, C.M., Shyu, K.K.: ‘A single-stage AC/DC converter based on zero voltage switching LLC resonant topology’, IET Proc. Electr. Power 1.588 V, respectively, and were in good agreement with the Appl., 2007, 1, (5), pp. 743–752 theoretical values presented in Fig. 5b. The AC resistances 3 Wai, R.J., Lin, C.Y., Liu, L.W., Chang, Y.R.: ‘High efficiency single- of the secondary winding of the transformer measured at stage bidirectional with multiple-input power sources’, IET Proc. 100, 200, and 400 kHz were 0.275, 0.706, and 1.404 V, Electr. Power Appl., 2007, 1, (5), pp. 763–777 respectively, and were in fairly good agreement with the 4 Lin, B.R., Huang, C.L., Li, M.Y.: ‘Novel zero voltage switching dual- switch forward converter with ripple current cancellation’, IET Proc. theoretical values presented in Fig. 5e. Electr. Power Appl., 2007, 1, (5), pp. 799–807 The parameters extracted from the constructed two- 5 Lin, B.R., Huang, C.L.: ‘Zero voltage switching active clamp buck- winding flyback transformer were used in the transformer boost stage C´ uk converter’, IET Proc. Electr. Power Appl., 2007, 1, for the simulation of a flyback converter in CCM. Figs. 7a (2), pp. 173–182 6 Bennett, E., Larson, S.C.: ‘Effective resistance to alternating currents of and b show the spectra of the primary and secondary multilayer windings’, Trans. AIEE, 1940, 59, pp. 1010–1016 winding currents for minimum and maximum DC input 7 Dowell, P.L.: ‘Effect of eddy currents in transformer windings’, IEE voltages, respectively, for the flyback transformer in CCM Proc., 1966, 113, (8), pp. 1387–1394

372 IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 & The Institution of Engineering and Technology 2011 doi: 10.1049/iet-pel.2010.0040 www.ietdl.org

8 Perry, M.P.: ‘Multiple layer series connected winding design for 26 Bartoli, M., Reatti, A., Kazimierczuk, M.K.: ‘Modeling iron- minimum losses’, IEEE Trans. Power Appar. Syst., 1979, 98, (1), powder inductors at high-frequencies’. IEEE Proc. Industry pp. 116–123 Applications Society Annual Meeting, Denver, USA, October 1994, 9 Venkatraman, P.S.: ‘Winding eddy current losses in switch mode power pp. 1225–1232 transformers due to rectangular wave currents’. Proc. POWERCON 11, 27 Bartoli, M., Noferi, N., Reatti, A., Kazimierczuk, M.K.: ‘Modeling Section A-1, 1984, pp. 1–11 winding losses in high-frequency power inductors’, J. Circuits, Syst. 10 Carsten, B.: ‘High frequency conductor losses in switchmode Comput., 1995, 5, (4), pp. 607–626 magnetics’. Proc. HPFC, May 1986, pp. 155–176 28 Bartoli, M., Noferi, N., Reatti, A., Kazimierczuk, M.K.: ‘Modeling litz- 11 Jongsma, J.: ‘Minimum loss transformer windings for ultrasonic wire winding losses in high-frequency power inductors’. IEEE Power frequencies–part 2: transformer winding design’. Philips Electronics Electronics Specialists Conf. Record, 1996, pp. 1690–1696 Application Bulletin E. A. B. vol. 35, pp. 211–226 29 Grandi, G., Kazimierzcuk, M.K., Massarini, A., Reggiani, U., 12 Jongsma, J.: ‘High-frequency ferrite power transformer and choke Sancineto, G.: ‘Model of laminated iron-core inductors’, IEEE Trans. design–part 3: transformer winding design’. Philips Electron – Comp. Magn., 2004, 40, (4), pp. 1839–1845 and Materials Tech. vol. 207, pp. 1–28 30 McLyman, W.T.: ‘Transformer and inductor design handbook’ (Marcel 13 Vandelac, J., Ziogas, P.D.: ‘A novel approach for minimizing high- Dekker, 2004, 3rd edn.) frequency transformer copper losses’, IEEE Trans. Power Electron., 31 Kazimierczuk, M.K., Sekiya, H.: ‘Design of AC resonant inductors 1988, 3, (3), pp. 166–176 using area product method’. Proc. IEEE First Energy Conversion 14 Rosa, J.: ‘Calculation of flux linkages in multiwinding transformers’. Conf., 2009, pp. 994–1001 IEEE Power Electronics Specialists Conf. Record, 1986, 86CH2310-1, 32 Sekiya, H., Kazimierczuk, M.K.: ‘Design of RF-choke inductors using pp. 639–644 core geometry coefficient’. Proc. Electrical Manufacturing and Coil 15 Coonrod, N.R.: ‘Transformer computer design aid for higher frequency Winding Conf., 2009 switching power supplies’. IEEE Power Electronics Specialists Conf. 33 Kazimierczuk, M.K.: ‘Pulse-width modulated DC–DC power Record, 1984, 84CH2000-8, pp. 257–267 converters’ (John Wiley & Sons, 2008) 16 Hurley, W.G., Gath, E., Breslin, J.G.: ‘Optimizing the AC resistance of 34 Murthy-Bellur, D., Kazimierczuk, M.K.: ‘Two-switch flyback PWM multilayer transformer windings with arbitrary current waveforms’, DC–DC converter in discontinuous-conduction mode’, Int. J. Circuit IEEE Trans. Power Electron., 2000, 15, (2), pp. 369–376 Theory Appl., (To be published), Early view published online: 17 Urling, A.M., Niemela, V.A., Skutt, G.R., Wilson, T.G.: ‘Characterizing April 2010, DOI: 10.1002/cta.672, http://onlinelibrary.wiley.com/doi/ high-frequency effects in transformer windings – a guide to several 10.1002/cta.672/pdf significant articles’. IEEE APEC, March 1989, pp. 373–385 35 Murthy-Bellur, D., Kazimierczuk, M.K.: ‘Two-switch flyback PWM 18 Kassakian, J.G., Schlecht, M.F.: ‘High-frequency high-density DC–DC converter in continuous-conduction mode’, Int. J. Circuit converters for distributed power-supply systems’, Proc. IEEE, 1988, Theory Appl., (To be published), Early view published online: 76, pp. 362–376 July 2010, DOI: 10.1002/cta.690, http://onlinelibrary.wiley.com/doi/ 19 Williams, R., Grant, D.A., Gowar, J.: ‘Multielement transformers for 10.1002/cta.690/pdf switched-mode power-supplies: toroidal designs’, IEE Proc., B, 1993, 36 Murthy-Bellur, D., Kazimierczuk, M.K.: ‘Harmonic winding loss in 140, (2), pp. 152–160 buck DC–DC converter for discontinuous conduction mode’, IET 20 Cheng, K.W.E., Evans, P.D.: ‘Calculation of winding losses in high- Power Electron., 2010, 3, (5), pp. 740–754 frequency toroidal inductors using multistrand conductors’, IEE Proc., 37 Murthy-Bellur, D., Kazimierczuk, M.K.: ‘Winding losses caused by B, Electric Power Appl., 1995, 142, (5), pp. 313–322 harmonics in high-frequency flyback transformers for pulse-width 21 Cheng, K.W.E., Evans, P.D.: ‘Calculation of winding losses in high- modulated dc–dc converters in discontinuous conduction mode’, IET frequency toroidal inductors using single strand conductors’, IEE Power Electron., 2010, 3, (5), pp. 804–817 Proc. B, Electric Power Appl., 1994, 141, (2), pp. 52–62 38 Kondrath, N., Kazimierczuk, M.K.: ‘Inductor winding loss owing to 22 Evans, P.D., Chew, W.M.: ‘Reduction of proximity losses in coupled skin and proximity effects including harmonics in non-isolated pulse- inductors’, Proc. Inst. Elect. Eng. B, 1991, 138, (2), pp. 51–58 width modulated dc–dc converters operating in continuous conduction 23 Hanselman, D.C., Peake, W.H.: ‘Eddy-current effects in slot-bound mode’, IET Power Electron., 2010, 3, (6), pp. 989–1000 conductors’, Proc. Inst. Elect. Eng. B, Electr. Power Appl., 1995, 142, 39 Kazimierczuk, M.K.: ‘High-frequency magnetic components’ (John pp. 131–136 Wiley & Sons, 2009) 24 Hurley, W.G., Wo¨lfle, W., Breslin, J.G.: ‘Optimized transformer design: 40 Steinmetz, C.P.: ‘On the law of hysteresis’, AIEE Trans., 1892, 9, inclusive of high frequency effects’, IEEE Trans. Power Electron., pp. 3–64 1998, 13, (4), pp. 651–659 41 Magnetics incorporated: ‘Curve fit equations for ferrite material’. 25 Petkov, R.: ‘Optimized design of a high-power high-frequency Technical Bulletin FC-S7, available at http://www.mag-inc.com/ transformer’, IEEE Trans. Power Electron., 1996, 11, (1), pp. 33–42 design/technical-documents, accessed July 2009, pp. 1–6

IET Power Electron., 2011, Vol. 4, Iss. 4, pp. 363–373 373 doi: 10.1049/iet-pel.2010.0040 & The Institution of Engineering and Technology 2011