Computer-Aided Design and Optimization of High … et al.: COMPUTER-AIDED DESIGN AND OPTIMIZATION OF HIGH-EFFICIENCY LLC SERIES RESONANT CONVERTER 3245 Fig. 1. Half-bridge LLC series

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 7, JULY 2012 3243

Computer-Aided Design and Optimization ofHigh-Efficiency LLC Series Resonant Converter

Ruiyang Yu, Godwin Kwun Yuan Ho, Bryan Man Hay Pong, Senior Member, IEEE,Bingo Wing-Kuen Ling, Senior Member, IEEE, and James Lam, Senior Member, IEEE

Abstract—High conversion efficiency is desired in switch modepower supply converters. Computer-aided design optimization isemerging as a promising way to design power converters. In thiswork a systematic optimization procedure is proposed to optimizeLLC series resonant converter full load efficiency. A mode solvertechnique is proposed to handle LLC converter steady-state solu-tions. The mode solver utilizes numerical nonlinear programmingtechniques to solve LLC-state equations and determine operationmode. Loss models are provided to calculate total component lossesusing the current and voltage information derived from the modesolver. The calculated efficiency serves as the objective function tooptimize the converter efficiency. A prototype 300-W 400-V to 12-V LLC converter is built using the optimization results. Details ofdesign variables, boundaries, equality/inequality constraints, andloss distributions are given. An experimental full-load efficiencyof 97.07% is achieved compared to a calculated 97.4% efficiency.The proposed optimization procedure is an effective way to designhigh-efficiency LLC converters.

Index Terms—Computer-aided design, efficiency, LLC resonantconverter, optimization, power converter.

NOMENCLATURE

a, b, c Curve fitting factor.aDF , bDF Curve fitting factor.Ae Effective cross-sectional area of transformer.Ae Lr Effective cross-sectional area of resonant

inductor.bxl Lower bound vector of design variables.bxu Upper bound vector of design variables.Bm X F Peak-to-peak swing of transformer flux

density.ΔBm X F Amplitude of transformer flux density swing.ΔBm Lr Amplitude of resonant inductor flux density

swing.Cr Value of resonant capacitor.

Manuscript received June 22, 2011; revised August 26, 2011 and October12, 2011; accepted November 23, 2011. Date of current version April 3, 2012.Recommended for publication by Associate Editor D. Xu.

R. Yu, G. K. Y. Ho, and B. M. H. Pong are with the Department of Electricaland Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail:[email protected]; [email protected]; [email protected]).

B. W.-K. Ling is with the School of Engineering, University of Lincoln,Lincolnshire, LN6 7TS, U.K. (e-mail: [email protected]).

J. Lam is with the Department of Mechanical Engineering, The University ofHong Kong, Hong Kong (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2011.2179562

dAWG Diameter of AWG wire in transformerprimary winding.

dLr AWG Diameter of AWG wire in resonant inductorwinding.

DF Dissipation factor.Eoff Turn-off energy consumed by primary

MOSFET.f Frequency.fr Resonant frequency (LrCr ).fs Switching frequency.F Normalized frequency.FR Ratio of AC–DC resistance.FR pri Transformer primary side FR.

FR sec Transformer secondary side FR.

FRn Lr Resonant inductor FRn.

hfoil Thickness of foils in transformer secondarywinding.

iLr Resonant inductor current.Ibase Base current for normalization.In pri nth harmonic component of primary RMS

current.In sec nth harmonic component of secondary RMS

current.ILr MAX Maximum resonant inductor current.Ioff Turn-off current of primary MOSFET.Irip in Input ripple current.Irip out Output ripple current.IRMS pri Primary side RMS current.IRMS sec Secondary side RMS current.jLr Normalized resonant inductor current.jLm Normalized magnetizing current.jout Normalized output current.jrec Normalized secondary rectified current.k Steinmetz coefficient.k1 Ratio of two resonant frequencies.koff Ratio of turn-off energy and turn-off voltage.Lr Resonant inductor value.Lm Nagnetizing inductor value.m1 ,m2 Normalized input/output voltage.mc Normalized resonant capacitor voltage.mLr Normalized resonant inductor voltage.mm Normalized transformer voltage.mm2 Normalized transformer voltage (mode

indicator).M Normalized conversion ratio.n Order of harmonic frequency.nC in Number of input capacitor parallelled.

0885-8993/$26.00 © 2011 IEEE

3244 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 7, JULY 2012

nC out Number of output capacitor parallelled.nlayer Number of layers in transformer primary

winding.nLr Number of turns in resonant inductor.nLr layer Number of layers in resonant inductor.nsample Number of samples in each switching cycle.np Number of transformer primary turns.ns Number of transformer secondary turns.p Number of layers in magnetic component

winding.Pcd pri Conduction loss of primary MOSFET.Pcd SR Conduction loss of synchronous rectifier.Pcu X F pri Copper loss of transformer primary side.Pcu X F sec Copper loss of transformer secondary side.Pcu Lr Copper loss of resonant inductor.Pcore X F Core loss of transformer.Pcore Lr Core loss of resonant inductor.PC r Loss of resonant capacitor.PC in Loss of input capacitor.PC out Loss of output capacitor.Pg SR Gate-drive loss of synchronous rectifier.Pg pri Gate-drive loss of primary MOSFET.Ploss Sum of all losses.Pout Output power.Psw pri Loss of primary MOSFET.Psw SR Switching-loss of synchronous rectifier.Qg pri Gate-drive charge of power MOSFET.Qg SR Gate-drive charge of secondary SR.Qoss SR Output capacitance charge of secondary SR.rL Normalized value of output load.RC r ESR of resonant capacitor.Rds pri On-state resistance of primary MOSFET.Rds SR On-state resistance of synchronous rectifier.RL Value of output load.RLr DC resistance of resonant inductor.RX F pri Transformer DC resistance of the primary

side.RX F sec Transformer DC resistance of the secondary

side.t Time.vc Voltage of resonant capacitor.vLm Voltage of magnetizing inductor.V1 Input voltage of LLC equivalent circuit.V2 Output voltage of LLC equivalent circuit.Vbase Base voltage in normalization.Vg pri Gate-drive voltage of primary MOSFET.Vg SR Gate-drive voltage of synchronous rectifier.Ve Lr Volume of inductor core.Ve X F Volume of transformer core.Vin Input voltage.Vout Output voltage.x Vector of design variables.Zbase Base impedance.α, β Normalized angles.αcore , βcore Steinmetz coefficients.γ Normalized half period of switching cycle.θ Normalized angle (time based).

λ Ratio of Lr/Lm .δ(n) Skin depth of nth harmonic frequency.ρcu Electrical resistivity of copper.μ0 Vacuum permeability.Ω Optimization constraint set.ω0 Resonant frequency (LrCr ).ω1 Resonant frequency (Lr + LM Cr ).

I. INTRODUCTION

ENERGY efficiency is a hot topic that has drawn the atten-tion of researchers and engineers for decades. Numerous

research works have focused on improving power converter effi-ciency. Computer-aided design optimization is one of the meth-ods used to achieve high-energy conversion efficiency, and it hasbeen applied widely in conventional PWM converter design.Early research work [1] utilized the sequential unconstrainedminimization technique (SUMT) or the augmented Lagrangian(ALAG) penalty function technique to optimize the convertermass. A practical converter optimization approach was devel-oped in [2] for industrial applications, which utilized the nonlin-ear optimization program to optimize converter design. Designoptimization of interleaved converter for automobile applica-tions was investigated in [3]. A Monte Carlo searching methodwas applied to handle a large number of design variables. Fuelcell system mass was minimized in [4] under a certain dura-tion constraint. The Pareto-front of power converter multiobjectoptimization was investigated by [5]. The Pareto-front of con-verter volume and efficiency were obtained, which means nofurther efficiency improvement can be achieved under certainconstraints, such as converter volume or mass. Converter vol-ume and efficiency were included in the weighted objectivefunction to determine the degree of optimized efficiency or vol-ume. The Pareto-front curve of power density versus efficiencyshowed that the optimized efficiency was limited by a certainvolume constraint. A similar optimization approach was appliedto phase-shift PWM converter design in [6] to achieve 99% ef-ficiency.

LLC series resonant converter is emerging to meet thehigh-efficiency requirements of offline converter and it is be-coming increasingly popular in industrial applications. It haszero-voltage switching (ZVS) at primary side and zero-currentswitching (ZCS) at secondary side. Design methodology of 1-MHz 1-kW LLC converter was investigated in [7]. Details ofdesign procedure were presented in a digital control LLC con-verter [8]. The LLC converter efficiency can be further im-proved by using synchronous rectifiers [9]–[11]. Actually, it ismore sensible to design and optimize the LLC converter andsynchronous rectifiers as an entire system. Adaptive controlmethodology was proposed to improve the performance of LLCconverter [12]. The application of LLC converter in photovoltaic(PV) system was developed in [13]. The advantage of high effi-ciency from light load to full load shows performance improve-ments of the entire PV system. Design procedures for wide rangeLLC converter and dead-time of LLC converter were presentedin [14], [15], respectively.

YU et al.: COMPUTER-AIDED DESIGN AND OPTIMIZATION OF HIGH-EFFICIENCY LLC SERIES RESONANT CONVERTER 3245

Fig. 1. Half-bridge LLC series resonant DC/DC converter.

So far, there is minimal work on the optimization of LLCconverter. The optimization of LLC converter is more difficultthan conventional PWM converters. This is because of the fol-lowing reasons: First, there are multiple modes of operations;each mode has different resonant characteristics. Second, thenonlinear behavior of LLC converter does not have closed-formsolutions.

One of the conventional methods used to predict LLC op-eration behavior is the fundamental harmonic approximationmethod [16]. However, this method only considers the fun-damental frequency harmonic and produces errors when theswitching frequency is not at resonant frequency. An improvedLLC model was proposed in [17] to present more accurate wave-forms. The key equations were solved by numerical method,but this LLC model still assumed that the resonant current issinusoidal. The steady-state solutions based on state-variableequation were developed in [18]. This method can accuratelypredict LLC resonant behaviors. However, the nonlinear equa-tions do not have closed-form solutions. With the developmentof numerical computational techniques, the present researchwork utilizes nonlinear programming techniques to solve LLCconverter steady-state equations. A mode solver is proposed toaccurately predict LLC resonant behaviors. Such mode solveris a numerical procedure that considers LLC resonances at dif-ferent modes. Hence, the proposed mode solver is suitable forhandling LLC design variables.

Loss models are presented to predict converter losses thatserve as the objective function to optimize LLC efficiency. Aprototype 400 V to 12 V/25 A LLC converter is built to verifyoptimization results. The measured efficiency of optimized LLCconverter is 97.07% at full load.

II. LLC CONVERTER MODELS

A. LLC Mode Solver

A mode solver is proposed to compute the multiple-modesteady-state operation of the LLC converter. The half-bridgeLLC converter topology is shown in Fig. 1. There are two res-onant inductors and a resonant capacitor in the resonant tank.Hence, the name LLC represents these three resonant elements.Power MOSFETs are applied as the half-bridge switches, whichare operated in complementary manner with nearly 50% duty.Output voltage is regulated by variable frequency control. Adead-time is applied during the transition of switching to achievezero voltage switching and to avoid cross conduction from high-side to low-side switches.

The proposed LLC mode solver serves as a function block inthe main optimization procedure. The input variables of the LLCsteady-state solver are the values of resonant parameters, suchas Lr , Cr , and Lm and the excitations, such as the switchingfrequency, load, and input/output voltage. The state equationsare solved numerically and the output of this function block arevectors containing particular waveform information of currentand voltage. The LLC converter has several modes of opera-tion. These modes include the continuous conduction mode be-low or above resonance, discontinuous conduction mode belowor above resonance, and cut-off mode. Continuous conductionmode is defined as a state in which the secondary diode con-ducts throughout the switching cycle. Discontinuous conductionmode is defined as the state in which secondary diode has cer-tain periods not conducting. The mode solver presented cantackle different modes, which are determined by the nonlinearrelationship of the switching frequency, load, and input/outputvoltage. The detailed procedures of the LLC mode solver are inFig. 2.

B. Normalization

The solver procedures start with normalization, as shown inFig. 2 (a1) and (a2). The resonant characteristics of the tankcircuit are normalized for the sake of uniformity. We use ω0 andω1 to denote the two resonant frequencies

ω0 =1√

LrCr

= 2πfr , ω1 =1

√(Lr + Lm )Cr

. (1)

The operation angle θ is given by

θ = ω0t. (2)

Denote F the ratio of two frequencies

F =fs

fr. (3)

A half period of switching cycle γ is defined by

γ =ω0

2fs=

π

F. (4)

The conversion ratio M is defined as

M =V2

V1. (5)

We define some normalized parameters in the following :

Vbase = V2 =np

nsVout ,m2 =

V2

Vbase= 1 (6)

m1 =1M

(7)

Zbase =√

Lr

Cr(8)

Ibase =Vbase

Zbase(9)

where Vbase is defined as the V2 so that m2 is normalizedto unity, and m1 is the normalized input voltage. The baseimpedance Zbase and base current Ibase are given by (8) and (9),respectively.


Fig. 2. Flow chart of LLC mode solver.

The normalized voltage on resonant capacitor mc(θ) andnormalized current through resonant inductor jLr (θ) are,respectively, given by

mc(θ) =vc( θ

ω0)

Vbase(10)

jLr (θ) =iLr ( θ

ω0)

Ibase. (11)

Similar expressions are applied to mm (θ), mm2(θ), mLr (θ)jLm (θ) and jout .

The ratio of two resonant inductance λ and the ratio of tworesonant frequencies k1 are, respectively, given by

λ =Lr

Lm=

mLr (θ)mm (θ)

(12)

k1 =ω1

ω0. (13)

The normalized output load resistance rL is defined as

rL =n2

pRL

n2sZbase

. (14)

C. Operation Below Resonant Frequency

1) Discontinuous Conduction Mode Below Resonance: IfF < 1, the LLC is assumed to operate in discontinuous con-duction mode below resonance (DCMB) first, as shown inFig. 2 (a1). DCMB is one of the popular designed operationmodes. In DCMB mode, the LLC converter voltage conversionratio M is larger than unity (M > 1). Typical waveforms inDCMB mode are shown in Fig. 3(b). The equivalent circuit ofDCMB mode in θ ∈ [0, α) is shown in Fig. 4(b). The dead-timetransition is ignored for simplified analyses. The state equationsare given by (15)

θ ∈ [0, α)⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

mc(θ) = [mc(0) − 1M + 1] cos(θ) + jLr (0) sin(θ)

+ 1M − 1 (15a)

mm (θ) = 1 (15b)jLr (θ) = [−mc(0) + 1

M − 1] sin(θ)+jLr (0) cos(θ) (15c)

jLm (θ) = jLm (0) + λθ. (15d)


Fig. 3. LLC operation modes.


Fig. 4. Equivalent circuits of LLC converter.

The equivalent circuit in θ ∈ [α, γ) is shown in Fig. 4(c). Thestate equations are given by (16)

θ ∈ [α, γ)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

mc(θ) = [mc(α) − 1M ] cos[k1(θ − α)]

+ jL r (α)k1

sin[k1(θ − α)] + 1M (16a)

mm (θ) = {[−mc(α) + 1M ] cos[k1(θ − α)]

− jL r (α)k1

sin[k1(θ − α)]}/(1 + λ) (16b)jLr (θ) = [−mc(α) + 1

M ]k1 sin[k1(θ − α)]+jLr (α) cos[k1(θ − α)] (16c)

jLm (θ) = jLr (θ). (16d)

The average output current jout is given by

jout =1γ

∫ γ

0|jLr (θ) − jLm (θ)|dθ

=1γ

∫ α

0[jLr (θ) − jLm (θ)]dθ

=1γ

{[− mc(0) +

1M

− 1](1 − cos α)

+ jLr (0) sin α − jLr (0)α − 12λα2

}. (17)

The steady-state solution in DCMB [jLr (0),mc(0), α,M ] canbe solved by⎧⎪⎨

⎪⎩

mc(0) + mc(γ) = 0 (18a)jLr (0) + jLr (γ) = 0 (18b)jLr (α) − jLm (α) = 0 (18c)joutrL − 1 = 0. (18d)

These four equations become the basis of the solver, andwhich adequately describe the waveforms of the resonant oper-ation. The initial condition mc(0) is equal to −mc(γ), as shownin Fig. 3(b), given by (18a). The same reasoning can be applied tojLr (0) and −jLr (γ) in (18b). The diode stops conducting at an-gle α, where the resonant current equals to the magnetizing cur-rent. Hence, (18c) is formulated that jLr (α) = jLm (α). Finally,the unity output voltage is equal to joutrL , given by (18d). Thesefour equations have four unknowns [jLr (0),mc(0), α,M ]. Twounknowns are the normalized boundary value of resonant induc-tor current jLr (0) and resonant capacitor voltage mc(0). Thethird unknown is the normalized time α that the secondary diode

stops conducting. The fourth unknown is the conversion ratioM . Since these unknowns do not have the analytical closed-form solution, the equations are solved by MATLAB functionfsolve(x), which is a numerical-based search function.

After solving the above four equations, the following proce-dures are carried out to validate the assumption of DCMB. Atthe time when jLr (θ) = jLm (θ) (the moment θ = 0 or intervalθ ∈ [α, γ] in DCMB), the voltage on Lm determines whetherthe diodes start to conduct or not. A mode indicator mm2(θ) isdefined as the normalized voltage on Lm , based on the equiv-alent circuit Fig. 4 (c), when θ = 0 or θ ∈ [α, γ]. According toKirchhoff’s Voltage Law, we obtain

mLr (θ) + mm2(θ) + mc(θ) = m1 . (19)

The solution of mm2(θ) can be derived by inserting (7) and (12)into (19). To simplify the analyses, we only consider the instants0 and γ

mm2(θ) =−mc(θ) + 1

M

1 + λ |θ=0,γ. (20)

(θ = 0): If |mm2(0) ≥ 1| (the output voltage is normalizedto 1), the secondary diode conducts and clamps the mm (0) to 1(DCMB true).

Otherwise, if |mm2(0)| < 1, the secondary diode is OFF andit is no longer DCMB (DCMB false) but in another mode,discontinuous conduction mode above and below resonance(DCMAB), as shown in Fig. 2 (c1) and Fig. 3(c). Since theLLC converter is assumed operating at DCMB at this moment,DCMAB should be considered later.

(θ = γ): At the end of DCMB first half cycle γ, if |mm2(γ)| ≤1, diode is OFF (DCMB true). Otherwise, if |mm2(γ)| > 1, theassumption of DCMB is violated (DCMB false).

Summaries are listed as below:Flow chart Fig. 2 (b1) shows, if |mm2(0)| ≥ 1 and

|mm2(γ)| ≤ 1, the assumption of DCMB is true.If |mm2(0)| < 1 and |mm2(γ)| < 1, the assumption of

DCMB is false then LLC converter is assumed to operate atDCMAB, as shown in Fig. 2 (c1).

If |mm2(0)| > 1 and |mm2(γ)| > 1, the assumption ofDCMB is false and then LLC converter is assumed to oper-ate at DCMB2, as shown in Fig. 2 (d1).


2) Discontinuous Conduction Mode Above or Below Reso-nance: Flow chart Fig. 2 (c1) shows that the converter may op-erate at DCMAB. Procedures to solve DCMAB are presentedlater. The secondary diodes do not conduct in θ ∈ [0, α) or[β, γ) when the LLC converter operates at DCMAB, as shownin Fig. 3(c). The equivalent circuit in DCMAB mode θ ∈ [0, α)and [β, γ) is the circuit (c) of Fig. 4. The equivalent circuit inθ ∈ [α, β) is the circuit (a) of Fig. 4. Five equations are formu-lated to solve DCMAB given by

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

mc(0) + mc(γ) = 0 (21a)jLr (0) + jLr (γ) = 0 (21b)jLr (α) − jLm (α) = 0 (21c)jLr (β) − jLm (β) = 0 (21d)joutrL − 1 = 0. (21e)

Similar to the procedure in DCMB equivalent circuit, themode indicator mm2(0) and mm2(γ) can be calculated by (20).

The assumption of DCMAB is true when |mm2(0)| < 1 and|mm2(γ)| < 1, as shown in Fig. 2 (e1).

3) Other Modes Below Resonant Frequency: Two othermodes below the resonant frequency are discontinuous con-duction mode below resonance “2” (DCMB2) and continuousconduction mode below resonance (CCMB). Typical operationwaveforms of the two modes are shown in Fig. 3(e) and (f).Although these two modes are not widely designed in the LLCconverter, their operations are still included in the solver for thesake of completeness. The blocks (d1), (f1), (g1), and (h1), inthe flow chart in Fig. 2, states the procedures to solve DCMB2and CCMB.

D. Operation Above Resonant Frequency

1) Continuous Conduction Mode Above Resonance(CCMA): If the F > 1, the LLC is assumed to operate in con-tinuous conduction mode above resonance (CCMA), as shownin Fig. 2 (a2). CCMA is a popular mode in LLC converter oper-ation. In this mode, the LLC converter voltage conversion ratiois less than unity (M < 1). Typical waveforms in CCMA modeare shown in Fig. 3(a). The equivalent circuit in CCMA modein θ ∈ [0, α) is the circuit (b) of Fig. 4. The state equations aregiven by

θ ∈ [0, α)⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

mc(θ) = [mc(0) − 1M − 1] cos(θ) + jLr (0) sin(θ)

+ 1M + 1 (22a)

mm (θ) = −1 (22b)jLr (θ) = [−mc(0) + 1

M + 1] sin(θ)+jLr (0) cos(θ) (22c)

jLm (θ) = jLm (0) − λθ. (22d)

The equivalent circuit in θ ∈ [α, γ) is the circuit (a) in Fig. 4.The state equations are presented as follows:

θ ∈ [α, γ)⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

mc(θ) = [mc(α) − 1M + 1] cos(θ − α)

+jLr (α) sin(θ − α) + 1M − 1 (23a)

mm (θ) = 1 (23b)jLr (θ) = [−mc(α) + 1

M − 1] sin(θ − α)+jLr (α) cos(θ − α) (23c)

jLm (θ) = jLm (α) + λθ. (23d)

The normalized average output current is given by

jout =1γ

∫ γ

0|jLr (θ) − jLm (θ)|dθ

=1γ

∫ α

0[jLm (θ) − jLr (θ)]dθ

+1γ

∫ γ

α

[jLr (θ) − jLm (θ)]dθ

=1γ

{αjLm (0) − 1

2λα2 −

[− mc(0) +

1M

+ 1]

× (1 − cos α) − jLr (0) sin α +[− mc(α) +

1M

− 1]

× [1 − cos(γ − α)] + jLr (α) sin(γ − α)

− jLm (α)(γ − α) − 12λ(γ − α)2

}. (24)

Four equations are formulated to solve CCMA given by⎧⎪⎨

⎪⎩

mc(0) + mc(γ) = 0 (25a)jLr (0) + jLr (γ) = 0 (25b)jLr (α) − jLm (α) = 0 (25c)joutrL − 1 = 0. (25d)

Similar validation procedures are carried out. The mm2(0),mm2(α) and mm2(γ) can be calculated by (20).

If |mm2(0)| ≥ 1 and |mm2(γ)| ≥ 1, and |mm2(α)| ≥ 1, theassumption of CCMA is true, as shown in flow chart Fig. 2 (b2).

Otherwise the assumption of CCMA is false [Fig. 2 (c2)].The assumption of operating mode changes to discontinuousconduction modes above resonance (DCMA) in this case, asshown in Fig. 2 (d2).

2) Other Modes above Resonance: DCMA and DCMAB aretwo modes operating above resonant frequency. Typical opera-tion waveforms are shown in Fig. 3(c) and (d). The blocks (d2),(f2), and (g2), in the flow chart in Fig. 2, state the procedures tosolve DCMA and DCMAB equations.

Table I is presented to summarize the LLC converter operationmodes, angles, and their equivalent circuits. Table II reveals thekey characteristics used for validating the operation modes.

III. LOSS MODELS

The current waveforms of LLC converter are determined bythe operation mode and calculated by the proposed mode solver.Current harmonics are calculated to predict losses. A numericalmethod is used to sample a switching cycle with nsample points.The current harmonics are calculated by fast Fourier transform,as shown in Fig. 5.


TABLE IOPERATION MODES OF LLC CONVERTER

TABLE IIVALIDATIONS OF LLC OPERATION MODES

Fig. 5. Transformer current waveform and harmonic components.

Fig. 6. PSPICE-simulated primary MOSFET turn-off loss.

A. Primary MOSFET

The most promising feature of the LLC converter is zerovoltage switching turn-on and small turn-off current for primaryside MOSFETs. A simple and effective MOSFET switchingloss model is proposed for the prediction of turn-off switchingloss at different turn-off currents, as shown in Fig. 6(a). Thisproposed model utilizes a curve-fitting method to record SPICEsimulation results of turn-off switching loss Eoff (Ioff ). The inputvoltage is fixed at 330, 365, and 400 V in SPICE simulation,as shown in Fig. 6(a). Eoff is nearly linearly increasing withVin from 330 to 400 V at a certain turn-off current level, asshown in Fig. 6(b). The parameter koff is defined as the ratioof Eoff increasing value from 330 to 400 V divided by thevoltage increasing value 70 V. The actual energy dissipatedduring switching is Eoff (Ioff , Vin)

Eoff (Ioff ) = aebIo f f + c (26)

koff (Ioff ) =Eoff (Ioff , 400 V) − Eoff (Ioff , 330 V)

400 V − 330 V(27)

Eoff (Ioff , Vin) = Eoff (Ioff , 330 V)

+ koff (Ioff )(Vin − 330). (28)

It should be noted that during turn-off, there are two currentsflowing through the MOSFET and the total energy value is


Eoff (Ioff , Vin ). One current is to charge the output capacitanceof MOSFET to Vin with the energy Eoff (0, Vin ), the other currentproduces energy dissipation (cannot be recovered) in the MOS-FET channel with the energy Eoff (Ioff , Vin ) − Eoff (0, Vin ).During the dead-time, the energy stored in the output capac-itance of MOSFET Eoff (0, Vin ) is recovered to the input capac-itor (the drain source voltage of MOSFET drops from Vin to 0,soft switching achieved).

The switching loss and conduction loss of the high side andthe low side primary MOSFETs (assuming the same type ofMOSFETs at the high side and the low side) are denoted asPsw pri(Ioff , Vin) and Pcd pri , respectively. The gate drivingloss of primary MOSFETs is denoted as Pg pri

Psw pri(Ioff , Vin) = 2fs [Eoff (Ioff , Vin ) − Eoff (0, Vin )]

(29)

Pcd pri = I2RMS priRds pri (30)

Pg pri = Qg priVg prifs. (31)

B. Isolation Transformer

Transformer design of LLC converter is an important tasktoward achieving high efficiency. Here, sandwich winding isimplemented in order to reduce the AC resistance of the trans-former. A center tap configuration is applied at secondary withcopper foils for high current low voltage applications. Magne-tizing inductance is integrated in the isolation transformer witha certain air gap. A typical transformer structure is shown inFig. 7. The primary and secondary DC resistance RX F pri andRX F sec can be directly calculated by the winding geometry.The skin depth of the nth harmonics frequency is given by

δ(n) =√

2ρcu

2πnfsμ0. (32)

The AC-to-DC resistance ratio FR at nth harmonic frequencyis calculated by Dowell’s equation [20], [21], given by

FR (n, p,X) = Xe2X − e−2X + 2 sin(2X)e2X + e−2X − 2 cos(2X)

+ 2Xp2 − 1

3eX − e−X − 2 sin(X)eX + e−X + 2 cos(X)

. (33)

We have: FR pri(n) = FR (n, p,X) is for primary round con-

ductors with p = nlayer , X =√

πdAW G2δ(n) [21].

FR sec(n) = FR (n, p,X) is for secondary foils with p = ns

2and X = h f o i l

δ(n) .The AC copper loss at each harmonic frequency is calculated

by summing the losses from DC to 32nd harmonics. The primaryside and secondary copper losses of the transformer are givenby

Pcu X F pri = RX F pri

32∑

n=0

FR pri(n)I2n pri (34)

Fig. 7. Winding structure of transformer.

Pcu X F sec = RX F sec

32∑

n=0

FR sec(n)I2n sec (35)

where In pri and In sec denote the nth order harmonic currentat the primary side and the secondary side of the transformer.Flux swing of the half-bridge LLC converter is bidirectional.The peak-to-peak flux density is given by

Bm X F =

∫ t s2

0 |vLm (t)|dt

npAe X F. (36)

The empirical Steinmetz equation [19] is applied to calculatethe core loss of the transformer, given by

Pcore X F = Ve X F kfα c o r es ΔBβc o r e

m X F (37)

where ΔBm X F = 12 Bm X F is the flux swing, and k, αcore ,

and βcore are the Steinmetz coefficients provided by the manu-facturer [22].

C. Resonant Inductor

A separate resonant inductor is applied in the LLC converter.The separate inductor is used because it simplifies the resonancedesign process. Integrated transformer may lead to totally dif-ferent loss models, designs, and optimization procedures.

The losses in the resonant inductor are copper loss and coreloss. The DC resistance of resonant inductor is calculated ac-cording to its geometry. Dowell’s equation (33) is also applied tocalculate AC resistance. FR Lr (n) = FR (n, p,X) is for reso-nant inductor, with p = nLr layer , X =

√πdL r AW G

2δ(n) . The copperloss of resonant inductor is given by

Pcu Lr = RLr

32∑

n=0

FR Lr (n)I2n pri. (38)

Core loss of resonant inductor is given by

Pcore Lr = Ve Lrkfα c o r es ΔBβc o r e

m Lr (39)

where ΔBm Lr = Lr IL r m a xnL r Ae L r

is the flux swing of resonantinductor.


Fig. 8. Current-driven synchronous rectifiers driver.

D. Synchronous Rectifier

Synchronous rectification (SR) is implemented at the sec-ondary side to achieve high efficiency at the low-voltage high-current output condition. The current driven synchronous recti-fier driving scheme [23], [24] is implemented. The SR driver isshown in Fig. 8. We assume that the SR works under a timingscheme that current does not flow through synchronous rectifierbody diode. The major losses for the synchronous rectifier arethe conduction loss, turn-off switching loss, and the gate-driveloss. Turn-off switching loss is the energy stored in the stray in-ductance and being dissipated by the circuit [25]. The simplifiedmodel for the turn-off loss and the gate-driving loss of SR aredenoted as Psw SR and Pg SR, , respectively. The conductionloss of SRs is denoted as Pcd SR

Psw SR =nsVinQoss SRfs

2np(40)

Pg SR = Qg SRVg SRfs (41)

Pcd SR = I2RMS secRds SR . (42)

E. Capacitors

1) Resonant Capacitor: The resonant capacitor in serieswith the power path carries high RMS current and high voltage.A low-loss capacitor is used to achieve high efficiency and lowtemperature. A metalized polypropylene capacitor is selectedbecause of its low dissipation factor and low cost. Typically, thedissipation factor (or loss angle tan δ) of polypropylene capac-itor increases with the increasing of frequency up to 10 MHz.Same as before, the curve-fitting method is applied to record thedissipation factor of the resonant capacitor Fig. 9

DF = aD F

f + bD F

(43)

RC r =DF

2πfCr(44)

PC r = I2RMS priRC r (45)

Fig. 9. Dissipation factor of resonant capacitor.

where DF is the dissipation factor with fitting parameteraDF = 0.03642 and bDF = 2.611. The equivalent series re-sistance (ESR) of the resonant capacitor RC r is calculated ac-cording to the dissipation factor and the capacitance, given by(44), and the loss of resonant capacitor PC r is given by (45).

2) Input/output capacitors: Attention should be paid to theoutput capacitor selection. The output ripple current of LLC con-verter is higher than that of PWM converters (such as forward,half-bridge, and Cuk converter). The actual ripple currents arecalculated by summing frequency harmonic components (DCcomponent excluded) from fundamental frequency to 32nd fre-quency, given by

I2rip in =

32∑

n=1

I2n pri (46)

I2rip out =

32∑

n=1

I2n sec (47)

Large capacitance to volume ratio and low cost make aluminumelectrolytic capacitors a suitable choice for the input/output ca-pacitor. One has to parallel sufficient number of output capac-itors to share ripple current. Low ESR series output capacitoris preferred to avoid excessive power dissipation. Such exces-sive power dissipation results in significant life degrading. Thepower dissipations of input/output capacitors are given by

PC in = I2rip in

RC in

nC in(48)

PC out = I2rip out

RC out

nC out. (49)

IV. OPTIMIZATION PROCEDURES

An optimization procedure is presented in this section. Theoptimization program in this paper is developed under MAT-LAB environment. The LLC efficiency optimization involvesnonlinear, constrained, continuous optimization problems. Thefmincon(x) function of MATLAB optimization toolbox is ap-plied as the optimizer to solve such problems. The “active-set” algorithm is used in the fmincon(x) function. Detailed


Fig. 10. Flow chart of LLC efficiency optimization procedure.

optimization procedures can be found in [26]. The aim of theoptimization is to minimize the loss at a certain loading condi-tion. The flow chart of the optimization procedure is presentedin Fig. 10. The characteristics of the power components are dis-crete, such as the primary MOSFETs, transformer core and bob-bin size. The continuous optimization methods cannot handlesuch discrete values, so we pre-select the discrete componentsat the discrete component selection stage. In the continuousoptimization stage, the discrete components and their relatedparameters are fixed.

Let x denote a vector containing all the design variables, suchas switching frequency, primary turns, secondary turns, value ofLr , Lm , and Cr , etc.

x = [fs, np , ns, Lm ,Cr , Lr , dAWG , nlayer , hfoil,

nLr , dAWG Lr , nLr layer ] (50)

The objective function Ploss(x) is the converter loss at fullload condition. The optimization problem is to minimize theloss Ploss(x) subject to constraints set Ω, given by

minx∈Ω

Ploss(x) (51)

where Ω is given by

Ω = {x|bxl ≤≤ x ≤≤ bxu , 0.3 − ΔBm Lr ≥ 0,

0.3 − ΔBm X F ≥ 0, Vout = 12}. (52)

The lower bound vector bxl and the upper bound vector bxu ofdesign variables give the searching range, where the expression“x ≥≥ bxl” denotes “x − bxl” to be a vector with non-negativeentries. “0.3 − ΔBm Lr ≥ 0” and “0.3 − ΔBm X F ≥ 0” de-note that the resonant inductor and the transformer do not satu-rate (0.3 is assumed to be the ferrite flux saturation level).

TABLE IIIOPTIMIZED RESULTS

TABLE IVLOSSES PROFILE

The output voltage Vout given by (53) is required to satisfythe equality constraint and given by

Vout =nsMVin

2np. (53)

This means that the output voltage should be regulated at12 V. This equality constraint is nonlinearly related to manydesign variables, such as fs , np , ns , Lr , Cr , Lm , and the op-eration mode. The optimizer algorithm searches the optimumresult that satisfies the constraint set.

V. OPTIMIZATION AND EXPERIMENTAL RESULTS

A. Optimization Results

The optimization program aims to optimize a 400 V inputvoltage, 12 V output voltage, and 25 A output current LLC reso-nant converter. The optimized design variables and lower/upperbounds are presented in Table III. The lower/upper bounds arepredefined. It can be seen that some of the design variablesconverge to their boundaries. These boundaries are limited byphysical factors such as size. This table also indicates thoseboundaries that can be improved to have even higher efficiency.

The loss distributions are also presented in Table IV. Theoptimized efficiency is calculated to be 97.4%, where the calcu-lated efficiency is (output power)/[(all losses)+(output power)].The loss table indicates the loss distribution and facilitates the


Fig. 11. Calculated waveforms at optimized efficiency.

Fig. 12. Prototype converter and thermal images. (a) Prototype LLC converter.(b) Thermal image of prototype converter.

thermal design for components. The table can also direct thechoice of individual components. The calculated waveforms ofthe optimized converter are presented in Fig. 11. The optimizedLLC converter operates in CCMA and the switching frequencyis only slightly higher than resonant frequency. The results arevery similar to previous research works [7], [8] that good effi-ciency of LLC converter occurs at resonant frequency.

B. Experimental Results

A prototype LLC converter is built as shown in Fig. 12(a). Theconverter is designed to be naturally cooled. The circuit parame-ters are listed in Table V. The synchronous rectifier modules are

TABLE VCOMPONENT LIST

Fig. 13. Experimental waveform of prototype converter.

placed vertically with the heat sinks in the original design. How-ever, it is placed horizontally and the heat sinks are removed inorder to take thermal inferred images. The thermal infrared im-age of the prototype converter operating at full load 12 V 25 A(for 2 h) is also shown in Fig. 12(b). The ambient temperature is25 ◦C and there is no air flow. The prototype LLC converter op-erates at CCMA during full-load conditions, shown in Fig. 13.The Ch1 of Fig. 13 is the drain-source voltage of primary lowside MOSFET. Ch2 and Ch3 are the drain-source voltages ofsecondary SRs. Ch4 is the resonant inductor current. The mea-sured efficiency is 97.07% at full load where the efficiency iscalculated by [(output voltage)*(output current)]/[(input volt-age)*(input current)]. The average input voltage, input current,and output voltage are measured by the DC voltage meters. Theaverage output current is measured by the DC current meter ofthe electronic load. The efficiency curve of prototype converteris shown in Fig. 14. The 50%-load-efficiency is higher than97% and 20%-load-efficiency is higher than 96%. The calcu-lated full-load loss is 7.99 W and the experimental full-load lossis (input power)-(output power), which is 9.05 W. The error ofloss calculation is 11.8%, which is calculated by [(experimentalloss)-(calculated loss)]/(experimental loss).


Fig. 14. Measured efficiency of prototype converter.

VI. CONCLUSION

In this research work a systematic optimization procedure isproposed to optimize the LLC converter full-load efficiency. Amode solver technique is proposed to handle the LLC convertersteady-state solutions. The mode solver utilizes numerical non-linear programming techniques to solve LLC state equationsand determine the operation mode. Loss models are providedto calculate the total component losses using the current andvoltage information derived from the mode solver. The calcu-lated efficiency serves as the objective function to optimize theconverter efficiency. A prototype 300-W 400-V to 12-V LLCconverter is built using the optimization results. The detailsof design variables, boundaries, equality/inequality constraints,and loss distributions are given. A measured full-load efficiencyof 97.07% is achieved compared to the calculated 97.4% effi-ciency. The proposed optimization procedure is an effective wayto design high-efficiency LLC converters.

REFERENCES

[1] S. Balachandran and F. C. Lee, “Algorithms for power converter designoptimization,” IEEE Trans. Aerosp Electron. Syst., vol. AES–17, no. 3,May 1981.

[2] R. B. Ridley, C. Zhou, and F. C. Lee, “Application of nonlinear design opti-mization for power converter components,” IEEE Trans. Power Electron.,vol. 5, no. 1, pp. 29–40, Jan. 1990.

[3] T. C. Neugebauer and D. J. Perreault, “Computer-Aided optimizationof DC/DC converters for automotive applications,” IEEE Trans. PowerElectron., vol. 18, no. 3, pp. 775–783, May 2003.

[4] N. Benavides and P. L. Chapman, “Mass-optimal design methodology forDC–DC converters in low-power portable fuel cell applications,” IEEETrans. Power Electron., vol. 23, no. 3, pp. 1545–1555, May 2008.

[5] J. W. Kolar, J. Biela, and J. Minibock, “Exploring the Pareto front ofmulti-objective single-phase PFC rectifier design optimization - 99.2%efficiency versus 7 kW/dm3 power density,” in Proc. IEEE Int. PowerElectron. Motion Control Conf., May, 2009, pp. 1–21.

[6] U. Badstuebner, J. Biela, and J. W. Kolar, “Design of an 99%-efficient,5kW, phase-shift PWM DC-DC converter for telecom applications,” inProc. IEEE Appl. Power Electron. Conf., 2010, pp. 773–780.

[7] B. Lu, W. Liu, Y. Liang, F. C. Lee, and J. D. van Wyk, “Optimal designmethodology for LLC resonant converter,” in Proc. IEEE Appl. PowerElectron. Conf. Expo., 2006, pp. 533–538.

[8] H. de Groot, E. Janssen, R. Pagano, and K. Schetters, “Design of a 1-MHzLLC resonant converter based on a DSP-driven SOI half-bridge powerMOS module,” IEEE Trans. Power Electron., vol. 22, no. 6, pp. 2307–2320, Nov. 2007.

[9] D. Fu, Y. Liu, F. C. Lee, and M. Xu, “A novel driving scheme for syn-chronous rectifiers in LLC resonant converters,” IEEE Trans. PowerElectron., vol. 24, no. 5, pp. 1321–1329, May 2009.

[10] X. Wu, G. Hua, J. Zhang, and Z. Qian, “A new current driven syn-chronous rectifier for series-parallel resonant (LLC) DC-DC converter,”IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 289–297, Jan. 2010.

[11] C. Zhao, M. Chen, G. Zhang, X. Wu, and Z. Qian, “A novel symmetricalrectifier configuration with low voltage stress and ultralow output-currentripple,” IEEE Trans. Power Electron., vol. 25, no. 7, pp. 1820–1831, Jun.2010.

[12] B. C. Kim, K. B. Park, C. E. Kim, B. H. Lee, and G. W. Moon, “LLCresonant converter with adaptive link-voltage variation for a high-power-density adapter,” IEEE Trans. Power Electron., vol. 25, no. 9, pp. 2248–2252, Sep. 2010.

[13] Z. Liang, R. Guo, J. Li, and A. Q. Huang, “A high-efficiency PV module-integrated DC/DC converter for PV energy harvest in FREEDM systems,”IEEE Trans. Power Electron., vol. 26, no. 3, pp. 897–909, Mar. 2011.

[14] R. Beiranvand, B. Rashidian, M. R. Zolghadri, and S. M. H. Alavi, “Op-timizing the normalized dead-time and maximum switching frequency ofa wide-adjustable-range LLC resonant converter,” IEEE Trans. PowerElectron., vol. 26, no. 2, pp. 462–472, Feb. 2011.

[15] R. Beiranvand, B. Rashidian, M. R. Zolghadri, and S. M. H. Alavi, “UsingLLC resonant converter for designing wide-range voltage source,” IEEETrans. Ind. Electron., vol. 58, no. 5, pp. 1746–1756, May 2011.

[16] R. L. Steigerwald, “A comparison of half-bridge resonant convertertopologies,” IEEE Trans. Power Electron., vol. 3, no. 2, pp. 174–1781,Apr. 1988.

[17] M. P. Foster, C. R. Gould, A. J. Gilbert, D. A. Stone, and C. M. Bingham,“Analysis of CLL voltage-output resonant converters using describingfunctions,” IEEE Trans. Power Electron., vol. 23, no. 4, pp. 1772–1781,Jul. 2008.

[18] J. F. Lazar and R. Martinelli, “Steady-state analysis of the LLC seriesresonant converter,” in Proc. IEEE Appl. Power Electron. Conf. Expo.,2001, vol. 2, pp. 728–735.

[19] C. P. Steinmetz, “On the law of hysteresis,” Trans. AIEE, vol. 9, pp. 3–64,1892.

[20] P. L. Dowell, “Effect of eddy currents in transformer windings,” IEEProc., vol. 113, no. 8, pp. 1387–1394, Aug. 1966.

[21] J. A. Ferreira, “Improved analytical modeling of conductive losses in mag-netic components,” IEEE Trans. Power Electron., vol. 9, no. 1, pp. 127–131, Jan. 1994.

[22] Ferroxcube application note, “Design of planar power transformers,” [On-line]. Available: www.ferroxcube.com.

[23] X. Xie, J. C. P. Liu, F. N. K. Poon, and M. H. Pong, “A novel high fre-quency current driven synchronous rectifier applicable to most switchingtopologies,” IEEE Trans. Power Electron., vol. 16, no. 5, pp. 635–648,Sep. 2001.

[24] F. N. K. Poon, J. C. P. Liu, and M. H. Pong, “Current driven synchronousrectifier with energy recovery using hysterisis driver,” U.S. Patent 6 597587 B1, Jul. 22, 2003.

[25] Infineon Application Note. (2009, Jun.). “Improving efficiency of syn-chronous rectification by analysis of the MOSFET power loss mecha-nism,” [Online]. Available: www.infineon.com.

[26] S. P. Han, “A globally convergent method for nonlinear programming,”J. Optim. Theory Appl., vol. 22, p. 297, 1977.

Ruiyang Yu was born in Sichuan, China, in 1985.He received the B.Eng. degree in electrical engineer-ing from Shandong University, Shandong, China, in2007, and M.Sc. degree in electrical engineering fromthe University of Hong Kong, in 2008. He is nowpursuing the Ph.D. degree in electrical engineeringat Power Electronics Laboratory, the University ofHong Kong.

His research interests include high effi-ciency power converter optimization, computer–aided design of power converter, and synchronous

rectification.


Godwin Kwun Yuan Ho received the B.Eng degreein electrical engineering form the University of HongKong, Hong Kong, in 2010, and is currently pursuingthe M.Phil degree at the Power Electronics Labora-tory, University of Hong Kong.

Bryan Man Hay Pong (M’84–SM’96) was born inHong Kong. He received the B.Sc. degree in elec-tronic and electrical engineering from the Universityof Birmingham, Birmingham, U.K., in 1983, and thePh.D. degree in power electronics from CambridgeUniversity, Cambridge, U.K. in 1987.

He worked with National Semiconductor Hong asa Senior Design Engineer and then a Chief DesignEngineer. He also worked with ASTEC Internationalas a Principal Engineer and a Division EngineeringManager. He is now an Associate Professor at the

University of Hong Kong, Hong Kong. He is In Charge of the Power Electron-ics Laboratory. His research interests include high efficiency and high reliabilitypower conversion, EMI reduction techniques, magnetic components, and otheraspects of switch mode power conversion. He has coinvented a number ofpatents. He also works with CET Opto Co. Ltd. on power conversion and light-ing products.

Bingo Wing-Kuen Ling (M’08-SM’08) received theB.Eng. (Hons) and M.Phil. degrees from the Depart-ment of Electronic and Computer Engineering, theHong Kong University of Science and Technology,in 1997 and 2000, respectively, and the Ph.D. degreefrom the Department of Electronic and InformationEngineering, the Hong Kong Polytechnic University,in 2003.

In 2004, he joined the Kings College, London, asa Lecturer. He has published more than fifty journalpapers. He is also the author of the textbook titled

Nonlinear Digital Filters: Analysis and Applications (Elsevier, 2007), and hasedited a book titled Control of Chaos in Nonlinear Circuits and Systems (WorldScientific Publishing, 2009). His research interests include theory and applica-tions of optimizations, symbolic dynamics, filter banks, and wavelets as wellas control theory. He has served as a Technical Committee member of severalIEEE international conferences as well as an Organizer of a special session inthe International Symposium on Communication Systems, Networks and Dig-ital Signal Processing, in 2008 and 2010. He has also served as a Guest Editorof a special issue on nonlinear circuits and systems in the Journal of Circuits,Systems and Signal Processing and on optimization for signal processing andcommunications in the American Journal of Engineering and Applied Science.He is currently a Guest Associate Editor of the International Journal of Bifur-cations and Chaos, and an Associate Editor of Circuits, Systems and SignalProcessing.

James Lam (M’89–SM’99) received a first classB.Sc. degree in mechanical engineering from the Uni-versity of Manchester, U.K., in 1983. He received theM.Phil. and Ph.D. degrees from the University ofCambridge, Cambridge, U.K., in 1985 and 1988, re-spectively.

He has been a Distinguished Visiting Fellow ofthe Royal Academy of Engineering. Prior to join-ing the University of Hong Kong in 1993, he was aLecturer at the City University of Hong Kong andat the University of Melbourne. He has held Guest

Professorships in many universities in China. He has research interests in modelreduction, robust control and filtering, delay, singular systems, Markovian jumpsystems, multidimensional systems, networked control systems, vibration con-trol, and biological networks.

Prof. Lam was awarded the Ashbury Scholarship, the A.H. Gibson Prize,and the H. Wright Baker Prize for his academic performance. He was a re-cipient of the Outstanding Researcher Award of the University of Hong Kong.He is a corecipient of the International Journal of Systems Science Prize PaperAward. He is a Chartered Mathematician, Chartered Scientist, a fellow of theInstitute of Mathematics and its Applications, and a fellow of the Institutionof Engineering and Technology. He is a Panel Member (Engineering) of theResearch Grants Council, HKSAR. In addition to serving as Subject Editor ofthe Journal of Sound and Vibration, he is also Associate Editor of Asian Journalof Control, International Journal of Systems Science, International Journal ofApplied Mathematics and Computer Science, IEEE TRANSACTIONS ON SIGNAL

PROCESSING, Journal of the Franklin Institute, Automatica, MultidimensionalSystems and Signal Processing, and is an editorial member of IET ControlTheory and Applications, Dynamics of Continuous, Discrete and Impulsive Sys-tems: Series B (Applications & Algorithms), and Proc. IMechE Part I: Journalof Systems and Control Engineering. He formerly served as Editor-in-Chief ofthe IEE Proceedings: Control Theory and Applications and was a member ofthe IFAC Technical Committee on Control Design.

Documents

Computer-Aided Design and Optimization of High … et al.: COMPUTER-AIDED DESIGN AND OPTIMIZATION OF HIGH-EFFICIENCY LLC SERIES RESONANT CONVERTER 3245 Fig. 1. Half-bridge LLC series