Optimal Design of LCL Harmonic Filters for Three-Phase PFC ...

© 2013 IEEE 

IEEE Transactions on Power Electronics, Vol. 28, No. 7, pp. 3114-3125, July 2013. 

Optimal Design of LCL Harmonic Filters for Three-Phase PFC Rectifiers 

J. Mühlethaler 

M. Schweizer 

R. Blattmann 

J. W. Kolar 

A. Ecklebe 

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3114 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 7, JULY 2013 

Optimal Design of LCL Harmonic Filters 

for Three-Phase PFC Rectifiers 

Jonas Mühlethaler, Student Member, IEEE, Mario Schweizer, Student Member, IEEE, Robert Blattmann, 

Johann W. Kolar, Fellow, IEEE, and Andreas Ecklebe, Member, IEEE 

Abstract—Inductive components such as line filter inductors or 

transformers occupy a significant amount of space in today’s power 

electronic systems, and furthermore, considerable losses occur in 

these components. A main application of inductive components is 

EMI filters, as, e.g., employed for the attenuation of switching frequency 

harmonics of power factor correction (PFC) rectifier systems. 

In this paper, a design procedure for the mains side LCL filter 

of an active three-phase rectifier is introduced. The procedure is 

based on a generic optimization approach, which guarantees a low 

volume and/or low losses. Different designs are calculated to show 

the tradeoff between filter volume and filter losses. The design procedure 

is verified by experimental measurements. Furthermore, an 

overall system optimization, i.e., an optimization of the complete 

three-phase PFC rectifier, is given. 

Index Terms—LCL filter, magnetic components, optimization. 

I. INTRODUCTION 

THE trend in power electronics research is toward 

higher efficiency and higher power density of converter 

systems. This trend is driven by cost considerations (e.g., material 

economies), space limitations, (e.g., for automotive applications), 

and increasing efficiency requirements (e.g., for telecom 

applications). The increase of the power density often affects 

the efficiency, i.e., a tradeoff between these two quality indices 

exists [1]. 

Inductive components occupy a significant amount of space in 

today’s power electronic systems, and furthermore, considerable 

losses occur in these components. In order to increase the power 

density and/or efficiency of power electronic systems, losses 

in inductive components must be reduced, and/or new cooling 

concepts need to be investigated. 

At the Power Electronic Systems Laboratory at ETH Zurich, 

a project has been initiated with the goal to establish comprehensive 

models of inductive power components which can be 

adapted to various geometric properties, operating conditions, 

and cooling conditions. These models will form the basis for the 

Manuscript received April 17, 2012; revised June 11, 2012; accepted July 17, 

2012. Date of current version December 24, 2012. Recommended for publication 

by Associate Editor J. Antenor Pomilio. 

J. Mühlethaler, M. Schweizer, and J. W. Kolar are with the Power Electronic 

Systems Laboratory, ETH Zurich, 8006 Zurich, Switzerland (e-mail: 

muehlethaler@lem.ee.ethz.ch; schweizer@lem.ee.ethz.ch; kolar@lem.ee.ethz. 

ch). 

R. Blattmann is with the ABB Switzerland, Ltd., CH-5400 Baden, 

Switzerland (e-mail: robert.blattmann@ch.abb.com). 

A. Ecklebe is with the Corporate Research Center, ABB Switzerland, Ltd., 

CH-5405 Baden-Dättwil, Switzerland (e-mail: andreas.ecklebe@ch.abb.com). 

Color versions of one or more of the figures in this paper are available online 

at http://ieeexplore.ieee.org. 

Digital Object Identifier 10.1109/TPEL.2012.2225641 

optimization of inductive components employed in key power 

electronics applications. 

Numerous papers have been written about how to find optimal 

designs of power electronic systems, whereas the goal of 

the according optimization task can be minimum costs, volume, 

losses, etc. Some papers investigated parts of a system, some 

others tried to model and optimize the overall system. For instance, 

in [2], an overall optimization of a two-FET forward 

converter is presented. For it, a two-stage optimization method 

to optimize the efficiency from light load to full load is proposed. 

Another approach to conduct an overall optimization is given 

in [3], where a flyback converter has been optimized. In [4], an 

overall optimization of a boost power factor correction (PFC) 

converter including the input electromagnetic interference filter 

is conducted. Examples of papers in which the focus is put only 

on parts of a system are, e.g., given in [5] and [6]. In [5], it 

is shown on the example of a buck converter, how the capacitors 

of a dc-link capacitor tank are selected optimally. In [6], a 

design methodology that finds the volumetric optimized inductors 

employed in single-phase PFC boost converters is given, 

whereas the corresponding optimum switching frequency and 

input current ripple is found. 

The aim of this paper is to use the recently established comprehensive 

models of inductive power components to optimally 

design LCL input filters for a three-phase PFC rectifier. The focus 

of the paper at hand is put on the optimal design of the LCL 

harmonic filters. LCL input filters are an attractive solution to 

attenuate switching frequency current harmonics of active voltage 

source rectifiers [7], [8]. In this paper, a design procedure 

for LCL filters based on a generic optimization approach is introduced 

guaranteeing low volume and/or low losses. Different 

designs are calculated showing the tradeoff between filter volume 

and filter losses. Furthermore, it is shown at the end of this 

paper how an overall system optimization, i.e., an optimization 

of the complete three-phase PFC rectifier (not only the filter), 

could be conducted. It is important to consider the complete 

system, since there are parameters that bring advantages for one 

subsystem but bring disadvantages for another. For instance, in 

the selected example of an LCL input filter, a higher switching 

frequency leads to lower volume, or, when keeping the volume 

constant, to lower losses of the inductive components. However, 

higher semiconductor switching losses are expected in the case 

of higher switching frequencies. 

In Section II, the three-phase PFC rectifier is introduced; in 

Section III, the applied models of the LCL filter components 

are discussed; and in Section IV, the optimization algorithm for 

the LCL filter is described. Simulation and experimental results 

0885-8993/$31.00 © 2012 IEEE

MÜHLETHALER et al.: OPTIMAL DESIGN OF LCL HARMONIC FILTERS FOR THREE-PHASE PFC RECTIFIERS 3115 

Fig. 1. 

(a) Three-phase PFC rectifier with LCL input filter. (b) Cross sections of inductors employed in the input filter. 

TABLE I 

SPECIFICATION OF THE THREE-PHASE PFC RECITIFIER 

is taken for the conductors. The windings are divided into two 

halves arranged on the two legs which lead to a more distributed 

winding structure. A more distributed winding structure has 

advantages such as better heat dissipation capabilities, lower 

inductor volume, etc. 

are given in Sections V and VI, respectively. In Section VII, the 

converter volume and losses are taken into consideration, i.e., 

an overall system optimization is performed. 

II. THREE-PHASE PFC RECTIFIER WITH INPUT FILTER 

The three-phase PFC rectifier investigated in this paper is 

shown in Fig. 1. The rectifier comprises in each phase a boost 

inductor L 2 , a damped LC filter L 1 /C/C d /R d , and a pair of 

switches with free-wheeling diodes. The load is assumed to 

be a dc current source. The considered operating point of the 

PFC rectifier is described in Table I. A space vector modulation 

(SVM) scheme with loss-optimal clamping has been implemented 

to have a fundamental displacement factor of cos φ =1. 

In the applied SVM scheme with loss-optimal clamping, a 

switching of the phase with the highest current is omitted; hence, 

the switching losses are reduced accordingly. The functionality 

and the detailed control of the used SVM scheme are described 

in [9]. The three-phase PFC rectifier with input filter has been 

simulated in MATLAB/Simulink, where the nonlinearity of the 

core material of the inductors, i.e., the change of the inductance 

value with changing current, is taken into account. 

The three inductors L 1,a , L 1,b , and L 1,c , and the three capacitors 

C a , C b , and C c in star arrangement, together with the three 

boost inductors L 2,a , L 2,b , and L 2,c , result in a third-order LCL 

low-pass filter between the mains and the switching stage. The 

capacitor/resistor branches C d,a /R d,a , C d,b /R d,b , and C d,c /R d,c 

are necessary to damp the resonance of the LC input filter. All 

inductors are assumed to have the same geometry, which is illustrated 

in Fig. 1(b). The cores are made of grain-oriented steel 

(M165-35S, lamination thickness 0.35 mm). Solid copper wire 

III. MODELING OF INPUT FILTER COMPONENTS 

The loss modeling is based on the framework introduced 

in [10], where a high level of accuracy has been achieved by 

combining the best state-of-the-art approaches and by embedding 

newly developed approaches into a novel loss calculation 

framework. In [10], the impact of the peak-to-peak flux density 

ΔB, frequency f, dc premagnetization H DC , temperature T , 

core shape, minor and major BH-loops, flux waveform, and 

material on the core loss calculation has been considered. In 

order to calculate the winding losses, formulas for round conductors 

and litz wires, each considering skin and proximity 

effects and also considering the influence of an air gap fringing 

field have been included. In the following, a discussion about the 

implemented models for the employed inductors [cf., Fig. 1(b)] 

is given. 

A. Calculation of the Inductance 

The inductance of an inductive component with N winding 

turns and a total magnetic reluctance R m,tot is calculated as 

L = N 2 

R m,tot 

. (1) 

Accordingly, the reluctance of each section of the flux path 

has to be derived first in order to calculate R m,tot . The total 

reluctance for a general inductor is calculated as a function 

of the core reluctances and air gap reluctances. The core and 

air gap reluctances can be determined applying the methods 

described in [11]. The reluctances of the core depend on the 

relative permeability μ r which is defined by the nonlinear BHrelation 

of the core material; hence, the reluctance is described as 

a function of the flux. Therefore, as the flux depends on the core 

reluctance and the reluctance depends on the flux, the system


Fig. 2. Reluctance model of the inductor shown in Fig. 1(b). It consists of one 

voltage source (representing the two separated windings), one air gap reluctance 

R g (representing the two air gaps), and one core reluctance R c . 

can only be solved iteratively by using a numerical method. In 

the case at hand, the problem has been solved by applying the 

Newton approach. 

The reluctance model of the inductor shown in Fig. 1(b) is 

illustrated in Fig. 2. It consists of one voltage source (representing 

the two separated windings), one air gap reluctance R g 

(representing the two air gaps), and one core reluctance R c .The 

fact that the flux density in core parts very close to the air gap is 

(slightly) reduced as the flux lines already left the core has been 

neglected. 

B. Core Losses 

The applied core loss approach is described in [10] in detail 

and can be seen as a hybrid of the improved–improved Generalized 

Steinmetz Equation i 2 GSE [12] and a loss map approach: 

a loss map is experimentally determined and the interpolation 

and extrapolation for operating points in between the measured 

values is then made with the i 2 GSE. 

The flux density waveform for which the losses have to be 

calculated is, e.g., simulated in a circuit simulator, where the 

magnetic part is modeled as a reluctance model. This simulated 

waveform is then divided into its fundamental flux waveform 

and into piecewise linear flux waveform segments. The loss 

energy is then derived for the fundamental and all piecewise 

linear segments, summed up and divided by the fundamental 

period length in order to determine the average core loss. The dc 

flux level of each piecewise linear flux segment is considered, as 

this influences the core losses [13]. Furthermore, the relaxation 

term of the i 2 GSE is evaluated for each transition from one 

piecewise linear flux segment to another. 

Another aspect to be considered in the core loss calculation 

is the effect of the core shape/size. By introducing a reluctance 

model of the core, the flux density can be calculated. Subsequently, 

for each core section with (approximately) homogenous 

flux density, the losses can be determined. In the case at 

hand, the core has been divided into four straight core sections 

and four corner sections. The core losses of the sections are then 

summed up to obtain the total core losses. 

C. Winding Losses 

The second source of losses in inductive components is the 

ohmic losses in the windings. The resistance of a conductor 

increases with increasing frequency due to eddy currents. Selfinduced 

eddy currents inside a conductor lead to the skin effect. 

Fig. 3. Cross section of the round conductor considered with a current in 

z-direction. The conductor is infinitely large in z-direction. 

Eddy currents due to an external alternating magnetic field, 

e.g., the air gap fringing field or the magnetic field from other 

conductors, lead to the proximity effect. 

The sum of the dc losses and the skin effect losses per unit 

length in round conductors [cf., Fig. 3(a)] can be calculated 

as [14] 

and 1 F R (f) = ξ 

4 √ 2 

with δ = 

1 √ πμ0 σf ,ξ = 

P S = R DC · F R (f) · Î2 (2) 

( ber0 (ξ)bei 1 (ξ) − ber 0 (ξ)ber 1 (ξ) 

ber 1 (ξ) 2 + bei 1 (ξ) 2 

− bei ) 

0(ξ)ber 1 (ξ)+bei 0 (ξ)bei 1 (ξ) 

ber 1 (ξ) 2 + bei 1 (ξ) 2 

(3) 

√ d 

2δ 

,R DC = 4 

σπd 

; δ is commonly 

2 

named the skin depth, f is the frequency, d is the conductor 

diameter, and σ is the electric conductivity of the conductor 

material. 

The proximity effect losses in round conductors [cf., Fig. 3(b)] 

per unit length can be calculated as [14] 

P P = R DC · G R (f) · Ĥ2 e (4) 

and 

G R (f) = − ξπ2 d 2 ( 

2 √ ber2 (ξ)ber 1 (ξ)+ber 2 (ξ)bei 1 (ξ) 

2 ber 0 (ξ) 2 + bei 0 (ξ) 2 

+ bei ) 

2(ξ)bei 1 (ξ) − bei 2 (ξ)ber 1 (ξ) 

ber 0 (ξ) 2 + bei 0 (ξ) 2 . (5) 

R DC is the resistance per unit length; hence, the losses P S and 

P P are losses per unit length as well. The external magnetic 

field strength H e of every conductor has to be known when 

calculating the proximity losses. In the case of an ungapped 

core and windings that are fully enclosed by core material, 

1-D approximations to derive the magnetic field exist. The most 

popular method is the Dowell method [16]. However, in the case 

of gapped cores, such 1-D approximations are not applicable as 

the fringing field of the air gap cannot be described in a 1-D 

manner. For the employed gapped cores, another approach has 

to be selected. The applied approach is a 2-D approach and is 

1 The solution for F R (and G R ) is based on a Bessel differential equation 

that has the form x 2 y ′′ + xy ′ +(k 2 x 2 − v 2 )y =0. With the general solution 

y = C 1 J v (kx)+C 2 Y v (kx), whereas J v (kx) is known as Bessel function 

of the first kind and order v and Y v (kx) is known as Bessel function of the 

second kind and order v [15]. Furthermore, to resolve J v (kx) into its real and 

imaginary part, the Kelvin functions can be used: J v (j 3 2 x) =ber v x + jbei v x.


temperature. The ambient temperature T A isassumedtobe 

constant at 25 ◦ C. 

The heat transfer due to convection is described with 

P = αA(T L − T A ) (7) 

where P is the heat flow, A the surface area, T L the body surface 

temperature (i.e., inductor temperature), and T A the fluid (i.e., 

ambient air) temperature. α is a coefficient that is determined 

by a set of characteristic dimensionless numbers, the Nusselt, 

Grashof, Prandtl, and Rayleigh numbers. Radiation has to be 

considered as a second important heat transfer mechanism and is 

described by the Stefan–Boltzmann law. Details about thermal 

modeling are not given here; the interested reader is referred 

to [18], from where the formulas used in this paper have been 

taken. 

Fig. 4. (a) Illustration of the method of images (mirroring). (b) Illustration of 

modeling an air gap as a fictitious conductor. 

Fig. 5. 

Thermal model with only one thermal resistance. 

described in detail in [10], where it has been implemented based 

on a previously presented work [17]. 

The magnetic field at any position can be derived as the 

superposition of the fields of each of the conductors. The impact 

of a magnetic conducting material can further be modeled with 

the method of images, where additional currents that are the 

mirrored version of the original currents are added to replace 

the magnetic material [17]. In case of windings that are fully 

enclosed by magnetic material (i.e., in the core window), a new 

wall is created at each mirroring step as the walls have to be 

mirrored as well. The mirroring can be continued to pushing 

the walls away. This is illustrated in Fig. 4(a). For this study, 

the mirroring has been stopped after the material was replaced 

by conductors three times in each direction. The presence of 

an air gap can be modeled as a fictitious conductor without 

eddy currents equal to the magneto-motive force across the air 

gap [17] as illustrated in Fig. 4(b). 

According to the previous discussion, the winding losses have 

to be calculated differently for the sections A and B illustrated 

in Fig. 1(b), as the mirroring leads to different magnetic fields. 

D. Thermal Modeling 

A thermal model is important when minimizing the filter volume, 

as the maximum temperature allowed is the limiting factor 

when reducing volume. The model used in this paper consists 

of only one thermal resistance R th and is illustrated in Fig. 5. 

The inductor temperature T L is assumed to be homogenous; it 

can be calculated as 

T L = T A + P loss R th (6) 

where P loss are the total losses occurring in the inductor, consisting 

of core and winding losses, and T A is the ambient 

E. Capacitor Modeling 

The filter and damping capacitors are selected from the EP- 

COS X2 MKP film capacitors series, which have a rated voltage 

of 305 V. The dissipation factor is specified as tan δ ≤ 1 W/kvar 

(at 1 kHz) [19], which enables the capacitor loss calculation. The 

capacitance density to calculate the capacitors volume can be 

approximated with 0.18 μF/cm 3 . The capacitance density has 

been approximated by dividing the capacitance value of several 

components by the according component volume. 

F. Damping Branch 

An LC filter is added between the boost inductor and the 

mains to meet a total harmonic distortion (THD) constraint. 

The additional LC filter changes the dynamics of the converter 

and may even increase the current ripple at the filter resonant 

frequency. Therefore, a C d /R d damping branch has been added 

for damping. In [20] and [21], it is described how to optimally 

choose C d and R d . Basically, there is a tradeoff between the 

size of damping capacitor C d and the damping achieved. For 

this study, C d = C has been selected as it showed to be a good 

compromise between additional volume needed and a reasonable 

damping achieved. Hence, the volume of the damping capacitor 

is the same as the volume of the filter capacitor. The 

value of the damping resistance that leads to optimal damping 

is then [20], [21] 

√ 

R d = 2.1 L 1 

C . (8) 

The C d /R d damping branch increases the reactive power consumption 

of the PFC rectifier system. Therefore, often other 

damping structures, such as the R f -L b series damping structure, 

are selected [21]. For this study, however, the C d /R d damping 

branch has been favored as its practical realization is easier 

and lower losses are expected. Furthermore, as will be seen in 

Section V, the reactive power consumption of the PFC rectifier 

system including the damped LC input filter is in the case at hand 

rather small, and, if necessary, it could be actively compensated 

by the rectifier.


The losses in the damping branch, which occur mainly in 

the resistors, are calculated and taken into consideration in the 

optimization procedure as well. 

IV. OPTIMIZATION OF THE INPUT FILTER 

The aim is to optimally design a harmonic filter of the introduced 

three-phase PFC rectifier. For the evaluation of different 

filter structures, a cost function is defined that weights the filter 

losses and filter volume according to the designer needs. In the 

following, the steps toward an optimal design are described. All 

steps are illustrated in Fig. 6. The optimization constraints are 

discussed first. 

A. Optimization Constraints and Conditions 

The high-frequency ripple in the current i 2,a/b/c is limited 

to the value I HF,pp,max , which is important as a too high 

I HF,pp,max , e.g., impairs controllability (for instance, an accurate 

current measurement becomes more difficult). Furthermore, 

the THD of the mains current is limited. In industry, a typical 

value for the THD that is required at the rated operating point is 

5 % [22]. In the design at hand, a THD limit of 4 % is selected in 

order to have some safety margin. Two other design constraints 

are the maximum temperature T max and the maximum volume 

V max the filter is allowed to have. A fixed switching frequency 

f sw is assumed. The dc-link voltage V DC and the load current 

I L of the converter are also assumed to be given and constant. 

All constraints/condition values for the current system are given 

in Fig. 6. 

B. Calculation of L 2,min 

The minimum value of the inductance L 2,min can be calculated 

based on the constraint I HF,pp,max as 

L 2,min = δ (100) · 

2 

3 V DC − √ 2V mains 

I HF,pp,max · f sw 

(9) 

with δ (100) = √ 3M 

2 

cos(π/6) the relative turn-on time of the 

space vector (100) when the current of phase a peaks. Equation 

(9) is based on the fact that, in case of a fundamental 

displacement factor of cos φ =1, the maximum current ripple 

I HF,pp,max occurs when the current reaches the peak value 

Î LF of the fundamental (at this instant, the voltage 2 3 V DC − 

√ 

2Vmains 

becomes 

L 2,min = 

is across the inductor). With M = 2√ 2V mains 

V DC 

,(9) 

√ 

3 

√ 

2|Vmains | 

V DC 

cos(π/6) · 

2 

3 V DC − √ 2V mains 

I HF,pp,max · f sw 

. 

(10) 

Fig. 6. Design procedure for three-phase LCL filters based on a generic optimization 

approach. 

C. Loss Calculation of Filter Components 

In the foregoing sections, it has been shown how an accurate 

loss modeling based on simulated current and voltage waveforms 

is possible. However, such a calculation based on simulated 

waveforms is time consuming and therefore, for an efficient 

optimization, simplifications have to be made. In Fig. 7, idealized 

current waveforms for each filter component of a phase are 

Fig. 7. 

Idealized current waveforms for each filter component.


illustrated. The current in L 1 is approximated as purely sinusoidal 

with a peak value of 

Î = 2 I L V 

√ DC 

(11) 

3 2Vmains 

where V mains is the RMS value of the mains-phase voltage. A 

possible reactive current is rather small and has been neglected. 

With the mains frequency f mains =50Hz, losses, volume, and 

temperature of L 1 can be calculated. 

The current in L 2 has a fundamental (sinusoidal) component, 

with an amplitude as calculated in (11) and a fundamental frequency 

of f mains =50Hz, and a superimposed ripple current. 

The ripple current is, for the purpose of simplification, in a 

first step considered to be sinusoidal with constant amplitude 

I HF,pp,max over the mains period. The losses for the fundamental 

and the high-frequency ripple are calculated independently, 

and then summed up. By doing this, it is neglected that core 

losses depend on the dc bias condition. 

The ripple current is assumed to be fully absorbed by the filter 

capacitor C, hence, with the given dissipation factor tan δ, the 

losses in the capacitor can be calculated as well. 

How the aforementioned simplifications affect the modeling 

accuracy will be discussed in Section V. 

D. Optimization Procedure 

After the optimization constraints and the simplifications chosen 

for the loss calculation have been described, the optimization 

procedure itself will be explained next. 

A filter design is defined by 

⎛ 

⎞ 

a L 1 

a L 2 

w L 1 

w L 2 

N L 1 

N L 2 

do 

X = L 1 

do L 2 

h L 1 

h L 2 

⎜ t L 

⎝ 

1 

t L 2 

⎟ 

⎠ 

ww L 1 

ww L 2 

d L 1 

d L 2 

(12) 

where all inductor parameters are defined as in Fig. 1(b). The 

subscripts L 1 and L 2 describe to which inductor each parameter 

corresponds. The capacitance value of the filter capacitor 

is calculated based on the L 1 value to guarantee that the THD 

constraint is met. The parameters in X are varied by an optimization 

algorithm to obtain the optimal design. The optimization is 

based on the MATLAB function fminsearch() that applies the 

downhill simplex approach of Nelder and Mead [23]. 

The optimization algorithm determines the optimal parameter 

values in X. A design is optimal when the cost function 

F = k Loss · q Loss · P + k Volume · q Volume · V (13) 

is minimized. k Loss and k Volume are weighting factors, q Loss 

and q Volume are proportionality factors, and P and V are the 

filter losses and filter volume, respectively. The proportionality 

factors are chosen such that, for a “comparable performance,” 

q Loss · P and q Volume · V are in the same range. 2 In the case 

at hand, this was achieved with q Loss =1/W and q Volume = 

3 · 10 4 /m 3 (these values differ a lot from case to case and, 

accordingly, are valid for the case at hand only). 

In order to calculate the volumes V L1 and V L2 of the inductors, 

their boxed volumes are calculated based on the parameters in 

X. 

The following steps are conducted to calculate the filter losses 

P and the filter volume V of (13) (cf., Fig. 6). 

1) Calculate the inductance value L 2 as a function of the 

current. Furthermore, the losses P L 2 

,thevolumeV L 2 

, 

and temperature T L 2 

of the boost inductors L 2 are calculated. 

In case a constraint cannot be met, the calculation is 

aborted and the design is discarded and a new design will 

be evaluated. 

2) Calculate the inductance value L 1 as a function of the 

current. Furthermore, the losses P L 1 

,thevolumeV L 1 

, and 

temperature T L 1 

of the filter inductors L 1 are calculated. In 

case a constraint cannot be met, the calculation is aborted 

and the design is discarded and a new design will be 

evaluated. 

3) For the purpose of simplification, the THD without filter 

is approximated with THD = I HF,pp,max /I (1),pp , where 

I (1),pp is the fundamental peak-to-peak value of the mains 

current. 3 Furthermore, it is assumed that the dominant harmonic 

content appears at f sw (this assumption is motivated 

by simulation results). The LC filter has then to attenuate 

the ripple current by 

( ) 

A = 20 log 10 I(1),pp THD max /I HF,pp,max (14) 

(in dB) at a frequency of f sw . Therewith, C is calculated 

as 

C = 1 

1 

L 1 ω0 

2 = 

(15) 

L 1 (2πf sw · 10 A 

40 dB ) 2 

where ω 0 is the filter cutoff frequency. The losses and 

the volume of the capacitors and damping resistors can 

then be calculated. In case a constraint cannot be met, the 

calculation is aborted and the design is discarded and a 

new design will be evaluated. 

4) The volume, temperature, and losses are now known and 

the cost function (13) can be evaluated. 

The optimal matrix X is found by varying the matrix parameters, 

evaluating these matrixes by repeating previous steps, and 

minimizing the cost function (13). After the optimal design is 

found, the algorithm quits the loop of Fig. 6. 

2 The filter losses are in the range of approximately 100 W ...250 W, whereas 

the filter volumes are in the range of approximately 0.001 m 3 ...0.01 m 3 . 

With q Loss =1/W andq Volume =3· 10 4 /m 3 , the volume range is lifted to 

30 ...300 and therewith becomes comparable to the losses. 

√ 

I 2 2 + I 2 3 + I 2 4 + ...+ I 2 n 

3 The THD is defined as THD = 

I 1 

,whereI n is the RMS 

value of the nth harmonic and I 1 is the RMS value of the fundamental current. 

Under the assumption that only the dominant harmonic content I HF,pp,max 

(which, for the purpose of simplification, is assumed to appear at one frequency) 

leads to a harmonic distortion, the THD becomes I HF,pp,max /I (1),pp , 

consequently.


Fig. 8. P –V Pareto front showing filter volumes V and filter losses P of 

different optimal designs. 

V. OPTIMIZATION OUTCOMES 

The optimization procedure leads to different filter designs 

depending on the chosen weighting factors k 1 and k 2 in (13), 

i.e., depending whether the aim of the optimization is more 

on reducing the volume V or more on reducing the losses P . 

Limiting factors are the maximum temperature T max (limits 

the volume from being too low) and a maximum volume V max 

(limits the efficiency from being too high). Different designs are 

shown by a P –V plot, i.e., a P –V Pareto front in Fig. 8; the 

tradeoff between losses and volume can be clearly identified. 

One design of Fig. 8 has been selected for further investigations. 

Particularly, a comparison between the (for the optimization 

procedure) simplified and the more elaborate calculation 

based on voltage/current waveforms from a circuit simulator 

has been made. The filter parameters of the selected 

design are detailed in Fig. 8. The circuit of the three-phase 

PFC rectifier with the selected input filter has been simulated 

in MATLAB/Simulink, and the simulated current and voltage 

waveforms have been taken to calculate the losses according 

to Section III. The results are given in Fig. 9. The THD constraint 

is met and the current I HF,pp,max is only insignificantly 

higher. The simplified loss calculation used for the optimization 

leads to an overestimation of the boost inductor losses. This is 

due to the fact that the maximum ripple current has been assumed 

to be constant over the mains period. The losses in the 

filter inductors, on the other hand, have been underestimated as 

any high-frequency ripple in the current through L 1 has been 

neglected in the simplified calculations. One could try to improve/change 

the simplifications made for the optimization and 

therewith improve the simplified loss calculation. However, the 

difference between the two calculation approaches has been 

considered as acceptable for this study. 

Another important design criteria is the achieved power factor. 

The reactive power consumption of the PFC rectifier system, 

including the damped LC input filter, is in the case at hand rather 

small (power factor = [real power]/[apparent power] = 0.998). 

So far, all results are based on simulations and calculations. 

The models have to be verified experimentally to prove the 

validity of the optimization procedure. In the following section, 

experimental results are shown. 

Fig. 9. Simulations and calculations to one selected design (cf., Fig. 8). 

(a) Filter inductor L 1,a . (b) Boost inductor L 2,a . 

VI. EXPERIMENTAL RESULTS 

Experimental measurements have been conducted to show 

that the previous introduced calculations are valid. The filter 

prototype built in the course of this study has been assembled 

of laminated sheets and coil formers of standard sizes, in order 

to keep the cost low. This avoids an exact implementation of an 

optimum as can be seen in Fig. 8 (the prototype built is not on the 

8 kHz-line). However, this does not impair the significance of the 

measurement results. Specifications, dimensions, and photos of 

the LCL filter built are given in Fig. 10. All measurements have 

been carried out with a Yokogawa WT3000 Precision Power 

Analyzer. 

The measurements have been conducted with the T-type converter 

introduced in [24]. In Fig. 11, a photo of the converter is 

given. The T-type converter is a three-level converter; however, 

a two-level operation is possible as well. A two-level operation 

with the same modulation scheme (optimal-loss clamping 

modulation scheme) as in the MATLAB simulation has been 

implemented. 

The results of the comparative measurements and simulations 

are given in Figs. 12 and 13. As can be seen, the calculated and 

measured loss values are very close to each other. The maximum 

current ripple in the actual system is (slightly) higher than 

in the simulation. This can be explained by the fact that in the 

simulation the inductance of the boost inductor is assumed to be 

constant over the full frequency range. However, in reality, the


Fig. 12. (a) Simulations and (b) measurements on one of the filter inductors 

L 1 of the implemented design. 

Fig. 10. Specifications, dimensions and photos of (a) one filter inductor L 1 , 

(b) all filter and damping capacitors/resistors C, C d ,andR d , and (c) one boost 

inductor L 2 . 

VII. OVERALL RECTIFIER OPTIMIZATION 

As can be seen in Fig. 8, an increase in switching frequency 

leads to lower filter losses and lower filter volumes. However, in 

return, an increase in switching frequency leads to higher switching 

losses in the converter semiconductors. In other words, it is 

important to consider the system as a whole in order to achieve 

a truly optimal design. In the following, first, a model for the 

converter is derived, which allows to determine a P –V Pareto 

front of the converter. Later, the tradeoff in the switching frequency 

is illustrated and the optimal frequency for the overall 

system is determined. 

Fig. 11. 

Photo of the PFC converter. 

effective inductance decreases with increasing frequency due 

to inductor losses and parasitic capacitances. The higher THD 

value can also be explained with the same effect since the filter 

inductance decreases with increasing frequency as well. The frequency 

behavior could be modeled analytically by representing 

the inductors as RLC networks. 

A. Overall Optimized Designs 

The Infineon Trench and Field Stop 1200 V IGBT4 series 

has been selected to determine the P –V pareto front of the converter. 

These semiconductors are the same as the ones employed 

in the converter of Fig. 11. A converter model has been set up, 

which determines the junction temperature of the transistors 

T j,T , the junction temperature of the diodes T j,D ,thevolume 

of the converter V , and the losses of the converter P . As input 

variables, it needs the transistor chip area A T , the diode chip 

area A D , the cooling system volume V CS , and the operating 

point defined by the switching frequency f sw , and peak-to-peak 

fundamental current I (1),pp . The high-frequency ripple current 

I HF,pp has, as long as I HF,pp ≪ I (1),pp , only a negligible impact 

on the losses and is therewith not taken into consideration.


where CSPI =15W/(K · L) can be approximated to be constant 

for a given cooling concept. The value 15 W/(K · L) has been 

calculated for the cooling concept of the converter shown in 

Fig. 11 according to [27]. According to [25], for the selected 

insulated-gate bipolar transistor (IGBT) series, the junction to 

sink surface thermal resistance R th,js can be approximated as a 

function of the chip area as 

K 

R th,js =23.94 

W · mm 2 · A−0.88 . (17) 

The switching and conduction losses per chip area have been 

extracted from data sheets. Different IGBTs with different current 

ratings, but from the same IGBT series have been analyzed. 

To determine the losses as a function of the chip area, the IGBT 

and diode chip area as a function of the nominal chip current I N 

has to be known. This chip area-current dependency has been 

taken from [25] and is for the transistor 

Fig. 13. (a) Simulations and (b) measurements on one of the boost inductors 

L 2 of the implemented design. 

Fig. 14. 

Illustration of a converter model. 

TABLE II 

CONSTRAINTS AND CONDITIONS FOR CONVERTER OPTIMIZATION 

The inputs and outputs of the converter model are illustrated in 

Fig. 14 and all optimization constraints and conditions are listed 

in Table II. This optimization based on the semiconductor chip 

area is motivated by previously presented works [25], [26]. 

The thermal resistance of the cooling system, i.e., the heat sink 

surface to ambient thermal resistance R th,sa , has been modeled 

with the cooling system performance index (CSPI) [27]; the 

R th,sa is then calculated as 

R th,sa = 

1 

CSPI · V CS 

(16) 

A T =0.95 mm2 

A · I N +3.2 mm 2 (18) 

and for the diode 

A D =0.47 mm2 

A · I N +3.6 mm 2 . (19) 

The switching losses scaled to the same current do not vary much 

with the chip size in the considered range, as a comparison of 

different data sheets of the selected IGBT series has shown; 

therefore, for this study, the switching loss energies have been 

considered as independent of the chip area. The switching loss 

energies have been extracted at two junction temperatures (25 ◦ C 

and 150 ◦ C) and, for intermediate temperatures, a linear interpolation 

has been made. These, from the data sheet extracted and 

interpolated, switching loss energies are then assigned to each 

switching instant in order to determine the switching losses. 

In return, the conduction losses depend substantially on the 

chip size area. This has been modeled as in [26], but with taking 

the impact of the temperature into consideration. The following 

equation describes the conduction losses 

P cond (A, i, T j,T /D )=V f · i + R on,N(T j,T /D ) · A N 

· i 2 

A 

(20) 

where V f is the forward voltage drop, R on,N (T j,T /D ) is the 

nominal on-resistance, A N is the nominal chip size area (to 

which R on,N (T j,T /D ) is taken from the data sheet), i is the current 

through the transistor, and A is the actual chip size area. The 

on-resistance has been extracted at two junction temperatures 

(25 ◦ C and 150 ◦ C) and, for intermediate on-resistance values, 

a linear interpolation has been made. The current i has been 

calculated for the chosen modulation scheme. Equation (20) is 

evaluated with different values for V f and R on,N , depending 

whether the conduction losses of the transistor or the diode are 

calculated. 

The converter volume is the sum of the cooling system volume, 

the dc-link capacitor volume, and the volume of the switching 

devices. The switching device volume has been calculated 

by multiplying the chip areas by a depth of 1 cm in order to 

approximate the transistor volume. The cooling system volume 

is known as it is a model input parameter. The dc-link capacitor


Fig. 15. P –V Pareto front showing converter volumes V and converter losses 

P of different optimal designs. 

C DC has been selected such that the dc-link voltage in case of 

an abrupt load drop from nominal load to zero load does not exceed 

a predefined value. The maximum dc-link voltage increase 

ΔV DC after the load drop can then be approximated as 

ΔV DC = 1 

√ 

3 2Vmains 

· 

C DC 2 V DC 

( 

Î2 

f sw 

+ 1 2Î2t max 

) 

where Î2 is the peak value of the phase current through the 

boost inductor L 2 and t max is the time difference between the 

moment where the system sampled the load drop (i.e., latest 

1/f sw after the load drop occurred, since the sampling interval 

is 1/f sw ) and the moment where the dc-link voltage peaks (i.e., 

V DC +ΔV DC is reached). The chosen value for the maximum 

voltage overshoot is ΔV DC =50V. The dc-link capacitors have 

been selected from the EPCOS MKP dc-link film capacitors 

series, which have a rated voltage of 800 V. The capacitor losses 

are low and, therewith, have been neglected. The capacitance 

density to calculate the capacitors volume can be approximated 

with 0.6 μF/cm 3 . 

The losses and volumes of other system parts, such as the DSP, 

auxiliary supply, gate driver, etc., have not been considered. 

An optimization algorithm has been set up similar to that of 

the filter. A design is characterized by the parameters A T , A D , 

and V CS . The optimization algorithm varies these parameter 

in order to find optimal designs. The optimization procedure 

leads to different designs depending whether the aim of the 

optimization is more on reducing the volume V or more on 

reducing the losses P . Different designs are shown by a P –V 

plot, i.e., a P –V Pareto front in Fig. 15; the tradeoff between 

losses and volume can be clearly identified. It becomes clear that 

a higher switching frequency leads to higher losses; therefore, 

there must exist an overall optimal switching frequency at which 

the system losses are minimized. 

Basically, one can combine the results from the filter and 

from the converter and, therewith, determine the overall system 

performance. The losses of loss optimized designs for different 

frequencies have been calculated and are shown in Fig. 16(a). 

In the case at hand, the optimal switching frequency is at approximately 

5–6 kHz. In Fig. 16(b), the losses of a volumetric 

optimized designs are given for different frequencies; it can be 

seen that the optimal switching frequency is in the same range. 

Fig. 16. (a) Losses of loss optimized designs for different frequencies. 

(b) Losses of volumetric optimized designs for different frequencies. 

Fig. 17. 

Volume of volumetric optimized designs for different frequencies. 

In case of loss-optimized designs, the major volume part 

comes from the filter volume. This becomes clear when Fig. 8 

is compared with Fig. 15: the maximum filter volume is much 

higher than the maximum converter volume. This has to do 

with the selected constraints; however, a further increase of 

the converter volume would not have a big impact in further 

decreasing the losses; therefore, it can be concluded that the 

constraints have been selected well. 

In Fig. 17, the volume of volumetric optimized designs for 

different frequencies is plotted. As both, the filter size and the 

converter size, decrease with increasing frequencies, a high frequency 

is favorable with respect to system size. The converter 

size decreases with increasing frequency because the dc-link 

capacitance reduces with increasing frequency. The fact that the 

volume decreases with increasing switching frequency is only 

true in a limited frequency range; above this frequency range, 

the losses become very high and the size of the components


increases again so that the heat can be dissipated. This effect is 

not visible here. 

VIII. CONCLUSION AND FUTURE WORK 

A design procedure for three-phase LCL filters based on a 

generic optimization approach is introduced guaranteeing low 

volume and/or low losses. The cost function, which characterizes 

a given filter design, allows a weighting of the filter losses 

and of the filter volume according to the designer’s need. Different 

designs have been calculated to show the tradeoff between 

filter volume and filter losses. Experimental results have shown 

that a very high loss accuracy has been achieved. To improve 

the THD and current ripple accuracy, the frequency behavior of 

the inductors could be modeled as well. 

As can be seen in Fig. 8, a higher switching frequency leads 

to lower filter volume, or, when keeping the volume constant, 

to lower filter losses. However, higher switching losses are expected 

in case of higher switching frequencies. Therefore, an 

overall system optimization, i.e., an optimization of the complete 

three-phase PFC rectifier including the filter, has been 

performed. Generally, it is important to consider the system 

to be optimized as a whole, since there are parameters that 

bring advantages for one subsystem while deteriorating another 

subsystem. 

ACKNOWLEDGMENT 

The authors would like to thank ABB Switzerland, Ltd., for 

giving them the opportunity to work on this very interesting 

project. 

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Jonas Mühlethaler (S’09) received the M.Sc. and 

Ph.D. degrees from the Swiss Federal Institute of 

Technology Zurich (ETHZ), Zurich, Switzerland, in 

2008 and 2012, respectively, both in electrical engineering. 

During his master studies, he focused on 

power electronics and electrical machines and involved 

on compensating torque pulsation in permanent 

magnet motors. During the Ph.D. studies, he 

was involved on modeling and multiobjective optimization 

of inductive power components. 

He is currently a Postdoctoral Fellow at Power 

Electronic Systems Laboratory, ETHZ. 

Mario Schweizer (S’09) was born in Switzerland on 

February 20, 1983. He studied electrical engineering 

at ETH Zurich, Zurich, Switzerland. During his studies, 

he focussed on power electronics, mechatronics 

and energy systems. He received the M.Sc. degree 

in May 2008. His Master thesis research involved 

implementation of a new control hardware and different 

control strategies for an ultra sparse matrix 

converter. He has been working toward the Ph.D. 

degree at the Power Electronic Systems Laboratory 

since July 2008. 

His current research interests include eco’intelligent drive systems and gridfriendly 

active rectifier interfaces.


Robert Blattmann received the M.Sc. degree in electrical 

engineering from the Swiss Federal Institute 

of Technology Zurich, Zurich, Switzerland, in 2011. 

During his master studies, he focused on power electronics 

and energy systems and was involved on a 

new concept for a magnetic bearing system. 

In 2012, he joined the Trainee Program of ABB 

Switzerland, Ltd., Baden, Switzerland. 

Group. 

Andreas Ecklebe (M’07) received the Dipl.-Ing. and 

Dr.-Ing. degrees from the Otto-von-Guericke University, 

Magdeburg, Germany, in 2002 and 2009, respectively, 

both in electrical engineering. 

From 2002 to 2004, he was with SMS Demag 

AG and Alstom Power Conversion, where he was involved 

in the design of technological control systems 

for large-scale industrial automation solutions. Since 

2008, he has been with ABB Corporate Research, 

Baden-Daettwil, Switzerland, where he is currently 

leading the Power Electronics Integration Research 

Johann W. Kolar (S’89–M’91–SM’04–F’10) received 

the M.Sc. and Ph.D. degrees (summa cum 

laude/promotio sub auspiciis praesidentis rei publicae) 

from the University of Technology Vienna, 

Vienna, Austria. 

Since 1984, he has been an Independent International 

Consultant in close collaboration with the University 

of Technology Vienna, in the fields of power 

electronics, industrial electronics and high performance 

drives. He has proposed numerous novel converter 

topologies and modulation/control concepts, 

e.g., the VIENNA Rectifier, the Swiss Rectifier, and the three-phase ac–ac 

sparse matrix converter. He has published more than 450 scientific papers in 

international journals and conference proceedings and has filed more than 85 

patents. He became a Professor and Head of the Power Electronic Systems 

Laboratory, Swiss Federal Institute of Technology (ETH) Zurich, on February 

1, 2001. His current research interests include ac–ac and ac–dc converter 

topologies with low effects on the mains, e.g., for data centers, more-electricaircraft 

and distributed renewable energy systems, and solid-state transformers 

for smart microgrid systems. Further main research interests include the realization 

of ultracompact and ultraefficient converter modules employing latest 

power semiconductor technology (SiC and GaN), micropower electronics and/or 

power supplies on chip, multidomain/scale modeling/simulation and multiobjective 

optimization, physical model-based lifetime prediction, pulsed power, 

and ultrahigh speed and bearingless motors. 

Dr. Kolar has been appointed the IEEE Distinguished Lecturer by the IEEE 

Power Electronics Society (PELS) in 2011. He received the IEEE Transactions 

Prize Paper Award of the IEEE Illuminating Engineering Society in 2005, of 

the IEEE Institute of Aeronautical Sciences in 2009 and 2010, the IEEE/ASME 

in 2010, and of the IEEE PELS in 2009, 2010, and 2011. He also received the 

Best Paper Award at the International Conference on Performance Engineering 

2007, the IAS 2008, the IECON 2009, the International Symposium on Parameterized 

and Exact Computation 2010 (for two papers), the Energy Conversion 

Congress and Exposition (ECCE) Asia 2011, and the ECCE USA 2011. Furthermore, 

he received the ETH Zurich Golden Owl Award 2011 for Excellence 

in Teaching and an Erskine Fellowship from the University of Canterbury, New 

Zealand, in 2003. He initiated and/or is the founder/cofounder of four spinoff 

companies targeting ultrahigh speed drives, multidomain/level simulation, 

ultracompact/efficient converter systems, and pulsed power/electronic energy 

processing. In 2006, the European Power Supplies Manufacturers Association 

awarded the Power Electronics Systems Laboratory of ETH Zurich as the Leading 

Academic Research Institution in Power Electronics in Europe. He is a 

member of the IEEJ and of International Steering Committees and Technical 

Program Committees of numerous international conferences in the field (e.g., 

Director of the Power Quality Branch of the International Conference on Power 

Conversion and Intelligent Motion). He is the founding Chairman of the IEEE 

PELS Austria and Switzerland Chapter and Chairman of the Education Chapter 

of the EPE Association. From 1997 to 2000, he has been serving as an Associate 

Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and 

since 2001 as an Associate Editor of the IEEE TRANSACTIONS ON POWER ELEC- 

TRONICS. Since 2002, he has been an Associate Editor of the Journal of Power 

Electronics of the Korean Institute of Power Electronics and a member of the 

Editorial Advisory Board of the IEEJ Transactions on Electrical and Electronic 

Engineering.

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