Optimization of the Main Inductor in a LCL Filter for

Optimization of the Main Inductor in a LCL Filter for Three Phase Active Rectifier Lixiang Wei, Richard A Lukaszewski Rockwell Automation – Allen Bradley, 6400 West Enterprise Drive, Mequon, WI, 53092, USA [email protected] Abstract — this paper evaluates the optimal design of the main inductor for LCL filter in a three phase active front end (AFE) rectifier. The objective is to evaluate the optimal inductance of the main inductance and its cost under various switching frequency, air flow speed, and inductance. The paper presents a detailed step by step optimization procedure to optimize main inductor with lamination materials. Then, the optimized designs under various conditions are compared and evaluated. In the end, one optimal sample is built up and tested to prove the effectiveness of the design concept.

conditions are evaluated. The concept of the optimization under 4kHz switching frequency condition is built up and tested effectively with experiment. II.

GENERAL DESRIPTION OF THE STUDIED SYSTEM AND CONSTRAINTS OF THE OPTIMIZATION DESIGN The equivalent three phase circuit for the AFE rectifier is shown in Figure.1. The following base values are introduced:

Keywords – Inductor optimization, LCL filter, Active rectifier

Vb = 2 3 ⋅ Vs

I. INTRODUCTION The PWM rectifiers are generally used in high performance adjustable speed drive (ASD) where regeneration or high quality input current is required. When some of the harmonic standards, such as IEEE-519 and IEC 61000-3-2/ IEC 61000-34 which limit the harmonic current of the power electronic converters, become more and more popular. It is believed to be one of the best options to replace the diode rectifier in these applications [1][2][3][4]. However, to make the PWM rectifier more competitive, reducing the overall cost of the AFE system is very important.

I b = (2 Pn ) /(3Vs )

Comparing to the uncontrolled diode rectifier, there are mainly three added cost for the PWM rectifier system, including input LCL filter (an input LC filter at the source side plus a main inductor at the converter side rectifier), IGBT module, and the control board. Among them, main inductor is one of the most costly components. To help reducing the overall cost, the optimization of the main inductor and the optimal selection of switching frequency of the AFE rectifier are very important. A rule of thumb value for the design of LCL filter is to choose the line side inductor as 3% per unit (PU) of the base inductance and the main inductance as 10% pu. This result was derived under rather low switching frequency. Nowadays, with the fast improvement of power devices, the switching frequency of the rectifier can be largely increased. Thus, it becomes more and more important to evaluate the cost of LCL filter under higher switching frequency [5]. This paper investigates the cost optimization of the main inductor using currently widely used low cost laminations (M6/Z11). The optimizations are evaluated with three factors: switching frequency fs, inductance of the inductor, and air cooling method. Finally, the optimized values under different

Z b = Vs2 Pn

Cb = 1 ωn Zb Lb = Z b ωn

(1)

where, Vs is the line to line rms voltage, Ȧn is the grid frequency and Pn is the active power absorbed by the converter at rated conditions. To simplify the analysis, the voltage, current, inductances L and capacitance C are all expressed in [pu] of the base value. Source+Impedance

Sap

Va

Lsa isa

Lfa

Vb

Lsb isb

Lfb

Vc

Lsc

Lfc

isc

Sbp

Scp

La

ia

N

Lb

ib ic

C1

Vdc

Inverter

Lc San

Sbn

Scn

P

Cfa Cfb Cfc Ra

Rb Rc

Figure 1. Three phase AFE rectifier with input LCL filter

Since the main objective of this paper is to optimize the cost of the main inductor, the control scheme of the AFE rectifier utilizes similar method as shown in reference [3]. Moreover, it is assumed that the filter capacitance is large enough and the ripple voltage of the capacitor voltage can be neglected. According to [5], the input current harmonics are still acceptable for IEEE 519 specification with a smaller inductance in the converter side. The cost of the main inductor will be evaluated for a 150hp regenerative converter under

0197-2618/07/$25.00 © 2007 IEEE Authorized licensed use limited to: UNIVERSIDADE FEDERAL DE MINAS GERAIS. Downloaded on February 9, 2009 at 18:13 from IEEE Xplore. Restrictions apply.

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various inductance (2% ~ 16% PU) and various switching frequencies (2kHz ~ 14kHz) will be evaluated. III.

OPTIMIZATION CONSIDERATIONS FOR THE MAIN INDUCTOR

The biggest concern of designing a high frequency inductor with lamination materials is the temperature limit. To reduce the operating temperature, the following factors are generally considered by •

Reduce the operating flux density of core materials

•

Adding multiple air gaps in the lamination to reduce air gap and proximity losses

•

Adding air duct between the coil and core, or between coil materials to increase cooling surface area

The fundamental current of the main inductor can be calculated as im1 =

Ib

(2)

η ⋅ PF

where, Ș is the efficiency of the converter and PF is the power factor of the AFE rectifier at the converter side.

However, there are numerous ways of optimizing the inductors and it is not possible to be covered in a single paper. In order to simplify the analysis, the following assumptions are made. 1) A three phase inductor as shown in Figure.2 will be studied; 2) Three evenly distributed air gap are added in each phase; 3) Air ducts with 3/8 inch thickness are added between the coil and core to increase the cooling surface area as shown Figure.2 (a) and (b);

(a)

4) The coil windings are selected as two or three paralleled rectangle windings, air ducts will be added between the paralleled windings as shown in Figure.2 (b); 5) Both natural and force air cooling conditions are studied. The air speed under forced cooling condition is 1.8 meter/second; 6) Two mounting straps are added in each phase to hold the lamination together. 7) While designing the inductor, standard three phase EI lamination dimensions as defined from TEMPEL steel will be used. The thickness of the core can be adjusted to optimize the overall inductor.

(b)

During the optimization, the following parameters are defined as variables: Core thickness: d Current density of the coil: J Operating flux density: Bm Inductance: Lm Switching frequency: fs Name of the lamination: x x is a number of the TEMPEL steel laminaton. Each number represents one set of the size for EI lamination shape. A. Basic equations for inductance calculation In the optimization, the inductor current is separated into two frequency components: fundamental and ripple current.

(c) Figure 2. The drawings of the optimized three phase inductor (a) 3D drawing of the inductor, (b) top view drawing of the inductor, (c) dimensions of the lamination

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The ripple current of the main inductor are generated by the PWM switching of the AFE rectifier. Even though various harmonics are generated in the inductor, the ripple current at switching frequency are the maximum and will only be considered during the optimization: im _ fs ( f s , Lm ) =

k f s ⋅ Lm

(3)

where, k is a constant and can be derived from either the simulation or real system test, im _ fs is the current amplitude of the main inductor at switching frequency. Then, the operating flux of the fundamental and ripple component can be calculated as: Bm1 = Bm ⋅

im1 ; im1 + im _ fs

Bm _ fs = Bm ⋅

Then, the overall core losses can be calculated as Pcore ( d , Bm , f s , x ) = Pbh ( d , Bm , f s , x ) + Pgap (d , Bm , f s , x )

(8) C. Winding losses The cross section of the windings can be calculated as Ac ( J ) =

I m1 J

(9)

To reduce the proximity losses of the windings, only one layer is selected for the windings. The height of the winding in each phase can be chosen as slightly lower than that of the window height. In the optimization, the height of the winging is H coil ( x) = G − 1.5cm

im _ fs

(4)

im1 + im _ fs

(10)

Since only one layer of the winding is selected, the thickness and height of the coil can then be calculated as:

where, Bm is the operating flux for the core material, Bm1 and Bm_fs are the operating flux density of fundamental and switching frequency condition respectively

hcoil (d , J , Bm , Lm , f s , x ) =

H coil ( x) Ns

The basic inductor equations used in the optimization program by neglecting the fringing factor of core lamination are:

t coil (d , J , Bm , Lm , f s , x ) =

Ac ( J ) hcoil

S c ( d , x) = E x ⋅ d N s (d , J , Bm , Lm , f s , x) =

Bm ⋅ Sc (d , x)

Rcoil (d , J , Bm , Lm , f s , x) =

B. Core losses The core losses of the inductor can be separated into two parts. One is the hysteresis losses in the lamination, it can be approximated as [6] [7] [8]: (6)

where, kc, Į, ȕ are three constants for the lamination, Vcore is the overall volume of the core material. The second part is the air gap losses generated by the fringing flux around the air gap [6][7][8]: 2 2 Pgap (d , Bm , f s , x) = k g E x ( f1Bm 1 + f s Bm _ fs )

ρcu ⋅ Lcoil Ac ( J )

(12)

(5)

where, S c (d , x ) is the core cross section area, Ex is the width of the core leg for lamination x, J is the current density of the coil winding, Lg is the overall air gap length. Ns is the turn of the winding.

Pbh (d , Bm , f s , x ) = kcVcore ( f1α Bmβ1 + f sα Bmβ _ fs )

where, hcoil and tcoil are the height and thickness of the coil respectively, Then, the resistance of winding for each phase can be calculated as

10 4 (im1 + im _ fs ) Lm

0.4π ⋅ N s ⋅ Sc ⋅ 10−8 Lg (d , J , Bm , Lm , f s , x) = Lc

(11)

(7)

where, Lcoil is the overall length of the coil winding for each phase, ρ cu is the resistivity of the copper winding under operating temperature Coil losses of fundamental components are relatively simple. Since the frequency is low, the proximity losses of the winding can be neglected. The dimensions of the windings are much lower than the skin depth. Then, the coils losses of the fundamental component in each phase can be calculated as Pcu1 (d , J , Bm , Lm , f s , x ) =

1 2 Rcoil ⋅ im 1 2

(13)

Under switching frequency, the skin depth is much lower than the height of the wind, the proximity losses of the winding has to be considered for the ripple current. By adding evenly distributed air gaps in each core, the per phase coil losses generated by the ripple current can be approximated as:

where, kg is the air gap losses constant for the lamination


Pcu _ fs (d , J , Bm , Lm , f s , x) =

$ Lm (d , J , Bm , Lm , f s , x ) = k m ⋅ (Wcu $Cu + Wcore $Core)

1 k p ⋅ Rcoil ⋅ im21 _ fs 2

(19) (14)

where, k p is the proximity factor of the winding at switching frequency. According to chapter.2 of [9], when multiple evenly distributed air gaps are applied in one leg, the proximity factor k p for rectangle wire can be calculated as:

where, Wcu and Wcore are the weight of the copper and core respectively, $Cu and $Core are the unit weight cost of the copper and core material, The cost of the $Cu and $Core are randomly picked up from the market in the optimization. km is a ratio between the inductor cost and the material cost. It varies with different supplier. IV.

ª § Ns 1 h k p (d , J , Bm , Lm , f s , x) = ⋅ coil ⋅ «2¨ « ¨ 3 δ cu 2 ⋅ N gap ¬« ©

· ¸ ¸ ¹

2

º + 1» » »¼

(15) where Ngap is the number of air gap in each leg, δ cu is the skin depth of the wind at switching frequency. Then, the total coil losses can be calculated by adding all above losses as: Pcu (d , J , Bm , Lm , f s , x) = Pcu _ fs + Pcu1

OPTIMIZATION RESULT

An optimization program combining all above equations was built up to optimize the cost of the main inductor. The overall program was written under Mathcad. The detailed optimization structure is shown in Figure.3. During the optimization, the maximum coil and core temperature is set as 160°C at 50°C ambient temperature, the maximum allowable operating flux density is set as 1.4T and the maximum window filling factor is set as 0.5. Bm < B max

(16) fs

D. Temperature rise There are three heat transfer mechanisms in the inductor: conduction, convection, and radiation. In the inductors optimized, there are air-duct added between the core and coil. Thus, the conduction heat transfer between the core and coils are neglected. The allowable maximum core and copper losses can be calculated similarly according to the following equation (Ch.6 of [9]).

(

)

4 q = εσAr ⋅ Tmax − Ta4 + hc Ac (Tmax − Ta )

max Trise < Trise

(17)

where, q is the allowable core/coil power losses of the inductor; Tmax is the maximum allowable operating temperature, Ta is the ambient temperature, Ar is the core/coil radiation surface area, Ac is the convection surface area of either core/coil; İ is the emissivity of the surface area, ı is the Stefan Boltzman constant, hc is the convection coefficient. According to [9], the convection coefficient can be approximated as hc = (3.33 + 4.8 ⋅ u∞ ) ⋅ L−0.288

0.02 < Lm < 0.12

(18)

where, L is the total distance of the boundary layer of the component. u ∞ is the velocity of the approaching flow. E. Cost formula It is assumed that majority of the materials cost of an inductor is copper and core materials. The overall cost of inductor can be assumed to be proportional to the copper and core cost for the inductor as

Figure 3. Optmization of main inductor Lm for regenerative LCL filter

A. Optimal cost of the inductor as a function of the inductance The optimal cost of the inductor as a function of the inductance value (from 0.02 to 0.16pu ) was first evaluated. In Figure.4 (a) and (b), the cost comparison of the inductor at 4kHz and 10kHz switching frequency under different cooling method are plotted respectively. In these two figures, the minimum cost of the inductor at the respective switching frequency is set as 1.0. It can be found from these two figures that the most cost effective inductance locates in 0.04~0.06pu under 10kHz and 0.07~0.12pu under 4kHz. The main reason for the cost increase under lower inductance area is due to the increase of ripple current component. This has largely increases the hysteresis losses in the core and the proximity losses in the coil. To prevent the inductor from over temperature, the core has to be over sized.


On the other hand, the primary reason for the cost increase under higher inductance area is due to the increase of inductance value. Under this condition, more materials have to be added to generate the correct inductance value. It can also be found from Figure.4 that the cost of the inductor reduces dramatically when forced air is provided. The air speed is another important factor for the optimization of the LCL filter and should be considered with system optimization.

forced air condition are defined as 1. This figure shows that the cost of the inductor reduces around 50% by increasing the switching frequency of the converter from 2kHz to 14kHz. The effectiveness of increasing the switching frequency on the cost reduction of the LCL filter is clearly verified. 0.14

0.12

fs = 10kHz Optimized inductance (uH)

uH(free air)

2.5

inductor cost comparison

2

1.5

0.10

uH(forced air)

0.08

0.06

0.04

1

0.02

10kHz, free air 10kHz, forced air

0.00

0.5

0

2

4

6

8

10

12

14

16

fs(kHz)

(a) 0 0.00

0.05

0.10

0.15

0.20

Lm(pu) 4

(a) 3.5

fs = 4k Hz inductor cost comaprison

inductor cost comparison

4 3.5

4kHz, free air 4kHz, forced air

3

free air

3

4.5

2.5

forced air

2.5

2

1.5

1

2 0.5

1.5 0

1

0

2

4

6

0 0.00

8

10

12

14

16

fs(kHz)

0.5

(b) 0.05

0.10

0.15

0.20

Lm(pu)

1.6

(b)

1.4 Bm:forced air Bm:free air

Figure 4. Optimal cost comparison of the inductor under various condition (a): fs = 10kHz, (b): fs=4kHz

1.2

Figure.5 (a) shows the optimized inductance as a function of the switching frequency. It shows that the optimal inductance of the inductor reaches 0.12pu when the switching frequency is 2kHz, however, it drops to 0.04pu when the switching frequency of the converter increases to 14kHz. Figure.5 (b) further shows the optimized cost comparison of the inductor as a function of the switching frequency. Under this figure, the optimized cost of the inductor at 14kHz and

Bm(T)

1

B. Optimal inductance design as a function of switching frequency In Figure.5, the optimal inductance, cost, and the operating flux density of the core of the main inductor under difference switching frequencies are provided.

0.8

0.6

0.4

0.2

0 0

2

4

6

8

10

12

14

16

fs(kHz)

(c) Figure 5. Optimal design of the inductor at various switching frequency (a): optimal inductance, (b): optimized cost of the main inductor, (c) maximum flux density of the inductor under optimized condition


Figure.5 (c) further shows the operating flux density of the core material under optimized condition. It shows clearly that the flux density of the core material has to be reduced dramatically to avoid temperature saturation under high switching frequency condition. V. TEST RESULT AND HARMONICS VERIFICATION Finally, an optimal design of the main inductor with natural convection under 4kHz switching frequency is built up. From Figure.4 (b) and Figure.5 (a), the inductance of the main inductor is selected as 0.09pu (432uH). Figure.6 shows the picture of the sample. It was tested on a 150hp regenerative drive. A 0.05pu capacitor with a capacitance of 80uF and the line side filter inductor with an inductance of 0.11mH are added to the LCL filter. The drive was connect to a direct line with 2MVA transformer and a short circuit ratio is higher than 100.

Figure 6. Picture of the sample of the optimized inductor under 4kHz condition ( 432uH )

Figure.7 shows the voltage and current waveform system of the system under full load condition. The waveforms shown from Ch1 to Ch4 are line side voltage, line current, main inductor voltage and converter respectively. The operating temperature of the inductor under 22°C ambient can be found from Figure.8. The line side input voltage and current THD measured is 3.1% and 4.9% respectively. VI.

CONCLUSION

The optimization of a main inductor for a LCL filter under various conditions was studied. The effectiveness of the optimization program was verified by a single sample. The following conclusion has been derived from this analysis. •

Studies have been done on the optimization of main inductor as a function of switching frequency and air speed. It can help the designed to make a trade off between the switching frequency and main inductance for an regenerative LCL filter;

•

The optimal inductance of the main inductor reduces when the switching frequency of the AFE rectifier increased. A roughly 50% of material cost reduction can be seen when the switching frequency changes from 2kHz to 14kHz. The benefit of increasing switching frequency on reducing the cost of magnetic component was clearly verified;

•

A dramatically cost reduction can also be seen when forced air cooling method are applied, however, the optimal inductance does not change when the air speed of inductor changes.

•

The overall optimization of the LCL filter and rectifier can largely reduce the overall cost of the system VII. REFERENCES

[1]

R. Wu, S. B. Dewan and G. R. Slemon: “ Analysis of an ac-to-dc voltage source converter using PWM with phase and amplitude control”. IEEE trans. On Industry Applications, Vol.27, No.3, march/April, 1991, pp.355-364;

Figure 7. Voltage and current waveform of the LCL filter study, (Ch1: line voltage: 500V/div; Ch2: line current: 200A/Div; Ch3: voltage of inductor La, 500V/Div; Ch4: current of inductor La: 200A/Div)

Figure 8. Steady state thermal image of the inductor under rated condtion [2]

[3]

S. Hansen, M. Malinowski, F. Blaabjerg, M Kazmierkowski: “Sensorless control strategies for PWM rectifiers”, proc. Of APEC 2000 conference, New Orleans (USA), February 2000; V. Blasko, V. Kaura, “ A novel control to actively damp resonance in input lc filter of a three phase voltage source converter”, IEEE Trans. On Industry Applications, Vol.33, No.2, 1997, pp.542-550;


[4]

S. Chandrasekaran, D. Borojevic, D.K. Lindner, “ input filter interaction in three phase AC-DC converters, Proc. of PESC’1999, Charleston, SC, USA, June/July, 1999, VOl.2, pp.987-992; [5] Marco Liserre, Frede Blaabjerg, and Steffan Hansen, “design and control of an LCL filter based three phase active rectifier”, proc. Of IEEE IAS 2001 conference; [6] Colonel Wm.T.Mclyman, “Magnetic core selection for transformers and inductors – A user’s guide to practice and specification”, Marcel and Dekker, Inc, New York and Basel, USA [7] Colonel Wm.T.Mclyman, “Transformer and inductor design handbook”, second edition, Marcel and Dekker, Inc, New York and Basel, USA [8] Reuben Lee, Donald Stephens, “Influence of core gap in design of current limit transformers”, IEEE Trans. On Magnetics, No.3, September, 1973, pp. 408-410 [9] Alex Van Den Bossche, Vencislav Cekov Valchev, “ Inductors and transformers for power electronics”, Taylor & Francis Group, Boca Raton, London, UK [10] Lixiang Wei, Richard A Lukaszewski, “ Analysis of mounting strap losses for high power high frequency inductors ”, to be published in the proc. of IAS’2007 conference , New Orleans, USA, 2007
