Active du/dt Filtering for Variable-Speed AC Drives

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Juha-Pekka Ström

ACTIVE DU/DT FILTERING FOR VARIABLE- SPEED AC DRIVES

Thesis for the degree of Doctor of Science (Technology) to be presented with due permission for public examination and criticism in the Auditorium 1382 at Lappeenranta Uni- versity of Technology, Lappeenranta, Finland, on the 17^th of December, 2009, at noon.

Acta Universitatis

Lappeenrantaensis

378

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Faculty of Technology

Lappeenranta University of Technology Finland

Reviewers Professor Heikki Tuusa

Laboratory of Power Electronics Tampere University of Technology Finland

Dr. Mika Sippola

Nidecon Technologies Oy Finland

Opponent Dr. Mika Sippola

Nidecon Technologies Oy Finland

ISBN 978-952-214-888-9 ISBN 978-952-214-889-6 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto

Digipaino 2009

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Abstract

Juha-Pekka Ström

Active du/dtFiltering for Variable-Speed AC Drives

Acta Universitatis Lappeenrantaensis 378

Dissertation, Lappeenranta University of Technology 127 p.

Lappeenranta 2009

ISBN 978-952-214-888-9, ISBN 978-952-214-889-6 (PDF) ISSN 1456-4491

An oscillating overvoltage has become a common phenomenon at the motor terminal in inverter-fed variable-speed drives. The problem has emerged since modern insulated gate bipolar transistors have become the standard choice as the power switch component in low- voltage frequency converter drives. The overvoltage phenomenon is a consequence of the pulse shape of inverter output voltage and impedance mismatches between the inverter, motor cable, and motor. The overvoltages are harmful to the electric motor, and may cause, for instance, insulation failure in the motor. Several methods have been developed to mitigate the problem. However, most of them are based on filtering with lossy passive components, the drawbacks of which are typically their cost and size.

In this doctoral dissertation, application of a new active du/dt filtering method based on a low-loss LC circuit and active control to eliminate the motor overvoltages is discussed. The main benefits of the method are the controllability of the output voltage du/dtwithin certain limits, considerably smaller inductances in the filter circuit resulting in a smaller physical component size, and excellent filtering performance when compared with typical traditional du/dt filtering solutions. Moreover, no additional components are required, since the active control of the filter circuit takes place in the process of the upper-level PWM modulation using the same power switches as the inverter output stage.

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Since additional switching is required in the output stage, additional losses are generated in the inverter as a result of the application of the method. Considerations on the application of the active du/dt filtering method in electric drives are presented together with experimental data in order to verify the potential of the method.

Keywords: Electric drive, output filter, active filter UDC 681.527.7 : 681.532.52

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Acknowledgments

The research documented in this work was carried out at the LUT Institute of Energy Tech- nology (LUT Energia) at Lappeenranta University of Technology (LUT) during the years 2006–2009. The research was funded by the Finnish Graduate School of Electrical Engineer- ing (GSEE), Vacon Plc., and Lappeenranta University of Technology.

The beginning of the active du/dtresearch came from Vacon Plc. during the fall 2006, as the author was invited to Vacon Plc. to discuss the topic of cable reflections and output filtering.

Especially the contribution of Dr. Hannu Sarén and Dr. Kimmo Rauma on the research topic is most sincerely acknowledged, as is also the valuable support by the Vacon Company during the research projects. Without you and Vacon Plc. this research would not have been possible.

I would like to thank the preliminary examiners of this dissertation, Professor Heikki Tuusa and Dr. Mika Sippola for their valuable comments on the manuscript. I am very grateful for your contribution and help in improving the thesis. I would like to express my gratitude to my supervisor, Professor Pertti Silventoinen, and to Dr. Julius Luukko and Dr. Mikko Kuisma for their valuable guidance and help during the process.

I am very grateful to Dr. Hanna Niemelä for her help in improving the language of the text.

I really appreciate your contribution, and your patience with my sometimes not-so-steady writing schedule. It has been a huge help in the writing.

I express my deepest thanks to my collegues, Mr. Juho Tyster and Mr. Juhamatti Korho- nen for their contribution, many ideas, and uncompromising attitude towards the active du/dt research. Your work for the development of the method, for the prototypes, and in the laboratory has really been worthy. I also thank for your help during the preparation of the manuscript.

I would like to thank all the people I have worked with at the Deparment of Electrical Engi- neering here at LUT during these years; especially those who have been spending all those coffee breaks – sometimes even the longer ones – around the coffee table. All the shared experiences when we have hit the road – in the name of science, of course – have always been something worth remembering. It has been a pleasure, thank you!

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Finally, I would like to express my deepest gratitude to my family; your support during all the rush, and your understanding for my absence during all those hours at work have been very important. This is for you Tiina, Pietu, and Neela; you are my all.

Lappeenranta, December 1^st, 2009

Juha-Pekka Ström

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List of Symbols and Abbreviations

Roman letters

A amplitude

A mains phase A

B mains phase B

C capacitance

C_DCLINK DC link capacitor

C mains phase C

c speed of light

f frequency

f_BW frequency bandwidth fc switching frequency f_osc oscillation frequency

G conductance

H system transfer function in Laplace plane U_out output voltage in Laplace plane

I current wave

i instantaneous current i_cm common mode current i_f filter current

IL load current

K active du/dtrise and fall time coefficients with respect to filter time constant

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l_c critical cable length L_f filter inductance L_L load inductance

L_m magnetizing inductance of an asynchronous machine L⁰_s transient inductance of an asynchronous machine L_rσ rotor leakage inductance of an asynchronous machine L_sσ stator leakage inductance of an asynchronous machine M number of charge periods

N number of charge pulses

n index in sum

P power

R resistance

R_loss equivalent loss resistance

R_r rotor resistance of an asynchronous machine R_s stator resistance of an asynchronous machine s Laplace variable,s=σ+jω

T period

t time

t₁ charge sequence pulse turn-off time, same ast_1/2

t₂ time at which the charge sequence is complete, same as active du/dt t_r tcorr load current correction pulse length

t_f fall time

tp cable propagation delay t_r rise time

t_1/2 instant at which the system output voltage is half the applied step amplitude u instantaneous voltage

u_A mains voltage phase A

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u_B mains voltage phase B ucm common mode voltage u_C mains voltage phase C U_DC DC link voltage u_inv inverter output voltage u_out output voltage

uU inverter output phase U voltage u⁰_U filtered inverter output phase U voltage u_V inverter output phase V voltage u⁰_V filtered inverter output phase V voltage u_W inverter output phase W voltage u⁰_W filtered inverter output phase W voltage U inverter output phase U

V voltage wave

V⁻ reflected voltage wave V⁺ incident voltage wave V_p voltage peak value V inverter output phase V W inverter output phase W XC capacitive reactance X_L inductive reactance

∆z differentially small increment in position

z position

Z₀ characteristic impedance Zc cable impedance Z_L load impedance Z_m motor impedance

Greek letters

α attenuation constant

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ε_eff effective dielectric constant experienced by electromagnetic wave at a certain dielectric configuration at a certain frequency

γ complex propagation constant Γi inverter reflection coefficient ΓL load reflection coefficient Γm motor reflection coefficient

λ wave length

µ permeability

ωn resonance frequency ω angular frequency

φL load reflection phase angle

δ_s skin depth

σ conductivity

νp propagation velocity

ζ damping factor

Subscripts

max maximum value

Other symbols

δ(t) Dirac delta function, impulse function ε(t) Heaviside step function

Acronyms

AC Alternating current DC Direct current DOL Direct-on-line

EMI Electromagnetic interference

ETD Ferroxcube ETD coil former cores and accessories product line FIR Finite impulse response digital filter

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FPGA Field programmable gate array

IEC International electrotechnical commission IGBT Insulated gate bipolar transistor

LC Electrical circuit consisting of an inductive and capacitive component MCMK Power cable type

NXP Vacon NX performance product line

PVC Polyvinyl chloride, insulating material used e.g. in power cables PWM Pulse width modulation

RLC Electrical circuit consisting of an inductive, capacitive, and resistive component TEM Transversal electromagnetic mode of wave propagation

VSI Voltage source inverter

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Chapter 1 Introduction

The consumption of energy has considerably increased in the European Union during the past years, despite the efforts of union-wide and national policies and programmes to increase energy efficiency. During the period of 1990–2005, the final electricity consumption in the EU-27 countries has increased by 29 %, at an annual growth rate of 1.8 %, and in Finland by 37 %, at an annual growth rate of 2.3 % (Eurostat, 2007). In the future, even more pressure will be put on cutting electricity consumption. The European Union has agreed on an inte- grated energy and environmental policy, and one of the main objectives is to save 20 % of the projected energy consumption by the year 2020.

In order to rise to the challenge, modern electronic power converters and control of electric power on the whole play a very important role in improving the energy efficiency of upcoming and present installations. A very typical and important application example of such a system is an electric drive, in which electrical power is converted into mechanical torque, or vice versa. To control the electromechanical conversion process, in many cases, an electronic power converter is essential in a modern electric drive. This is achieved by controlling the output voltage and frequency of the power converter to match the demands of the application.

This leads to improvements in energy efficiency, especially for instance in pump, fan, and compressor applications, and also in the control of the process in general, when compared with a noncontrolled drive. This is one of the main reasons why power-converter-controlled electric drives have established themselves in the industry during the past decades. This has taken place especially in the low-voltage segment (under one thousand volts) in both low- and high-power drives, because of the rapid development of low-voltage semiconductor power switches. Typically, the power converter in an electric drive is called a frequency converter, and the drive is called a variable-speed or a variable-frequency drive.

A major part of the produced electricity, over 40 % in the EU-27 countries, is consumed in the industry (Bertoldi and Atanasiu, 2007), for example in the above-mentioned, numerous electric drives. Even though frequency-converter-controlled electric drives have been applied especially in new electric drive installations, even more energy could be saved by installing a

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frequency converter to all suitable electric drives. In the industry, of all the installed electric drives, induction motor drives are widely used in the industry, and are generally considered very reliable. However, the switched-mode operation of the frequency converter may cause adverse effects in the drive, as will be discussed later in this chapter. Therefore, various filtering solutions have been introduced to be used in conjunction with converter drives in order to mitigate the effects.

In this dissertation, a new output filtering method consisting of a passive LC filter circuit and active control is developed for induction motor drives. Compared with more traditional, completely passive approaches, the filtering performance is improved. Further, the size of the electrical components is decreased in terms of both electrical and physical dimensions, thereby improving the integrability of the filter, decreasing filter losses, and reducing the cost of the actual filter. However, extra losses are introduced as a result of extra switching of the output stage. The method is verified by both a theoretical analysis and measurements for induction motor drives utilizing a modern frequency converter that uses fast switching IGBT power switches. The focus of this dissertation is on the development and feasibility study of the method, while the actual implementation on a real electric drive still requires further development.

The work documented in this doctoral dissertation focuses on induction motor drives only, because of their large number of installations in the industry. However, a frequency converter can as well be applied to generator and synchronous motor drives. The number of converter drives is also likely to increase in the future, because of the significant improvements in energy efficiency for example in the above-mentioned motor drives, and in decentralized and renewable energy production.

Further, there is no reason why the method should not be applicable also to other types of machines suitable for converter drives, because the developed output filtering method is inde- pendent of the electric motor properties present in the drive. Only the output voltage is shaped to achieve a more motor-friendly behavior by decreasing the du/dtvalue of the transients. The filtering method does not interfere with the upper-level control of the drive, because the control of the filter circuit can be carried out as the lowest level of modulation. The developed method may even improve the control performance, since the motor terminal voltage, and therefore motor flux, can now be accurately predicted, because the harmful cable oscillation is eliminated when the method is applied.

1.1 Background and motivation of the work

The voltage source inverter (VSI) based on insulated gate bipolar transistors (IGBT) applying pulse width modulation (PWM) has established as the frequency converter in low-voltage AC drives. As a result of the remarkable advancements in the semiconductor power switch device generations, the switching losses have reduced significantly. This has made it possible for example to reduce the sizes of cooling profiles and device enclosures and to use higher switching frequencies. Using a higher switching frequency results in a more sinusoidal motor

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1.1 Background and motivation of the work 17

current with less ripple and less copper loss. However, both the switching losses in the inverter output stage and the iron losses caused by eddy current losses in the motor increase as a function of switching frequency (Mohan et al., 2003).

The basic operating principle of a voltage source inverter using pulse width modulation is presented in Figure 1.1.

0 0.005 0.01 0.015 0.02

−500 0 500

Time [s]

a)

Voltage [V]

0 0.005 0.01 0.015 0.02

−500 0 500

Time [s]

b)

Voltage [V]

Figure 1.1. Normal sine wave, 50 Hz, 400 V, three-phase, phase-to-phase AC voltages available from the standard European grid are shown in Figure 1.1a. In Figure 1.1b, the same voltages are constructed from rectified AC voltage applying pulse width modulation (PWM). The modulated voltage consists of switched voltage pulses, which are modulated according to the reference, which is in this case the 50 Hz three-phase sine voltages.

Figure 1.1a shows the standard 50 Hz, three-phase sine AC voltages available from the standard European grid. These are the voltage waveforms for which most electric motors are designed. However, in an electric drive using a power converter, the output voltage waveform is quite different from the sine wave, as shown in Figure 1.1b. Because of the requirement to be able to control the output frequency and voltage, the electrical power available from the grid has to be constructed by using an inverter to produce the desired output voltage properties. In the most common case, the output voltage is produced using pulse width modulation from rectified AC voltage, in which the width of the voltage pulse is modulated according to the reference voltage. Therefore, the output voltage consists of steep rising and falling edges of the DC voltage, instead of true sine wave behavior. This has a remarkable effect on the frequency content of the output voltage. The properties of the output voltage edges depend on the properties of the power switches used in the output stage of the inverter.

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Although the IGBT has clear advantages when set against previous generations of semiconductor power switches, the remarkable advances in the switching times also manifest certain drawbacks far more clearly than the older generations of power switches: The rise and fall switching times of an IGBT are very short, at present in the order of tens of nanoseconds at best, and therefore the rate of change, namely du/dt, in the inverter output voltage pulse is very high. Hence, the output voltage contains a broad range of frequencies, including a lot of high-frequency components (Skibinski et al., 1999). In the industry, a typical length of the interconnecting cable is tens or hundreds of meters, which is substantial compared with the wavelength of the high-frequency components present in the fast transient voltages.

This leads to voltage reflections resulting in transient overvoltages at the motor terminals and electromagnetic oscillation in the motor cable (Persson, 1992) and (Saunders et al., 1996).

In order to suppress these effects of the fast switching transients, passive lowpass filtering is typically applied to the output voltage to narrow the frequency spectrum of the motor voltage below the natural oscillation frequency of the motor cable.

These effects have been mitigated by using many different passive filtering topologies, which are typically somewhat large in size and therefore expensive, but not very effective in all respects. The active du/dtfiltering method presented in this dissertation is based on a passive LC filter circuit and active control of the filter using pulse width modulation: each transient or edge in the fundamental modulation of the inverter has to be supplemented with additional edge modulation to provide control for the filter circuit to produce output voltage of the desired shape in a controlled way. Both the guidelines of pulse width modulation and the behavior of a passive LC circuit are commonly known and documented, whereas combining these in the output filtering of an electric drive has novelty value.

However, there are publications considering active du/dtcontrol in the inverter output voltage, see (Idir et al., 2006) and (Kagerbauer and Jahns, 2007). In these, the analysis is carried out from a different point of view, for example EMI reduction, and the switching transition speed of the power switch is reduced to decrease the EMI produced by the inverter output stage.

Therefore, filtering is implemented on a totally different basis than the work carried out in this study. By using the method presented in the publications for output filtering of the drive, where the required rise and fall times are in the order of microseconds, as discussed later in Chapter 3, significant switching losses would be generated, and therefore the methods are not beneficial for conducting output du/dtfiltering.

1.2 Objective of the work

The main objective of the study was to develop an efficient source filter solution for electric drives, in terms of both electrical performance and size. In this study, the goal is achieved by active control of the filter circuit, which is based on fast control of the circuit and fast switching properties of the modern semiconductor power switches. This results in better electrical performance, but also in both electrically and physically smaller filter components.

This in turn provides better electrical performance of the output filter and savings both in terms of the filter size and cost, and therefore better integrability of the output filter. The

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1.3 Outline of the thesis 19

inductor in particular is a costly component in a traditional passive du/dt filter, and it is the component, in which major cost savings can be achieved in the total cost of the filter.

Furthermore, the method presented will benefit from the development of faster and more efficient power switch components, for example the development of silicon-carbide (SiC) technology for power switches. In addition, an advantage of active du/dtis that the faster the components are and the less switching loss is generated, the more beneficial it will be for the developed output filtering method.

1.3 Outline of the thesis

This doctoral dissertation studies output filtering needs arising from the switching-mode operation of a motor driven with a frequency converter. This is mainly a result of the advancements in semiconductor power switch transition times between the conducting and noncon- ducting states. Existing output filtering solutions and the problems caused by the fast transitions are discussed, and a new output filtering method to be used in a frequency converter applying fast power switches is introduced. The theoretical background for the method is developed, and the feasibility of the method is verified by implementing it in a real induction motor drive, which consists of a standard industrial frequency converter with a custom-built control and an induction motor.

The rest of the dissertation is divided into the following chapters:

Chapter 2 gives general information about the background and the problems evolved in frequency-converter-fed electric drives as a result of the development of the power switch components. Common solutions to the problems presented in the literature are also discussed in brief.

Chapter 3 discusses output filtering of a frequency-converter-fed electric drive and introduces issues to be taken into account in the design of output filtering for a certain electric motor drive. The developed active output filtering method is presented, and the theory for application of the method is provided. Design considerations for the implementation of the method are presented.

Chapter 4 introduces issues related to the developed active output filtering method in an actual electric drive. Guidelines are given for solving these issues, when the output filtering method is applied to a drive. Measurements using a prototype equipment are presented. The objective of the measurements is to show that the theory developed is feasible and the narrow pulses required by the method are in fact achievable in standard industrial electric drive hardware. Considerations especially for the selection of components are presented.

Chapter 5 concludes the work covered in this dissertation and discusses the results obtained.

The usability of the results is evaluated and suggestions for future work are given.

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1.4 Scientific contribution

The scientific contributions of this doctoral dissertation are:

• Development of a new active output filtering method, which consists of a passive LC circuit and a specific control of the circuit in order to produce voltage slopes of designed length to suppress the effects of fast transients in an electric drive.

• Formulation of the theoretical background for the application of the active du/dtfiltering method in an electric drive.

• Development of guidelines for the filter component value selection and the basis for the corresponding control sequences for the application of the method in an electric drive.

• A method is introduced for correction of the error caused by the load current of the motor present in the drive.

• The method is proven to be a potential output du/dt filtering solution by a series of experimental measurements.

The author has published research results related to the subjects covered in the dissertation as a co-author in the following publications:

1) J.-P. Ström, J. Tyster, J. Korhonen, K. Rauma, H. Sarén and P. Silventoinen, "Active du/dtFiltering for Variable Speed AC drives," in13^thEuropean Conference on Power Electronics and Applications, EPE 2009, 8–10 September, Barcelona, Spain, 2009, (Ström et al., 2009).

2) J. Korhonen, J.-P. Ström, J. Tyster, H. Sarén, K. Rauma and P. Silventoinen, "Control of an Inverter Output Active du/dt Filtering Method", inThe 35^thAnnual Conference of the IEEE Industrial Electronics Society, IECON 2009, 3–5 November, Porto, Portugal, 2009, (Korhonen et al., 2009).

3) J. Tyster, M. Iskanius, J.-P. Ström, J. Korhonen, K. Rauma, H. Sarén and P. Silventoinen,

"High-speed gate drive scheme for three-phase inverter with twenty nanosecond minimum gate drive pulse," in13^thEuropean Conference on Power Electronics and Appli- cations, EPE 2009, 8–10 September, Barcelona, Spain, 2009, (Tyster et al., 2009).

4) J.-P. Ström, H. Eskelinen and P. Silventoinen, "Manufacturability and Assembly Aspects of an Advanced Cable Gland Design for an Electrical Motor Drive,"Intl. Journal of Design Engineering, Vol. 1, Issue 4, 2009.

5) J.-P. Ström, M. Koski, H. Muittari and P. Silventoinen, "Analysis and filtering of common mode and shaft voltages in adjustable speed AC drives," in12^thEuropean Conference on Power Electronics and Applications, EPE 2007, 2–5 September, Aalborg, Denmark, 2007.

6) J.-P. Ström, M. Koski, H. Muittari and P. Silventoinen, "Transient Over-Voltages in PWM Variable Speed AC Drives - Modeling and Analysis," inNordic Workshop on Power and Industrial Electronics, 12–14 June, Lund, Sweden, 2006.

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1.4 Scientific contribution 21

J.-P. Ström has been the primary author in publications 1 and 4–6. The background research for publications 1–3 has been done together by J.-P. Ström, Mr. J. Korhonen, and Mr. J.

Tyster. The prototype used in the measurements of publications 1–2 was developed by Mr.

J. Tyster and Mr. J. Korhonen. The prototype used in publication 3 was developed by Mr. J.

Tyster and Mr. M. Iskanius. Measurements for publications 1–3 were carried out by the first authors of the corresponding publications.

Background research for publication 4 was carried out by the authors. The research on the manufacturability and assembly aspects in publication 4 was carried out by Dr. H. Eskelinen.

The cable gland prototypes were constructed by the Department of Mechanical Engineering at Lappeenranta University of Technology and the measurements were carried out by J.-P.

Ström.

For publication 5, background research was carried out by Ms. H. Muittari. Filter prototype construction and the measurements were carried out by J.-P. Ström and Ms. H. Muittari. For publication 6, J.-P. Ström was in the major role in the background research, measurements and writing, with the help of the co-authors.

The author is designated as a co-inventor in the following patents or patent applications considering the subjects presented in this dissertation:

FI Patent 119669 B "Jännitepulssin rajoitus". Patent granted Jan 30 2009, (Sarén et al., 2009).

EU Patent application 08075493.0 - 1242 "Limitation of voltage pulse". Application filed May 19 2008, (Sarén et al., 2008a).

US Patent application 20080316780 "Limitation of voltage pulse". Application filed Dec 25 2008, (Sarén et al., 2008b).

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Chapter 2 Cable-reflection-induced terminal overvoltages in variable-speed drives

Along with the development of new power semiconductor switching components and identi- fication of the side effects produced by the frequency converters applying these components, the topic of cable reflection has been under extensive research, and numerous scientific publications can be found considering both the phenomenon itself and various means to mitigate its effects. Some key publications on cable reflections are for example (Persson, 1992) and (Saunders et al., 1996).

As presented in the introductory chapter, three-phase motors are controlled by means of variable voltage and frequency, and in a very typical case, this is implemented by using a switching-mode DC to a three-phase AC converter, typically a voltage source inverter (VSI) applying pulse width modulation (PWM). The energy from the utility source is rectified into a DC link capacitor by using a rectifying bridge, and the DC link capacitor acts as the low- impedance voltage source for the inverter bridge.

The AC voltage is formed from the DC link voltage by the inverter bridge as a series of pulses, which have a constant amplitude – neglecting the DC link fluctuations – and a varying width, the output of the phases being connected either to the positive or negative DC link rail; therefore, the phase-to-phase voltage between two phases can be either the positive or negative DC bus voltage. A schematic of a main circuit of a frequency converter is shown in Figure 2.1. Further, a possible output filter connection is shown along with a typical motor common-mode current path.

In order to keep the losses produced in the switching operation of a single power semiconductor component in the inverter bridge to a minimum, the transition time between the on- and off-states (and vice versa) of the switching component should be made as short as possible.

This is because the voltage across the component is larger than the on-state saturation voltage

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CL

M

D C - D C +

u 'U u 'V

u 'W

O u t p u t f i l t e r ( i f a p p l i e d )

C D C L I N K u U u V u W u C M

iC M

U V W

u Au Bu C

Figure 2.1. Frequency converter main circuit. Power from the grid is rectified into the DC link. The motor AC voltage of variable frequency and voltage is generated from the DC link voltage using the three-phase inverter bridge shown. A possible output du/dtfilter, and a typical motor common-mode current path are also presented.

of the component and a possible current flowing through the component will generate power loss (heat) during the transition according to the following equation

P= 1 T Z

uidt. (2.1)

On this account, the transitions in the voltage pulses generated by the DC to AC converter in the adjustable speed drive are kept as short as possible, leading to the fact that the steepness of the edges of the voltage pulses is high. In an inverter power switch component generally applied, that is, the insulated gate bipolar transistor (IGBT), the transition time between the states is at fastest in the order of tens of nanoseconds, as can be seen for instance in the next section. In addition to the benefits presented above, the fast switching voltage transient and thereby the output voltage of the inverter contains a lot of high-frequency components as a byproduct of the switching mode operation. The frequency components beside the base frequency of the electric drive are by definition unnecessary and even harmful to the operation of the drive, but are not irrelevant for the operation of the drive. This is the key difference between the voltage waveforms in a traditional direct-on-line (DOL) and VSI-converter-fed drives.

The switching transients occuring in the inverter are – and have to be – fast, when compared with the fundamental and switching frequencies. Therefore, the output voltage waveform contains in addition to the fundamental base frequency, switching frequency, and their har- monics, high-frequency components resulting from the steep voltage pulse edges extending up to the megahertz range (Skibinski et al., 1999). If the speed of propagation in the motor cable is for example in the order of 0.5c, see for example (Skibinski et al., 1997; Ahola, 2003), the wavelength of a 50 Hz signal is in the order of thousands of kilometers, whereas the wavelength of a signal of 1 MHz is only 300 meters.

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2.1 Frequency spectrum of the output voltage of a typical three-phase switching mode

inverter 25

Hence, the lengths of a typical motor cable, which are in the order of tens to a few hundred meters, are substantial compared with the high-frequency components present in the inverter output voltage. Therefore, each switching in the inverter output stage induces a traveling wave into the motor cable, and the transmission line theory must be applied in the analysis of the behavior of the traveling waves in the motor cable (Persson, 1992); see Chapter 4 for measurements of the propagation speed for the MCMK power cables used in the measurements of this dissertation.

This also sets special requirements for the motor cabling and the insulations in the electric motor, because the motor and the motor cable are typically designed for low operating frequencies, and also the effects caused by the high frequency content in the output voltage must be taken into account in a converter drive, for example the overvoltages caused by wave reflections, as will be discussed later in this chapter.

2.1 Frequency spectrum of the output voltage of a typical three-phase switching mode inverter

As presented in (Skibinski et al., 1999), the output voltage of a pulse-width-modulated (PWM) voltage source inverter can be approximated as a series of trapezoids of varying width, and the frequency spectrum of the signal can be approximated by means of Fourier analysis (Zhong et al., 1998). An example of an inverter output voltage and corresponding differential-mode voltage spectrum presented in (Skibinski et al., 1999) are shown in Fig- ures 2.2a and 2.2b.

tr

T = 1 / f_c t [ s ] f [ H z ]

Uphase [V] fc fB W

U D C

Uphase [dB] 0 - 1 0 0 - 2 0 0

a ) b )

Figure 2.2. a) Inverter phase output voltage and b) corresponding voltage spectrum. In this example, from (Skibinski et al., 1999), the switching frequency fcis 500 Hz, the duty cycle 50 % andtr200 ns.

The frequency axis is logarithmic.

The main parameters that the spectral width of the signal depends on are the rise timet_rand the switching frequency f_c. According to (Zhong et al., 1998), the theoretical spectrum is flat until f_c, and it begins to attenuate after this frequency by 20 dB/decade and after f_BWby

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40 dB/decade. Therefore, f_BWcan be used as a rough approximate for the spectral width of the inverter output voltage waveform (Skibinski et al., 1999):

f_BW≈ 1

πt_r. (2.2)

When IGBT power switches with typical transition times between 50 and 400 ns (Saunders et al., 1996; IEC, 2007) are employed in the inverter output stage, the frequency spectrum of the output voltage extends up to the radio frequency region, from hundreds of kilohertz up to several megahertz. As an example, the rise and fall times and the calculated bandwidth estimate using (2.2) for some Semikron Semitrans packaged IGBT modules are presented in Table 2.1. These modules are selected as an example, because they fit in the Vacon NXP series frame size 6 industrial frequency converter, which is also used in the prototype equipment and tests. The total switching energy at the rated, continuous collector current is also presented.

Table 2.1. Rise and fall times, the total switching energies and the calculated bandwidth estimates of some Semikron Semitrans packaged IGBT modules, as stated by the manufacturer

Module Typical Typical Total switching Bandwidth

type rise time fall time energy estimate

tr t_f @100 A Eq. (2.2)

Semikron SKM

100GB123D 70 ns 70 ns 27 mJ 4.5 MHz

1200 V Standard Semikron SKM

100GB125DN 40 ns 20 ns 22 mJ 16 MHz

1200 V Ultra fast Semikron SKM

100GB176D 40 ns 145 ns 100 mJ 10.6 MHz

1700 V Trench

Spectrum measurements of an inverter output voltage are presented for example in (Skibinski et al., 1999), in which the spectral width was found to reach up to the megahertz range. In the example, rise time was 200 ns and the spectral width was more than 1 MHz.

2.2 Overvoltages caused by switching transients

In a centralized industrial installation, typical motor feeding cable lengths vary from tens of meters up to a few hundred meters. Unless the converter is installed immediately next to

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2.2 Overvoltages caused by switching transients 27

the motor, the motor cable has to be regarded as a transmission line, if the electric drive is converter fed. In this case, the voltages and currents are not only functions of time, but have to be regarded also as functions of position along the motor cable. This is because the inverter output voltage contains frequency components that have wavelengths in the order of the motor cable length, as pointed out above. As a consequence, voltage and current oscillations may occur along the power cable. Providing that the physical length of the motor cable lengthl is less thanλ/16 of a frequency component in the output voltage, the voltages and currents can be assumed to be constant along the transmission line, and hence no transmission line analysis is required, nor cable oscillations or overvoltage caused by it have to be taken into account.λ is the wave length of a certain frequency in the cable. Equation (2.2) can be used to roughly approximate the spectral bandwidth of the inverter output voltage.

The transmission line theory, see (Heaviside, 1893, 1899), which describes the propagation of an electromagnetic wave along a transmission line, has been succesfully applied to the analysis of cable oscillations and voltage reflections, as pointed out above. In general, the motor cable consists of several phase conductors and a ground conductor, since the three- phase AC system is used in most installations. Therefore, the motor cable is generally a multiconductor transmission line.

However, the motor cable is typically presented as a two-wire transmission line model, because the analysis is simplified and the use of multiple phase transmission line models is avoided. In addition, since the cable oscillation phenomenon takes place at each transition of the inverter output stage, the use of a one-phase equivalent circuit is justified. Nonetheless, the limitations of the simplification have to be taken into account in the analysis: only one phase can be considered at a time, and the other phases have to be assumed stationary and in a steady state during the analysis.

In the two-wire transmission line model, the electromagnetic wave is assumed to propagate in the pure transverse electromagnetic (TEM) mode. However, in an actual motor cable, the mode of propagation is not pure TEM, as the wave also has small longitudinal components, for example because of the finite conductivity of the conductors. In practice, the structures of the fields are similar to pure TEM, and the wave can be approximated as a TEM wave (Ahola, 2003). This kind of a propagation mode is called a quasi-TEM mode.

2.2.1 Transmission line properties of the motor feeder cable in an elec- tric drive

The electrical length of the transmission line depends on the phase velocity (propagation velocity) and the frequency of the electromagnetic wave. The relation between the phase speed, νp, the wave length,λ, and the frequency, f, of the wave is described by the fundamental equation:

νp=λf. (2.3)

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The equivalent circuit of a two-wire transmission line of an infinitesimal length∆zis presented in Figure 2.3, which consists of the distributed parameters inductance, capacitance, resistance, and conductance per unit length of the transmission line. These parameters describe the properties of the transmission line and depend on the geometry and the dielectrics used in the physical conductor. Series inductance describes the self-inductivity of the conductor, capacitance refers to the natural capacitance in the proximity of the conductors, resistance represents the resistive losses caused by the finite conductivity of the conductor, and finally, conductance describes the losses owing to the conductivity and the dielectric losses caused by the polarization of dipoles in the insulating material. The inductance and capacitance rep- resent delay, whereas resistance and conductance express losses (or attenuation) along the transmission line. A transmission line of finite length can be thought to consist of a group of elements, as presented in Figure 2.3, connected in series.

+ - i ( z , t ) D z

v ( z , t ) v ( z + D z , t ) + -

R D z L D z G D z C D z i ( z + D z , t )

Figure 2.3. Equivalent circuit of a two-wire TEM transmission line of an infinitesimal length. R, L, G, andCare the distributed resistance, inductance, conductance, and capacitance of the line per unit length. The voltagesvand currentsiindicated in the figure describe the voltages and currents in the transmission line atzand∆zat the time instantt.

It can be derived that on a transmission line of this kind, the voltages and currents may vary not only as a function of time, but also as a function of positionz, according to the telegra- pher’s equations (Heaviside, 1899). The voltages and currents consist of a superposition of incident and reflected waves. Therefore, standing waves may occur on the line. The properties of the transmission line are defined by the complex propagation constantγ and the characteristic impedanceZ0. The propagation constant is defined by the equation (Collin, 1992) p. 88

γ=p

(R+jωL) (G+jωC) =α+jβ, (2.4)

whereα is the attenuation constant, andβ is the propagation constant, which describe the damping and the wavelength as a function of the length of the transmission line with the distributed circuit parameters resistanceR, inductanceL, conductanceGand capacitanceC per unit length. The characteristic impedance of a transmission line is defined as (Collin, 1992) p. 88

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Z₀= s

R+jωL

G+jωC. (2.5)

If the transmission line is assumed lossless or the losses are negligible, the characteristic impedance can be approximated with the following equation:

Z₀= rL

C. (2.6)

The characteristic impedance defines the relation between the amplitudes of the corresponding voltage and the current waves on the transmission line, thus

Z₀(z) =V(z)

I(z), (2.7)

for every positionz. In general, all the distributed parameters are functions of frequency, and therefore the propagation constant and the characteristic impedance are also frequency dependent.

The propagation velocity of the wave can be calculated using the following equation:

νp=ω β = 1

√ε µ = 1

√LC, (2.8)

whereε andµ depend on the dielectric material used in the power cable. The propagation velocity of the wave depends only on the properties of the dielectric materials, if the currents propagate only along the surface of the conductor. However, because of the finite conductivity of the conductor, the currents flow also inside the conductive material. The current distribution on the cross-section of the conductor at a certain frequency is described by skin depth, which depends on the angular frequencyω, permeabilityµ, and conductivity σ as follows (Wheeler, 1942)

δs= s

2

ω µ σ. (2.9)

The skin-effect also increases the resistive losses, because the current density near the surface of the conductor increases, increasing the ac resistance of the conductor. In addition, the proximity effect increases the ac resistance even further in budled cables.

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2.2.2 Transmission line discontinuities

A discontinuity along a transmission line means a change in the characteristic impedance.

The characteristic impedance is the proportion of voltage and current waves. At a discontinuity, part of the incident power passes through the interface while part reflects back to the original direction, because the potential has to be equal at the point of mismatch. Generally, any variation in the geometry of the dielectrics along a propagation path causes a change in the characteristic impedance and therefore a reflection. Typically, a change in the characteristic impedance is a consequence of a mismatched load impedance at the end of a transmission line, or changes in the type of the transmission lines along the propagation path. In addition, connectors, connections, and junction boxes typically employed in electrical power engineering cause a significant change in the geometry of the propagation path and therefore in the characteristic impedance.

The relationship between the incident wave and the reflected wave depends on the difference of the characteristic impedances at the discontinuity: the greater the difference, the more of the incident wave is reflected. The reflection coefficient is defined at the impedance mismatch as follows (Heaviside, 1899), p. 375:

Γ_L=V⁻

V⁺=Z0−ZL

Z₀+Z_L=|Γ_L| ·e^jφ^L, (2.10) whereV⁺is the incident wave,V⁻is the reflected wave at the discontinuity,Z0is the characteristic impedance of the transmission line, andZL is the loading impedance seen at the discontinuity in the direction of the incident wave.|Γ_L|defines the magnitude of the reflected wave andφLdefines the phase angle shift of the reflected wave with respect to the incident wave at the mismatch point. If the transmission line is perfectly matched,Z_L=Z₀, no reflection takes place, as can be seen from the above equation. If the transmission line is terminated to a short circuit (Z_L=0) or an open circuit (Z_L=∞), all the incident wave is reflected at a phase angle of 0 or 180 degrees, correspondingly. During the transient, the electric motor resembles an open circuit at the end of the motor cable, leading to an in-phase reflection and overvoltage as an outcome of the superposition of the incident and reflected waves.

The voltages and currents can be written as a function of the length of the motor cable applying the reflection coefficient as follows:

V(z) =V₀⁺e^jγz 1+ΓLe^−j2γz

(2.11)

I(z) =V₀⁺

Z₀e^jγz 1−Γ_Le^−j2γz

. (2.12)

The above equations show that if the transmission line is not terminated at the characteristic

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impedanceZ₀, the amplitudes of the voltage and current waves become functions of position, and standing waves occur at the transmission line.

2.2.3 Discontinuities in a typical inverter-fed electric drive

The main factors contributing to the overvoltages are the magnitude and rise time of the output voltage pulses, the interconnecting power cable length, the motor characteristic impedance, and the impedance mismatch between the characteristic impedances of the cable and the motor.

As the inverter output stage is operated, switching transient injects an incident voltage wave in the interconnecting power cable that propagates toward the electric motor. In the inverter-fed electric drive, there are at least two significant impedance mismatches between the inverter output stage and the motor: the interfaces between the inverter and the motor cable and between the cable and the motor, if the cable is solid and there are no additional connections along the cable. Because of the geometry and the construction, the characteristic impedances of the motor and the motor cable are usually significantly mismatched.

The amplitude of the reflected wave in proportion to the incident wave is defined by the voltage reflection coefficientΓmat the motor terminal:

Γm=Z_m−Z_c

Z_m+Z_c, (2.13)

whereZmis the motor characteristic impedance andZcis the characteristic impedance of the interconnecting power cable. The maximum peak voltage at the motor terminal expressed using (2.13) results in (Saunders et al., 1996)

Vp

_(z=l) = (1+Γ_m)·UDC, (2.14) where the amplitude of the incident wave equals the amplitude of the voltage at the drive output,UDC, and the motor reflection coefficient isΓm. Because the impedance of the motor resembles an open end compared with a typical cable impedance, the incident wave is reflected back in-phase from the interface of the motor and the cable. Therefore, the voltage reflection can cause overvoltages up to twice the bus voltage at the motor terminal. The overvoltage may degrade the insulation and potentially produce destructive stress on the insulation system of the motor. Typically, the voltage is not evenly distributed in the stator winding; a major part of the voltage is across the first few coil rounds before the voltage distribution is balanced in the winding. Furthermore, the faster the transient, the more of the voltage occurs across the first coil round, which adds to the stress caused to the insulation of the stator winding (Suresh et al., 1999), (Hwang et al., 2005), and (IEC, 2007).

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The load reflection coefficient at the motor end depends on the size of the motor. As the size of the motor increases, the reflection coeffient decreases, for example because of larger stray capacitances, and the theoretical maximum value of the overvoltage decreases from the double voltage. Typically, in the literature, the reflection coefficient is reported to vary between 0.65 and 0.95, which causes a theoretical overvoltage of 1.65 to 1.95 times the DC link voltage. Typical motor reflection coefficients for various motor sizes are presented for example in (Saunders et al., 1996) and (Skibinski et al., 1998). However, it has to be taken into account that the motor impedance, and the load reflection coefficient, similarly as other transmission line parameters, are frequency dependent.

As the incident voltage wave is reflected back from the motor terminal, the reflected wave starts to propagate back towards the frequency converter. A new reflection takes place at the interface of the motor cable and the inverter, the magnitude of which depends on the reflection coefficient at that interface. The reflection coefficient can be obtained from Eq.

(2.10), if the characteristic impedances are known. The characteristic impedance of the cable can be determined by measurements as presented in (Ahola, 2003). The impedance of the output stage depends on the state of the switches; measurements of the inverter output stage impedances as a function of switching state are presented for example in (Kosonen, 2008).

Generally, the reflection coefficient of the inverter end is approximated as a short circuit in the literature, because the DC link capacitor and freewheeling diodes are assumed to act as a short circuit to the steep-edged switched voltages (Skibinski et al., 1997, 1998).

The voltage wave is reflected from the inverter towards the motor, but now out of phase, because the reflection coefficientΓ_i≈ −1. The voltage wave remains in the motor cable re- flecting back and forth between the inverter and the motor, and after each switching transient, a decaying cable oscillation may build up. The frequency of the cable oscillation depends on the propagation velocity of the wave and the length of the motor cable. The oscillation decays mainly as a result of the high-frequency attenuation of the cable, and also if the motor reflection coefficient is smaller than one, part of the incident wave is transmitted to the motor.

The propagation delay of the incident wave depends on the propagation speed of the wave in the cable and the cable length. Therefore, the frequency of the cable oscillation can be solved as follows (Skibinski et al., 1997):

f_osc= 1 4t_p=νp

4l, (2.15)

wheret_pis the propagation delay of the cable,νpthe propagation velocity, andlthe length of the cable.

The cable oscillation frequency and decaying time are also important factors in the origin of overvoltages that are greater than the theoretical maximum of twice the voltage for a single transition discussed so far. If a new transient occurs before the oscillation caused by the previous transient has decayed, overvoltages above twice the DC link voltage are also possible, see (Skibinski et al., 1997). This condition is called double pulsing.

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2.3 Critical cable length 33

Yet another important factor in the origin of overvoltages greater than twice the DC link voltage is called polarity reversal, where two of the inverter phases are switched from opposite states at the same time.

The key in reducing the overvoltage at the motor end is to slow down the rising and falling times of the modulated voltage pulses according to the cable length, see (Persson, 1992).

The longer the feeding motor cable, the longer the rise or fall time should be. The switching time can be prolonged by slowing down the switching operation of the semiconductor power switch, as previously mentioned, or by filtering. Slowing down the power switch generates excessive switching losses, and therefore it is not an optimal solution. Different filtering solutions will be discussed in more detail later in this chapter. Further, a conventional LC filter can be used to produce rising and falling slopes of desired length, if the active control is used, as will be shown in the next chapter.

2.3 Critical cable length

As presented earlier, a propagation delay is introduced to the incident voltage and current waves by the motor cable. The rise time of the injected voltage affects the maximum value of the overvoltage. If the propagation delay is smaller than half the rise time, the voltage wave reflected from the inverter end reduces the overvoltage at the motor end before it has reached its full value. This is the definition for the critical cable length in an electric drive (Persson, 1992), and full overvoltage is induced by the voltage reflection at this cable length. The key in mitigating the motor-end overvoltage is to increase the critical cable length by decreasing the du/dtin the voltage injected to the cable. The critical cable length is defined as

l_c=t_r

2·νp, (2.16)

wheret_ris the rise time of the voltage pulse andνpthe propagation velocity in the motor cable.

2.4 Fundamental properties of second-order systems

Systems that can be described by second-order differential equations are called second-order systems, such as most output filtering circuits are. The differential equation of the second- order system in the general form is

A f(t) =d²y

dt²+2ζ ω_ndy

dt +ω_n²y, (2.17)

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whereωnis the undamped resonance frequency of the system andζ is the damping factor, which describes how the system responds to a step input. The resonance frequency is the natural frequency at which the output of the system resonates if not damped. Critical damping (ζ =1) provides the fastest system response in the absence of overshoot. The greater the damping factor is, the slower the system responds to the input. A damping factor below the critical value provides a faster system response, but in this case there is overshoot in the output, the amount of which depends on how close the damping factor is to zero. If the damping factor is zero, the oscillation at the system output does not decay, and the amount of overshoot is equal to the magnitude of the input step. Hence, an undamped system resonates between zero and two times the input step at the natural frequency of the system. The step responses of second-order systems with various damping factors are presented in Figure 2.4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Step response of a second order system as a function of damping factor

ζ=0 ζ=0.2 ζ=0.5 ζ=1 ζ=2 ζ=5

Figure 2.4. Second-order system step response for various damping factorsζwith a constant undamped resonance frequencyωn.

Typically, in a passive output filter the inductance and capacitance define the resonance frequency of the filter. In addition to these, the damping factor is defined by the resistance of the circuit. In a passive filter design, and in filter design on the whole, the step response of the filter circuit is an important design consideration, in addition to the frequency response of the system.

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2.5 Typical output filtering solutions 35

2.5 Typical output filtering solutions

The reflection from the motor and motor cable interface can cause exceeding of the motor impulse voltage rating, which is harmful to the insulation system of the motor. Furthermore, in addition to the differential-mode line-to-line voltages, steep common mode voltages of high du/dt are coupled to the motor as a result of the operating principle of the two-level inverter, see for example (Skibinski et al., 1999). These phase-to-ground common-mode voltages have been shown to cause a high-frequency current in the grounding system of the drive and are a major cause of shaft voltages, which are among the factors causing bearing currents (Erdman et al., 1996; von Jouanne et al., 1998).

The overvoltages and adverse effects caused by voltage reflections in electrically long cables have been mitigated by applying various different filtering solutions: output reactors, output filters, such as sine wave and du/dtfilters, and cable terminators.

2.5.1 Output du/dt filters

The most typical solutions are different kinds of passive output filtering approaches, in which the du/dtof the output voltage is decreased. A very typical du/dtfilter, see Figure 2.5, consists of a series inductance and a parallel capacitance, and the losses in the circuit are tuned in order to obtain the desired transient output response for the drive (Finlayson, 1998). This kind of a system consisting of inductance, capacitance, and resistance is generally a second order system.

CL

M

D C - D C +

u '_U u 'V

u '_W

L C f i l t e r

C

D C L I N K u U u V u W

Figure 2.5. Schematic of a conventional du/dtoutput filter. Damping resistors or equivalent losses in the inductors are not illustrated in the figure.

However, since a second order system itself is a resonance circuit, it easily becomes a source of overvoltage and oscillation instead of the inverter-power cable electric motor resonator, if not sufficiently damped. The du/dtis decreased according to theLCconstant value, but in order to obtain a good transient response, damping is necessary, which means losses. In the

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design procedure of the passive du/dt, the most essential design parameters are the resonanve frequencyωnand the damping factorζ. In addition to the transient response, the frequency plane response is important. At the cable oscillation frequencies the filter is designed for, the filter attenuation should be maximized. The du/dtfilter design is a compromise between these key features. Also, if the resonance frequency is below the inverter switching frequency f_c, the filter is a sinewave filter, and if it is above this frequency, it is called a du/dtfilter. Designs with resonance frequencies close to possible switching frequencies should be avoided, since a strong filter resonance is induced.

Output du/dt filters based on inductors and capacitors have been introduced for example in (Finlayson, 1998), (Moreira et al., 2005), (Moreira et al., 2002), (Rendusara and Enjeti, 1998), (Rendusara and Enjeti, 1997), (Sozey et al., 2000), (Palma and Enjeti, 2002), (von Jouanne and Enjeti, 1997), (von Jouanne et al., 1996b), and (Steinke, 1999), and sinewave filters in (Skibinski, 2002) and (Skibinski, 2000).

2.5.2 Output du/dt filters with a clamping diode circuit

Some of the output filters use clamping diodes to limit the overshoot in the filter circuit to the positive and negative DC link voltage, see Figure 2.6. The clamping diodes are effective in preventing the filter oscillation, but they provide an alternative path for the reactive motor current, which is thus not seen by the current measurements of the output phases. As a result, part of the low-du/dt LC resonance sine wave is fed to the motor cable, but the natural LC overshoot is removed by the clamping circuit. However, current spikes through the diodes are introduced along with losses. The current amplitude of the current spikes can be decreased by adding resistance between the clamping circuit and the DC link, but at the expense of losses.

CL

M

D C - D C +

u '_U u 'V

u '_W L C f i l t e r a n d c l a m p i n g c i r c u i t

C D C L I N K u U u V u W

Figure 2.6. Schematic of a conventional du/dt output filter with clamping diodes. The natural LC overshoot is removed by the clamping circuit.

Filters utilizing clamping diodes are presented in (Moreira et al., 2002), and (Habetler et al., 2002), and a clamping filter to be placed at the motor end in (Chen and Xu, 1998).

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2.5 Typical output filtering solutions 37

2.5.3 Motor terminal cable terminators

Cable terminators have also been used to mitigate the overvoltages (Skibinski, 1996; Chen and Xu, 1998; Moreira et al., 2005). These are based on the fact that if the transmission line is terminated to the characteristic impedanceZ₀, no reflection takes place. In these solutions, the terminating resistors are chosen close to the assumed cable characteristic impedance via a capacitive coupling interface. The actual terminating impedances are the terminator and the electric motor in parallel. However, the impedance of the motor is assumed to be far higher than the cable characteric impedance, and therefore the effect of the motor on the terminating impedance can be neglected (Skibinski, 1996), which is also a typical case in reality. The cable terminator, see Figure 2.7, is very effective in limiting the overvoltage in the motor terminal, but it does not limit the du/dtvalue, creates power loss, since resistors in the order of the cable characteristic impedance are used (in the order of 10² Ω), even if capacitive coupling is used.

CR

M

D C - D C +

u '_U u '_V

u 'W

R C f i l t e r

C

D C L I N K u U u V u W

Figure 2.7. Schematic of a cable terminator. The motor cable is matched to the characteristic impedance using a terminator via a capacitive coupling interface. The purpose of the interface is to reduce losses in the circuit.

2.5.4 Summary on typical topologies

Drawbacks of the typical filtering solutions are often their large physical size, resulting in dif- ficulties in the integrability. As can be seen from the well-known equation for the resonance frequency of a second-order system, the lower is the resonance frequency, the greater the component values are. A more thorough summary of the commonly used filtering solutions in frequency-converter-fed electric drives is provided for example in (Moreira et al., 2005).

Active du/dt Filtering for Variable-Speed AC Drives

Juha-Pekka Ström