Hybrid magnet - field winding solutions for the exciters of synchronous generators

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Dipartimento di Ingegneneria dell'Energia, dei Sistemi,

del Territorio e delle Costruzioni

Corso di Laurea Magistrale in Ingengeria Elettrica

Hybrid Magnet - Field Winding

Solutions for Exciters of Synchronous

Generators

Supervisors:

Prof. Dr. Paolo Bolognesi - University of Pisa

Dr. Stefano Nuzzo - University of Nottingham

Student:

Giovanni Decuzzi

Academic Year 2016/2017

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List of Figures

Figure 1.Placement of the air inlet, outlet and exciter in a typical small-mid size gen-set ... 10

Figure 2: Classical excitation systems including by brushed DC machines ... 15

Figure 3:Modern brushless excitation system [Courtesy of CGT] ... 17

Figure 4: Modern static excitation system ... 18

Figure 5: Self-Excited system CGT ... 20

Figure 6: Isotropic (on the left) and non-isotropic SG (on the right)... 21

Figure 7: Stator (on the left) and rotor with the diodes plate (on the right) of the considered exciter. .... 23

Figure 8:Connection between rotor windings and diode bridge ... 24

Figure 9: Typical AC input to the rectifier and DC output to the main rotor. ... 24

Figure 10: CGT Automatic Voltage Regulator ... 25

Figure 11: Generalized equivalent circuit ... 34

Figure 12. Block diagram of balance energy ... 34

Figure 13.Spatial coordintes, and picture showing a very general configuration of an electrical machine [20] ... 41

Figure 14. High-level flowchart of the code implemented in Matlab ... 46

Figure 15: Equivalent winding function of phase 1 of the rotor ... 47

Figure 16:Stator equivalent winding function ... 47

Figure 17: Equivalent permeability function ... 48

Figure 18:Self inductances of the rotor windings ... 49

Figure 19: Self inductance of stator winding ... 49

Figure 20: Mutual inductances between rotor phases. ... 50

Figure 21: Mutual inductances between stator and rotor phases ... 50

Figure 22:No-load curve ... 51

Figure 23.Comparison between analytical and FE results ... 52

Figure 24. B-H curve of ferromagnetic material used for the rotor and stator cores ... 58

Figure 25. 2-D model of the studied exciter ... 59

Figure 26. Exciter circuit coupled to the FE model to simulate load conditions. ... 59

Figure 27. Exciter circuit coupled to the FE model to simulate no-load (left) and short-circuit conditions (right). ... 60

Figure 28: FE exciter model, with a special view on the imposed boundary conditions ... 60

Figure 29:Mesh of the exciter model ... 61

Figure 30.Comparison between FE and experimental no-load curves. ... 65

Figure 31. Comparison between FE and experimental short-circuit curves ... 66

Figure 32.B-H curve of a magnet material ... 69

Figure 33.Magnetic open circuit... 70

Figure 34. Operating point of PMs ... 71

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Figure 36. Airgap magnetic flux density at Iecc=0.9A (no-load operation of the main SG). ... 74

Figure 37.Embedded [13] and superficial PM ... 75

Figure 38: Equivalent permeability function of the hybrid exciter ... 78

Figure 39: Null-current equivalent m.m.f of hybrid exciter ... 78

Figure 40: Comparison between analytical and FE results... 79

Figure 41. FE model of the proposed hybrid exciter, with zoom on a PM ... 80

Figure 42. Particular of the mesh in the airgap and in the PM layers. ... 81

Figure 43: FE model of hybrid exciter ... 81

Figure 44: Simulated current waveform in the field winding of the main generator, operating at rated (400V) no-load condition. ... 82

Figure 45:Field map of the classic exciter at no-load operation. ... 84

Figure 46:Field map of hybrid exciter at no-load operation ... 84

Figure 47:Field map of classic exciter at full-load operation ... 85

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List of Tables

Table 1:Magnetic characteristics of SmCo 18/30 ... 77

Table 2: Comparison at no-load operation of the main generator ... 83

Table 3: Comparison at full-load operation ... 85

Table 4: Comparison at 50% of the full-load operation. ... 86

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ABSTRACT

The use of the classical, wound-field synchronous generator in the field of power generation, as opposed to a permanent magnet machine, is driven by the need of controlling the field winding voltage and current in such a way to guarantee a constant voltage at the generator’s armature terminals. In a traditional generating set, this function is accomplished by a relatively complex feedback control system which tipically comprises an excitation system and an automatic voltage regulator. Due to their excellent and proven performance capability, such classical generating sets have seen only strictly incremental improvements in the last 60 years or so. However, today, there is an interest in revamping their design and development, partly due to the ever increasing efficiency and reliability requirements and partly due to advances in materials and manufacturing techniques.

This thesis investigates the feasibility of a hybrid permanent magnet - field winding excitation for the exciters of synchronous generators. This design solution aims at improving the efficiency of the whole gen-sets, while also increasing the reliability of the overall system. As vessel for studying the proposed concept, the exciter of an alternator in the power range of 400kVA is investigated. Advanced analytical and finite-element models are first implemented and then validated against experimental results. The validated tools are then used to achieve optimal design solutions, whose performance are finally compared to the original exciter design.

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1 INTRODUCTION

The wound-field synchronous generator (SG) is by far the type of electric machine most used for power generation. Its advent in the energy sector stems from the development of long distance power transmission systems, in parallel with the improvements in the field of the insulating materials (which permitted higher voltage levels) of the late 19th century. Nowadays, although SGs are still mainly designed and used for the production of electricity ranging from very large scale (e.g. nuclear power plants) to very small scale (e.g. alternators used in cars), they are employed also as motors in other applications, such as naval propulsion, large pumps and compressors, etc.

The long history of utilization of the SG in the field of power generation is due to its high reliability and efficiency, its well consolidated and durable technology and its proven effectiveness in the role. Besides these aspects, one of the main advantages of this classical electrical machine consists in the ease of control and regulation of the output voltage generated at its armature terminals, achieved thanks to the presence of the field winding (as opposed to permanent magnets) in the rotor. In fact, the field winding of such machines is fed by an excitation system, consisting either in another electrical machine or in a static converter. The set made up of the alternator, the excitation system and the automated voltage regulator (AVR) represents the so-called generating set (gen-set).

Being such an estabilished and successful technology, all the advances registered in the field of power generation in the last century, including the generators and their whole gen-sets, have been very strictly incremental. However, the recent push towards more efficient systems and the even more stringent regulations in terms of power quality are leading the genset’s manufacturers and, more in general, the research community to adapt the design of such classical systems to these new demanding requirements. The trend towards the adoption of improved technologies is also enabled by recent advancments in materials, manufacturing techniques and also by the signiﬁcant improvements in simulation software and computational power available.

In this context, also Cummins Generator Technologies (CGT), one of the largest manufacturers of SGs in the world, has invested huge resources and effort in renewing the design of the gen-set components, with the aim of improving the performance and reducing the costs associated to the manufacturing of such systems. Stemming from

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these needs, CGT has funded several research projects to be developed from the Power Electronics, Machined and Control group of the Department of Electrical and Electronic Engineering of the University of Nottingham. This thesis was then developed in collaboration with the PEMC group within the above framework, proposing and investigating an original solution presently not employed by CGT and seldom seen even in other contexts.

1.1 Project description

CGT is a company that has nearly one hundred years of experience in the design and manufacturing of power gen-sets, with a broad portfolio of products ranging from 0.5kVA up to 20MVA. Amongst these, a very important market product from a business perspective is represented by the so-called HC4, a 400kVA SG to which CGT has dedicated huge efforts aiming to meet the efficiency and power quality requirements discussed above. Although the main focus is on to the alternator itself, in general the company aims at improving the performance of the entire gen-set, which features a classical brushless solution including an auxiliary generator whose armature is located on the rotor and supplies the field winding through a rotating rectifier.

This work aims to investigate potential improvements in the design of excitation sysmets for SGs, with the above mentioned HC4 system used as a case study for the application of the proposed concepts.

1.1.1 Challenges

The gen-set considered in this project consists of a main electrical machine, namely the alternator, whose field winding is fed by a brushless excitation system, made up of an “inside-out”, smaller SG, namely the exciter, and a rotating diode rectifier. In turn, the exciter field winding is fed by the AVR, whose main task is therefore that of providing the required power necessary to maintain the main machine terminal voltage at its rated value. More details about the major features and the operation of the system under analysis will be provided in section 2.3.

Besides the functional differences between the main alternator and the exciter, it is very important to mention already at this stage that these electrical machines are very

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different also from a constructive point of view. In particular, in Cummins gen-sets, the exciter has the same radial development as the main SG, meaning that the machines have the same outer diameter. This results on the other hand in a design configuration of the exciter much more compact in the axial direction than that of the SG. This solution aims at achieving a very compact structure of the gen-sets by minimizing the overall lemgth of the frame hosting all the components. However, this configuration comes at the cost of obstructing the air path towards the main machine, since the exciter is placed in the path of the cold air inlet, as shown in Fig. 1.

Figure 1.Placement of the air inlet, outlet and exciter in a typical small-mid size gen-set

This clearly results in a reduction of the thermal performance of the machine. One possible way to overcome this aspect may be that of reducing the exciter radial dimension, which in turn would result in an increased cooling air flow towards the SG. Besides the above considerations, in more general terms, it is also important to notice that the excitation loss plays a significant role in determining the overall efficiency of the gen-set. In fact, the experience gained by CGT in testing SGs and associated components leads to consider the excitation loss contributor to be around the 10% of the main rotor copper loss at full-load operation of the alternator. It is then clear that a reduction of the exciter total loss would lead to increase the efficiency of the entire system.

Although the above confirms that the excitation systems of SGs are of paramount importance in determining the overall performance and efficiency of the gen-sets, most of the aspects relative to this inside-out generator are often overlooked and neglected.

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Considering the above, it is felt that there is room for improving the efficiency of the whole system by re-designing the exciters of SGs. As as a significant practical example, the exciter of a 400kVA alternator (i.e. the model HC4) was considered as a case study for this thesis.

1.1.2 Proposed solution

With the advent of permanent magnet (PM) materials and the advancements in driving techniques, the use of PM machines has rapidly increased in many industrial applications, where they have replaced wound-field motors mainly due to the following reasons:

1) no electrical energy is required to sustain the operation of the excitation system and thus there are no excitation (Joule) losses, which results in a substantial increase of the efficiency;

2) higher torque and/or output power per volume than when using electromagnetic excitation, especially when powerful rare earth magnets are used;

3) better dynamic performance than motors with electromagnetic excitation; 4) simplification of construction and maintenance; reduction of prices for some

types of machines.

Considering on one side the advantages deriving from the use of PMs and, on the other hand, the persistent need of permitting the adjustment of the output voltage provided by the machine during operation, a hybrid magnet - field winding solution is proposed to overcome the challenges highlighted in Section 1.1.1. In other words, this solution lies in producing the excitation field of the exciter through both a field winding and PMs. In this way, it may be possible to provide the basic, continuous contribution to the main generator field winding through the PMs, while accomplishing the voltage regulation requirements through the exciter excitation windings, opportunely optimized according to the new configuration.

1.2 Aims and objectives

As highlighted in the previous introductory section, the main objective of this work is to investigate the feasibility of a hybrid PM - field winding excitation for the exciters

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of SGs, with the aim of improving efficiency and performance of the whole gen-sets. This study is performed by developing and implementing advanced analytical and FE models for the electromagnetic analysis that, once validated, will be then used to seek for the optimal hybrid solution. A brushless exciter preserntly used in a current CGT market product (i.e. a 400kVA salient-pole alternator) is considered as a significant case study.

More specifically, in order to address the requirements identified as challenges in Section 1.1.1, it is proposed to provide via PMs the basic quote of the field of the auxiliary generator, i.e. the value which is required to supply the field winding of the main generator with the current that will make the armature voltage of the SG equal to its rated value when the machine operates at no-load and rated speed. The reasons behind the choice of this specific operating point is that, when the SG is used to supply an equivalent ohmic-inductive load, keeping the rated voltage at rated speed will require to increase the SG excitation current at any non-null load. This implies that also the excitation field in the auxiliary generator shall be equal or more intense than the basic condition at any time, meaning that the field winding will be only required to support, and never contrast, the PMs. Such condition is thus interesting since the magnets are less likely to risk incurring in demagnetization, whereas the current fed to the field winding of the exciter should be only positive, meaning that for the AVR a simpler and cheaper power stage can be emplyed, similar to the one presently used. Besides the aspects related to the efficiency requirements, another very important advantage that can potentially derive from the use of PMs in the exciters of SGs is related to the reliability of the overall system. In fact, the excitation field produced by the PMs would guarantee a seamless excitation to the hybrid exciter, even when any unforeseen event that would interrupt the field winding supply occurs.

In summary, the main aim of this project is to investigate the proposed hybrid excitation solution, consolidate the improvements in terms of performance and reliability and, possibly, reduce the manufacturing cost that can potentially result from a downsizing of the considered exciter. In order to achieve this aim, a number of objectives has been identified:

• Achieving a comprehensive understanding of wound-field SGs, excitation systems and AVRs, with special focus on the application at hand.

• Developing comprehensive and advanced models for the detailed analysis of the platform in question.

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• Achieving a comprehensive knowledge of hard magnetic materials, aimed at appropriately select the PMs for the exciter under investigation.

• Developing appropriate analytical and simulation models of the exciter permitting to carry out an in-depth analysis of the proposed hybrid configuration referring to the considered case study.

1.3 Thesis outline

The thesis structure is organised in such a way to report all the logical passages and steps of the methodology followed towards the completion of the objectives of this project.

Chapter 1 introduces the project by highlighting relevant applications and challenges and by proposing, at a very high-level, a possible design solution. Aims and objectives of this work are also presented in this introductory chapter.

In Chapter 2, these concepts are strengthened through a review of the methods most commonly implemented as excitation systems for SGs. Particular attention is given to the brushless excitation system, as it represents the solution utilised in the gen-set under investigation. This chapter is concluded by providing a comprehensive description of the major features of the 400kVA gen-set.

In Chapter 3, a purely analytical model for the analysis of the considered exciter is presented, implemented and validated against finite-element (FE) analysis. Chapter 4 details the features of a 2-D FE model built for the in-depth electromagnetic analysis of the exciter. The developed model is then validated through a comparative exercise with available experimental data.

Once validated, the models developed in Chapters 3 and 4 are used in Chapter 5 to achieve an optimal, hybrid PM – field winding excitation for the considered exciter. The final configuration is conclusively compared with the existing topology for proving the feasibility of the proposed concept.

Chapter 6 concludes this thesis by presenting overall discussions and further work to be done in the related applications.

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2 LITERATURE REVIEW

This chapter presents a review of the methods most commonly used in gen-sets for power plants, consisting in brush, brushless and static excitation systems. Further applications and different solutions are also described with the aim of highlighting that the design aspects related to these well-consolidated systems are often overlooked and thus need more attention.

Also, a full understasting and description of the main components that make a gen-set up is provided. To this end, special focus is given to the SG, the exciter and the AVR used in Cummins gen-set, as they are used in this thesis as vessels to overcome the challenges highlighted in Section 1.1.1 and to achieve the objectives outlined in Section 1.2.

2.1 Introduction

The main reasons for providing an excitation system on salient-pole synchronous generators are well-known and are summarised by [1]. In brief, due to the elevated voltage variations that occur when a SG is loaded, a compensation system is always needed and this is done in practice by opportunely regulate the excitation current. Therefore, in order to maintain a constant voltage at the machine’s armature terminals, when the load conditions of the alternator varies, the field current has to be “adjusted” in such a way to produce a no-load e.m.f necessary to provide, at on-load operation, the desired voltage at the machine’s terminals.

In the next sections, the methods most commonly used in classical gen-sets to accomplish this regulation are discussed.

2.2 Various excitation systems

The excitation and the voltage regulation systems in SGs have been extensively studied over the years, due to their very important role for the proper operation of these classical, wound-field, machines. It is however worth to recall the basic concepts relative to the methods that have been hystorically implemented for the purposes highlighted above. These are briefly described below in the next sections.

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2.2.1 Brushes

The first and most widespread (for hystorical reasons) excitation system for SGs make use of a DC machine (dynamo). Usually, this is used in combination with a shunt resistor. This solution consists in adjusting the field winding voltage of the main generator through regulation of the field winding voltage of the dynamo.

The connection between the armature winding of the dynamo and the excitation winding of the SG is realized through a system of rings and brushes.The two machines are connected on the same mechanical shaft, resulting in the same rotational speed. This type of excitation system is named ”coaxial excitation”.

The electrical connection is realised by means of a ring system rigidly welded on the mechanical axis, and system of brushes that craw and thus feed the SG field winding. In this system, the excitation of the DC machine is performed through a “derived” field rheostat, meaning that the regulation of the excitation field of the main alternator is directly linked to this field rheostat.

Another solution that is often implemented is comprised of two dynamos: 1) an auxiliary dynamo, which is used for regulation and 2) a secondary dynamo, which feeds the main generation field winding Fig.2 shows the schemes of the two solutions discussed above.

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The main difference between the systems described above is related to the response times of the two solutions. In fact, the excitation circuit being equipped with a considerable inductance does not generally allow rapid variations of the current flowing through it.

In other words, the solution with the two dynamos is used when high performance is not required for the excitation systems. In fact, in this case, the dynamics of all the three electrical machines must be taken into account, meaning that a longer response time is needed compared to the other solution. One major drawback of the solutions presented in this section is the significant resistive voltage drop caused by the system of rings and brushes. Also, these need periodical mintenence and replacement.

Although the excitation with brushes is still widely in use, due to the drawbacks presented above, in many applications the brushless [2-4] and the static excitation systems are coming to fore [5]. These are presented in the next sections.

2.2.2 Brushless excitation

With the advent of the power electronics and the need of overcoming the challenges related to the slip rings and the brushes, many applications are equipped nowadays with brushless excitation systems.

In the simplest configuration, the brushless excitation system consists of an AVR, an exciter and a rotating diode rectifier system, mounted on the same mechanical shaft of the main SG. The main advantages over the dynamo excitation system include 1) a reduction of the resistive voltage drops, 2) a reduction of maintenance times and costs of the brushes, 3) a reduction of the response time of the regulation system.

In the brushless excitation system, [6-7], the field winding current of the main generator is rectified by the diode rectifier bridge, whose AC link is connected to the exciter rotor terminals. The output voltage control of the generator is performed via a feedback control system by the AVR, which in turn provides a DC voltage to the exciter field winding.

In the so-called self-excited configuration, the AVR is directly powered by the residual voltage available at the stator terminals of the machine, taking advantage of the remanence present in the ferromagnetic stractures. A more reliable system consists in powering the AVR through an additional machine, which is usually represented by a

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relatively small PM generator. This system is known as a separately-excited configuration [2], [8] and its main advantages over the self-excited systems include the followings: 1) sustained short-circuit under fault conditions, 2) unaffected by waveform distortion caused by non-linear load, 3) powerful voltage build-up system on initial run-up, meaning that it does not rely upon the residual magnetism.

Figure 3:Modern brushless excitation system [Courtesy of CGT]

An example of a separately-excited system is shown in Fig. 3, where a typical gen-set configuration is illustrated.

2.2.3 Static Excitation

Static excitation systems do not have moving parts. In the whole system, the only rotating part is represented by the main generator [5]. The static excitation system is used in applications where fast transiest responses are required. The system usually comprises a power transformer, a controlled thyristor rectifier, the electronic AVR and a de-excitation unit. The excitation power is taken from the output terminals of the generator, whose output voltages are rectified and fed to the machine field winding via collector and rings. This system can provide a high ceiling voltage and, as previously mentioned, extremely fast voltage response times.

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Figure 4: Modern static excitation system

The actual regulation is performed through an AC-DC converter, which is fed by the power grid or by the main generator itself. The static converter tipically consists of a three-phase thyristor bridge, whose control circuit allows to change the average DC value of the rectified voltage.

Besides the fast response, an additional advantage is related to the relatively low cost in comparison with a rotating excitation system, while the main drawback is due to the presence of rings and brushes, which require frequent maintenance.

2.2.4 Further consideration

The excitation systems analyzed above are typically used in gen-sets of medium-to-large size and, in general terms, for power plant applications. There exists a number of additional configurations that are used in automotive and aerospace applications, where compact excitation systems are requiredFor example, a hybrid excitation synchronous machine is used for a hybrid vehicles, in combination with a thermic motor. Similar configurationa are employed for island operation [9] and aircraft applications [10],[11]. This solution presents a complex structure using three integrated machines. In such configuration, a PM machine operates as a generator which supplies AC power to a rectifier/chopper set. The chopper is then connected to a stationary excitation coil of a second machine integrated into the system. The three-phase ac output voltage is then rectified (with a rotating rectifier) and finally applied to the main generator field. This structure allows one to control the excitation of the main generator without brushes, but

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this system configuration comes however at the cost of high complexity, elevated prices and need of maintenance.

Furthermore, in recent years, the advancements in the computational resources as well as in the development of more and more powerful magnetic materials, has led to re-consider the aspects related to all the components of the classical gen-set [12], including the exciter. In this context, it is worth mentioning that a magnet – field winding excitation for the exciters of SGs has been proposed by [13]. This solution is comprised of magnets embedded inside the stator teeth. Although the basic concepts are similar to those presented in this work, however updates and variation of the solution proposed by [13] are investigated here.

In particular, it is possible to locate the PMs in such a way that a minimum disruption of the existing exciter design is achieved. The details related to the proposed, hybrid concept will be discussed in Chapter 5. In order to set the bases for the concepts, models and analyses that will be presented in the next chapters, a detailed description of the gen-set under study is given in the next section.

2.3 The considered gen-set

The excitation system considered in this thesis work is a “self-excited” brushless configuration of an alternator in the 400kVA power range, manufactured by CGT. The whole system is shown in Fig.5, where the absence of the PM generator can be noted when compared to Fig.3 of Section 2.2.2.

The gen-set is made up of two rotating electrical machines (the exciter and the SG) coupled on the same mechanical axis, a three-phase rotating diode bridge and the AVR.

The set of all the elements which the gen-set is comprised of are all embedded within the case of the SG.

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Figure 5: Self-Excited system CGT

In this case, the SG residual voltage of the machine is used to power the AVR. This voltage is then rectified and controlled by the AVR to supply the exciter field winding in such a way that the voltage level at the generator terminals is maintained constant. The “inside-out” machine that acts as the exciter produces a three-phase system of voltages at its rotor terminals, which in turn are rectified by the diode bridge which feeds the field winding of the generator, which finally closes the loop by generating the desired output voltage.

In the following sections, more details on the the operation of each element constituting the genset will be discussed.

2.3.1 The Synchronous generator

The SG is a rotating electrical machine, operating in AC current, whose name derives from the fact that the rotational speed is the same as the rotating magnetic field. It represents a reverisible machine, in the sense that the flux of the power can be bidirectional and the machine can work as generator or motor. In this work, the operation of the machine as a generator is considered.

The SG is made up of two magnetic structure facing the air gap: 1) the rotating part, namely the rotor, where the field winding is usually placed for the generation of a stationary magnetic field and 2) the stationary part, namely the stator, which typically

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hosts a three-phase winding in which the voltage is induced, namely the armature winding.It is worth to say that other configurations are often used according to the applications (including armature on the rotor and field winding on the stator, poly-phase systems, etc…), however the vast majority of the SGs are of the type described above, including the platform of interest in this thesis. In relation to the the reluctance of the air gap, SGs can present an isotropic (constant reluctance) or an anisotropic (variable reluctance) rotor, better known as non-salient and salient pole machines, respectively. For the sake of completeness, these two most utilized configurations are shown in Fig.6.

Figure 6: Isotropic (on the left) and non-isotropic SG (on the right).

In both cases, the excitation winding, driven by DC current, creates a f.m.m. into the air gap and a magnetic field that produce alternating poles in the ferromagnetic structures. The angular distance between the axis of a north-pole and a south-pole measured by the air gap is equal to polar pich τ given in (1), where R is the airgap radius and p is the number of pole pairs.:

𝜏 =𝜋 ∗ 𝑅 𝑝

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The spatial development of the excitation field along the periphery of the air gap is more or less sinusoidal. In a non-salient pole SG, this is achived by distributing properly the slots in the rotor structure, while in a salient-pole machine this is usually done by opportunely shaping the polar expansions.

When the rotor rotates at constant angular speed ωm, the excitation field and the rotor

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three-phase armature windinga symmetric system of three-three-phase f.e.m., sinusoidal in time with electrical pulsation ω, given by (2).

𝜔 = 𝑝 ∗ 𝜔𝑚 (2)

If the stator feeds a balanced load, a three-phase system of currents (with pulsation ω) flow in the stator windings. This flow produces a stator rotating field. If the number of pole pairs of the stator is equal to the number of rotor pole pairs, the stator field rotates with angular speed ωc=ω/p=ωm. Therefore, the stator and rotor fields have no relative speed between them and the resulting field is then a field rotating at the same rotor speed.

Having given a brief, general description of the main operation of a SG, it is worth to introduce the major features of the SG considered in this thesis, namely the HC4. The machine has a rated apparent power of 400kVA, designed to provide a rated voltage of 400V, at 50Hz and with a 0.8 power factor.

The HC4 is a salient-pole alternator with tow rotor pole pairs and a classic stator laminated structure. The rotor is also equipped with a damper cage symmetrically displaced around the polar axes.

The rotor coils are connected in series, each coil is wound around the pole in reversed direction with the aim of producing the required polarities. The stator structure, on the other hand, has 48 slots, which host a three-phase winding, which are skewed by one slot pitch along the axial direction to achieve a “cleaner” shape of the output waveforms.

2.3.2 The Exciter

The exciter is the second rotating electrical machine that makes up the gen-set shown in Fig.5. This section first explains the general operation of a brushless exciter and then presents the CGT’s realization solution.

As well as the main SG, the exciter works as a generator in a gen-set. Contrarily to the main alternator, however, the exciter field winding is on the stationary ferromagnetic structure of the machine, while its armature is placed in the rotor slots. For this reason, it is considered as an “inside-out” generator.

The main operation of the exciter is very similar to that of a classical SG. Therefore the concepts outlined above in Section 2.3.1 are not repeated here. The different roles that the main alternator and its exciter cover in the gen-set operation make evident that there

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would be a dimensional difference between the two machines. In fact, the HC4 exciter has an actual length that is the 10% of that of the main generator, while keeping the same outer diameter. This makes its design quite compact: the ratio between the exciter axial length and its outer stator diameter is equal to 0.1.

Going more in detail into the description of the exciter features, it is worth to highlight that

1) The stator structure, in order to accommodate the excitation winding, has 14 relatively small salient poles, around which the coils are wound, resulting in the qualitative picture shown in Fig.7.

2) There are 250 turns per stator coil.

3) The rotor structure, unlike the stator, accommodates a three-phase winding. On the inner profile of the rotor, there are 42 slots, with a single-layer layout, meaning that there are 14 slots per phase. In other words, the rotor is featured with 1 slot-per-pole-phase, with a full-pitch configuration.

4) Each rotor coil is made up of 7 turns per phase.

Figure 7: Stator (on the left) and rotor with the diodes plate (on the right) of the considered exciter.

From a functional point of view, it can be affirmed that the exciter is a magnetic power amplifier supplying the main rotor current. The exciter, in order to provide the desired DC voltage value to the main generator field winding, is equipped with a diode three-phase bridge which is mounted on the machine shaft and rotates synchronously with the rotor [14-17]. Each rotor phase is connected to two diodes, as observed in Fig.7 and Fig.8. If a three-phase system of alternating voltage is feeding the diode rectifier, such

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as in the scheme illustrated in Fig.8, then at any instant in time two diodes of different levels and legs conduct. Is simplistic terms, these are the diode with the most positive potential on its anode and that with the most negative potential on its cathode. This layout permits to obtain the rectified voltage shown in Fig.9 (at the bottom) at the main rotor terminals.

Figure 8:Connection between rotor windings and diode bridge

Figure 9: Typical AC input to the rectifier and DC output to the main rotor.

2.3.3 The AVR

The last main component to analyse whitin the whole gen-set is the AVR. In simplistic terms, this is an electronic converter that serves for the voltage regulation. By regulating the DC voltage of the exciter, it becomes possible to adjust the output voltage of the main generator. Several ways of implementing an AVR are achieved in different applications. However, in this section, the AVR proposed by CGT and utilized in the considered gen-set is described.

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Cummins’ AVRs are simply made up of a silicon controlled rectifier (SCR), as shown in Fig.10, where the basic operating principle can be also observed.

Figure 10: CGT Automatic Voltage Regulator

In a self-excited system, such as that analysed in this section, the AC power supply to the AVR is given by the machine armature itself. The SCR then rectifies the AC input according to the scheme of Fig.10, where a typical half-wave controlled DC voltage is shown. The latter feeds the exciter field winding in such a way to guarantee a constant output voltage (at the fixed pre-set level) irrespective of load or speed change. To do this, the AVR compares a reference voltage value with the actual, measured output voltage and automatically adjusts the excitation level. This represents a closed loop voltage control system.

The AVR solution adopted by CGT is extremely easy and economical. However, it is possible to improve the waveform of the input voltage of the excitation winding of the exciter. It is common, in fact, to use a three-phase diode bridge of the type described in Section 2.3.2 [7], at the expense of slightly increasing the overall costs.

2.4 Summary

This chapter discusses the excitation systems normally used in power gen-sets and some realization solutions proposed by the recent literature. It is evident that the choice of the most appropriate excitation system is strictly dependent on the type of application to be considered and, above all, by the voltage regulation requirements of the whole system. In the second part of the chapter, all the components of the considered CGT gen-set is presented, with special attention given to the self-excited excitation system, as it

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represents the component targeted as vehicle to investigate the solution proposed in this work.

From the literature review carried out in this chapter, it is clear that there is room for improving the design of such inside-out small alternator and that the achievements of better performance can be highly beneficial for the whole gen-set.

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3 THE ANALYTICAL MODEL

The equivalent circuit methodology has been widely exploited for the theoretical analysis of the electromagnetic devices and in particular, represents a classic approach for modelling the phenomena internally occurring in electrical machines. In this chapter, an advancded and generalized theoretical model for the analysis of electromagnetic devices is proposed ad then properly adopted to specific considered machines [17]. Once the analytical model is adapted to the exciter, it is important to validate the results. For this reasons, the chapter is concluded by comparingthe analyticall-evaluated results with those obtained through an in-detail FE analysis of the exciter under study.

3.1 Model introduction

An analytical model for the electromagnetic analysis of the exciter described in section 2.3.2 is presented. This analytical treatment is general, in the sense that it can be used for the analysis of any type of electo-magneto-mechanical device [14-15], [18-19]. This generalized analysis aims at identifying the links between the electromagnetic and mechanical phenomena existing within the device, without specyfing in a first instance the type of the device we are analyzing. This permits to introduce basic quantities and to draw important general conclusions for the analysis of electromechanical devices. For this analysis a matrix notation will be used in order to obtain general and compact results. For the mechanical aspects, the Lagrangian approach will be used.

3.2 Model assumptions

The basic assumptions that are necessary to list downbefore describing the generalized model are the following:

• The device is analyzed by considering only the electro-magneto-mechanical aspects. The other possible relevant phenomena, i.e. thermal phenomena, are then supposed to be either negligible or treatable as variables slowly altering the parameters of the model.

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• The phenomena are observed by a reference systems, even arbitrary, by which the geometrical configuration, taken instantaneously, is uniquely defined by our device. The configuration of the considered device is univocally sigled out by the values of a finite set of 𝑛𝑚 ∈ 𝑁 independent variables gruped in a vector 𝜒̅(𝑡)which represents the generalized position.

• The high frequency effects are considered negligible into the device.

• The device is made up of 𝑛𝑒 ∈ 𝑁 independent windings. Each winding consists of a geometrically closed path (whith terminals very close to each other and located in areas where the magnetic field is negligible), without intermediate branches.

• The currents flowing into the windings are represented by the vector 𝑖̅(𝑡), having dimension equal to 𝑛_𝑒.

• Secondary phenomena, such as those relative to the hysteresis and the eddy currents, are negligible.

The above hypotheses imply that the windings constitute well-defined current paths, linking well-defined magnetic fluxes since the magnetic field map is independent on the selected reference frame deriving univocally from the currents values, the geometrical configuration of the device and the magnetic characteristics of its material.

3.3 Problem definition

From the hypotheses made in the previous section it is shown that the “magnetic state” of the device is uniquely defined by means the two vector 𝑖̅(𝑡) and 𝜒̅(𝑡). The two vectors take the name of “state variables” and define instant by instant the magnetic field map of the device.

3.3.1 Mechanical analysis

As the device under analysis is subject to mechanical-type phenomena, a general description of how to deal with them is given below.

Considering that our device consists of 𝑛_𝑝 material points, which have a well defined 𝑦_𝑘

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𝜒 ∈ 𝐷_𝜒 ⊆ ℝ𝑛𝑚_{0 < 𝑛}

𝑚 ≤ 3 ∗ 𝑛𝑝 constituting a valid set of lagrangian coordinates for the systems.

By recallingthat the derivative of the position with respect to the time is a speed, and the derivative of the speed is an acceleration, it is possible to express the speed and the acceleration of the material points in the Lagrangian form, as given below in (3) and (4). 𝑑 𝑑𝑡𝑌̅(𝜒̅(𝑡)) = 𝛿𝑌̅(𝜒̅) 𝛿𝜒̅𝑇 ∗ 𝜂̅(𝑡) (3) 𝑑2 𝑑𝑡2𝑌̅(𝜒̅(𝑡)) = 𝛿𝑌̅(𝜒̅) 𝛿𝜒̅𝑇 ∗ 𝑑 𝑑𝑡𝜂̅(𝑡) + 𝑑 𝑑𝑡( 𝛿𝑌̅(𝜒̅) 𝛿𝜒̅𝑇 ∗ (𝜒̅(𝑡))) ∗ 𝜂̅(𝑡) (4) In these expressions, 𝜂̅(𝑡) represents the vector of the generalized speed 𝜂̅(𝑡) = 𝑑

𝑑𝑡𝜒̅(𝑡), while the vector function 𝑌̅(𝜒̅)=[𝑦̅̅̅(𝜒̅)]. _𝑘

It is possible to express the second Newton low for the whole set of points as in (5), where m is a diagonal matrix formed by the masses of the material points and 𝐹̅(𝑡) = [𝐹𝑘]. 𝐹̅(𝑡) = 𝑚 ∗ 𝑑 2 𝑑𝑡2𝑌̅(𝜒̅(𝑡)) (5) 𝑚 = [ 𝑚1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑚_𝑛𝑝 ]

Considering the total power 𝑃_𝐺(𝑡) developed for any group G of forces, the expression (6) can be derived, where 𝑊̅_𝐺(𝑡) is the wrench vector or the generalized forces vector given in (7). 𝑃_𝐺(𝑡) = ∑ 𝐹̅𝑇 𝐺𝑘(𝑡) ∗ 𝑑 𝑑𝑡 𝑛𝑝 𝑘=1 𝑦_𝑘 ̅̅̅(𝑡) = 𝑊̅𝑇_𝐺(𝑡) ∗ 𝜂̅(𝑡) (6) 𝑊̅_𝐺(𝑡) = ∑𝜕𝑦̅𝑘 𝑇_(𝜒̅) 𝜕𝜒̅ (𝜒̅(𝑡)) ∗ 𝐹̅ 𝑇 𝐺𝑘(𝑡) 𝑛𝑝 𝑘=1 (7)

The expression of the wrench can be exploited as follows in (8).

𝑊̅ (𝑡) = 𝑊̅_𝐸(𝑡) − 𝑊̅_𝑀_{(𝑡) − 𝑊}̅_𝐴_(𝑡) ₍₈₎ In such expression We, Wm, Wa are the electromagnetic, mechanical and apparent parts of the total wrench.

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The most important term for the sake of this work is 𝑊̅_𝐸(𝑡). In order to investigate this quantity, the generalized inertial matrix is defined as in (9), while the electromagnetic component of the wrench can be then expressed as in (10).

𝐽(𝜒̅) =𝛿𝑌̅ 𝑇_(𝜒̅) 𝛿𝜒̅ ∗ 𝑚 ∗ 𝛿𝑌̅(𝜒̅) 𝛿𝜒̅𝑇 = ∑ 𝑚𝑘∗ 𝜕𝑦̅_𝑘𝑇(𝜒̅) 𝜕𝜒̅ ∗ 𝜕𝑦̅_𝑘(𝜒̅) 𝜕𝜒̅ 𝑛𝑝 𝑘=1 (9) 𝑊̅𝐸(𝑡) = 𝐽(𝜒̅) ∗ 𝑑2 𝑑𝑡2𝜒̅(𝑡) (10)

As a conclusion, the electromagnetic wrench can be defined as the quantity that expresses the equivalent effect of all the distributions of elementary forces acting on the material points of the considered device.

3.3.2 Electromagnetic analysis

This section defines the electromagnetic quantities of interest in this thesis. To do this, it is first defined a flux state function, which depends on the device magnetic state and from the two state variables defined above

𝜓̅(𝑡) =

𝑐 𝑐_{𝜓̅(𝑖̅(𝑡), 𝜒̅(𝑡)) ∀𝑡}

(11) Through a pseudo-linear decomposition, 𝑐𝜓̅(𝑡) (as defined in (11)) can be separated in two terms: the “null-current fluxes” 𝜓̅𝐶 ₀(𝜒̅)and the “incremental fluxes” 𝜓̅𝐶 _𝐼(𝑖̅, 𝜒̅), defined in (12) and (13) respectively.

𝜓̅( 𝐶 _{𝑖̅, 𝜒̅) = 𝜓̅} 0( 𝐶 _{𝜒̅) + 𝜓̅} 𝐼( 𝐶 _{𝑖̅, 𝜒̅)} ₍₁₂₎ 𝜓̅₀( 𝐶 _{𝜒̅) = 𝜓̅(}𝐶 _{0, 𝜒̅)} ₍₁₃₎ 𝜓̅_𝐼( 𝐶 _{𝑖̅, 𝜒̅) = 𝜓̅(}𝐶 _{𝑖̅, 𝜒̅) − 𝜓̅} 0( 𝐶 _𝜒̅) (14) The flux function vector (12) has the same dimension of the number of windings 𝑛𝑒, i.e. 𝐶𝜓̅(𝑖̅, 𝜒̅): ℝ𝑛𝑒,𝑛𝑚⟶ ℝ𝑛𝑒

The null-current flux does not depend on the vector 𝑖̅. This term will be different from zero in all those devices in which, with null currents in the windings, presents a contribution in term of flux, i.e. when PMs are present inside the device.

An “incremental inductance” matrix state function 𝐿𝑐 _𝐼

can be defined as the quantity linking 𝑐𝜓̅_𝐼(𝑖̅, 𝜒̅) and 𝑖̅. This is given in (15).

𝜓̅𝐼(𝑖̅, 𝜒̅) =

𝑐 _𝐿

𝐼

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Such definition is not univocal, unless either 𝑛_𝑒=1 or 𝑐𝐿_𝐼 is independent on 𝑖̅. The flux state function variation with respect to the current vector, at fixed position, is given below in (16), where the new state function of the matrix of “differential inductance”

𝐿_𝐷 𝑐

is introduced. In (16), 𝑐𝐿_𝑐 represents a new matrix state function, namely the “complementary inductances” and this is given in (17).

𝐿_𝐷 𝑐 _{(𝑖̅, 𝜒̅) =}𝜕 𝜓̅(𝑖,̅ 𝜒̅)𝑐 𝜕𝑖̅𝑇 = 𝜕 𝜓𝑐̅̅̅(𝑖̅, 𝜒̅)_𝐼 𝜕𝑖̅𝑇 = 𝐿𝐼 𝑐 _{(𝑖̅, 𝜒̅) − 𝐿} 𝑐( 𝑐 _{𝑖̅, 𝜒̅)} ₍₁₆₎ 𝐿_𝐷 𝑐 _{: ℝ}𝑛𝑒,𝑛𝑚_{⟶ ℝ}𝑛𝑒∗𝑛𝑒 𝐿_𝑐 𝑐 _{(𝑖̅, 𝜒̅) = −}𝜕 𝐿𝑐 𝐼(𝑖,̅ 𝜒̅) 𝜕𝑖̅𝑇 ∗ [ 𝑖̅ ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑖̅ ] (17)

In a magnetically linear scenery, the matrix of complementary inductances is null, and therefore the 𝑐𝐿_𝐼 is equal to 𝑐𝐿_𝐷. In this case, their common form is named “absolute inductance” matrix.

Having defined the link existing between the flux state function and the currents vector, in a similar way the flux vector is studied below according to its dependency on the vector of the generalized positions 𝜒̅.

Therefore, the derivative of 𝑐𝜓̅(𝑖̅, 𝜒̅)with respect to 𝜒̅, at fixed currents, permits to introduce a new state function called “motional coefficients” vector (18):

𝑀 𝑐 _{(𝑖̅, 𝜒̅) =} 𝜕 𝜓̅( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕𝜒̅𝑇 = 𝜕 𝜓̅𝐶 ₀(𝜒̅) 𝜕𝜒̅𝑇 + 𝜕 𝜓̅𝐶 _𝐼(𝑖̅, 𝜒̅) 𝜕𝜒̅𝑇 = 𝑀0 𝑐 _{(𝜒̅) + 𝑀} 𝐼 𝑐 _{(𝑖̅, 𝜒̅) (18)} By exploiting (18), the two terms 𝑐𝑀₀(𝜒̅) and 𝑀𝑐 𝐼(𝑖̅, 𝜒̅), respectively named “null-current motional coefficients” and “incremental motional coefficients” are obtained. The functions 𝑐𝐿𝐷(𝑖̅, 𝜒̅), 𝐿𝑐 𝐼(𝑖̅, 𝜒̅), 𝐿𝑐 𝑐(𝑖̅, 𝜒̅), 𝑀𝑐 0(𝜒̅) and 𝑀𝑐 𝐼(𝑖̅, 𝜒̅) are defined as “secondary state functions”, as they depend on the “primary state function” 𝜓̅(𝑐 _{𝑖̅, 𝜒̅).}

3.3.3 Wrench

The analysis performed in the previous section for the flux state function can also be replicated for the wrench state function. The electromagnetic wrench, as well as the flux, also depends instant by instant on the field map of the device. The electromagnetic wrench state function is defined by (19).

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𝑊̅_𝐸(𝑡) = 𝑊𝑐̅_𝐸_{(𝑖̅(𝑡), 𝜒̅(𝑡)) ∀𝑡}

(19) Through the same pseudo-linear decomposition concept used in the previous section, two terms can be identified; these are given by the expressions (21) and (22). The vector of incremental wrench can be defined by introducing the matrix of the “incremental wrench coefficients”, as in (23). 𝑊̅_𝐸( 𝐶 _{𝑖̅, 𝜒̅) = 𝑊}_̅ 𝐸0( 𝑐 _{𝜒̅) + 𝑊}_̅ 𝐸𝐼 𝑐 _{(𝑖̅, 𝜒̅)} (20) 𝑊̅_𝐸0( 𝑐 _{𝜒̅) = 𝑊}_̅ 𝐸 𝑐 _{(0, 𝜒̅)} (21) 𝑊̅_𝐸𝐼( 𝐶 _{𝑖̅, 𝜒̅) = 𝑊}_̅ 𝐸(𝑖̅, 𝑐 _{𝜒̅) − 𝑊}_̅ 𝐸0 𝑐 _(𝜒̅) (22) 𝑊̅_𝐸𝐼( 𝐶 _{𝑖̅, 𝜒̅) = 𝐾} 𝐼( 𝑐 _{𝑖̅, 𝜒̅) ∗ 𝑖̅} ₍₂₃₎

By considering the vector of the currents to be fixed, it is possible to define the variation of the wrench with respect to the currents, as given below in (24).

𝐾_𝐷( 𝑐 _{𝑖̅, 𝜒̅) =}𝜕 𝑊𝑐̅𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇 = 𝜕 𝑊𝑐̅_𝐸𝐼(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇 = 𝜕 𝐾𝑐 _𝐼(𝑖̅, 𝜒̅) ∗ 𝑖̅ 𝜕𝑖̅𝑇 = 𝐾𝑐 _𝐼(𝑖̅, 𝜒̅) + 𝐾𝑐 _𝑐(𝑖̅, 𝜒̅) (24)

In (24), 𝐶𝐾_𝐷(𝑖̅, 𝜒̅) represents the state function of the “matrix of differential wrench coefficients”: 𝐾𝐶 _𝐷(𝑖̅, 𝜒̅):ℝ𝑛𝑒,𝑛𝑚_{⟶ ℝ}𝑛𝑚∗𝑛𝑒_{. Therefore the} _𝐾

𝑐(

𝑐 _{𝑖̅, 𝜒̅) is a matrix of} complementary coefficient of wrench.

As can be seen from (24), secondary state functions are similar to each other. Indeed, the mathematical analysis following for the flux state function is equal the one followed for the wrench state function. Furthermore, the mathematical considerations for the

𝐿_𝐼

𝑐 _{(𝑖̅, 𝜒̅) matrix are also valid for the 𝐾} 𝐼(

𝑐 _{𝑖̅, 𝜒̅) matrix.}

The variation of wrench with respect to the vector of position, at fixed currents, is expressed by (25), where 𝑐𝑆_𝐷(𝜒̅) is the electromechanical differential stiffness, consisting in a null-current and incremental component, i.e. 𝑐𝑆_𝐷0(𝜒̅) and 𝑆𝑐 _𝐷𝐼(𝑖̅, 𝜒̅) respectively. 𝑆𝐷( 𝑐 _{𝑖̅, 𝜒̅) =}𝜕 𝑊𝑐̅𝐸(𝑖̅, 𝜒̅) 𝜕𝜒̅𝑇 = 𝜕 𝑊𝑐̅_𝐸0(𝜒̅) 𝜕𝜒̅𝑇 + 𝜕 𝑊𝑐̅_𝐸𝐼(𝑖̅, 𝜒̅) 𝜕𝜒̅𝑇 = 𝑆𝑐 𝐷0(𝜒̅) + 𝑆𝑐 𝐷𝐼(𝑖̅, 𝜒̅) (25)

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3.4 Elecrtical equation

Having defined the physical quantities, it becomes possible to derive the electrical equations at the terminals of each winding. The “motor” convention is adopted for the analysis, as described by [17].

The voltage at the terminals of the windings is given by two contributions. The first is the resistive voltage due to the Joule effect, the second one is the electromagnetic voltage induced in the windings.

𝑣̅(𝑡) = 𝑣̅𝑅(𝑡) + 𝑣̅𝐸(𝑡) (26)

With respect to the resistive voltage contribution, a new state function 𝑣̅_𝑅(𝑡) can be introduced. Analogously to the previously performed discussion, the pseudo-linear decomposition can be applied, leading to the following expression:

𝑣̅_𝑅(𝑡) = 𝑣̅𝑐 _𝑅(𝑖̅(𝑡), 𝜒̅(𝑡)) = 𝑣̅𝑐 _𝑅0(0, 𝜒̅(𝑡)) + 𝑣̅𝑐 _𝑅𝐼(𝑖̅(𝑡), 𝜒̅(𝑡)) (27) In practical applications, the null-current term 𝑐𝑣̅_𝑅0(0, 𝜒̅(𝑡)) of this equation is always null. Only the incremental term will then be considered. An incremental resistance state function 𝐶𝑅𝐼 is introduced in (28), as the quantity linking the above defined state function 𝑐𝑣̅_𝑅𝐼(𝑖̅(𝑡), 𝜒̅(𝑡)) and the state variable 𝑖̅. (28) expresses the generalized Ohm’s law.

𝑣̅_𝑅𝐼(

𝑐 _{𝑖̅(𝑡), 𝜒̅(𝑡)) = 𝑅} 𝐼

𝐶 _{(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗ 𝑖̅(𝑡)} ₍₂₈₎ The electromagnetic voltage induced in each coil may be determinated univocally by applying the Faraday’s law in integral form. This leads to (29) for expressing 𝑣̅_𝐸(𝑡), where it can be observed the further separation of this term in two contributions, i.e. 𝑣̅𝐸𝐿(𝑡) and 𝑣̅𝐸𝑀(𝑡), whose derivation is also given below. These represent the “inductive” electromagnetic force (30) and the “motional” electromagnetic force (31), respectively. 𝑣̅_𝐸(𝑡) = 𝑑 𝑑𝑡𝜓̅(𝑡) = 𝑑 𝑑𝑡 𝜓̅(𝑖̅(𝑡), 𝜒̅(𝑡)) = 𝑐 _𝑣̅ 𝐸𝐿(𝑡) + 𝑣̅𝐸𝑀(𝑡) (29) 𝑣̅𝐸𝐿(𝑡) = 𝜕 𝜓̅(𝑖̅, 𝜒̅)𝑐 𝜕𝑖̅𝑇 (𝑡) ∗𝑑𝑖̅(𝑡) 𝑑𝑡 = 𝐿𝑐 𝑐 _{(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗}𝑑𝑖̅(𝑡) 𝑑𝑡 (30) 𝑣̅_𝐸𝑀(𝑡) =𝜕 𝜓̅(𝑖̅, 𝜒̅) 𝑐 𝜕𝜒̅𝑇 (𝑡) ∗𝑑𝑖̅(𝑡) 𝑑𝑡 = 𝑀 𝑐 _{(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗}𝑑𝜒̅(𝑡) 𝑑𝑡 (31)

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The complete electrical equation, at the terminal of any type of device, is finally provided in (32). 𝑣̅(𝑡) = 𝑅𝐶 _𝐼(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗ 𝑖̅(𝑡) + 𝐿𝑐 _𝑐(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗𝑑𝑖̅(𝑡) 𝑑𝑡 + 𝑀𝑐 (𝑖̅(𝑡), 𝜒̅(𝑡)) ∗𝑑𝜒̅(𝑡) 𝑑𝑡 (32)

Figure 11: Generalized equivalent circuit

In conclusion, in Fig.11, the electrical equivalent circuit deriving from the approach presented above is shown.

3.5 Energy analysis

In order to perform the study aimed at understanding the link between electromagnetic and mechanical phenomena, the energy conservation principle is used and the relevant balance is shown in the scheme of Fig.12.

Pe Pt Pm Pu

P Pp Pr Ph Pn

Pc

Figure 12. Block diagram of balance energy

𝑣̅𝐸𝑀(𝑡) 𝑣̅𝐸(𝑡) 𝑣̅(𝑡) 𝑣̅𝑅(𝑡) 𝑣̅𝐸𝐿(𝑡) 𝑅𝐼( 𝐶 _{𝑖̅, 𝜒̅)} _𝐿 𝐷 𝑐 _{(𝑖̅, 𝜒̅)} 𝑀 𝑐 _{(𝑖̅, 𝜒̅)} 𝑖̅(𝑡) Ee Em Et

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In Fig.12, the top branch is made up of 1) the electric power P, 2) the electromagnetic power Pe, 3) the transformed power Pt, 4) the mechanical power Pm and 5) the output

power Pu. The power loss Pp links the top branch with the bottom one, which is made

up of 1) the resistive power Pr, 2) the cooling power Pc, and 3) the heat-dissipated power

Pe. In the same scheme, 1) represents the electromagnetic energy Ee, 2) the mechanical

energy Em and 3) the thermal energy Et.

By correlating the electromagnetic and the transformed powers (of mechanical origin) the following expression can be derived, as given in (33).

𝑃𝐸(𝑡) − 𝑃𝑇(𝑡) =

𝑑𝐸_𝐸(𝑡)

𝑑𝑡 (33)

The electromagnetic energy stored inside the generic device under analysis depends instantaneously on its field map. In (34), the new state function 𝑐𝐸𝐸 is instroduced. In (35), P(t) is exploited and split in two terms, namely the resistive and electromagnetic power contributions. These terms, i.e. PR and PE, are exploited in (36) and (37) and, in

turn, PE can be separated in the inductive and motional components, as given in (38)

and (39) respectively. 𝐸𝐸(𝑡) = 𝐸𝑐 𝐸(𝑖̅(𝑡), 𝜒̅(𝑡)) ∀𝑡 (34) 𝑃(𝑡) = ∑ 𝑣𝑘(𝑡) ∗ 𝑖𝑘(𝑡) = 𝑖̅(𝑡)𝑇 𝑛𝑒 𝑘=1 ∗ 𝑣̅(𝑡) = 𝑃𝑅(𝑡) + 𝑃𝐸(𝑡) (35) 𝑃_𝑅(𝑡) = 𝑖̅(𝑡)𝑇_{∗ 𝑣̅} 𝑅(𝑡) = 𝑖̅(𝑡)𝑇∗ 𝑅𝐶 𝐼(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗ 𝑖̅(𝑡) (36) 𝑃_𝐸(𝑡) = 𝑖̅(𝑡)𝑇_{∗ 𝑣̅} 𝐸(𝑡) = 𝑃𝐸𝐿(𝑡) + 𝑃𝐸𝑀(𝑡) (37) 𝑃_𝐸𝐿(𝑡) = 𝑖̅(𝑡)𝑇_{∗ 𝑣̅} 𝐸𝐿(𝑡) = 𝑖̅(𝑡)𝑇∗ 𝐿𝑐 𝐷(𝑖̅(𝑡), 𝜒̅(𝑡)) ∗ 𝑑𝑖̅(𝑡) 𝑑𝑡 (38) 𝑃_𝐸𝑀(𝑡) = 𝑖̅(𝑡)𝑇_{∗ 𝑣̅} 𝐸𝑀(𝑡) = 𝑖̅(𝑡)𝑇∗ 𝑀𝑐 (𝑖̅(𝑡), 𝜒̅(𝑡)) ∗ 𝑑𝜒̅(𝑡) 𝑑𝑡 (39)

The transformed power can be expressed as in (40), while the derivative of the electromagnetic energy is given in (41). By using and manipulating (38-41) it becomes possible to obtain the elegant expression given in (42).

(37)

𝑃_𝑇(𝑡) = 𝑊̅_𝐸𝑇_∗𝑑𝜒̅(𝑡) 𝑑𝑡 (40) 𝑑𝐸_𝐸(𝑡) 𝑑𝑡 = ( 𝜕 𝐸𝑐 _𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇 𝑡 ∗𝑑𝑖̅(𝑡) 𝑑𝑡 ) + ( 𝜕 𝐸𝑐 _𝐸(𝑖̅, 𝜒̅) 𝜕𝜒̅𝑇 𝑡 ∗𝑑𝜒̅(𝑡) 𝑑𝑡 ) (41) [𝜕 𝐸𝐸( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕𝑖̅𝑇 − 𝑖̅𝑇 𝐿𝐷 𝑐 _{(𝑖̅, 𝜒̅)]} (𝑡) ∗𝑑𝑖̅(𝑡) 𝑑𝑡 + [ 𝑊𝑐 _𝐸(𝑖̅, 𝜒̅)𝑇+𝜕 𝐸𝐸( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕𝜒̅𝑇 − 𝑖̅ 𝑇 𝑐_𝑀_{(𝑖̅, 𝜒̅)]} (𝑡) ∗𝑑𝜒̅(𝑡) 𝑑𝑡 = 0 (42)

Considering the above, it can be affirmed that a suitable vector state function have to result orthogonal to the time derivative of the state variables for any possible trajectory in the state space. As these trajectories can pass by any given state with any time derivative, this clearly implies that the above state function must be null.

In order to make the first member of (42) equal to zero, the two terms written in square brackets must be null. As a result, it becomes possible to separately analyze them. Therefore, the first term can be written as in (43). By defining the “electromagnetic coenergy” state function as in (44), it is possible to re-write the flux state function and this expression is given in (45). The latter means that the coenergy function is the scalar potential for the flux state function in current state

𝜕 𝐸𝑐 𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇 = 𝑖̅𝑇∗ 𝐿𝐷 𝑐 _{(𝑖̅, 𝜒̅) =} 𝜕 𝜕𝑖̅𝑇(𝑖̅𝑇∗ 𝜓̅(𝑖,̅ 𝜒̅) 𝑐 _{) − 𝜓̅(𝑖,̅ 𝜒̅)}𝑐 𝑇 ₍₄₃₎ 𝐶_𝐸( 𝑐 _{𝑖̅, 𝜒̅) = (𝑖̅}𝑇_{∗ 𝜓̅(𝑖,̅ 𝜒̅)}𝑐 _{− 𝐸} 𝐸( 𝑐 _{𝑖̅, 𝜒̅))} (44) 𝜓̅(𝑖,̅ 𝜒̅) 𝑐 ₌𝜕 𝐶𝑐 𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅ (45)

By recalling the definition of 𝑐𝐿𝐷(𝑖̅, 𝜒̅) and of 𝑀(𝑖,̅ 𝜒̅)𝑐 , it is possible to obtain the following expressions, as reported in (46) and (47).

𝐿_𝐷 𝑐 _{(𝑖̅, 𝜒̅) =}𝜕 𝜓̅(𝑖,̅ 𝜒̅) 𝑐 𝜕𝑖̅𝑇 = 𝜕2 𝑐𝐶𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇_𝜕𝑖̅ = 𝐿𝐷 𝑐 _{(𝑖̅, 𝜒̅)}𝑇 ₍₄₆₎

(38)

𝑀(𝑖,̅ 𝜒̅) 𝑐 ₌ 𝜕 𝜓̅(𝑖,̅ 𝜒̅) 𝑐 𝜕𝜒̅𝑇 = 𝜕2 _𝐶 𝐸( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕𝜒̅𝑇_𝜕𝑖̅ (47)

A similar analysis can be carried out for the second term in brackets of (42), leading to the expression given below in (48). Similarly, it can be affirmed that the coenergy function is the scalar potential also for the electromagnetic wrench function in the generalised position space. As well as for 𝑐𝐿_𝐷(𝑖̅, 𝜒̅) and of 𝑀(𝑖,̅ 𝜒̅)𝑐 , 𝑐𝐾_𝐷(𝑖̅, 𝜒̅) and

𝑆𝐷

𝑐 _{(𝑖̅, 𝜒̅) can be exploited resulting in the expressions reported in (49) and (50).} 𝑊̅_𝐸 𝑐 _{(𝑖̅, 𝜒̅) =} 𝜕 𝜕𝜒̅(𝑖̅ 𝑇_{∗ 𝜓}𝑐 _{(𝑖̅, 𝜒̅)) −}𝜕 𝐸𝐸( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕𝜒̅ = 𝜕 𝐶𝑐 𝐸(𝑖̅, 𝜒̅) 𝜕𝜒̅ (3) 𝐾𝐷 𝑐 _{(𝑖̅, 𝜒̅) =}𝜕 𝑊𝑐̅𝐸(𝑖,̅ 𝜒̅) 𝜕𝑖̅𝑇 = 𝜕2 𝑐𝐶_𝐸(𝑖̅, 𝜒̅) 𝜕𝑖̅𝑇_𝜕𝜒̅ = 𝑀 𝑐 _{(𝑖̅, 𝜒̅)}𝑇 ₍₄₉₎ 𝑆𝐷 𝑐 _{(𝑖̅, 𝜒̅) =}𝜕 𝑊𝑐̅𝐸(𝑖,̅ 𝜒̅) 𝜕𝑖̅𝑇 = 𝜕2 𝑐𝐶_𝐸(𝑖̅, 𝜒̅) 𝜕𝜒̅𝑇_𝜕𝜒̅ = 𝑆𝐷 𝑐 _{(𝑖̅, 𝜒̅)}𝑇 ₍₅₀₎

Having re-written the equation of the secondary state functions, it is possible to define the Hessian matrix of the coenergy state function, as reported in (51).

ℋ[ 𝐶𝑐 𝐸(𝑖̅, 𝜒̅)] = 𝜕2 _𝐶 𝐸( 𝑐 _{𝑖̅, 𝜒̅)} 𝜕[𝑖̅𝑇 _𝜒̅𝑇_{]𝜕 [}𝑖̅ 𝜒̅] = [ 𝐿𝐷 𝑐 _{(𝑖̅, 𝜒̅)} 𝑐_{𝑀(𝑖,̅ 𝜒̅)} 𝐾𝐷 𝑐 _{(𝑖̅, 𝜒̅)} _𝑆 𝐷 𝑐 _{(𝑖̅, 𝜒̅)}] ₍₄₎ It can be concluded that the link between the electromagnetic and mechanical aspects of the considered device is expressed through the coenergy state function. This strong relation can be clearly observed thoughtout the definition of the differential wrench coefficients, which is the same as that of the motion voltage coefficients.

For the sake of coherency with the studies performed on the state functions introduced in the previous sections, the same pseudo-linear decomposition is applied to the coenergy, with the aim of separating the null-current effect from the incremental one, as given in (53) and (54).

The equation for the electromagnetic energy, coenery and wrench functions can be written in the integral form between two states (A and B) featuring the same position 𝜒̅ .

𝐶_𝐸( 𝐶 _{𝑖̅, 𝜒̅) = 𝐶} 𝐸0( 𝑐 _{𝜒̅) + 𝐶} 𝐸𝐼 𝑐 _{(𝑖̅, 𝜒̅)} ₍₅₂₎ 𝐶𝐸0( 𝑐 _{𝜒̅) = 𝐶} 𝐸 𝑐 _{(0, 𝜒̅)} (53) 𝐶_𝐸𝐼( 𝐶 _{𝑖̅, 𝜒̅) = 𝐶} 𝐸(𝑖̅, 𝑐 _{𝜒̅) − 𝐶} 𝐸0 𝑐 _(𝜒̅) ₍₅₄₎