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Elektrik makinalarında uzay harmoniklerinin uzay fazörleri ile bilgisayar destekli analizi

Elektrik makinalarında uzay harmoniklerinin uzay fazörleri ile bilgisayar destekli analizi

##### Dosyalar

##### Tarih

1997

##### Yazarlar

Kocabaş, Derya Ahmet

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Alternatif akım makinalannda hava aralığındaki ampersanm dağılımının harmoniklerinin incelenmesi sırasında kullanılan Fourier metodu oldukça sıkıcı, uzun, zor ve hata yapma olasılığı yüksek bir yöntemdir. Harmonikler sargı dağılımının bir fonksiyonu olduğundan, sargı konusu ana hatlarıyla bilinmelidir. Sargı faktörü, ilk bakışta sargının kalitesini ortaya koyduğundan belirlenmesi ve her harmonik için hesaplanması esastır. Mmk eğrisinin harmoniklerinin incelenmesinde kullanılan klasik metotlarda, çözümü basitleştirmek amacıyla belirli yaklaşıklıklar kabul edilir. Fakat bu yaklaşıklıklar işlem karmaşasını ortadan kaldırmaya yetmez. Bu tezde, bu karmaşanın üstesinden gelebilmek ve zamandan tasarruf edebilmek amacıyla harmonik analizinin bilgisayar programlan ile yapılabileceği gösterilmiştir. Burada kullanılan metotların işlerlikleri ve çabuklukları gözlemlenmiştir. Yazılan programlar ile bir sargı yapısına ilişkin mmk harmonikleri ve sargı faktörleri hesaplanabilmekte ve sargının, dolayısıyla tasarlanacak olan makinanın kalitesi hakkında bir ön bilgi verilebilmektedir.

This study represents some methods for harmonic analysis in alternative current windings. First of all, alternative current windings are dealed with and the special charecteristics of a winding are described. Then the space distribution of the magnetomotive force in an electrical machine is observed. For different kinds of distributions of coils, the harmonic components of the magnetomotive force are calculated. Also, some information about the space phasor theory is given. Furthermore, a rapid method in time domain and an easy method in matrix transform according to the beginning the space phasor theory are explained. In the first metod, harmonic components of magnetomotive force for one coil in a phase, for all coils of a phase and space harmonics of the resultant magnetomotive force for all phases are dealed. The resultant formula is obtained for the magnetomotive force distributions explained above. After these, harmonic winding factor is calculated. Using the space phasor theory, complex winding factor is given. In this last method, the space harmonics of magnetomotive force is calculted too. In the next section, some simulation programs for calculation of harmonic components of magmetomotive force for any alternative current winding are introduced. The programs are written for the given two methods above. Last section represents the simulation results for an ordinary winding and an improved winding. The results are compared and the usage of the programs is understood. The space distribution of the magnetomotive force depends on the distributions of the coils around the periphery. The rotating field in the air gap is produced by the stator windings. The kinds and the characteristic quantities of the windings must be well known to minimize the harmonics in the air gap. The characteristic quantities of a winding are the number of the phases, the number of the slots around the periphery and the number of the poles. The other quantities of the winding can be calculated from given ones. Also the kind of the winding must be given too. The drawn winding scheme must be controlled according to the number of the poles. The quality of a winding is measured with the winding factor (kw). The winding factor effects the induced electromotor force, so it must be calculated. It varies for the different harmonics. It is a function of chording of a coil, the form of the flux around the air gap and the spread of the slots around the armature. ix In alternative current machines, windings are situated in slots round the periphery of the armature. If the circumferantial distance of one slot pitch in the air gap is traversed, the alternation of mmf will be that of the amper-conductors in the slots. The total change of mmf with several slots will be the sum of the amper-conductors in each slots. The mmf wave, whatever it's shape, repeats its general form every pole pitch and may therefore be expressed as a series of sine and cosine functions. Of course, the series will vary with the shape of the mmf wave. The analysis for the actual distribution of mmf may be complicated. For this reason, a number of approximations may be made to the actual waveshape. Where there is a distributed winding with a large number of coils uniformly spread round the periphery, the mmf distribution may be taken as being triangular. The harmonics move with a velocity inversely proportional to their order; since the harmonic pole pitch is also inversely propotional to the order of the harmonics; but however it is shown in related section that the space harmonics of mmf generate only fundamental frequency voltages in the armature conductors. For solving complicated problems of space harmonics in electrical machines, space phasor theory was developed. The space phasor theory is a general method for developing equations of electrical machines under both steady and transient conditions. The winding can have any number of phases and can be asymmetrical. The air gap doesn't need to be uniform. Naturally, the theory is also suitable for deriving the well-known basic equations of symmetrical machines with constant air gap, if only the main wave is taken into consideration. However, this theory has not been developed for elemantary problems which have been solved by other methods. In electrical machines, it is very hard and cumbersome to observe the change of electromagnetic quantities by using the sinusoidal expressions. Mathematical process gets harder if the harmonic components are dealed with. In this way, it will make easier the mathematical process by defining the quantities changing sinusoidally in the space of the machine like magnetic flux density, flux as a complex number in the complex plane. It is very convenient to use polar coordinates in this theory that the electrical machines have a cylinderical construction. The angular current density of stator (A) is a function of time and polar coordinate a. This is a periodic but non-sinusoidal wave and can be expressed by a sum of infinite number of sine waves. The resultant flux density wave of one phase of the stator, by the way space phasor, can be obtained by calculating the sum of the flux density waves of all slots that the winding is distributed. After these the complex winding factor (kwn) can be calculated. The usual method of evaluating the harmonic components the mmf wave of a polyphase winding is to perform Fourier analysis. This can, however, be a long and tedious process, especially for stepped waveforms. Further, it is common practice to perform the analysis for two limitting current-vector positions, to provide a check on the results. The first method described here is expectionally simple to apply, and makes use of the well-known standard winding factors. It is shown that, by a simple substitions, the magnitudes of the fourier fundamental and harmonic components of the mmf in the air gap can be found directly, although no indication is given of their relative phase angles. The mathematics of the proof has been fitted as closely as possible to the conditions prevailing in a real machine winding, whilst at the same time maintaining generality of treatment. The value of the method is exemplified in the last section by carrying out the analysis for a mmf wave of some complexity. The ease of the method is at once apparent. By the ortodox method of Fourier analysis, both the magnitudes and relative phase angles of the harmonic components of a mmf wave can be found. But in practice, it is usually only the magnitudes which are significant, because a study of the relative values of these will indicate, for example, whether or not an induction motor is likely to crawl or to be noisy in operation. At the beginning of the theory, two functions are defined for two different but related quantities. One of them is the space distribution of the mmf produced by the conductors carrying current. The other one is the distribution of the conductors producing the mmf themselves. These functions are developed for one coil and also all coils of the phase. To get to the second step, mmf function is written in the form of the sum of the mmf waves of all coils by sigma function. After these, enduced emf is derived in two different ways to obtain the winding factor.

This study represents some methods for harmonic analysis in alternative current windings. First of all, alternative current windings are dealed with and the special charecteristics of a winding are described. Then the space distribution of the magnetomotive force in an electrical machine is observed. For different kinds of distributions of coils, the harmonic components of the magnetomotive force are calculated. Also, some information about the space phasor theory is given. Furthermore, a rapid method in time domain and an easy method in matrix transform according to the beginning the space phasor theory are explained. In the first metod, harmonic components of magnetomotive force for one coil in a phase, for all coils of a phase and space harmonics of the resultant magnetomotive force for all phases are dealed. The resultant formula is obtained for the magnetomotive force distributions explained above. After these, harmonic winding factor is calculated. Using the space phasor theory, complex winding factor is given. In this last method, the space harmonics of magnetomotive force is calculted too. In the next section, some simulation programs for calculation of harmonic components of magmetomotive force for any alternative current winding are introduced. The programs are written for the given two methods above. Last section represents the simulation results for an ordinary winding and an improved winding. The results are compared and the usage of the programs is understood. The space distribution of the magnetomotive force depends on the distributions of the coils around the periphery. The rotating field in the air gap is produced by the stator windings. The kinds and the characteristic quantities of the windings must be well known to minimize the harmonics in the air gap. The characteristic quantities of a winding are the number of the phases, the number of the slots around the periphery and the number of the poles. The other quantities of the winding can be calculated from given ones. Also the kind of the winding must be given too. The drawn winding scheme must be controlled according to the number of the poles. The quality of a winding is measured with the winding factor (kw). The winding factor effects the induced electromotor force, so it must be calculated. It varies for the different harmonics. It is a function of chording of a coil, the form of the flux around the air gap and the spread of the slots around the armature. ix In alternative current machines, windings are situated in slots round the periphery of the armature. If the circumferantial distance of one slot pitch in the air gap is traversed, the alternation of mmf will be that of the amper-conductors in the slots. The total change of mmf with several slots will be the sum of the amper-conductors in each slots. The mmf wave, whatever it's shape, repeats its general form every pole pitch and may therefore be expressed as a series of sine and cosine functions. Of course, the series will vary with the shape of the mmf wave. The analysis for the actual distribution of mmf may be complicated. For this reason, a number of approximations may be made to the actual waveshape. Where there is a distributed winding with a large number of coils uniformly spread round the periphery, the mmf distribution may be taken as being triangular. The harmonics move with a velocity inversely proportional to their order; since the harmonic pole pitch is also inversely propotional to the order of the harmonics; but however it is shown in related section that the space harmonics of mmf generate only fundamental frequency voltages in the armature conductors. For solving complicated problems of space harmonics in electrical machines, space phasor theory was developed. The space phasor theory is a general method for developing equations of electrical machines under both steady and transient conditions. The winding can have any number of phases and can be asymmetrical. The air gap doesn't need to be uniform. Naturally, the theory is also suitable for deriving the well-known basic equations of symmetrical machines with constant air gap, if only the main wave is taken into consideration. However, this theory has not been developed for elemantary problems which have been solved by other methods. In electrical machines, it is very hard and cumbersome to observe the change of electromagnetic quantities by using the sinusoidal expressions. Mathematical process gets harder if the harmonic components are dealed with. In this way, it will make easier the mathematical process by defining the quantities changing sinusoidally in the space of the machine like magnetic flux density, flux as a complex number in the complex plane. It is very convenient to use polar coordinates in this theory that the electrical machines have a cylinderical construction. The angular current density of stator (A) is a function of time and polar coordinate a. This is a periodic but non-sinusoidal wave and can be expressed by a sum of infinite number of sine waves. The resultant flux density wave of one phase of the stator, by the way space phasor, can be obtained by calculating the sum of the flux density waves of all slots that the winding is distributed. After these the complex winding factor (kwn) can be calculated. The usual method of evaluating the harmonic components the mmf wave of a polyphase winding is to perform Fourier analysis. This can, however, be a long and tedious process, especially for stepped waveforms. Further, it is common practice to perform the analysis for two limitting current-vector positions, to provide a check on the results. The first method described here is expectionally simple to apply, and makes use of the well-known standard winding factors. It is shown that, by a simple substitions, the magnitudes of the fourier fundamental and harmonic components of the mmf in the air gap can be found directly, although no indication is given of their relative phase angles. The mathematics of the proof has been fitted as closely as possible to the conditions prevailing in a real machine winding, whilst at the same time maintaining generality of treatment. The value of the method is exemplified in the last section by carrying out the analysis for a mmf wave of some complexity. The ease of the method is at once apparent. By the ortodox method of Fourier analysis, both the magnitudes and relative phase angles of the harmonic components of a mmf wave can be found. But in practice, it is usually only the magnitudes which are significant, because a study of the relative values of these will indicate, for example, whether or not an induction motor is likely to crawl or to be noisy in operation. At the beginning of the theory, two functions are defined for two different but related quantities. One of them is the space distribution of the mmf produced by the conductors carrying current. The other one is the distribution of the conductors producing the mmf themselves. These functions are developed for one coil and also all coils of the phase. To get to the second step, mmf function is written in the form of the sum of the mmf waves of all coils by sigma function. After these, enduced emf is derived in two different ways to obtain the winding factor.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1997

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Social Sciences, 1997

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Social Sciences, 1997

##### Anahtar kelimeler

Alternatif akım motorları,
Fourier analizi,
Harmonikler,
Harmonics,
Fourier analysis,
Electric machinery