Fundamentals of AC Machinery Revised October 6, 2008 4. Fundamentals of AC Machinery 1
AC Machines: We begin this study by first looking at some commonalities that eist for all machines, then look at specific machines such as synchronous machines induction machines DC machines 4. Fundamentals of AC Machinery 2
AC Machines: Motors: AC Electrical Energy Mechanical Energy We apply a current to a wire in the presence of an applied magnetic field which produces a torque this is motor action. Just like when two permanent magnets are near each other, a force is eerted which makes their magnetic fields want to line up. Generators: e Mechanical ca Energy AC Electrical ca Energy We move a wire in the presence of an applied magnetic field which induces a voltage generator action. In each case, the applied magnetic field for AC machines is produced by a field winding. 4. Fundamentals of AC Machinery 3
AC Machines: Two major classes of AC machines: Synchronous Machines are motors and generators whose field current (or current in the rotor windings) is supplied by a direct connection via a rotating contact to a separate stationary power source. Induction Machines are motors o and generators e whose field current (current in the rotor windings) is induced in the rotor by magnetic induction (transformer action) and by the relative motion of the rotor and stator. We will consider each machine in detail. First we look at the basic commonalities that govern both types of machines. 4. Fundamentals of AC Machinery 4
In rotating machines, voltages are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor. By any of these methods, the flu linking a specific coil is changed cyclically, and a time-varying voltage is generated. A set of such coils connected together is typically referred to as an armature winding. 4. Fundamentals of AC Machinery 5
In general, the term armature winding is used to refer to a winding or a set of windings on a rotating machine which carry ac currents. In ac machines such as synchronous or induction machines, the armature winding is typically on the stationary portion of the motor referred to as the stator, and these windings may also be referred to as stator windings. Terminology: The stationary part of a machine is called the stator, while the part that rotates is called to rotor. The field circuits of most synchronous and induction machines are located on their rotors. 4. Fundamentals of AC Machinery 6
With rare eceptions, the armature winding of a synchronous machine is on the stator, t and the field winding is on the rotor. The field winding is ecited by direct current conducted to it by means of stationary carbon brushes which contact rotating slip rings or collector rings. 4. Fundamentals of AC Machinery 7
A Simple Generator A highly idealized analysis of this machine would assume a sinusoidal distribution of magnetic flu in the air gap. 4. Fundamentals of AC Machinery 8
AC Machine Basics Instead of rotating a conductor in a fied magnetic field as we did in an earlier eample (see Set 1, Slides 66 and onward), suppose we now (somehow) rotate the magnetic field inside a fied conductor. i B i e X 4. Fundamentals of AC Machinery 9
AC Machine Basics For a uniform magnetic flu density B, the total flu linking the circuit is, for a constant rotational speed, and the induced voltage is B da BAcos t BAcos t d e BAsint dt 4. Fundamentals of AC Machinery 10
AC Machine Basics It is difficult to have a uniform rotating magnetic field like the one considered so as to achieve a sinusoidal voltage. Instead, practical machines attempt to create a sinusoidally distributed rotating magnetic field in the air gap between the rotor and stator. As we will later see, this produces the desired sinusoidal induced voltage. 4. Fundamentals of AC Machinery 11
Rotating Magnetic Field Fundamental principle of AC machine operation: If a threephase set of currents flows in a three-phase winding, it will produce a rotating magnetic field. 4. Fundamentals of AC Machinery 12
a Three-Phase Windings c b bb a cc EMPTY STATOR 4. Fundamentals of AC Machinery 13
Convention: Current is positive if it flows into the unprimed and out of the primed winding. c aa H aa t b Magnetic Field produced by winding aa bb a cc EMPTY STATOR 4. Fundamentals of AC Machinery 14
aa c H bb t b Magnetic Field produced by winding bb bb a cc EMPTY STATOR 4. Fundamentals of AC Machinery 15
aa c H cc t b Magnetic Field produced by winding cc bb a cc EMPTY STATOR 4. Fundamentals of AC Machinery 16
Recall Set 1, Slide 14 C Hd I H Ni H enclosed c c B B H H Ni c NiA Bda i A i B H i c i i i i i i i Ni c 4. Fundamentals of AC Machinery 17
Add all three fields aa c y b bb a cc EMPTY STATOR H aa t Hbb t H H cc 4. Fundamentals of AC Machinery 18 t
B B aa t Bbb t Bcc t B t B t 120 B t 240 aa 0 bb 1 3 B aa t Bbb t Bbb t y 2 2 1 3 B cc t Bc c t y 2 2 1 1 B aa t Bbb t Bcc t 2 2 cos 120 180 cos 240 180 3 3 B bb t Bcc t y 2 2 cc y 05 0.5 sin 120 0.866 180 0.5 sin 240 180 4. Fundamentals of AC Machinery 19 0.866
1 1 B B aa t Bbb t Bcc t 2 2 3 3 B bb t Bcc t y 2 2 B H i 1 1 B B sin sin 120 sin 240 M t t t 2 2 3 3 BM sin t 120 sin t 240 2 2 y 4. Fundamentals of AC Machinery 20
Let Mathcad do the algebra... sint 1 sin t 120 2 180 1 sin t 240 2 180 3 2 sint 3 2 sin t 120 180 3 2 sin t 240 180 3 2 cos t 4. Fundamentals of AC Machinery 21
1 1 B BM sint sin t 120 sin t 240 2 2 3 3 BM sin t 120 sin t 240 2 2 3 3 BM sint BM cost y 2 2 y 4. Fundamentals of AC Machinery 22
3 B B sin t cos t y t t 2 M t y t sint cost 0 0 1 /2 1 0 0-1 3/2-1 0 t B M 3 2 B M t B M t t B M 0 2 The magnetic flu has constant magnitude (1.5 B M ) but rotates in time with angular velocity. 4. Fundamentals of AC Machinery 23
The winding has produced an effective (rotating) magnet with a north and a south pole, and is hence termed a two pole winding. a c N B S S b B S 1.5B M b a c 4. Fundamentals of AC Machinery 24
The magnetic poles complete one complete mechanical rotation for each electrical cycle of applied current. The mechanical speed of rotation of the magnetic field in revolutions per second is equal to the electrical frequency in Hertz, f f, two poles electrical mechanical 4. Fundamentals of AC Machinery 25
Note the order of the windings, acbacb a c b B S bb a cc 4. Fundamentals of AC Machinery 26
Now suppose that we repeat the pattern: a c b a c b a c b a c b a2 c 1 b b 2 1 c 1 a 2 a 1 c 2 2 a 1 b c 2 b 1 4. Fundamentals of AC Machinery 27
Here s how a phase is wound: a 2 a 2 a 1 a a 2 b 2 a2 c 1 b 1 c 1 a a 1 a a a 1 a 2 a 2 a 1 c 2 b 2 a 1 c 2 b 1 a 1 4. Fundamentals of AC Machinery 28
The tet illustrates windings as follows: a c b c a b c a b c bb a c Back end of coil c a b c a 1 2 b 1 1 c 1 1 a 2 1 b2 2 c 2 2 2 a b a c b c counterclockwise a b 4. Fundamentals of AC Machinery 29
Recall: The winding has produced an effective (rotating) magnet with a north and a south pole, and is hence termed a two pole winding. c a b a c b a c b N S S N B S b a c 4. Fundamentals of AC Machinery 30
acbacb acbacb S N S N a 2 b 2 a2 c 1 S B b 1 B N c 1 a 1 c 2 a 1 b c 2 2 b 1 N S 4. Fundamentals of AC Machinery 31
The tet illustrates windings as follows: c a b c a b c B B B B c S a N S b c a N 2 a 1 2 b 1 1 c 1 1 a 2 1 b2 2 c 2 2 b Back end of coil a c b c counterclockwise a b 4. Fundamentals of AC Machinery 32
S 90 o mechanical 180 o mechanical 180 o 360 o N electrical N S electrical S N N S S N N S Though a full 360 o of electrical rotation has occurred a pole has mechanically progressed by only 180 o, hence f electrical 2 f mechanical 4. Fundamentals of AC Machinery 33
The magnetic poles complete one complete mechanical rotation for two electrical cycles of applied current. For the four-pole winding, the electrical frequency of the current in Hertz is twice the mechanical frequency of rotation. f 2 f, four poles electrical mechanical P felectrical fmechanical P poles 2 Generalization:, For n m mechanical revolutions per minute (rpm), f electrical np m, 120 P poles 1 cycle cycle = 60 second minute 4. Fundamentals of AC Machinery 34
The windings have produced two effective (rotating) magnets with two north and two south poles, and is hence termed a four pole winding. a 2 b 2 a2 c 1 S mech b 1 mech B mech N mech c 1 a 1 c 2 2 N S mech a 1 b c 2 b 1 4. Fundamentals of AC Machinery 35
S N S N N S 2pole 4pole 4. Fundamentals of AC Machinery 36
Interesting: If the current in any two of the three coils is swapped, the direction of the magnetic field s rotation will be reversed. This means that the direction of an AC motor can be revered by switching the connection on any two of the three coils. The verification of this fact is left as an eercise (it s done for you in the book). 4. Fundamentals of AC Machinery 37
Magnetomotive Force and Flu Distribution on AC Machines So far we ve considered only an empty stator. The direction of the magnetic flu density produced by the coils was assumed perpendicular to the plane of the coil, determined via the right-hand-rule. In a real machine (i.e., one with a rotor) things are a bit more complicated. In a real machine there is a ferromagnetic rotor with a (small) gap between the rotor and the stator. The rotor can be a simple cylinder or something more complicated. 4. Fundamentals of AC Machinery 38
Stator Stator Gap S Rotor N Gap Rotor The cylindrical rotor is called a non-salient pole machine. The non-cylindrical rotor is called a salient pole machine. 4. Fundamentals of AC Machinery 39
variable spacing air gap For the salient pole machine, note how the gap width varies with angle. Since the total reluctance is dominated by the air gap, is possible to design a shape that produces a flu density which varies approimately as cos ma cos 4. Fundamentals of AC Machinery 40
Magnetomotive Force and Flu Distribution on AC Machines Two observations, as per our earlier discussion of magnetic circuits: only the reluctance of the gap will be non-negligible the magnetic flu density vector B will be the shortest path possible or perpendicular to both the rotor and stator (path of least reluctance). 4. Fundamentals of AC Machinery 41
Magnetomotive Force and Flu Distribution on AC Machines Consider a non-salient pole machine: We wish to produce a sinusoidal voltage in the stator windings. How do we do this? 4. Fundamentals of AC Machinery 42
Magnetomotive Force and Flu Distribution on AC Machines If the induced potential (emf) at the terminals is to be sinusoidal, then the (magnitude) of the flu density vector B must vary sinusoidally along the surface of the gap. Recall again the result from Note Set 1 (Slide 14) that the flu is: Ni o A BA HA o A Y Y e gap gap gap This sinusoidal variation will happen if H and in turn, the mmf Y vary sinusoidally along the surface as shown on the net slide... 4. Fundamentals of AC Machinery 43
Magnetomotive Force and Flu Distribution on AC Machines This is the ideal case that we would like to achieve: B B sin ma sin 4. Fundamentals of AC Machinery 44
10 10 Magnetomotive Force and Flu Distribution on AC Machines A simple way to accomplish this is to allow the number of turns of the stator winding to vary in an approimate sinusoidal id fashion as shown, but this is rarely, even never done in practice. 3 3 7 7 7 7 3 3 nc Nc cos d e n N dt 10 10 induced c c d cos dt Number of conductors n c in each slot Y 0 180 Approimate sinusiodal mmf distribution 360 4. Fundamentals of AC Machinery 45
Distributed Windings How it s really done. Most armatures have a distributed winding where the coils are spread over a number of slots around the air-gap periphery. The individual coils are connected so that the result is a magnetic field having the same number of poles as the field winding. 4. Fundamentals of AC Machinery 46
MMF of Distributed Windings The study of the magnetic fields of distributed windings can be approached by eamining the magnetic field produced by a winding consisting of a single N-turn coil (called a concentrated winding) which spans 180 electrical degrees, as shown 180 4. Fundamentals of AC Machinery 47
MMF of Distributed Windings A coil which spans 180 electrical degrees is known as a full-pitch winding (as opposed to a fractional pitch winding). 180 4. Fundamentals of AC Machinery 48
MMF of Distributed Windings The magnetic field produced by the current in the coil is shown by the dashed lines. Since the permeability of the armature and field iron is much greater than that of air, assume that all the reluctance of the magnetic circuit is in the air gap. Magnetic ais of stator coil 4. Fundamentals of AC Machinery 49
MMF of Distributed Windings From symmetry it is evident that the magnetic field intensity H ag in the air gap at angle a under one pole is the same in magnitude as that at angle a + under the opposite pole, but the fields are in the opposite direction. Remember that H is always perpendicular to the stator/rotor surface though the diagram doesn t really show it. Magnetic ais of stator coil H ˆ ag Hag a a H ˆ ag Hag a 4. Fundamentals of AC Machinery 50
MMF of Distributed Windings Around any of the closed paths shown by the flu lines the mmf is Ni. Since all the reluctance of this magnetic circuit is in the air gap, the mmf drops associated with the magnetic circuit inside the iron can be neglected. Magnetic ais of stator coil a BA Y e Y e gap gap 4. Fundamentals of AC Machinery 51
MMF of Distributed Windings Since the air-gap fields H ag on opposite sides of the rotor are equal in magnitude but opposite in direction, so will the air-gap mmf. Since each flu line crosses the air gap twice, the mmf drop across the air gap equals half of the total or Ni/2. The air-gap mmf fdistribution ib i is a step-like distribution ib i of amplitude Ni/2. On the assumption of narrow slot openings, the mmf jumps abruptly by Ni in crossing from one side to the other of a coil. This is shown on the net slide... 4. Fundamentals of AC Machinery 52
MMF of Distributed Windings Note how the mmf flips here and here 4. Fundamentals of AC Machinery 53
MMF of Distributed Windings Unwound actual mmf in gap Ni 0 2 2 Ni 2 3 a Rotor Surface X X Stator Surface Gap 4. Fundamentals of AC Machinery 54
MMF of Distributed Windings Unwound actual mmf in gap sinusoidal fundamental Ni 0 2 2 Ni 2 Y gap fund 4 Ni 2 3 cos a a Rotor Surface X X Stator Surface Gap 4. Fundamentals of AC Machinery 55
MMF of Distributed Windings Concentrated Winding (one phase) Rotor Surface Stator Surface X a Distributed Winding (one phase) Rotor Surface a X X X X X Stator Surface 4. Fundamentals of AC Machinery 56
Distributed Windings - A distributed winding consists of coils distributed in several slots. This helps reduce the harmonics which, in general, are quite undesirable, e, leaving only the fundamental. Phase a of fthe armature winding of a simplified two-pole, three-phase ac machine. Phases b and c occupy the empty slots. The windings of the three phases are identical and are located with their magnetic aes 120 degrees apart. 4. Fundamentals of AC Machinery 57
Distributed Windings - 4. Fundamentals of AC Machinery 58
Distributed Windings Note that the mmf will be less for a distributed winding than would be for a concentrated winding. 4. Fundamentals of AC Machinery 59
Distributed Windings Fractional Pitch Windings further serve to reduce the harmonic content, but these will not be considered in this course. FULL PITCH FRACTIONAL PITCH 4. Fundamentals of AC Machinery 60
Non-salient pole machines Distributed Rotor Windings In a similar manner the rotor windings are also distributed to reduce higher harmonics. 4. Fundamentals of AC Machinery 61
Induced Voltage in AC Machines Just as a three-phase set of currents can produce a rotating magnetic field, a rotating magnetic field can produce a three-phase set of voltages in the coils of a stator. 4. Fundamentals of AC Machinery 62
Induced Voltage in AC Machines Consider first a single coil. Stator c Gap d Rotor e id inducedd Stator Winding b a 4. Fundamentals of AC Machinery 63
Induced Voltage in AC Machines Consider first a single coil, and consider a rotating rotor with a sinusoidally distributed magnetic field in the center of a stationary coil. How this magnetic field is generated will be discussed later. Specifically assume that the magnitude of the flu density in the air gap varies sinusoidally with mechanical angle while the direction of B is radial this is the ideal the ideal situation. This angle will be measured with respect to the peak rotor flu density. All this is shown as... 4. Fundamentals of AC Machinery 64
Induced Voltage in AC Machines A snapshot in time: B B M The magnetic flu density measured around the rotor is: B B cos M Rotor magnetic flu ais. Here is measured with respect to the rotor. 4. Fundamentals of AC Machinery 65
Induced Voltage in AC Machines But since the rotor is rotating, the magnetic flu density vector in the gap at any angle measured with respect to the stator is B B M B B cost M 4. Fundamentals of AC Machinery 66
Induced Voltage in AC Machines Recall that the induced voltage in the wire is given by v e v B induced where is the velocity of the wire relative to the magnetic field and is the length of the conductor in the magnetic field. Recall that points to the positive end. v B When we derived this equation (in Note Set 1), the wire was moving and the magnetic field was stationary. Here the magnetic field is moving and the wire is stationary, so some modification is needed. 4. Fundamentals of AC Machinery 67
Lorentz Force Law induced e B v wire + + + F qv B v + + + v e induced F B B into page _ F qv B Recall Note Set 1 4. Fundamentals of AC Machinery 68 B into page Recall Note Set 1
Induced Voltage in AC Machines We establish a moving reference frame in that we move (rotate) with the magnetic field. Then to us the magnetic field is stationary and the coil will appear to go by us with an apparent velocity of v relative and the original equation applies. This is captured by the following picture... 4. Fundamentals of AC Machinery 69
Induced Voltage in AC Machines B v relative Note here that B is shown as being directed radially outward. This is because its actual direction will be obtained with the from the factor B B cost M v relative lti v B B is really in the other direction here since = at this point and cos t cos t B 4. Fundamentals of AC Machinery 70
Induced Voltage in AC Machines B v B v 0 relative c v relative v B M 180 B e induced d v relative b a Bt e vb t ba vb t 4. Fundamentals of AC Machinery 71
Induced Voltage in AC Machines This is why we want a radial B v B, vb ba e vb t vb t vb M cos t vb cos t 180 vbm M cost 4. Fundamentals of AC Machinery 72
Induced Voltage in AC Machines B B M v relative e induced d v relative v c B relative B b a v 180 relative B 4. Fundamentals of AC Machinery 73
Induced Voltage in AC Machines v ebc vb B e bc 0 from the picture similarly, eda 0 4. Fundamentals of AC Machinery 74
Induced Voltage in AC Machines v B B e dc vb B B M v relative v 180 relative B e induced d a B vb c v relative 4. Fundamentals of AC Machinery 75 b
Induced Voltage in AC Machines e vb dc vb vb vb M M cost cos t 0 vb cos M t 4. Fundamentals of AC Machinery 76
Induced Voltage in AC Machines e vb cos t ba M c d e induced a b e vb cost dc M e e e 2vB cost induced ba dc M 4. Fundamentals of AC Machinery 77
Induced Voltage in AC Machines e 2vB cost induced M v r for an N C - turn coil, e 2rB cost induced AB cos t, A 2r M M cos t, AB M e N cost induced S loop area 4. Fundamentals of AC Machinery 78
Induced Voltage in AC Machines For three coils, each with N C -turns, and place around the rotor 120 o apart, the magnitude of each voltage will the same as for the case of a single coil but will differ in phase by 120 o,or e N cost aa bb S e N t cos 120 S e N cos t 240 cc S hence ceat three-phase set of currents can generate e a uniform rotating magnetic field in the stator, and a uniform rotating magnetic field can generate a three-phase set of voltages in the stator as claimed, 4. Fundamentals of AC Machinery 79
Induced Voltage in AC Machines The peak voltage in any phase is E N N f ma S S2 E 2N f rms Note that the voltage at the terminals of the machine will depend on whether the stator is Y or connected. For a -connected machine, V E 2N f phase rms S while for ay-connected machine, V 3E 6N f phase rms S S 4. Fundamentals of AC Machinery 80
Induced Torque in AC Machines In AC machines there are normally two magnetic fields one from the rotor and one from the stator. It is the interaction between these two fields that produces the torque in the machine. Consider the following simple single coil machine... 4. Fundamentals of AC Machinery 81
Induced Torque in AC Machines Consider a sinusoidal stator flu distribution and a single coil of wire mounted on the rotor. The stator flu distribution is, B B sin S S B S How much torque is produced? i i 4. Fundamentals of AC Machinery 82
Induced Torque in AC Machines How much torque is produced? Conductor 1: F i B ind 1 ib S r F ind 1 sin ind 1 1 rib S sin counterclockwise r 2 F F ind 1 r F ind2 1 B S 4. Fundamentals of AC Machinery 83
Induced Torque in AC Machines How much torque is produced? Conductor 2: F i B ind 2 ind ib 2 S r F sin ind 2 2 rib S sin F ind 1 r 1 2 F ind2 r B S counterclockwise 4. Fundamentals of AC Machinery 84
Induced Torque in AC Machines The torque on the rotor is: ind 2riB sin S counterclockwise 4. Fundamentals of AC Machinery 85
Induced Torque in AC Machines But the current in the rotor produces its own magnetic field, whose direction is determined via the right-hand-rule. 180 i B R B S B H i R R 4. Fundamentals of AC Machinery 86
Induced Torque in AC Machines Note the net magnetic ais for the stator. B sin B S BSma S 4. Fundamentals of AC Machinery 87
Induced Torque in AC Machines 2riB sin ind counterclockwise Sma 2r B B sin sin sin sin ma BR BS BB R Ssin BB R Ssin ind R S B H i R R i B B S B B R 4. Fundamentals of AC Machinery 88
Induced Torque in AC Machines 2 sin ind ribs counterclockwise 2rBB sin ind R S BB counterclockwise B B ind R S This is a general result though qualitative in nature. It applies to any AC machine and will be used to develop a qualitative understanding of torque in AC machines. 4. Fundamentals of AC Machinery 89
Induced Torque in AC Machines This can be put into a form that will be useful later. The net magnetic field in the machine is the vector sum of the rotor and stator fields, B S B net B R B B B net R S 4. Fundamentals of AC Machinery 90
Induced Torque in AC Machines Bnet BR BS BS Bnet BR B B B B B B B ind R S R net R R net B B ind R net 4. Fundamentals of AC Machinery 91
Induced Torque in AC Machines A qualitative Eample Consider the simple (salient pole) machine whose fields are rotating in a counterclockwise direction. B R B S B B ind R S The torque is clockwise, via the right-hand-rule, opposite the direction of rotation, so this machine is a generator. 4. Fundamentals of AC Machinery 92
We now will look at specific types of machines and obtain circuit models for them. 4. Fundamentals of AC Machinery 93