Electrical Machines and Energy Systems: Operating Principles (Part 1) SYED A Rizvi

Transcription

1 Electrical Machines and Energy Systems: Operating Principles (Part 1) SYED A Rizvi

2 AC Machines Operating Principles: Rotating Magnetic Field The key to the functioning of AC machines is the rotating magnetic field. We will start the discussion of rotating magnetic field in the context of AC motors first. Application of rotating magnetic field in synchronous AC machines will be discussed next, followed by a detailed analysis of synchronous motors and generators. Finally, the application of rotating magnetic field in induction AC machines will be discussed. 2

3 AC Machines Operating Principles: Rotating Magnetic Field Assumptions: The stator of the machine has three identical windings, which are 120 mechanical (spatially) degrees apart (see the figure on the right). The windings are prepared such that a positive voltage applied to a conductor will create a North pole (N) on that side of the core. The windings are powered by a three phase source with a sequence of abc. 3

4 AC Machines Operating Principles: Rotating Magnetic Field The three windings are connected in a Y configuration. An equivalent circuit of the windings is shown below: 4

5 AC Machines Operating Principles: Rotating Magnetic Field We will use phase a as the reference for electrical degrees (timing). Accordingly, the three phase voltages can be described mathematically as follows: V a tt = VVVV SSSSSS ωωωω (1) V b tt = VVVV SSSSSS (ωωωω 120 oo ) (2) V c tt = VVVV SSSSSS (ωωωω 240 oo ) (3) Let s denote the magnetic field intensity at the peak voltage V p by B p. 5

6 AC Machines Operating Principles: Rotating Magnetic Field Accordingly, the instantaneous magnetic field intensities at any time t due to the three phase voltages can be described mathematically as follows: B a tt = BBpp SSSSSS ωωωω (4) B b tt = BBpp SSSSSS (ωωωω 120 oo ) (5) B c tt = BBpp SSSSSS (ωωωω 240 oo ) (6) Note that Eqs. 4-6 provide magnitude of the three magnetic field intensities at any given time; however, they are physically 120 o apart from each other. We will refer to them as the phase magnetic field intensities. 6

7 AC Machines Operating Principles: Rotating Magnetic Field Therefore, each of the three phase magnetic field intensities at any given time t can be represented by a vector in space domain with a specific magnitude and the orientation (see Eqs. 7-9 below). B as tt = BBpp SSSSSS ωωωω 0 o (7) B bs tt = BBpp SSSSSS (ωωωω 120 oo ) 120 oo (8) B cs tt = BBpp SSSSSS (ωωωω 240 oo ) 240 o (9) 7

8 AC Machines Operating Principles: Rotating Magnetic Field The resultant magnetic field intensity (B net ) at any given time t can be found by computing the three phase magnetic field intensities at time t and adding them taking into account their spatial location (vector addition in space domain). We will compute B net for one full cycle of line voltage (360 electrical degrees) starting with ωωωω = 30 o with 60 o intervals. 8

9 AC Machines Operating Principles: Rotating Magnetic Field 9

10 AC Machines Operating Principles: Rotating Magnetic Field 10

11 AC Machines Operating Principles: Rotating Magnetic Field 11

12 AC Machines Operating Principles: Rotating Magnetic Field 12

13 AC Machines Operating Principles: Rotating Magnetic Field 13

14 AC Machines Operating Principles: Rotating Magnetic Field 14

15 AC Machines Operating Principles: Rotating Magnetic Field Note that: The three phase power with the sequence abc was applied to the stator windings in a CCW fashion. The net magnetic field intensity (B net ) had a constant magnitude during the entire cycle of the source voltage (360 electrical degrees). It rotated, effectively as a pair of north-south (N-S) poles, 360 mechanical degrees CCW during this time (see figure on the right). 15

16 AC Machines Operating Principles: Rotating Magnetic Field We will refer to a machine with this kind of windings as a two-pole machine. In the case of a two-pole machine, the mechanical speed of rotation (f m : revolutions per second) is the same as the electrical frequency (f e : cycles per second or Hz). The rotation of magnetic field can be reversed (made CW) by applying the three phase power with the sequence abc to the stator windings in a CW fashion. It is equivalent to swapping any two phases while powering up the stator windings. 16

17 AC Machines Operating Principles: Rotating Magnetic Field If the stator windings are constructed such that phases windings are located 60 machinal degrees apart, it will create two rotating sets of N-S poles as shown in the figure on the right. This machine will be referred to as a four-pole machine. The B net will now rotate only half the revolution (mechanical) in one cycle of electrical frequency. 17

18 AC Machines Operating Principles: Rotating Magnetic Field In general, the mechanical speed of rotation of B net is related to electrical frequency as follows: nn mm = ff ee 120 PP (10) Alternatively, ff ee = nn mm PP 120 (11) where, n m = mechanical speed of rotation in revolutions per minute (RPM) f e = electrical frequency in Hz P = number of poles in the machine 18

19 AC Machines Operating Principles: Synchronous Motor In synchronous motors, the stator of the motor has a rotating magnetic field. The rotor is wound to produce a stationary magnetic field through the rotor windings by powered by an external D.C. source. However, the rotor is free to rotate with respect to its axle. Since the opposite magnetic poles attract each other, the rotor will rotate in an attempt to lock its magnetic poles with the opposing poles of the stator s rotating magnetic field (not as simple, as discussed later). However, the rotor will lag the stator field by a mechanical angle α. In this way, the rotor continuously chases stator s magnetic field rotating with the same speed as that of the stator (in synchronism of the stator field). 19

20 AC Machines Operating Principles: Synchronous Motor The mechanical angle α depends on the torque on the rotor s shaft and also referred to as the torque angle. The rotor continues to rotate in synchronism of the stator field regardless of the load (torque) on the rotor s shaft (within certain limits, of course). However, as the torque on the rotor s shaft increases, the torque angle α increases as well until it reaches 90 o. At that point, the rotor falls out of synchronism with the stator s magnetic field and eventually stops rotating. The maximum torque that the machine can provide (the torque corresponding to α = 90 o ) is called pullout torque (typically three times the full load torque of the machine). 20

21 AC Machines Operating Principles: Synchronous Motor The figure on the right shows the interaction of the magnetic fields of the stator and rotor of the machine, which results in an induce torque τ induced in the rotor. Mathematically, τ induced = k(br BBs) = k(bb rr BB ss ) ssssss(α) (12) Direction of rotation of the induce torque τ induced? CCW! 21

22 AC Machines Operating Principles: Synchronous Motor An increase in the load on the rotor s shaft results in a momentary slow down in rotor s speed causing an increase in the torque angle α. However, an increase in α results in increasing the induced torque (Eq. 13), which in turn, increases rotor s speed to synchronous speed. As can be seen from Eq. (12), the maximum induced torque is obtained at α = 90 o. For α > 90 o, the induced torque decreases as α increases causing a further slow down in rotor s speed and eventually stopping the motor. What happens when α > 180 o? Before we develop an equivalent circuit for the synchronous motor, we need to consider the impact of the rotating magnetic field of the rotor on the windings of the stator. Before we do that, let s consider the operation of the synchronous generator (we ll come back to motor). 22

23 AC Machines Operating Principles: Synchronous Generator A synchronous motor or generator is essentially the same machine except that they have opposite direction of power flow. That is, in the synchronous motor the input is electrical power and the output is the mechanical power (torque to drive a load). On the other hand, in the synchronous generator the input is mechanical power (a prime mover to rotate the rotor) that creates a time varying magnetic flux ( ). The output is the electrical power (induced voltage in stator windings based on Faraday s law of electromagnetic induction). The induced voltage in each phase of the stator winding is given by EE AA = K ddd dddd (13) where K depends upon machine construction. 23

24 AC Machines Operating Principles: Synchronous Generator It can be seen from Eq. (13) that magnitude of the generated voltage is proportional to magnetic flux of the rotating field and thus to the field current (assuming a fixed rotational speed of the PM). With the generator under no load condition, there is no current in the stator windings and, therefore, B s = 0. Accordingly, the induced torque, τ induced = 0. When a load is connected to the generator, the current through the stator windings create B s. Note that B s will lag B R by an angle α. The interaction of the magnetic field now produces τ induced, according to Eq. (12). What is direction of the induced torque τ induced? Opposite the direction of rotation of the rotor (Prime Mover)! 24

25 AC Machines Operating Principles: Synchronous Generator Figure below shows the equivalent circuit of the generator (per phase). 25

26 AC Machines Operating Principles: Synchronous Generator E A = induced voltage in the armature (in volts) V P = phase voltage available at the terminals (in volts) V F = field voltage (D.C.) applied to the field windings (volts) E A = induced voltage in the armature (in volts) I A = armature current (in amps) I F = field current (in amps) R A = armature resistance (in ohms) X A = armature reactance (in ohms) R F = field resistance (variable, in ohms) X A = field reactance (in ohms) 26

27 AC Machines Operating Principles: Synchronous Generator In the following analysis, we will use the phase voltage as the reference with a phase angle of 0 o. That is, V P = V P 0 o E A = E A α o I A = I A ɵ o and EE AA = VV PP + II AA (R A + jjx A ) (14) Note that the figure on the right shows the phaser diagram of a generator with a lagging power factor. 27

28 AC Machines Operating Principles: Synchronous Generator The output power of the generator (per phase) is given by P out = VV PP II AA cos θθ (15) In general R A << X A and therefore, can be ignored. Rewriting Eq. (14) under this condition would yield EE AA = VV PP + II AA (jjx A ) (16) or or EE AA = VV PP 0 o + II AA ɵ XX AA 90 o (17) EE AA = VV PP 0 o + II AA XX AA ɵ + 90 o (18) 28

29 AC Machines Operating Principles: Synchronous Generator or EE AA = VV PP + (II AA XX AA ) cos θθ + 90 o + jj(( II AA XX AA ) sin θθ + 90 o ) (19) noting cos θθ + 90 o = sin θθ (20) and therefore, Eq. (19) can be rewritten as sin θθ + 90 o = cos θθ (21) EE AA = VV PP II AA XX AA sin θθ + jj( II AA XX AA ) cos θθ (22) 29

30 AC Machines Operating Principles: Synchronous Generator but comparing Eqs. (22) and (23), we get EE AA = EE AA cos αα + jjee AA sin αα (23) ( II AA XX AA ) cos θθ = EE AA sin αα (24) or II AA cos θθ = EE AA sin αα XX AA (24) Substituting LHS of Eq. (24) into Eq. (15), we get P out = VV PPEE AA sin αα XX AA (25) 30

31 AC Machines Operating Principles: Synchronous Generator The figure below shows the phaser diagram of a generator with a unity power factor. 31

32 AC Machines Operating Principles: Synchronous Generator The figure below shows the phaser diagram of a generator with a leading power factor. 32

33 AC Machines Operating Principles: Synchronous Motor Missing piece: The induced voltage in the stator windings, E S. The rotor s rotating magnetic field induces a voltage in the stator windings of the motor in the same way that the rotating magnetic did in the armature windings in the case of the synchronous generator. Note that the rotor s magnetic field is rotating in synchronism with the rotating magnetic field of the stator. Therefore, the frequency of the induced voltage E S is the same as that of the applied voltage V P at the terminals of the motor. However, E S lags V P by an angle α. We now have enough information to develop the equivalent circuit of the synchronous motor, which is essentially the same as that of the synchronous generator except that the current now flows from machine s terminals to the stator windings. 33

34 AC Machines Operating Principles: Synchronous Motor The figure below shows the equivalent circuit of the synchronous motor (per phase). 34

35 AC Machines Operating Principles: Synchronous Motor In the following analysis, we will assume the motor is operating with a lagging power factor and use the phase voltage as the reference with a phase angle of 0 o. That is, V P = V P 0 o E A = E A α o I A = I A ɵ o and EE AA = VV PP II AA (R A + jjx A ) (26) or VV PP = EE AA + II AA (R A + jjx A ) (27) 35

36 AC Machines Operating Principles: Synchronous Motor Accordingly, total power delivered to the motor can be expressed by Eq. (25). That is, P total = 3VV PPEE AA sin αα XX AA (28) noting we get P total = ττ iiiiiiiiiiiiii ωω mm (29) ττ iiiiiiiiiiiiii = 3VV PP EE AA sin αα XX AA ωω mm (30) 36

37 AC Machines Operating Principles: Synchronous Motor Accordingly, the pull out torque can be expresses as ττ pppppppppppppp = 3VV PP EE AA XX AA ωω mm (33) 37

38 AC Machines Operating Principles: Synchronous Motor The figure below shows the phaser diagram of a synchronous motor with a unity power factor. 38

39 AC Machines Operating Principles: Synchronous Motor The figure below shows the phaser diagram of a motor with a leading power factor. 39

40 AC Machines Operating Principles: Synchronous Motor Start up issues: Let s assume that the magnetic fields of the stator and rotor are as shown in the figure on the right with the stator s magnetic field rotating CCW. The rotor will experience an induced torque in CCW direction. So far, everything seems to be in order: the rotor should start rotating CCW under the induced torque and chase the rotating magnetic field of the stator. 40

41 AC Machines Operating Principles: Synchronous Motor In a half cycle of electrical frequency (1/120 of a second or about 2.7 ms), the stator s magnetic field would rotate such that its orientation becomes exactly opposite to the one at the start (see figure on the right). However, the rotor would barely move from its initial position in that amount of time due to inertia. So the rotor s magnet field would remain at the same orientation as it was at the beginning of the process. 41

42 AC Machines Operating Principles: Synchronous Motor However, the induced torque on the rotor is now CW. In this way, the induced torque on the rotor would keep reversing direction twice in every electrical cycle, making it practically impossible for the rotor to rotate. In fact, we would need a separate process to put the rotor in motion without powering up its field windings. Once the rotor gains close to the synchronous speed, the rotor s field windings can be powered up to run the synchronous motor normally. 42

43 AC Machines Operating Principles: Induction Motor An Induction motor also uses a rotating magnet field generated by the stator. The rotor of the induction motor, however, does not require a separate source to establish rotor s magnetic field. The rotor s magnetic field is established through the voltage in the rotor s windings induced by the stator s rotating magnetic field. Induction motors use two kind of rotors: (1) squirrel-cage rotor and (2) wound rotor. The squirrel-cage rotor has slots carved into the face of the rotor, which carry metal bars. Those bars are short circuited at either end. A wound rotor has three-phase windings in the rotor, which are the mirror image of the stator s windings. The windings are available through brush and slip ring mechanism to add extra resistance in the windings if desired. Otherwise, they are shorted at the brushes. 43

44 AC Machines Operating Principles: Induction Motor The induced voltage in the rotor windings of an induction motor s is given by Eq. (14) (fundamentals). E = vv BB. ll (34) where, BB = is the flux density (in AA Weber/meter2 : Wb/m 2 or Telsa: T) l = active length of the conductor in the magnetic field (m) v = relative speed of the conductor (m/s) w.r.t. the stator s magnetic field 44

45 AC Machines Operating Principles: Induction Motor The figure on the right shows the orientation of the stator s magnetic field along with that of the rotor s magnetic field. The rotor s magnetic field is created by the induced voltage (the dot and cross represent the current out of the paper and in the paper, respectively). The direction of the induced torque is CCW. 45

46 AC Machines Operating Principles: Induction Motor The figure on the right shows the orientation of the stator s magnetic field along with that of the rotor s magnetic field after half of the electrical cycle. Note that as B s reverses so do the induced voltage, current causing B r to reverse its orientation as well. The direction of the induced torque is still CCW. 46

47 AC Machines Operating Principles: Induction Motor Induction motor uses rotating magnet field through the stator. The rotor of the induction motor, however, does not require a separate source to develop rotor s magnetic field. 47

48 AC Machines Operating Principles: Induction Motor Induction motor uses rotating magnet field through the stator. The rotor of the induction motor, however, does not require a separate source to develop rotor s magnetic field. 48

49 AC Machines Operating Principles: Induction Motor Induction motor uses rotating magnet field through the stator. The rotor of the induction motor, however, does not require a separate source to develop rotor s magnetic field. 49