Introduction to Rotating MachinesAC MACHINES

Synchronous Machines

the armature winding of a synchronous machine is on the stator. the field winding is on the rotor. The field winding is excited by direct current conducted to it by means of stationary carbon brushes which contact rotating slip rings or collector rings. The conductors forming the coil sides (a and a() are parallel to the shaft of the machine and are connected in series by end connections (not shown in the figure).

The rotor is turned at a constant speed by a source of mechanical power connected to its shaft.

Fig. 4.4 A simple, two-pole (salient pole), single-phase synchronous generator.

Figure 4.5 (a) Space distribution of flux density (B). (b) corresponding waveform of the generated voltage (e) for the single-phase generator of Fig. 4.4.

Figure 4.6 A simple, four-pole, single-phase synchronous generator.

Figure 4.7 Space distribution of the air-gap flux density in an idealized, four-pole synchronous generator.

The electrical frequency fe of the voltage generated in a synchronous machine:

The electrical frequency of the generated voltage in radians per second is

Figure 4.8 Elementary two-pole cylindrical-rotor field winding.For the production of a set of three voltages phase-displaced by 120 electrical degrees in time, a minimum of three coils phase-displaced 120 electrical degrees in space must be used.

Figure 4.12 Three-phase generators (a) Two-pole (b) Four pole (c) Y connections of the windings.

Induction Machines The stator winding of an induction machine is excited with alternating currents. Rotor currents are produced by induction, i.e., transformer action. The induction machine may be regarded as a generalized transformer in which electric power is transformed between rotor and stator together with a change of frequency and flow of mechanical power. It is seldom used as a generator (except in wind-power applications). The stator windings are essentially the same as those of a synchronous machine. The rotor windings are electrically short-circuited and frequently have no external connections (squirrel-cage rotor).

Figure 4.15 Typical induction-motor speed-torque characteristic.DC Machines

The armature winding of a dc generator is on the rotor with current conducted from it by means of carbon brushes. The field winding is on the stator and is excited by direct current.

Space distribution of air-gap flux density in an elementary dc machine

waveform of voltage between brushesMMF of distributed windingsdistributed winding: coils are spread over a number of slots around the air-gap periphery.

winding consisting of a single N-turn coil which spans 180 electrical degrees (full-pitch):

The air-gap mmf:

MMF of AC Machines:The rectangular air-gap mmf wave of the concentrated two-pole, full-pitch coil can be resolved into a Fourier series comprising a fundamental component and a series of odd harmonics. The fundamental component Fag1 is (sinusoidal space wave)

Amplitude: (peak aligned with the magnetic axis of the coil)

Distributed winding: coils distributed in several slots.The winding is arranged in two layers, each full-pitch coil of Nc turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away. This two-layer arrangement simplifies the geometric problem of getting the end turns of the individual coils past each other.

air-gap mmf of phase a alone

The distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil.

For a distributed multipole winding having Nph series turns per phase, the fundamental mmf is

where kw : winding factor; takes into account the distribution of the winding.

The factor kwNph is the effective series turns per phase for the fundamental mmf. The peak amplitude of this mmf wave is

Example 4.1The phase-a two-pole armature winding of Fig. 4.20a can be considered to consist of 8 Nc-turn, full-pitch coils connected in series, with each slot containing two coils. There are a total of 24 armature slots, and thus each slot is separated by 360/24 = 15. Assume angle a is measured from the magnetic axis of phase (a) such that the four slots containing the coil sides labeled (a) are at a = 67.5, 82.5, 97.5, and 112.5. The opposite sides of each coil are thus found in the slots found at -112.5, -97.5 , -82.5 and -67.5 , respectively. Assume this winding to be carrying current ia.(a) Write an expression for the space-fundamental mmf produced by the two coils whose sides are in the slots at a=112.5 and -67.5.

(b) Write an expression for the space-fundamental mmf produced by the two coils whose sides are in the slots at a = 67.5 and -112.5.(c) Write an expression for the space-fundamental mmf of the complete armature winding.(d) Determine the winding factor kw for this distributed winding.a. The magnetic axis of this pair of coils is at a = (112.5 - 67.5)/2 = 22.5 and that the total ampere-turns in the slot is equal to 2Ncia. The mmf produced by this pair of coils is

b.

Magnetic Fields in Rotating MachinesUniform air-gap:

It is assumed that the magnetic field intensity H in the air gap has radial component only and has constant magnitude across the air gap.

The fundamental space-harmonic component of Hag can be found directly from the fundamental component Fag1 Distributed winding: Nph : the total number of series turns/phase.

ROTATING MMF WAVES IN AC MACHINESMMF Wave of a Single-Phase Winding

When this winding is excited by a sinusoidally varying current in time at electrical frequency e

Then, the mmf distribution becomes

Using

: rotating mmf waves

MMF Wave of a Polyphase Winding

Balanced three-phase currents (phase sequence abc):

For phases (b) and (c), whose axes are at ae = 120 and ae = -120, respectively,

The total mmf is the sum of the contributions from each of the three phases

In (a) t = 0 , ia is maximum (ia = Im), hence standing mmf wave due to phase (a) has maximum peak in the direction of its magnetic axis; ib and ic are equal and negative (ib = ic = (Im/2), hence their mmfs are equal in magnitude and opposite to their magnetic axes.In (b) t = /3e , ic is negative maximum (ic = (Im), hence standing mmf wave due to phase (c) has maximum peak in the direction opposite to its magnetic axis; ia and ib are equal and positive (ia = ib = Im/2).

In (c) t = 2/3e , ib is maximum (ib = Im), hence standing mmf wave due to phase (b) has maximum peak in the direction of its magnetic axis; ia and ic are equal and negative.GENERATED VOLTAGE

AC Machines

The field winding produces radial space-fundamental air-gap flux of peak flux density Bpeak. For uniform airgap

When the rotor poles are in line with the magnetic axis of a stator phase, the flux linkage with a stator phase winding is , where is the air-gap flux per pole. Magnetic flux density as a function of angular position in the air-gap, measured from the magnetic axis of the field winding

l : axial length of the stator/rotor iron. If the rotor rotates at constant angular speed m , the flux-linkage of phase-a winding

Voltage induced in phase a:

The polarity of this induced voltage is such that if the stator coil were short-circuited, the induced voltage would cause a current to flow in the direction that would oppose any change in the flux linkage of the stator coil.First term: transformer voltage, present only when the amplitude of the air-gap flux wave changes with timeSecond term: speed voltage generated as a result of the relative motion of the air-gap flux and the stator.

In normal steady-state operation, the amplitude of the air-gap flux wave is constant ( the first term is zero. The term electromotive force (emf) is used for the speed voltage. For constant air-gap flux,

DC Machines

AC voltages are induced in the armature winding coils as the coils rotate through the flux created by the dc field winding on the stator. The commutator provides full rectification of the AC voltage.

Voltage between the brushes.

The average value of the brush voltage:

TORQUE IN NONSALIENT-POLE MACHINES

Coupled-Circuit Viewpoint

Fig. 4.34Assumptions:

mmf waves can be approximated by space sinusoids. the reluctances of the stator and rotor iron are negligible. the stator and rotor self-inductances Lss and Lrr are constant.

The stator-to-rotor mutual inductance depends on the electrical angle me between the magnetic axes of the stator and rotor windings.

The stator and rotor flux linkages:

The terminal voltages are:

When the rotor rotates at speed me: Electromagnetic torque: Co-energy

The negative sign means that the electromechanical torque acts in the direction to bring the magnetic fields of the stator and rotor into alignment.Example 4.6Consider the elementary two-pole, two-winding machine of Fig. 4.34. Its shaft is coupled to a mechanical device which can be made to absorb or deliver mechanical torque over a wide range of speeds. This machine can be connected and operated in several ways. For this example, let us consider the situation in which the rotor winding is excited with direct current Ir and the stator winding is connected to an ac source which can either absorb or deliver electric power. Let the stator current be

where t = 0 is arbitrarily chosen as the moment when the stator current has its peak value.

(a) Derive an expression for the magnetic torque developed by the machine as the speed is varied by control of the mechanical device connected to its shaft.(b) Find the speed at which average torque will be produced if the stator frequency is 60 Hz.(c) With the assumed current-source excitations, what voltages are induced in the stator and rotor windings at synchronous speed (m = e)?Solution:(a) For a two-pole machine,

Let

the clockwise angular velocity impressed on the rotor by the mechanical drive.

( : the angular position of the rotor at t = 0.

(b) The time average of the torque is zero except when . For this synchronous case

Average torque:

A nonzero average torque will also be produced when .

(c) Voltage induced on the stator winding:

Magnetic Field Viewpoint

Simplified two-pole machine:

Fs , Fr : space phasors representing the stator and the rotor mmf waves in the air-gap.

Fsr : space phasor representing the resultant mmf wave.

The magnitude of the resultant mmf:

The resultant radial Hag field is a sinusoidal space wave with peak value Hag,peak

Let

Coenergy density in the air-gap is . Therefore, the total coenergy in the air-gap is

Torque for a two-pole machine:

For a P-pole machine:

The angle is the space angle between the stator and rotor mmf phasors in elec. rad. The minus sign means that the fields tend to align themselves. From the phasor diagram

Example 4.8An 1800-r/min, four-pole, 60-Hz synchronous motor has an air-gap length of 1.2 mm. The average diameter of the air-gap is 27 cm, and its axial length is 32 cm. The rotor winding has 786 turns and a winding factor of 0.976. Assuming that thermal considerations limit the rotor current to 18 A, estimate the maximum torque and power output one can expect to obtain from this machine.Solution

Assume that the maximum air-gap flux density is 1.5 T.

Power generated by the machine:

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