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- ELECTRICAL MACHINES - III - EM 3
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**Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP**- 1 Topics
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Rotor of water wheel generator consists of salient poles. Poles are built with thin silicon steel laminations of 0.5mm to 0.8 mm thickness to reduce eddy current laminations. The laminations are clamped by heavy end plates and secured by studs or rivets. For low speed rotors poles have the bolted on construction for the machines with little higher peripheral speed poles have dove tailed construction as shown in Figs. Generally rectangular or round pole constructions are used for such type of alternators. However the round poles have the advantages over rectangular poles. Generators driven by water wheel turbines are of either horizontal or vertical shaft type. Generators with fairly higher speeds are built with horizontal shaft and the generators with higher power ratings and low speeds are built with vertical shaft design. Vertical shaft generators are of two types of designs (i) Umbrella type where in the bearing is mounted below the rotor. (ii) Suspended type where in the bearing is mounted above the rotor. In case of turbo alternator the rotors are manufactured form solid steel forging. The rotor is slotted to accommodate the field winding. Normally two third of the rotor periphery is slotted to accommodate the winding and the remaining one third unslotted portion acts as the pole. Rectangular slots with tapering teeth are milled in the rotor. Generally rectangular aluminum or copper strips are employed for filed windings. The field windings and the overhangs of the field windings are secured in place by steel retaining rings to protect against high centrifugal forces. Hard composition insulation materials are used in the slots which can with stand high forces, stresses and temperatures. Perfect balancing of the rotor is done for such type of rotors. Damper windings are provided in the pole faces of salient pole alternators. Damper windings are nothing but the copper or aluminium bars housed in the slots of the pole faces. The ends of the damper bars are short circuited at the ends by short circuiting rings similar to end rings as in the case of squirrel cage rotors. These damper windings are serving the function of providing mechanical balance; provide damping effect, reduce the effect of over voltages and damp out hunting in case of alternators. In case of synchronous motors they act as rotor bars and help in self starting of the motor. Relation between Speed and Frequency: In the previous course on induction motors it is established that the relation between speed and frequency and number of poles is given by Frequency f = P x N /120 Hz Windings in Alternators: In case of three phase alternators the following types of windings are employed. (i) Lap winding, (ii) wave winding and (iii) Mush winding. Based on pitch of the coil (i) full pitched (ii) short pitched windings Based on number of layers (i) Single layer (ii) Double layer Figure: 1.3. Single layer winding 3

Figure: 1.4. Double layer winding EMF Equation of an alternator: Consider the following Φ = flux per pole in wb P = Number of poles Ns = Synchronous speed in rpm f = frequency of induced emf in Hz Z = total number of stator conductors Zph = conductors per phase connected in series Tph = Number of turns per phase Assuming concentrated winding, considering one conductor placed in a slot According to Faradays Law electromagnetic induction, The average value of emf induced per conductor in one revolution eavg = d /dt eavg = Change of Flux in one revolution/ Time taken for one revolution Change of Flux in one revolution = p x Time taken for one revolution = 60/Ns seconds Hence eavg = (p x ) / ( 60/Ns) = p x x Ns / 60 We know f = PNs /120 hence PNs /60 = 2f Hence eavg = 2 f volts Hence average emf per turn = 2 x 2 f volts = 4 f volts If there are Tph, number of turns per phase connected in series, then average emf induced in T ph turns is Eph, avg = Tph x eavg = 4 f ø Tph volts Hence RMS value of emf induced E = 1.11 x Eph, avg = 1.11 x 4 f ø Tph volts = 4.44 f ø Tph volts This is the general emf equation for the machine having concentrated and full pitched winding. In practice, alternators will have short pitched winding and hence coil span will not be 180 0, but on or two slots short than the full pitch. Pitch Factor: Pitch factor Kp= emf induced in a short pitched coil/ emf induced in a full pitched coil = (2E cos α/2 )/ 2E Kp = cos α/2 where α is called chording angle. Distribution Factor: Even though we assumed concentrated winding in deriving emf equation, in practice an attempt is made to distribute the winding in all the slots coming under a pole. Such a winding is called distributed winding. In concentrated winding the emf induced in all the coil sides will be same in magnitude and in phase with each other. In case of distributed winding the magnitude of emf will be same but the emfs induced in each coil side will not be in phase with each other as they are distributed in the slots under a pole. Hence the total emf will not be same as that in concentrated winding but will be equal to the vector sum of the emfs induced. Hence it will be 4

less than that in the concentrated winding. Now the factor by which the emf induced in a distributed winding gets reduced is called distribution factor and defined as the ratio of emf induced in a distributed winding to emf induced in a concentrated winding. Distribution factor Kd = emf induced in a distributed winding/ emf induced in a concentrated winding = vector sum of the emf/ arithmetic sum of the emf Let E = emf induced per coil side m = number of slots per pole per phase, n = number of slots per pole β = slot angle = 180/n The emf induced in concentrated winding with m slots per pole per phase = mE volts. Fig below shows the method of calculating the vector sum of the voltages in a distributed winding having a mutual phase difference of β. When m is large curve ACEN will form the arc of a circle of radius r. From the figure below AC = 2 x r x sin β/2 Hence arithmetic sum = m x 2r sinβ/2 Now the vector sum of the emfs is AN as shown in figure below = 2 x r x sin mβ/2 Hence the distribution factor Kd = vector sum of the emf / arithmetic sum of the emf = (2r sin mβ/2) / (m x 2r sin β/2) Kd = ( sin mβ/2) / (m sin β/2) Figure: 1.5. Calculation of vector sum In practical machines the windings will be generally short pitched and distributed over the periphery of the machine. Hence in deducing the emf equation both pitch factor and distribution factor has to be considered. Hence the general emf equation including pitch factor and distribution factor can be given as EMF induced per phase = 4.44 f Tph x KpKd volts Eph = 4.44 KpKd f Tph vlolts Hence the line Voltage EL = √3 x phase voltage = √3 Eph Harmonics: When the uniformly sinusoidal distributed air gap flux is cut by either the stationary or rotating armature sinusoidal emf is induced in the alternator. Hence the nature of the waveform of induced emf and current is sinusoidal. But when the alternator is loaded waveform will not continue to be sinusoidal or becomes non sinusoidal. Such non sinusoidal wave form is called complex wave form. By using Fourier series representation it is possible to represent complex non sinusoidal waveform in terms of series of sinusoidal components called harmonics, whose frequencies are integral multiples of fundamental wave. The fundamental wave form is one which is having the frequency same as that of complex wave. 5

The waveform, which is of the frequency twice that of the fundamental is called second harmonic. The one which is having the frequency three times that of the fundamental is called third harmonic and so on. These harmonic components can be represented as follows. Fundamental: e1 = Em1 Sin (ωt ± θ1) 2nd Harmonic e2 = Em2 Sin (2ωt ± θ2) 3rd Harmonic e3 = Em3 Sin (3ωt ± θ3) 5th Harmonic e5 = Em5 Sin (5ωt ± θ5) etc. In case of alternators as the field system and the stator coils are symmetrical the induced emf will also be symmetrical and hence the generated emf in an alternator will not contain any even harmonics. Slot Harmonics: As the armature or stator of an alternator is slotted, some harmonics are induced into the emf which is called slot harmonics. The presence of slot in the stator makes the air gap reluctance at the surface of the stator non uniform. Since in case of alternators the poles are moving or there is a relative motion between the stator and rotor, the slots and the teeth alternately occupy any point in the air gap. Due to this the reluctance or the air gap will be continuously varying. Due to this variation of reluctance ripples will be formed in the air gap between the rotor and stator slots and teeth. This ripple formed in the air gap will induce ripple emf called slot harmonics. Minimization of Harmonics: To minimize the harmonics in the induced waveforms following methods are employed: 1. Distribution of stator winding. 2. Short Chording 3. Fractional slot winding 4. Skewing 5. Larger air gap length. Effect of Harmonics on induced emf: The harmonics will affect both pitch factor and distribution factor and hence the induced emf. In a well designed alternator the air gap flux density distribution will be symmetrical and hence can be represented in Fourier series as follows. B = Bm1sin ωt + Bm3 sin 3ωt + Bm5sin 5ωt + ................... The emf induced by the above flux density distribution is given by e = Em1sin ωt + Em3 sin 3ωt + Em5sin 5ωt + ................... The RMS value of the resultant voltage induced can be given as Eph = √ [(E1)2 + (E3)2 + (E5)2 + …………… (En)2] And line voltage ELine = √3 x Eph Effect of Harmonics of pitch and distribution Factor: The pitch factor is giventh by Kp = cos α/2, where α is the chording angle. For any harmonic say n harmonic the pitch factor is given by Kpn = cos nα/2 The distribution factor isth given by Kd = (sin mβ/2) / (m sin β/2) For any harmonic say n harmonic the distribution factor is given by Kdn = (sin m nβ/2) / (m sin nβ/2) Operation of Alternators: Similar to the case of DC generator, the behavior of a Synchronous generator connected to an external load is different than that at no-load. In order to understand the performance of the Synchronous generator when it is loaded, consider the flux distributions in the machine when the armature also carries a current. Unlike in the DC machine in alternators the emf peak and the current peak will not occur in the same coil due to the effect of the power factor of the load. The current and the induced emf will be at their peaks in the same coil only for upf loads. For zero power factor lagging loads, the current reaches its peak in a coil which falls behind that coil wherein the induced emf is at its peak by 90 electrical degrees or half a pole-pitch. Likewise for zero power factor leading loads, the current reaches its peak in a coil which is ahead of that coil wherein the induced emf is at its peak by 90 electrical degrees or half a pole-pitch. For simplicity, assume the resistance and leakage reactance of the stator windings to be negligible. Also assume the magnetic circuit to be linear i.e. the flux in the magnetic circuit is deemed to be proportional to the resultant ampere-turns - in other words the machine is operating in the linear portion of the magnetization characteristics. Thus the emf induced is the same as the terminal voltage, and the phase-angle between current and emf is determined only by the power factor (pf) of the external load connected to the synchronous generator. 6

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