Presentation on theme: "3. Slip and Rotor frequency of IM"— Presentation transcript:
13. Slip and Rotor frequency of IM The revolving magnetic field of the stator of 3 phase IMrotates in a given direction (clockwise/counter basedon the phase sequence of supplied voltage) withsynchronous speed Ns.The rotor of 3 phase induction motor starts running inthe direction of rotating magnetic field.At standstill, rotor conductors are being cut by rotatingflux at synchronous speed. At this instant, frequency f2of rotor emf and current is equal to the supplyfrequency f1.
2The difference in speeds depends upon the load on the motor . When the rotor rotates at a speed nr in the direction of rotating field, the relative speed between synchronously rotating stator flux and rotor conductors becomes:-In practice the rotor never succeeds in catching up with the stator field.If it really did so, then, there would be no relative speed between the two; hence no rotor emf, no rotor current, and so no torque to maintain rotation.That is why, in the rotor windings emf and current are induced and the rotor develops torque .The difference in speeds depends upon the load on the motor .Where, ns - Slip speedNs - synch.speednr – rotor speed
3Therefore, frequency of the rotor emf is:- ,rpsSlip is defined asThus, the frequency of the rotor emf and current in an induction motor f2 = Sf1; and f2 is called the slip frequency.We know that,
4Rotor emf, current and power of IM At standstill, relative speed between magnetic field of the stator and rotor conductors is synchronous speed of the field Ns and slip S = 1. Let the per phase induced emf in rotor circuit at this moment be E2.Stator field speedRotorspeedRelativeslipRotor emfNs1E20.4Ns0.6Ns0.60.6E20.8Ns0.2Ns0.20.2E2In general, for any value of slip S, the per phase generated emf in the rotor conductors is SE2
5Rotor leakage reactance at standstill is:- Rotor leakage reactance at any slip:-Rotor leakage impedance at standstill:-Rotor leakage impedance at any slip is:-
6Per phase rotor current at standstill:- Per phase rotor current at any slip:-Per phase power input to rotor is:-Where,pg – the power transferred from stator to rotor across the air gap.θ - power factor angle of the rotor which I2 lugs E2.θ - arc tan
11Where, - rotor ohmic loss - internal mechanical powerdeveloped in the rotor
12Thus from rotor Equivalent circuit of IM, ohmic loss in the rotor circuit is equal to slip times power input to the rotor. i.e.Internal Torque (gross) developed per phase is:_Where,
13windage) OR, Power output at the shaft can be obtained from Pg Pshaft = PM – Pmech loss (friction & windage)Pshaft = Pg – (rotor ohmic loss + fric. &windage)
14Output or shaft torque is – If the stator input is known, then, air gap power is –
15Losses and efficiency of 3-phase IM There are two major losses in 3-phase IM.1) Fixed losses (Rotational losses)2) Variable lossesFixed losses – are usually taken to be constant at a constantspeed over the normal working range of the IM;even though friction losses may vary slightly withload and speed. Fixed losses are composed of:-a) Core losses - are a function of speed and flux- Hyteresis loss –-- Eddy current loss –
17where, – is constant for a given iron used. - flux density raised to Steinmetzexponent, and the value ofx is 1.6 –– frequency of flux- volume of iron used.– eddy current constant for theconductive material used.– thickness of the conductive material used.b) Mechanical loss- Friction losses ( bearing & brush frictionin WRIMs)- Windage losses.
18Under usual working conditions, the rotor current of 3-phase IMs is of very low frequency. As a result, rotor core losses which are almost proportional to frequency squared are negligible.Fixed losses (rotational losses) of IMs can be obtained directly by performing no-load test on the IM.Fixed losses = (Pinput– I2R stator loss) at no-load
19Variable losses – The value of these losses vary with load variation and are composed of:- a) Stator ohmic lossesb) Rotor ohmic lossesc) Brush contact losses for WRIMs onlyd) Stray load losses – are losses due to step harmonics, skin effect losses in stator conductors and iron losses in structural parts of machines.The stator and rotor ohmic losses (total ohmic loss) can be obtained from block rotor test.The brush contact loss for WRIMs and stray losses are very small & can be taken as 0.5 – 1% of the total losses.
20Power flow diagram of IM. Power inputPin = √3VlIlcosθRotor input powerShaft powerMech. power developedPgPMRotor core lossRotor I2R lossStator core lossFriction and windage lossStator I2R loss
21Efficiency of IM.Actually, the IM experiences a change in rotor speed with load as well as a change in the rotor frequency resulting from the speed change. i.e. neither the rotational losses (which are a functions of speed and frequency ) nor the stator and rotor electrical losses ( which are functions of load) are constant.However, the efficiency of a 3-phase induction motor is:-
22Phasor diagram of IMAn IM is similar to transformers in many respects.IM at stand stillApplication of 3-phase balanced voltage at frequency f1to the stator winding causes the development of rotating magnetic field; which induces rotor and stator emfs.Line frequency f1 appears in E2 also, because rotor is at stand still.The emf ratio for the IM is:-
23Where, Kw1 and Kw2 are stator and rotor winding distribution factors. Stator and rotor windings which obviously possess resistances and reactance are treated as transformer primary and secondary windings.Thus, the phasor diagram and equivalent circuit of a three-phase IM are almost similar to those in a trans-r. The main differences are;- Appearance of winding factors Kw1 and Kw2 in thevoltage ratio in IMs.- The no-load current in IM varies from 30-50% of full load current, where as in transformers it varies from 2-6% of the full load current. This is because, the resultant flux in IM completes its path through high reluctance air-gap.
24Phasor diagram of IM at standstill. E1E2=I2Z2I2I2r2jI2x2фImIc-I2I1V1= -E1I1r1jI1x1Ioθ1θ2
25Pasor diagram of IM at full load slip V1jI1x1I1r1V1= -E1I1θ1θ0I0IcI2r2фImθ2I2E2jI2x2E1
26Observe that,the power factor angle at stand still condition is large. i.e. the stator power factor at starting is very poor.At normal operating slip, θ1 is small. The rotor voltage equation now becomes SE2 = I2(r2 + jX2) and it is illustrated in the phasor diagram.θ1