ENGNG2024 Electrical Engineering  E Levi, 2002 1 INDUCTION MACHINES 1. PRELIMINARY CONSIDERATIONS Consider an electric machine with six windings. Stator and rotor are of cylindrical cross-section and three windings are situated on stator while the remaining three windings are on the rotor, as shown in Fig. 1. Both stator and rotor windings are displaced in space for 120 degrees electrical. In this electromechanical converter a continual electromechanical energy conversion may take place provided that, if angular frequency of stator currents is ω s and angular frequency of rotor currents is ω r , rotor speed is rs ωωω −= . Note that, according to the condition of average torque existence, this is the only possible correlation between stator and rotor frequency and frequency of rotation, when both stator and rotor carry AC currents. Then the developed torque becomes time independent and equal to the average torque value. This type of the electromechanical energy converter is called asynchronous machine (or induction machine; the origin of this name will become clearer later), because the rotor rotates with speed ω , while the stator revolving field rotates with synchronous speed ω s . Rotor currents form a revolving field as well, which rotates with angular velocity ω r with respect to rotor, while with respect to stator its angular velocity is sr ωωω =+ , i.e. synchronous speed. Note that creation of the rotating field in stator is enabled by displacing in space the three windings of the stator by 120 degrees and by feeding these three windings with a system of three phase currents with mutual phase displacement of 120 degrees. Rotor b -c Stator Air-gap -a a c -b Rotor winding Fig.1 – Cross-section of a three-phase induction machine. Before proceeding further into discussion of operating principles and analytical theory of induction machines, let us here briefly review the main constructional features of induction machines. Stator of induction machines, together with its three-phase winding, completely corresponds to the stator of synchronous machines. This means that the stator core is assembled of laminated iron sheets. An appropriate number of iron sheets are put together, thus forming stator core of the necessary length. Laminated iron sheets are insulated in order to reduce eddy-current losses in the iron core. Such a design of iron core is always utilised for those parts of electric machine where flux density and magnetic flux is time varying. The sheets are mutually isolated in order to prevent formation of a circuit for eddy-current flow from sheet to sheet. The iron core material is silicon alloyed. Addition of silicon reduces hysteresis ENGNG2024 Electrical Engineering  E Levi, 2002 2 losses in the iron core. Simultaneously, electric resistivity of the material is increased, thus giving rise to a substantial decrease in eddy-current losses as well. Windings of induction machine are placed into slots, which may be of open shape, or semi-closed. Semi-closed slots are usually selected for power ratings up to 200 kW; above 200 kW slots are open. Rotor core in induction machines is of laminated structure as well, because in rotor windings flows AC current, giving rise to time-varying flux density in the rotor (in synchronous machines rotor frequency is zero and rotor can be manufactured using a solid piece of ferromagnetic material). Rotor winding is placed into rotor slots in one of the two different ways. Subdivision of induction machines into two categories is normally done in conjunction with the way in which the rotor winding is formed. Rotor winding may be wound three-phase winding and such a machine is called wound-rotor induction machine or slip ring induction machine. The latter name stems from the fact that three terminals of the rotor phases are in wound-rotor machine brought out of the machine and connected to three slip rings (one for each phase), while the remaining three ends of the phase windings are connected into star neutral point and they remain inside the machine. Wound-rotor induction machine is illustrated in Fig. 2. The purpose of slip rings is beyond the scope of interest here. Slip rings are mechanically fixed to rotor and they rotate together with rotor. Three carbon brushes are mounted on the stator (one for each phase) and brushes slip along the rings as they rotate, thus establishing an electric contact between the stationary world and the rotating rotor. An electric approach to rotor winding (which rotates together with rotor) is enabled in this way. In other words, electric energy can be either brought or taken away from rotor winding during machine operation through this assembly, which consists of rotating slip rings and fixed brushes. Brushes are connected with leads to rotor winding terminals in the terminal box of the machine. Slip rings can be short-circuited and then rotor winding becomes a three-phase short- circuited winding. When the slip rings are short-circuited, brushes are raised and detached from slip rings. This is the normal operating state of a slip-ring induction machine. The second type of induction machines are so-called squirrel-cage induction machines. Rotor winding is in this case cast into slots and it is formed of solid aluminium or copper bars. The both ends of bars are electrically connected through end-rings. The winding manufactured in this way resembles a squirrel cage and this explains the origin of the name. The winding is shown in Fig. 3. Note that rotor winding is in this case always short-circuited and there is no possibility of electrical approach to the winding. In other words, electric energy cannot be either brought or taken away from the rotor winding. A winding formed in this way is essentially an n-phase winding, where the number of phases n equals number of bars. However, such an n-phase winding can be always substituted with an equivalent three-phase winding for all analytical considerations. As already stated at the beginning of this section, there are rotor currents in the rotor winding whose frequency is ω r . We have just seen that in a squirrel-cage induction machine there is no electrical access to the rotor winding. Similarly, the normal operating regime of a slip-ring induction machine is with rotor winding short-circuited; thus no electrical access is possible to the rotor winding and the question which arises is how do we get currents in rotor windings when we cannot approach the winding and connect an appropriate electric source. 2. OPERATING PRINCIPLES OF INDUCTION MACHINES Consider an induction machine with a three phase winding on the stator and an equivalent three phase short-circuited winding on the rotor. Let the stator winding be connected to the utility supply which provides three phase balanced system of AC voltages. These voltages will cause appropriate three phase currents to flow through stator the winding. ENGNG2024 Electrical Engineering  E Levi, 2002 3 The currents will give rise to formation of Tesla’s revolving field in the air gap of the machine. Given the supply frequency of stator voltages and currents f s in [Hz], the revolving field will rotate in space (in the cross section of the machine) at the angular frequency of ss f πω 2= .As both stator and rotor are initially stationary, the revolving field cuts conductors of both stator and rotor windings, causing induced electromotive forces in the windings to occur. slip ring b rush winding terminal A B C Three-phase winding Slip rings Brushes Terminals Shaft Fig. 2 – Schematic representation of the rotor of a slip ring induction machine and the physical appearance. In stator phase a winding a counter electromotive force is induced and it balances the applied voltage, the difference in their rms value being caused by voltage drop on the winding resistance and leakage reactance and being equal to a couple of percents of the applied voltage rms value. An electromotive force is induced in the rotor winding as well and its direction is shown in Fig. 4. As the rotor winding is short-circuited, the induced electromotive force will causeacurrentintherotorconductorsI r , whose real component has the same direction as induced emf. As the conductor, which carries current I r , is in the magnetic field, a magnetic force F will be created. This force will cause rotor to start rotating in the direction of stator ENGNG2024 Electrical Engineering  E Levi, 2002 4 revolving field rotation. The same happens in all the rotor conductors and sum of all individual multiples of rotor radius and force gives overall electromagnetic torque in the machine. Summarising, when stator winding of a three-phase induction machine is connected to the mains, electromagnetic induction causes currents in rotor windings and a torque is created which pulls rotor into rotation in the direction of rotation of the stator revolving field. This implies that transfer of electric energy from stator to rotor is realised exclusively by electromagnetic induction; therefore asynchronous machines are called induction machines. Rotor winding bars End ring Fig. 3 – Squirrel-cage winding of a squirrel cage induction machine. . ω s Rotating field force force N S Fig. 4 – Creation of an electromagnetic torque in an induction machine. Rotor can never reach synchronous speed of rotation. Rotation of rotor at synchronous speed implies that rotor rotates synchronously with revolving field. In that case there is no relative motion between stator revolving field and rotor and no electromotive force can be induced in the rotor windings. Consequently, no current can flow in the rotor winding if the ENGNG2024 Electrical Engineering  E Levi, 2002 5 speed is synchronous and no electromagnetic force can be generated. Therefore at synchronous speed the developed torque in an induction machine equals zero. As certain amount of torque is always necessary in a machine that operates as a motor in order to cover mechanical losses, induction motor has to operate with certain amount of developed torque even when it is not loaded at the shaft. Thus the rotor in motoring regime can never attain synchronous speed, i.e. it can never catch with the revolving field. When the motor is unloaded, it runs under no-load conditions and the amount of torque that is needed is determined with mechanical losses (windage losses and friction losses in bearings). The torque that describes mechanical losses is small, thus indicating that the induction motor will have the highest possible speed when it runs unloaded; this is so-called no-load speed and it is only slightly smaller from synchronous speed. Let us summarise the above given explanations: connection of three-phase stator winding of an induction machine at standstill to a voltage source causes current flow in stator windings; these currents give rise to production of revolving field; revolving filed cuts conductors of both stator and rotor windings; emf is induced in stator and it provides voltage balance to supply voltage; emf is induced in rotor as well and it causes current flow through short-circuited rotor winding; an electromagnetic force is created which acts on every rotor conductor, leading to the creation of the electromagnetic torque which pulls rotor into rotation; the direction of rotation is the same as the direction of rotation of stator revolving field; when a steady-state is established, rotor rotates with angular velocity equal to rs ωωω −= ; rotor currents create another revolving field whose absolute speed equals synchronous speed. Therefore, the torque is consequence of mutual interaction between stator and rotor revolving fields. At synchronous speed rotor currents become zero and electromagnetic torque disappears. Windings are by the virtue of their construction of resistive-inductive nature. Reactive power has to be provided for magnetisation of iron cores and air gap between stator and rotor. The question is how this reactive power is provided in induction machines. The machine does not contain any capacitances that could produce reactive energy. The only electrical connection with outside world is the connection of the stator winding to the supply, as the rotor winding is short-circuited. This means that there is no source of reactive power available inside an induction machine. Therefore induction machine has always to absorb reactive energy from the supply. Under all the possible operating conditions induction machine will act as a reactive energy consumer. As there is no rotor winding connected to another electric source, as is the case in synchronous machines, there is no way of exciting the induction machine in a manner similar to synchronous machines. This is one of the main reasons why induction machine is mainly utilised in motoring regime, while synchronous machine is used for generation purposes. When an induction machine is applied as a generator, reactive power has to be either taken from the power system or to be provided by a static VAr compensator (e.g., capacitor bank). As already emphasised, during motoring induction machine has to rotate slower than the revolving field, even under no-load conditions. The angular velocity of the rotor is given with rs ωωω −= . Revolving fields of stator and rotor rotate with angular velocity ω s .The difference between rotor speed and synchronous speed is characterised with the so-called slip. The slip is expressed either as a percentage value of the synchronous speed or as a per unit non-dimensional quantity. It is usually calculated out of the speeds given in [rpm] in the following way: [%]100or[p.u] s s s s n nn s n nn s − = − = (1) where: ENGNG2024 Electrical Engineering  E Levi, 2002 6 n s - synchronous mechanical speed, which is a function of the number of magnetic pole pairs P and which is correlated with synchronous electrical speed 60f s as P f n s s 60 = (2) For 50 Hz supply synchronous speeds are P1234 n s [rpm] 3000 1500 1000 750 n - asynchronous mechanical speed of rotation of induction machine shaft. Note that definition of the slip and the values are the same regardless of whether speeds in [rpm] or angular speeds in [rad/s] are used. Slip during normal operation of induction machines is in the range 10% to 2% for induction machines with power ratings in the range 1 kW 100 kW. The value of the slip that corresponds to the rated operating conditions, when speed is n n , will be denoted as s n .Indexn will in general always define the rated (nominal) operating condition of the machine. Let us now investigate correlation between stator and rotor frequencies with respect to newly introduced notion of slip. From slip definition of (1) it follows that ()() rs ss ss ss s Psnnn nnsn ωωω ωωω π += += += −= timesameat theSince 1260bydivided (3) it follows that the rotor frequency is determined with srsr sffs == ωω (4) Example 1: A 4-pole, 3-phase induction machine is fed from 50 Hz supply and operates in steady state with slip equal to 0.03. Determine the rotor speed and frequency of rotor currents. Solution: [Hz]5.15003.0 [rpm]14551500)03.01(1 [rpm]15002/5060 60 === =−== === xsff x-s)n(n x P f n sr s s s Example 2: A 60 Hz induction motor has one pole pair and runs at 3150 rpm. Calculate the synchronous speed and slip in per unit and in percents. Solution: Note the this is an American machine, since the frequency is 60 Hz. Hence %5.2100]p.u.[[%] 025.03600/)31503600(/)(]p.u.[ [rpm]36001/6060 60 == =−=−= === xss nnns x P f n ss s s ENGNG2024 Electrical Engineering  E Levi, 2002 7 According to (4), frequency of the current in rotor is slip times frequency of stator currents. For 50 Hz stator frequency and operating slips of 10% to 2% in an induction machine rotor frequency is only 5….1 Hz. Consequently, as the losses in the iron core are proportional to frequency and frequency squared, it follows that rotor iron losses are going to be negligibly small and that the major part of the iron loss will take place in stator. The total iron loss in an induction machine is for this reason always assumed to take place in stator only. According to the slip definition, equation (1), slip is a variable determined with the speed of rotation. This implies that frequency of rotor currents is, according to (4), a variable as well, proportional to the slip. Characteristic slip values in motoring operation are: n = 0 rotor at standstill s = 1 0<n<n 0 rotor rotates, machine is loaded 1 > s > s 0 n=n 0 <n s no-load, machine is unloaded s = s 0 n=n n <n 0 rotor rotates, rated load s = s n Normal operating range of induction machines in steady-states is in the speed range between rated speed and no-load speed, the actual operating speed being dependent on the load torque that the motor is driving. Suppose now that a source of mechanical energy is connected to the induction machine shaft and that the mechanical power provided by mechanical source is exactly equal to the power which describes mechanical losses (i.e. mechanical source provides torque to overcome mechanical loss torque). Then the speed of rotation will become equal to synchronous, as the mechanical loss torque is equated by torque of the prime mover. Simultaneously the induction machine torque will become equal to zero. Therefore at synchronous speed 00 === es Tsnn Suppose now that the power provided by the prime mover increases. The induction machine then enters generating regime. Note that only real power will be generated, while the reactive power is still absorbed. For generation 00 <<> es Tsnn This means that in generation speed is above synchronous speed, slip is negative and the machine’s electromagnetic torque is negative as well. In contrast to this, in motoring slip and torque are positive since the reason for rotation is the machine’s electromagnetic torque. A representation of induction machine operating modes is given in Fig. 5. Motoring Generating 10-0.5s 0n s 1.5n s n Fig. 5 – Schematic representation of induction machine operating modes, in terms of slip and speed of rotation. Given the slip s in a steady state, speed can be determined from equation (1) as ENGNG2024 Electrical Engineering  E Levi, 2002 8 [] () [] () [rpm]100/%1 [rpm]p.u.1 s s nsn nsn −= −= (5) Taking into account that during motoring slip is within the range from 1 to 0, it is obvious that frequency of rotor currents varies as a function of slip in the range between stator frequency and zero. At standstill rotor frequency equals stator frequency, while at synchronous speed rotor frequency becomes equal to zero. Therefore it follows that frequency of rotor currents and voltages is determined with slip. 3. ANALYTICAL THEORY OF INDUCTION MACHINE OPERATION It can be shown that a balanced three-phase induction machine fed from symmetrical sinusoidal three-phase supply can be treated in terms of per-phase equivalent circuit in steady state. However, such a derivation is pretty involved and time consuming. As only steady states under symmetrical supply conditions are of interest here, complex representatives of AC sinusoidal quantities may be used (phasors). Furthermore, for the purpose of steady-state analysis of a balanced induction machine fed from symmetrical source, the whole analysis can be performed by utilising per phase representation with complex phasors. Such an approach is utilised in what follows. Let stator winding be connected to mains, which provide symmetrical three-phase voltages and let rotor be at standstill, so that rotor speed is zero. Frequency of stator voltages and currents is f s . An electromotive force will be induced in rotor winding that will cause current to circulate around the rotor winding. As the rotor is at standstill, slip equals one and the frequency in rotor winding equals stator frequency. When the rotor is at standstill, difference between the speed of the rotating field and the rotor speed is of maximum value and equals synchronous speed. This speed difference determines the induced emf, since the emf is directly proportional to the speed at which the conductors are cut by the field (i.e. to the difference between the synchronous speed and the rotor speed). Once when rotor rotates at certain speed, the speed at which conductors are cut by the rotating field will be determined with ssr sf πωωω 2=−= and will be smaller than at standstill. When the rotor is at standstill let the induced electromotive force in one rotor phase E is identified with index rl. Its existence will cause current flow and rotor currents produce corresponding revolving field and flux. One part of the flux dissipates around the rotor winding (leakage flux), while major part links with stator windings contributing to the mutual flux. Current flow through rotor winding, caused by induced emf, is opposed by the resistance of the rotor winding and leakage reactance (which describes leakage flux). The value of the rotor leakage reactance is again frequency dependent. At standstill rotor frequency, rotor leakage reactance and modulus of rotor current are 22 22 rlr rl rl rsrrrl ssr XR E I LfLfX fsff + = == == γγγ ππ (6) Suppose now that the rotor starts rotating, so that slip becomes smaller than 1 since speed is greater than one. According to the fundamental expression for induced electromotive force due to the relative movement of a conductor with respect to flux density, induced emf is proportional to the relative speed of conductor with respect to flux density. As the rotor rotates with certain speed, while revolving fields rotate with synchronous speed, relative speed ENGNG2024 Electrical Engineering  E Levi, 2002 9 of the revolving field with respect to the rotor conductors is equal to the rotor angular frequency. This means that induced electromotive force at slip s is proportional to frequency of rotor currents. Consequently, at any other speed different from zero, () 2 2 22222 1 1 rr rl r rlr rl rr r r rlr rlr rlr XsR E I XsR sE XR E I sXX sEE ssEE γ γγ γγ + = + = + = = == ≠= (7) The expression for modulus of the rotor current in (7) enables the following phasor equation (phasors are identified with a bar over the symbol) to be written: r rl r r rl rlr rl r r r IjXI s R E EsIjsXIRE γ γ +=− −=+=− (8) Equation (8) enables construction of the rotor per-phase equivalent circuit, shown in Fig. 6. Let us consider now voltage balance for one stator phase winding. Stator phase winding is characterised with resistance and stator leakage reactance. Note that for stator rotating field always cuts the conductors at the same, synchronous speed. Hence the frequency of the stator is constant (50 Hz) and the induced emf is proportional to this fixed frequency regardless of the speed of rotation of the rotor. The induced emf exists in each stator phase and it holds balance to the applied stator voltage. Following the same approach as for the rotor phase, one can immediately write the phasor voltage equation for one stator phase as: ss s s s EIjXIRV ++= γ (9) Corresponding equivalent circuit for one stator phase is shown in Fig. 7. jX γ rl I r E rl R r /s Fig. 6 - Rotor per-phase equivalent circuit R s jX γ s I s VE s Fig. 7 – Stator per-phase equivalent circuit. By combining Figures 6 and 7, resulting complete equivalent circuit can be constructed. It is shown in Fig. 8 and is described with the following two voltage phasor equations: ss s s s EIjXIRV ++= γ (10) ENGNG2024 Electrical Engineering  E Levi, 2002 10 r rl r r rl IjXI s R E γ +=− (11) R s jX γ s I s VE s jX γ rl I r E rl R r /s Fig. 8 – Induction machine per-phase equivalent circuit. Two circuits shown in Fig. 8 apply to two different voltage levels, since the induced emf in stator is in general different from the induced emf in rotor. It is therefore not possible to directly connect them. In order to be able to put the two circuits together, it is necessary to apply transformer theory, which means that rotor voltage needs to be referred to stator voltage level (as the secondary is referred to primary using the transformation ratio in transformers). Correlation between the stator and rotor induced emf is established at standstill through the transformation ratio m. Transformation ratio is defined as rls rl EEEEm s // == (12) and in an induction machine it is dependent on design features. Rotor voltage equation (11) is multiplied next with the transformation ratio r rl r r rl r rl r r rl IjmXI s mR Em IjXI s R E γ γ +=− •+=− m/ (13) and new fictitious rotor variables, referred to stator, are then introduced respecting the condition that power in terms of original variables and in terms of new (primed) variables has to be the same: mII IEIE EmE rr rrrr rlrl = = = ' '' ' (14) Rotor current and induced emf in (13) are replaced next with the new rotor current and new induced emf ''' 2 2 r rl r r rl IXjmI s Rm E γ +=− (15) Finally, new rotor parameters (primed symbols) are introduced and the equation is brought into final form of rlrlrr r rl r r rl XmXRmR IjXI s R E γγ γ 22 '' ''' ' ' == +=− (16) [...]... power appears across the magnetising reactance) 4 NO-LOAD AND LOCKED-ROTOR REGIMES OF AN INDUCTION MACHINE No-load and mechanical short-circuit (locked-rotor, rotor at standstill) are two important regimes of an induction machine which occur during normal operation and which are performed as standard tests on induction machines as well When performed as tests, these two tests enable calculation of the parameters... characteristic state of an induction motor is mechanical short-circuit, which is in no way connected to, and thus is not to be confused with, electric short-circuit Mechanical short-circuit simply denotes an induction machine whose stator winding is supplied with threephase symmetrical voltages and whose rotor does not rotate (i.e the rotor is at standstill) This state is met whenever an induction motor is... time in normal operation of induction machines As the rotor is at standstill, s = 1, n = 0, and therefore Pout = 0 η =0 Te ≠ 0 (23) Note that although there is no electromechanical conversion as the rotor is at standstill, torque does not equal zero When the motor starts running-up (s = 1) and the voltage equals rated value, this being the normal starting condition for induction motors, torque has... would cause very high stator current to flow In some machines this value would be even ten times greater than rated current Consequently, the test is performed with voltage value that is just sufficient to cause rated stator current to flow The value of the voltage is for machines of low power rating above one third of the rated voltage, while for machines of high power rating it is in between one sixth... Note that it is not possible to separate total leakage reactance into stator and rotor part in an exact manner For vast majority of induction machines it can be assumed however that stator and rotor leakage reactance are mutually equal Example 7: A squirrel cage three-phase induction motor has the following parameters: R s = 0.5 Ω R 'r = 0.25 Ω ' X γs = X γr = 0.4 Ω The motor is four-pole, three-phase,... its typical value is five to eight times the rated motor current The issue is elaborated in more detail in the next section Mechanical short-circuit is at the same time a test which is performed on induction machines During the test rotor is prevented from rotating and it is forced to remain at standstill This test is most frequently termed as locked rotor test (it is for this reason that index l was... Equivalent per-phase circuit of an induction machine with included iron loss representation Note that the addition of the equivalent iron loss resistance in Fig 11 changes the node current balance equation From Fig 11 (19) I Fe + I m = I s + I r ' Note as well that the fact that rotor parameters in the circuit of Fig 11 are referred to stator is irrelevant, since in a cage induction machine it is anyway... once more that an induction motor, when connected to the supply, will start running-up from mechanical short-circuit state This state will last shortly, because the speed will start increasing very soon after the motor is switched on During that short time interval a stator current flows which is significantly greater than the rated current Example 5: A three-phase squirrel-cage induction motor with... to the machine) At the same time, the reactive power drawn from the mains is basically the same as it is when machine delivers its mechanical output (real) power equal to rated No-load current in induction machines is in the range from 30% to 80% of the rated current This is significantly higher than for transformers, the reason being the existence of the air gap that requires large reactive power for... of the starting current From this circuit I st = 5 Vn ( ph ) Z in = Vn ( ph ) ( R s + Rr ' ) 2 + (X γs + X γrl ') 2 = 240 0.75 2 + 0.8 2 = 288.6 A TORQUE CHARACTERISTIC OF AN INDUCTION MACHINE As a starting point in derivation of induction machine torque characteristic it is convenient to utilise per-phase equivalent circuit, which is for convenience redrawn in Fig 14, for an arbitrary speed, i.e slip . features of induction machines. Stator of induction machines, together with its three-phase winding, completely corresponds to the stator of synchronous machines. . operating state of a slip-ring induction machine. The second type of induction machines are so-called squirrel-cage induction machines. Rotor winding is in