Torque-Slip (speed) Characteristics of Induction Motor (IM)

 

Instructional Objectives

 

• Derivation of the expression for the gross torque developed as a function of slip (speed) of Induction motor

 

• Sketch the above characteristics of torque-slip (speed), explaining the various features

 

• Derive the expression of maximum torque and the slip (speed) at which it occurs

 

• Draw the above characteristics with the variation in input (stator) voltage and rotor resistance

 

Introduction

In the previous, i.e. third, lesson of this module, starting with the formulas for the induced emfs per phase in both stator and rotor windings, the equivalent circuit per phase of the three-phase induction motor (IM), has been derived. The relation between the rotor input, rotor copper loss and rotor output (gross) are derived next. Finally, the various losses − copper losses (stator/rotor), iron loss (stator) and mechanical loss, including the determination of efficiency, and also power flow diagram, are presented. In this lesson, firstly, the torque-slip (speed) characteristics of IM, i.e., the expression of the gross torque developed as a function of slip, will be derived. This is followed by the sketch of the different characteristics, with the variations in input (stator) voltage and rotor resistance, along with the features. Lastly, the expression of maximum torque developed and the slip (speed) at which it occurs, are derived.

Keywords: The equivalent circuit per phase of IM, gross torque developed, torque-slip (speed) characteristics, maximum torque, slip at maximum torque, variation of the characteristics with changes in input (stator) voltage and rotor resistance.

Gross Torque Developed

The current per phase in the rotor winding (the equivalent circuit of the rotor, per phase is shown in Fig. 31.1) is (as given in earlier lesson (#31))

222222222)()/()()(xsrExsrEsIrr+=⋅+⋅=

Please note that the symbols used are same as given in the earlier lesson.

In a similar way, the output power (gross) developed (W) is the loss in the fictitious resistance in the equivalent circuit as shown earlier, which is

()[]()[][]()[]2222222222222220)()()1(3)()(/)1(3/)1(3xsrssrExsrssrEsssrIPrr⋅+−⋅⋅⋅=⋅+−⋅⋅⋅=−⋅⋅=

The motor speed in rps is srnsn⋅−=)1(

The motor speed (angular) in rad/s is srsωω⋅−=)1(

The gross torque developed in mN⋅ is

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()[]()[]22222222222200)()(23)()()1()1(3xsrnsrExsrsssrEPTsrsrr⋅+⋅⋅⋅=⋅+−−⋅⋅⋅==πωω

The synchronous speed (angular) is ssn⋅=πω2

The input power to the rotor (or the power transferred from the stator via air gap) is the loss in the total resistance (), which is sr/2

()()()()[]()[]222222222222222)()(3)()(/3/3xsrsrExsrsrEssrIPrri⋅+⋅⋅⋅=⋅+⋅⋅⋅=⋅⋅=

The relationship between the input power and the gross torque developed is 0TPsi⋅=ω

So, the input power is also called as torque in synchronous watts, or the torque is siPTω/0=

Torque-slip (speed) Characteristics

The torque-slip or torque-speed characteristic, as per the equation derived earlier, is shown in Fig. 32.1. The slip is )/(1/)(/)(srsrssrsnnnnns−=−=−=ωωω. The range of speed, is between 0.0 (standstill) and (synchronous speed). The range of slip is between 0.0 () and 1.0 (rnsnsrnn=0.0=rn).

For low values of slip, )(22xsr⋅>>. So, torque is

()()22222203)(3rsErsrETsrsr⋅⋅=⋅⋅⋅⋅=ωω

This shows that , the characteristic being linear. The following points may be noted. The output torque developed is zero (0.0), at sT∝00.0=s, or if the motor is rotated at synchronous speed (). This has been described in lesson No. 30, when the srnn=

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principle of operation was presented. Also, the slip at full load (output torque ) is normally 4-5% (), the full load speed of IM being 95-96% of synchronous speed (flT)(0=05.004.0−=flsssflflrnnsn⋅−=−=)96.005.0()1()().

For large values of slip, )(22xsr⋅<<. So, torque is

()()222222220)(3)(3xsrExssrETsrsr⋅⋅⋅=⋅⋅⋅⋅⋅=ωω

This shows that, , the characteristic being hyperbolic. The starting torque (, or ) developed, along with starting current, is discussed later. )/1(0sT∝0.1=s0.0=rn

So, starting from low value of slip (), at which torque is proportional to slip, whereas for large values of slip (0.0>s0.1<s), torque is inverse proportional to slip, both being derived earlier. In the characteristic shown, it may be observed that torque reaches a maximum value, which can be obtained in the following way. The relation between torque and slip is

[] )()(222220xsrsrKT⋅+⋅⋅= where, ()srEKω/32⋅=

or, []⎟⎟⎠⎞⎜⎜⎝⎛⋅+=⋅⋅⋅+=2222222220)(1)()(1rxssrKsrKxsrT

To determine the maximum value of torque () in terms of slip, the minimum value of its inverse () need be determined from the relation, 0T0/1T

0)(11222220=⎟⎟⎠⎞⎜⎜⎝⎛+−=⎟⎟⎠⎞⎜⎜⎝⎛rxsrKTdsd

from which ⎟⎟⎠⎞⎜⎜⎝⎛=22222)()(xrs or, 22/xrs= .

Please note that, for motoring condition as shown earlier, slip, s is positive (+ve), as . At this slip, , srnn<mss=22xsrm⋅=. This may be termed as slip at maximum torque. The motor speed is [smmrnsn)1()(−=]. This value of slip is small, for normal wound rotor (or slip ring) IM, without any additional resistance inserted in the rotor circuit. This value is higher in the case of squirrel cage IM.

Substituting the value of s, the maximum value of torque is

222021)(32xExKTsrm⋅⋅⋅=⋅=ω

which shows that it is independent of . The maximum torque is also termed as 2rpull-out torque. If the load torque on the motor exceeds this value, the motor will stall, i,e. will come to standstill condition.

The values of maximum torque and the slip at that torque, can be obtained by using

0.0)(0=Tdsd

which is not shown here.

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It may be observed from the torque-slip characteristic (Fig. 32.1), or described earlier, that the output torque developed increases, if the slip increases from 0.0 to , or the motor speed decreases from to . This ensures stable operation of IM in this region (), for constant load torque. But the output torque developed decreases, if the slip increases from to 1.0, or the motor speed decreases from .to zero (0.0). This results in unstable operation of IM in this region (), for constant load torque. However, for fan type loads with the torque as (), stable operation of IM is achieved in this region (mssnmrn)(mss<<0.0msmrn)(0.1<<ssm2)(rLnT∝0.1<<ssm).

Starting Current and Torque

The starting current (rotor) is

22222)()()(xrEIrst+=

as slip at starting () is 1.0, which is the same at standstill (or stalling condition). The magnitude of the induced voltage per phase in the stator winding is nearly same as input voltage per phase fed to the stator, if the voltage drop in the stator impedance, being small, is neglected, i.e. . As shown in the earlier lesson (#31), the ratio of the induced emfs per phase in the stator and rotor winding can be taken as the ratio of the effective turns in two windings, i.e. 0.0=rnssEV≈′′=rsrsTTEE//, where and . The winding factor for the stator winding is swssTkT=′rwrrTkT=′psdswskkk⋅=. Same formula is used for the above factor in the rotor winding, assuming it to be wound rotor one.

The starting current in the stator winding can be shown as )/()()(′′⋅=srstrstsTTII, neglecting the no load current. This current is normally large, much greater than full load current. This current is reduced by using starters in both types (cage and wound rotor) of IM, which will be taken up in the next lesson.

The starting torque in is mN⋅

()[]2222222220)()()(3/)(3)(xrrErITsrsstst+⋅⋅⋅=⋅⋅=ωω

This expression is obtained substituting 0.1=s in the expression of derived earlier. If the starter is used, the starting torque is also reduced, as is the case with starting current. 0T

Torque-slip (speed) Characteristics,

with variation in input (stator) voltage and rotor circuit resistance

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The set of torque-slip characteristics with variation in input (stator) voltage is shown in Fig. 32.2a. The point to note that the torque at a given slip decreases with the decrease in input (stator) voltage, as . The characteristics shown are for decreasing stator voltages (). The speed decreases or the slip increases with constant load torque, as the input (stator) voltage decreases. The region for stable operation with constant load torque remains same (20VT∝321VVV>>mss<<0.0), as given earlier. But again, stable operation can be obtained in the region (0.1<<ssm), with fan type loads with the torque as (). Another problem is that the maximum or pull-out torque decreases as , where V is input (stator) voltage, which is a drawback with constant load torque operation.. 2)(rLnT∝20)(VTm∝

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The set of torque-slip characteristics with variation in rotor circuit resistance is shown in Fig. 32.2b. The characteristics shown are for increasing rotor circuit resistances (). The point to note that, the maximum torque remains same for all the characteristics. This has been shown earlier that the maximum torque depends on rotor reactance only, but not on rotor circuit resistance. Only the slip at maximum torque increases with the increase in rotor circuit resistance. So, for constant load torque operation, the slip increases or the speed decreases with the increase in rotor circuit resistance. The motor efficiency decreases, as the rotor copper loss increases with the increase in slip. The load torque remains same, but the output power decreases, as the speed decreases. Also, it may be observed that the starting torque increases with the increase in rotor circuit resistance, with the total rotor circuit resistance lower than rotor reactance. The starting torque is equal to the maximum torque, when the total rotor circuit resistance is equal to rotor reactance. If the rotor circuit resistance is more than rotor reactance, the starting torque decreases. 4322RRRr>>>

In this lesson − the fourth one of this module, the expression of gross torque developed, as a function of slip (speed), in IM has been derived first. The sketches of the different torque-slip (speed) characteristics, with the variations in input (stator) voltage and rotor resistance, are presented, along with the explanation of their features. Lastly, the expression of maximum torque developed and also the slip, where it occurs, have been derived. In the next lesson, the various types of starters used in IM will be presented, along with the need of the starters, followed by the comparison of the starting current and torque developed using the starters.

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Different Types of Starters for Induction Motor (IM)

 

 

Instructional Objectives

 

• Need of using starters for Induction motor

 

• Two (Star-Delta and Auto-transformer) types of starters used for Squirrel cage Induction motor

 

• Starter using additional resistance in rotor circuit, for Wound rotor (Slip-ring) Induction motor

 

Introduction

In the previous, i.e. fourth, lesson of this module, the expression of gross torque developed, as a function of slip (speed), in IM has been derived first. The sketches of the different torque-slip (speed) characteristics, with the variations in input (stator) voltage and rotor resistance, are presented, along with the explanation of their features. Lastly, the expression of maximum torque developed and also the slip, where it occurs, have been derived. In this lesson, starting with the need for using starters in IM to reduce the starting current, first two (Star-Delta and Auto-transformer) types of starters used for Squirrel cage IM and then, the starter using additional resistance in rotor circuit, for Wound rotor (Slip-ring) IM, are presented along with the starting current drawn from the input (supply) voltage, and also the starting torque developed using the above starters.

Keywords: Direct-on-Line (DOL) starter, Star-delta starter, auto-transformer starter, rotor resistance starter, starting current, starting torque, starters for squirrel cage and wound rotor induction motor, need for starters.

Direct-on-Line (DOL) Starters

Induction motors can be started Direct-on-Line (DOL), which means that the rated voltage is supplied to the stator, with the rotor terminals short-circuited in a wound rotor (slip-ring) motor. For the cage rotor, the rotor bars are short circuited via two end rings. Neglecting stator impedance, the starting current in the stator windings is (see lesson 32) is

22221)()()(′+′′=xrEIrst

where,

Starting current in the motor (stator) ==′=aIIIststst/)()()(221

Effective turns ratio between stator and rotor windings =′′=rTa/T s

Input voltage per phase to the motor (stator) ==′=rrsEaEE

Induced emf per phase in the rotor winding =rE

Rotor resistance in terms of stator winding ==′222rar

Rotor reactance at standstill in terms of stator winding ==′222xax

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The input voltage per phase to the stator is equal to the induced emf per phase in the stator winding, as the stator impedance is neglected (also shown in the last lesson (#32)).

In the formula for starting current, no load current is neglected. It may be noted that the starting current is quite high, about 4-6 times the current at full load, may be higher, depending on the rating of IM, as compared to no load current.

The starting torque is ([]210)()(ststIT∝), which shows that, as the starting current increases, the starting torque also increases. This results in higher accelerating torque (minus the load torque and the torque component of the losses), with the motor reaching rated or near rated speed quickly.

Need for Starters in IM

The main problem in starting induction motors having large or medium size lies mainly in the requirement of high starting current, when started direct-on-line (DOL). Assume that the distribution line is starting from a substation (Fig. 33.1), where the supply voltage is constant. The line feeds a no. of consumers, of which one consumer has an induction motor with a DOL starter, drawing a high current from the line, which is higher than the current for which this line is designed. This will cause a drop (dip) in the voltage, all along the line, both for the consumers between the substation and this consumer, and those, who are in the line after this consumer. This drop in the voltage is more than the drop permitted, i.e. higher than the limit as per ISS, because the current drawn is more than the current for which the line is designed. Only for the current lower the current for which the line is designed, the drop in voltage is lower the limit. So, the supply authorities set a limit on the rating or size of IM, which can be started DOL. Any motor exceeding the specified rating, is not permitted to be started DOL, for which a starter is to be used to reduce the current drawn at starting.

Starters for Cage IM

The starting current in IM is proportional to the input voltage per phase () to the motor (stator), i.e. , where, sV()sstEI∝1ssEV≈, as the voltage drop in the stator impe-dance is small compared to the input voltage, or ssEV=, if the stator impedance is neglected. This has been shown earlier. So, in a (squirrel) cage induction motor, the

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starter is used only to decrease the input voltage to the motor so as to decrease the starting current. As described later, this also results in decrease of starting torque.

Star-Delta Starter

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This type is used for the induction motor, the stator winding of which is nominally delta-connected (Fig. 33.2a). If the above winding is reconnected as star (Fig. 33.2b), the voltage per phase supplied to each winding is reduced by )577.0(3/1. This is a simple starter, which can be easily reconfigured as shown in Fig. 33.2c. As the voltage per phase in delta connection is , the phase current in each stator winding is (), where is the impedance of the motor per phase at standstill or start (stator impedance and rotor impedance referred to the stator, at standstill). The line current or the input current to the motor is [()sVssZV/sZssstZVI/)3(1⋅=], which is the current, if the motor is started direct-on-line (DOL). Now, if the stator winding is connected as star, the phase or line current drawn from supply at start (standstill) is [3/)/(ssZV], which is (2)3/1(3/1=) of the starting current, if DOL starter is used. The voltage per phase in each stator winding is now (.3/sV). So, the starting current using star-delta starter is reduced by 33.3%. As for starting torque, being proportional to the square of the current in each of the stator windings in two different connections as shown earlier, is also reduced by (2)3/1(3/1=), as the ratio of the two currents is (3/1), same as that (ratio) of the voltages applied to each winding as shown earlier. So, the starting torque is reduced by 33.3%, which is a disadvantage of the use of this starter. The load torque and the loss torque, must be lower than the starting torque, if the motor is to be started using this starter. The advantage is that, no extra component, except that shown in Fig. 33.2c, need be used, thus making it simple. As shown later, this is an auto-transformer starter with the voltage ratio as 57.7%. Alternatively, the starting current in the second case with the stator winding reconnected as star, can be found by using star-delta conversion as given in lesson #18, with the impedance per phase after converting to delta, found as (sZ⋅3),

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and the starting current now being reduced to () of the starting current obtained using DOL starter, with the stator winding connected in delta. 3/1

Auto-transformer Starter

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An auto-transformer, whose output is fed to the stator and input is from the supply (Fig. 33.3), is used to start the induction motor. The input voltage of IM is , which is the output voltage of the auto-transformer, the input voltage being . The output voltage/input voltage ratio is sVx⋅sVx, the value of which lies between 0.0 and 1.0 (). Let be the starting current, when the motor is started using DOL starter, i.e applying rated input voltage. The input current of IM, which is the output current of auto-transformer, is 0.10.0<<x()stI1()stIx1⋅ , when this starter is used with input voltage as . The input current of auto-transformer, which is the starting current drawn from the supply, is , obtained by equating input and output volt-amperes, neglecting losses and assuming nearly same power factor on both sides. As discussed earlier, the starting torque, being proportional to the square of the input current to IM in two cases, with and without auto-transformer (i.e. direct), is also reduced by , as the ratio of the two currents is sVx⋅()stIx12⋅2xx, same as that (ratio) of the voltages applied to the motor as shown earlier. So, the starting torque is reduced by the same ratio as that of the starting current. If the ratio is , both starting current and torque are %)80(8.0=x

%)64(64.0)8.0(22==x times the values of starting current and torque with DOL starting, which is nearly 2 times the values obtained using star-delta starter. So, the disadvantage is that starting current is increased, with the result that lower rated motor can now be started, as the current drawn from the supply is to be kept within limits, while the advantage is that the starting torque is now doubled, such that the motor can start against higher load torque. The star-delta starter can be considered equivalent to an auto-transformer starter with the ratio, %)7.57(577.0=x. If %)70(7.0=x, both starting current and torque are times the values of starting current and torque with DOL starting, which is nearly 1.5 times the values obtained using star-delta starter. By varying the value of the voltage ratio x of the auto-transformer, the %)50(5.049.0)7.0(22≈==x

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values of the starting current and torque can be changed. But additional cost of auto-transformer with intermittent rating is to be incurred for this purpose.

Rotor Resistance Starters for Slip-ring (wound rotor) IM

In a slip-ring (wound rotor) induction motor, resistance can be inserted in the rotor circuit via slip rings (Fig. 33.4), so as to increase the starting torque. The starting current in the rotor winding is

22222)()()(xRrEIextrst++=

where = Additional resistance per phase in the rotor circuit. extR

The input (stator) current is proportional to the rotor current as shown earlier. The starting current (input) reduces, as resistance is inserted in the rotor circuit. But the starting torque, [] increases, as the total resistance in the rotor circuit is increased. Though the starting current decreases, the total resistance increases, thus resulting in increase of starting torque as shown in Fig. 32.2b, and also obtained by using the expression given earlier, for increasing values of the resistance in the rotor circuit. If the additional resistance is used only for starting, being rated for intermittent duty, the resistance is to be decreased in steps, as the motor speed increases. Finally, the external resistance is to be completely cut out, i.e. to be made equal to zero (0.0), thus leaving the slip-rings short-circuited. Here, also the additional cost of the external resistance with intermittent rating is to be incurred, which results in decrease of starting current, along with increase of starting torque, both being advantageous. Also it may be noted that the cost of a slip-ring induction is higher than that of IM with cage rotor, having same power rating. So, in both cases, additional cost is to be incurred to obtain the above advantages. This is only used in case higher starting torque is needed to start IM with high load torque. It may be observed from Fig. 32.2b that the starting torque increases till it reaches maximum value, i.e. ([])()(3)(2220extststRrIT+⋅⋅=mstTT)()(00<), as the external resistance in the rotor circuit is increased, the range of total resistance being [222)(xRrrext<+<].

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The range of external resistance is between zero (0.0) and (22rx−). The starting torque is equal to the maximum value, i.e. (mstTT)()(00=), if the external resistance inserted is equal to (). But, if the external resistance in the rotor circuit is increased further, i.e. [], the starting torque decreases (22rx−)(22rxRext−>mstTT)()(00<). This is, because the starting current decreases at a faster rate, even if the total resistance in the rotor circuit is increased.

In this lesson − the fifth one of this module, the direct-on-line (DOL) starter used for IM, along with the need for other types of starters, has been described first. Then, two types of starters − star-delta and auto-transformer, for cage type IM, are presented. Lastly, the rotor resistance starter for slip-ring (wound rotor) IM is briefly described. In the next (sixth and last) lesson of this module, the various types of single phase induction motors, along with the starting methods, will be presented.

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Starting Methods for Single-phase Induction Motor

 

Instructional Objectives

 

• Why there is no starting torque in a single-phase induction motor with one (main) winding in the stator?

 

• Various starting methods used in the single-phase induction motors, with the intro-duction of additional features, like the addition of another winding in the stator, and/or capacitor in series with it.

 

Introduction

In the previous, i.e. fifth, lesson of this module, the direct-on-line (DOL) starter used in three-phase IM, along with the need for starters, has been described first. Two types of starters − star-delta, for motors with nominally delta-connected stator winding, and auto-transformer, used for cage rotor IM, are then presented, where both decrease in starting current and torque occur. Lastly, the rotor resistance starter for slip-ring (wound rotor) IM has been discussed, where starting current decreases along with increase in starting torque. In all such cases, additional cost is to be incurred. In the last (sixth) lesson of this module, firstly it is shown that there is no starting torque in a single-phase induction motor with only one (main) winding in the stator. Then, the various starting methods used for such motors, like, say, the addition of another (auxiliary) winding in the stator, and/or capacitor in series with it.

Keywords: Single-phase induction motor, starting torque, main and auxiliary windings, starting methods, split-phase, capacitor type, motor with capacitor start/run.

Single-phase Induction Motor

The winding used normally in the stator (Fig. 34.1) of the single-phase induction motor (IM) is a distributed one. The rotor is of squirrel cage type, which is a cheap one, as the rating of this type of motor is low, unlike that for a three-phase IM. As the stator winding is fed from a single-phase supply, the flux in the air gap is alternating only, not a synchronously rotating one produced by a poly-phase (may be two- or three-) winding in the stator of IM. This type of alternating field cannot produce a torque (), if 0.0)(0=stT

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the rotor is stationery (0.0=rω). So, a single-phase IM is not self-starting, unlike a three-phase one. However, as shown later, if the rotor is initially given some torque in either direction (0.0≠rω), then immediately a torque is produced in the motor. The motor then accelerates to its final speed, which is lower than its synchronous speed. This is now explained using double field revolving theory.

Double field revolving theory

When the stator winding (distributed one as stated earlier) carries a sinusoidal current (being fed from a single-phase supply), a sinusoidal space distributed mmf, whose peak or maximum value pulsates (alternates) with time, is produced in the air gap. This sinusoidally varying flux (φ) is the sum of two rotating fluxes or fields, the magnitude of which is equal to half the value of the alternating flux (2/φ), and both the fluxes rotating synchronously at the speed, (Pfns/)2(⋅=) in opposite directions. This is shown in Fig. 34.2a. The first set of figures (Fig. 34.1a (i-iv)) show the resultant sum of the two rotating fluxes or fields, as the time axis (angle) is changing from °=0θ to )180(°π. Fig. 34.2b shows the alternating or pulsating flux (resultant) varying with time or angle.

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The flux or field rotating at synchronous speed, say, in the anticlockwise direction, i.e. the same direction, as that of the motor (rotor) taken as positive induces emf (voltage) in the rotor conductors. The rotor is a squirrel cage one, with bars short circuited via end rings. The current flows in the rotor conductors, and the electromagnetic torque is produced in the same direction as given above, which is termed as positive (+ve). The other part of flux or field rotates at the same speed in the opposite (clockwise) direction, taken as negative. So, the torque produced by this field is negative (-ve), as it is in the clockwise direction, same as that of the direction of rotation of this field. Two torques are in the opposite direction, and the resultant (total) torque is the difference of the two torques produced (Fig. 34.3). If the rotor is stationary (0.0=rω), the slip due to forward (anticlockwise) rotating field is 0.1=fs. Similarly, the slip due to backward rotating field is also . The two torques are equal and opposite, and the resultant torque is 0.0 (zero). So, there is no starting torque in a single-phase IM. 0.1=bs

But, if the motor (rotor) is started or rotated somehow, say in the anticlockwise (forward) direction, the forward torque is more than the backward torque, with the resultant torque now being positive. The motor accelerates in the forward direction, with the forward torque being more than the backward torque. The resultant torque is thus positive as the motor rotates in the forward direction. The motor speed is decided by the load torque supplied, including the losses (specially mechanical loss).

Mathematically, the mmf, which is distributed sinusoidally in space, with its peak value pulsating with time, is described as θcospeakFF=, θ (space angle) measured from the winding axis. Now, tFFpeakωcosmax=. So, the mmf is distributed both in space and time, i.e. tFFωθcoscosmax⋅=. This can be expressed as,

)(cos)2/()(cos)2/(maxmaxtFtFFωθωθ+⋅+−⋅=,

which shows that a pulsating field can be considered as the sum of two synchronously rotating fields (ssnπω2=). The forward rotating field is, )(cos)2/(maxtFFfωθ−⋅=, and the backward rotating field is, )(cos)2/(maxtFFbωθ+⋅=. Both the fields have the

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same amplitude equal to , where is the maximum value of the pulsating mmf along the axis of the winding. )2/(maxFmaxF

When the motor rotates in the forward (anticlockwise) direction with angular speed (rrnπω2=), the slip due to the forward rotating field is,

)/(1/)(srsrsfsωωωωω−=−=, or sfrsωω)1(−=.

Similarly, the slip due to the backward rotating field, the speed of which is sω−(), is,

bsrsrsbss−=+=+=2)/(1/)(ωωωωω,.

The torques produced by the two fields are in opposite direction. The resultant torque is,

bfTTT−=

It was earlier shown that, when the rotor is stationary, bfTT= , with both 0.1==bfss, as 0.0=rω or . Therefore, the resultant torque at start is 0.0 (zero). 0.0=rn

Starting Methods

The single-phase IM has no starting torque, but has resultant torque, when it rotates at any other speed, except synchronous speed. It is also known that, in a balanced two-phase IM having two windings, each having equal number of turns and placed at a space angle of (electrical), and are fed from a balanced two-phase supply, with two voltages equal in magnitude, at an angle of , the rotating magnetic fields are produced, as in a three-phase IM. The torque-speed characteristic is same as that of a three-phase one, having both starting and also running torque as shown earlier. So, in a single-phase IM, if an auxiliary winding is introduced in the stator, in addition to the main winding, but placed at a space angle of (electrical), starting torque is produced. The currents in the two (main and auxiliary) stator windings also must be at an angle of , to produce maximum starting torque, as shown in a balanced two-phase stator. Thus, rotating magnetic field is produced in such motor, giving rise to starting torque. The various starting methods used in a single-phase IM are described here. °90°90°90°90

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Resistance Split-phase Motor

The schematic (circuit) diagram of this motor is given in Fig. 34.4a. As detailed earlier, another (auxiliary) winding with a high resistance in series is to be added along with the main winding in the stator. This winding has higher resistance to reactance () ratio as compared to that in the main winding, and is placed at a space angle of from the main winding as given earlier. The phasor diagram of the currents in two windings and the input voltage is shown in Fig. 34.4b. The current () in the auxiliary winding lags the voltage (V) by an angle, aaXR/°90aIaφ, which is small, whereas the current () in the main winding lags the voltage (V) by an angle, mImφ, which is nearly . The phase angle between the two currents is (°90aφ−°90), which should be at least . This results in a small amount of starting torque. The switch, S (centrifugal switch) is in series with the auxiliary winding. It automatically cuts out the auxiliary or starting winding, when the motor attains a speed close to full load speed. The motor has a starting torque of 100−200% of full load torque, with the starting current as 5-7 times the full load current. The torque-speed characteristics of the motor with/without auxiliary winding are shown in Fig. 34.4c. The change over occurs, when the auxiliary winding is switched off as given earlier. The direction of rotation is reversed by reversing the terminals of any one of two windings, but not both, before connecting the motor to the supply terminals. This motor is used in applications, such as fan, saw, small lathe, centrifugal pump, blower, office equipment, washing machine, etc. °30

Capacitor Split-phase Motor

The motor described earlier, is a simple one, requiring only second (auxiliary) winding placed at a space angle of from the main winding, which is there in nearly all such motors as discussed here. It does not need any other thing, except for centrifugal switch, as the auxiliary winding is used as a starting winding. But the main problem is °90

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low starting torque in the motor, as this torque is a function of, or related to the phase difference (angle) between the currents in the two windings. To get high starting torque, the phase difference required is (Fig. 34.5b), when the starting torque will be proportional to the product of the magnitudes of two currents. As the current in the main winding is lagging by °90mφ, the current in the auxiliary winding has to lead the input voltage by aφ, with (°=+90amφφ). aφ is taken as negative (-ve), while mφ is positive (+ve). This can be can be achieved by having a capacitor in series with the auxiliary winding, which results in additional cost, with the increase in starting torque, The two types of such motors are described here.

Capacitor-start Motor

The schematic (circuit) diagram of this motor is given in Fig. 34.5a. It may be observed that a capacitor along with a centrifugal switch is connected in series with the auxiliary winding, which is being used here as a starting winding. The capacitor may be rated only for intermittent duty, the cost of which decreases, as it is used only at the time of starting. The function of the centrifugal switch has been described earlier. The phasor diagram of two currents as described earlier, and the torque-speed characteristics of the motor with/without auxiliary winding, are shown in Fig. 34.5b and Fig. 34.5c respectively. This motor is used in applications, such as compressor, conveyor, machine tool drive, refrigeration and air-conditioning equipment, etc.

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Capacitor-start and Capacitor-run Motor

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In this motor (Fig. 34.6a), two capacitors − for starting, and for running, are used. The first capacitor is rated for intermittent duty, as described earlier, being used only for starting. A centrifugal switch is also needed here. The second one is to be rated for continuous duty, as it is used for running. The phasor diagram of two currents in both cases, and the torque-speed characteristics with two windings having different values of capacitors, are shown in Fig. 34.6b and Fig. 34.6c respectively. The phase difference between the two currents is (sCrC°>+90amφφ) in the first case (starting), while it is for second case (running). In the second case, the motor is a balanced two phase one, the two windings having same number of turns and other conditions as given earlier, are also satisfied. So, only the forward rotating field is present, and the no backward rotating field exists. The efficiency of the motor under this condition is higher. Hence, using two capacitors, the performance of the motor improves both at the time of starting and then running. This motor is used in applications, such as compressor, refrigerator, etc. °90

Beside the above two types of motors, a Permanent Capacitor Motor (Fig. 34.7) with the same capacitor being utilised for both starting and running, is also used. The power factor of this motor, when it is operating (running), is high. The operation is also quiet

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and smooth. This motor is used in applications, such as ceiling fans, air circulator, blower, etc.

Shaded-pole Motor

A typical shaded-pole motor with a cage rotor is shown in Fig. 34.8a. This is a single-phase induction motor, with main winding in the stator. A small portion of each pole is covered with a short-circuited, single-turn copper coil called the shading coil. The sinusoidally varying flux created by ac (single-phase) excitation of the main winding induces emf in the shading coil. As a result, induced currents flow in the shading coil producing their own flux in the shaded portion of the pole.

Let the main winding flux be tmωφφsinmax=

where

(flux component linking shading coil) scmmφφ=

+ mφ′ (flux component passing down the air-gap of the rest of the pole)

The emf induced in the shading coil is given by

dtdescmscφ= (since single-turn coil) tscωωφcosmax=

Let the impedance of the shading coil be scscscscXjRZ+=∠θ

The current in the shading coil can then be expressed as

()[])(cos/maxscscscsctZiθωωφ−=

The flux produced by is sci

)(cos1maxscscscscsctRZRiθωφωφ−=×=

where reluctance of the path of =Rscφ

As per the above equations, the shading coil current () and flux (scIscφ) phasors lag behind the induced emf () by angle scEscθ ; while the flux phasor leads the induced emf () by . Obviously the phasor scE°90mφ′ is in phase with . The resultant flux in the shaded pole is given by the phasor sum scmφ

scscmspφφφ+=

as shown in Fig. 34.8b and lags the flux mφ′ of the remaining pole by the angle α. The two sinusoidally varying fluxes mφ′ and spφ′ are displaced in space as well as have a time phase difference (α), thereby producing forward and backward rotating fields, which produce a net torque. It may be noted that the motor is self-starting unlike a single-phase single-winding motor.

It is seen from the phasor diagram (Fig. 34.8b) that the net flux in the shaded portion of the pole (spφ) lags the flux (mφ′) in the unshaded portion of the pole resulting in a net torque, which causes the rotor to rotate from the unshaded to the shaded portion of the pole. The motor thus has a definite direction of rotation, which cannot be reversed.

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The reversal of the direction of rotation, where desired, can be achieved by providing two shading coils, one on each end of every pole, and by open-circuiting one set of shading coils and by short-circuiting the other set.

The fact that the shaded-pole motor is single-winding (no auxiliary winding) self-starting one, makes it less costly and results in rugged construction. The motor has low efficiency and is usually available in a range of 1/300 to 1/20 kW. It is used for domestic fans, record players and tape recorders, humidifiers, slide projectors, small business machines, etc. The shaded-pole principle is used in starting electric clocks and other single-phase synchronous timing motors.

In this lesson − the sixth and last one of this module, firstly, it is shown that, no starting torque is produced in the single-phase induction motor with only one (main) stator winding, as the flux produced is a pulsating one, with the winding being fed from single phase supply. Using double revolving field theory, the torque-speed characteristics of this type of motor are described, and it is also shown that, if the motor is initially given some torque in either direction, the motor accelerates in that direction, and also the torque is produced in that direction. Then, the various types of single phase induction motors, along with the starting methods used in each one are presented. Two stator windings − main and auxiliary, are needed to produce the starting torque. The merits and demerits of each type, along with their application area, are presented. The process of production of starting torque in shade-pole motor is also described in brief. In the next module consisting of seven lessons, the construction and also operation of dc machines, both as generator and motor, will be discussed. Main winding Squirrel-cage rotor Stator Shading coil sc sp m sc=+φφφ Fig. 34.8(a): Shaded-pole motor (single-phase induction type) 'mφ

Version 2 EE IIT, Kharagpur EscIscΦscΦspθscα sc mΦ ' mΦ Fig. 34.8(b): Phasor diagram of the fluxes in shaded=pole motor

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