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Chapter 6: Polyphase Induction Machines

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

Study on the behavior of polyphase induction machines:

The analysis begins with the development of single-phase equivalent circuits.

The general form is suggested by the similarity of an induction machine to a transformer.
The equivalent circuits can be used to study the electromechanical characteristics of an induction machine as well as the loading presented by the machine on its supply source.

§6.1 Introduction to Polyphase Induction Machines

An induction machine is one in which alternating current is supplied to the stator directly and to the rotor by induction or transformer action from the stator.

The stator winding is excited from a balanced polyphase source and produces a magnetic field in the air gap rotating at synchronous speed.
The rotor winding may one of two types.

A wound rotor is built with a polyphase winding similar to, and wound with the same number of poles as, the stator. The rotor terminals are available external to the motor.
A squirrel-cage rotor has a winding consisting of conductor bars embedded in slots in the rotor iron and short-circuited at each end buy conducting end rings. It is the most commonly used type of motor in sizes ranging from fractional horsepower on up.

The difference between synchronous speed and the rotor speed is commonly referred to as the slip of the rotor. The fractional slip s is

s=ns−nns

s=ns−nns size 12{s= { {n rSub { size 8{s} } - n} over {n rSub { size 8{s} } } } } {}

(6.1)

The slip is often expressed in percent.
n : rotor speed in rpm

n=(1−s)ns

n=(1−s)ns size 12{n= \( 1 - s \) n rSub { size 8{s} } } {}

(6.2)

ωmωm size 12{ω rSub { size 8{m} } } {}: mechanical angular velocity

ωm=(1−s)ωs

ωm=(1−s)ωs size 12{ω rSub { size 8{m} } = \( 1 - s \) ω rSub { size 8{s} } } {}

(6.3)

frfr size 12{f rSub { size 8{r} } } {}: the frequency of induced voltages, the slip frequency

fr=sfe

fr=sfe size 12{f rSub { size 8{r} } = ital "sf" rSub { size 8{e} } } {}

(6.4)

–A wound-rotor induction machine can be used as a frequency changer.

The rotor currents produce an air-gap flux wave that rotates at synchronous speed and in synchronism with that produced by the stator currents.

With the rotor revolving in the same direction of rotation as the stator field, the rotor currents produce a rotating flux wave rotating at snssns size 12{ ital "sn" rSub { size 8{s} } } {} with respect to the rotor in the forward direction.
With respect to the stator, the speed of the flux wave produced by the rotor currents (with frequency sfcsfc size 12{ ital "sf" rSub { size 8{c} } } {} ) equals

sns+n=sns+ns(1−s)=ns

sns+n=sns+ns(1−s)=ns size 12{ ital "sn" rSub { size 8{s} } +n= ital "sn" rSub { size 8{s} } +n rSub { size 8{s} } \( 1 - s \) =n rSub { size 8{s} } } {}

(6.5)

Because the stator and rotor fields each rotate synchronously, they are stationary with respect to each other and produce a steady torque, thus maintaining rotation of the rotor. Such torque is called an asynchronous torque.
Equation (4.81) T=−π2poles22ΦsrFrsinδrT=−π2poles22ΦsrFrsinδr size 12{T= - { {π} over {2} } left ( { { ital "poles"} over {2} } right ) rSup { size 8{2} } Φ rSub { size 8{ ital "sr"} } F rSub { size 8{r} } "sin"δ rSub { size 8{r} } } {}can be expressed in the form

T=−KIrsinδr

T=−KIrsinδr size 12{T= - ital "KI" rSub { size 8{r} } "sin"δ rSub { size 8{r} } } {}

(6.6)

Ir size 12{I rSub { size 8{r} } } {}

: the rotor current

δr

δr size 12{δ rSub { size 8{r} } } {}

: the angle by which the rotor mmf wave leads the resultant air-gap mmf wave

Fig. 6.1 shows a typical polyphase squirrel-cage induction motor torque-speed curve. The factors influencing the shape of this curve can be appreciated in terms of the torque equation.

Figure 1

Figure 6.1 Typical induction-motor torque-speed

curve for constant-voltage, constant-frequency operation

Under normal running conditions the slip is small: 2 to 10 percent at full load.
The maximum torque is referred to as the breakdown torque.
The slip at which the peak torque occurs is proportional to the rotor resistance.

§6.2 Currents and Fluxes in Polyphase Induction Machines

§6.3 Induction-Motor Equivalent Circuit

Only machines with symmetric polyphase windings exited by balanced polyphase voltages are considered. It is helpful to think of three-phase machines as being Y-connected.

Stator equivalent circuit:

Vˆ1=Eˆ2+Iˆ1(R1+jX1)

Vˆ1=Eˆ2+Iˆ1(R1+jX1) size 12{ { hat {V}} rSub { size 8{1} } = { hat {E}} rSub { size 8{2} } + { hat {I}} rSub { size 8{1} } \( R rSub { size 8{1} } + ital "jX" rSub { size 8{1} } \) } {}

(6.7)

Vˆ1=

Vˆ1= size 12{ { hat {V}} rSub { size 8{1} } ={}} {}

Stator line-to-neutral terminal voltage

Eˆ2=

Eˆ2= size 12{ { hat {E}} rSub { size 8{2} } ={}} {}

Counter emf (line-to-neutral) generated by the resultant air-gap flux

Iˆ1=

Iˆ1= size 12{ { hat {I}} rSub { size 8{1} } ={}} {}

Stator current

R1=

R1= size 12{R rSub { size 8{1} } ={}} {}

Stator effective resistance

X1=

X1= size 12{X rSub { size 8{1} } ={}} {}

Stator leakage reactance

Figure 2

Figure 6.2 Stator equivalent circuit for a polyphase induction motor.

Rotor equivalent circuit:

Z2=Eˆ2Iˆ2

Z2=Eˆ2Iˆ2 size 12{Z rSub { size 8{2} } = { { { hat {E}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {}

(6.8)

{}

Z2s=Eˆ2sIˆ2sNeff2(ErotorIrotor)=Neff2Zrotor

Z2s=Eˆ2sIˆ2sNeff2(ErotorIrotor)=Neff2Zrotor size 12{Z rSub { size 8{2s} } = { { { hat {E}} rSub { size 8{2s} } } over { { hat {I}} rSub { size 8{2s} } } } N rSub { size 8{ ital "eff"} } rSup { size 8{2} } \( { {E rSub { size 8{ ital "rotor"} } } over {I rSub { size 8{ ital "rotor"} } } } \) =N rSub { size 8{ ital "eff"} } rSup { size 8{2} } Z rSub { size 8{ ital "rotor"} } } {}

(6.9)

: the slip-frequency leakage impedance of the equivalent rotor

Zrotor

Zrotor size 12{Z rSub { size 8{ ital "rotor"} } } {}

: the slip-frequency leakage impedance

Z2s=Eˆ2sIˆ2s=R2+jsX2

Z2s=Eˆ2sIˆ2s=R2+jsX2 size 12{Z rSub { size 8{2s} } = { { { hat {E}} rSub { size 8{2s} } } over { { hat {I}} rSub { size 8{2s} } } } =R rSub { size 8{2} } + ital "jsX" rSub { size 8{2} } } {}

(6.10)

R2=

R2= size 12{R rSub { size 8{2} } ={}} {}

Referred rotor resistance

sR2=

sR2= size 12{ ital "sR" rSub { size 8{2} } ={}} {}

Referred rotor leakage reactance at slip frequency

X2=

X2= size 12{X rSub { size 8{2} } ={}} {}

Referred rotor leakage reactance at stator frequency fc

fc size 12{f rSub { size 8{c} } } {}

Figure 3

Figure 6.3 Rotor equivalent circuit for a polyphase induction motor at slip frequency.

Iˆ2s=Iˆ2

Iˆ2s=Iˆ2 size 12{ { hat {I}} rSub { size 8{2s} } = { hat {I}} rSub { size 8{2} } } {}

(6.11)

E2s=sE2

E2s=sE2 size 12{E rSub { size 8{2s} } = ital "sE" rSub { size 8{2} } } {}

(6.12)

Eˆ2s=sEˆ2

Eˆ2s=sEˆ2 size 12{ { hat {E}} rSub { size 8{2s} } =s { hat {E}} rSub { size 8{2} } } {}

(6.13)

Eˆ2sIˆ2s=sEˆ2Iˆ2=Z2s=R2+jsX2

Eˆ2sIˆ2s=sEˆ2Iˆ2=Z2s=R2+jsX2 size 12{ { { { hat {E}} rSub { size 8{2s} } } over { { hat {I}} rSub { size 8{2s} } } } = { {s { hat {E}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } =Z rSub { size 8{2s} } =R rSub { size 8{2} } + ital "jsX" rSub { size 8{2} } } {}

(6.14)

Z2=Eˆ2Iˆ2=R2s+jX2

Z2=Eˆ2Iˆ2=R2s+jX2 size 12{Z rSub { size 8{2} } = { { { hat {E}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } = { {R rSub { size 8{2} } } over {s} } + ital "jX" rSub { size 8{2} } } {}

(6.15)

Fig. 6.4 shows the single-phase equivalent circuit.

Figure 4

Figure 6.4 Single-phase equivalent circuit for a polyphase induction motor.

§6.4 Analysis of the Equivalent Circuit

The single-phase equivalent circuit can be used to determine a wide variety of steady-state performance characteristics of polyphase induction machines.

PgapPgap size 12{P rSub { size 8{ ital "gap"} } } {}: the total power transferred across the air gap from the stator

Protor

Protor size 12{P rSub { size 8{ ital "rotor"} } } {}

: the total rotor ohmic loss

Pgap=nphI22(R2s)

Pgap=nphI22(R2s) size 12{P rSub { size 8{ ital "gap"} } =n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } \( { {R rSub { size 8{2} } } over {s} } \) } {}

(6.16)

Protor=nphI2ssR2

Protor=nphI2ssR2 size 12{P rSub { size 8{ ital "rotor"} } =n rSub { size 8{ ital "ph"} } I rSub { size 8{2s} } rSup { size 8{s} } R rSub { size 8{2} } } {}

(6.17)

Protor=nphI22R2

Protor=nphI22R2 size 12{P rSub { size 8{ ital "rotor"} } =n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } R rSub { size 8{2} } } {}

(6.18)

Pmech=Pgap−Protor=nphI22(R2s)−nphI22R2

Pmech=Pgap−Protor=nphI22(R2s)−nphI22R2 size 12{P rSub { size 8{ ital "mech"} } =P rSub { size 8{ ital "gap"} } - P rSub { size 8{ ital "rotor"} } =n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } \( { {R rSub { size 8{2} } } over {s} } \) - n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } R rSub { size 8{2} } } {}

(6.19)

Pmech=nphI22R2(1−ss)

Pmech=nphI22R2(1−ss) size 12{P rSub { size 8{ ital "mech"} } =n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } R rSub { size 8{2} } \( { {1 - s} over {s} } \) } {}

(6.20)

Pmech=(1−s)Pgap

Pmech=(1−s)Pgap size 12{P rSub { size 8{ ital "mech"} } = \( 1 - s \) P rSub { size 8{ ital "gap"} } } {}

(6.21)

Of the total power delivered across the air gap to the rotor, the fraction 1  s is converted to mechanical power and the fraction s is dissipated as ohmic loss in the rotor conductors.
When power aspects are to be emphasized, the equivalent circuit can be redrawn in the manner of Fig. 6.5.

Figure 5

Figure 6.5 Alternative form of equivalent circuit.

Consider the electromechanical torque Tmech

Tmech size 12{T rSub { size 8{ ital "mech"} } } {}

Pmech=ωmTmech=(1−s)ωsTmech

Pmech=ωmTmech=(1−s)ωsTmech size 12{P rSub { size 8{ ital "mech"} } =ω rSub { size 8{m} } T rSub { size 8{ ital "mech"} } = \( 1 - s \) ω rSub { size 8{s} } T rSub { size 8{ ital "mech"} } } {}

(6.22)

Tmech=Pmechωm=Pgapωs=nphI22(R2/s)ωs

Tmech=Pmechωm=Pgapωs=nphI22(R2/s)ωs size 12{T rSub { size 8{ ital "mech"} } = { {P rSub { size 8{ ital "mech"} } } over {ω rSub { size 8{m} } } } = { {P rSub { size 8{ ital "gap"} } } over {ω rSub { size 8{s} } } } = { {n rSub { size 8{ ital "ph"} } I rSub { size 8{2} } rSup { size 8{2} } \( R rSub { size 8{2} } /s \) } over {ω rSub { size 8{s} } } } } {}

(6.23)

ωs=4πfepoles=2polesωe

ωs=4πfepoles=2polesωe size 12{ω rSub { size 8{s} } = { {4πf rSub { size 8{e} } } over { ital "poles"} } = left [ { {2} over { ital "poles"} } right ]ω rSub { size 8{e} } } {}

(6.24)

Pshaft=Pmech−Prot

Pshaft=Pmech−Prot size 12{P rSub { size 8{ ital "shaft"} } =P rSub { size 8{ ital "mech"} } - P rSub { size 8{ ital "rot"} } } {}

(6.25)

Tshaft=Pshaftωm=Tmech−Trot

Tshaft=Pshaftωm=Tmech−Trot size 12{T rSub { size 8{ ital "shaft"} } = { {P rSub { size 8{ ital "shaft"} } } over {ω rSub { size 8{m} } } } =T rSub { size 8{ ital "mech"} } - T rSub { size 8{ ital "rot"} } } {}

(6.26)

Figure 6

Figure 6.6 Equivalent circuits with the core-loss resistance Rc

Rc size 12{R rSub { size 8{c} } } {}

neglected .

§6.5 Torque and Power by Use of Thevenin’s Theorem

Considerable simplification will be obtained from application of Thevenin’s network theorem to the induction-motor equivalent circuit.

Figure 7

Figure 6.7 (a) General linear network and

(b) its equivalent at terminals ab by Thevenin’s theorem.

Figure 8

Figure 6.8 Induction-motor equivalent circuits simplified by Thevenin’s theorem.

Vˆ1,eq=Vˆ1jXmR1+j(X1+Xm)

Vˆ1,eq=Vˆ1jXmR1+j(X1+Xm) size 12{ { hat {V}} rSub { size 8{1, ital "eq"} } = { hat {V}} rSub { size 8{1} } left [ { { ital "jX" rSub { size 8{m} } } over {R rSub { size 8{1} } +j \( X rSub { size 8{1} } +X rSub { size 8{m} } \) } } right ]} {}

(6.27)

Z 1, eq = R 1, eq + jX 1, eq = ( R 1 + jX 1 ) in parallel jX m

Z 1, eq = R 1, eq + jX 1, eq = ( R 1 + jX 1 ) in parallel jX m size 12{Z rSub { size 8{1, ital "eq"} } =R rSub { size 8{1, ital "eq"} } + ital "jX" rSub { size 8{1, ital "eq"} } = \( R rSub { size 8{1} } + ital "jX" rSub { size 8{1} } \) " in parallel " ital "jX" rSub { size 8{m} } } {}

Z1,eq=Vˆ1jXm(R1+jX1)R1+j(X1+Xm)

Z1,eq=Vˆ1jXm(R1+jX1)R1+j(X1+Xm) size 12{Z rSub { size 8{1, ital "eq"} } = { hat {V}} rSub { size 8{1} } { { ital "jX" rSub { size 8{m} } \( R rSub { size 8{1} } + ital "jX" rSub { size 8{1} } \) } over {R rSub { size 8{1} } +j \( X rSub { size 8{1} } +X rSub { size 8{m} } \) } } } {}

(6.28)

Iˆ2=Vˆ1,eqZ1,eq+jX2+R2/s

Iˆ2=Vˆ1,eqZ1,eq+jX2+R2/s size 12{ { hat {I}} rSub { size 8{2} } = { { { hat {V}} rSub { size 8{1, ital "eq"} } } over {Z rSub { size 8{1, ital "eq"} } + ital "jX" rSub { size 8{2} } +R rSub { size 8{2} } /s} } } {}

(6.29)

Tmech=1ωsnphV1,eq2(R2/s)(R1,eq+(R2/s))2+(X1,eq+X2)2

Tmech=1ωsnphV1,eq2(R2/s)(R1,eq+(R2/s))2+(X1,eq+X2)2 size 12{T rSub { size 8{ ital "mech"} } = { {1} over {ω rSub { size 8{s} } } } left [ { {n rSub { size 8{ ital "ph"} } V rSub { size 8{1, ital "eq"} } rSup { size 8{2} } \( R rSub { size 8{2} } /s \) } over { \( R rSub { size 8{1, ital "eq"} } + \( R rSub { size 8{2} } /s \) \) rSup { size 8{2} } + \( X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} } \) rSup { size 8{2} } } } right ]} {}

(6.30)

The general shape of the torque-speed or torque-slip curve with motor connected to a constant-voltage, constant-frequency source is shown in Figs. 6.9 and 6.10.

Figure 9

Figure 6.9 Induction-machine torque-slip curve showing braking, motor, and generator regions.

Figure 10

Figure 6.10 Computed torque, power, and current curves for the 7.5-kW motor in Exps 6.2 and 6.3.

Maximum electromechanical torque will occur at a value of slip smax Tsmax T size 12{s rSub { size 8{"max T"} } } {}for which

R2smaxT=R1,eq2+(X1,eq+X2)2

R2smaxT=R1,eq2+(X1,eq+X2)2 size 12{ { {R rSub { size 8{2} } } over {s rSub { size 8{"max"T} } } } = sqrt {R rSub { size 8{1, ital "eq"} } rSup { size 8{2} } + \( X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} } \) rSup { size 8{2} } } } {}

(6.31)

smaxT=R2R1,eq2+(X1,eq+X2)2

smaxT=R2R1,eq2+(X1,eq+X2)2 size 12{s rSub { size 8{"max"T} } = { {R rSub { size 8{2} } } over { sqrt {R rSub { size 8{1, ital "eq"} } rSup { size 8{2} } + \( X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} } \) rSup { size 8{2} } } } } } {}

(6.32)

Tmax=1ωs0.5nphV1,eq2R1,eq+R1,eq2+(X1,eq+X2)2

Tmax=1ωs0.5nphV1,eq

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