electric machine Chapter 6 Polyphase Induction Machines

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Chapter 6 Polyphase Induction Machines

 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
is s

s
s
nn
s
n
−
= (6.1)
 The slip is often expressed in percent.

: rotor speed in rpm n

(
)
s
nsn
−
=
1 (6.2)

m
ω
: mechanical angular velocity
(
)
sm
s
ω
ω
−
=
1 (6.3)

r
f
: the frequency of induced voltages, the slip frequency
re
f
sf
=
(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
with respect to the rotor in the
forward direction.
s
sn
 With respect to the stator, the speed of the flux wave produced by the rotor currents
(with frequency
) equals
e
sf
(
)
sss
1sn n sn n s n
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.

1
 Equation (4.81)
2
sr r r
poles
sin
22
T
π
F
δ
⎛⎞
=− Φ
⎜⎟
⎝⎠
can be expressed in the form
r
sinTKI
r
δ
=
− (6.6)

r
I
: the rotor current

r
δ
: the angle by which the rotor mmf wave leads the resultant air-gap mmf wave
 Fig. 6.4 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 6.4 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:

(
)
11121
ˆˆˆ
jXRIEV ++=
(6.8)

1
2
1
1
1
ˆ
Stator line-to-neutral terminal voltage
ˆ
Counter emf (line-to-neutral) generated
b
y the resultant air-gap flux
ˆ
Stator current
Stator effective resistance
Stator leakage reactance
V
E
I
R
X
=
=
=
=
=


2


Figure 6.7 Stator equivalent circuit for a polyphase induction motor.

 Rotor equivalent circuit:
2
2
2
ˆ
ˆ
I
E
Z =
(6.9)
22
2s rotor
2s eff eff rotor
2s rotor
?
?
EE
Z
NN
II
⎛⎞
== =
⎜⎟
⎝⎠
Z
(6.10)

2s
Z
: the slip-frequency leakage impedance of the equivalent rotor

rotor
Z
: the slip-frequency leakage impedance
2s
2s 2 2
2s
ˆ
ˆ
E
Z
RjsX
I
==+
(6.11)

2
R
= Referred rotor resistance

= Referred rotor leakage reactance at slip frequency
2
sR

2
X
= Referred rotor leakage reactance at stator frequency
e
f



Figure 6.8 Rotor equivalent circuit for a polyphase induction motor at slip frequency.


22
ˆˆ
II
s
= (6.12)
22
sEE
s
=
(6.13)
22
ˆˆ
EsE
s
= (6.14)
222
2
2
2
2
ˆ
ˆ
ˆ
ˆ
jsXRZ
I
Es
I
E
s
s
s
+===
(6.15)
2
2
2
2
2
ˆ
ˆ
jX
s
R
I
E
Z +==
(6.16)


3
 Fig. 6.9 shows the single-phase equivalent circuit.

Figure 6.9 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.
 : the total power transferred across the air gap from the stator
gap
P
rotor
P : the total rotor ohmic loss
⎟
⎠
⎞
⎜
⎝
⎛
=
s
R
InP
2
2
2phgap
(6.17)
2
2
2phrotor
RInP
s
= (6.18)
2
2
2phrotor
RInP = (6.19)
2
2
2ph
2
2
2phrotorgapmech
RIn
s
R
InPPP −
⎟
⎠
⎞
⎜
⎝
⎛
=−= (6.20)
⎟
⎠
⎞
⎜
⎝
⎛
−
=
s
s
RInP
1
2
2
2phmech
(6.21)
(
)
gapmech
1 PsP
−
=
(6.22)
rotor gap
PsP
=
(6.23)
 Of the total power delivered across the air gap to the rotor, the fraction is
converted to mechanical power and the fraction
is dissipated as ohmic loss in the
rotor conductors.
1 s−
s
 When power aspects are to be emphasized, the equivalent circuit can be redrawn in
the manner of Fig. 6.10.


Figure 6.10 Alternative form of equivalent circuit.


4



Consider the electromechanical torque .
mech
T
(
)
mechmechmech
1 TsTP
sm
ω
ω
−
=
=
(6.24)
(
)
s
sRInP
P
T
ωωω
/
2
2
2ph
s
gap
m
mech
mech
=== (6.25)
e
e
s
f
ω
π
ω
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
==
poles
2
poles
4
(6.26)
rotmechshaft
PPP
−
=
(6.27)
rotmech
m
shaft
shaft
TT
P
T −==
ω
(6.28)


Figure 6.11 Equivalent circuits with the core-loss resistance
Rc neglected corresponding to
(a) Fig. 6.9 and (b) Fig. 6.10.


5

6


§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 6.12 (a) General linear network and
(b) its equivalent at terminals ab by Thevenin’s theorem.


Figure 6.13 Induction-motor equivalent circuits simplified by Thevenin’s theorem.

()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++
=
m
m
XXjR
jX
VV
11
1eq1,
ˆˆ
(6.29)
(
)
1,eq 1,eq 1,eq 1 1
in parallel with
m
Z
RjX RjX jX=+ =+ (6.30)
(
)
()
m
m
XXjR
jXRXj
VZ
++
+
=
11
11
1eq1,
ˆ
(6.31)
sRjXZ
V
I
/
ˆ
ˆ
22eq1,
eq,1
2
++
=
(6.32)
(
)
()
()()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+++
=
2
2eq1,
2
2eq1,
2
2
eq1,ph
mech
/
/
1
XXsRR
sRVn
T
s
ω
(6.33)

 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. 6.14 and 6.15.

7

Figure 6.14 Induction-machine torque-slip curve showing braking, motor, and generator regions.


Figure 6.15 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 for which
maxT
s
()
2
2
2
1,eq 1,eq 2
maxT
R
RXX
s
=++
(6.34)
()
2
maxT
2
2
1,eq 1,eq 2
R
s
RXX
=
++
(6.35)
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+++
=
2
2eq1,
2
eq1,eq1,
2
eq1,
max
5.0
1
XXRR
Vn
T
ph
s
ω
(6.36)


8


9


Figure 6.16 Induction-motor torque-slip curves showing effect of changing rotor-circuit resistance.

§6.5 Parameter Determination from No-Load and Blocked-Rotor
Tests


 The equivalent-circuit parameters needed for computing the performance of a poly-phase
induction motor under load can be obtained from the results of a no-load test, a blocked-rotor
test, and measurement of the dc resistances of the stator windings.

§6.6.1 No-Load Test

 Like the open-circuit test on a transformer, the no-load test on an induction motor
gives information with respect to exciting current and no-load losses.

§6.6.2 Blocked-Rotor Test

 Like the short-circuit test on a transformer, the blocked-rotor test on an induction
motor give information with respect to the leakage impedances.


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