Unformatted text preview: 91 Chapter 9
Introduction to AC Machines Exit 2001 by N. Mohan Print TOC " ! 92 Introduction
❏ Primary AC motor drives
x Induction motors
x Permanent Magnet Brushless (Synchronous Motors)
Both have similar stators but different rotor construction b − axis ❏ Stator windings produce
sinusoidal field distribution ib
2π / 3
2π / 3
2π / 3 ia a − axis ic
c − axis Exit 2001 by N. Mohan TOC " ! 93 Sinusoidallydistributed Stator Windings
dθ θ
3' 4' 5' ia 6' 2' 1' 7'
1 7
6 θ =0
magnetic axis
of phase a θ ia 2
5 4 3 [no. of Conductor density ˆ
ns (θ ) = ns sin(θ ) Total N s = ∫0 ns (θ ) dθ = π ⇒ ns (θ ) = Exit 2001 by N. Mohan π
∫0 Ns
sin(θ )
2 conductors / rad ] 0 < θ < π ˆ
ˆ
ns sin(θ ) dθ = 2ns 0 <θ <π TOC " ! 94 Airgap Field Distribution ξ H a (θ ) = − H a (θ + π )
H a (θ ) l g − H a (θ + π ) l g = 2H a (θ )l g
(negative sign because line of integration points
inwards at θ + π )
2H a (θ ) l g = dξ θ
θ =π π
∫0 θ =0
µ =∞ Ha ns (θ + ξ ) ia dξ
N
π
2 H a (θ ) ! g = s ia ∫0 sin(θ + ξ ) dξ = N s ia cos (θ )
2
N
Fa , H a , Ba (θ )
⇒ H a (θ ) = s ia cos (θ )
2! g θ +π
at time 't' stator surface Ns
ia cos (θ )
2! g µo N s Ba (θ ) = µo H a (θ ) = i cos (θ ) 2! g a N
Fa (θ ) = ! g H a (θ ) = s ia cos (θ )
2 H a (θ ) = − π
2 rotor surface Fa , H a , Ba (θ ) θ
0 π
2 t1 3π
2 π t2 t3 2π t4 θ Field quantities have different
magnitudes and units but same shape
Exit 2001 by N. Mohan TOC " ! 95 b − axis
∠120 o Threephase sinusoidallydistributed
stator windings
b ib b ib b' Ba (θ ) 0
a − axis
o
a ' i a ∠0
a ic 1 ia 1
a
Bb (θ ) 0 c' c θ 1
0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 1 ic
c 1 Bc (θ ) c − axis
∠240 o 0
1
1 ❏ Example Bs (θ ) µ Ni
Ba (θ ) = o s a cosθ = 0.628 cosθ Wb / m 2
2! g
Bb (θ ) = −0.440 × cos θ − 1200 Wb / m 2 θ θ 0
1 θ (
)
Bc (θ ) = −0.188 × cos (θ − 2400 ) Wb / m 2 Bs (θ ) = Ba( θ ) + Bb( θ) + Bc θ = 0.967 cos θ − 13.03°
( )
(
) Exit 2001 by N. Mohan TOC " ! 96 Space Vector to Represent Sinusoidal Distributions
b − axis b − axis ib = − ""#
Fc (t ) ""#
Fa (t )
ic = − ""#
Fb (t ) At time ‘t’ θ Fs (t ) ""#
Fs (t )
""#
Fc (t )
""#
a − axis
Fa (t ) ""#
Fb (t ) a − axis
ia = + c − axis c − axis ❏ Properties of sine(cosine) Fs (θ ) x sum of two sine = sine
x integral/derivative of sine = sine 0 ❏ Complex number representation π
2 π 3π
2 stator surface
θ
2π 5π rotor surface
2 ""#
Ns
N
i a (t ) cos( θ ) ⇔ Fa ) = s i a (t ) ∠0 o
Fa (θ , t ) =
(t
2
2
""
#
""
#
N
Ns
o ;
Similarly, Fb ( t ) = ib (t ) ∠120
Fc ( t ) = s ic (t ) ∠240 o
2
"" ""# 2 "" ""
#
#
#
ˆs ∠θ F
And
Fs = Fa + Fb + Fc = F ❏ Similar expressions for B and H
Exit 2001 by N. Mohan s TOC " ! 97 Example
N
Threephase, sinusoidallydistributed stator with s = 50 turns
2
At time t , ia = 10 A, ib = −10 A and ic = 0 A
"#
Find F
"#
N
F (t ) = s ia ∠00 + ib∠1200 + ic ∠2400
2 (
)
= 50 {10 + ( −10 ) [cos 1200 + j sin 1200 ](+ )0 "#
F (t ) = 50 × 17.32∠ − 30 o = 866 ∠ − 30 o A ⋅ turns
Fs (θ ) } [cos 2400 + j sin 2400 ] at t phase 'a' magnetic axis
−30 o 30 o θ θ =0 ""#
Fs (t ) "
#
If l g = 1.5 mm and µ m =∞, find B (t ).
"
#
B(t ) = 0.73∠ − 30 o T
Exit 2001 by N. Mohan TOC " ! 98 Space Vectors Representation of
Combined Phase Currents and Voltages
❏ Mathematical concept b − axis
∠120 o b At time t
"
#
is (t ) = ia (t )∠00 + ib (t )∠1200 + ic (t )∠2400
ˆ
= I s (t )∠θ i (t ) b' a − axis
o
a ' i a ∠0
a s #
v s (t ) = va (t )∠00 + vb (t )∠1200 + vc (t )∠2400
ˆ
= Vs (t )∠θ vs (t ) Exit 2001 by N. Mohan ib ic c' c c − axis
∠240 o TOC " ! "
#
Physical interpretation of is (t ) #
""
#
Ns "
Ns
0 Ns
0 Ns
0
is (t ) =
ia (t )∠0 +
ib (t )∠120 +
ic (t )∠240 = Fs (t )
2
2
2 %
$%&%' $% ""#
&%% $% ""#
' 2 %
&%%
'
""#
Fa (t )
F (t )
Fc (t )
at time t
#
ˆs (t ) b
#
F (t )
F
ˆ
⇒ I s (t ) =
is (t ) = s
Ns 2
Ns 2 magnetic axis of
hypothetical winding
ˆ
with current I s and θ is (t ) = θ Fs (t )
""
#
"
#
Fs (t ) and is (t ) are collinear
""
#
#
N s µo "
Bs (t ) =
is (t )
2l g ❏ Magnetic filed is produced by
combined effect of ia , ib and ic
but could equivalently be
produced by hypothetical
"
#
winding current is (t ) at θ is
❏ helps in obtaining expression
for torque
Exit 2001 by N. Mohan 99 a − axis
ˆ
Is phase b
magnetic axis
o phase a
magnetic axis 30
"#
is
""#
Bs
phase c
magnetic axis TOC " ! 910 Space Vector Components:
Finding Phase Currents from Current Space Vector
b − axis
∠120 o #
3
Re is ∠00 = ia + Re ib ∠1200 + Re ib ∠2400 = ia %
%
$% &%% $% &%% 2
'
'
1
− ib
2 1
− ic
2 1
− ic
2 c − axis
∠240 o b − axis
"
#
o
2
2ˆ
θ − 120 o ⇒ ib (t ) = Re(is ∠ − 120 ) = I s cos is 3
3 #
is ∠ − 2400 = Re ia ∠ − 2400 + Re ib∠ − 2400 + ic = 3 ic
Re $%% &%% $%% &%% 2
'
' 1
− ia
2 1
− ib
2 "
#
o
2
2ˆ
θ − 240 o ⇒ ic (t ) = Re(is ∠ − 240 ) = I s cos is 3
3 Exit 2001 by N. Mohan at time t ib b' " o
#
2
2ˆ
⇒ ia (t ) = Re(is ∠0 ) = I s cosθ is
3
3
#
3
Re is ∠ − 1200 = Re ia ∠ − 1200 + ib + Re ic ∠1200 = ib $%% &%% %
'
$% &%% 2
'
1
− ia
2 b a − axis
o
a ' i a ∠0
a ic c' c ia + ib + ic = 0
projection × 2
= ic (t )
3 θ "#
is (t ) "#
" is
$%&%' a − axis 2
projection × = ia (t )
3
projection × c − axis 2
= ib (t )
3 TOC " ! 911 Balanced Sinusoidal SteadyState
Excitation (Rotor OpenCircuited)
b imb ima (t ) ωt 0 a @ ( ) @ ωt = ωt = π ωt = 0 c ❏ imc (t ) ˆ
Im ima imc imb (t ) 2π
@ωt =
3 ( @ ωt = π
3 ωt = 0 4π
3 @ ) ωt = 5π
3 ˆ
ˆ
ˆ
ima = I m cos ωt;
imb = I m cos ωt − 1200 ; imc = I m cos ωt − 2400
""#
ˆ
ims (t ) = I m cos ω t∠00 + cos(ω t − 1200 )∠1200 + cos(ω t − 2400 )∠2400 ""#
3ˆ
ˆ
ˆ
⇒ ims (t ) = I ms ∠ω t
where I ms = I m
"""#
# 2
Ns "
ˆ
ˆ
Rotating MMF Fms (t ) = is (t ) = Fms ∠ω t where Fms = 3 N s Iˆm = N s Iˆms
2
2 2
2
""""
# µo N s ""#
& Flux density Bms (t ) = ims (t ) !g 2 ❏ Constant amplitude
Exit 2001 by N. Mohan TOC " ! 912 Relation Between Space Vectors and
Phasors
ima (t ) ❒ Time domain
0 imb (t ) ˆ
Im imc (t )
ωt ωt ˆ
ia (t ) = I m cos (ω t − α )
α
ωt = 0 ❒ Phasor
ˆ
I a = I m∠ − α ˆ
I ma = I m ∠ − α ❒ Space Vector
""#
ims t =0 ""#
ims Exit 2001 by N. Mohan @ t =0 3ˆ
ˆ
ˆ
= I ms ∠ − α ; I ms = I m
2 ❒ Space Vector
t =0 ref α α a − axis """
#
ˆ
ims = I ms ∠ − α ⇔ phasor ⇔ 3
I ma
2
TOC " ! 913 Voltages in the stator windings
b Emc imb
+ eb
−
− − e
a ec
+ + Ema a ema (t ) = Lm """
#
ims I ma +
Ema jω Lm − imc
c I mc I mb ima @ t =0
"""#
ems I ma Emb d
ima (t ) etc.
dt Where the three phase magnetizing inductance (2 pole), Lm = 3 πµo rl N s 2 lg 2 """#
""#
#
N s """"
3
⇒ ems (t ) = jω Lm ims (t ) = jω π rl Bms (t )
2 2 Exit 2001 by N. Mohan TOC 2 " ! 914 Example
va (t ) = 120 2 cos ω t v phase b
magnetic axis ""
#
vs (t ) vb (t ) = 120 2 cos (ω t − 120° ) 30 o vc (t ) = 120 2 cos (ω t − 240° )
# 3
vs = × 120 2∠300 = 254.56 ∠300 V
2 ""#
ims = ""
#
im (t )
phase c
magnetic axis phase a
magnetic axis 60 o """
#
Bm (t ) #
vs
254.56 ∠(300 − 900 )
=
= 0.869∠ − 600 A
jω Lm
2π × 60 × 0.777 #
0
−7
"""" µ N i
#
o s ms = 4π × 10 × 50 × 0.869 ∠ − 60 = 0.055∠ − 600 Wb / m 2
Bms =
2! g
10 −3 Exit 2001 by N. Mohan TOC " ! 915 Summary
❏ Draw the threephase axis in the motor crosssection. Also,
draw the three phasors Va , Vb and Va in a balanced sinusoidal
steady state. Why is the phaseb axis ahead of the phasea
axis by 120 degrees, but Vb lags Va by 120 degrees?
❏ Ideally, what should be the field (F, H, and B) distributions
produced by each of the three stator windings? What is the
direction of this field in the air gap? What direction is
considered positive and what is considered negative?
❏What should the conductordensity distribution in a winding
be in order to achieve the desired field distribution in the air
gap? Express the conductordensity distribution ns (θ ) for
phasea.
Exit 2001 by N. Mohan TOC " ! 916 Summary
❏ How is sinusoidal distribution of conductor density in
a phase winding approximated in practical machines with
only a few slots available to each phase?
❏ How are the three field distributions (F, H, and B) related
to each other, assuming that there is no magnetic saturation
in the stator and the rotor iron?
❏ What is the significance of the magnetic axis of any phase
winding?
❏ Mathematically express the field distributions in the air gap
due to ia as a function of θ . Repeat this for ib and ic .
❏ What do the phasors V and I ""#
denote? What are the
""#
meanings of the space vectors Ba (t ) and Bs (t ) at time t,
assuming that the rotor circuit is electrically opencircuited?
❏ What is the constraint on the sum of the stator currents?
Exit 2001 by N. Mohan TOC " ! 917 Summary
❏ What are physical interpretations of various stator
winding inductances?
❏ Why is the perphase inductance Lm greater than the single
phase inductance Lm,1− phase by a factor of 3/2?
❏ What are the characteristics of space vectors which represent
the field distributions Fs (θ ), H s (θ ) and Bs (θ ) at a given
time? What notations are used for these space vectors?
Which axis is used as a reference to express them
mathematically in this chapter?
❏ Why does a dc current through a phase winding produce a
sinusoidal fluxdensity distribution in the air gap?
❏ How are the terminal phase voltages and currents combined
for representation by space vectors?
❏ What is the physical interpretation of the stator current
"#
space vector is (t ) ?
Exit 2001 by N. Mohan TOC " ! 918 Summary
❏ With no excitation or currents in the rotor,#are """" the #
all # """"
of
"""
space vectors associated with the stator ims (t ), Fms (t ), Bms (t )
collinear (oriented in the same direction)?
""
#
"#
❏ In ac machines, a stator space vector vs (t ) or is (t ) consists
of a unique set of phase components. What is the condition on
which these components are based?
❏ Express the phase voltage components in terms of the stator
voltage space vector.
❏ Under threephase balanced sinusoidal condition with no
rotor currents, and neglecting the stator winding resistances
Rs and the leakage inductance Lls for simplification,
answer the following questions: (a) What is the speed at
which all of the space vectors rotate?
Exit 2001 by N. Mohan TOC " ! 919 Summary
(b) How is the peak flux density related to the magnetizing
currents? Does this relationship depend on the frequency f
of the excitation? If the peak flux density is at its rated
value, then what about the peak value of the magnetizing
currents? (c) How do the magnitudes of the applied
voltages depend on the frequency of excitation, in order to
keep the flux density constant (at its rated value for example)?
❏ What is the relationship between space vectors and phasors
under balanced sinusoidal operating conditions? Exit 2001 by N. Mohan TOC " ! ...
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 Fall '06
 Scalzo
 Vector graphics, Stator, N. Mohan

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