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Unformatted text preview: 101 Chapter 10
Sinusoidal Permanent Magnet
AC Drives,
LoadCommutatedInverter
Synchronous Motor Drives,
and Synchronous Generators
Exit 2001 by N. Mohan Print TOC " ! 102 PermanentMagnet AC (PMAC) Drives
Power
Processing Utility Control
input Unit Controller Sinusoidal ia
ib
ic PMAC Load Position
sensor motor θ m (t ) ❏ System level operation similar to DC machines but without
brushes  sometimes called Brushless DC Drives
❏ Motor essentially a synchronous machine whose field flux
is provided by permanent magnets
Exit 2001 by N. Mohan TOC " ! 103 Structure of PermanentMagnet
Synchronous Machines
b − axis
!!"
Br (t ) ib θm
N
S ia a − axis ic θ m (t )
a − axis c − axis ❏ Permanent Magnet rotor
❏ Sinusoidally distributed stator windings
Exit 2001 by N. Mohan TOC " ! 104 Principle of Operation
❏ Magnets shaped to produce sinusoidal
flux density distribution b − axis
!"
is !!
"
ˆ
Br (t ) = Br ∠θ m (t ) θ is (t ) = θ m (t ) + 900 • δ = 90o maximizes torque/ampere
• Stator current space vector is
controlled so that it leads the peak
rotor flux by 90 degrees
Exit 2001 by N. Mohan a' θm δ = 90oN ❏ Controlled stator currents
"
• is (t ) controlled by PPU
(controlling ia (t ), ib (t ), and ic (t ) ) such that
"
ˆ
is (t ) = I s (t ) ∠θ is (t ), where ib !!"
Br ia a − axis S ic a
c − axis ˆ
Is !"
is !!"
Br θm N a − axis
S TOC " ! 105 Torque Calculation
!"
is dξ f em = Bli Using ξ
S dTem (ξ ) = r ˆ
Br cos ξ
$&%
% ' ˆ
Is Ns
ˆ
cos ξ ⋅ dξ
⋅
#
⋅ Is ⋅
$ &%
% '
2%
$%&%%
' flux density at ξ cond .length !!"
Br N diff no. of cond . at ξ ξ =π / 2 Tem Ns ˆ ˆ π / 2 N
ˆ ˆ
r #Br I s ∫ cos 2 ξ ⋅ dξ = π s r #Br I s
= 2 × ∫ dTem (ξ ) = 2
2 2 ξ =−π / 2
−π / 2 N
ˆ
ˆ
⇒ Tem = k I s , where the machine torque constant, kT = π s r # Br
2
T ˆ
❏ Torque is proportional to I s alone, just as in dc motors
with constant field excitation. Hence the name
BrushLess DC
Exit 2001 by N. Mohan TOC " ! Similarity Between DC Motor and
Brushless DC Motor
ωm φf
N S (stationary) S 106 !"
is δ = 90 o
N ωm !!"
Br φa
(stationary) BrushLess DC motor drive
!!"
• Br produced by rotor magnets
• Stationary φ f produced
and rotates with the rotor
by stator windings
!"
• φa produced by rotating rotor • is produced by stator winding
currents and is made to rotate
windings and is made
at rotor speed by the action of
stationary by commutator
the PPU
action
Exit
TOC
" !
2001 by N. Mohan
DC motor 107 Mechanical System
Tem
Motor Load TL dω m Tem − TL
αm =
=
dt
J eq
t ω m (t ) = ω m (0 ) + ∫o α m (τ ) ⋅ dτ
t θ m (t ) = θ m (0 ) + ∫o ω m (τ ) ⋅ dτ
Exit 2001 by N. Mohan TOC " ! 108 Calculation of the Reference
* *
*
Values: ia , ib and ic
❏ Reference values are generated by the controller based on
desired torque output and rotor position
❏ Reference values tell the PPU what stator currents to deliver
❏ Starting with the desired torque and known rotor position, the
desired stator currents are found as follows:
*
(Tem ,θ m ) * T (t )
ˆ*
I s (t ) = em
kT θ i* (t ) = θ m (t ) + 900
s
"
ˆ*
is* (t ) = I s (t )∠θ i* (t )
s
Exit 2001 by N. Mohan !
"
* * *
→ (is ) → (ia , ib , ic ) "
2
2 ˆ*
*
ia (t ) = Re is* (t ) = I s (t )cosθ i* (t )
s 3
3 2 "*
2 ˆ*
*
ib (t ) = Re is (t )∠ − 1200 = I s (t )cos(θ i* (t ) − 1200 )
s 3
3 "
2
2 ˆ*
*
ic (t ) = Re is* (t )∠ − 2400 = I s (t )cos(θ i* (t ) − 2400 )
s 3
3 TOC " ! 109 Example
kT = 0.5 Nm/A
To produce a counter clockwise
holding torque of 5 Nm at θ m = 45° !!"
Br b − axis ib !"
is 135 o ˆ T
I s = em = 10 A
kT 45o
N ia θ is = θ m + 900 = 1350 a − axis S ic "
ˆ
is = I s ∠θ is = 10∠1350 c − axis ia = 2ˆ
I s cosθ is = −4.71 A
3 ic = 2ˆ
I s cos(θ is − 2400 ) = −1.73 A
3 ib = 2ˆ
I s cos(θ is − 1200 ) = 6.44 A
3 Stator currents are dc in this example.
Exit 2001 by N. Mohan TOC " ! 1010 Induced EMF in Stator Windings under
Balanced Sinusoidal Steady State
!!"
1. Br (t ) rotates with an instantaneous speed of ω m (t ) . This
rotating fluxdensity distribution cuts the stator windings to
induce a backemf. !"
2. The rotating fluxdensity distribution due to rotating is (t )
space vector induces an emf in the stator windings. Exit 2001 by N. Mohan TOC " ! 1011 Induced EMF in the!! !!!
Stator Windings due
"
"
to Rotating Br (ems ,!!" )
B
r !!!
"
"
3
N s !!!!
ems (t ) = jω ( π r # ) Bms (t ) (Eq. 941)
2
2
with substitutions in the current case:
!!!
"
!!"
!! (t ) = jω ( 3 π r # N s ) B (t )
"
ems , B
m
r
r
2
2
Voltage Constant: ωm t =0 !!!!!!!
"
ems, !!"
B a − axis r N ωm N ˆ V Nm = kT = π r # s Br
kE 2 rad / s A !!!
"
!! (t ) = j 3 k ω ∠θ (t ) = 3 k ω ∠(θ (t ) + 90 o )
ems , B"
E m
m
E m
m
r
2
2
Exit 2001 by N. Mohan TOC " ! 1012 Induced EMF in the Stator Windings due
! !!!
"
"
to Rotating is (ems ,i!" )
s !!!
"
!!!
"
ems (t ) = jω Lm ims (t ) (Eq. 940) !!!!!!
"
ems, i!" ωm t =0 s !"
is with substitutions in the current case:
!!
"
!"
! (t ) = jω L i (t )
"
es ,i
m m s
s ˆ
= ω m Lm I s ∠(θ m (t ) + 90 o + 90 o )
$%&%'
θ is (t ) a − axis N
!!"
Br ωm Lm : Magnetizing inductance Exit 2001 by N. Mohan TOC " ! 1013 Net induced EMF in the stator windings
!!!
"
ems (t )
ωm t =0 ( !!!"
ems ) !!!!!!
"
ems, i!"
s !"
is !!!!!!! a − axis
"
ems, !!"
B
r Ema Ema, i!"
s !!"
Ema, B N r !!"
Br ωm Space vector diagram Ia ref − axis Phasor diagram for phasea !!!
"
!!!
"
!!!
"
!!" (t ) + e , !" (t )
ems (t ) = ems , B
ms i
r s !!!
"
!
"
3
o
ems (t ) = k Eω m ∠(θ m (t ) + 90 ) + jω m Lm is (t )
2
2
Ema = k Eω m ∠(θ m (t ) + 90 o ) + j ω m Lm I a
3
Exit 2001 by N. Mohan TOC " ! 1014 PerPhase Equivalent Circuit
L Ia
I a ( jω m Ls ) Va + I a Rs Rs (%%%% s
)%%%%
*
Lls
Lm
+ + Ema, i!" −
s Va r Ia − !!"
Ema, B Ema, i!" − !!"
Ema, B +
r s − ref − axis ˆ " = 2 E " =k E ω m = E fa
ˆ
ˆ
Ea ,B
ms ,Br
r
3
Ls = L#s + Lm Lm : Magnetizing inductance
Lls : Stator leakage inductance Va = E fa + jω m Ls I a + Rs I a Rs can often be ignored
Exit 2001 by N. Mohan TOC " ! 1015 Controller and Power Processing Unit
Power
Processing Utility Control
input Unit Controller Sinusoidal ia
ib
ic PMAC Load Position
sensor motor θ m (t ) ❏ Controller determines desired phase currents based on
desired torque and motor position Exit 2001 by N. Mohan TOC " ! 1016 Hysterisis current control
actual current
reference current
t 0 *
Tem + phase a Vd
− 1
kT *
ia (t ) ˆ*
Is *
ib (t )
*
ic (t ) + Σ
−
ia (t ) q A (t ) θ m (t ) Exit 2001 by N. Mohan TOC " ! 1017 LoadCommutatedInverter (LCI)
Supplied Synchronous Motor Drives
Id If Ld
ac line
input Linecommutated
converter Loadcommutated
inverter Synchronous
motor ❏ High power levels
❏ Field windings on rotor carrying a dc current
❏ Thyristor PPU needed at these power levels
❏ DClink between utility and inverter is a nearly constant
current ( I d ) rather than a constant voltage ( Vd ) as in
previous circuits
❏ Inverter thyristors commutated by load (synchronous
motor)
Exit 2001 by N. Mohan TOC " ! 1018 Synchronous Generators
❏ Generally larger sizes
❏ Directly connected to utility without PPU
❏ Threephase winding on stator  DC field winding on rotor
❏ Angle between rotor flux and stator flux not necessarily 90o
allowing generator to sink or source VARS Exit 2001 by N. Mohan TOC " ! 1019 PerPhase Model and PowerAngle
Characteristics
Pem steady state
stability limit Ia
+
ˆ
E fa = E f ∠δ jX s − generator
mode
+
ˆ
Va = Va ∠0 o −90 o 90 o δ −
motoring
mode ❏ Total 3phase power
ˆ ˆ
3 Ef V
Pem =
sin δ
2 Xs 0 steady state
stability limit ❏ For angles between 90o and +90o rotor speed remains locked to
line frequency
❏ When the machine is asked to either supply or absorb to much power
the angle will move outside the ±90 o range. In this situation the rotor
will no longer be synchronized to the line and will either speed up out
of control or slow down. In either case excessive currents should trip the
circuit breakers.
Exit 2001 by N. Mohan TOC " ! 1020 Adjusting Reactive Power and Power
Factor
E fa E fa E fa jX s I a 90 o jX s I a δ
Ia Va I a ,q δ Va I a ,q { δ jX s I a Ia
Va Ia 90 o ❏ Unity Power Factor Operation
For every operating condition there is one value of field current that will
cause the generator to deliver only real power. ❏ Overexcitation
Increasing field current causes generator to supply more reactive power. ❏ Underexcitation
When field current is decreased below the value for Unity Power Factor
operation, the generator will absorb reactive power.
Exit 2001 by N. Mohan TOC " ! 1021 Summary
❏ List various names associated with the PMAC drives and the
reasons behind them.
❏ Draw the overall block diagram of a PMAC drive. Why
must they operate in a closedloop?
❏ How do sinusoidal PMAC drives differ from the ECM
drives described in Chapter 7?
❏ Ideally, what are the fluxdensity distributions produced
by the rotor and the stator phase windings?
❏ What does the space vector represent?
❏ In PMAC drives, why at all times is the space vector
placed 90 degrees ahead of the space vector in the intended
direction of rotation?
❏ Why do we need to measure the rotor position in PMAC
drives?
Exit 2001 by N. Mohan TOC " ! 1022 Summary
❏ What does the electromagnetic torque produced by a
PMAC drive depend on?
❏ How can regenerative braking be accomplished in PMAC
drives?
❏ Why are PMAC drives called selfsynchronous? How is the
frequency of the applied voltages and currents determined?
Are they related to the rotational speed of the shaft?
❏ In a ppole PMAC machine, what is the angle of the space
vector in relation to the phasea axis, for a given ?
❏ What is the frequency of currents and voltages in the stator
circuit needed to produce a holding torque in a PMAC drive?
❏ In calculating the voltages induced in the stator windings
of a PMAC motor, what are the two components that are
superimposed? Describe the procedure and the expressions.
Exit 2001 by N. Mohan TOC " ! 1023 Summary
❏ Does in the perphase equivalent circuit of a PMAC machine
have the same expression as in Chapter 9? Describe the
differences, if any.
❏ Draw the perphase equivalent circuit and describe its
various elements in PMAC drives.
❏ Draw the controller block diagram and describe the hysteresis
control of PMAC drives.
❏ What is an LCIsynchronous motor drive? Describe it briefly.
❏ For what purpose are lineconnected synchronous generators
used?
❏ Why are there problems of stability and loss of synchronism
associated with lineconnected synchronous machines?
❏ How can the power factor associated with synchronous
generators be made to be leading or lagging?
Exit 2001 by N. Mohan TOC " ! ...
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 Fall '06
 Scalzo

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