Download Induction Machine Speed Control

Induction Machine
Speed Control
Master Thesis in Electronics
th
August 30 , 2007
By
Lars-Göran Andersson
[email protected]
Mälardalen University
Department of Computer Science and Electronics
Supervisor
Magnus Jansson
Bombardier Transportation
Company Supervisor
Lars Hörnlund
LSI Svenska AB
Examiner
Mikael Ekström
Mälardalen University
Department of Computer Science and Electronics
Department of Computer Science and Electronics
Page 2 of 28
Abstract
This thesis work finds and presents an alternative method of motor speed control to the
voltage control method currently used by LSI Svenska AB, the constant Volt/Hertz method
with sinusoidal Pulse With Modulation (PWM) and shows the advantages and disadvantages
that is possible to achieve with this new method compared with the currently used method.
Acknowledgement
I really appreciate the guidance and support given by my supervisor Magnus Jansson and
my company supervisor Lars Hörnlund they have been quite dynamic through out my thesis
work and contributed with quite invaluable ideas which really helped me to progress I
acknowledge the Department of Computer Science and Electronics for providing enough
resources to help me complete this thesis work.
Last but not the least I am thankful to my loved ones, my friends and colleagues who
supported me during my time at Mälardalen University.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 3 of 28
Table of Contents
1
2
3
4
5
6
7
Introduction ........................................................................................................................ 5
1.1
Background ................................................................................................................ 5
Theoretical Background ..................................................................................................... 5
2.1
AC motor market ........................................................................................................ 5
2.2
Induction motor theoretical basics ............................................................................. 5
2.2.1
Dynamical model ............................................................................................... 5
2.2.2
Power performance ............................................................................................ 6
2.3
Speed control systems for Induction Motors.............................................................. 8
Analysis of problem ........................................................................................................... 9
3.1
Comparison current ↔ new method .......................................................................... 9
3.1.1
Properties method currently used ....................................................................... 9
3.1.2
New method requirements ................................................................................. 9
3.2
Model / method ........................................................................................................ 10
3.2.1
Creating a schematic in Pspice ......................................................................... 10
3.2.2
Studying different control techniques .............................................................. 10
Solution ............................................................................................................................ 11
4.1
Deciding what model to be used .............................................................................. 11
4.1.1
Three-phase voltage source inverters with sinusoidal PWM ........................... 12
4.2
Building a model ...................................................................................................... 16
4.2.1
Simulation with Simulink / Matlab .................................................................. 16
4.3
Analysis of results .................................................................................................... 21
4.4
Future work .............................................................................................................. 22
4.4.1
Implementation with Digital Signal Processor (DSP)...................................... 22
Summary and conclusions ................................................................................................ 22
References ........................................................................................................................ 23
Appendix .......................................................................................................................... 24
7.1
Simulink model ........................................................................................................ 24
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 4 of 28
List of figures
Figure 1. Per phase equivalent circuit of polyphase Induction Machine ................................... 6
Figure 2. Power flow in an induction motor .............................................................................. 8
Figure 3. Typical motor torque - load characteristic .................................................................. 8
Figure 4. Schematic LSI speed regulator ................................................................................. 10
Figure 5. 3-phase inverter......................................................................................................... 11
Figure 6. Three-phase inverter VLL/Vd as a function of ma ....................................................... 15
Figure 7. Three-phase sinusoidal PWM waveforms and harmonic spectrum .......................... 15
Figure 8. Voltage - frequency relation Induction Machine ...................................................... 16
Figure 9. Induction machine..................................................................................................... 18
Figure 10. Dynamic T-equivalent circuit for the induction motor ........................................... 19
Figure 11. 3-phase PWM converter with DC-link ................................................................... 19
Figure 12. Simulation of Induction Machine start ................................................................... 21
Figure 13. Main Simulink model ............................................................................................. 24
Figure 14. Discrete PWM Generator 4 pulses.......................................................................... 24
Figure 15. IGBT Bridge ........................................................................................................... 25
Figure 16. Three-Phase Induction Machine (IM main)............................................................ 25
Figure 17. IM Sub. 1 ................................................................................................................ 26
Figure 18. IM Sub. 2 ................................................................................................................ 26
Figure 19. IM Sub. 3 ................................................................................................................ 27
Figure 20. IM Sub. 3.1 ............................................................................................................. 27
Figure 21. IM Sub. 3.2 ............................................................................................................. 28
List of tables
Table 1. Harmonics of VLL for a large and odd mf (mf a multiple of 3) .................................... 14
Table 2. Comparison of Adjustable Frequency Drives ............................................................ 20
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 5 of 28
1 Introduction
1.1 Background
The method for Induction Machine (IM) speed control currently used by LSI Svenska AB
in Falun is limited to a maximum of four amperes current delivered to the motor due to
excessive heat build up inside the speed control unit housing. They would like to increase the
maximum allowed current of the speed control unit up to sixteen amperes thereby making it
possible to sell their products to a wider range of customers with different needs for example
control of pumps and more powerful fans, so here it is important to minimize the losses
created in the machine/motor. If possible they would also want to decrease the number of
steps (time) needed in the production line for each unit.
2 Theoretical Background
2.1 AC motor market
Market analysis shows that most of all industrial motor applications uses AC induction
motors. The reasons for this include high robustness, reliability, low price and high efficiency,
η > 90% is preferred in order to reduce costs of operation and to maximize long term profit
gains for the user. However, the use of induction motors also has its disadvantages, these lie
mostly in its difficult controllability, due to its complex mathematical model, its non linear
behavior during saturation effect and the electrical parameter oscillation which depends on the
physical influence of the temperature.
2.2 Induction motor theoretical basics
2.2.1 Dynamical model
In the stationary reference frame (αβ-coordinates), the dynamic model of a 3-phase
induction motor can be described as
d  i s   a11
 
dt  r  a21

a  j   i   b 
 v
a  j     0 
12
r
22
r
s
s
(2.1)
s
r
where:
a
11
a
b
21
s
1 

  Rs 
 L
 r
s



L

m




a
12
L
 L L
m
s
a
r

22

r
r
1

r
1
L
s
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Rs , Rr
Ls , Lr
Lm
ωr
τr
ρ
σ
Page 6 of 28
: stator, rotor resistance per phase respectively
: stator, rotor inductance per phase respectively
: magnetizing inductance per phase
: rotor angular speed
: rotor time constant (= Lr / Rr)
: Lm / σ Ls Lr
: leakage constant (= 1 – Lm2 / Ls Lr)
The input and state variables are as follows,
stator current
: is = isα + j isβ
stator voltage
: vs = vsα + j vsβ
rotor flux
: Φr = Φrα + j Φrβ
2.2.2 Power performance
Figure 1. Per phase equivalent circuit of polyphase Induction Machine
Where:
U1 = stator terminal voltage
E1 = stator emf generated by resultant air-gap flux
R1 = stator effective resistance
X1 = stator leakage reactance
Rm = iron core-loss resistance
Xm = magnetizing reactance
R'2 = rotor effective resistance referred to stator
X'2 = rotor leakage reactance referred to stator
urb = e.m.f due to the saturable iron bridges in the rotor slots
I0 = sum of magnetizing I0X and core-loss I0R current components
I1 = stator current
I´2 = rotor current referred to stator
Some of the important steady-state performance characteristics of a polyphase induction
motor include the variation of current, speed, and losses as the load-torque requirements
change, and the starting and maximum torque. Performance calculations can be made from
the equivalent circuit. All calculations can be made on a per-phase basis, assuming balanced
operation of the machine. Total quantities can be obtained by using an appropriate
multiplying factor.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 7 of 28
The equivalent circuit of (Fig. 1), is usually employed for the analysis. The core losses, most
of which occur in the stator, as well as friction, windage, and stray-load losses are included in
efficiency calculations. The power-flow diagram for an induction motor is given in (Fig. 2), in
which m1 is the number of stator phases, Φ1 is the power-factor angle between U1 and I1, Φ2 is
the power-factor angle between E1 and I2´, T is the internal electromagnetic torque developed,
ωs is the synchronous angular velocity in mechanical radians per second, and ωm is the actual
mechanical rotor speed given by ωs(1 - S).
The total power Pg in watts transferred across the air gap from the stator is the difference
between the electrical power input Pi and the stator copper loss. Pg is thus the total rotor input
power, which is dissipated in the resistance R2´ / S of each phase so that
R
P  m I '2 S
2
g
1
'
2
 T s
(2.2)
where T is the internal electromagnetic torque developed by the machine, and ωs is the
synchronous angular velocity in mechanical radians per second. Subtracting the total rotor
copper loss, which is m1(I2´)2R2´=SPg, from (Eq. 2.2) for Pg, we get the internal mechanical
power developed:
Pm  Pg 1  S   T  m  m1
  R 1 S S 
'
I2
2
'
2
(2.3)
This much power is absorbed by a resistance of R2´(1-S)/S, which corresponds to the load.
From (Eq. 2.3), we can see that, of the total power delivered to the rotor, the fraction (1-S) is
converted to mechanical power and the fraction S is dissipated as rotor copper loss. We can
conclude then that an induction motor operating at high slip values will be inefficient.
The total rotational losses including the core losses can be subtracted from Pm to obtain
the mechanical power output Po that is available in mechanical form at the shaft for useful
work:
(2.4)
Po  Pm  Prot  T o m

The per-unit efficiency of the induction motor is then given by

P
P
o
(2.5)
i
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 8 of 28
Figure 2. Power flow in an induction motor
2.3 Speed control systems for Induction Motors
The angular speed in rad / s of an induction AC machine mechanical speed is given by
  1 s
m
(2.6)
s
this shows us that there are two possibilities for speed regulation of an induction motor,
altering the slip s (typical < 5%), or the synchronous speed ωs . When the motor is connected
to mains with constant frequency the speed will be determined by the point of intersection
between the motor and the load torque characteristic
Figure 3. Typical motor torque - load characteristic
Ignoring the stator resistance and the magnetizing impedance in the equivalent circuit model
of an induction motor, which is usually a valid approximation for motors, gives a simple
equation for the shaft torque in Nm
T
e
 2T e max
ss
s s
m
(2.7)
m
where the maximal torque and corresponding slip is given by
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 9 of 28
2
T
e max

3U1
2 s X 
(2.8)
respectively
'
s R
X
2
m
(2.9)
3 Analysis of problem
3.1 Comparison current ↔ new method
Try to find a suitable method making it possible to fulfill the needs of LSI Svenska AB.
3.1.1 Properties method currently used
Benefits:
 Simple circuit
 Few components
 Low component cost
 Reliable
Disadvantages:
 Excessive heat build-up
 Limited current (due to heat and losses)
 Needs extra hardware mounted on the machine
3.1.2 New method requirements
 16 A current supplied to the IM
 Higher starting torque




3-phase output from inverter
Limited power loss (less heat)
Must handle EMC
Must compile with regulations
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 10 of 28
3.2 Model / method
3.2.1 Creating a schematic in Pspice
C1
100nF
B1
V1
F1
1
L1
2
1
F2
2
1.0mH
6.3A
128
V2
1Vac
0Vdc
D1
C4
IM
IM
68nF
R3
10k
P1
1Meg
P2
2.2Meg
R2
10k
R1
C2
330
150nF
P1 Potentiometer for speed regulation
P2 Potentiometer for min current adjustment
Figure 4. Schematic LSI speed regulator
3.2.2 Studying different control techniques
 VOLTAGE CONTROL (the model used by LSI Svenska AB)
Unfortunately this is not an effective control. As the voltage decreases, the torque
decreases (the torque developed in an induction motor is proportional to the square of the
terminal voltage). Practically this is confined to 80-100% control.
 FREQUENCY CONTROL
This is by far the most efficient way to control the speed. However, one has to make sure
that the machine does not saturate. Since the flux is proportional to V/f, this control has to
assure that the magnitude of the voltage is proportional to the speed. Power electronic circuits
are best suited for this kind of control.
 VECTOR CONTROL
The magnetizing current always lags (inductive) the voltage by 90° and the torque
producing current is always in phase with the voltage. In vector control the magnetizing
current (Id) is controlled in one control loop and the torque producing current (Iq) in another.
The two vectors Id and Iq which are always 90° apart, are then added (vector sum) and sent to
the modulator, which turns the vector information into a rotating PWM modulated 3-phase
system with the correct frequency and voltage. This will reduce torque pulsation and a robust
control with fast dynamic response for the induction motor is achieved.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 11 of 28
 CHANGING STATOR POLES
For a stator witch has several independent windings, one can connect them is series for
starting, essentially building N*poles. The speed of the machine will be reduced by the same
factor. As the machine speed increases, one can switch the stator connection to a parallel
connection, hence reducing the amount of poles and hence accelerating the machine. This
method is simple, but can really accommodate only 2 speeds.
 ROTOR RESISTANCE
As seen for the starting, one can insert a variable resistance in the rotor (slip rings) and
hence cause the developed torque to vary, hence control the speed.
 DOUBLY FED MOTOR
A special application can be to inject a current in the rotor. Hence the air gap flux will
depend upon the difference of frequency between stator and rotor currents, and therefore the
speed can be varied by varying the rotor frequency.
 KRAMER CIRCUIT
With the method of variable resistor in the rotor circuit, a lot of power is dissipated in this
additional resistor. With the Kramer method, one takes the rotor windings, and feed a 3-phase
rectifier. This DC voltage is then fed through an inverter back to the source. Here only the
component losses are accounted for. The excess power not transformed in mechanical torque
will be fed back to the source.
4 Solution
4.1 Deciding what model to be used
The most frequently used three-phase inverter circuit consists of three legs, one for each
phase, as shown in (Fig. 5). The output of each leg, for example VAN (with respect to the
negative dc bus), depends only on Vd and the switch status the output voltage is independent
of the output load current since one of the two switches in a leg is always on at any instant.
Here, I ignore the blanking time required in practical circuits by assuming the switches to be
ideal. Therefore, the inverter output voltage is independent of the direction of the load current.
Figure 5. 3-phase inverter
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 12 of 28
4.1.1 Three-phase voltage source inverters with sinusoidal PWM
Most motors are designed to for sine wave AC supply and the inverter output should be as
near to sinusoidal as possible. It is therefore best to choose the control wave with sine shape
to give a PWM pattern in which the pulse width is sinudosially modulated throughout the half
cycle. Pulse-width-modulated three-phase inverters shape and control the three-phase output
voltages in magnitude and frequency with an essentially constant input voltage Vd. To obtain
balanced three-phase output voltages in a three-phase PWM inverter, the same triangular
voltage waveform is compared with three sinusoidal control voltages that are 120° out of
phase, as shown in (Fig. 7) (which is drawn for modulation factor mf = 15).
m
f

f
f
tri
(4.1)
control
It should also be noted from (Fig. 7b) that an identical amount of average DC component
is present in the output voltages VAN and VBN, which are measured with respect to the negative
DC bus. These dc components are canceled out in the line-to-line voltages, for example in VAB
shown in (Fig. 7b).
In a three-phase inverter, only the harmonics in the line-to-line voltages are of concern.
The harmonics in the output of any one of the legs, for example VAN in (Fig. 7b), only the odd
harmonics exist as sidebands, centered around mf and its multiples, provided mf is odd. Only
considering the harmonic at mf (the same applies to its odd multiples), the phase difference
between the mf harmonic in VAN and VBN is (120 mf)°. This phase difference will be equivalent
to zero (a multiple of 360°) if mf is odd and a multiple of 3. As a consequence, the harmonic
at mf is suppressed in the line-to-line voltage VAB, The same argument applies in the
suppression of harmonics at the odd multiples of mf if mf is chosen to be an odd multiple of 3
(where the reason for choosing mf to be an odd multiple of 3 is to keep mf odd and, hence,
eliminate even harmonics). Thus, some of the dominant harmonics in the one-leg inverter can
be eliminated from the line-to-line voltage of a three-phase inverter.
Sinusoidal PWM considerations are summarized as follows:
1. Small mf (mf ≤ 21): To eliminate the even harmonics, a synchronized PWM should be
used and mf should be an odd integer. Moreover, mf should be a multiple of 3 to cancel
out the most dominant harmonics in the line-to-line voltage.
2. Large mf (mf > 21): The amplitudes of subharmonics due to asynchronous PWM are
small at large values of mf. Therefore, at large values of mf the asynchronous PWM
can be used where frequency of the triangular waveform is kept constant, whereas the
frequency of vcontrol varies, resulting in noninteger values of mf (so long as they are
large). However, if the inverter is supplying a load such as an ac motor, the
subharmonics at zero or close to zero frequency, even though small in amplitude, will
result in large currents that will be highly undesirable. Therefore, the asynchronous
PWM should be avoided.
3. Overmodulation (ma > 1.0): Regardless of the value of mf, the conditions pertinent to a
small mf should be observed.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 13 of 28
Modulation Index
The average output voltage can be affected by changing the modulation index, defined as
voltage ratio between the fundamental of the control wave to the fundamental of the nonmodulated carrier wave as ma modulation index (fig. 7a)
m
a

V
V
control(1)
(4.2)
tri(1)
Linear Modulation (ma < 1.0)
In the linear region (ma < 1.0), the fundamental-frequency component in the output
voltage varies linearly with the amplitude modulation ratio ma (fig. 6). The peak value of the
fundamental frequency component in volts in one of the inverter legs is
(Vˆ AN )  m V2
1
d
(4.3)
a
Therefore, the line-to-line rms voltage at the fundamental frequency, due to 120° phase
displacement between phase voltages (ma ≤ 1.0), can be written as
V
LL

3
(Vˆ AN )1  3 maV d  0.612 maV d
2
2 2
(4.4)
The harmonic content for the phase to phase voltage will be strongly dependent on the
modulation index ma. A Fourier analysis on the output phase-to-phase waveform and in
general the harmonic spectra will be given from
h 4
V  n


An m1  n d sin  2

h 4
V  n


Bn m1  n d sin  2

 


 cos n m  
2 


(4.5a)
 


 sin n m  
2 


(4.5b)
With n=kmf ± m k= 1, 2, 3, ... and m can be odd or even. These rms harmonic voltages are
listed in Table 1 below.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
ma
Page 14 of 28
0.2
0.4
0.6
0.8
1.0
0.122
0.010
0.245
0.037
0.367
0.080
0.116
0.200
0.227
0.027
0.085
0.007
0.096
0.124
0.029
0.005
0.021
0.490
0.135
0.005
0.192
0.008
0.108
0.064
0.064
0.051
0.010
0.612
0.195
0.011
0.111
0.020
0.038
0.096
0.042
0.073
0.030
f
1
mf ± 2
mf ± 4
2mf ± 1
2mf ± 5
3mf ± 2
3mf ± 4
4mf ± 1
4mf ± 5
4mf ± 7
0.100
Table 1. Harmonics of VLL for a large and odd mf (mf a multiple of 3)
Overmodulation (ma > 1.0)
In PWM overmodulation, the peak of the control voltage is allowed to exceed the peak of
the triangular waveform. Unlike the linear region, in this mode of operation the fundamentalfrequency voltage magnitude does not increase proportionally with mu. This is shown in Fig.
3, where the rms value of the fundamental-frequency line-to-line voltage VLL is plotted as a
function of ma. Similar to a single-phase PWM, for sufficiently large values of ma, the PWM
degenerates into a square-wave inverter waveform. This results in the maximum value of VLL
equal to 0.78 Vd (fig. 6).
In the overmodulation region compared to the region with ma < 1.0, more sideband
harmonics appear centered around the frequencies of harmonics mf and its multiples.
However, the dominant harmonics may not have as large amplitude as with ma < 1.0.
Therefore, the power loss in the load due to the harmonic frequencies may not be as big in the
overmodulation region as the presence of additional sideband harmonics would suggest.
Depending on the nature of the load and on the switching frequency, the loss due to these
harmonics in overmodulation may be even less than those in the linear region of the PWM.
Disadvantages with sinusoidal PWM
The biggest disadvantages with sinusoidal PWM are that the liner range is relatively
small. There are many ways to extend it
 A third harmonic is added to the sine wave to make the waveform more flat topped.
Adding a third harmonic does not constitute a problem as the third harmonic and
multiples thereby will not been seen in the line-to-line voltage.
 The sine wave is replaced by a trapezoid or staircase wave to a flat top reference wave.
 The carrier wave is only applied during the first and last 60◦ intervals per half cycle,
e.g. 0◦ to 60◦ and 120◦ to 180◦.
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 15 of 28
Figure 6. Three-phase inverter VLL/Vd as a function of ma
Figure 7. Three-phase sinusoidal PWM waveforms and harmonic spectrum
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 16 of 28
To me the frequency control model with flux proportional to V/f and voltage proportional
to the speed seems to bee the best solution. Building a solution with a rectifier, a DC link with
LP-filter and a three-phase Pulse With Modulated (PWM) inverter feeding the motor. At first
testing this approach with a computer model built in Simulink (Matlab) making simulations of
the solution possible.
Figure 8. Voltage - frequency relation Induction Machine
4.2 Building a model
4.2.1 Simulation with Simulink / Matlab
4.2.1.1 Dynamic model for the IM
Let us first consider the stator circuit. The resistance Rs of the stator winding is (for all
practical purposes) equal in all three phases. From the law of induction it follows that the part
of the stator voltage which is not dissipated in the stator resistance will build up a flux in the
stator winding. Hence, with vss as the stator voltage space vector, the following relation must
hold:
v R i
s
s
s
s s

d
dt
s
s
0
(4.6)
where iss and ψss are the space vectors for stator current and stator flux linkage respectively.
The rotor circuit, with winding resistance Rr, can be treated in a similar way. Suppose that the
rotor is observed from a coordinate system (rotor coordinates) which rotates with the same
speed as the rotor ωr. Let us denote rotor coordinates with superscript "r". As the coordinate
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 17 of 28
system is rotor-fixed, there will be no induced voltage due to the rotation, so the same relation
as for the stator must hold, but with “s → r”:
v R i
r
s
r
r s

d
r
r
dt
0
(4.7)
Here vrr, irr and ψrr are the rotor voltage, current, and flux space vectors respectively. But the
rotor winding is short-circuited, so vrr = 0. Now, let us transform irr and ψrr to stationary
coordinates. This is a αβ transformation using the rotor position θr = ∫ ωr dt:
i e i ,  e
s
j
r
r
r
s
r
r
j
r
r
r
(4.8)
Equation (4.7) is transformed as
s
j
d  e  r 
r
j
s
0  Rr e  r i r  
0
dt

  j r s  
d

e  r
s
j
s
j
 Rr e  r ir    j  r e  r 
0
r
dt




j  r  Rr ir 
s
s
r
d
(4.9)
s
r
dt
0
The induction motor is thus described by the following equations:
d
s
s
dt
d
dt
 vs  Rs i s
s
s
( stator)
(4.10)
s
r
 j  r  Rr ir
s
r
s
(rotor )
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 18 of 28
Figure 9. Induction machine
Let us now find a relation between the stator and rotor flux linkages. The rotor winding is
referred to the stator, i.e., the rotor winding is represented by coils in the α and β directions,
cf. (Fig. 9). Assuming linear magnetic conditions, the air gap flux ψαs can then be expressed
as

s

 Lm im ,
i  i i
s
s
s
s
m
s
r
(4.11)
where Lm is the mutual inductance between the stator and the rotor, which is also called the
magnetizing inductance, and ims, is the magnetizing current. The stator flux is the sum of the
air gap flux and the stator leakage flux, the latter which under linear magnetic conditions is
proportional to the stator current only. Similar reasoning for the rotor flux yields
 L i L i
 L i L i
s
s
m m
s
s
s
m m
r
s
sl s
(4.12)
s
rl r
where Lsl and Lrl are the stator and rotor leakage inductances, respectively. The leakage
inductances are typically 10% of Lm or less. Alternatively, with Ls = Lm + Lsl and Lr = Lm +
Lrt as the stator and rotor self-inductances, respectively, the relations can be expressed as
 L i L i
 L i L i
s
s
s
s
s s
m r
s
s
s
r
m s
r r
(4.13)
Combining (4.12) with (4.10), assuming constant inductances, yields
s
v R i L
s
s
s
s s
d is
sl
dt
s
 Lm
j   R i  L
r
s
s
r
r r
rl
d im
dt
0
dt
(4.14)
s
s
d ir
 Lm
d im
dt
0
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Induction Machine Speed Control
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Department of Computer Science and Electronics
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These equations describe the dynamic equivalent circuit depicted in (Fig. 10). As there are
three inductances configured in a “T”, this is known as the T-equivalent circuit.
Figure 10. Dynamic T-equivalent circuit for the induction motor
4.2.1.2 Simulink model
I developed an AC/AC converter with DC-link in Simulink / Matlab (Appendix 7.1) with
a diode rectifier and a 3-phase PWM inverter controlling both the frequency and magnitude of
the voltage output. The induction machine model was based on the equations of the dynamic
induction machine model. For generation of PWM pulses the technique shown in Fig. (7.a-b)
was used comparing sinusoidal control voltage (at the desired output frequency and
proportional to the output voltage magnitude) with a triangular waveform at a selected
switching frequency.
Figure 11. 3-phase PWM converter with DC-link
The harmonics in the output voltage appears as sidebands of the switching frequency and
its multiples in a PWM inverter. Therefore a high switching frequency results in an essentially
sinusoidal current (plus a superimposed small ripple at a high frequency) in the motor.
Since the ripple current through the dc bus capacitor is at the switching frequency, the
input dc source impedance seen by the inverter would be smaller at higher switching:
frequencies. Therefore, a small value of capacitance suffices in PWM inverters, but this
capacitor must be able to carry the ripple current. A small capacitance across the diode
rectifier also results in a better input current waveform drawn from the utility source.
However, care should be taken in not letting the voltage ripple in the dc bus voltage become
too large, which would cause additional harmonics in the voltage applied to the motor.
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
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In a PWM inverter output voltage, since the harmonics are at a high frequency, the ripple
in the motor current is usually small due to high leakage reactances at these frequencies. Since
these high-frequency voltage harmonics can have as high or even higher amplitude compared
to the fundamental-frequency component, the iron losses (eddy current and hysteresis in the
stator and the rotor iron) dominate. In fact, the total losses due to harmonics may even be
higher with a PWM inverter than with a square-wave inverter. This comparison would of
course depend on the motor design class, magnetic material property, and switching
frequency. Because of these additional harmonic losses, it is generally recommended that a
standard motor with a 5-10% higher power rating be used.
In a PWM drive, the pulsating torqueses developed are small in amplitude and are at high
frequencies (compared to the fundamental). Therefore, as shown in (Eq. 4.10), they produce
little speed pulsations because of the motor inertia.
Amplitude of speedrippl e  k
amplitude of torqueripple
ripple frequency  inertia
Parameter
Input power factor
P
WM
+
Square
Wave
-
(4.10)
C
SI
-
Torque pulsation
+
-
-
+
-
+
-
+
+
Multi motor capability
Regeneration
+
Short-circuit protection
-
-
+
+
Open-circuit protection
Ability to handle undersized
motor
Ability to handle oversized
motor
Efficiency at low speeds
Size and weight
+
+
+
+
-
-
-
-
+
+
+
+
-
Ride-trough capability
+
-
-
Table 2. Comparison of Adjustable Frequency Drives
___________________________________________________________
Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 21 of 28
4.2.1.3 Simulation results
Figure 12. Simulation of Induction Machine start
4.3 Analysis of results
The volt hertz (V/f) Pulse-Width-Modulation model seems to fulfill perhaps not all but
most of the requirement stated earlier in this document (3.1.2). Making an increase in the
current supplied to the induction machine up to 16 amperes possible without big losses in the
speed controller, thereby decreasing heat inside the enclosure. The 3-phase output gives a
better feed to the induction machine without extra components needed on the motor and also
produces a higher starting torque and reduced speed pulsation amplitude.
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 22 of 28
4.4 Future work
4.4.1 Implementation with Digital Signal Processor (DSP)
Traditionally motor control was designed with analog components; they are easy to
design and can be implemented with relatively inexpensive components. However, there are
several drawbacks with analog systems. Aging and temperature can bring about component
variation causing the system to need regular adjustment, as the parts count increase the
reliability of the system decreases. Analog components raise tolerance issues and upgrades
are difficult as the design is hardwired. Digital systems offer improvements over analog
designs. Drift is eliminated since most functions are performed digitally, upgrades can easily
be made in software and part count is also reduced since digital systems can handle several
functions on chip.
Digital Signal Processors go on further to provide high speed, high resolution and sensor
less algorithms in order to reduce system costs. Providing a more precise control to achieve
better consumption or radiation performances often means performing more calculations, the
use of some 1-cycle multiplication & addition instructions included in a DSP speeds-up
calculations.
Generally fixed point DSPs are preferred for motor control for two reasons. Firstly, fixed
point DSPs cost much less than the floating point DSPs. Secondly, for most application a
dynamic range of 16 bits is enough. If and when needed, the dynamic range can be increased
in a fixed-point processor by doing floating-point calculations in software.
5 Summary and conclusions
It seem to me that switching from “volt control” to “frequency control (volt / hertz)”
method would make it possible to achieve the increase in current supplied to the induction
motor requested by LSI Svenska AB without excessive heat buildup inside the speed
controller housing. 3-phase output from the PWM inverter will reduce speed pulsation and
produce a higher starting torque making it possible to reduce the size of the motor. There will
be no need for manual adjustment of each unit, but the component cost and technical
complicity are going in the wrong direction.
For the future Digital Signal Processors may be the best solution providing high speed,
high resolution and sensor less algorithms in order to reduce system costs. Upgrades can
easily be made in software and part count is also reduced since digital systems can handle
several functions on chip.
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 23 of 28
6 References
I. Sadarangani, C., “Electrical Machines”,
Royal Institute of Technology, Stockholm, Sweden, (2000)
II. Mohan N, Undeland, T. M. and Robbins, W. P., “Power Electronics”,
John Wiley & Sons Inc., USA, (2003)
III. Harnefors, L., “Control of Power Electronic Converters and Variable-Speed Drives”,
Mälardalen University, Västerås, Sweden, (1999)
IV. Slemon, G. R, “Electric Machines and Drives”,
Addison-Wesley Publishing Company, USA, (1992)
V. Sarma, M. S, “Electrical Machines, Steady-State Theory and Dynamic Performance”,
West Publishing Company, USA, (1994)
VI. El-Hawary, M. E, “Principled of Electric Machines with Power Electronic App.”,
John Wiley & Sons Inc, USA, (2002)
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 24 of 28
7 Appendix
7.1 Simulink model
Mag
In
Vab f und.
Phase
Terminator1
Discrete
Fourier
Selector
Mag
In
Discrete
Fourier1
+
v
-
Is f und.
Phase
Terminator
Scope
Fundamental
Vab
Vab
Tm
Tr1a
is_abc
Tr1a
A
Tr1b
Tr1b
Tr2a
Tr2a
Tr2b
Tr2b
A
Te
B
B
C
Vd-
Discrete
PWM Generator
4 pulses
Vd+
is_abc
is_abc
C1
IGBT Bridge
DC 400V
wr
Te
Scope
IM
wr
3 Phase IM
5.77e-4*u^2
Fcn
0
I_Sw1 I_Sw2
Multimeter
Scope
I_Sw1 I_Sw2
Continuous
pow ergui
Figure 13. Main Simulink model
1
Tr1a
Sign
2
Tr1b
Sinref
???
Triangular Wave
Sine Wave
To Workspace4
1
Constant
Tri
To Workspace5
3
Tr2a
4
Sign1
Tr2b
Figure 14. Discrete PWM Generator 4 pulses
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
2
B
Page 25 of 28
1
A
Tr1b
2
Tr1a
1
g
g
E
E
C
Tr2b
4
Tr2a
3
C
IGBT1
IGBT2
D1
D2
g
g
E
E
C
C
IGBT3
4
Vd+
IGBT4
D3
D4
3
Vd-
Figure 15. IGBT Bridge
A
1
B
is_abc
2
Va
C
Vb
3
Vc
Sub1
+
v
-
Vsre
Vab
Vs
+
v
-
is_abc
Re(u)
Vbc
is_abc
Im(u)
Vsim
wr
Tm
3
wr
Sub2
1
1
Tm
Te
Sub3
2
Te
Figure 16. Three-Phase Induction Machine (IM main)
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 26 of 28
s
+
1
is_abc
-
T erminator
1
Va
2
Vb
s
3
-
+
Vc
Figure 17. IM Sub. 1
1
Vab
f(u)
u K
f(u)
u K
2
Vbc
2/3
Gain
f(u)
1
Vs
u K
Figure 18. IM Sub. 2
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Induction Machine Speed Control
Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 27 of 28
f(u)
1
is_abc
f(u)
[wr]
Re(u)
Im(u)
[fisr]
Re
Im
is
FIs
[Te]
3
Te
[fisi]
Re(u)
Im(u)
2
wr
ir
[firr]
Re
Im
FIr
[Te]
Te
[firi]
Sub1
1
Vsre
Vsre
isre
rs
rs
rs
2
[fisr]
Vsim
Vsim
[fisi]
isim
[firr]
Out
irre
[firi]
rr
rr
[wr]
rr
irim
3
Tm
Tm
J
J
J
p
Sub2
[p]
p
Figure 19. IM Sub. 3
Product1
Lm
Lr
Procuct2
Lr
is
Divide
In1 Out1
f(u)
Ls
1
Product5
1
FIs
Conjugate
Product3
Ls
Im(u)
f(u)
[p]
Fcn1
3
Te
2
Divide1
ir
Product4
2
FIr
Figure 20. IM Sub. 3.1
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Lars-Göran Andersson Mälardalen University
Department of Computer Science and Electronics
Page 28 of 28
1
Vsre
2
3
isre
rs
5
f(u)
4
Vsim
f(u)
isim
6
7
irre
rr
[wr]
f(u)
x' = Ax+Bu
y = Cx+Du
[firi]
1
Out
[p]
8
irim
[wr]
f(u)
[firr]
[Te]
9
Tm
f(u)
10
J
Figure 21. IM Sub. 3.2
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