ELSEVIER
Electric Power Systems Research 31 (1994) 203 216
ELECTRIC
POWER
SW$'l'lm$
RE$1 I OI,I
Applications of the current state space model in analyses of
saturated induction machines
E. Levi
School of Electrical and Electronic Engineering, Liverpool John Moores University, Byrom Street, Liverpool L3 3AF, UK
Accepted 26 July 1994
Abstract
Main flux saturation in induction machines plays an important role in a number of operating regimes and therefore has to be
accounted for in the process of modelling. The current state space model of a saturated induction machine has gained substantial
popularity recently as an excellent tool for transient and steady-state time domain analysis of induction motor drives when the
main flux saturation has to be considered. This paper discusses different applications of the model. First, the self-excitation process
in single-cage and double-cage (deep-bar) induction generators is addressed. Switch-off and reclosing transients of an induction
motor with capacitor reactive power compensation are dealt with next. Two inverter fed induction motor drives, namely, the
current-source and resonant parallel inverter fed induction machines, are elaborated further. The last application described here
is the rotor flux oriented control of an induction machine. The elaborated cases are illustrated with simulation and experimental
results.
Keywords: Induction machines; Saturation modelling
I. Introduction
Classical general theory of electric machines assumes that the non-linear magnetizing curve of a machine can be approximated by a straight line through
the origin. Thus, the magnetizing (main, air-gap) flux
is supposed to vary linearly with the magnetizing current. This assumption turns out to be completely unsatisfactory for modelling and simulation of a number
of steady-state and transient regimes of both induction
and synchronous machines. F r o m a historical point of
view, the need to incorporate somehow the real shape
of the magnetizing curve within the frame of the general theory first occurred in studies of salient-pole synchronous generators [1]. With regard to induction
machines fed from the mains, the only operating
regimes where a need for saturation modelling has
been experienced in the past are switch-off and reclosing transients of an induction m o t o r with reactive
power compensation [2]. The influence of main flux
saturation on other transients of induction machines
fed from the mains supply is in general small, the
exception being the reclosing transient of an uncompensated machine with a large phase difference between the mains voltage and the induced electromotive
0378-7796/94/$07.00 ,~ 1994 Elsevier Science S.A. All rights reserved
SSDI 0378-7796(94~00887-A
force [3]. For example, a detailed study of a start-up
process [4] shows that the main flux saturation effect
can be fully neglected. However, developments in converter fed induction m o t o r drives and in applications
of induction machines as generators in autonomous
power systems have led to a significant increase in
interest in the possibilities of including the main flux
saturation effect in models of induction machines. The
reason for such a trend is rather simple: these new
technologies have introduced new applications of induction machines where the omission of the main flux
saturation effect from the model leads to either inaccurate results, or, even worse, makes simulation and
performance prediction impossible.
All the existing methods of saturation modelling
utilize the assumption that the stator and rotor flux
components along the d and q axes can be resolved
into sums of leakage flux and main flux components.
Saturation of leakage and main flux paths is treated
independently further on. Main flux saturation is far
more important than leakage flux saturation. Saturation of leakage paths is usually of interest in analyses
associated with induction motors fed from the mains,
where during start-up and reversing transients large
currents lead to significant saturation of the leakage
204
E. Levi~Electric Power Systems Research 31 (1994) 203 216
paths. F r o m the modelling point of view, leakage flux
saturation is a simple task and the models that account for it are readily available and simple to use
[5,6].
In general, two c o m m o n approaches to main flux
saturation modelling can be identified. The first utilizes d,q-axis flux components as state space variables,
while the other relies on currents as state space variables. The latter approach is the subject of this paper.
Although the flux state space model is slightly simpler
to use, its main drawback is that it conceals the effect
of cross-saturation which is, however, implicitly accounted for. On the other hand, the current state
space model contains explicit terms that describe the
cross-saturation effect and therefore a better physical
insight into the overall behaviour of saturated induction machines is achieved. Still, it should be noted
that the corresponding flux state space model of a
saturated induction machine may be utilized instead,
without any loss of accuracy in the simulation results.
The cross-saturation effect is the usual way of denoting the mutual influence which occurs between two
perpendicular axes, even in a uniform-air-gap machine
under saturated conditions, and is purely due to main
flux saturation. More specifically, a change of saturation level in one axis causes a corresponding change
of flux level in the other, perpendicular, axis, and this
is usually named cross-saturation or cross-magnetization. The cross-saturation effect was recognized a long
time ago [1], but it was not until recently that it
became the subject of wider interest. Cross-saturation
between two perpendicular axes in smooth-air-gap
machines has been studied in detail in a number of
papers [7-16]. The existence of this effect has been
verified in different ways, applying a variety of theoretical and experimental approaches. The approach in
Ref. [9] relies on magnetic co-energy, while that in
Ref. [10] uses different starting premises but yields the
same results. The finite-element method enabled a detailed study of the electromagnetic fields in the machine and proved the existence of cross-saturation as
well [13]. Sophisticated experiments, specially designed
in order to achieve an experimental p r o o f of the
cross-saturation effect, are analysed in Ref. [8,12,15].
In simulation examples [17,18] which refer to an induction generator's self-excitation, the influence of
cross-saturation is separated from other effects and
shown explicitly.
The idea of modelling a saturated induction machine with currents as state space variables stems
from the work of Kovfics [19-21], where a special
common reference frame fixed to the magnetizing current (or main flux) space vector is selected. This current state space model is derived by means of space
vector theory and is a special case of a more general
model of a saturated induction machine in an arbi-
trary common reference frame [22]. An important
feature of the current state space model of a saturated induction machine in a common frame fixed to
the magnetizing current space vector is the absence of
the terms which describe cross-saturation, this being a
consequence of the specific choice of the common reference frame. The above-mentioned general current
state space model in an arbitrary frame of reference
[22,23] contains explicit terms which account for
cross-saturation. The origin of the more general
model can be found in an earlier paper by Kovfics
[24]. Unfortunately, due to an incorrect mathematical
derivation, the resultant model of a saturated induction machine given in Ref. [24] is not correct. The
original intention to develop a saturated induction
machine model in Ref. [24] came from the need to
model and simulate the behaviour of a selfexcited induction generator. The behaviour of an induction generator which operates in an autonomous
power system and is equipped with capacitors which
provide reactive power has been brought into the
scope of research due to the development of small
autonomous electric energy plants. The self-excitation
process of such an induction generator cannot be
modelled and simulated unless the main flux saturation is encompassed by the model. The standard linear models of induction machines simply fail in this
case.
The generalized analysis of main flux saturation
effects in orthogonal-axis models of electric machines
[25-28] presents as a final result a general model of
saturated electric machines, with currents as state
space variables, which accommodates induction, synchronous and DC machines. Therefore, it easily reduces to the saturated induction machine model of
Refs. [22,23]. Consequently, the current state space
model of a saturated induction machine can be regarded as just a special case of the general current
state space model of a saturated electric machine.
The current state space model of saturated induction machines for small-signal analysis [29] is just a
special case of the corresponding large-signal model
discussed so far. This model is commonly applied to
determine the stability limits of induction machines
under different operating conditions [30- 34].
The large-signal current state space model of saturated induction machines has recently been developed
for double-cage [18,35] and deep-bar [36] induction
machines as well. In these cases the principle of main
flux saturation modelling remains the same as for
single-cage machines but the overall complexity of the
model increases significantly due to addition of new
differential equations.
The influence of main flux saturation on the behaviour of an induction machine fed from the mains
is, as has already been stated, very small and can
E. Levi/Electric Power Systems Research 31 (1994) 203-216
usually be neglected. A similar conclusion holds true
if the induction machine is fed from a voltage-source
inverter, provided that the voltage to frequency ratio
is held constant and that the magnetizing (mutual)
inductance value in the linear model corresponds to
the steady-state saturated value of the magnetizing inductance at the operating point of the magnetizing
curve. If the machine is fed from a current source,
the main flux saturation effect tends to be much more
pronounced [18] and influences the drive behaviour
more significantly.
Summarizing the cases where main flux saturation
has to be accounted for, it may be stated that the
need for a saturated induction machine model arises
in two distinct types of application. The first type
encompasses all the situations where the induction
machine is not fed from a power electronic converter,
there is a capacitance or three-phase bank of capacitors connected to the machine terminals and mains
voltages are usually absent for one reason or another.
More specifically, self-excitation of a single-cage
[18,37-39] or double-cage [40,41] induction generator
with terminal capacitors, operation of a compensated
induction generator in an autonomous power system
[18,38], switch-off and reclosing transients of a compensated induction motor [2,42], and three-phase
symmetrical or single-phase asymmetrical capacitor
braking of induction motors [18,35] all belong to this
type of application. One specific case where mains
voltages are present and there is still a need for inclusion of main flux saturation in the model is subsynchronous resonance instability in an induction motor
supplied through a capacitor compensated feeder [18].
Here the inclusion of the saturation effect is important only if the unstable operation does take place.
The second type includes all the converter fed induction motor variable-speed drives where the source
may be treated as a current source. The currentsource inverter fed induction machine [43,44] and the
resonant parallel inverter fed induction machine [4547] are two typical cases with simple control algorithms, while a field oriented induction machine fed
from a current controlled pulse-width modulator
(PWM) inverter represents a further example [48-51],
this time with a rather sophisticated control structure.
It is worth noting here that in the latter case the
current state space model of a saturated induction
machine has given excellent physical insight into the
mechanism of the influence of saturation on the behaviour of vector controlled induction machines. As
shown in Refs. [48,50], decoupled rotor flux and
torque control in a rotor flux oriented induction machine is lost during transient operation in the constant-flux region purely due to the cross-saturation
effect. However, it is worth noting that compensation
of the main flux saturation effect in vector controlled
205
induction machines is most easily achieved by application of the flux state space model of a saturated
induction machine [52,53].
Most of the applications listed above are elaborated in this paper. More specifically, self-excitation
in single-cage and double-cage induction generators,
switch-off and reclosing transients of a compensated
induction motor, induction motor drives fed from
current-source inverters and from resonant parallel inverters, and finally the current fed rotor flux oriented
induction machine are discussed. The second section
presents a brief overview of the current state space
model of a saturated induction machine. The third
section deals with self-excitation in induction generators and transients associated with the compensated
induction motor. The fourth section is devoted to
variable-speed drives and encompasses the three previously mentioned cases. Emphasis in the paper is
placed on simulation and experimental results, rather
than on detailed mathematical modelling. A more detailed theoretical treatment is available in the references listed above for all the applications discussed.
2. Review of a saturated induction machine model
with currents as state space variables
It is assumed that the machine has P pairs of
poles, mechanical losses are neglected, power invariant transformation is applied and the model is given
in an arbitrary frame of reference whose angular velocity is O~a, in matrix form as
v = L di/dt + Bi
( 1)
Te - TL = ( J / P ) dco~/dt
(2)
Eq. (1) can be expressed in state space form, with
currents as state space variables, as
d i / d t = L-~v -- L - I B i
(la)
The matrices and vectors contained in Eq. (1), as well
as the expression for the induction torque Te in Eq.
(2), differ depending on whether the machine is single
cage or double cage. For this reason these two cases
are addressed separately.
2.1. Single-cage induction m a c h i n e
The matrices and vectors of Eq. (1) and the torque
are given for the single-cage machine by the following
expressions:
V = [V~s, % , 0, 0]'
i = [ids, iqs, iar, iqr] t
(3)
206
E. Levi~Electric Power Systems Research 31 (1994)203 216
[~
s
B =
--OOa(Lm + Lw)
a(tm + Lw)
R~
--(0)
- °)r)Lm
[
a --
¢o~)Lm
0
Lv~ + Lddm Ldq
L=
Lddm
Lv~ + Lqqm Ldq
Ldq
Lddm
L Ldq
Ldq
Lqqm
0
-oaLm
o)~,Lm
0
-(toa - ~r)(Lm + L w)
R~
R~
((oa - O)r)(Lm + Lv~)
Ldq
Lqqm
(4)
]
(5)
Lvr -~ Lddm Ldq
Ldq
L°Ir -- Lqqm
T e = PLm(idriqs -- iqrids )
(6)
Indices s and r denote stator and rotor parameters and
variables, respectively, index 7 stands for leakage inductances, and index m defines parameters and variables
associated with magnetizing flux and magnetizing current. The inductance terms in Eq. (5) are determined as
both the steady-state magnetizing inductance and the
dynamic inductance are variables and depend on the
instantaneous value of the magnetizing current and the
corresponding value of the magnetizing flux:
Lddm = L cos2# + L~ sin2#
(7a)
Lqqm = L sin2# + Lm cos2#
(7b)
The magnetizing current and magnetizing flux d,q-axis
components are
Ldq = (L -- Lm) COS # sin #
(7c)
idm=ids-{-idr,
iqm=iqs+iqr
(lla)
~/dm = Lm idm,
IPqm = Lmiqm
(llb)
where L is the dynamic (tangent, slope) inductance
defined as
L =dqJm/dim
where
L = L(im)
(8)
ff/m= (Odin2 -~ ~/qm2)1/2,
i m = ( i d m 2 + i q m 2 ) '/2
(10)
The model given in Eqs. (1)-(11) is the current state
space model which completely describes the single-cage
saturated induction machine.
Lm is the steady-state saturated magnetizing inductance
Lm = ~lm/im
where
2.2. Double-cage induction machine
L m = Lm(im)
(9)
and # is the angle between the d axis of the common
reference frame and the magnetizing current (flux)
space vector whose real and imaginary components are
idm and iqm (@dm and ~/q,n), respectively. Due to the
non-linearity of the magnetizing curve of the machine,
The matrices and vectors present in Eq. (1) take the
following form for the case of the double-cage induction machine:
13: [Uds, Vqs, 0, 0, 0, 0] t
i = [ids, /',.is, id,.1, iqrl, /'dr2, iq,.2] i
-R~
e)~L~
B=
- co~L~
R~
0
-
eJ, Lm
(o~l L m
0
0
e)aLm
--O)slL m Rrl + R~
-(Osl(Lm + Lvrl +Lmr)
O)slLm
0
O)sl(Lm 4- Lvrl + tmr )
Rrl + R c
0
-- O)slL m R~
--¢~si(Lm + Lmr)
(o~lLrn
0
O)sl(Lm + Lmr)
R~
0
Rc
O)sl(Lm + Lmr)
Rr2 + R~
~Osl(Lm + Lvr2 + Lmr)
(12)
-
oJaLm
0
-eOsl(Lm + Lm,.)
Re
-~)~I(Lm + Lw2 + L,~,~)
R~2 + Rc
(13)
-LvS+ Lddm Ldq
Ldq
Lvs -{- Lqqm
Lddm
Ldq
L =
Ldq
Lqqm
Lddm
Ldq
Ldq
Lqqm
Lddm
Ldq
Ldq
Lddm
Lqqm
Ldq
Lvrl -~- Lmr ~- Lddm Ldq
Lmr + Lddm
Ldq
Lvrl + Lmr + Lqqm Ldq
Lmr -- Lddm
Ldq
Lvr2 + Lmr ~- tdd m
Ldq
Lmr -]- Lddm
Ldq
Ldq
Lqqm
Ldq
Lmr -t- Lqqm
Ldq
Lyr2 + Lmr + Lqqm
(]4)
E. Levi~Electric Power Systems Research 31 (1994) 203 216
0.80
E
mted
~.emt/ng
potnt
207
cage and double-cage induction generators, respectively. The third case encompasses switch-off and reclosing transients of an induction motor with a parallel
reactive power compensator consisting of a three-phase
capacitor bank.
PsIm
0,60
3.1. Self-excitation in single-cage induction generators
O.dO
Eo.20 /
_J
0.1~
o 6 ......
I
'o'.~ .......
1:66 . . . . . . .
Im
I'.~ .......
2:6o
Fig. I. Magnetizing curve of Machine A, together with the steadystate magnetizing inductance and dynamic inductance variation
(Psim _-- ~m; q',~ and Im are given in terms of r.m.s, values).
It is assumed that the induction generator operates at
constant speed and that there is no load connected to the
output stator terminals. A three-phase capacitor bank is
connected in parallel to the machine. Initiation of selfexcitation is based on remanent magnetism and therefore an initial non-zero value of the magnetizing current
is needed in the model in order to start self-excitation.
This value is easily obtainable by running the machine
at constant speed and measuring the remanent voltage.
The additional equations that describe the capacitor
bank are
dvds/dt = (J)aUqs ids/C
(17a)
dvqs/dt = - ( D a / ) d s - - iqs/C
(17b)
-
Te = PL,,[(idr, + idr2)iqs -- (iqr, + iqr2)ids]
(15)
In Eqs. (12)-(15) index 1 denotes the upper rotor cage,
while index 2 stands for the lower rotor cage. The slip
angular frequency is introduced in Eq. (13) as ogsj=
~oa - ~ o r and the inductance term L~ as Ls = Lv~ + L m.
Eqs. ( 7 ) - ( 1 0 ) remain the same as for the single-cage
machine. The magnetizing current components given in
Eq. ( l l a ) for the single-cage machine now become
idm = ids + idr, + idr2,
iqm = iqs + iqr] + iqr2
(16)
The resistance R c in Eq. (13) is the common end-ring
resistance between the two cages and may or may not
be present depending on the construction of the machine. Inductance L m r denotes the mutual leakage inductance between the two rotor cages.
The data for all the induction machines employed in
the simulation and experiments are given in the Appendix. The magnetizing curve of one of the machines
(Machine A) is shown here, together with variations of
the steady-state magnetizing inductance and dynamic
inductance (Fig. 1). By inspecting Fig. 1 one easily
concludes that the difference between the steady-state
magnetizing inductance and the dynamic inductance is
considerable over a major portion of the magnetizing
curve; consequently, inductance Ldq , defined in Eq. (7),
which describes the cross-saturation phenomenon, will be
non-zero as well and every attempt to ignore cross-saturation will lead to incorrect simulation results [17,18].
3. Induction machine with reactive power compensation
Three different cases are discussed in this section, the
first and the second being self-excitation in single-
-
As constant-speed operation is analysed here, the equation of mechanical motion (2) may be omitted. Hence,
Eqs. (la), ( 3 ) - ( 1 1 ) and (17) constitute the required
mathematical model.
The star connection of the capacitor bank and induction generator stator windings was analysed experimentally and by simulation for Machine A with a per-phase
capacitance of 25 jaF. The value of the capacitance was
chosen to provide steady-state operation at the rated
point on the magnetizing curve. The speed was held
constant at a synchronous value of 50 Hz ( 1500 rpm).
Fig. 2 shows the experimentally recorded phase voltage
and current build-up (note that in Fig. 2(a) the steady
state has not yet been reached; the steady-state voltage
peak value equals 307.2 V). The corresponding simulation results are displayed in Fig. 3. Comparison of the
experimental and simulation results shows pretty good
agreement.
3.2. Self-excitation in double-cage induction generators
The model needed in this case encompasses Eqs.
(la), (7)-(10), ( l l b ) and (12)-(17). Once more selfexcitation at constant speed was analysed under noload conditions and Machine B was utilized in simulations and experiments. The machine stator windings
and capacitor bank were delta connected this time. The
capacitor value was chosen in the same way as for the
single-cage machine. The experimental and simulation
results of the voltage build-up at a speed of 3180 rpm
with capacitors of 24.5 ~tF capacitance are summarized
in Fig. 4. Once more the results are in satisfactory
agreement, although the voltage build-up is slightly
208
E, Levi / Electric Power Systems Research 31 (1994) 203 216
400.00
200.00
o
o
>
~
0.00
rQ
I -2~.oo
2
(a)
-400,00
0.
~' .... ~6b'.~6'"%~'.66'"'~66'.&'"'~66'.66'"~b~.0o
time - t ( m s )
3.00
<
v
oJ 1.oo
~-1.~
Fig. 2. Experimentally recorded phase voltage and current build-up
during single-cage induction generator self-excitation (current recorded
as voltage drop on a resistor of 0.02 fl resistance).
•
(b)
I'l'lIVi'll''llJ
200.00
''lll'''ll,',l,'lll
,oo.oo
time
faster in the initial part of the transient in the experiment compared with that obtained by the simulation.
3.3. Switch-off and reclosing transients of an induction
motor with parallel reactive power compensation
In this example a single-cage induction motor operates on the mains and its reactive power consumption is
compensated by a three-phase capacitor bank connected
in parallel to the stator terminals. During normal operation the capacitor voltages are dictated by the mains;
however, if the machine is switched off from the supply,
capacitor equations are again needed. Therefore the
model of the machine consists of Eqs. (la), (2) (11) and
(17) for the analysis of the switch-off transients (a
single-cage machine is discussed here), while during
start-up and for analysis of the reclosing transients Eq.
(17) is omitted from the model. It is worth noting that
the switch-off transient will fully correspond to the
so-called symmetrical capacitor braking [18] if the machine is not reconnected to the supply, provided that the
value of the capacitors is appropriately selected and zero
initial conditions are used for capacitor voltages.
-
6o0.00
I
'~68'.6~" ~ . o o
t (ms)
Fig. 3. Simulation results for phase voltage and current build-up during
single-cage induction generator self-excitation.
If the machine is overcompensated by the capacitor
bank, switch-off may lead to overvoltages at the machine terminals. The amount of overvoltage and its
duration will depend on the capacitance of the capacitors, the drive inertia and the load torque. The inclusion of saturation ensures correct prediction of the
overvoltages, as an overestimate of the voltage peaks is
obtained if the saturation is neglected. The induction
machine used in this analysis is again Machine A, it
runs unloaded and its stator winding and capacitor
bank are star connected. The value of the capacitor
which provides compensation of the induction motor's
reactive power consumption is 25 pF, the same as was
used in Section 3.1. With this capacitor value there will
be no overvoltages at the machine terminals during the
switch-off transient, as the experimentally recorded
waveform shown in Fig. 5 clearly demonstrates.
However, if the machine is overcompensated, overvoltage will occur. Both simulations and experiments
209
E. Levi~Electric Power Systems Research 31 (1994) 203-216
400.00 "In
@
0
I
>
I~D - 2 0 0 . 0 0
(a)
t - time [.,]
,
1
,VVv
'
'
4oooo-]'
1 I
(a)-600.00 ooo
0
!
0
f \
\
t/\
\,
/
°'°t-
( d 3°
0
o°
x..../
I
100.00
(b)
t - time [B]
-100.00
LoolVVVVVVVV!V
>
-500.00 :{ .........
0.00
(b)
10
, .......
0.10
I
I
I,, , ....
0.20
,, ....
t - ti~
(s)
, .........
0.30
, .........
0.40
,,
0.50
a
!
(c)
t - time [s]
Fig. 4. Experimental and simulation results of voltage build-up in a
self-excited double-cage induction generator operating at a speed of
3180 rpm, with capacitors of 24.5 laF: (a) experimental, initial portion
of the transient (400 V/div., 100 ms/div.); (b) experimental, voltage in
steady-state (400 V/div., 5 ms/div.); (c) simulation, complete voltage
build-up.
l~lilllI
I I I I I I II I~
llvllllll!llllllllllllllll
!IIHI!
IIII[I
I illlfl;lll
vlllllf
I I I I I I I I111
I I I I I I
vavlvlt Ill f!V!Vlf!
Illlillllllvivllll
Illl!l~
Ihl I I!
i ~Ylv ~
I
t
,
Fig. 5. Experimentally recorded induction motor voltage during
switch-off with capacitors of 25 laF per phase.
Fig. 6. Voltage waveforms during the switch-off transient with capacitors of 50 laF per phase: (a) simulation with main flux saturation
neglected; (b) simulation with main flux saturation accounted for
(reclosing transient included); (c) experimental.
were p e r f o r m e d using a capacitance of 50/aF per phase
(double the value that provides full compensation).
Switch-off was p e r f o r m e d first and the mains voltages
were reconnected later on. The c o n s t a n t - p a r a m e t e r
model with saturation neglected predicts an overvoltage
of almost 30% (Fig. 6(a)) during the switch-off transient. If the saturated machine model is applied, the
predicted overvoltage is much smaller (Fig. 6(b)) and
corresponds to that obtained experimentally (Fig. 6(c)).
The instant of switch-off is indicated in Fig. 6(a) and
6(b) by the b r o k e n line (the second b r o k e n line in Fig.
6(b) corresponds to the instant of mains reconnection).
Current w a v e f o r m s for the same value of capacitance
(50 p.F) are given in Fig. 7, where the experimentally
recorded currents during the switch-off (Fig. 7(a)) and
210
E. Levi~Electric Power Systems Research 31 (1994)203 216
4. Variable-speed converter fed induction motor drives
Main flux saturation plays an important role in cases
when the variable-speed induction motor drive is fed
from a current source. Induction motor drives fed from
a current-source inverter and a resonant parallel inverter are discussed in the first two subsections. The
third subsection deals with the field oriented control of
an induction motor fed from an ideal current controlled
PWM inverter.
In
?:
4. I. Current-source inverter f e d induction motor drive
The configuration of the current-source inverter
(CSI) fed induction machine is shown in Fig. 8. A
three-phase thyristor bridge rectifier supplies, through a
DC link inductance, a standard autosequentially cornmutated three-phase CSI which consists of six thyristots, six diodes and six commutating capacitors. It is
assumed that the CSI operates with independent output
frequency control (i.e. without slip frequency control).
Consequently, steady-state operation takes place on the
stable part of the torque-slip curve and the machine
operates in the highly saturated region of the magnetizing curve. Steady-state time domain analysis is performed here with the ultimate goal of predicting the
steady-state voltage and torque of the machine. Ripple
in the DC current and the commutation interval may or
may not be disregarded in the analysis, depending on
the desired accuracy. The approach here assumes that
the DC link current may be treated as constant. However, commutation overlap is accounted for in such a
way that the machine phase currents have a trapezoidal
waveform. As the machine is current fed, the stator
phase and consequently the d,q-axis currents are
known. Thus the stator d,q-axis currents and their
derivatives act as inputs to the model, while voltages
become outputs. The machine model (Eqs. (1) (11))
has to be modified accordingly.
The machine used in this analysis is Machine C. Fig.
9 summarizes experimental and simulation results for
steady-state operation at a frequency equal to 40 Hz
with a DC link current of 12.2 A and a load torque of
W
Z
10.00
D
0.00
-10.00
r
-20.00
(c)
0.00
. . . . . . . . .
,
. . . . . .
0.10
t
-
time
,
. . . . . . . . .
0.20
(s>
,
. . . . . . . . .
0,50
i
. . . . . . . . .
0.40
i
. . . .
0,50
Fig. 7. Experimental and simulation results for the motor current
with per phase capacitors equal to 50 gF: (a) experimental, switch-off
transient; (b) experimental, reclosing transient; (c) simulation using
the saturated machine model (start-up, switch-off and reclosing transients).
reclosing transients (Fig. 7(b)) are shown together with
the simulated waveform for the whole operation consisting of the start-up, switch-off and reclosing transients (Fig. 7(c)). The simulated waveform was
obtained using the saturated machine model. Figs. 6
and 7 demonstrate very good agreement between simulation and experimental results.
ia
3x380
~-'
L
i
d
[b
50 Hz
[
Fig. 8. Simplified schematic representation of a CSI fed induction machine.
fo
.i
Induction
machine
I
c
E. Levi/Electric' Power Systems Research 31 (1994)203 216
V
r
400
-200
[
24(
200
0
5601
560
V
224
k
/
K
-288
-400
I
I -336
-464
.600
-576
j
(a)
Z
li
474,3 338
'
i
4g
v-
/
0
-48
"88
v
(100%)--500V
<476 -736
(b)
,I
46
g
-4g
',j
Te (100%)=25Nm
However, the predicted values of the overvoltages are
dependent on the current approximation during the
commutation interval only; considering the very simple
current approximation utilized in the simulation, it can
be stated that the agreement between the experimental
and simulation commutation overvoltages is fairly
good.
4.2. Parallel resonant inverter fed induction motor drive
-t32
.176 ~ ' ~
211
ur (100~) =250s-1
-80
(c)
Fig. 9. Steady-state operation of the CSI fed induction motor drive at
40 Hz, with a DC link current of 12.2 A and a load torque of 12 N m:
(a) experimentally obtained voltage waveform; (b) phase current
waveform (the input in the simulation) and simulated voltage wave*
form; (c) torque and speed obtained by simulation.
12 N m. The experimentally recorded waveform of the
steady-state voltage is shown in Fig. 9(a). Values of the
commutation overvoltages and steady-state peak values
are given for better comparison with simulation results.
The voltage waveform obtained by simulation is displayed in Fig. 9(b), together with the phase a input
current, while the simulated induction motor torque
and speed waveforms are depicted in Fig. 9(c). Comparison of the experimental and simulated voltage
waveforms shows very good agreement between the
steady-state voltage values (the peak value in the simulation is _+279 V, while the experimental value varies
slightly between _+273V and _+279V); this was
achieved by utilizing the saturated machine model. As
far as the commutation overvoltages are concerned, the
difference between the experimental and simulation values is quite significant in some commutation intervals.
The drive under consideration consists of a lowvoltage three-phase voltage supply (an autotransformer
operating with constant-voltage output), a three-phase
diode bridge, a DC link inductance and capacitance, a
three-phase thyristor inverter and a three-phase capacitor bank connected in parallel to the terminals of the
induction motor (Fig. 10). A suitable choice of the DC
link inductance and capacitors at the AC side of the
inverter leads to resonant operation of the circuit; the
current in the DC link is discontinuous and the inverter
operates as a zero-switching-current resonant inverter.
Two thyristors in the inverter conduct during the conduction intervals. If the maximum required output frequency is 50 Hz, the resonant frequency of the circuit
has to be set above 150 Hz. It should be noted that the
drive operation is possible over the entire designed
frequency range with a constant DC link voltage. The
motor's voltage/frequency ratio is reasonably constant.
However, due to the absence of feedback, instability
results if a step increase in the load torque occurs; some
feedback has therefore to be introduced in order to
stabilize the drive operation.
Steady-state operation of the drive takes place on the
saturated part of the magnetizing curve and simulation
is not possible unless a model which accounts for main
flux saturation is utilized. As the single-cage machine
(Machine A) is used, the model that describes the
machine is once more given by Eqs. (la) and (2)-(11).
The input into the model is the DC link voltage Vd
across capacitance Ca, which is assumed to be constant.
The equation of the DC link circuit is
did~dr
=
(V d -
vi)/L d
(18)
where vi is the inverter input voltage and is a linear
combination of the products of switching functions and
induction motor terminal voltages. As there is a capacitor bank connected to the machine terminals, Eq. (17)
is needed once more. However, it is more convenient in
this case to use the capacitor bank equations in the
original phase domain rather than Eq. (17).
The capacitor bank and stator windings were delta
connected in the experiments and simulations, the load
torque had a fan characteristic and steady-state operation at 40 Hz was investigated (the relevant data of the
rig are given in the Appendix). The resonant frequency
was approximately equal to 200 Hz. The simulation
E. Levi / Electric Power Systems Research 31 (1994) 203 216
212
Diode
rectifieP
,,50
o
Hz
~
d
1d
~
Vd
id
I
Induction
machine
Vi
o
C
low
vo 1t age
]
|
C
Fig. 10. Resonant parallel inverter fed induction motor drive.
20C
v %
(a) t,c,
I00.
o ),5
-1(11
2: I'-
~
5
-200
ia]43i°tfA]
!
0
-I
r
-2
-3
-4
01
%5
(c)
o~2s
0$2s-Fig. I I. Simulation results of the resonant parallel inverter fed
induction motor drive operation at 40 Hz: (a) line-to-linevoltage; (b)
motor line and phase currents; (c) torque; (d) DC link current.
results are shown in Fig. 1 1 and include motor line-toline voltage v, motor line current ial and phase current
i,, motor torque Te and DC link current id.
The corresponding experimental results are shown in
Fig. 12. The agreement between the experimental and
simulation results is remarkably good. However, it
should be noted that the DC voltage in the experiment
Fig. 12. Experimentally recorded waveforms in a resonant parallel
inverter fed induction motor drive at f = 40 Hz and Va = 64 V: (a)
voltage; (b) motor line current; (c) motor phase current.
E. Levi/Electric Power Systems Research 31 (1994) 203-216
213
m
inverter [ ]
,i
Do
.
•
~_~
T M L-
with
li
local
current
___J
'1
sl
] -e
' feedback I
i
~r
= slip frequency
KI
m
T)
2
Fig. 13. Indirect rotor flux oriented induction motor drive (two-pole machine is illustrated).
was equal to 64 V, while that employed in the simulation was only 44 V. These voltages produced identical
DC link current waveforms with identical peak values
and consequently the motor voltage and currents obtained by experiment and in the simulation have the
same waveforms with the same values. The difference in
the DC voltage values required to produce the same
waveforms in the simulation as in the experiment is
attributed to the fact that all the losses in the DC link,
inverter and AC part of the circuit prior to the motor
are neglected in the simulation.
q-axis reference current. The consequence of cross-saturation will be a reduction in the actual rotor flux and
motor torque with respect to the reference values and
an unwanted transient in the response.
Two operating regimes were simulated in order to
illustrate the influence of saturation on the operation of
the drive. In both cases the value of the magnetizing
I
I
4.3. Vector controlled induction motor drives
Among the three possible choices of vector control
(orientation along the stator, air-gap or rotor flux space
vector), the one chosen for discussion here is the rotor
flux oriented control, which is still the predominant
type in actual realizations. Rotor flux oriented control
may be realized in a number of different ways. Either a
current-fed or a voltage-fed machine, with either an
indirect type of orientation or with one of the direct
schemes, where estimation of the rotor flux space vector
is performed, may be selected depending on the application. The scheme discussed here is the simplest one, the
so-called indirect rotor flux oriented control (Fig. 13).
The machine is once more represented by model equations (la) and ( 2 ) - ( 1 1 ) with the same change already
mentioned in Section 4.1. As the machine is treated as
current fed, the stator-axis currents and their derivatives are known and act as inputs to the model. A
PWM inverter with local current feedback is taken as
ideal, so that reference currents equal actual phase
currents. There are two distinct operating regimes
where main flux saturation will play an important role.
The first is operation in the field-weakening region,
where a change in the saturation level in the machine
happens simply because of a change in the desired flux
level. The other situation is operation in the constantflux region, with a constant rotor flux reference value.
In this case, a change of saturation level in the machine
will occur purely due to the cross-saturation effect
during dynamic regimes with high values of the stator
1
I
I
I
I
I
f
I
%*
T*e
(a)
@~ =~ rn/1.33
L •" =L
=T
T•
•
mn
on
@r: 0+1 Vs
Te:-20+20 Nm t: O+O.t s
l
I
I
I
I
I
It
I
I
/~
I
I
I
I
T;
I
I
~*
-T e
%
(b)
=
~r:
I
~kr~
L
n
=
L
T
mn
o
=
7T
en
0+1 VS To:-40+40 Nm t: 0+60
I
I
I
It
l
I
I
ms
I
Fig. 14. Saturation effects in an indirect rotor flux oriented induction
machine operating in the torque mode: (a) application of a rated step
torque command in the field-weakening region; (b) application of a
high step torque command in the constant-flux region.
214
E. Levi/Electric Power Systems Research 31 (1994) 203 216
inductance in the controller was set to rated (i.e.
Lm = Lm,) and the speed loop was left open, so that the
independent control variable is the reference torque and
the drive operates in the torque mode. Machine A was
used once more. The first transient applied was a step
torque c o m m a n d equal to the rated torque while the
machine operated in the field-weakening region with a
rotor flux c o m m a n d equal to 75% of the rated. The
c o m m a n d e d and actual torques and the rotor flux
waveforms are shown in Fig. 14(a) (an asterisk denotes
the c o m m a n d e d quantities). As can be seen from the
figure, prior to the application of the step torque comm a n d the drive operates under no-load conditions and
there is a large discrepancy between the actual and
c o m m a n d e d rotor flux. This is a consequence of the
value of the magnetizing inductance used in the controller being inadequate for the operating flux level.
Once the torque c o m m a n d is applied, there are unwanted transients in both the torque and flux responses.
Experimental results regarding the influence of saturation on operation in the field-weakening region are
available in Ref. [49].
The second transient is concerned with operation
in the constant rated rotor flux region. A large torque
command, equal to seven times the rated torque, was
first applied and then removed. The responses are illustrated in Fig. 14(b). Due to the cross-saturation effect
there is a significant drop in the actual rotor
flux and the torque produced by the machine is smaller
than the c o m m a n d e d one. It is worth noting that, to the
best of the author's knowledge, experimental p r o o f of
the cross-saturation effect in rotor flux oriented induction machines has not been provided yet.
The majority of cases studied here involve a singlecage induction machine. However, if the actual machine
in any of the applications is double cage or deep bar,
the corresponding model of a saturated double-cage
machine should be used. The same degree of accuracy
in the simulation results should be achieved with regard
to the influence of the main flux saturation on the
machine's behaviour.
Appendix
Induction machine and drive data
Machine A
Pn=0.75 kW,
V~=380 V/220 V,
cos ~o = 0.72,
f = 50 Hz,
I , = 2 . 1 A/3.6 A
n, = 1390 rpm
Star/delta connected stator winding
Index n denotes rated values
Parameters:
R~ = 10 ~,
R~ = 6.3 f~
Lmn = 0.42119 H,
L~ = L:~ + Lm~ = 0.464257 H
Lr = L.~r+ L ..... = 0.46226 H,
J = 0.00442 kg m z
Magnetizing curve approximation in terms of phase
r.m.s, values:
L = dTm/dI m
~um -- 0.86427 x 0.599761mlm 121 J,
Additional data for Section 4.2
C = 100 gF,
L d = 4.2 mH,
T~. = 0.025 x 10 30)r2
Machine B (double-cage induction machine)
5. Conclusion
Pn = 7.5 kW,
Main flux saturation in induction machines is of
p a r a m o u n t importance in a number of everyday applications and, if successful analysis is to be performed,
it has to be included in the model of induction machines.
This paper has concentrated on the current state space
model of saturated single-cage and double-cage (deepbar) induction machines and its applications in the
simulation of induction machines and the variable-speed
induction m o t o r drive. The majority of practical situations where a need for a saturated machine model exists
are addressed. The applications studied include self-excitation of single-cage and double-cage induction generators, switch-off and reclosing transients of induction
motors with reactive power compensation, variablespeed drives fed from current-source and parallel resonant inverters and finally vector controlled highperformance induction m o t o r drives. A large number of
simulation and experimental results are included and
these are found to be in very good agreement.
cos ~0 = 0.9,
Vn = 380 V,
f = 50 Hz,
In = 14.7 A
nn = 2905 rpm
Delta connected stator winding
Parameters:
Rs = 1.9685 fL
Rrl = 2.82 fL
Rc = 0.649 fL
Lmr -= 2.79 m H
L ..... = 0 . 4 4 9 7 7 H ,
Rr2 = 1.36 f~
L S = L Y ~ + L .... = 0 . 4 6 H
L~l =- LyrL + Lmn + Lmr = 0.45256 H
Lr2 = L.frz + Lmn + Lmr = 0.46057 H
Magnetizing curve approximation in terms of phase
r.m.s, values:
7Jm
~LmoI m
((a + b/Im + c/I~ 2) -~
a = 0.67905,
b = 0.067911
c = 0.94346,
Lm0 = 0.59 H
I m < 1.25 A
Im > 1.25 A
E. Levi / Electric Power Systems Research 31 (1994) 203-216
d ~ m _ yLm0
L _ m
dim
I m < 1.25 A
[~m2(b/Im 2 + 2C/Im 3)
I m > 1.25 A
Machine C
P, = 3 kW,
V, = 220 V,
f = 50 Hz,
nn = 1400 r p m
1,1 = 6.3 A
Delta connected stator winding
Parameters:
R s = 1.54 ~ ,
Lys = 6.6526 m H
R, = 2.22 fL
Lyr = 6.6526 m H ,
J = 0.21 kg m 2
D C link i n d u c t a n c e Ld = 28.5 m H
C o m m u t a t i n g c a p a c i t a n c e C = 8 ~tF
M a g n e t i z i n g curve a p p r o x i m a t i o n in terms o f p h a s e
r.m.s, values:
Magnetizing
current Im
Magnetizing
flUX ~m
Dynamic
inductance L
0.00-1.10
1.10-1.85
1.85 3.60
3.60 6.10
6.10 50.0
0.3181m
0.2271m +
0.1031m +
0.0681m +
0.0441m +
0.318
-0.1841 m + 0.52
-0.0541 m + 0.28
-0.0161 m +0.14
0.044
0.10
0.33
0.45
0.60
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