A GENERALISED OPERATIONAL EQUIVALENT CIRCUIT OF
INDUCTION MACHINES FOR TRANSIENT/DYNAMIC STUDIES
UNDER DIFFERENT OPERATING CONDITIONS
S. S. Murthy
Department of Electrical Engineering
Indian Institute of Technology
New Delhi-110 016. (India)
Abstract. The paper derives a generalised operational
equivalent circuit for induction machines starting from
fundamental principles in any arbitrary rotating reference
frame. Directly measurable parameters are used in this
equivalent circuit From this generalised model, the
equations for any reference frame are derivable as particular
cases. The model presented is versatile as it can be used for
any operating condition and any asymmetrical connection or
switching and for both 3-phase and 2-phase wound machines.
The equivalent circuit provides space phase vectors and the
relevant expressions can be used to develop software for on
line control of motors through field orientations. Different
possible modes are considered in this paper to derive
relevant equation from the general model to illustrate its
potential.
1. INTRODUCTION
Induction machine modelling has continuously
attracted the attention of researchers not only because such
machines are made and used in largest numbers but also due
to their varied modes of operation both under steady and
dynamic states. Increased use of Power Electronic controllers
with such machines makes appropriate modelling and
parameter identification crucial both for working control
strategies and performance prediction.
Induction machines operate in both motoring and
generating modes. Till recently motoring operation was
almost universal. But recent exploitation of renewable energy
systems such as wind and small hydro has led to use of grid
connected induction generators driven by wind and hydro
turbines.
While steady state operation of 3-phase motors at
fixed balanced voltage, frequency and load is common,
operations at unbalanced voltages and asymmetrical winding
connection (1-phase, 2-phase, 3-phase) are often prevalent
needing appropriate models to analyse. The power devices
used for motor control often cause asymmetrical switching of
windings. Transient studies under run-up, reswitching,
braking, load change and short circuit conditions too require
suitable analytical models. Modelling under induction
generator operation driven by wind and hydro turbines has
recently assumed importance. Field oriented or vector control
requires dynamic modelling under different reference frames
for real time transformations. Machines with asymmetrical
0-7803-2795-0
•622
windings need special equations. This paper presents a
generalised model with an operational equivalent circuit from
which specific equations for different operating modes can be
derived by substituting appropriate constraints. The concepts
of generalised rotating field theory and symmetrical
component theory have been used effectively for steady state
operation under unbalanced voltages or asymmetrical
connections [1,2,3]. Mathematically, though not physically,
analogous concepts of instantaneous symmetrical components
and operational equivalent circuits have been shown to be
effective in analysing transients under similar conditions [4-9].
But these were confined to a stationary reference frame with
symmetrical stator and rotor windings.
The concepts of space vectors of voltage, current and
flux-linkages [10,11] used in field orientation and field
acceleration method of control of motors have a direct
relation to instantaneous symmetrical components
representing dynamically changing complex quantities which
can be denoted by voltage, current and flux linkage vectors.
Further, the equivalent circuit is an easily understandable
concept for a student working on induction machines, wherein
the parameters can be determined by test or design, It is
possible to incorporate non linear variation of these
parameters under different operating modes dynamically in a
program. For example studies on capacitor self excitation
require simulation of non-linear magnetising reactance. This
is true in any situation involving flux control. It has been
experienced by the author that operational equivalent circuit
leads to model equations under asymmetrical operation more
elegantly compared to alternative methods using direct d-q
variables.
The generalised operational equivalent circuit
presented in the paper relates to an arbitrary rotating
reference frame such that specific equation for any reference
frame can be evolved as a particular case. The concept may
be extended for asymmetrical 2-phase or 3-phase winding
configurations, which may also include zero sequence circuit.
2. DERIVATION OF GENERALISED
EQUIVALENT CIRCUIT
A symmetrical induction machine can be considered
to have 3 phase symmetrical stator windings sa, sb, sc and
equivalent short circuited 3 phase symmetrical rotor windings
ra, rb, re (Fig. 1) or 2 phase symmetrical windings sa, sb and
(b)
Fig.1 Winding Orientation of a Symmetrical
Transformed Windings on
Induction Machine
Arbitrary Rotating Frame
(a) 3-phase
(b) 2-phase
ra, rb at quadrature. Let R» R, be resistance of each stator
(2)
and rotor winding and L^ L, be self inductances. Let M^,
M,,, be mutual inductance between any two windings in stator
and rotor respectively. Let M be the mutual inductance
where the time derivative operator p=—
between a stator and a rotor winding when they are aligned.
Let the rotor speed be u, elect, rad/s. Let us consider an
If we denote [ V ] , P ' ] , [R«], [L'] as the
arbitrarily rotating reference frame at «o elect, rad/s on
corresponding matrices in the transformed plane it is possible
which d and q axes at quadrature are fixed (Fig, 2). If 6 is
to show that
the angle between sa and ra axis at any instant, we may
assume that all inductances expect those between stator and
(3)
rotor windings are independent of e. Neglecting space
harmonics the mutual inductance between stator and rotor
where
windings may be considered to be varying sinusoidally with ©.
Transformations are resorted in machine theory to make the
inductance parameters independent of 6 in the transformed
(3a)
plane.
In the arbitrarily rotating frame both stator and rotor
quantities are transformed to sd, sq, so and rd, rq> ro windings
using the transformation matrix,
[C]
ICJ
(1)
0
where
an
Starting from the original [R], [L] & [G] matrices and
using the transformation of (1) we get [ R ' ] , [L'J & [G'] as
given in Appendix-I, from which eq. (3) leads to the following
transformed equations.
-dnd
cos8-
(la)
(4b)
JS) -^(6 A -L
3
3
c«<e.-e)
,/2
(4c)
42
(lb)
3J
(4d)
i
Here eo = wo t and © = «, t
Denoting [V], [I], [R], [L] as voltage, current,
resistance and inductance matrices and [G] as derivative of
[L] with 0, the basic volt-ampere equation is
623
(4d)
(4Q
Same holds true for currents and rotor voltages.
Since negative sequence quantities are complex
conjugates of positive sequence ones, only the latter would
suffice to define the system.
Following relations on inductances are valid
LK - Msm - L,, + (3/2) L ^
K - M ra - L,r + (3/2) L ^
Defining operator />'•.£.«—_
where L^, Ljr are stator and rotor leakage inductance
per phase all referred to stator turns, L^ is the mutual
inductance of all windings referred to stator turns. At the
base radian frequency eo, following reactance can be defined
Stator leakage reactance/phase, x,,=<i> L^
p.u. speed of rotor v=—-.
Rotor leakage reactance/phase (ref. to stator) xJr"<«>£|r
p.u. speed of rotating frame
Magnetizing reactance x,,, = u> (3/2) L ra
The reactances xb, x,r and x^, are the standard
parameters of the per phase equivalent circuit of an induction
machine defined at the base frequency. Referring all the
quantities to stator turns, eq. (4) now simplifies to
x,.
x_
v a =—
eq. (5) through (6) yields,
(7a)
x.
(7b)
(5a)
The above equations can be represented by the
operational equation circuit of Fig. 3, with the following
notations
: voltage, current
.
v, i
: Resistance, reactance
R, x
(5b) v, v 0 : p.u. rotor speed and speed of rotating frame
: operator, (1/co) (d/dt)
p'
: base radian frequency
<•>
(5c) Subscripts :
X
ls
CO
(»„-«,)
(<•>.-<•>
s, r
: stator, rotor
1, m : leakage, magnetising
Superscripts :
+ , - : positive, negative sequence.
The general equivalent circuit is derivable from
symmetrical 3-phase or 2-phase machine and only the
<5d) transformation matrices [Cffl] and [CJ of {1) would differ as
also the inductance parameters.
For negative sequence quantities eqs. (7a) & (7b) can
be modified by replacing the superscript " + ' by "-" and the
operator j by -j. Same changes apply for Fig. 3 to obtain the
negative sequence operational circuit.
It is possible to write a zero sequence operational
equivalent circuit using zero sequence voltage equation of
eq. (5) as shown in Fig. 4
(5f)
Generalised Operational equivalent Circuit
Using eq. (5) we may attempt to derive a generalised
operational equivalent circuit, by effecting so called
instantaneous symmetrical component transformation, by
defining [4-7]
General torque expression is
2.1
Iq. terms of transformed quantities
On substitution from eqs. (1)
n
1 ; 0
1
1 -/ 0
0 0 1
(6)
(8)
Referring all terms to stator turns, general torque
624
»|rP
"Is"
(b) Roto'
( o ) stator
fig.4 Zero Sequence Operational Equivalent Circuit
Fig.3 Generalised Operational Equivalent Circuit
R*
x
lsp>
"Ir"'
expression in terms of sequence quantities is
(9)
MODEL WITH PARTICULAR REFERENCE
FRAMES
In practice we need model equations in different
frames; for example the d-q axes fixed to stator, rotor or
rotating at synchronous speed. The operational equivalent
circuit for each frame can be obtained by simple substitution
illustrated below.
3.
Fig.5 Operational Equivalent Circuit for
Stator Reference Frame
3.1 Stationary reference frame
Here 0>o = 0, eo = 0, vo = 0, the matrices [CJ,
[CJ of eq. [1] are simplified such that [CJ becomes a
constant matrix leaving only [C J dependent on e. With these
substitution, the operational equivalent circuit is as in Fig.5,
from which the relevant volt-ampere and torque expression
can be obtained.
Fig.6 Operational Equivalent Circuit for
Rotor Reference Frame
3.2 Rotor reference frame
Here d-q axes are fixed to rotor and &0~Q, &>o = &>r,
vo = v. The matrices [CJ, [CJ of eq. [1] are simplified such
that [CJ becomes a constant matrix leaving only [CJ
dependent on 8. With these substitution the operational
equivalent circuit is as in Fig. 6, from which the relevant voltampere and torque expressions can be obtained.
33 Synchronously rotating reference frame
Here &>o = to, vo = 1. Here both [CJ and [CJ of
eq. (1) are 6 dependent. Defining p.u. slip s = 1 - v, and
with the above substitution the operational equivalent circuit
is as in Fig. 7, from which the relevant volt-ampere and
torque expression can be obtained.
4.
SOME IMPORTANT INTERPRETATIONS OF
THE EQUIVALENT CIRCUIT
Before looking at some special applications of the
general circuit, it is important to identify and interpret some
special features.
With due idealisation, it contains measurable
resistance and reactance parameters, whose nonlinear
variation with operating conditions such as temperature, flux,
current and frequency can be accounted for.
The terms v,*, i,* and ir* are complex numbers often
changing with time. They can also be interpreted as space
phasors adoptable for field orientation or field acceleration
through real time modelling [10,11]. From the general
Fig.7 Operational Equivalent Circuit for
Synchronously Rotating Reference Frame
equivalent circuit appropriate model equations for developing
the software for .field orientation/acceleration can be
developed along with relevant transformations for the chosen
reference frame. The current vectors i,+ and ir+ respectively
represent net stator and rotor fields. By controlling the real
and imaginary components of i,+ independently, the stator
flux can be oriented. The resultant flux is caused by
V = i,+ + ir+
The magnitude of i,+, i / , i^,* and angle between
them will have a bearing on torque or power development as
per eq.(9). The above space phasors'are either rotating or
stationary under steady state, depending on the frame chosen.
They rotate at synchronous speed (each with fixed magnitude)
with stator frame and slip speed with rotor frame. They are
stationary at synchronously rotating frame. Further detailing
of these concepts is beyond the scope of this paper. Equations
(1) & (6) help in obtaining space phasors from original
machine voltages and currents.
625
5.
APPLICATION OF THE GENERAL
EQUIVALENT CIRCUIT TO PARTICULAR CASES
The general operational equivalent circuit has been
found to be useful in deriving model equations for almost all
practical cases. Only a few cases are presented here for
illustration.
5.1 Transients ander motoring, generating and braking
Transients in induction motors and generators can be
estimated through suitable modelling. The operational
equivalent circuit with stationary frame can be employed. For
the given voltage and torque conditions currents and speed
can be determined by deriving the expressions for derivatives
of currents p'[i] and speed as given in Appendix-II where
subscripts x and y denote real and imaginary parts of complex
quantities. By choosing suitable initial conditions of [i] and v,
transient response with time can be computed at different
time steps using Runge-Kutta method. Motor transients under
run-up, reswitcaing and load change can be obtained by
setting suitable initial conditions and effecting the transient
input change. Typical transients in induction generator relate
to aided starting, (k is started as motor with additional torque
provided by the prime mover) change of input power and
switching the generator to the grid. Transient under counter
current braking or d.c. injection are obtained by injecting
respective voltage or current at the operating speed v.
Throughout the transient process resistance and reactance
parameters need to be suitably chosen or changed if required.
• ! sb
sb
Fig.8 2-Phase Motor With 1-Phase Supply
\ = Z, i,
If the zero sequence component is present
v.° = Z,° i.°
From eqs. (10, 11, 12) we get
V =(—
1)|
(12b)
(12c)
13)
2
which can be shown by the equivalent circuit of Fig.9
52
SYMMETRICAL INDUCTION MOTOR WITH 1PHASE SUPPLY
There are several practical situations wherein a
symmetrical induction motor (3-phase or 2-phase) is
connected to a 1-phase supply. Dynamic models are
obtainable from the general equivalent circuit as follows.
52.1
2-phase symmetrical induction motor with 1-phase
supply across one winding
Fig.8 shows a 2-phase machine' with symmetrical
btator windings sa and sb (with symmetrical rotor). The sa
winding is connected to a single phase supply, such that the
applied voltage v = vM Further i,,, = 0.
In terms of sequence voltage vs
following relations hold
v
v v
(
/ >
and
the
(10)
Fig.9 Operational Equivalent Circuit With 1-Phase Supply
3-phase induction motor with 1-phase supply across
one winding
Fig. 10 shows a 3-phase star connected induction
motor (with symmetrical rotor) with 1-phase supply across
one winding.
Here
(14)
Since i^, = 0,
'«
'»
r-•
»
(11)
From the operational circuit of Fig. 4, we may write
V = W
(15)
(12)
where Z, + = operational positive sequence
impedance.
Similarly we may write for negative sequence
Note that zero sequence component is also present
here. Combining eqs. (12, 14, 15), we get
626
(17)
Further
(18)
v/3
Combining (12,. 17, 18), we get
V = (Z, + + Z>) I,,
Fig.10 3-Phase Motor With 1-Phase Supply
^ - j - ^ .
leading to the operational circuit of Fig. 13.
There are several practical situations when such a
single-phasing operation occurs such as sudden disconnection
of a supply line and in-line-thyristor switching. The modelling
of such systems using the operational circuits has been
reported[6].
(16)
which can be shown by the equivalent circuit of
Rg.lL
52.4
3-phase delta connected motor with 1-phase supply
Similar to star connected motor, modelling and
analysis of a delta connected motor with 1-phase supply
becomes necessary e.g. sudden line disconnection or use of inline thyristor for voltage control.
For the connection of Fig. 14 following terminal
relations hold
v
which yield
Fig. 11 Operational Equivalent Circuit of 3-Phase Motor
With 1-Phase Supply
Leading to the operational circuit of Fig. 15.
Logically equation (13) & (16) and circuits of Figs.8
& 10 must be same. While stator leakage impedance matches
by including the zero sequence impedance, air gap
impedances appear to differ. It is important to note that the
air gap impedance parameters x,,,, x^ and x,, in Fig. 9 and
Fig.ll are not numerically same, being referred to 2-phase
and 3-phase systems respectively, The parameters of the 2phase system would be (2/3) times those of the 3-phase
system, as the transformation matrices of eq. (1) differ.
3-phase star connected induction motor with 1phase supply across 2-windings
For the connection shown in Fig. 12, where 1-phase
supply is connected across both sa and sb phases.
0
- 0, vK
5.2.5
3-phase motor with 1-phase supply with phase
converter
It is possible to operate a 3-phase star or delta
connected motor with 1-phase supply using a static phase
conyertor, which is normally a capacitor across two of the
terminals. Dynamic modelling and analysis can be elegantly
handled using operational equivalent circuits.
If a capacitor of reactance x,. at base frequency is
connected across the terminals a, c of Fig. 12, the following
terminal relations hold
v - VM 1- v* = 0
5.23
1
sb
Methodology of modelling and analysis has
been detailed in an earlier paper [7] , using the equivalent
circuit, for both star and delta connected cases.
627
S3
Analysis under capacitor self excitation
Induction machines driven at any speed experience
self-excitation and operate under generating or braking modes
with sufficient terminal capacitors. Under balanced conditions
the operational equivalent circuit of Fig. 3 can be used. It is
important to note that xm in Z, + is a variable parameter
changing from unsaturated to saturated value as the voltage
builds up. The- loop equation
'so
need to be solved with suitable initial conditions, to obtain
the response. For determining the voltage build-up at a fixed
speed as a self excited induction generator this equation
should be solved numerically keeping v constant and choosing
a small initial value of current. The value of x,,, has to be
continuously changed during integration. To analyse under
capacitor braking, the same equations have to be solved but
choosing suitable initial values before braking and allowing
the speed to change due to mechanical factors and the
braking torque.
Capacitor - excited generator or braking action is
prevalent [12] even with a single capacitor connected across
two of its terminals. This is similar to replacing the voltage
source with a capacitor in Fig. 12 which leads to the circuit of
Fig. 16, with the operational impedance xjp' replacing v,
such that
Fig.13 Equivalent Circuit for Connection of Fig.12
"5b
Hg.14 Delta Connected Motor on 1-Phase Supply
This equation can be solved for both generating and
braking modes but employing suitable value of x^, in each
iteration.
T
Fig.12 1-Phase Supply Across 2-Phases
Other cases with stator frame of reference
A few other cases can be cited wherein the
operational circuit with stator frame of reference may be
tried. Two-phase induction motor with 1-phase supply is one
such case with a capacitor for phase shift. In this paper the
cases with asymmetrical windings have not been considered
wherein the windings may have unequal turns or arbitrary
phase displacements. It is hoped that the concepts enunciated
in this paper may be extended to such cases too similar to
steady state operation with gene~alised rotating field theory
[1]. The transformation through the general operational
circuit can be employed for real time-on line modelling for
inverter fed motor control with field orientation. Necessary
equation to effect intelligent control can be evolved.
Z<f
Fig.15 Equivalent Circuit for Delta Connected Motor with 1Phase Supply
5.4
P'
Z*(P)
Fig.16 Equivalent Circuit With Capacitor Self-Excitation
628
5J5
Modelling with rotor unbalance
Unbalance in rotor is prevalent mostly in slip ring
motors due to asymmetrical switching or unbalanced external
impedances connected across rotor terminals. Power
electronMTconvertors in rotor circuits for slip energy control
may often result in rotor unbalance. Start-up of slip ring
motor with unbalanced external starter resistors is another
practical case. The modelling through operational circuit in
the rotor frame of reference can be effected in such cases
especially under asymmetrical/unbalanced conditions.
•-** Only one case of single-phasing in rotor circuit when
onw of the three star connected rotor phases is open circuited
is considered here, while other cases can be taken up for
further study.
The corresponding rotor circuit is shown in Fig. 17.
The operational circuit of Kg. 5 with rotor frame can be
written with voltage sources across rotor terminals as in
Fig.18
V+ and vr' can be written from Fig. 5 and its
complement as
•xj'a:*o
M
xwp%+Q
(22b)
+
Denoting the complex terms i, and i," in terms of
real and imaginary parts as
Z-i.+K
(23a)
and substituting in (21) through (20), (22), (23), we get
(24)
writing
the stator voltage equation from the stator current loop of
Fig. 5 can be simplified as
V * V < * * « > V v < V * J l > ^ V ^ v V« (26)
Fig.17 1-Phasing in Rotor
Equations (24 - 26) are the governing volt-ampere
equations with single-phasing in rotor. Combined with
appropriate torque equation, total dynamics can be studied
through numerical methods. [6-8]
6.
Fig.18 Equivalent Circuit for Fig.17
Following relations holds true for Fig. 17
=0
(19)
rt>
T,
which yield
(20a)
(20b)
CONCLUSIONS
The paper presents a generalised operational
equivalent circuit for induction machines for different possible
operating conditions specially when external asymmetries are
involved. The general circuit for an arbitrarily rotating
reference frame yields specific models for a particular frame
with simple substitution. Several practical cases including
stator and rotor asymmetries can be elegantly handled using
this general circuit as demonstrated in the paper.
Considerable scope exists towards varied applications of the
proposed model and also for extending the same for machines
involving internal winding asymmetries. The non-linear
parameters can also be easily accounted for a dynamic
simulation. The circuit can also be used to explain field
orientation.
7.
1.
V3
2.
629
REFERENCES
J. B. Brown and C. S. Jha, " Generalised Rotating Field Theory of
Polyphase Induction Motors and its Relationship with Symmetrical
Component Theory ", Proc. IEE, Vol. 109A, pp 59-69, Feb. 1962.
J. E. Brown and C S. Jha," The Starting of a 3-phase Induction Motor
Connected to a 1-phase Supply System ", Proc. IEE, Vol. 106A, pp Ig3190, April 1959.
3.
C. S. Jha, " The Starting of Single-phase Induction Motors having
Asymmetrical Stator Windings not in Quadrature ", Proc. IEE, Vol.
_ 109A, pp 47-58, Feb. 1962
4. J. E. Brown, P. Vas and S. S. Murthy, * Instantaneous Operational
Equivalent Circuits of 3-phase Motors having Current Displacement
Rotors ", Proc. International Conference on Electrical Machines,
Athens, Sept. 1980.
5. S. S. Murthy, Bhim Singh and A. K. Tandon, " Dynamic Models for
Transient Analysis of Induction Machines with Asymmetrical Winding
Connections *, Electrical Machines and Elcctromechanics (USA) Vol.
6, No. 6, 1981, pp 479-492.
6. S. & Murthy and GJF. Berg, " A New Approach to Dynamic Modelling
and Transient Analysis of SCR Controlled Induction Motors *, IEEE
Trans, on Power Apparatus & System, No. 9 SepL 1982, pp 3140-3150.
7. S. S. Morthy G. J. Berg, Bhim Singh, C. S. Jha and B. P. Singh, '
Transient Analysis of a 3-phase Induction Motor with Single-phase
Supply ", (paper No, 82wm 228-5), IEEE Trans, on Power Apparatus
& Systems, VoL 102, Jan. 1983, pp 28-37.
8. S. S. Murthy and G. J. Berg, " Transient Analysis of Induction Motor
with Plugging, Operation ", Canadian Electrical Engineering Journal,
VoL 9(1), 1984, pp 22-28.
9. S. S. Murthy and G. J. Berg, " Improved Dynamic Model for the
Transient Analysis of SCR Controlled Induction Motors ", J.I.E (India),
VoL 66, Oct. 1985, pp 57-65.
10. Sakae Yama Mura," A.C. Motors for High Performance Applications
", Marcel Deckker, 1986.
11. P. Vas," Vector Control of A . C Machines ", Clarendon Press, Oxford,
1990.
11 S . & Murthy, G. J. Berg, C. S. Jha, and A. K. Tandon, " A Novel
Method of Multistage Dynamic Braking of Three Phass Induction
Motors ", IEEE Trans, on Industry Applications, April 1984. pp. 329334.
Appeedis-I
The matrices R ' , L' and G' of eq. (3a) on substitution
yields
Appendix-II
In Fig. 5 v, + , i, + , i,+ are complex quantities and can be
written as
Relevant loop equations for
derivative of current as [6,7]
where
0
V
0
x,
0
0 xm
*r
0
0 x_ 0 x.
The time of derivative of speed is [6.7]
2<*H
R, R, R, Rr
•M^
and i/ yield the time
where Te is developed electromagnetic torque from eq. 9,
TL is the load torque and H is the inertia constant.
0
-M
Ln-M
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-M
0
-~M
0
0 -(
0
0
0
0
0
630