ROTOR VOLTAGE INFLUENCE ON THE CHARACTERISTICS
OF A DOUBLY FED INDUCTION MACHINE
ŽARKO MILKIĆ1, ĐUKAN VUKIĆ, SAŠA ŠTATKIĆ
Key words: Doubly fed induction motor, Rotor voltage, Speed regulation.
We present an analysis of rotor voltage influence on the characteristics of a wound rotor
induction motor, the speed of which is regulated by double feeding. Stator windings of
the motor are fed from the power system, while the feeding for the rotor ones is
provided by a cycloconverter. Expressions for all characteristic variables are derived on
the basis of a mathematical model defined for particular operating conditions. Rotor
voltage regulation laws, which ensure optimal motor operation according to
predetermined criteria and apply to the entire regulated speed range, are defined from
the obtained expressions.
1. INTRODUCTION
Double feeding is one of the ways for regulating the speed of a wound rotor
induction motor. Stator windings of such motors are usually fed from the power
system, while the feeding of the rotor ones is realised by a frequency converter
(Fig. 1). Semiconductor converter with direct frequency conversion – the socalled cycloconverter – represents the most convenient solution for a variable
frequency source [1].
Characteristics of a doubly fed induction motor are significantly influenced by
the rotor voltage value [2, 3]. Hence, by defining an appropriate voltage regulation
law, optimal motor performance according to a predefined criterion can be achieved
over the entire regulated speed range. Choice of such a criterion depends on several
factors: motor rating, regulated speed range, cycloconverter characteristics, and
power system demands [4]. Specifically, for motors with large power rating, such
as those employed as drives in centrifugal pumps, ventilators, compressors and
other similar devices, motor operation with minimum stator current is advantageous
[5]. In this regime, a motor takes the required reactive power from the rotor side only
1
University of Priština, Faculty of Technical Sciences, Kosovska Mitrovica, Kneza Miloša 7, 38220
Kosovska Mitrovica, Serbia; E-mail: zarkomilkic@yahoo.com
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 59, 3, p. 249–258, Bucarest, 2014
250
Žarko Milkić, Đukan Vukić, Saša Štatkić
2
[6, 7]. This reactive power (Q2) is significantly lower than the reactive power required
in standard operation mode (Q1), since it is proportional to the slip of the motor
(Q2 = s Q1), which is the main advantage of this speed regulation method. On the
other hand, modes of motor operation with either minimum rotor current or
maximum efficiency are significant for small power motors [8].
According to the aforestated, motors are herein investigated in four different
operation modes, which correspond to:
1) minimum stator current (cosφ1=1);
2) minimum rotor current (cosφ2=1);
3) maximum efficiency (η = ηmax);
4) stator to rotor current ratio equal to that of a standard operation mode
(optimal efficiency mode of an induction motor with standard construction,
operating in the double feeding regime).
Fig. 1 – Power supply and control scheme of a doubly fed induction motor.
Starting from the mathematical model for a doubly fed induction motor in the
synchronous operating mode [9, 10], appropriate expressions for all characteristic
variables are derived in the paper. In order to determine the rotor voltage regulation
law for all previously stated operation modes, an adequate computer code is
developed and the corresponding results are presented.
2. MATHEMATICAL MODEL FOR A DOUBLY FED INDUCTION
MOTOR
Depending on the manner of frequency regulation, a doubly fed induction
motor can operate in either the synchronous or the asynchronous mode. In the
synchronous mode, analyzed in the present paper, independent regulation of rotor
voltage and frequency is enabled, which results in very good regulation characteristics.
3
Rotor voltage influence on the induction machine characteristics
251
Speed regulation is possible both above and under the synchronous speed, as well
as in both directions [1, 4]. Angular velocity is defined by
n=
60
( f1 ∓ f 2 ) .
Π
(1)
In this case, the induction motor operates as a synchronous machine, and the
change of load causes the change of the load (torque) angle.
When considering the characteristics of a doubly fed induction motor in the
synchronous operating mode, it is most convenient to define the mathematical
model with regard to a synchronous reference frame, i.e. one that rotates at the
synchronous speed ω1 (equal to stator angular frequency). When the reference
frame is so chosen, the mathematical model can be represented by the following set
of complex equations [1–3]
U 1 = R1 I 1 + d Ψ1 / dt + jω1 Ψ1 ,
(2)
U 2 = R2 I 2 + d Ψ 2 / dt + j(ω1 − ω)Ψ 2 ,
(3)
Ψ1 = Ls I 1 + Lm I 2 ,
(4)
Ψ 2 = Lm I 1 + Lr I 2 .
(5)
Using the differentiating operator p, the previous set of equations can be
transformed to the following form
u1 = r1i1 + ( p + j)ψ1 ,
(6)
u 2 = r2 i 2 + ( p + js )ψ 2 ,
(7)
ψ1 = x1 i1 + xm i 2 ,
(8)
ψ 2 = xm i1 + x2 i 2 ,
(9)
seeing that the slip s is defined as
s = f 2 / f1 = (ω1 − ω) / ω1 .
(10)
For the stationary mode of operation (p = 0) these become
u 1 = r1 i1 + jψ1 ,
(11)
u 2 = r2 i 2 + jsψ 2 .
(12)
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Žarko Milkić, Đukan Vukić, Saša Štatkić
4
Fig. 2 – Equivalent circuit of a doubly fed induction motor.
Assuming that the stator voltage vector (phasor) coincides with the reference
axis, complex expressions for stator and rotor voltages become
0
u1 = u1 e j0 ,
(13)
u 2 = u2 e -jα .
(14)
Fig. 3 – Phasor diagram for a doubly fed induction motor.
Starting from the equivalent circuit (Fig. 2) and the phasor diagram (Fig. 3)
for a doubly fed induction motor in synchronous operation mode, the relation
which connects the angle between rotor and stator voltage vectors (α) to the angle
between rotor axis and the stator voltage vector (δ, also called “the load angle” in
synchronous machine theory), is obtained as
δ = α −β ,
where β is defined by β = atan
r2 x1 − sr1 x2
(
r1r2 − s xm2 − x1 x2
(15)
)
.
5
Rotor voltage influence on the induction machine characteristics
253
3. MATHEMATICAL EXPRESSIONS FOR CHARACTERISTIC
VARIABLES
Solving the set of equations formed by (8), (9), (11) and (12), while taking
into account relations (13), (14) and (15), expressions for all characteristic variables of
a doubly fed induction motor in the investigated operation mode are obtained.
The meanings of new variables introduced in expressions that follow are
x1 = xsγ + xm , x2 = xrγ + xm , c1 = xm x1 , c2 = xm x2 , c3 = 1 − c1c2 ,
x11 = c3 x1 , x22 = c3 x2 , k1 = r1r2 − sx11 x2 , k 2 = sr1 x2 + r2 x1 , u = u2 u1 ,
a = r1r2 + sx1 x2 − sxm2 , b = sr1 x2 − r2 x1 .
Stator and rotor fluxes are respectively given by
ψ =
1
u1
k12
+ k22
{⎡⎣ x1 ( r2 k1 + x22 sk2 ) + ur1 x2 c2 ( k1 cos α − k2 sin α )⎤⎦ +
(16)
}
+ j ⎡⎣ x1 ( x22 sk1 − r2 k2 ) − ur1 x2 c2 ( k2 cos α + k1 sin α ) ⎤⎦ ,
ψ =
2
u1
k12
{
+ k22
{{r x c k + ux
2 1 1 1
2
}
⎡⎣( r1k1 + x11k2 ) cos α + ( x11k1 − r1k2 ) sin α ⎤⎦ +
}}
(17)
+ j − r2 x1c1k2 + ux2 ⎡⎣( x11k1 − r1k2 ) cos α − ( r1k1 + x11k2 ) sin α⎤⎦ .
Stator and rotor currents have active and reactive components
i1 =
u1
k12
+ k22
i1 = i1a + ji1r ,
(18)
i 2 = i2 a + ji2 r ,
(19)
{⎡⎣ r2 k1 + sx2 k2 − uxm ( k2 cos α + k1 sin α)⎤⎦ +
(20)
}
+ j ⎡⎣ sx2 k1 − r2 k2 − uxm ( k1 cos α − k2 sin α ) ⎤⎦ ,
i2 =
u1
k12
{
+ k22
{{−sx
m k2
}
+ u ⎡⎣ ( r1k1 + x1k2 ) cos α + ( x1k1 − r1k2 ) sin α ⎤⎦ +
}}
(21)
+ j − sxm k1 + u ⎡⎣( x1k1 − r1k2 ) cos α − ( r1k1 + x1k2 ) sin α⎤⎦ .
The corresponding absolute values of these currents are
i1 = u1
r22 + s 2 x22 + u 2 xm2 − 2uxm (sx2 cos α + r2 sin α )
,
k12 + k 22
(22)
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Žarko Milkić, Đukan Vukić, Saša Štatkić
(
6
)
s 2 xm2 + u 2 r12 + x12 − 2usxm ( x1 cos α − r1 sin α )
.
k12 + k22
i2 = u1
(23)
Electromagnetic torque can be expressed as
mem = m1 + m2 + m3 ,
(24)
where m1 and m2 are the asynchronous components, while m3 is the synchronous
component of the electromagnetic torque, yeilding
u12 ⎛
sr x 2 − u 2 r1 xm2 + uxm a 2 + b 2 sin δ ⎞⎟ .
2
2 ⎜ 2 m
⎝
⎠
k1 + k2
mem =
(25)
Stator and rotor active powers are, respectively
[
]
u12
[(r2 k1 + sx2 k2 ) − uxm (k2 cos α + k1 sin α )] ,
k12 + k22
]
u12
u 2 (r1k1 + x1k2 ) − usxm (k2 cos α − k1 sin α ) .
2
2
k1 + k 2
*
p1 = Re u1 ⋅ i1 =
[
*
[
p2 = Re u 2 ⋅ i 2 =
(26)
]
(27)
Stator and rotor reactive powers are, respectively
[
]
u12
[(sx2 k1 − r2k2 ) − uxm (k1 cos α − k2 sin α )] ,
k12 + k22
]
u12
u 2 (x1k1 − r1k 2 ) − usxm (k1 cos α + k 2 sin α ) .
2
2
k1 + k 2
*
q1 = I m u1 ⋅ i1 =
[
*
q2 = I m u 2 ⋅ i 2 =
[
(28)
]
(29)
(r2 k1 + sx2 k2 ) − uxm (k2 cos α + k1 sin α )
,
)[r22 + s 2 x22 + u 2 xm2 − 2uxm (sx2 cos α + r2 sin α )]
(30)
Stator and rotor power factors are
cos ϕ1 =
(
cos ϕ2 =
(
k12
k12
+
k 22
+
k 22
u (r1k1 + x1k 2 ) − sxm (k 2 cos α − k1 sin α )
)[s x
2 2
m
(
)
]
+ u 2 r12 + x12 − 2usxm (x1 cos α − r1 sin α )
.
(31)
Motor efficiency is given by
η=
P
,
P + ΣPγ
(32)
7
Rotor voltage influence on the induction machine characteristics
η=
Pem (1 − s ) − Pfv
(
Pem (1 − s ) − Pfv + PCus + PFes + PCur + PFer + Pfv
),
255
(33)
where P is the useful active power, and ΣPγ is the sum of all active losses in the
doubly fed induction motor. When relative units are considered pem = mem, which
finally leads to
mem (1 − s ) − Pfvn (1 − s )
1.5
η=
mem (1 − s ) + r1i12 + PFes + r2i22 + sp Hkr + s 2 p Fkr
.
(34)
4. ROTOR VOLTAGE REGULATION
Relations given in the previous section make it possible to define a certain
rotor voltage regulation law, which ensures optimal motor operation according to
predetermined criteria and applies to the entire regulated speed range.
To this end, it is necessary to solve a set of transcendental quadratic
equations, which can be obtained from the above expressions. This can be
accomplished by applying various graphical and analytical methods, but such
approaches are rather inexpedient. For that reason, a computer code for numerical
solving of the said set of equations is developed. It enables a rotor voltage
regulation law to be defined for the following four modes:
1) motor operation with minimum stator current, corresponding to the case
when motor takes the required reactive power from the rotor side only;
2) motor operation with minimum rotor current, corresponding to the case
when motor takes the required reactive power from the stator side only;
3) motor operation with maximum efficiency;
4) motor operation with stator to rotor current ratio equal to that of the
standard operation mode (i1 = i2), when optimal efficiency of an induction motor
with standard construction operating in the double feeding regime is achieved.
For any of the listed operation modes, induction motor parameters should be
known, while the analysis of motor operation with maximum efficiency requires
additional knowledge of the mechanical losses (friction and windage losses), stator
core losses, rotor hysteresis and eddy current losses. Separate recognition of
hysteresis losses and eddy current losses for the rotor is necessary because they
depend on frequency in different ways. In the present paper, identification of losses
is performed according to the method presented in [6].
5. TEST EXAMPLE
In order to verify the validity of the developed code, tests were performed on
a wound rotor induction motor, used as a centrifugal pump drive (model:
ZPD112M-4, manufactured by SEVER, Subotica, Serbia). Motor rating is 3.7 kW,
256
Žarko Milkić, Đukan Vukić, Saša Štatkić
8
and its parameters in per unit are: r1 = 0.047, r2 = 0.080, xsγ = 0.090, xrγ = 0.090 and
xm = 1.73. The experiments provided values for the following: rotor voltage in
maximum efficiency mode, stator core losses, mechanical losses, hysteresis and
eddy current components of rotor core losses. Results obtained at f = 50 Hz
frequency are: PFes=57 W, Pfv=106 W, PHkr=21.78 W and PFkr=31.80 W.
The range of angular velocity change is 0.8⋅nn < n < 1.2⋅nn. The
corresponding slip range is 0.2 > s > -0.2. Rotor voltage regulation laws are given
as functions of the slip in Figs. 4.a)…d) for each of the motor operation modes
stated in the previous section. The change of rotor voltage results in a change of the
phase angle between stator and rotor voltage vectors (phasors), as well as in the
change of the load angle. Dependencies of the load angle δ on the slip are also
shown in the Figs. 4.a)…d) for the four investigated operation modes.
6. CONCLUSION
The influence of rotor voltage on the characteristics of a wound rotor
induction motor, the speed of which is regulated by double feeding, has been
analyzed. Starting from the mathematical model for a doubly fed induction motor
in synchronous operating mode, expressions have been derived for all
characteristic variables.
A rotor voltage regulation law is defined with the aim of achieving optimal
operation of the motor in the whole regulated speed range, according to previously
defined criteria. Since the choice of these criteria depends on the motor rating,
regulated speed range, cycloconverter characteristics, and power system demands,
four particularly interesting modes of motor operation have been investigated:
minimum stator current mode, minimum rotor current mode, maximum efficiency
mode, and the mode with stator to rotor current ratio like in the standard mode of
operation.
a)
b)
9
Rotor voltage influence on the induction machine characteristics
c)
257
d)
Fig. 4 – Rotor voltage regulation law and load angle versus slip for motor operation with:
a) minimum stator current; b) minimum rotor current; c) maximum efficiency; d) stator to rotor
current ratio like that of the standard operation mode.
In order to determine rotor voltage regulation laws for all stated operation
modes, an appropriate computer code for numerical solving of a set of
transcendental quadratic equations has been developed. Results obtained for the
dependences of the rotor voltage regulation law and of the load angle on slip have
been presented.
APPENDIX
p – differentiating operator
Π – number of pole pairs
r1, r2 – stator, rotor resistance per phase
xsγ, xrγ – stator, rotor leakage reactance
xm – magnetizing reactance
x1, x2 – stator, rotor reactance per phase
Ls, Lr – stator, rotor inductance
Lm – magnetizing inductance
i1, i2 – per unit stator, rotor current
I1, I2 – absolute unit stator, rotor current
u1, u2 – stator, rotor voltage
f1, f2 – stator, rotor frequency
Ψ1, Ψ2 – stator, rotor magnetic flux
δ – load angle
α – the angle between stator and rotor
voltage vectors (phasors)
Received on November 2, 2013
s – slip
N – rotor speed
ω – rotor angular velocity
ω1 – synchronous angular velocity
M – electromagnetic torque
p1, p2 – stator, rotor active power
q1, q2 – stator, rotor reactive power
pfvn – mechanical losses at rated speed
pFes – stator core losses
pHkr – hysterezis losses in rotor core (at
locked rotor)
pFkr – eddy current losses in rotor core (at
locked rotor)
η – motor efficiency
cosφ1, cosφ2 – stator, rotor power factor
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Žarko Milkić, Đukan Vukić, Saša Štatkić
10
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