Method and apparatus of determining the stator flux estimate of an electric machine

The invention relates to a method of determining the estimate of the stator flux of an electric machine, comprising the determination of a stator flux estimate (.PHI..sub.sest) of the electric machine as a time integral of the difference between the voltage (u.sub.s) supplied to the stator and the product of the stator current (i.sub.s) and the stator resistance estimate (R.sub.sest), whereby the stator resistance estimate (R.sub.sest) of the electric machine is determined by correcting the initial value or another earlier value of the stator resistance on the basis of the stator current component (i.sub.d.sup.s, i.sub.d.sup..delta., i.sub.d.sup.r) parallel to the stator flux estimate (.PHI..sub.sest), the air gap flux estimate (.PHI..sub..delta.est) or the rotor flux estimate (.PHI..sub.rest). To be able to follow changes in the stator resistance of the electric machine in different operational conditions of the machine, said current component (i.sub.d.sup.s, i.sub.d.sup..delta., i.sub.d.sup.r) is compared with a set value (i.sub.d *) for the current component in question, the stator resistance estimate (R.sub.sest) being changed on the basis of the result of this comparison. (FIG. 4).

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to a method and apparatus of determining the stator 
flux estimate of an electric machine, comprising the determination of 
a stator current taken by the electric machine; 
a voltage supplied to the stator of the electric machine, 
a stator resistance estimate of the electric machine; and 
a stator flux estimate of the electric machine as a time integral of the 
difference between the voltage supplied to the stator and the product of 
the stator current and the stator resistance estimate, whereby the stator 
resistance estimate of the electric machine is determined by correcting 
the initial value or another earlier value of the stator resistance on the 
basis of the stator current component parallel to the stator flux 
estimate, the air gap flux, estimate or the rotor flux estimate. 
Related art As is known, the flux produced in the stator windings of an 
electric machine can be determined by calculating the time integral of the 
voltage supplied to the stator windings. However, the voltage causing the 
flux is not directly the voltage u.sub.s supplied to the poles of the 
winding but a voltage e.sub.s obtained by subtracting the resistive 
voltage drop R.sub.s . i.sub.s of the stator from the voltage u.sub.s. 
EQU e.sub.s =u.sub.s -R.sub.s . i.sub.s ( 1) 
When determining the flux of the stator side, the proportion of the voltage 
developing the flux and the voltage drop to the supply voltage has to be 
known. As the stator current is measured, and the voltage is the product 
of the current and the resistance according to Ohm's law, the only 
remaining unknown parameter is the resistance of the stator circuit. The 
estimated stator flux is thus obtained from the formula 
##EQU1## 
where .PHI..sub.sest =the estimated stator flux 
u.sub.s =the stator voltage 
i.sub.s =the stator current 
R.sub.sest =the estimated stator resistance 
When the electric machine is operated in the constant flux range, the basic 
frequency w.sub.s of the machine is substantially directly proportional to 
the supply voltage u.sub.s. The effective value of the voltage is 
therefore low at low frequencies and high at high frequencies. Since the 
voltage drop, however, is not dependent on the frequency but merely on the 
stator resistance and current, the proportion of the voltage drop to the 
supply voltage increases with decreasing supply frequency. As a result, 
the influence of the stator resistance is very significant at low 
frequencies and therefore it is necessary to know its value the more 
accurately the lower the voltage supplied to the machine is. 
The torque developed by the electric machine is determined by the cross 
product of the flux and the current. Utilizing the stator flux and stator 
current, the torque can be represented in the form 
EQU T=k.sub.1 . .PHI..sub.s .times.i.sub.s ( 3) 
where 
T=the electric torque caused by the machine 
k.sub.1 =a constant coefficient 
.PHI..sub.s =the stator flux 
i.sub.s =the stator current 
Equivalently, the torque estimate T.sub.est of the machine is obtained by 
utilizing the estimated flux .PHI..sub.sest and the stator current 
i.sub.s. 
EQU T.sub.est =k.sub.1 . .PHI..sub.sest .times.i.sub.s ( 4) 
The vector diagram of FIG. 1 of the attached drawings illustrates the 
influence of an error occurring in the value of the stator resistance 
estimate R.sub.sest on the estimate T.sub.est of the torque developed by 
the electric machine. If the estimated stator resistance R.sub.sest of 
Formula (1) is smaller than an actual value R.sub.s, the angle between the 
actual stator current i.sub.s and stator flux .PHI..sub.s is smaller than 
the angle between the stator current and the estimated stator flux 
.PHI..sub.sest, whereby the actual developed torque is smaller than the 
torque T.sub.est calculated by the control system. As a result, the 
electric machine does not develop the desired torque. The smaller the 
supply voltage, the greater the difference between the calculated and the 
actual value of the torque will be. 
Previously, there has not been any efficient method by means of which the 
stator flux of an electric machine could be estimated appropriately even 
at a low supply voltage, taking into account the stator resistance and its 
changes merely by means of an estimating method based on the measurement 
of the stator current and stator voltage. 
SUMMARY OF THE INVENTION 
The object of the present invention is to provide a method of determining 
the stator flux estimate, which method is able to follow changes in the 
stator resistance during operation without feedback information from the 
secondary side. Therefore all necessary information has to be derived from 
the stator voltage and stator current. 
This is achieved by means of a method of the invention which is 
characterized in that said current component is compared with a set value 
of the current component in question, the stator resistance estimate being 
changed on the basis of the result of this comparison. 
By means of the method, it is possible to continuously follow changes in 
the stator resistance during the operation of the machine and to take them 
into account when calculating the torque developed by the machine. A 
restriction of the method is that, in theory, it is not operable at a zero 
supply frequency. At high frequencies the accuracy is decreased but this 
is insignificant as the voltage drop of the stator is relatively small due 
to the high supply voltage, and the stator resistance can be assumed to be 
equal to zero without impairing the efficiency of the control.

FIG. 2a shows the vector diagram of a cage induction motor in motor 
operation. This vector diagram illustrates the dependence between the 
stator flux .PHI..sub.s, the air gap flux .PHI..sub..delta. and the rotor 
flux .PHI..sub.r. On the basis of the equations (1) and (2) the stator 
flux .PHI..sub.s can be calculated by integrating a voltage e.sub.s the 
fundamental harmonic of which can be represented by a vector rotating at a 
constant frequency w.sub.s. 
EQU e.sub.sest1 =e.sub.sest1 .multidot.e.sup.j.multidot.w s.sup..multidot.t (5) 
When the time integral is calculated from this equation the fundamental 
harmonic .PHI..sub.sest1 of the stator flux will be 
##EQU2## 
It is to be seen that the fundamental harmonic .PHI..sub.sest1 of the flux 
is also a vector rotating at the frequency w.sub.s and having a phase 
shift of -90.degree. with respect to the fundamental harmonic e.sub.sest1 
of the voltage. In the steady state, the above applies not only to 
fundamental harmonics but also to total values. 
By forming the scalar product of the flux and the current and by dividing 
it by the absolute value of the flux, a value i.sub.d is obtained which 
represents the stator current component parallel to the calculated flux. 
Thus the current component i.sub.d.sup.s parallel to the stator flux 
estimate .PHI..sub.sest is 
##EQU3## 
Equivalently, the component i.sub.d.sup..delta. parallel to the air gap 
estimate .PHI..sub..delta.est is 
##EQU4## 
where L.sub..sigma.s is the stray inductance of the stator, and 
L.sub..sigma.s i.sub.s represents the stray flux of the stator. 
Furthermore, the component i.sub.d.sup.r parallel to the rotor flux 
estimate .PHI..sub.rest is 
##EQU5## 
where L.sub.s is the stator inductance, .sigma. is the total dispersion 
coefficient and .sigma.L.sub.s is the short-circuit inductance of the 
machine. 
FIG. 3 illustrates the influence of an increase in the stator resistance 
R.sub.s on the stator current component i.sub.d.sup.s parallel to the 
estimated stator flux in motor operation. In the diagram of FIG. 3, the 
arrows indicate the directions of change of the different vectors. 
When the actual stator resistance R.sub.s increases, the real part of the 
input impedance of the electric machine increases and the stator current 
vector i.sub.s rotates towards the voltage vector u.sub.s and its absolute 
value is decreased. Similarly, the angle between the current i.sub.s and 
the estimated stator flux .PHI..sub.sest is increased. As a consequence, 
the current component i.sub.d.sup.s parallel to the estimated stator flux 
is decreased. The same applies to the stator current component parallel to 
the estimated air gap flux or the estimated rotor flux, respectively, 
which can be judged from the vector diagram of FIG. 2 as well as from the 
equations (8) and (9). This information on the dependence between the 
change of the actual stator resistance, the stator current and the change 
of the current component parallel to the estimated stator, air gap or 
rotor flux, respectively, is utilized according to the invention in the 
determination of the estimate of the stator resistance. In other words, 
when the current component in question is observed to decrease when the 
machine operates as a motor, and the set value i.sub.d.sup.* remains 
unchanged, the estimate of the stator resistance has to be increased in 
order to allow for the increase in the actual stator resistance, which 
originally caused the change in the current component. In motor operation, 
the peak value of the idle current given by the motor manufacturer or the 
peak value of the fundamental harmonic of the idle current measured at a 
high frequency, for instance, can be used as a rough set value 
i.sub.d.sup.* for the stator current component i.sub.d.sup.s parallel to 
the estimated stator flux. 
FIG. 4 is a block diagram illustrating the determination of the stator flux 
estimate .PHI..sub.sest, utilizing the stator current component 
i.sub.d.sup.s parallel to the estimated stator flux. In the block diagram 
of FIG. 4, a motor 1 is supplied from a three-stage supply. Both the 
current taken by the motor 1 and the operating voltage of the motor are 
determined from this supply in vector form. To simplify the block diagram, 
the current vector is indicated with the reference i.sub.s and the voltage 
vector with the reference u.sub.s. In other words, they represent the 
rotational vectors of the stator current and the stator voltage. The 
stator current i.sub.s is first applied to a multiplier 2 in which it is 
multiplied by the stator resistance estimate R.sub.sest. The stator 
resistance estimate can be the initial value of the stator resistance 
estimate, which can be set to zero or determined by a dc measurement, for 
instance, or it may be a value given in advance by the manufacturer of the 
electric machine. Alternatively, the stator resistance estimate can be the 
value obtained for the stator resistance in a preceding calculation 
operation. The product R.sub.sest . i.sub.s is applied from the multiplier 
2 to a summer 3, where it is added with a negative sign to the stator 
voltage u.sub.s to produce the voltage of Eq. (1) in the output of the 
summer 3. This voltage is applied to an integrator 4, which produces the 
time integral of this voltage in accordance with Eq. (2) to obtain the 
stator flux estimate .PHI..sub.sest at the output. This stator flux 
estimate is applied, on the one hand, to a multiplier 5 where it is 
multiplied by the stator current i.sub.s and, on the other hand, to a unit 
6 which forms its absolute value. The outputs of the units 5 and 6, in 
turn, are applied to a divider 7, which in accordance with Eq. (7) 
calculates the stator current component i.sub.d.sup.s parallel to the 
estimated stator flux. This current component is added with a negative 
sign to the set value i.sub.d.sup.* of the current in question in a summer 
8. In this way, a difference value .DELTA.i.sub.d representing the 
magnitude of the change of the current component in question with respect 
to its set value is obtained in the output of the summer 8. This 
difference value .DELTA.i.sub.d is applied to a controller 9 which may be, 
e.g., a PID type controller which changes the initial value of the stator 
resistance in connection with the first calculation operation, or a value 
obtained from a preceding calculation operation, in proportion to the 
change of the difference value .DELTA.i.sub.d with respect to the 
preceding calculation operation. In this way a stator resistance estimate 
R.sub.sest corrected according to the invention is obtained in the output 
of the controller 9. The stator flux estimate .PHI..sub.sest, for 
instance, can form the actual output value of the block diagram of FIG. 4. 
The torque estimate T.sub.est of the machine can then be calculated from 
the stator flux estimate in accordance with Eq. (4). 
It is to be noted that the block diagram of FIG. 4 shows only one example 
of the realization of the method of the invention. As mentioned above, the 
calculation could also be based on the utilization of the stator current 
component parallel to the air gap flux estimate or the rotor flux 
estimate. The block diagram itself could naturally be realized with other 
type of operational blocks than those shown.