1. 9.11. FLUX OBSERVERS FOR DIRECT VECTOR CONTROL WITH MOTION SENSORS
The motor stator or airgap flux space phasor amplitude lma and its instantaneous position - qer+ga* - with respect to stator phase a axis have to be computed on line, based on measured motor voltages, currents and, when available, rotor speed. The torque may be calculated from flux and current space phasors and thus once the flux is computed and stator currents measured, the torque problem is solved.
Open loop flux observers
Open loop flux observers are based on the voltage model or on the current model. Voltage model makes use of stator voltage equation in stator coordinates (from (9.32) with w1 = 0):
(9.95)
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2. From (9.37) with a = Lm / Lr: (9.96)
Both stator flux, , and rotor flux, , space phasors, may thus, in principle, be calculated based on and measured. The corresponding signal flow diagram is shown in figure 9.29.
Figure Voltage - model open loop flux observer (stator coordinates)
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3. The corresponding signal flow diagram is shown in figure 9.30.
On the other hand the current model for the rotor flux space phasor is based on rotor equation in rotor coordinates (wb = wr):
(9.97)
Two coordinate transformations - one for current and other for rotor flux - are required to produce results in stator coordinates. This time the observer works even at zero frequency but is very sensitive to the detuning of parameters Lm and tr due to temperature and magnetic saturation variation. Besides, it requires a rotor speed or position sensor. Parameter adaptation is a solution.
The corresponding signal flow diagram is shown in figure 9.30.
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4. 9.11.2. Closed loop flux observers
Figure Current control open loop flux observer
More profitable, however, it seems to use closed loop flux observers.
Closed loop flux observers
Figure Close loop voltage and current model rotor flux observer
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5. Many other flux observers have been proposed
Many other flux observers have been proposed. Among them, the third flux (voltage) harmonic estimator [21] and Gopinath observer [22], model reference adaptive and Kalman filter observers.
They all require notable on line computation effort and knowledge of induction motor parameters. Consequently they seem more appropiate when used together with speed observers for sensorless induction motor drives as shown in the corresponding paragraph, to follow after the next case study.
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6. 9.12. INDIRECT VECTOR SYNCHRONOUS CURRENT CONTROL
WITH SPEED SENSOR - A CASE STUDY
The simulation results of a vector control system with induction motor based on d.c. current control - are now given. The simulation of this drive is implemented in MATLAB - SIMULINK. The motor model was integrated in two blocks, the first represents the current and flux calculation module in d - q axis (figure 9.32), the second represents the torque, speed and position computing module (figure 9.33).
The motor used for this simulation has the following parameters: Pn = 1100W, Vnf = 220V, 2p = 4, rs = 9.53W, rr = 5.619W, Lsc = 0.136H, Lr = 0.505H, Lm = 0.447H, J = kgfm2.
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7. Figure 9.32. The indirect vector current control system
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8. Figure 9.33. The motor space phasor (d, q) model
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9. Figure 9.34. Speed transient response
The following figures represent the speed, torque, current and flux responses, for the starting process and with load torque applied at 0.4s. The value of load torque is 4Nm.
Figure Speed transient response
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10. Figure 9.35. Torque response
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11. Figure 9.36. Phase current waveform
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12. Figure 9.37. Stator flux amplitude
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13. Figure 9.38. Rotor flux amplitude
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14. 9.13. FLUX AND SPEED OBSERVERS IN SENSORLESS DRIVES
Sensorless drives are becoming predominant when only up to 100 to 1 speed control range is required even in fast torque response applications (1-5ms for step rated torque response).
Performance criteria
To assess the performance of various flux and speed observers for sensorless drives the following performance criteria have become widely accepted:
steady state error;
torque response quickness;
low speed behaviour (speed range);
sensitivity to noise and motor parameter detuning;
complexity versus performance.
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15. 9.13.2. A classification of speed observers
The basic principles used for speed estimation (observation) may be classified as:
A. Speed estimators
B. Model reference adaptive systems
C. Luenberger speed observers
D. Kalman filters
E. Rotor slot ripple
With the exception of rotor slot ripple all the other methods imply the presence of flux observers to calculate the motor speed.
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16. Speed estimators
Speed estimators are in general based on the classical definition of rotor speed :
(9.98)
where w1 is the rotor flux vector instantaneous speed and (Sw1) is the rotor flux slip speed. w1 may be calculated in stator coordinates based on the formula:
(9.99)
or (9.100)
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17. This seems to be a problem with most speed observers.
are to be determined from a flux observer (see figure 9.32, for example). On the other hand the slip frequency (Sw1), (9.23), is:
(9.101)
Notice that is strongly dependent only on rotor resistance rr as Lm / Lr is rather independent of magnetic saturation. Still rotor resistance is to be corrected if good precision at low speed is required. This slip frequency value is valid both for steady state and transients and thus is estimated quickly to allow fast torque response.
Such speed estimators may work even at 20rpm although dynamic capacity of torque disturbance rejection at low speeds is limited.
This seems to be a problem with most speed observers.
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18. 9.13.4. Model reference adaptive systems (MRAS)
MRASs are based on comparision of two estimators. One of them does not include speed and is called the reference model. The other, which contains speed, is the adjustable model. The error between the two is used to derive an adaption model that produces the estimated speed for the adjustable model.
To eliminate the stator resistance influence, the airgap reactive power qm [25] is the output of both models:
(9.102)
(9.103)
The rotor flux magnetization current equation in stator coordinates is ((9.15) with w1 = 0):
(9.104)
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19. Now the speed adaptation mechanism is:
(9.105)
The signal flow diagram of the MRAS obtained is shown in figure 9.39.
Figure MRAS speed estimator based on airgap reactive power error
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20. Figure 9.40. a.) Zero frequency b.) low speed operation
The effect of the rotor time constant tr variation persists and influences the speed estimation. However if the speed estimator is used in conjunction with indirect vector current control at least rotor field orientation is maintained as the same (wrong!) tr enters also the slip frequency calculator.
The MRAS speed estimator does not contain integrals and thus works even at zero stator frequency (d.c. braking) (figure 9.40.a) and does not depend on stator resistance rs. It works even at 20rpm (figure 9.40.b) [25].
Figure a.) Zero frequency b.) low speed operation
of MRAS speed estimator
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21. 9.13.5. Luenberger speed observers
First the stator current and the rotor flux are calculated through a full order Luenberger observer based on stator and rotor equations in stator coordinates:
(9.106)
with (9.107)
The full order Luenberger observer writes:
(9.110)
The matrix G is chosen such that the observer is stable.
(9.111)
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22. The speed estimator is based on rotor flux and estimators: (9.113)
In essence the speed estimator is based on some kind of torque error.
If the rotor resistance rr has to be estimated an additional high reference current ida* is added to the reference flux current ids*. Then the rotor resistance may be estimated [26] as:
(9.114)
Remarkable results have been obtained this way with minimum speed down to 30rpm.
The idea of an additional high frequency (10 times rated frequency) flux current may be used to determine both the rotor speed and rotor time constant tr [27].
Extended Kalman filters for speed and flux observers [28] also claim speed estimation at rpm though they require considerable on line computation time.
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23. 9.13.6. Rotor slots ripple speed estimators
The rotor slots ripple speed estimators are based on the fact that the rotor slotting openings cause stator voltage and current harmonics ws1,2 related to rotor speed , the number of rotor slot Nr and synchronous speed :
(9.115)
Band pass filters centered on the rotor slot harmonics are used to separate and thus calculate from (9.115). Various other methods have been proposed to obtain and improve the transient performance. The response tends to be rather slow and thus the method, though immune to machine parameters, is mostly favorable for wide speed range but for low dynamics (medium - high powers) applications [27].
For more details on sensorless control refer to [30].
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24. 9.14. DIRECT TORQUE AND FLUX CONTROL (DTFC)
DTFC is a commercial abbreviation for the so called direct self control proposed initially [ ] for induction motors fed from PWM voltage source inverters and later generalized as torque vector control (TVC) in [4] for all a.c. motor drives with voltage or current source inverters.
In fact, based on the stator flux vector amplitude and torque errors sign and relative value and the position of the stator flux vector in one of the 6 (12) sectors of a period, a certain voltage vector (or a combination of voltage vectors) is directly applied to the inverter with a certain average timing.
To sense the stator flux space phasor and torque errors we need to estimate the respective variables. So all types of flux (torque) estimators or speed observers good for direct vector control are also good for DTFC. The basic configurations for direct vector control and DTFC are shown on figure 9.41.
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25. Electric Drives

26. Figure 9.41. a.) Direct vector current control b.) DTFC control
As seen from figure 9.41 DTFC is a kind of direct vector d.c. (synchronous) current control.
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27. DTFC principle
Though figure 9.41 uncovers the principle of DTFC, finding how the T.O.S. is generated is the way to a succesful operation.
Selection of the appropiate voltage vector in the inverter is based on stator equation in stator coordinates:
(9.117)
By integration:
(9.118)
In essence the torque error eT may be cancelled by stator flux acceleration or deceleration. To reduce the flux errors, the flux trajectories will be driven along appropriate voltage vectors (9.118) that increase or decrease the flux amplitude.
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28. a.) Stator flux space phasor trajectory
Figure 9.42.
a.) Stator flux space phasor trajectory
b.) Selecting the adequate voltage vector in the first sector (-300 to +300)
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29. The complete table of optimal switching, TOS, is shown in table 9.2.
Table 9.2. Basic voltage vector selection for DTFC
As expected the torque response is quick (as in vector control) but it is also rotor resistance independent above 1 - 2Hz (figure 9.43) [2].
Figure TVC torque response
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30. 9.15. DTFC SENSORLESS: A CASE STUDY
The simulation results of a direct torque and flux control drive system for induction motors are presented. The example was implemented in MATLAB - SIMULINK. The motor model was integrated in two blocks, first represents the current and flux calculation module in d - q axis, the second represents the torque, speed and position computing module (figure 9.44).
Figure 9.44.The DTFC system
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31. Figure The I.M. model
The motor used for this simulation has the following parameters: Pn = 1100W, Unf = 220V, 2p = 4, rs = 9.53W, rr = 5.619W, Lsc = 0.136H, Lr = 0.505H, Lm = 0.447H, J = kgfm2.
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32. Figure 9.46. Speed and torque estimators
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33. Electric Drives

34. Figure 9.47. Speed transient response (measured and estimated)
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35. Figure 9.48. Phase current waveform (steady state)
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36. Electric Drives

37. Figure 9.49. Torque response (measured and estimated)
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38. Figure 9.50. Stator flux amplitude
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39. Figure 9.51. Rotor flux amplitude
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